Reinforced Concrete: Plates¶
CalcpadCE worksheets in this section derive the bending moments of two-way reinforced concrete slabs on rectangular panels with assorted edge restraint conditions, using the moment coefficients of yield-line theory applied to the short and long span aspect ratio.
The series covers the full set of nine boundary cases that a designer encounters in a typical floor: the simply supported panel and the fully restrained panel at the extremes, the panels with one longer edge restrained, one shorter edge restrained, two adjacent edges restrained, both longer edges restrained and both shorter edges restrained, and the inverse cases of a panel with an unrestrained longer edge or an unrestrained shorter edge.
Each sheet returns the design moments at the centre of the span and along the restrained edges, and follows up with the required reinforcement areas in both directions for the chosen concrete and steel grades.
Fully Restrained Slab Panel¶
Two-way slab panel with all four edges fully restrained: span and support moment coefficients tabulated against the aspect ratio, with the required reinforcement in both directions.
'<small>According to <strong>Eurocode</strong>: EN 1992-1-1</small>
'<div style="max-width:180mm;">
'<img alt="f-span-cl.png" class="side" style="height:180pt;" src="../../Images/structures/rc/spans/f-span-cl.png">
'<h4>Dimensions</h4>
'Clear spans(<i>l</i><sub>x</sub> < <i>l</i><sub>y</sub>)
l_x_cl = ?'m,'l_y_cl = ?'m
'Beam width along <i>x</i> -'b_y = ?'cm
'Beam width along <i>y</i> -'b_x = ?'cm
'Slab thickness -'h = ?'cm
'Concrete cover -'c = ?'cm
#post
'Design span lengths
l_x = l_x_cl + min(b_x/100;h/100)'m
l_y = l_y_cl + min(b_y/100;h/100)'m
'Span ratio -'k = l_y/l_x
#if k < 1
'Span length <i>l</i><sub>y</sub> should be greater than <i>l</i><sub>x</sub>
#else
#show
'Design spans lengths and edge moments in adjacent panels
'<img alt="f-span.png" class="side"style="height:150pt;" src="../../Images/structures/rc/spans/f-span.png">
'<table><tr><td rowspan="2">
'Parallel to <i>x</i> - <br />
'</td><td>
l_x1 = ?'m;
'</td><td>
M_Ed_xt1_ = ?'kN·m/m
'</td></tr><tr><td>
l_x2 = ?'m;
'</td><td>
M_Ed_xt2_ = ?'kN·m/m
'</td></tr></table>
'<!--
#if k ≤ 2
'<-->
'<table><tr><td rowspan="2">
'Along <i>y</i> -
'</td><td>
l_y1 = ?'m;
'</td><td>
M_Ed_yt1_ = ?'kN·m/m
'</td></tr><tr><td>
l_y2 = ?'m;
'</td><td>
M_Ed_yt2_ = ?'kN·m/m
'</td></tr></table>
'<!--
#end if
'<-->
'(in absence of contiguous span <i>l</i> = 0)
'<h4>Loads</h4>
'Characteristic uniform loads
'Self weight -'g_sw = 0.25*h'kN/m²
'Permanent - 'g_k = ?'kN/m²
'Variable -'q_k = ?'kN/m²
'Partial safety factor -'γ_G = 1.35';'γ_Q = 1.5
#post
'Design uniform loads
p = (g_sw + g_k)*γ_G + q_k*γ_Q'kN/m²
'<div class="fold">
'<p><b>Beam loads</b></p>
'(as uniform loads)
#if k ≤ 2
'<table><tr><td rowspan="2">
'Parallel to <i>x</i> - <br />
'</td><td>
g_k_x = (g_sw + g_k)*l_x_cl/4'kN/m
'</td></tr><tr><td>
q_k_x = q_k*l_x_cl/4'kN/m
'</td></tr><tr><td rowspan="2">
'Parallel to <i>y</i> - <br />
'</td><td>
g_k_y = (g_sw + g_k)*l_x_cl*(l_y_cl - l_x_cl/2)/(2*l_y_cl)'kN/m
'</td></tr><tr><td>
q_k_y = q_k*l_x_cl*(l_y_cl - l_x_cl/2)/(2*l_y_cl)'kN/m
'</td></tr></table>
#else
'Parallel to <i>y</i> -'q_y = p*l_x_cl/2'kN/m
#end if
'</div>
#show
'<h4>Material properties</h4>
'<p><b>Concrete</b> [EN 1992-1-1, Table 3.1]</p>
'Characteristic compressive cylinder strength -'f_ck = ?'MPa
'Partial safety factors for concrete -'γ_c = 1.5','α_cc = ?
#post
'Design compressive cylinder strength -'f_cd = α_cc*f_ck/γ_c'MPa
'Factor defining the effective height of the compression zone -'λ = 0.8'
'Factor defining the effective strength - 'η = 1.0
'Mean value of axial tensile strength -'f_ctm = 0.30*f_ck^(2/3)'MPa
#show
'<p><b>Steel</b></p>
'Characteristic yield strength -'f_yk = ?'MPa
#post
'Partial safety factor for steel - 'γ_s = 1.15
'Design yield strength - 'f_yd = f_yk/γ_s'MPa
'Limit relative compressive zone depth -'ξ_lim = 0.45
#show
'<p><b>Bar diameters</b></p>
'- Bottom parallel to <i>x</i> -'d_bxb = ?'mm, parallel to <i>y</i> -'d_byb = ?'mm
'- Top parallel to <i>x</i> -'d_bxt = ?'mm, parallel to <i>y</i> -'d_byt = ?'mm
#post
'<div class="fold">
'<h4>Static calculations</h4>
'<p><b>Shear forces and bending moments coefficients</b></p>
#if l_y > 2*l_x
l_y/l_x'>'2'- One-way slab
'For the actual static scheme
'- for moments in mid-span-'β_xb1 = 0.04167
'- for moments over supports -'β_xt = 0.08333
'- for shear forces -'β_vx = 0.5
'For simply supported slab
'- for span moments -'β_xb2 = 0.125
#else if l_x ≡ l_y
l_y/l_x'- Two-way square slab
'For the actual static scheme
'- for moments in mid-span -'β_xb1 = 0.024';'β_yb1 = 0.024
'- for moments over supports -'β_xt = 0.031';'β_yt = 0.031
'- for shear forces -'β_vx = 0.33';'β_vy = 0.33
'For simply supported slab
'- for span moments -'β_xb2 = 0.055';'β_yb2 = 0.055
#else
'For the actual static scheme
'- for shear forces
'<table class="bordered centered">
'<tr><th rowspan="2"></th><th style="width:100mm;" colspan="9">Short span coefficients</th><th style="width:30mm;" rowspan="2">Long span coefficients</th></tr>
'<tr><th><i>k</i> =</th><th>1.0</th><th>1.1</th><th>1.2</th><th>1.3</th><th>1.4</th><th>1.5</th><th>1.75</th><th>2.0</th></tr>
'<tr><th>V</th><td><i>β</i><sub>vx</sub> = </td><td>0.33</td><td>0.36</td><td>0.39</td><td>0.41</td><td>0.43</td><td>0.45</td><td>0.48</td><td>0.50</td><td>'β_vy = 0.33'</td></tr></table>
#if k < 1.1
β_vx = 0.33 + (0.36 - 0.33)*(k - 1)/0.1
#else if k < 1.2
β_vx = 0.36 + (0.39 - 0.36)*(k - 1.1)/0.1
#else if k < 1.3
β_vx = 0.39 + (0.41 - 0.39)*(k - 1.2)/0.1
#else if k < 1.4
β_vx = 0.41 + (0.43 - 0.41)*(k - 1.3)/0.1
#else if k < 1.5
β_vx = 0.43 + (0.45 - 0.43)*(k - 1.4)/0.1
#else if k < 1.75
β_vx = 0.45 + (0.48 - 0.45)*(k - 1.5)/0.25
#else if k < 2
β_vx = 0.48 + (0.5 - 0.48)*(k - 1.75)/0.25
#else if k ≡ 2
β_vx = 0.5
#end if
'- for moments in mid-span
'<table class="bordered centered">
'<tr><th rowspan="2"></th><th style="width:100mm;" colspan="9">Short span coefficients</th><th style="width:30mm;" rowspan="2">Long span coefficients</th></tr>
'<tr><th><i>k</i> =</th><th>1.0</th><th>1.1</th><th>1.2</th><th>1.3</th><th>1.4</th><th>1.5</th><th>1.75</th><th>2.0</th></tr>
'<tr><th>+M</th><td><i>β</i><sub>xb1</sub> = </td><td>0.024</td><td>0.028</td><td>0.032</td><td>0.035</td><td>0.037</td><td>0.040</td><td>0.044</td><td>0.048</td><td>'β_yb1 = 0.024'</td></tr>
'<tr><th>-M</th><td><i>β</i><sub>xt</sub> = </td><td>0.031</td><td>0.037</td><td>0.042</td><td>0.046</td><td>0.050</td><td>0.053</td><td>0.059</td><td>0.063</td><td>'β_yt = 0.032'</td></tr></table>
#if k < 1.1
β_xb1 = 0.024 + (0.028 - 0.024)*(k - 1)/0.1
#else if k < 1.2
β_xb1 = 0.028 + (0.032 - 0.028)*(k - 1.1)/0.1
#else if k < 1.3
β_xb1 = 0.032 + (0.035 - 0.032)*(k - 1.2)/0.1
#else if k < 1.4
β_xb1 = 0.035 + (0.037 - 0.035)*(k - 1.3)/0.1
#else if k < 1.5
β_xb1 = 0.037 + (0.04 - 0.037)*(k - 1.4)/0.1
#else if k < 1.75
β_xb1 = 0.04 + (0.044 - 0.04)*(k - 1.5)/0.25
#else if k < 2
β_xb1 = 0.044 + (0.048 - 0.044)*(k - 1.75)/0.25
#else if k ≡ 2
β_xb1 = 0.048
#end if
'- for moments over supports
#if k < 1.1
β_xt = 0.031 + (0.037 - 0.031)*(k - 1)/0.1
#else if k < 1.2
β_xt = 0.037 + (0.042 - 0.037)*(k - 1.1)/0.1
#else if k < 1.3
β_xt = 0.042 + (0.046 - 0.042)*(k - 1.2)/0.1
#else if k < 1.4
β_xt = 0.046 + (0.05 - 0.046)*(k - 1.3)/0.1
#else if k < 1.5
β_xt = 0.05 + (0.053 - 0.05)*(k - 1.4)/0.1
#else if k < 1.75
β_xt = 0.053 + (0.059 - 0.053)*(k - 1.5)/0.25
#else if k < 2
β_xt = 0.059 + (0.063 - 0.059)*(k - 1.75)/0.25
#else if k ≡ 2
β_xt = 0.063
#end if
'For simply supported slab
' - for moments in mid-span
'<table class="bordered centered">
'<tr><th style="width:100mm;" colspan="9">Short span coefficients</th><th style="width:30mm;" rowspan="2">Long span coefficients</th></tr>
'<tr><th><i>k</i> =</th><th>1.0</th><th>1.1</th><th>1.2</th><th>1.3</th><th>1.4</th><th>1.5</th><th>1.75</th><th>2.0</th></tr>
'<tr><td><i>β</i><sub>xb2</sub> = </td><td>0.055</td><td>0.065</td><td>0.074</td><td>0.081</td><td>0.087</td><td>0.092</td><td>0.103</td><td>0.111</td><td>'β_yb2 = 0.056'</td></tr></table>
#if k < 1.1
β_xb2 = 0.055 + (0.065 - 0.055)*(k - 1)/0.1
#else if k < 1.2
β_xb2 = 0.065 + (0.074 - 0.065)*(k - 1.1)/0.1
#else if k < 1.3
β_xb2 = 0.074 + (0.081 - 0.074)*(k - 1.2)/0.1
#else if k < 1.4
β_xb2 = 0.081 + (0.087 - 0.081)*(k - 1.3)/0.1
#else if k < 1.5
β_xb2 = 0.087 + (0.092 - 0.087)*(k - 1.4)/0.1
#else if k < 1.75
β_xb2 = 0.092 + (0.103 - 0.092)*(k - 1.5)/0.25
#else if k < 2
β_xb2 = 0.103 + (0.111 - 0.103)*(k - 1.75)/0.25
#else if k ≡ 2
β_xb2 = 0.111
#end if
#end if
'<p><b>Loads for accounting alternation in adjacent spans</b></p>
p_1 = γ_G*(g_sw + g_k) + γ_Q*q_k/2'kN/m²
p_2 = γ_Q*q_k/2'kN/m²
'<p><b>Bending moments</b></p>
'Span moment (including load alternation)
M_Ed_xb = β_xb1*p_1*l_x^2 + β_xb2*p_2*l_x^2'kN·m/m
#if k ≤ 2
M_Ed_yb = β_yb1*p_1*l_x^2 + β_yb2*p_2*l_x^2'kN·m/m
#end if
'Support moment
M_Ed_xt = β_xt*p*l_x^2'kN·m/m
#if k ≤ 2
M_Ed_yt = β_yt*p*l_x^2'kN·m/m
#end if
'<p><b>Average moments at supports from the adjacent spans </b></p>
M_Ed_xt1 = (M_Ed_xt1_*l_x1 + M_Ed_xt*l_x)/(l_x1 + l_x)'kN·m/m
M_Ed_xt2 = (M_Ed_xt2_*l_x2 + M_Ed_xt*l_x)/(l_x2 + l_x)'kN·m/m
#if k ≤ 2
M_Ed_yt1 = (M_Ed_yt1_*l_y1 + M_Ed_yt*l_y)/(l_y1 + l_y)'kN·m/m
M_Ed_yt2 = (M_Ed_yt2_*l_y2 + M_Ed_yt*l_y)/(l_y2 + l_y)'kN·m/m
#end if
'<p><b>Shear forces</b></p>
#if k ≤ 2
V_Ed_x = β_vx*p*l_x'kN/m
V_Ed_y = β_vy*p*l_x'kN/m
#else
V_Ed_x = β_vx*p*l_x'kN/m
#end if
'<p><b>Edge support moments</b></p>
M_Ed_xte1 = M_Ed_xt1 - V_Ed_x*min(b_x/200;h/200)'kN·m/m
M_Ed_xte2 = M_Ed_xt2 - V_Ed_x*min(b_x/200;h/200)'kN·m/m
#if k ≤ 2
M_Ed_yte1 = M_Ed_yt1 - V_Ed_y*min(b_y/200;h/200)'kN·m/m
M_Ed_yte2 = M_Ed_yt2 - V_Ed_y*min(b_y/200;h/200)'kN·m/m
#end if
'</div><div class="fold">
'<h4>Bending design</h4>
'<p><b>Bottom reinforcement parallel to <i>x</i></b></p>
'Effective cross section depth -'d_xb = h - c - d_bxb/20'cm
'Relative design bending moment-'m_Ed = 10*M_Ed_xb/(d_xb^2*η*f_cd)
'Compression zone depth -'x = d_xb/λ*(1 - sqr(1 - 2*m_Ed))'cm
'Relative compression zone depth -'ξ = x/d_xb
'Lever arm of internal forces -'z = d_xb - 0.5*λ*x'cm
'Area of required reinforcement -'A_sxb = M_Ed_xb*10^3/(z*f_yd)'cm²/m
'Reinforcement ratio -'ρ_xb = A_sxb/(d_xb*100)
#if ξ > ξ_lim
'<p class="err">'ξ'>'ξ_lim'- Reinforcement in compression zone is required.</p>
#end if
#hide
#if k ≤ 2
'<p><b>Bottom reinforcement parallel to <i>y</i></b></p>
'Effective cross section depth-'d_yb = h - c - d_bxb/10 - d_byb/20'cm
'Relative design bending moment -'m_Ed = 10*M_Ed_yb/(d_yb^2*η*f_cd)
'Compression zone depth -'x = d_yb/λ*(1 - sqr(1 - 2*m_Ed))'cm
'Relative compression zone depth -'ξ = x/d_yb
'Lever arm of internal forces -'z = d_yb - 0.5*λ*x'cm
'Area of required reinforcement -'A_syb = M_Ed_yb*10^3/(z*f_yd)'cm²/m
'Reinforcement ratio -'ρ_yb = A_syb/(d_yb*100)
#if ξ > ξ_lim
'<p class="err">'ξ'>'ξ_lim'- Reinforcement in compression zone is required</p>
#end if
#end if
'<p><b>Top reinforcement parallel to <i>x</i> за <i>M</i><sub>Ed_xtel</sub></b></p>
'Effective cross section depth-'d_xt1 = h - c - d_bxt/20'cm
'Relative design bending moment -'m_Ed = 10*M_Ed_xte1/(d_xt1^2*η*f_cd)
'Compression zone depth -'x = d_xt1/λ*(1 - sqr(1 - 2*m_Ed))'cm
'Relative compression zone depth -'ξ = x/d_xt1
'Lever arm of internal forces-'z = d_xt1 - 0.5*λ*x'cm
'Area of required reinforcement -'A_sxt1 = M_Ed_xte1*10^3/(z*f_yd)'cm²/m
'Reinforcement ratio -'ρ_xt1 = A_sxt1/(d_xt1*100)
#if ξ > ξ_lim
'<p class="err">'ξ'>'ξ_lim'- Reinforcement in compression zone is required.</p>
#end if
'<p><b>Top reinforcement parallel to <i>x</i> за <i>M</i><sub>Ed_xte2</sub></b></p>
'Effective cross section depth-'d_xt2 = h - c - d_bxt/20'cm
'Relative design bending moment-'m_Ed = 10*M_Ed_xte2/(d_xt2^2*η*f_cd)
'Compression zone depth-'x = d_xt2/λ*(1 - sqr(1 - 2*m_Ed))'cm
'Relative compression zone depth -'ξ = x/d_xt2
'Lever arm of internal forces-'z = d_xt2 - 0.5*λ*x'cm
'Area of required reinforcement-'A_sxt2 = M_Ed_xte2*10^3/(z*f_yd)'cm²/m
'Reinforcement ratio -'ρ_xt2 = A_sxt2/(d_xt2*100)
#if ξ > ξ_lim
'<p class="err">'ξ'>'ξ_lim'- Reinforcement in compression zone is required.</p>
#end if
#if k ≤ 2
'<p><b>Top reinforcement parallel to <i>y</i> за <i>M</i><sub>Ed_yte1</sub></b></p>
'Effective cross section depth -'d_yt1 = h - c - d_bxt/10 - d_byt/20'cm
'Relative design bending moment -'m_Ed = 10*M_Ed_yte1/(d_yt1^2*η*f_cd)
'Compression zone depth -'x = d_yt1/λ*(1 - sqr(1 - 2*m_Ed))'cm
'Relative compression zone depth-'ξ = x/d_yt1
'Lever arm of internal forces -'z = d_yt1 - 0.5*λ*x'cm
'Area of required reinforcement -'A_syt1 = M_Ed_yte1*10^3/(z*f_yd)'cm²/m
'Reinforcement ratio -'ρ_yt1 = A_syt1/(d_yt1*100)
#if ξ > ξ_lim
'<p class="err">'ξ'>'ξ_lim'- Reinforcement in compression zone is required.</p>
#end if
'<p><b>Top reinforcement parallel to <i>y</i> за <i>M</i><sub>Ed_yte2</sub></b></p>
'Effective cross section depth -'d_yt2 = h - c - d_bxt/10 - d_byt/20'cm
'Relative design bending moment-'m_Ed = 10*M_Ed_yte2/(d_yt2^2*η*f_cd)
'Compression zone depth -'x = d_yt2/λ*(1 - sqr(1 - 2*m_Ed))'cm
'Relative compression zone depth -'ξ = x/d_yt2
'Lever arm of internal forces-'z = d_yt2 - 0.5*λ*x'cm
'Area of required reinforcement-'A_syt2 = M_Ed_yte2*10^3/(z*f_yd)'cm²/m
'Reinforcement ratio -'ρ_yt2 = A_syt2/(d_yt2*100)
#if ξ > ξ_lim
'<p class="err">'ξ'>'ξ_lim'- Reinforcement in compression zone is required.</p>
#end if
#end if
#post
'<p><b>The rest of the reinforcement is calculated similarly.</b></p>
'</div><div class="fold">
'<h4>Reinforcement detailing</h4>
'<p><b>Bottom reinforcement</b></p>
'<p class="ref">[EN 1992-1-1, § 9.2.1.1 (1)]</p>
'Minimum reinforcement ratio
ρ_min = max(0.26*f_ctm/f_yk;0.0013)
'<p class="ref">[EN 1992-1-1, § 9.2.1.1 (3)]</p>
'Maximum reinforcement ratio -'ρ_max = 0.04
#if ρ_xb < ρ_min
'<p><i>ρ</i><sub>xb</sub> < <i>ρ</i><sub>min</sub>. Minimum reinforcement is assumed -'A_sxb = ρ_min*d_xb*100'cm²/m
#else if ρ_xb > ρ_max
'<p class="err">Reinforcement ratio is greater than maximum:'ρ_xb'>'ρ_max'</p>
#end if
#if k > 2
'<p class="ref">[EN 1992-1-1, § 9.3.1.1 (2)]</p>
'Minimum bottom reinforcement parallel to y
A_syb = 0.2*A_sxb'cm²/m
#else
'<!--'
#if ρ_yb < ρ_min
'<p><i>ρ</i><sub>yb</sub> < <i>ρ</i><sub>min</sub>. Minimum reinforcement is assumed -'A_syb = ρ_min*d_yb*100'cm²/m
#else if ρ_yb > ρ_max
'<p class="err">Reinforcement ratio is greater than maximum:'ρ_yb'>'ρ_max'</p>
#end if
'<-->
#end if
'Required bar spacing
s_xb = floor(π*d_bxb^2/(4*A_sxb))'cm
#if k > 2
'<p class="ref">[EN 1992-1-1, § 9.3.1.1 (3)]</p>
'Minimum bottom reinforcement parallel to y
s_yb = min(3*h;40)'cm
'Required bar diameter for bottom reinforcement parallel to y
d_byb = max(ceiling(sqr(4*A_syb*s_yb/π));6)'mm
#else
'<!--'s_yb = floor(π*d_byb^2/(4*A_syb))'cm-->
#end if
'<p class="ref">[EN 1992-1-1, § 9.3.1.1 (3)]</p>
'Maximum bar spacing
s_max = min(2*h;25)'cm
#if s_xb > s_max
'<p><i>s</i><sub>xb</sub> > <i>s</i><sub>max</sub>. Assume's_xb = s_max'cm</p>
#end if
#if (k ≤ 2)*(s_yb > s_max)
'<!--<p><i>s</i><sub>yb</sub> > <i>s</i><sub>max</sub>. Assume's_yb = s_max'cm</p>-->
#end if
#hide
'<p><b>Top reinforcement</b></p>
#if ρ_xt1 < ρ_min
'<p><i>ρ</i><sub>xt1</sub> < <i>ρ</i><sub>min</sub>. Minimum reinforcement is assumed -'A_sxt1 = ρ_min*d_xt1*100'cm²/m
#else if ρ_xt1 > ρ_max
'<p class="err">Reinforcement ratio is greater than maximum:'ρ_xt1'>'ρ_max'</p>
#end if
#if ρ_xt2 < ρ_min
'<p><i>ρ</i><sub>xt2</sub> < <i>ρ</i><sub>min</sub>. Minimum reinforcement is assumed -'A_sxt2 = ρ_min*d_xt2*100'cm²/m
#else if ρ_xt2 > ρ_max
'<p class="err">Reinforcement ratio is greater than maximum:'ρ_xt2'>'ρ_max'</p>
#end if
#if k ≤ 2
#if ρ_yt1 < ρ_min
'<p><i>ρ</i><sub>yt1</sub> < <i>ρ</i><sub>min</sub>. Minimum reinforcement is assumed -'A_syt1 = ρ_min*d_yt1*100'cm²/m
#else if ρ_yt1 > ρ_max
'<p class="err">Reinforcement ratio is greater than maximum:'ρ_yt1'>'ρ_max'</p>
#end if
#if ρ_yt2 < ρ_min
'<p><i>ρ</i><sub>yt2</sub> < <i>ρ</i><sub>min</sub>. Minimum reinforcement is assumed -'A_syt2 = ρ_min*d_yt2*100'cm²/m
#else if ρ_yt2 > ρ_max
'<p class="err">Reinforcement ratio is greater than maximum:'ρ_yt2'>'ρ_max'</p>
#end if
#end if
'Required bar spacing
s_xt1 = floor(π*d_bxt^2/(4*A_sxt1))'cm
s_xt2 = floor(π*d_bxt^2/(4*A_sxt2))'cm
#if k ≤ 2
s_yt1 = floor(π*d_byt^2/(4*A_syt1))'cm
s_yt2 = floor(π*d_byt^2/(4*A_syt2))'cm
#end if
'<p class="ref">[EN 1992-1-1, § 9.3.1.1 (3)]</p>
'Maximum bar spacing
s_max = min(2*h;25)'cm
#if s_xt1 > s_max
'<p><i>s</i><sub>xt1</sub> > <i>s</i><sub>max</sub>. It is assumed's_xt1 = s_max'cm</p>
#end if
#if s_xt2 > s_max
'<p><i>s</i><sub>xt2</sub> > <i>s</i><sub>max</sub>. It is assumed's_xt2 = s_max'cm</p>
#end if
#if k ≤ 2
#if s_yt1 > s_max
'<p><i>s</i><sub>yt1</sub> > <i>s</i><sub>max</sub>. It is assumed's_yt1 = s_max'cm</p>
#end if
#if s_yt2 > s_max
'<p><i>s</i><sub>yt2</sub> > <i>s</i><sub>max</sub>. It is assumed's_yt2 = s_max'cm</p>
#end if
#end if
#post
'<p><b>The rest of the reinforcement is detailed similarly.</b></p>
'</div>
'<h4>Results</h4>
#val
'<table class="bordered centered">
'<tr><th>Reinforcement</th><th>M<sub>Ed</sub><br>kN·m/m</th><th>A<sub>s</sub><br>cm<sup>2</sup>/m</th><th>ρ, %</th><th>Selected</tr>
'<tr><td>Bottom parallel to <i>x</i></td><td>'M_Ed_xb'</td><td>'A_sxb'</td><td>'A_sxb/d_xb'%</td><td>Ø'd_bxb'/'s_xb'</td></tr>
#if k ≤ 2
'<tr><td>Bottom parallel to <i>y</i></td><td>'M_Ed_yb'</td><td>'A_syb'</td><td>'A_syb/d_yb'%</td><td>Ø'd_byb'/'s_yb'</td></tr>
#else
'<tr><td>Bottom parallel to <i>y</i></td><td></td><td></td><td></td><td>Ø'd_byb'/'s_yb'</td></tr>
#end if
'<tr><td>Top parallel to <i>x</i> - left side</td><td>'M_Ed_xte1'</td><td>'A_sxt1'</td><td>'A_sxt1/d_xt1'%</td><td>Ø'd_bxt'/'s_xt1'</td></tr>
'<tr><td>Top parallel to <i>x</i> - right side</td><td>'M_Ed_xte2'</td><td>'A_sxt2'</td><td>'A_sxt2/d_xt2'%</td><td>Ø'd_bxt'/'s_xt2'</td></tr>
#if k ≤ 2
'<tr><td>Top parallel to <i>y</i> - down side</td><td>'M_Ed_yte1'</td><td>'A_syt1'</td><td>'A_syt1/d_yt1'%</td><td>Ø'd_byt'/'s_yt1'</td></tr>
'<tr><td>Top parallel to <i>y</i> - up side</td><td>'M_Ed_yte2'</td><td>'A_syt2'</td><td>'A_syt2/d_yt2'%</td><td>Ø'd_byt'/'s_yt2'</td></tr>
#end if
'</table>
#end if
#show
'</div> 4 6 25 25 14 2 0 0 0 0 0 0 0 0 2.5 2 25 0.85 500 8 8 6 6
Clear spans(lx < ly)
lx_cl = 4 m, ly_cl = 6 m
Beam width along x - by = 25 cm
Beam width along y - bx = 25 cm
Slab thickness - h = 14 cm
Concrete cover - c = 2 cm
Design span lengths
lx = lx_cl + min(bx100; h100) = 4 + min(25100; 14100) = 4.14 m
ly = ly_cl + min(by100; h100) = 6 + min(25100; 14100) = 6.14 m
Span ratio - k = lylx = 6.144.14 = 1.48
Design spans lengths and edge moments in adjacent panels
|
Parallel to x - |
lx1 = 0 m; |
MEd_xt1_ = 0 kN·m/m |
|
lx2 = 0 m; |
MEd_xt2_ = 0 kN·m/m |
|
Along y - |
ly1 = 0 m; |
MEd_yt1_ = 0 kN·m/m |
|
ly2 = 0 m; |
MEd_yt2_ = 0 kN·m/m |
(in absence of contiguous span l = 0)
Characteristic uniform loads
Self weight - gsw = 0.25 · h = 0.25 · 14 = 3.5 kN/m²
Permanent - gk = 2.5 kN/m²
Variable - qk = 2 kN/m²
Partial safety factor - γG = 1.35 ; γQ = 1.5
Design uniform loads
p = ( gsw + gk ) · γG + qk · γQ = ( 3.5 + 2.5 ) · 1.35 + 2 · 1.5 = 11.1 kN/m²
Beam loads
(as uniform loads)
|
Parallel to x - |
gk_x = ( gsw + gk ) · lx_cl4 = ( 3.5 + 2.5 ) · 44 = 6 kN/m |
|
qk_x = qk · lx_cl4 = 2 · 44 = 2 kN/m | |
|
Parallel to y - |
gk_y = ( gsw + gk ) · lx_cl · (ly_cl − lx_cl2)2 · ly_cl = ( 3.5 + 2.5 ) · 4 · (6 − 42)2 · 6 = 8 kN/m |
|
qk_y = qk · lx_cl · (ly_cl − lx_cl2)2 · ly_cl = 2 · 4 · (6 − 42)2 · 6 = 2.67 kN/m |
Concrete [EN 1992-1-1, Table 3.1]
Characteristic compressive cylinder strength - fck = 25 MPa
Partial safety factors for concrete - γc = 1.5 , αcc = 0.85
Design compressive cylinder strength - fcd = αcc · fckγc = 0.85 · 251.5 = 14.17 MPa
Factor defining the effective height of the compression zone - λ = 0.8
Factor defining the effective strength - η = 1
Mean value of axial tensile strength - fctm = 0.3 · fck23 = 0.3 · 2523 = 2.56 MPa
Steel
Characteristic yield strength - fyk = 500 MPa
Partial safety factor for steel - γs = 1.15
Design yield strength - fyd = fykγs = 5001.15 = 434.78 MPa
Limit relative compressive zone depth - ξlim = 0.45
Bar diameters
- Bottom parallel to x - dbxb = 8 mm, parallel to y - dbyb = 8 mm
- Top parallel to x - dbxt = 6 mm, parallel to y - dbyt = 6 mm
Shear forces and bending moments coefficients
For the actual static scheme
- for shear forces
| Short span coefficients | Long span coefficients | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| k = | 1.0 | 1.1 | 1.2 | 1.3 | 1.4 | 1.5 | 1.75 | 2.0 | ||
| V | βvx = | 0.33 | 0.36 | 0.39 | 0.41 | 0.43 | 0.45 | 0.48 | 0.50 | βvy = 0.33 |
βvx = 0.43 + ( 0.45 − 0.43 ) · ( k − 1.4 ) 0.1 = 0.43 + ( 0.45 − 0.43 ) · ( 1.48 − 1.4 ) 0.1 = 0.447
- for moments in mid-span
| Short span coefficients | Long span coefficients | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| k = | 1.0 | 1.1 | 1.2 | 1.3 | 1.4 | 1.5 | 1.75 | 2.0 | ||
| +M | βxb1 = | 0.024 | 0.028 | 0.032 | 0.035 | 0.037 | 0.040 | 0.044 | 0.048 | βyb1 = 0.024 |
| -M | βxt = | 0.031 | 0.037 | 0.042 | 0.046 | 0.050 | 0.053 | 0.059 | 0.063 | βyt = 0.032 |
βxb1 = 0.037 + ( 0.04 − 0.037 ) · ( k − 1.4 ) 0.1 = 0.037 + ( 0.04 − 0.037 ) · ( 1.48 − 1.4 ) 0.1 = 0.0395
- for moments over supports
βxt = 0.05 + ( 0.053 − 0.05 ) · ( k − 1.4 ) 0.1 = 0.05 + ( 0.053 − 0.05 ) · ( 1.48 − 1.4 ) 0.1 = 0.0525
For simply supported slab
- for moments in mid-span
| Short span coefficients | Long span coefficients | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| k = | 1.0 | 1.1 | 1.2 | 1.3 | 1.4 | 1.5 | 1.75 | 2.0 | |
| βxb2 = | 0.055 | 0.065 | 0.074 | 0.081 | 0.087 | 0.092 | 0.103 | 0.111 | βyb2 = 0.056 |
βxb2 = 0.087 + ( 0.092 − 0.087 ) · ( k − 1.4 ) 0.1 = 0.087 + ( 0.092 − 0.087 ) · ( 1.48 − 1.4 ) 0.1 = 0.0912
Loads for accounting alternation in adjacent spans
p1 = γG · ( gsw + gk ) + γQ · qk2 = 1.35 · ( 3.5 + 2.5 ) + 1.5 · 22 = 9.6 kN/m²
p2 = γQ · qk2 = 1.5 · 22 = 1.5 kN/m²
Bending moments
Span moment (including load alternation)
MEd_xb = βxb1 · p1 · lx2 + βxb2 · p2 · lx2 = 0.0395 · 9.6 · 4.142 + 0.0912 · 1.5 · 4.142 = 8.84 kN·m/m
MEd_yb = βyb1 · p1 · lx2 + βyb2 · p2 · lx2 = 0.024 · 9.6 · 4.142 + 0.056 · 1.5 · 4.142 = 5.39 kN·m/m
Support moment
MEd_xt = βxt · p · lx2 = 0.0525 · 11.1 · 4.142 = 9.99 kN·m/m
MEd_yt = βyt · p · lx2 = 0.032 · 11.1 · 4.142 = 6.09 kN·m/m
Average moments at supports from the adjacent spans
MEd_xt1 = MEd_xt1_ · lx1 + MEd_xt · lxlx1 + lx = 0 · 0 + 9.99 · 4.140 + 4.14 = 9.99 kN·m/m
MEd_xt2 = MEd_xt2_ · lx2 + MEd_xt · lxlx2 + lx = 0 · 0 + 9.99 · 4.140 + 4.14 = 9.99 kN·m/m
MEd_yt1 = MEd_yt1_ · ly1 + MEd_yt · lyly1 + ly = 0 · 0 + 6.09 · 6.140 + 6.14 = 6.09 kN·m/m
MEd_yt2 = MEd_yt2_ · ly2 + MEd_yt · lyly2 + ly = 0 · 0 + 6.09 · 6.140 + 6.14 = 6.09 kN·m/m
Shear forces
VEd_x = βvx · p · lx = 0.447 · 11.1 · 4.14 = 20.52 kN/m
VEd_y = βvy · p · lx = 0.33 · 11.1 · 4.14 = 15.16 kN/m
Edge support moments
MEd_xte1 = MEd_xt1 − VEd_x · min(bx200; h200) = 9.99 − 20.52 · min(25200; 14200) = 8.55 kN·m/m
MEd_xte2 = MEd_xt2 − VEd_x · min(bx200; h200) = 9.99 − 20.52 · min(25200; 14200) = 8.55 kN·m/m
MEd_yte1 = MEd_yt1 − VEd_y · min(by200; h200) = 6.09 − 15.16 · min(25200; 14200) = 5.03 kN·m/m
MEd_yte2 = MEd_yt2 − VEd_y · min(by200; h200) = 6.09 − 15.16 · min(25200; 14200) = 5.03 kN·m/m
Bottom reinforcement parallel to x
Effective cross section depth - dxb = h − c − dbxb20 = 14 − 2 − 820 = 11.6 cm
Relative design bending moment- mEd = 10 · MEd_xbdxb2 · η · fcd = 10 · 8.8411.62 · 1 · 14.17 = 0.0464
Compression zone depth - x = dxbλ · ( 1 − √ 1 − 2 · mEd ) = 11.60.8 · ( 1 − √ 1 − 2 · 0.0464 ) = 0.689 cm
Relative compression zone depth - ξ = xdxb = 0.68911.6 = 0.0594
Lever arm of internal forces - z = dxb − 0.5 · λ · x = 11.6 − 0.5 · 0.8 · 0.689 = 11.32 cm
Area of required reinforcement - Asxb = MEd_xb · 103z · fyd = 8.84 · 10311.32 · 434.78 = 1.8 cm²/m
Reinforcement ratio - ρxb = Asxbdxb · 100 = 1.811.6 · 100 = 0.00155
The rest of the reinforcement is calculated similarly.
Bottom reinforcement
[EN 1992-1-1, § 9.2.1.1 (1)]
Minimum reinforcement ratio
ρmin = max(0.26 · fctmfyk; 0.0013) = max(0.26 · 2.56500; 0.0013) = 0.00133
[EN 1992-1-1, § 9.2.1.1 (3)]
Maximum reinforcement ratio - ρmax = 0.04
Required bar spacing
sxb = floor(π · dbxb24 · Asxb) = floor(3.14 · 824 · 1.8) = 27 cm
[EN 1992-1-1, § 9.3.1.1 (3)]
Maximum bar spacing
smax = min ( 2 · h; 25 ) = min ( 2 · 14; 25 ) = 25 cm
sxb > smax. Assume sxb = smax = 25 cm
The rest of the reinforcement is detailed similarly.
| Reinforcement | MEd kN·m/m | As cm2/m | ρ, % | Selected |
|---|---|---|---|---|
| Bottom parallel to x | 8.84 | 1.8 | 0.15% | Ø8/25 |
| Bottom parallel to y | 5.39 | 1.44 | 0.13% | Ø8/25 |
| Top parallel to x - left side | 8.55 | 1.72 | 0.15% | Ø6/16 |
| Top parallel to x - right side | 8.55 | 1.72 | 0.15% | Ø6/16 |
| Top parallel to y - down side | 5.03 | 1.48 | 0.13% | Ø6/19 |
| Top parallel to y - up side | 5.03 | 1.48 | 0.13% | Ø6/19 |
Restrained Adjacent Edges¶
Two-way slab panel with two adjacent edges restrained and the other two simply supported, typical of a corner panel of a floor system.
'<small>According to <strong>Eurocode</strong>: EN 1992-1-1</small>
'<div style="max-width:180mm;">
'<img alt="sffs-span-cl.png" class="side" style="height:180pt;" src="../../Images/structures/rc/spans/sffs-span-cl.png">
'<h4>Dimensions</h4>
'Clear spans (<i>l</i><sub>x</sub> < <i>l</i><sub>y</sub>)
l_x_cl = ?'m,'l_y_cl = ?'m
'Beam width along<i>x</i> -'b_y = ?'cm
'Beam width along <i>y</i> -'b_x = ?'cm
'Slab thickness -'h = ?'cm
'Concrete cover -'c = ?'cm
#post
'Design span lengths
l_x = l_x_cl + min(b_x/100;h/100)'m
l_y = l_y_cl + min(b_y/100;h/100)'m
'Span ratio -'k = l_y/l_x
#if k < 1
'Span length <i>l</i><sub>y</sub> should be greater than <i>l</i><sub>x</sub>
#else
#show
'Design spans lengths and edge moments in adjacent panels
'<img alt="sffs-span.png" class="side" style="height:140pt;" src="../../Images/structures/rc/spans/sffs-span.png">
'<table><tr><td>
'Along <i>x</i> - <br />
'</td><td>
l_x2 = ?'m;
'</td><td>
M_Ed_xt2_ = ?'kN·m/m
'</td></tr></table>
'<!--
#if k ≤ 2
'<-->
'<table><tr><td>
'Along <i>y</i> -
'</td><td>
l_y2 = ?'m;
'</td><td>
M_Ed_yt2_ = ?'kN·m/m
'</td></tr></table>
'<!--
#end if
'<-->
'(in absence of contiguous span <i>l</i> = 0)
'<h4>Loads</h4>
'Characteristic uniform loads
'Self weight -'g_sw = 0.25*h'kN/m²
'Permanent -'g_k = ?'kN/m²
'Variable -'q_k = ?'kN/m²
'Partial safety factor -'γ_G = 1.35';'γ_Q = 1.5
#post
'Design uniform loads
p = (g_sw + g_k)*γ_G + q_k*γ_Q'kN/m²
'<div class="fold">
'<p><b>Beam loads</b></p>
'(as uniform loads)
#if k ≤ 2
'<table><tr><td rowspan="2">
'Along <i>x</i> - <br />
'</td><td>
g_k_x = (g_sw + g_k)*l_x_cl/4'kN/m
'</td></tr><tr><td>
q_k_x = q_k*l_x_cl/4'kN/m
'</td></tr><tr><td rowspan="2">
'Along <i>y</i> - <br />
'</td><td>
g_k_y = (g_sw + g_k)*l_x_cl*(l_y_cl - l_x_cl/2)/(2*l_y_cl)'kN/m
'</td></tr><tr><td>
q_k_y = q_k*l_x_cl*(l_y_cl - l_x_cl/2)/(2*l_y_cl)'kN/m
'</td></tr></table>
#else
'Along <i>y</i> - <br />'q_y = p*l_x_cl/2'kN/m
#end if
'</div>
#show
'<h4>Material properties</h4>
'<p><b>Concrete</b> [EN 1992-1-1, Table 3.1]</p>
'Characteristic compressive cylinder strength -'f_ck = ?'MPa
'Partial safety factors for concrete -'γ_c = 1.5','α_cc = ?
#post
'Design compressive cylinder strength -'f_cd = α_cc*f_ck/γ_c'MPa
'Factor defining the effective height of the compression zone -'λ = 0.8'
'Factor defining the effective strength - 'η = 1.0
'Mean value of axial tensile strength -'f_ctm = 0.30*f_ck^(2/3)'MPa
#show
'<p><b>Steel</b></p>
'Characteristic yield strength -'f_yk = ?'MPa
#post
'Partial safety factor for steel - 'γ_s = 1.15
'Design yield strength - 'f_yd = f_yk/γ_s'MPa
'Limit relative compressive zone depth -'ξ_lim = 0.45
#show
'<p><b>Bar diameters</b></p>
'- Bottom parallel to <i>x</i> -'d_bxb = ?'mm, parallel to <i>y</i> -'d_byb = ?'mm
'- Top parallel to <i>x</i> -'d_bxt = ?'mm, parallel to <i>y</i> -'d_byt = ?'mm
#post
'<div class="fold">
'<h4>Static calculations</h4>
'<p><b>Shear forces and bending moments coefficients</b></p>
#if l_y > 2*l_x
l_y/l_x'>'2'- One-way slab
'For the actual static scheme
'- for moments in mid-span-'β_xb1 = 0.0703
'- for moments over supports -'β_xt = 0.125
'- for shear forces -'β_vx1 = 0.375';'β_vx2 = 0.625
'For simply supported slab
'- for span moments -'β_xb2 = 0.125
#else if l_x ≡ l_y
l_y/l_x'- Two-way square slab
'For the actual static scheme
'- for moments in mid-span-'β_xb1 = 0.036';'β_yb1 = 0.034
'- for moments over supports -'β_xt = 0.047';'β_yt = 0.045
'- for shear forces -'β_vx1 = 0.26';'β_vy1 = 0.26';'β_vx2 = 0.40';'β_vy2 = 0.40
'For simply supported slab
'- for span moments -'β_xb2 = 0.055';'β_yb2 = 0.055
#else
'For the actual static scheme
'- for shear forces
'<table class="bordered centered">
'<tr><th rowspan="2"></th><th style="width:100mm;" colspan="9">Short span coefficients</th><th style="width:30mm;" rowspan="2">Long span coefficientsth></tr>
'<tr><th><i>k</i> =</th><th>1.0</th><th>1.1</th><th>1.2</th><th>1.3</th><th>1.4</th><th>1.5</th><th>1.75</th><th>2.0</th></tr>
'<tr><th>V</th><td><i>β</i><sub>vx1</sub> = </td><td>0.26</td><td>0.29</td><td>0.31</td><td>0.33</td><td>0.34</td><td>0.35</td><td>0.38</td><td>0.40</td><td>'β_vy1 = 0.26'</td></tr>
'<tr><th>V</th><td><i>β</i><sub>vx2</sub> = </td><td>0.40</td><td>0.44</td><td>0.47</td><td>0.50</td><td>0.52</td><td>0.54</td><td>0.57</td><td>0.60</td><td>'β_vy2 = 0.40'</td></tr></table>
#if k < 1.1
β_vx1 = 0.26 + (0.29 - 0.26)*(k - 1)/0.1
β_vx2 = 0.40 + (0.44 - 0.40)*(k - 1)/0.1
#else if k < 1.2
β_vx1 = 0.29 + (0.31 - 0.29)*(k - 1.1)/0.1
β_vx2 = 0.44 + (0.47 - 0.44)*(k - 1.1)/0.1
#else if k < 1.3
β_vx1 = 0.31 + (0.33 - 0.31)*(k - 1.2)/0.1
β_vx2 = 0.47 + (0.50 - 0.47)*(k - 1.2)/0.1
#else if k < 1.4
β_vx1 = 0.33 + (0.34 - 0.33)*(k - 1.3)/0.1
β_vx2 = 0.50 + (0.52 - 0.50)*(k - 1.3)/0.1
#else if k < 1.5
β_vx1 = 0.34 + (0.35 - 0.34)*(k - 1.4)/0.1
β_vx2 = 0.52 + (0.54 - 0.52)*(k - 1.4)/0.1
#else if k < 1.75
β_vx1 = 0.35 + (0.38 - 0.35)*(k - 1.5)/0.25
β_vx2 = 0.54 + (0.57 - 0.54)*(k - 1.5)/0.25
#else if k < 2
β_vx1 = 0.38 + (0.40 - 0.38)*(k - 1.75)/0.25
β_vx2 = 0.57 + (0.60 - 0.57)*(k - 1.75)/0.25
#else if k ≡ 2
β_vx1 = 0.40
β_vx2 = 0.60
#end if
'- for moments in mid-span
'<table class="bordered centered">
'<tr><th rowspan="2"></th><th style="width:100mm;" colspan="9">Short span coefficients</th><th style="width:30mm;" rowspan="2">Long span coefficients</th></tr>
'<tr><th><i>k</i> =</th><th>1.0</th><th>1.1</th><th>1.2</th><th>1.3</th><th>1.4</th><th>1.5</th><th>1.75</th><th>2.0</th></tr>
'<tr><th>+M</th><td><i>β</i><sub>xb1</sub> = </td><td>0.036</td><td>0.042</td><td>0.047</td><td>0.051</td><td>0.055</td><td>0.059</td><td>0.065</td><td>0.070</td><td>'β_yb1 = 0.034'</td></tr>
'<tr><th>-M</th><td><i>β</i><sub>xt</sub> = </td><td>0.047</td><td>0.056</td><td>0.063</td><td>0.069</td><td>0.074</td><td>0.078</td><td>0.087</td><td>0.093</td><td>'β_yt = 0.045'</td></tr></table>
#if k < 1.1
β_xb1 = 0.036 + (0.042 - 0.036)*(k - 1)/0.1
#else if k < 1.2
β_xb1 = 0.042 + (0.047 - 0.042)*(k - 1.1)/0.1
#else if k < 1.3
β_xb1 = 0.047 + (0.051 - 0.047)*(k - 1.2)/0.1
#else if k < 1.4
β_xb1 = 0.051 + (0.055 - 0.051)*(k - 1.3)/0.1
#else if k < 1.5
β_xb1 = 0.055 + (0.059 - 0.055)*(k - 1.4)/0.1
#else if k < 1.75
β_xb1 = 0.059 + (0.065 - 0.059)*(k - 1.5)/0.25
#else if k < 2
β_xb1 = 0.065 + (0.07 - 0.065)*(k - 1.75)/0.25
#else if k ≡ 2
β_xb1 = 0.07
#end if
'- for moments over supports
#if k < 1.1
β_xt = 0.047 + (0.056 - 0.047)*(k - 1)/0.1
#else if k < 1.2
β_xt = 0.056 + (0.063 - 0.056)*(k - 1.1)/0.1
#else if k < 1.3
β_xt = 0.063 + (0.069 - 0.063)*(k - 1.2)/0.1
#else if k < 1.4
β_xt = 0.069 + (0.074 - 0.069)*(k - 1.3)/0.1
#else if k < 1.5
β_xt = 0.074 + (0.078 - 0.074)*(k - 1.4)/0.1
#else if k < 1.75
β_xt = 0.078 + (0.087 - 0.078)*(k - 1.5)/0.25
#else if k < 2
β_xt = 0.087 + (0.093 - 0.087)*(k - 1.75)/0.25
#else if k ≡ 2
β_xt = 0.093
#end if
'For simply supported slab
' - for moments in mid-span
'<table class="bordered centered">
'<tr><th style="width:100mm;" colspan="9">Short span coefficients</th><th style="width:30mm;" rowspan="2">Long span coefficients</th></tr>
'<tr><th><i>k</i> =</th><th>1.0</th><th>1.1</th><th>1.2</th><th>1.3</th><th>1.4</th><th>1.5</th><th>1.75</th><th>2.0</th></tr>
'<tr><td><i>β</i><sub>xb2</sub> = </td><td>0.055</td><td>0.065</td><td>0.074</td><td>0.081</td><td>0.087</td><td>0.092</td><td>0.103</td><td>0.111</td><td>'β_yb2 = 0.056'</td></tr></table>
#if k < 1.1
β_xb2 = 0.055 + (0.065 - 0.055)*(k - 1)/0.1
#else if k < 1.2
β_xb2 = 0.065 + (0.074 - 0.065)*(k - 1.1)/0.1
#else if k < 1.3
β_xb2 = 0.074 + (0.081 - 0.074)*(k - 1.2)/0.1
#else if k < 1.4
β_xb2 = 0.081 + (0.087 - 0.081)*(k - 1.3)/0.1
#else if k < 1.5
β_xb2 = 0.087 + (0.092 - 0.087)*(k - 1.4)/0.1
#else if k < 1.75
β_xb2 = 0.092 + (0.103 - 0.092)*(k - 1.5)/0.25
#else if k < 2
β_xb2 = 0.103 + (0.111 - 0.103)*(k - 1.75)/0.25
#else if k ≡ 2
β_xb2 = 0.111
#end if
#end if
'<p><b>Loads for accounting alternation in adjacent spans</b></p>
p_1 = γ_G*(g_sw + g_k) + γ_Q*q_k/2'kN/m²
p_2 = γ_Q*q_k/2'kN/m²
'<p><b>Bending moments</b></p>
'Span moment (including load alternation)
M_Ed_xb = β_xb1*p_1*l_x^2 + β_xb2*p_2*l_x^2'kN·m/m
#if k ≤ 2
M_Ed_yb = β_yb1*p_1*l_x^2 + β_yb2*p_2*l_x^2'kN·m/m
#end if
'Support moment
M_Ed_xt = β_xt*p*l_x^2'kN·m/m
#if k ≤ 2
M_Ed_yt = β_yt*p*l_x^2'kN·m/m
#end if
'<p><b>Average moments at supports from the adjacent spans</b></p>
M_Ed_xt2 = (M_Ed_xt2_*l_x2 + M_Ed_xt*l_x)/(l_x2 + l_x)'kN·m/m
#if k ≤ 2
M_Ed_yt2 = (M_Ed_yt2_*l_y2 + M_Ed_yt*l_y)/(l_y2 + l_y)'kN·m/m
#end if
'<p><b>Shear forces</b></p>
#if k ≤ 2
V_Ed_x1 = β_vx1*p*l_x'kN/m
V_Ed_x2 = β_vx2*p*l_x'kN/m
V_Ed_y1 = β_vy1*p*l_x'kN/m
V_Ed_y2 = β_vy2*p*l_x'kN/m
#else
V_Ed_x1 = β_vx1*p*l_x'kN/m
V_Ed_x2 = β_vx2*p*l_x'kN/m
#end if
'<p><b>Edge support moments</b></p>
M_Ed_xte2 = M_Ed_xt2 - V_Ed_x2*min(b_x/200;h/200)'kN·m/m
#if k ≤ 2
M_Ed_yte2 = M_Ed_yt2 - V_Ed_y2*min(b_y/200;h/200)'kN·m/m
#end if
'</div><div class="fold">
'<h4>Bending design</h4>
'<p><b>Bottom reinforcement parallel to <i>x</i></b></p>
'Effective cross section depth -'d_xb = h - c - d_bxb/20'cm
'Relative design bending moment-'m_Ed = 10*M_Ed_xb/(d_xb^2*η*f_cd)
'Compression zone depth -'x = d_xb/λ*(1 - sqr(1 - 2*m_Ed))'cm
'Relative compression zone depth -'ξ = x/d_xb
'Lever arm of internal forces -'z = d_xb - 0.5*λ*x'cm
'Area of required reinforcement -'A_sxb = M_Ed_xb*10^3/(z*f_yd)'cm²/m
'Reinforcement ratio-'ρ_xb = A_sxb/(d_xb*100)
#if ξ > ξ_lim
'<p class="err">'ξ'>'ξ_lim'- Reinforcement in compression zone is required.</p>
#end if
#hide
#if k ≤ 2
'<p><b>Bottom reinforcement parallel to <i>y</i></b></p>
'Effective cross section depth -'d_yb = h - c - d_bxb/10 - d_byb/20'cm
'Relative design bending moment -'m_Ed = 10*M_Ed_yb/(d_yb^2*η*f_cd)
'Compression zone depth -'x = d_yb/λ*(1 - sqr(1 - 2*m_Ed))'cm
'Relative compression zone depth-'ξ = x/d_yb
'Lever arm of internal forces-'z = d_yb - 0.5*λ*x'cm
'Area of required reinforcement -'A_syb = M_Ed_yb*10^3/(z*f_yd)'cm²/m
'Reinforcement ratio -'ρ_yb = A_syb/(d_yb*100)
#if ξ > ξ_lim
'<p class="err">'ξ'>'ξ_lim'-Reinforcement in compression zone is required.</p>
#end if
#end if
'<p><b>Top reinforcement parallel to <i>x</i> за <i>M</i><sub>Ed_xte2</sub></b></p>
'Effective cross section depth -'d_xt2 = h - c - d_bxt/20'cm
'Relative design bending moment -'m_Ed = 10*M_Ed_xte2/(d_xt2^2*η*f_cd)
'Compression zone depth -'x = d_xt2/λ*(1 - sqr(1 - 2*m_Ed))'cm
'Relative compression zone depth -'ξ = x/d_xt2
'Lever arm of internal forces -'z = d_xt2 - 0.5*λ*x'cm
'Area of required reinforcement -'A_sxt2 = M_Ed_xte2*10^3/(z*f_yd)'cm²/m
'Reinforcement ratio -'ρ_xt2 = A_sxt2/(d_xt2*100)
#if ξ > ξ_lim
'<p class="err">'ξ'>'ξ_lim'- Reinforcement in compression zone is required.</p>
#end if
#if k ≤ 2
'<p><b>Top reinforcement parallel to <i>y</i> за <i>M</i><sub>Ed_yte2</sub></b></p>
'Effective cross section depth -'d_yt2 = h - c - d_bxt/10 - d_byt/20'cm
'Relative design bending moment -'m_Ed = 10*M_Ed_yte2/(d_xt2^2*η*f_cd)
'Compression zone depth -'x = d_yt2/λ*(1 - sqr(1 - 2*m_Ed))'cm
'Relative compression zone depth -'ξ = x/d_yt2
'Lever arm of internal forces-'z = d_yt2 - 0.5*λ*x'cm
'Area of required reinforcement -'A_syt2 = M_Ed_yte2*10^3/(z*f_yd)'cm²/m
'Reinforcement ratio-'ρ_yt2 = A_syt2/(d_yt2*100)
#if ξ > ξ_lim
'<p class="err">'ξ'>'ξ_lim'- Reinforcement in compression zone is required.</p>
#end if
#end if
#post
'<p><b>The rest of the reinforcement is calculated similarly.</b></p>
'</div><div class="fold">
'<h4>Reinforcement detailing</h4>
'<p><b>Bottom reinforcement</b></p>
'<p class="ref">[EN 1992-1-1, § 9.2.1.1 (1)]</p>
'Minimum reinforcement ratio
ρ_min = max(0.26*f_ctm/f_yk;0.0013)
'<p class="ref">[EN 1992-1-1, § 9.2.1.1 (3)]</p>
'Maximum reinforcement ratio-'ρ_max = 0.04
#if ρ_xb < ρ_min
'<p><i>ρ</i><sub>xb</sub> < <i>ρ</i><sub>min</sub>. Minimum reinforcement is assumed -'A_sxb = ρ_min*d_xb*100'cm²/m
#else if ρ_xb > ρ_max
'<p class="err">Reinforcement ratio is greater than maximum:'ρ_xb'>'ρ_max'</p>
#end if
#if k > 2
'<p class="ref">[EN 1992-1-1, § 9.3.1.1 (2)]</p>
'Minimum bottom reinforcement parallel to y
A_syb = 0.2*A_sxb'cm²/m
#else
'<!--'
#if ρ_yb < ρ_min
'<p><i>ρ</i><sub>yb</sub> < <i>ρ</i><sub>min</sub>. Minimum reinforcement is assumed -'A_syb = ρ_min*d_yb*100'cm²/m
#else if ρ_yb > ρ_max
'<p class="err">Reinforcement ratio is greater than maximum:'ρ_yb'>'ρ_max'</p>
#end if
'<-->
#end if
'Required bar spacing
s_xb = floor(π*d_bxb^2/(4*A_sxb))'cm
#if k > 2
'<p class="ref">[EN 1992-1-1, § 9.3.1.1 (3)]</p>
'Minimum bottom reinforcement parallel to y
s_yb = min(3*h;40)'cm
'Required bar diameter for bottom reinforcement parallel to y
d_byb = max(ceiling(sqr(4*A_syb*s_yb/π));6)'mm
#else
'<!--'s_yb = floor(π*d_byb^2/(4*A_syb))'cm-->
#end if
'<p class="ref">[EN 1992-1-1, § 9.3.1.1 (3)]</p>
'Maximum bar spacing
s_max = min(2*h;25)'cm
#if s_xb > s_max
'<p><i>s</i><sub>xb</sub> > <i>s</i><sub>max</sub>. Assume's_xb = s_max'cm</p>
#end if
#if (k ≤ 2)*(s_yb > s_max)
'<!--<p><i>s</i><sub>yb</sub> > <i>s</i><sub>max</sub>. Assume's_yb = s_max'cm</p>-->
#end if
#hide
'<p><b>Top reinforcement</b></p>
#if ρ_xt2 < ρ_min
'<p><i>ρ</i><sub>xt2</sub> < <i>ρ</i><sub>min</sub>. Minimum reinforcement is assumed -'A_sxt2 = ρ_min*d_xt2*100'cm²/m
#else if ρ_xt2 > ρ_max
'<p class="err">Reinforcement ratio is greater than maximum:'ρ_xt2'>'ρ_max'</p>
#end if
#if k ≤ 2
#if ρ_yt2 < ρ_min
'<p><i>ρ</i><sub>yt2</sub> < <i>ρ</i><sub>min</sub>. Minimum reinforcement is assumed -'A_syt2 = ρ_min*d_yt2*100'cm²/m
#else if ρ_yt2 > ρ_max
'<p class="err">Reinforcement ratio is greater than maximum:'ρ_yt2'>'ρ_max'</p>
#end if
#end if
'Required bar spacing
s_xt2 = floor(π*d_bxt^2/(4*A_sxt2))'cm
#if k ≤ 2
s_yt2 = floor(π*d_byt^2/(4*A_syt2))'cm
#end if
'<p class="ref">[EN 1992-1-1, § 9.3.1.1 (3)]</p>
'Maximum bar spacing
s_max = min(2*h;25)'cm
#if s_xt2 > s_max
'<p><i>s</i><sub>xt2</sub> > <i>s</i><sub>max</sub>. Assume's_xt2 = s_max'cm</p>
#end if
#if (k ≤ 2)*(s_yt2 > s_max)
'<p><i>s</i><sub>yt2</sub> > <i>s</i><sub>max</sub>. Assume's_yt2 = s_max'cm</p>
#end if
#post
'<p><b>The rest of the reinforcement is detailed similarly.</b></p>
'</div>
'<h4>Results</h4>
#val
'<table class="bordered centered">
'<tr><th>Reinforcement</th><th>M<sub>Ed</sub><br>kN·m/m</th><th>A<sub>s</sub><br>cm<sup>2</sup>/m</th><th>ρ, %</th><th>Selected</tr>
'<tr><td>Bottom parallel to<i>x</i></td><td>'M_Ed_xb'</td><td>'A_sxb'</td><td>'A_sxb/d_xb'%</td><td>Ø'd_bxb'/'s_xb'</td></tr>
#if k ≤ 2
'<tr><td>Bottom parallel to <i>y</i></td><td>'M_Ed_yb'</td><td>'A_syb'</td><td>'A_syb/d_yb'%</td><td>Ø'd_byb'/'s_yb'</td></tr>
#else
'<tr><td>Bottom parallel to <i>y</i></td><td></td><td></td><td></td><td>Ø'd_byb'/'s_yb'</td></tr>
#end if
'<tr><td>Top parallel to <i>x</i> - right side</td><td>'M_Ed_xte2'</td><td>'A_sxt2'</td><td>'A_sxt2/d_xt2'%</td><td>Ø'd_bxt'/'s_xt2'</td></tr>
#if k ≤ 2
'<tr><td>Top parallel to <i>y</i> - up side</td><td>'M_Ed_yte2'</td><td>'A_syt2'</td><td>'A_syt2/d_yt2'%</td><td>Ø'd_byt'/'s_yt2'</td></tr>
#end if
'</table>
#end if
#show
'</div> 4 6 25 25 14 2 0 0 0 0 2.5 2 25 0.85 500 8 8 6 6
Clear spans (lx < ly)
lx_cl = 4 m, ly_cl = 6 m
Beam width alongx - by = 25 cm
Beam width along y - bx = 25 cm
Slab thickness - h = 14 cm
Concrete cover - c = 2 cm
Design span lengths
lx = lx_cl + min(bx100; h100) = 4 + min(25100; 14100) = 4.14 m
ly = ly_cl + min(by100; h100) = 6 + min(25100; 14100) = 6.14 m
Span ratio - k = lylx = 6.144.14 = 1.48
Design spans lengths and edge moments in adjacent panels
|
Along x - |
lx2 = 0 m; |
MEd_xt2_ = 0 kN·m/m |
|
Along y - |
ly2 = 0 m; |
MEd_yt2_ = 0 kN·m/m |
(in absence of contiguous span l = 0)
Characteristic uniform loads
Self weight - gsw = 0.25 · h = 0.25 · 14 = 3.5 kN/m²
Permanent - gk = 2.5 kN/m²
Variable - qk = 2 kN/m²
Partial safety factor - γG = 1.35 ; γQ = 1.5
Design uniform loads
p = ( gsw + gk ) · γG + qk · γQ = ( 3.5 + 2.5 ) · 1.35 + 2 · 1.5 = 11.1 kN/m²
Beam loads
(as uniform loads)
|
Along x - |
gk_x = ( gsw + gk ) · lx_cl4 = ( 3.5 + 2.5 ) · 44 = 6 kN/m |
|
qk_x = qk · lx_cl4 = 2 · 44 = 2 kN/m | |
|
Along y - |
gk_y = ( gsw + gk ) · lx_cl · (ly_cl − lx_cl2)2 · ly_cl = ( 3.5 + 2.5 ) · 4 · (6 − 42)2 · 6 = 8 kN/m |
|
qk_y = qk · lx_cl · (ly_cl − lx_cl2)2 · ly_cl = 2 · 4 · (6 − 42)2 · 6 = 2.67 kN/m |
Concrete [EN 1992-1-1, Table 3.1]
Characteristic compressive cylinder strength - fck = 25 MPa
Partial safety factors for concrete - γc = 1.5 , αcc = 0.85
Design compressive cylinder strength - fcd = αcc · fckγc = 0.85 · 251.5 = 14.17 MPa
Factor defining the effective height of the compression zone - λ = 0.8
Factor defining the effective strength - η = 1
Mean value of axial tensile strength - fctm = 0.3 · fck23 = 0.3 · 2523 = 2.56 MPa
Steel
Characteristic yield strength - fyk = 500 MPa
Partial safety factor for steel - γs = 1.15
Design yield strength - fyd = fykγs = 5001.15 = 434.78 MPa
Limit relative compressive zone depth - ξlim = 0.45
Bar diameters
- Bottom parallel to x - dbxb = 8 mm, parallel to y - dbyb = 8 mm
- Top parallel to x - dbxt = 6 mm, parallel to y - dbyt = 6 mm
Shear forces and bending moments coefficients
For the actual static scheme
- for shear forces
| Short span coefficients | Long span coefficientsth> | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| k = | 1.0 | 1.1 | 1.2 | 1.3 | 1.4 | 1.5 | 1.75 | 2.0 | ||
| V | βvx1 = | 0.26 | 0.29 | 0.31 | 0.33 | 0.34 | 0.35 | 0.38 | 0.40 | βvy1 = 0.26 |
| V | βvx2 = | 0.40 | 0.44 | 0.47 | 0.50 | 0.52 | 0.54 | 0.57 | 0.60 | βvy2 = 0.4 |
βvx1 = 0.34 + ( 0.35 − 0.34 ) · ( k − 1.4 ) 0.1 = 0.34 + ( 0.35 − 0.34 ) · ( 1.48 − 1.4 ) 0.1 = 0.348
βvx2 = 0.52 + ( 0.54 − 0.52 ) · ( k − 1.4 ) 0.1 = 0.52 + ( 0.54 − 0.52 ) · ( 1.48 − 1.4 ) 0.1 = 0.537
- for moments in mid-span
| Short span coefficients | Long span coefficients | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| k = | 1.0 | 1.1 | 1.2 | 1.3 | 1.4 | 1.5 | 1.75 | 2.0 | ||
| +M | βxb1 = | 0.036 | 0.042 | 0.047 | 0.051 | 0.055 | 0.059 | 0.065 | 0.070 | βyb1 = 0.034 |
| -M | βxt = | 0.047 | 0.056 | 0.063 | 0.069 | 0.074 | 0.078 | 0.087 | 0.093 | βyt = 0.045 |
βxb1 = 0.055 + ( 0.059 − 0.055 ) · ( k − 1.4 ) 0.1 = 0.055 + ( 0.059 − 0.055 ) · ( 1.48 − 1.4 ) 0.1 = 0.0583
- for moments over supports
βxt = 0.074 + ( 0.078 − 0.074 ) · ( k − 1.4 ) 0.1 = 0.074 + ( 0.078 − 0.074 ) · ( 1.48 − 1.4 ) 0.1 = 0.0773
For simply supported slab
- for moments in mid-span
| Short span coefficients | Long span coefficients | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| k = | 1.0 | 1.1 | 1.2 | 1.3 | 1.4 | 1.5 | 1.75 | 2.0 | |
| βxb2 = | 0.055 | 0.065 | 0.074 | 0.081 | 0.087 | 0.092 | 0.103 | 0.111 | βyb2 = 0.056 |
βxb2 = 0.087 + ( 0.092 − 0.087 ) · ( k − 1.4 ) 0.1 = 0.087 + ( 0.092 − 0.087 ) · ( 1.48 − 1.4 ) 0.1 = 0.0912
Loads for accounting alternation in adjacent spans
p1 = γG · ( gsw + gk ) + γQ · qk2 = 1.35 · ( 3.5 + 2.5 ) + 1.5 · 22 = 9.6 kN/m²
p2 = γQ · qk2 = 1.5 · 22 = 1.5 kN/m²
Bending moments
Span moment (including load alternation)
MEd_xb = βxb1 · p1 · lx2 + βxb2 · p2 · lx2 = 0.0583 · 9.6 · 4.142 + 0.0912 · 1.5 · 4.142 = 11.94 kN·m/m
MEd_yb = βyb1 · p1 · lx2 + βyb2 · p2 · lx2 = 0.034 · 9.6 · 4.142 + 0.056 · 1.5 · 4.142 = 7.03 kN·m/m
Support moment
MEd_xt = βxt · p · lx2 = 0.0773 · 11.1 · 4.142 = 14.71 kN·m/m
MEd_yt = βyt · p · lx2 = 0.045 · 11.1 · 4.142 = 8.56 kN·m/m
Average moments at supports from the adjacent spans
MEd_xt2 = MEd_xt2_ · lx2 + MEd_xt · lxlx2 + lx = 0 · 0 + 14.71 · 4.140 + 4.14 = 14.71 kN·m/m
MEd_yt2 = MEd_yt2_ · ly2 + MEd_yt · lyly2 + ly = 0 · 0 + 8.56 · 6.140 + 6.14 = 8.56 kN·m/m
Shear forces
VEd_x1 = βvx1 · p · lx = 0.348 · 11.1 · 4.14 = 16.01 kN/m
VEd_x2 = βvx2 · p · lx = 0.537 · 11.1 · 4.14 = 24.66 kN/m
VEd_y1 = βvy1 · p · lx = 0.26 · 11.1 · 4.14 = 11.95 kN/m
VEd_y2 = βvy2 · p · lx = 0.4 · 11.1 · 4.14 = 18.38 kN/m
Edge support moments
MEd_xte2 = MEd_xt2 − VEd_x2 · min(bx200; h200) = 14.71 − 24.66 · min(25200; 14200) = 12.98 kN·m/m
MEd_yte2 = MEd_yt2 − VEd_y2 · min(by200; h200) = 8.56 − 18.38 · min(25200; 14200) = 7.27 kN·m/m
Bottom reinforcement parallel to x
Effective cross section depth - dxb = h − c − dbxb20 = 14 − 2 − 820 = 11.6 cm
Relative design bending moment- mEd = 10 · MEd_xbdxb2 · η · fcd = 10 · 11.9411.62 · 1 · 14.17 = 0.0626
Compression zone depth - x = dxbλ · ( 1 − √ 1 − 2 · mEd ) = 11.60.8 · ( 1 − √ 1 − 2 · 0.0626 ) = 0.939 cm
Relative compression zone depth - ξ = xdxb = 0.93911.6 = 0.0809
Lever arm of internal forces - z = dxb − 0.5 · λ · x = 11.6 − 0.5 · 0.8 · 0.939 = 11.22 cm
Area of required reinforcement - Asxb = MEd_xb · 103z · fyd = 11.94 · 10311.22 · 434.78 = 2.45 cm²/m
Reinforcement ratio- ρxb = Asxbdxb · 100 = 2.4511.6 · 100 = 0.00211
The rest of the reinforcement is calculated similarly.
Bottom reinforcement
[EN 1992-1-1, § 9.2.1.1 (1)]
Minimum reinforcement ratio
ρmin = max(0.26 · fctmfyk; 0.0013) = max(0.26 · 2.56500; 0.0013) = 0.00133
[EN 1992-1-1, § 9.2.1.1 (3)]
Maximum reinforcement ratio- ρmax = 0.04
Required bar spacing
sxb = floor(π · dbxb24 · Asxb) = floor(3.14 · 824 · 2.45) = 20 cm
[EN 1992-1-1, § 9.3.1.1 (3)]
Maximum bar spacing
smax = min ( 2 · h; 25 ) = min ( 2 · 14; 25 ) = 25 cm
The rest of the reinforcement is detailed similarly.
| Reinforcement | MEd kN·m/m | As cm2/m | ρ, % | Selected |
|---|---|---|---|---|
| Bottom parallel tox | 11.94 | 2.45 | 0.21% | Ø8/20 |
| Bottom parallel to y | 7.03 | 1.53 | 0.14% | Ø8/25 |
| Top parallel to x - right side | 12.98 | 2.64 | 0.23% | Ø6/10 |
| Top parallel to y - up side | 7.27 | 1.54 | 0.14% | Ø6/18 |
Restrained Both Longer Edges¶
Two-way slab panel with both longer edges restrained and the two shorter edges simply supported, typical of an interior panel of a one-way-spanning floor strip.
'<small>According to <strong>Eurocode</strong>: EN 1992-1-1</small>
'<div style="max-width:180mm;">
'<img alt="fsfs-span-cl.png" class="side" style="height:180pt;" src="../../Images/structures/rc/spans/fsfs-span-cl.png">
'<h4>Dimensions</h4>
'Clear spans (<i>l</i><sub>x</sub> < <i>l</i><sub>y</sub>)
l_x_cl = ?'m,'l_y_cl = ?'m
'Beam width along <i>x</i> -'b_y = ?'cm
'Beam width along <i>y</i> -'b_x = ?'cm
'Slab thickness -'h = ?'cm
'Concrete cover -'c = ?'cm
#post
'Design span lengths
l_x = l_x_cl + min(b_x/100;h/100)'m
l_y = l_y_cl + min(b_y/100;h/100)'m
'Span ratio -'k = l_y/l_x
#if k < 1'-->
'Span length <i>l</i><sub>y</sub> should be greater than <i>l</i><sub>x</sub>
#else
#show
'Design spans lengths and edge moments in adjacent panels
'<img alt="fsfs-span.png" class="side" style="height:140pt;" src="../../Images/structures/rc/spans/fsfs-span.png">
'<table><tr><td rowspan="2">
'Parallel to <i>x</i> - <br />
'</td><td>
l_x1 = ?'m;
'</td><td>
M_Ed_xt1_ = ?'kN·m/m
'</td></tr><tr><td>
l_x2 = ?'m;
'</td><td>
M_Ed_xt2_ = ?'kN·m/m
'</td></tr></table>
'(in absence of contiguous span - <i>l</i> = 0)
'<h4>Loads</h4>
'Characteristic uniform loads
'Self weight -'g_sw = 0.25*h'kN/m²
'Permanent -'g_k = ?'kN/m²
'Variable -'q_k = ?'kN/m²
'Partial safety factor -'γ_G = 1.35';'γ_Q = 1.5
#post
'Design uniform loads
p = (g_sw + g_k)*γ_G + q_k*γ_Q'kN/m²
'<div class="fold">
'<p><b>Beam loads</b></p>
'(as uniform loads)
#if k ≤ 2
'<table><tr><td rowspan="2">
'Parallel to <i>x</i> - <br />
'</td><td>
g_k_x = (g_sw + g_k)*l_x_cl/4'kN/m
'</td></tr><tr><td>
q_k_x = q_k*l_x_cl/4'kN/m
'</td></tr><tr><td rowspan="2">
'Parallel to <i>y</i> - <br />
'</td><td>
g_k_y = (g_sw + g_k)*l_x_cl*(l_y_cl - l_x_cl/2)/(2*l_y_cl)'kN/m
'</td></tr><tr><td>
q_k_y = q_k*l_x_cl*(l_y_cl - l_x_cl/2)/(2*l_y_cl)'kN/m
'</td></tr></table>
#else
'Parallel to <i>y</i> -'q_y = p*l_x_cl/2'kN/m
#end if
'</div>
#show
'<h4>Material properties</h4>
'<p><b>Бетон</b> [EN 1992-1-1, EN 1992-1-1, Table 3.1]</p>
'Characteristic compressive cylinder strength -'f_ck = ?'MPa
'Partial safety factors for concrete -'γ_c = 1.5','α_cc = ?
#post
'Design compressive cylinder strength -'f_cd = α_cc*f_ck/γ_c'MPa
'Factor defining the effective height of the compression zone -'λ = 0.8'
'Factor defining the effective strength - 'η = 1.0
'Mean value of axial tensile strength -'f_ctm = 0.30*f_ck^(2/3)'MPa
#show
'<p><b>Steel</b></p>
'Characteristic yield strength -'f_yk = ?'MPa
#post
'Partial safety factor for steel - 'γ_s = 1.15
'Design yield strength - 'f_yd = f_yk/γ_s'MPa
'Limit relative compressive zone depth -'ξ_lim = 0.45
#show
'<p><b>Bar diameters</b></p>
'- Bottom parallel to <i>x</i> -'d_bxb = ?'mm, parallel to <i>y</i> -'d_byb = ?'mm
'- Top parallel to <i>x</i> -'d_bxt = ?'mm
#post
'<div class="fold">
'<h4>Static calculations</h4>
'<p><b>Shear forces and bending moments coefficients</b></p>
#if l_y > 2*l_x
l_y/l_x'>'2'- One-way slab
'For the actual static scheme
'- for moments in mid-span -'β_xb1 = 0.04167
'- for moments over supports -'β_xt = 0.08333
'- for shear forces -'β_vx = 0.5
'For simply supported slab
'- for span moments -'β_xb2 = 0.125
#else if l_x ≡ l_y
l_y/l_x'- Two-way square slab
'For the actual static scheme
'- for moments in mid-span -'β_xb1 = 0.034';'β_yb1 = 0.034
'- for moments over supports -'β_xt = 0.046
'- for shear forces -'β_vx = 0.26';'β_vy = 0.40
'For simply supported slab
'- for span moments -'β_xb2 = 0.055';'β_yb2 = 0.055
#else
'For the actual static scheme
'- for shear forces
'<table class="bordered centered">
'<tr><th rowspan="2"></th><th style="width:100mm;" colspan="9">Short span coefficients</th><th style="width:30mm;" rowspan="2">Long span coefficients</th></tr>
'<tr><th><i>k</i> =</th><th>1.0</th><th>1.1</th><th>1.2</th><th>1.3</th><th>1.4</th><th>1.5</th><th>1.75</th><th>2.0</th></tr>
'<tr><th>V</th><td><i>β</i><sub>vx</sub> = </td><td>0.26</td><td>0.30</td><td>0.33</td><td>0.36</td><td>0.38</td><td>0.40</td><td>0.44</td><td>0.47</td><td>'β_vy = 0.40'</td></tr></table>
#if k < 1.1
β_vx = 0.26 + (0.30 - 0.26)*(k - 1)/0.1
#else if k < 1.2
β_vx = 0.30 + (0.33 - 0.30)*(k - 1.1)/0.1
#else if k < 1.3
β_vx = 0.33 + (0.36 - 0.33)*(k - 1.2)/0.1
#else if k < 1.4
β_vx = 0.36 + (0.38 - 0.36)*(k - 1.3)/0.1
#else if k < 1.5
β_vx = 0.38 + (0.40 - 0.38)*(k - 1.4)/0.1
#else if k < 1.75
β_vx = 0.40 + (0.44 - 0.40)*(k - 1.5)/0.25
#else if k < 2
β_vx = 0.44 + (0.47 - 0.44)*(k - 1.75)/0.25
#else if k ≡ 2
β_vx = 0.47
#end if
'- for moments in mid-span
'<table class="bordered centered">
'<tr><th rowspan="2"></th><th style="width:100mm;" colspan="9">Short span coefficients</th><th style="width:30mm;" rowspan="2">Long span coefficients</th></tr>
'<tr><th><i>k</i> =</th><th>1.0</th><th>1.1</th><th>1.2</th><th>1.3</th><th>1.4</th><th>1.5</th><th>1.75</th><th>2.0</th></tr>
'<tr><th>+M</th><td><i>β</i><sub>xb1</sub> = </td><td>0.034</td><td>0.038</td><td>0.040</td><td>0.043</td><td>0.045</td><td>0.047</td><td>0.050</td><td>0.053</td><td>'β_yb1 = 0.034'</td></tr>
'<tr><th>-M</th><td><i>β</i><sub>xt</sub> = </td><td>0.046</td><td>0.050</td><td>0.054</td><td>0.057</td><td>0.060</td><td>0.062</td><td>0.067</td><td>0.070</td><td>-</td></tr></table>
#if k < 1.1
β_xb1 = 0.034 + (0.038 - 0.034)*(k - 1)/0.1
#else if k < 1.2
β_xb1 = 0.038 + (0.040 - 0.038)*(k - 1.1)/0.1
#else if k < 1.3
β_xb1 = 0.040 + (0.043 - 0.040)*(k - 1.2)/0.1
#else if k < 1.4
β_xb1 = 0.043 + (0.045 - 0.043)*(k - 1.3)/0.1
#else if k < 1.5
β_xb1 = 0.045 + (0.047 - 0.045)*(k - 1.4)/0.1
#else if k < 1.75
β_xb1 = 0.047 + (0.050 - 0.047)*(k - 1.5)/0.25
#else if k < 2
β_xb1 = 0.050 + (0.053 - 0.050)*(k - 1.75)/0.25
#else if k ≡ 2
β_xb1 = 0.053
#end if
'- for moments over supports
#if k < 1.1
β_xt = 0.046 + (0.050 - 0.046)*(k - 1)/0.1
#else if k < 1.2
β_xt = 0.050 + (0.054 - 0.050)*(k - 1.1)/0.1
#else if k < 1.3
β_xt = 0.054 + (0.057 - 0.054)*(k - 1.2)/0.1
#else if k < 1.4
β_xt = 0.057 + (0.060 - 0.057)*(k - 1.3)/0.1
#else if k < 1.5
β_xt = 0.060 + (0.062 - 0.060)*(k - 1.4)/0.1
#else if k < 1.75
β_xt = 0.062 + (0.067 - 0.062)*(k - 1.5)/0.25
#else if k < 2
β_xt = 0.067 + (0.070 - 0.067)*(k - 1.75)/0.25
#else if k ≡ 2
β_xt = 0.070
#end if
'For simply supported slab
' - for moments in mid-span
'<table class="bordered centered">
'<tr><th style="width:100mm;" colspan="9">Short span coefficients</th><th style="width:30mm;" rowspan="2">Long span coefficients</th></tr>
'<tr><th><i>k</i> =</th><th>1.0</th><th>1.1</th><th>1.2</th><th>1.3</th><th>1.4</th><th>1.5</th><th>1.75</th><th>2.0</th></tr>
'<tr><td><i>β</i><sub>xb2</sub> = </td><td>0.055</td><td>0.065</td><td>0.074</td><td>0.081</td><td>0.087</td><td>0.092</td><td>0.103</td><td>0.111</td><td>'β_yb2 = 0.056'</td></tr></table>
#if k < 1.1
β_xb2 = 0.055 + (0.065 - 0.055)*(k - 1)/0.1
#else if k < 1.2
β_xb2 = 0.065 + (0.074 - 0.065)*(k - 1.1)/0.1
#else if k < 1.3
β_xb2 = 0.074 + (0.081 - 0.074)*(k - 1.2)/0.1
#else if k < 1.4
β_xb2 = 0.081 + (0.087 - 0.081)*(k - 1.3)/0.1
#else if k < 1.5
β_xb2 = 0.087 + (0.092 - 0.087)*(k - 1.4)/0.1
#else if k < 1.75
β_xb2 = 0.092 + (0.103 - 0.092)*(k - 1.5)/0.25
#else if k < 2
β_xb2 = 0.103 + (0.111 - 0.103)*(k - 1.75)/0.25
#else if k ≡ 2
β_xb2 = 0.111
#end if
#end if
'<p><b>Loads for accounting alternation in adjacent spans</b></p>
p_1 = γ_G*(g_sw + g_k) + γ_Q*q_k/2'kN/m²
p_2 = γ_Q*q_k/2'kN/m²
'<p><b>Bending moments</b></p>
'Span moment (including load alternation)
M_Ed_xb = β_xb1*p_1*l_x^2 + β_xb2*p_2*l_x^2'kN·m/m
#if k ≤ 2
M_Ed_yb = β_yb1*p_1*l_x^2 + β_yb2*p_2*l_x^2'kN·m/m
#end if
'Support moment
M_Ed_xt = β_xt*p*l_x^2'kN·m/m
'<p><b>Average moments at supports from the adjacent spans</b></p>
M_Ed_xt1 = (M_Ed_xt1_*l_x1 + M_Ed_xt*l_x)/(l_x1 + l_x)'kN·m/m
M_Ed_xt2 = (M_Ed_xt2_*l_x2 + M_Ed_xt*l_x)/(l_x2 + l_x)'kN·m/m
'<p><b>Shear forces</b></p>
#if k ≤ 2
V_Ed_x = β_vx*p*l_x'kN/m
V_Ed_y = β_vy*p*l_x'kN/m
#else
V_Ed_x = β_vx*p*l_x'kN/m
#end if
'<p><b>Edge support moments</b></p>
M_Ed_xte1 = M_Ed_xt1 - V_Ed_x*min(b_x/200;h/200)'kN·m/m
M_Ed_xte2 = M_Ed_xt2 - V_Ed_x*min(b_x/200;h/200)'kN·m/m
'</div><div class="fold">
'<h4>Bending design</h4>
'<p><b>Bottom reinforcement parallel to <i>x</i></b></p>
'Effective cross section depth -'d_xb = h - c - d_bxb/20'cm
'Relative design bending moment -'m_Ed = 10*M_Ed_xb/(d_xb^2*η*f_cd)
'Compression zone depth -'x = d_xb/λ*(1 - sqr(1 - 2*m_Ed))'cm
'Relative compression zone depth -'ξ = x/d_xb
'Lever arm of internal forces -'z = d_xb - 0.5*λ*x'cm
'Area of required reinforcement -'A_sxb = M_Ed_xb*10^3/(z*f_yd)'cm²/m
'Reinforcement ratio -'ρ_xb = A_sxb/(d_xb*100)
#if ξ > ξ_lim
'<p class="err">'ξ'>'ξ_lim'- Reinforcement in compression zone is required.</p>
#end if
#hide
#if k ≤ 2
'<p><b>Bottom reinforcement parallel to <i>y</i></b></p>
'Effective cross section depth -'d_yb = h - c - d_bxb/10 - d_byb/20'cm
'Relative design bending moment -'m_Ed = 10*M_Ed_yb/(d_yb^2*η*f_cd)
'Compression zone depth -'x = d_yb/λ*(1 - sqr(1 - 2*m_Ed))'cm
'Relative compression zone depth -'ξ = x/d_yb
'Lever arm of internal forces -'z = d_yb - 0.5*λ*x'cm
'Area of required reinforcement -'A_syb = M_Ed_yb*10^3/(z*f_yd)'cm²/m
'Reinforcement ratio -'ρ_yb = A_syb/(d_yb*100)
#if ξ > ξ_lim
'<p class="err">'ξ'>'ξ_lim'- Reinforcement in compression zone is required.</p>
#end if
#end if
'<p><b>Top reinforcement parallel to <i>x</i> за <i>M</i><sub>Ed_xtel</sub></b></p>
'Effective cross section depth -'d_xt1 = h - c - d_bxt/20'cm
'Relative design bending moment -'m_Ed = 10*M_Ed_xte1/(d_xt1^2*η*f_cd)
'Compression zone depth -'x = d_xt1/λ*(1 - sqr(1 - 2*m_Ed))'cm
'Relative compression zone depth -'ξ = x/d_xt1
'Lever arm of internal forces -'z = d_xt1 - 0.5*λ*x'cm
'Area of required reinforcement -'A_sxt1 = M_Ed_xte1*10^3/(z*f_yd)'cm²/m
'Reinforcement ratio -'ρ_xt1 = A_sxt1/(d_xt1*100)
#if ξ > ξ_lim
'<p class="err">'ξ'>'ξ_lim'- Reinforcement in compression zone is required.</p>
#end if
'<p><b>Top reinforcement parallel to <i>x</i> за <i>M</i><sub>Ed_xte2</sub></b></p>
'Effective cross section depth -'d_xt2 = h - c - d_bxt/20'cm
'Relative design bending moment -'m_Ed = 10*M_Ed_xte2/(d_xt2^2*η*f_cd)
'Compression zone depth -'x = d_xt2/λ*(1 - sqr(1 - 2*m_Ed))'cm
'Relative compression zone depth -'ξ = x/d_xt2
'Lever arm of internal forces -'z = d_xt2 - 0.5*λ*x'cm
'Area of required reinforcement -'A_sxt2 = M_Ed_xte2*10^3/(z*f_yd)'cm²/m
'Reinforcement ratio -'ρ_xt2 = A_sxt2/(d_xt2*100)
#if ξ > ξ_lim
'<p class="err">'ξ'>'ξ_lim'- Reinforcement in compression zone is required.</p>
#end if
#post
'<p><b>The rest of the reinforcement is calculated similarly.</b></p>
'</div><div class="fold">
'<h4>Reinforcement detailing</h4>
'<p><b>Bottom reinforcement</b></p>
'<p class="ref">[EN 1992-1-1, § 9.2.1.1 (1)]</p>
'Minimum reinforcement ratio
ρ_min = max(0.26*f_ctm/f_yk;0.0013)
'<p class="ref">[EN 1992-1-1, § 9.2.1.1 (3)]</p>
'Maximum reinforcement ratio -'ρ_max = 0.04
#if ρ_xb < ρ_min
'<p><i>ρ</i><sub>xb</sub> < <i>ρ</i><sub>min</sub>. Minimum reinforcement is assumed -'A_sxb = ρ_min*d_xb*100'cm²/m</p>
#else if ρ_xb > ρ_max
'<p class="err">Reinforcement ratio is greater than maximum:'ρ_xb'>'ρ_max'</p>
#end if
#if k > 2
'<p class="ref">[EN 1992-1-1, § 9.3.1.1 (2)]</p>
'Minimum bottom reinforcement parallel to y
A_syb = 0.2*A_sxb'cm²/m
#else
'<!--'
#if ρ_yb < ρ_min
'<p><i>ρ</i><sub>yb</sub> < <i>ρ</i><sub>min</sub>. Minimum reinforcement is assumed -'A_syb = ρ_min*d_yb*100'cm²/m</p>
#else if ρ_yb > ρ_max
'<p class="err">Reinforcement ratio is greater than maximum:'ρ_yb'>'ρ_max'</p>
#end if
'<-->
#end if
'Required bar spacing
s_xb = floor(π*d_bxb^2/(4*A_sxb))'cm
#if k > 2
'<p class="ref">[EN 1992-1-1, § 9.3.1.1 (3)]</p>
'Minimum bottom reinforcement parallel to y
s_yb = min(3*h;40)'cm
'Required bar diameter for bottom reinforcement parallel to y
d_byb = max(ceiling(sqr(4*A_syb*s_yb/π));6)'mm
#else
'<!--'s_yb = floor(π*d_byb^2/(4*A_syb))'cm-->
#end if
'<p class="ref">[EN 1992-1-1, § 9.3.1.1 (3)]</p>
'Maximum bar spacing
s_max = min(2*h;25)'cm
#if s_xb > s_max
'<p><i>s</i><sub>xb</sub> > <i>s</i><sub>max</sub>. Assume's_xb = s_max'cm</p>
#end if
#if (k ≤ 2)*(s_yb > s_max)
'<!--<p><i>s</i><sub>yb</sub> > <i>s</i><sub>max</sub>. Assume's_yb = s_max'cm</p>-->
#end if
#hide
'<p><b>Top reinforcement</b></p>
#if ρ_xt1 < ρ_min
'<p><i>ρ</i><sub>xt1</sub> < <i>ρ</i><sub>min</sub>. Minimum reinforcement is assumed -'A_sxt1 = ρ_min*d_xt1*100'cm²/m</p>
#else if ρ_xt1 > ρ_max
'<p class="err">Reinforcement ratio is greater than maximum:'ρ_xt1'>'ρ_max'</p>
#end if
#if ρ_xt2 < ρ_min
'<p><i>ρ</i><sub>xt2</sub> < <i>ρ</i><sub>min</sub>. Minimum reinforcement is assumed -'A_sxt2 = ρ_min*d_xt2*100'cm²/m</p>
#else if ρ_xt2 > ρ_max
'<p class="err">Reinforcement ratio is greater than maximum:'ρ_xt2'>'ρ_max'</p>
#end if
'Required bar spacing
s_xt1 = floor(π*d_bxt^2/(4*A_sxt1))'cm
s_xt2 = floor(π*d_bxt^2/(4*A_sxt2))'cm
'<p class="ref">[EN 1992-1-1, § 9.3.1.1 (3)]</p>
'Maximum bar spacing
s_max = min(2*h;25)'cm
#if s_xt1 > s_max
'<p><i>s</i><sub>xt1</sub> > <i>s</i><sub>max</sub>. It is assumed's_xt1 = s_max'cm</p>
#end if
#if s_xt2 > s_max
'<p><i>s</i><sub>xt2</sub> > <i>s</i><sub>max</sub>. It is assumed's_xt2 = s_max'cm</p>
#end if
#post
'<p><b>The rest of the reinforcement is detailed similarly</b></p>
'</div>
'<h4>Results</h4>
#val
'<table class="bordered centered">
'<tr><th>Reinforcement</th><th>M<sub>Ed</sub><br>kN·m/m</th><th>A<sub>s</sub><br>cm<sup>2</sup>/m</th><th>ρ, %</th><th>Selected</tr>
'<tr><td>Bottom parallel to <i>x</i></td><td>'M_Ed_xb'</td><td>'A_sxb'</td><td>'A_sxb/d_xb'%</td><td>Ø'd_bxb'/'s_xb'</td></tr>
#if k ≤ 2
'<tr><td>Bottom parallel to <i>y</i></td><td>'M_Ed_yb'</td><td>'A_syb'</td><td>'A_syb/d_yb'%</td><td>Ø'd_byb'/'s_yb'</td></tr>
#else
'<tr><td>Bottom parallel to <i>y</i></td><td></td><td></td><td></td><td>Ø'd_byb'/'s_yb'</td></tr>
#end if
'<tr><td>Top parallel to <i>x</i> - left side</td><td>'M_Ed_xte1'</td><td>'A_sxt1'</td><td>'A_sxt1/d_xt1'%</td><td>Ø'd_bxt'/'s_xt1'</td></tr>
'<tr><td>Top parallel to <i>x</i> - right side</td><td>'M_Ed_xte2'</td><td>'A_sxt2'</td><td>'A_sxt2/d_xt2'%</td><td>Ø'd_bxt'/'s_xt2'</td></tr>
'</table>
#end if
#show
'</div> 4 6 25 25 14 2 0 0 0 0 2.5 2 25 0.85 500 8 8 6
Clear spans (lx < ly)
lx_cl = 4 m, ly_cl = 6 m
Beam width along x - by = 25 cm
Beam width along y - bx = 25 cm
Slab thickness - h = 14 cm
Concrete cover - c = 2 cm
Design span lengths
lx = lx_cl + min(bx100; h100) = 4 + min(25100; 14100) = 4.14 m
ly = ly_cl + min(by100; h100) = 6 + min(25100; 14100) = 6.14 m
Span ratio - k = lylx = 6.144.14 = 1.48
Design spans lengths and edge moments in adjacent panels
|
Parallel to x - |
lx1 = 0 m; |
MEd_xt1_ = 0 kN·m/m |
|
lx2 = 0 m; |
MEd_xt2_ = 0 kN·m/m |
(in absence of contiguous span - l = 0)
Characteristic uniform loads
Self weight - gsw = 0.25 · h = 0.25 · 14 = 3.5 kN/m²
Permanent - gk = 2.5 kN/m²
Variable - qk = 2 kN/m²
Partial safety factor - γG = 1.35 ; γQ = 1.5
Design uniform loads
p = ( gsw + gk ) · γG + qk · γQ = ( 3.5 + 2.5 ) · 1.35 + 2 · 1.5 = 11.1 kN/m²
Beam loads
(as uniform loads)
|
Parallel to x - |
gk_x = ( gsw + gk ) · lx_cl4 = ( 3.5 + 2.5 ) · 44 = 6 kN/m |
|
qk_x = qk · lx_cl4 = 2 · 44 = 2 kN/m | |
|
Parallel to y - |
gk_y = ( gsw + gk ) · lx_cl · (ly_cl − lx_cl2)2 · ly_cl = ( 3.5 + 2.5 ) · 4 · (6 − 42)2 · 6 = 8 kN/m |
|
qk_y = qk · lx_cl · (ly_cl − lx_cl2)2 · ly_cl = 2 · 4 · (6 − 42)2 · 6 = 2.67 kN/m |
Бетон [EN 1992-1-1, EN 1992-1-1, Table 3.1]
Characteristic compressive cylinder strength - fck = 25 MPa
Partial safety factors for concrete - γc = 1.5 , αcc = 0.85
Design compressive cylinder strength - fcd = αcc · fckγc = 0.85 · 251.5 = 14.17 MPa
Factor defining the effective height of the compression zone - λ = 0.8
Factor defining the effective strength - η = 1
Mean value of axial tensile strength - fctm = 0.3 · fck23 = 0.3 · 2523 = 2.56 MPa
Steel
Characteristic yield strength - fyk = 500 MPa
Partial safety factor for steel - γs = 1.15
Design yield strength - fyd = fykγs = 5001.15 = 434.78 MPa
Limit relative compressive zone depth - ξlim = 0.45
Bar diameters
- Bottom parallel to x - dbxb = 8 mm, parallel to y - dbyb = 8 mm
- Top parallel to x - dbxt = 6 mm
Shear forces and bending moments coefficients
For the actual static scheme
- for shear forces
| Short span coefficients | Long span coefficients | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| k = | 1.0 | 1.1 | 1.2 | 1.3 | 1.4 | 1.5 | 1.75 | 2.0 | ||
| V | βvx = | 0.26 | 0.30 | 0.33 | 0.36 | 0.38 | 0.40 | 0.44 | 0.47 | βvy = 0.4 |
βvx = 0.38 + ( 0.4 − 0.38 ) · ( k − 1.4 ) 0.1 = 0.38 + ( 0.4 − 0.38 ) · ( 1.48 − 1.4 ) 0.1 = 0.397
- for moments in mid-span
| Short span coefficients | Long span coefficients | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| k = | 1.0 | 1.1 | 1.2 | 1.3 | 1.4 | 1.5 | 1.75 | 2.0 | ||
| +M | βxb1 = | 0.034 | 0.038 | 0.040 | 0.043 | 0.045 | 0.047 | 0.050 | 0.053 | βyb1 = 0.034 |
| -M | βxt = | 0.046 | 0.050 | 0.054 | 0.057 | 0.060 | 0.062 | 0.067 | 0.070 | - |
βxb1 = 0.045 + ( 0.047 − 0.045 ) · ( k − 1.4 ) 0.1 = 0.045 + ( 0.047 − 0.045 ) · ( 1.48 − 1.4 ) 0.1 = 0.0467
- for moments over supports
βxt = 0.06 + ( 0.062 − 0.06 ) · ( k − 1.4 ) 0.1 = 0.06 + ( 0.062 − 0.06 ) · ( 1.48 − 1.4 ) 0.1 = 0.0617
For simply supported slab
- for moments in mid-span
| Short span coefficients | Long span coefficients | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| k = | 1.0 | 1.1 | 1.2 | 1.3 | 1.4 | 1.5 | 1.75 | 2.0 | |
| βxb2 = | 0.055 | 0.065 | 0.074 | 0.081 | 0.087 | 0.092 | 0.103 | 0.111 | βyb2 = 0.056 |
βxb2 = 0.087 + ( 0.092 − 0.087 ) · ( k − 1.4 ) 0.1 = 0.087 + ( 0.092 − 0.087 ) · ( 1.48 − 1.4 ) 0.1 = 0.0912
Loads for accounting alternation in adjacent spans
p1 = γG · ( gsw + gk ) + γQ · qk2 = 1.35 · ( 3.5 + 2.5 ) + 1.5 · 22 = 9.6 kN/m²
p2 = γQ · qk2 = 1.5 · 22 = 1.5 kN/m²
Bending moments
Span moment (including load alternation)
MEd_xb = βxb1 · p1 · lx2 + βxb2 · p2 · lx2 = 0.0467 · 9.6 · 4.142 + 0.0912 · 1.5 · 4.142 = 10.02 kN·m/m
MEd_yb = βyb1 · p1 · lx2 + βyb2 · p2 · lx2 = 0.034 · 9.6 · 4.142 + 0.056 · 1.5 · 4.142 = 7.03 kN·m/m
Support moment
MEd_xt = βxt · p · lx2 = 0.0617 · 11.1 · 4.142 = 11.73 kN·m/m
Average moments at supports from the adjacent spans
MEd_xt1 = MEd_xt1_ · lx1 + MEd_xt · lxlx1 + lx = 0 · 0 + 11.73 · 4.140 + 4.14 = 11.73 kN·m/m
MEd_xt2 = MEd_xt2_ · lx2 + MEd_xt · lxlx2 + lx = 0 · 0 + 11.73 · 4.140 + 4.14 = 11.73 kN·m/m
Shear forces
VEd_x = βvx · p · lx = 0.397 · 11.1 · 4.14 = 18.23 kN/m
VEd_y = βvy · p · lx = 0.4 · 11.1 · 4.14 = 18.38 kN/m
Edge support moments
MEd_xte1 = MEd_xt1 − VEd_x · min(bx200; h200) = 11.73 − 18.23 · min(25200; 14200) = 10.46 kN·m/m
MEd_xte2 = MEd_xt2 − VEd_x · min(bx200; h200) = 11.73 − 18.23 · min(25200; 14200) = 10.46 kN·m/m
Bottom reinforcement parallel to x
Effective cross section depth - dxb = h − c − dbxb20 = 14 − 2 − 820 = 11.6 cm
Relative design bending moment - mEd = 10 · MEd_xbdxb2 · η · fcd = 10 · 10.0211.62 · 1 · 14.17 = 0.0526
Compression zone depth - x = dxbλ · ( 1 − √ 1 − 2 · mEd ) = 11.60.8 · ( 1 − √ 1 − 2 · 0.0526 ) = 0.783 cm
Relative compression zone depth - ξ = xdxb = 0.78311.6 = 0.0675
Lever arm of internal forces - z = dxb − 0.5 · λ · x = 11.6 − 0.5 · 0.8 · 0.783 = 11.29 cm
Area of required reinforcement - Asxb = MEd_xb · 103z · fyd = 10.02 · 10311.29 · 434.78 = 2.04 cm²/m
Reinforcement ratio - ρxb = Asxbdxb · 100 = 2.0411.6 · 100 = 0.00176
The rest of the reinforcement is calculated similarly.
Bottom reinforcement
[EN 1992-1-1, § 9.2.1.1 (1)]
Minimum reinforcement ratio
ρmin = max(0.26 · fctmfyk; 0.0013) = max(0.26 · 2.56500; 0.0013) = 0.00133
[EN 1992-1-1, § 9.2.1.1 (3)]
Maximum reinforcement ratio - ρmax = 0.04
Required bar spacing
sxb = floor(π · dbxb24 · Asxb) = floor(3.14 · 824 · 2.04) = 24 cm
[EN 1992-1-1, § 9.3.1.1 (3)]
Maximum bar spacing
smax = min ( 2 · h; 25 ) = min ( 2 · 14; 25 ) = 25 cm
The rest of the reinforcement is detailed similarly
| Reinforcement | MEd kN·m/m | As cm2/m | ρ, % | Selected |
|---|---|---|---|---|
| Bottom parallel to x | 10.02 | 2.04 | 0.18% | Ø8/24 |
| Bottom parallel to y | 7.03 | 1.53 | 0.14% | Ø8/25 |
| Top parallel to x - left side | 10.46 | 2.11 | 0.18% | Ø6/13 |
| Top parallel to x - right side | 10.46 | 2.11 | 0.18% | Ø6/13 |
Restrained Both Shorter Edges¶
Two-way slab panel with both shorter edges restrained and the two longer edges simply supported, typical of an end panel oriented across the short span.
'<small>According to <strong>Eurocode</strong>: EN 1992-1-1</small>
'<div style="max-width:180mm;">
'<img alt="sfsf-span-cl.png" class="side" style="height:190pt;" src="../../Images/structures/rc/spans/sfsf-span-cl.png">
'<h4>Dimensions</h4>
'Clear spans (<i>l</i><sub>x</sub> < <i>l</i><sub>y</sub>)
l_x_cl = ?'m,'l_y_cl = ?'m
'Beam width along <i>x</i> -'b_y = ?'cm
'Beam width along <i>y</i> -'b_x = ?'cm
'Slab thickness -'h = ?'cm
'Concrete cover -'c = ?'cm
#post
'Design span lengths
l_x = l_x_cl + min(b_x/100;h/100)'m
l_y = l_y_cl + min(b_y/100;h/100)'m
'Span ratio -'k = l_y/l_x
#if k < 1
'Span length <i>l</i><sub>y</sub> should be greater than <i>l</i><sub>x</sub>
#else
#show
'Design spans lengths and edge moments in adjacent panels
'<img alt="sfsf-span.png" class="side" style="height:140pt;" src="../../Images/structures/rc/spans/sfsf-span.png">
'<!--
#if k ≤ 2
'<-->
'<table><tr><td rowspan="2">
'Along <i>y</i> -
'</td><td>
l_y1 = ?'m;
'</td><td>
M_Ed_yt1_ = ?'kN·m/m
'</td></tr><tr><td>
l_y2 = ?'m;
'</td><td>
M_Ed_yt2_ = ?'kN·m/m
'</td></tr></table>
'<!--
#end if
'<-->
'(in absence of contiguous span - <i>l</i> = 0)
'<h4>Loads</h4>
'Characteristic uniform loads
'Self weight -'g_sw = 0.25*h'kN/m²
'Permanent - 'g_k = ?'kN/m²
'Variable -'q_k = ?'kN/m²
'Partial safety factor -'γ_G = 1.35';'γ_Q = 1.5
#post
'Design uniform loads
p = (g_sw + g_k)*γ_G + q_k*γ_Q'kN/m²
'<div class="fold">
'<p><b>Beam loads</b></p>
'(as uniform loads)
#if k ≤ 2
'<table><tr><td rowspan="2">
'Along <i>x</i> - <br />
'</td><td>
g_k_x = (g_sw + g_k)*l_x_cl/4'kN/m
'</td></tr><tr><td>
q_k_x = q_k*l_x_cl/4'kN/m
'</td></tr><tr><td rowspan="2">
'Along <i>y</i> - <br />
'</td><td>
g_k_y = (g_sw + g_k)*l_x_cl*(l_y_cl - l_x_cl/2)/(2*l_y_cl)'kN/m
'</td></tr><tr><td>
q_k_y = q_k*l_x_cl*(l_y_cl - l_x_cl/2)/(2*l_y_cl)'kN/m
'</td></tr></table>
#else
'Along <i>y</i> -'q_y = p*l_x_cl/2'kN/m
#end if
'</div>
#show
'<h4>Material properties</h4>
'<p><b>Concrete</b> [ EN 1992-1-1, Table 3.1]</p>
'Characteristic compressive cylinder strength -'f_ck = ?'MPa
'Partial safety factors for concrete -'γ_c = 1.5','α_cc = ?
#post
'Design compressive cylinder strength -'f_cd = α_cc*f_ck/γ_c'MPa
'Factor defining the effective height of the compression zone -'λ = 0.8'
'Factor defining the effective strength - 'η = 1.0
'Mean value of axial tensile strength -'f_ctm = 0.30*f_ck^(2/3)'MPa
#show
'<p><b>Steel</b></p>
'Characteristic yield strength -'f_yk = ?'MPa
#post
'Partial safety factor for steel - 'γ_s = 1.15
'Design yield strength - 'f_yd = f_yk/γ_s'MPa
'Limit relative compressive zone depth -'ξ_lim = 0.45
#show
'<p><b>Bar diameters</b></p>
'- Bottom parallel to <i>x</i> -'d_bxb = ?'mm, parallel to <i>y</i> -'d_byb = ?'mm
'- Top parallel to <i>x</i> -'d_bxt = ?'mm, parallel to <i>y</i> -'d_byt = ?'mm
#post
'<div class="fold">
'<h4>Static calculations</h4>
'<p><b>Shear forces and bending moments coefficients</b></p>
#if l_y > 2*l_x
l_y/l_x'>'2'- One-way slab
'For the actual static scheme
'- for moments in mid-span -'β_xb1 = 0.125
'- for shear forces-'β_vx = 0.5
'For simply supported slab
'- for span moments -'β_xb2 = 0.125
#else if l_x ≡ l_y
l_y/l_x'- Two-way square slab
'For the actual static scheme
'- for moments in mid-span -'β_xb1 = 0.034';'β_yb1 = 0.034
'- for moments over supports -'β_yt = 0.045
'- for shear forces-'β_vx = 0.26';'β_vy = 0.40
'For simply supported slab
'- for span moments -'β_xb2 = 0.055';'β_yb2 = 0.055
#else
'For the actual static scheme
'- for shear forces
'<table class="bordered centered">
'<tr><th rowspan="2"></th><th style="width:100mm;" colspan="9">Short span coefficients</th><th style="width:30mm;" rowspan="2">Long span coefficients</th></tr>
'<tr><th><i>k</i> =</th><th>1.0</th><th>1.1</th><th>1.2</th><th>1.3</th><th>1.4</th><th>1.5</th><th>1.75</th><th>2.0</th></tr>
'<tr><th>V</th><td><i>β</i><sub>vx</sub> = </td><td>0.26</td><td>0.30</td><td>0.33</td><td>0.36</td><td>0.38</td><td>0.40</td><td>0.44</td><td>0.47</td><td>'β_vy = 0.40'</td></tr></table>
#if k < 1.1
β_vx = 0.26 + (0.30 - 0.26)*(k - 1)/0.1
#else if k < 1.2
β_vx = 0.30 + (0.33 - 0.30)*(k - 1.1)/0.1
#else if k < 1.3
β_vx = 0.33 + (0.36 - 0.33)*(k - 1.2)/0.1
#else if k < 1.4
β_vx = 0.36 + (0.38 - 0.36)*(k - 1.3)/0.1
#else if k < 1.5
β_vx = 0.38 + (0.40 - 0.38)*(k - 1.4)/0.1
#else if k < 1.75
β_vx = 0.40 + (0.44 - 0.40)*(k - 1.5)/0.25
#else if k < 2
β_vx = 0.44 + (0.47 - 0.44)*(k - 1.75)/0.25
#else if k ≡ 2
β_vx = 0.47
#end if
'- for moments in mid-span
'<table class="bordered centered">
'<tr><th rowspan="2"></th><th style="width:100mm;" colspan="9">Short span coefficients</th><th style="width:30mm;" rowspan="2">Long span coefficients</th></tr>
'<tr><th><i>k</i> =</th><th>1.0</th><th>1.1</th><th>1.2</th><th>1.3</th><th>1.4</th><th>1.5</th><th>1.75</th><th>2.0</th></tr>
'<tr><th>+M</th><td><i>β</i><sub>xb1</sub> = </td><td>0.034</td><td>0.046</td><td>0.056</td><td>0.065</td><td>0.072</td><td>0.078</td><td>0.091</td><td>0.100</td><td>'β_yb1 = 0.034'<BR>'β_yt = 0.045'</td></tr></table>
#if k < 1.1
β_xb1 = 0.034 + (0.046 - 0.034)*(k - 1)/0.1
#else if k < 1.2
β_xb1 = 0.046 + (0.056 - 0.046)*(k - 1.1)/0.1
#else if k < 1.3
β_xb1 = 0.056 + (0.065 - 0.056)*(k - 1.2)/0.1
#else if k < 1.4
β_xb1 = 0.065 + (0.072 - 0.065)*(k - 1.3)/0.1
#else if k < 1.5
β_xb1 = 0.072 + (0.078 - 0.072)*(k - 1.4)/0.1
#else if k < 1.75
β_xb1 = 0.078 + (0.091 - 0.078)*(k - 1.5)/0.25
#else if k < 2
β_xb1 = 0.091 + (0.100 - 0.091)*(k - 1.75)/0.25
#else if k ≡ 2
β_xb1 = 0.100
#end if
'For simply supported slab
' - for moments in mid-span
'<table class="bordered centered">
'<tr><th style="width:100mm;" colspan="9">Short span coefficients</th><th style="width:30mm;" rowspan="2">Long span coefficients</th></tr>
'<tr><th><i>k</i> =</th><th>1.0</th><th>1.1</th><th>1.2</th><th>1.3</th><th>1.4</th><th>1.5</th><th>1.75</th><th>2.0</th></tr>
'<tr><td><i>β</i><sub>xb2</sub> = </td><td>0.055</td><td>0.065</td><td>0.074</td><td>0.081</td><td>0.087</td><td>0.092</td><td>0.103</td><td>0.111</td><td>'β_yb2 = 0.056'</td></tr></table>
#if k < 1.1
β_xb2 = 0.055 + (0.065 - 0.055)*(k - 1)/0.1
#else if k < 1.2
β_xb2 = 0.065 + (0.074 - 0.065)*(k - 1.1)/0.1
#else if k < 1.3
β_xb2 = 0.074 + (0.081 - 0.074)*(k - 1.2)/0.1
#else if k < 1.4
β_xb2 = 0.081 + (0.087 - 0.081)*(k - 1.3)/0.1
#else if k < 1.5
β_xb2 = 0.087 + (0.092 - 0.087)*(k - 1.4)/0.1
#else if k < 1.75
β_xb2 = 0.092 + (0.103 - 0.092)*(k - 1.5)/0.25
#else if k < 2
β_xb2 = 0.103 + (0.111 - 0.103)*(k - 1.75)/0.25
#else if k ≡ 2
β_xb2 = 0.111
#end if
#end if
'<p><b>Loads for accounting alternation in adjacent spans </b></p>
p_1 = γ_G*(g_sw + g_k) + γ_Q*q_k/2'kN/m²
p_2 = γ_Q*q_k/2'kN/m²
'<p><b>Bending moments</b></p>
'Span moment (including load alternation)
M_Ed_xb = β_xb1*p_1*l_x^2 + β_xb2*p_2*l_x^2'kN·m/m
#if k ≤ 2
M_Ed_yb = β_yb1*p_1*l_x^2 + β_yb2*p_2*l_x^2'kN·m/m
'Support moment
M_Ed_yt = β_yt*p*l_x^2'kN·m/m
#end if
#if k ≤ 2
'<p><b>Average moments at supports from the adjacent spans</b></p>
M_Ed_yt1 = (M_Ed_yt1_*l_y1 + M_Ed_yt*l_y)/(l_y1 + l_y)'kN·m/m
M_Ed_yt2 = (M_Ed_yt2_*l_y2 + M_Ed_yt*l_y)/(l_y2 + l_y)'kN·m/m
#end if
'<p><b>Shear forces</b></p>
#if k ≤ 2
V_Ed_x = β_vx*p*l_x'kN/m
V_Ed_y = β_vy*p*l_x'kN/m
#else
V_Ed_x = β_vx*p*l_x'kN/m
#end if
#if k ≤ 2
'<p><b>Edge support moments</b></p>
M_Ed_yte1 = M_Ed_yt1 - V_Ed_y*min(b_y/200;h/200)'kN·m/m
M_Ed_yte2 = M_Ed_yt2 - V_Ed_y*min(b_y/200;h/200)'kN·m/m
#end if
'</div><div class="fold">
'<h4>Bending design</h4>
'<p><b>Bottom reinforcement parallel to <i>x</i></b></p>
'Effective cross section depth-'d_xb = h - c - d_bxb/20'cm
'Relative design bending moment -'m_Ed = 10*M_Ed_xb/(d_xb^2*η*f_cd)
'Compression zone depth -'x = d_xb/λ*(1 - sqr(1 - 2*m_Ed))'cm
'Relative compression zone depth -'ξ = x/d_xb
'Lever arm of internal forces-'z = d_xb - 0.5*λ*x'cm
'Area of required reinforcement -'A_sxb = M_Ed_xb*10^3/(z*f_yd)'cm²/m
'Reinforcement ratio -'ρ_xb = A_sxb/(d_xb*100)
#if ξ > ξ_lim
'<p class="err">'ξ'>'ξ_lim'- Reinforcement in compression zone is required. </p>
#end if
#hide
#if k ≤ 2
'<p><b>Bottom reinforcement parallel to <i>y</i></b></p>
'Effective cross section depth -'d_yb = h - c - d_bxb/10 - d_byb/20'cm
'Relative design bending moment -'m_Ed = 10*M_Ed_yb/(d_yb^2*η*f_cd)
'Compression zone depth -'x = d_yb/λ*(1 - sqr(1 - 2*m_Ed))'cm
'Relative compression zone depth-'ξ = x/d_yb
'Lever arm of internal forces -'z = d_yb - 0.5*λ*x'cm
'Area of required reinforcement -'A_syb = M_Ed_yb*10^3/(z*f_yd)'cm²/m
'Reinforcement ratio -'ρ_yb = A_syb/(d_yb*100)
#if ξ > ξ_lim
'<p class="err">'ξ'>'ξ_lim'- Reinforcement in compression zone is required.</p>
#end if
'<p><b>Top reinforcement parallel to <i>y</i> за <i>M</i><sub>Ed_yte1</sub></b></p>
'Effective cross section depth -'d_yt1 = h - c - d_byt/10'cm
'Relative design bending moment -'m_Ed = 10*M_Ed_yte1/(d_yt1^2*η*f_cd)
'Compression zone depth -'x = d_yt1/λ*(1 - sqr(1 - 2*m_Ed))'cm
'Relative compression zone depth -'ξ = x/d_yt1
'Lever arm of internal forces -'z = d_yt1 - 0.5*λ*x'cm
'Area of required reinforcement -'A_syt1 = M_Ed_yte1*10^3/(z*f_yd)'cm²/m
'Reinforcement ratio -'ρ_yt1 = A_syt1/(d_yt1*100)
#if ξ > ξ_lim
'<p class="err">'ξ'>'ξ_lim'- вReinforcement in compression zone is required.</p>
#end if
'<p><b>Top reinforcement parallel to <i>y</i> за <i>M</i><sub>Ed_yte2</sub></b></p>
'Effective cross section depth -'d_yt2 = h - c - d_byt/10'cm
'Relative design bending moment -'m_Ed = 10*M_Ed_yte2/(d_yt2^2*η*f_cd)
'Compression zone depth-'x = d_yt2/λ*(1 - sqr(1 - 2*m_Ed))'cm
'Relative compression zone depth -'ξ = x/d_yt2
'Lever arm of internal forces-'z = d_yt2 - 0.5*λ*x'cm
'Area of required reinforcement -'A_syt2 = M_Ed_yte2*10^3/(z*f_yd)'cm²/m
'Reinforcement ratio -'ρ_yt2 = A_syt2/(d_yt2*100)
#if ξ > ξ_lim
'<p class="err">'ξ'>'ξ_lim'- Reinforcement in compression zone is required.</p>
#end if
#end if
#post
'<p><b>The rest of the reinforcement is calculated similarly.</b></p>
'</div><div class="fold">
'<h4>Reinforcement detailing </h4>
'<p><b>Bottom reinforcement</b></p>
'<p class="ref">[EN 1992-1-1, § 9.2.1.1 (1)]</p>
'Minimum reinforcement ratio
ρ_min = max(0.26*f_ctm/f_yk;0.0013)
'<p class="ref">[EN 1992-1-1, § 9.2.1.1 (3)]</p>
'Maximum reinforcement ratio -'ρ_max = 0.04
#if ρ_xb < ρ_min
'<p><i>ρ</i><sub>xb</sub> < <i>ρ</i><sub>min</sub>. Minimum reinforcement is assumed -'A_sxb = ρ_min*d_xb*100'cm²/m
#else if ρ_xb > ρ_max
'<p class="err">Reinforcement ratio is greater than maximum:'ρ_xb'>'ρ_max'</p>
#end if
#if k > 2
'<p class="ref">[EN 1992-1-1, § 9.3.1.1 (2)]</p>
'Minimum bottom reinforcement parallel to y
A_syb = 0.2*A_sxb'cm²/m
#else
'<!--'
#if ρ_yb < ρ_min
'<p><i>ρ</i><sub>yb</sub> < <i>ρ</i><sub>min</sub>. Minimum reinforcement is assumed -'A_syb = ρ_min*d_yb*100'cm²/m
#else if ρ_yb > ρ_max
'<p class="err">Reinforcement ratio is greater than maximum:'ρ_yb'>'ρ_max'</p>
#end if
'<-->
#end if
'Required bar spacing
s_xb = floor(π*d_bxb^2/(4*A_sxb))'cm
#if k > 2
'<p class="ref">[EN 1992-1-1, § 9.3.1.1 (3)]</p>
'Minimum bottom reinforcement parallel to y
s_yb = min(3*h;40)'cm
'Required bar diameter for bottom reinforcement parallel to y
d_byb = max(ceiling(sqr(4*A_syb*s_yb/π));6)'mm
#else
'<!--'s_yb = floor(π*d_byb^2/(4*A_syb))'cm-->
#end if
'<p class="ref">[EN 1992-1-1, § 9.3.1.1 (3)]</p>
'Maximum bar spacing
s_max = min(2*h;25)'cm
#if s_xb > s_max
'<p><i>s</i><sub>xb</sub> > <i>s</i><sub>max</sub>. Assume's_xb = s_max'cm</p>
#end if
#if (k ≤ 2)*(s_yb > s_max)
'<!--<p><i>s</i><sub>yb</sub> > <i>s</i><sub>max</sub>. Assume's_yb = s_max'cm</p>-->
#end if
#hide
#if k ≤ 2
'<p><b>Top reinforcement</b></p>
#if ρ_yt1 < ρ_min
'<p><i>ρ</i><sub>yt1</sub> < <i>ρ</i><sub>min</sub>. Minimum reinforcement is assumed -'A_syt1 = ρ_min*d_yt1*100'cm²/m
#else if ρ_yt1 > ρ_max
'<p class="err">Reinforcement ratio is greater than maximum:'ρ_yt1'>'ρ_max'</p>
#end if
#if ρ_yt2 < ρ_min
'<p><i>ρ</i><sub>yt2</sub> < <i>ρ</i><sub>min</sub>. Minimum reinforcement is assumed -'A_syt2 = ρ_min*d_yt2*100'cm²/m
#else if ρ_yt2 > ρ_max
'<p class="err">Reinforcement ratio is greater than maximum:'ρ_yt2'>'ρ_max'</p>
#end if
#end if
'Required bar spacing
#if k ≤ 2
s_yt1 = floor(π*d_byt^2/(4*A_syt1))'cm
s_yt2 = floor(π*d_byt^2/(4*A_syt2))'cm
#end if
'<p class="ref">[EN 1992-1-1, § 9.3.1.1 (3)]</p>
'Maximum bar spacing
s_max = min(2*h;25)'cm
#if (k ≤ 2)*(s_yt1 > s_max)
'<p><i>s</i><sub>yt1</sub> > <i>s</i><sub>max</sub>. Assume's_yt1 = s_max'cm</p>
#end if
#if (k ≤ 2)*(s_yt2 > s_max)
'<p><i>s</i><sub>yt2</sub> > <i>s</i><sub>max</sub>. Assume's_yt2 = s_max'cm</p>
#end if
#post
'<p><b>The rest of the reinforcement is detailed similarly.</b></p>
'</div>
'<h4>Results</h4>
#val
'<table class="bordered centered">
'<tr><th>Reinforcement</th><th>M<sub>Ed</sub><br>kN·m/m</th><th>A<sub>s</sub><br>cm<sup>2</sup>/m</th><th>ρ, %</th><th>Selected</tr>
'<tr><td>Bottom parallel to <i>x</i></td><td>'M_Ed_xb'</td><td>'A_sxb'</td><td>'A_sxb/d_xb'%</td><td>Ø'd_bxb'/'s_xb'</td></tr>
#if k ≤ 2
'<tr><td>Bottom parallel to <i>y</i></td><td>'M_Ed_yb'</td><td>'A_syb'</td><td>'A_syb/d_yb'%</td><td>Ø'd_byb'/'s_yb'</td></tr>
#else
'<tr><td>Top parallel to <i>y</i></td><td></td><td></td><td></td><td>Ø'd_byb'/'s_yb'</td></tr>
#end if
#if k ≤ 2
'<tr><td>Top parallel to <i>y</i> - down side </td><td>'M_Ed_yte1'</td><td>'A_syt1'</td><td>'A_syt1/d_yt1'%</td><td>Ø'd_byt'/'s_yt1'</td></tr>
'<tr><td>Top parallel to <i>y</i> - up side </td><td>'M_Ed_yte2'</td><td>'A_syt2'</td><td>'A_syt2/d_yt2'%</td><td>Ø'd_byt'/'s_yt2'</td></tr>
#end if
'</table>
#end if
#show
'</div> 4 6 25 25 14 2 0 0 0 0 2.5 2 25 0.85 500 8 8 6 6
Clear spans (lx < ly)
lx_cl = 4 m, ly_cl = 6 m
Beam width along x - by = 25 cm
Beam width along y - bx = 25 cm
Slab thickness - h = 14 cm
Concrete cover - c = 2 cm
Design span lengths
lx = lx_cl + min(bx100; h100) = 4 + min(25100; 14100) = 4.14 m
ly = ly_cl + min(by100; h100) = 6 + min(25100; 14100) = 6.14 m
Span ratio - k = lylx = 6.144.14 = 1.48
Design spans lengths and edge moments in adjacent panels
|
Along y - |
ly1 = 0 m; |
MEd_yt1_ = 0 kN·m/m |
|
ly2 = 0 m; |
MEd_yt2_ = 0 kN·m/m |
(in absence of contiguous span - l = 0)
Characteristic uniform loads
Self weight - gsw = 0.25 · h = 0.25 · 14 = 3.5 kN/m²
Permanent - gk = 2.5 kN/m²
Variable - qk = 2 kN/m²
Partial safety factor - γG = 1.35 ; γQ = 1.5
Design uniform loads
p = ( gsw + gk ) · γG + qk · γQ = ( 3.5 + 2.5 ) · 1.35 + 2 · 1.5 = 11.1 kN/m²
Beam loads
(as uniform loads)
|
Along x - |
gk_x = ( gsw + gk ) · lx_cl4 = ( 3.5 + 2.5 ) · 44 = 6 kN/m |
|
qk_x = qk · lx_cl4 = 2 · 44 = 2 kN/m | |
|
Along y - |
gk_y = ( gsw + gk ) · lx_cl · (ly_cl − lx_cl2)2 · ly_cl = ( 3.5 + 2.5 ) · 4 · (6 − 42)2 · 6 = 8 kN/m |
|
qk_y = qk · lx_cl · (ly_cl − lx_cl2)2 · ly_cl = 2 · 4 · (6 − 42)2 · 6 = 2.67 kN/m |
Concrete [ EN 1992-1-1, Table 3.1]
Characteristic compressive cylinder strength - fck = 25 MPa
Partial safety factors for concrete - γc = 1.5 , αcc = 0.85
Design compressive cylinder strength - fcd = αcc · fckγc = 0.85 · 251.5 = 14.17 MPa
Factor defining the effective height of the compression zone - λ = 0.8
Factor defining the effective strength - η = 1
Mean value of axial tensile strength - fctm = 0.3 · fck23 = 0.3 · 2523 = 2.56 MPa
Steel
Characteristic yield strength - fyk = 500 MPa
Partial safety factor for steel - γs = 1.15
Design yield strength - fyd = fykγs = 5001.15 = 434.78 MPa
Limit relative compressive zone depth - ξlim = 0.45
Bar diameters
- Bottom parallel to x - dbxb = 8 mm, parallel to y - dbyb = 8 mm
- Top parallel to x - dbxt = 6 mm, parallel to y - dbyt = 6 mm
Shear forces and bending moments coefficients
For the actual static scheme
- for shear forces
| Short span coefficients | Long span coefficients | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| k = | 1.0 | 1.1 | 1.2 | 1.3 | 1.4 | 1.5 | 1.75 | 2.0 | ||
| V | βvx = | 0.26 | 0.30 | 0.33 | 0.36 | 0.38 | 0.40 | 0.44 | 0.47 | βvy = 0.4 |
βvx = 0.38 + ( 0.4 − 0.38 ) · ( k − 1.4 ) 0.1 = 0.38 + ( 0.4 − 0.38 ) · ( 1.48 − 1.4 ) 0.1 = 0.397
- for moments in mid-span
| Short span coefficients | Long span coefficients | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| k = | 1.0 | 1.1 | 1.2 | 1.3 | 1.4 | 1.5 | 1.75 | 2.0 | ||
| +M | βxb1 = | 0.034 | 0.046 | 0.056 | 0.065 | 0.072 | 0.078 | 0.091 | 0.100 | βyb1 = 0.034 βyt = 0.045 |
βxb1 = 0.072 + ( 0.078 − 0.072 ) · ( k − 1.4 ) 0.1 = 0.072 + ( 0.078 − 0.072 ) · ( 1.48 − 1.4 ) 0.1 = 0.077
For simply supported slab
- for moments in mid-span
| Short span coefficients | Long span coefficients | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| k = | 1.0 | 1.1 | 1.2 | 1.3 | 1.4 | 1.5 | 1.75 | 2.0 | |
| βxb2 = | 0.055 | 0.065 | 0.074 | 0.081 | 0.087 | 0.092 | 0.103 | 0.111 | βyb2 = 0.056 |
βxb2 = 0.087 + ( 0.092 − 0.087 ) · ( k − 1.4 ) 0.1 = 0.087 + ( 0.092 − 0.087 ) · ( 1.48 − 1.4 ) 0.1 = 0.0912
Loads for accounting alternation in adjacent spans
p1 = γG · ( gsw + gk ) + γQ · qk2 = 1.35 · ( 3.5 + 2.5 ) + 1.5 · 22 = 9.6 kN/m²
p2 = γQ · qk2 = 1.5 · 22 = 1.5 kN/m²
Bending moments
Span moment (including load alternation)
MEd_xb = βxb1 · p1 · lx2 + βxb2 · p2 · lx2 = 0.077 · 9.6 · 4.142 + 0.0912 · 1.5 · 4.142 = 15.01 kN·m/m
MEd_yb = βyb1 · p1 · lx2 + βyb2 · p2 · lx2 = 0.034 · 9.6 · 4.142 + 0.056 · 1.5 · 4.142 = 7.03 kN·m/m
Support moment
MEd_yt = βyt · p · lx2 = 0.045 · 11.1 · 4.142 = 8.56 kN·m/m
Average moments at supports from the adjacent spans
MEd_yt1 = MEd_yt1_ · ly1 + MEd_yt · lyly1 + ly = 0 · 0 + 8.56 · 6.140 + 6.14 = 8.56 kN·m/m
MEd_yt2 = MEd_yt2_ · ly2 + MEd_yt · lyly2 + ly = 0 · 0 + 8.56 · 6.140 + 6.14 = 8.56 kN·m/m
Shear forces
VEd_x = βvx · p · lx = 0.397 · 11.1 · 4.14 = 18.23 kN/m
VEd_y = βvy · p · lx = 0.4 · 11.1 · 4.14 = 18.38 kN/m
Edge support moments
MEd_yte1 = MEd_yt1 − VEd_y · min(by200; h200) = 8.56 − 18.38 · min(25200; 14200) = 7.27 kN·m/m
MEd_yte2 = MEd_yt2 − VEd_y · min(by200; h200) = 8.56 − 18.38 · min(25200; 14200) = 7.27 kN·m/m
Bottom reinforcement parallel to x
Effective cross section depth- dxb = h − c − dbxb20 = 14 − 2 − 820 = 11.6 cm
Relative design bending moment - mEd = 10 · MEd_xbdxb2 · η · fcd = 10 · 15.0111.62 · 1 · 14.17 = 0.0787
Compression zone depth - x = dxbλ · ( 1 − √ 1 − 2 · mEd ) = 11.60.8 · ( 1 − √ 1 − 2 · 0.0787 ) = 1.19 cm
Relative compression zone depth - ξ = xdxb = 1.1911.6 = 0.103
Lever arm of internal forces- z = dxb − 0.5 · λ · x = 11.6 − 0.5 · 0.8 · 1.19 = 11.12 cm
Area of required reinforcement - Asxb = MEd_xb · 103z · fyd = 15.01 · 10311.12 · 434.78 = 3.1 cm²/m
Reinforcement ratio - ρxb = Asxbdxb · 100 = 3.111.6 · 100 = 0.00268
The rest of the reinforcement is calculated similarly.
Bottom reinforcement
[EN 1992-1-1, § 9.2.1.1 (1)]
Minimum reinforcement ratio
ρmin = max(0.26 · fctmfyk; 0.0013) = max(0.26 · 2.56500; 0.0013) = 0.00133
[EN 1992-1-1, § 9.2.1.1 (3)]
Maximum reinforcement ratio - ρmax = 0.04
Required bar spacing
sxb = floor(π · dbxb24 · Asxb) = floor(3.14 · 824 · 3.1) = 16 cm
[EN 1992-1-1, § 9.3.1.1 (3)]
Maximum bar spacing
smax = min ( 2 · h; 25 ) = min ( 2 · 14; 25 ) = 25 cm
The rest of the reinforcement is detailed similarly.
| Reinforcement | MEd kN·m/m | As cm2/m | ρ, % | Selected |
|---|---|---|---|---|
| Bottom parallel to x | 15.01 | 3.1 | 0.27% | Ø8/16 |
| Bottom parallel to y | 7.03 | 1.53 | 0.14% | Ø8/25 |
| Top parallel to y - down side | 7.27 | 1.52 | 0.13% | Ø6/18 |
| Top parallel to y - up side | 7.27 | 1.52 | 0.13% | Ø6/18 |
Restrained Longer Edge¶
Two-way slab panel with one longer edge restrained and the other three simply supported, typical of an edge panel oriented along the long span.
'<small>According to <strong>Eurocode</strong>: EN 1992-1-1</small>
'<div style="max-width:180mm;">
'<img alt="ssfs-span-cl.png" class="side" style="height:170pt;" src="../../Images/structures/rc/spans/ssfs-span-cl.png">
'<h4>Dimensions</h4>
'Clear spans (<i>l</i><sub>x</sub> < <i>l</i><sub>y</sub>)
l_x_cl = ?'m,'l_y_cl = ?'m
'Beam width along <i>x</i> -'b_y = ?'cm
'Beam width along <i>y</i> -'b_x = ?'cm
'Slab thickness -'h = ?'cm
'Concrete cover -'c = ?'cm
#post
'Design span lengths
l_x = l_x_cl + min(b_x/100;h/100)'m
l_y = l_y_cl + min(b_y/100;h/100)'m
'Span ratio -'k = l_y/l_x
#if k < 1
'Span length <i>l</i><sub>y</sub> should be greater than <i>l</i><sub>x</sub>
#else
#show
'Design spans lengths and edge moments in adjacent panels
'<img alt="ssfs-span.png" class="side" style="height:140pt;" src="../../Images/structures/rc/spans/ssfs-span.png">
'<table><tr><td>
'Parallel to <i>x</i> - <br />
'</td><td>
l_x2 = ?'m;
'</td><td>
M_Ed_xt2_ = ?'kN·m/m
'</td></tr></table>
'(in absence of contiguous span - <i>l</i> = 0)
'<h4>Loads</h4>
'Characteristic uniform loads
'Self weight -'g_sw = 0.25*h'kN/m²
'Permanent -'g_k = ?'kN/m²
'Variable -'q_k = ?'kN/m²
'Partial safety factor -'γ_G = 1.35';'γ_Q = 1.5
#post
'Design uniform loads
p = (g_sw + g_k)*γ_G + q_k*γ_Q'kN/m²
'<div class="fold">
'<p><b>Beam loads</b></p>
'(as uniform loads)
#if k ≤ 2
'<table><tr><td rowspan="2">
'Parallel to <i>x</i> - <br />
'</td><td>
g_k_x = (g_sw + g_k)*l_x_cl/4'kN/m
'</td></tr><tr><td>
q_k_x = q_k*l_x_cl/4'kN/m
'</td></tr><tr><td rowspan="2">
'Parallel to <i>y</i> - <br />
'</td><td>
g_k_y = (g_sw + g_k)*l_x_cl*(l_y_cl - l_x_cl/2)/(2*l_y_cl)'kN/m
'</td></tr><tr><td>
q_k_y = q_k*l_x_cl*(l_y_cl - l_x_cl/2)/(2*l_y_cl)'kN/m
'</td></tr></table>
#else
'Parallel to <i>y</i> -'q_y = p*l_x_cl/2'kN/m
#end if
'</div>
#show
'<h4>Material properties</h4>
'<p><b>Concrete</b> [EN 1992-1-1, Table 3.1]</p>
'Characteristic compressive cylinder strength -'f_ck = ?'MPa
'Partial safety factors for concrete -'γ_c = 1.5','α_cc = ?
#post
'Design compressive cylinder strength -'f_cd = α_cc*f_ck/γ_c'MPa
'Factor defining the effective height of the compression zone -'λ = 0.8
'Factor defining the effective strength - 'η = 1.0
'Mean value of axial tensile strength -'f_ctm = 0.30*f_ck^(2/3)'MPa
#show
'<p><b>Steel</b></p>
'Characteristic yield strength -'f_yk = ?'MPa
#post
'Partial safety factor for steel - 'γ_s = 1.15
'Design yield strength - 'f_yd = f_yk/γ_s'MPa
'Limit relative compressive zone depth -'ξ_lim = 0.45
#show
'<p><b>Bar diameters</b></p>
'- Bottom parallel to <i>x</i> -'d_bxb = ?'mm, parallel to <i>y</i> -'d_byb = ?'mm
'- Top parallel to <i>x</i> -'d_bxt = ?'mm
#post
'<div class="fold">
'<h4>Static calculations</h4>
'<p><b>Shear forces and bending moments coefficients</b></p>
#if l_y > 2*l_x
l_y/l_x'>'2'- One-way slab
'For the actual static scheme
'- for moments in mid-span -'β_xb1 = 0.0703
'- for moments over supports -'β_xt = 0.125
'- for shear forces -'β_vx1 = 0.375';'β_vx2 = 0.625
'For simply supported slab
'- for span moments -'β_xb2 = 0.125
#else if l_x ≡ l_y
l_y/l_x'- Two-way square slab
'For the actual static scheme
'- for moments in mid-span -'β_xb1 = 0.043';'β_yb1 = 0.043
'- for moments over supports -'β_xt = 0.057
'- for shear forces -'β_vx1 = 0.30';'β_vx2 = 0.45';'β_vy = 0.29
'For simply supported slab
'- for span moments -'β_xb2 = 0.055';'β_yb2 = 0.055
#else
'For the actual static scheme
'- for shear forces
'<table class="bordered centered">
'<tr><th rowspan="2"></th><th style="width:100mm;" colspan="9">Short span coefficients</th><th style="width:30mm;" rowspan="2">Long span coefficients</th></tr>
'<tr><th><i>k</i> =</th><th>1.0</th><th>1.1</th><th>1.2</th><th>1.3</th><th>1.4</th><th>1.5</th><th>1.75</th><th>2.0</th></tr>
'<tr><th>V</th><td><i>β</i><sub>vx1</sub> = </td><td>0.30</td><td>0.32</td><td>0.34</td><td>0.35</td><td>0.36</td><td>0.37</td><td>0.39</td><td>0.41</td><td rowspan="2">'β_vy = 0.29'</td></tr>
'<tr><th>V</th><td><i>β</i><sub>vx2</sub> = </td><td>0.45</td><td>0.48</td><td>0.51</td><td>0.53</td><td>0.55</td><td>0.57</td><td>0.60</td><td>0.63</td></tr></table>
#if k < 1.1
β_vx1 = 0.30 + (0.32 - 0.30)*(k - 1)/0.1
β_vx2 = 0.45 + (0.48 - 0.45)*(k - 1)/0.1
#else if k < 1.2
β_vx1 = 0.32 + (0.34 - 0.32)*(k - 1.1)/0.1
β_vx2 = 0.48 + (0.51 - 0.48)*(k - 1.1)/0.1
#else if k < 1.3
β_vx1 = 0.34 + (0.35 - 0.34)*(k - 1.2)/0.1
β_vx2 = 0.51 + (0.53 - 0.51)*(k - 1.2)/0.1
#else if k < 1.4
β_vx1 = 0.35 + (0.36 - 0.35)*(k - 1.3)/0.1
β_vx2 = 0.53 + (0.55 - 0.53)*(k - 1.3)/0.1
#else if k < 1.5
β_vx1 = 0.36 + (0.37 - 0.36)*(k - 1.4)/0.1
β_vx2 = 0.55 + (0.57 - 0.55)*(k - 1.4)/0.1
#else if k < 1.75
β_vx1 = 0.37 + (0.39 - 0.37)*(k - 1.5)/0.25
β_vx2 = 0.57 + (0.60 - 0.57)*(k - 1.5)/0.25
#else if k < 2
β_vx1 = 0.39 + (0.41 - 0.39)*(k - 1.75)/0.25
β_vx2 = 0.60 + (0.63 - 0.60)*(k - 1.75)/0.25
#else if k ≡ 2
β_vx1 = 0.41
β_vx2 = 0.63
#end if
'- for moments in mid-span
'<table class="bordered centered">
'<tr><th rowspan="2"></th><th style="width:100mm;" colspan="9">Short span coefficients</th><th style="width:30mm;" rowspan="2">Long span coefficients</th></tr>
'<tr><th><i>k</i> =</th><th>1.0</th><th>1.1</th><th>1.2</th><th>1.3</th><th>1.4</th><th>1.5</th><th>1.75</th><th>2.0</th></tr>
'<tr><th>+M</th><td><i>β</i><sub>xb1</sub> = </td><td>0.043</td><td>0.048</td><td>0.053</td><td>0.057</td><td>0.060</td><td>0.063</td><td>0.069</td><td>0.074</td><td rowspan="2">'β_yb1 = 0.044'</td></tr>
'<tr><th>-M</th><td><i>β</i><sub>xt</sub> = </td><td>0.057</td><td>0.065</td><td>0.071</td><td>0.076</td><td>0.081</td><td>0.084</td><td>0.092</td><td>0.098</td></tr></table>
#if k < 1.1
β_xb1 = 0.043 + (0.048 - 0.043)*(k - 1)/0.1
#else if k < 1.2
β_xb1 = 0.048 + (0.053 - 0.048)*(k - 1.1)/0.1
#else if k < 1.3
β_xb1 = 0.053 + (0.057 - 0.053)*(k - 1.2)/0.1
#else if k < 1.4
β_xb1 = 0.057 + (0.06 - 0.057)*(k - 1.3)/0.1
#else if k < 1.5
β_xb1 = 0.063 + (0.06 - 0.063)*(k - 1.4)/0.1
#else if k < 1.75
β_xb1 = 0.063 + (0.069 - 0.063)*(k - 1.5)/0.25
#else if k < 2
β_xb1 = 0.069 + (0.074 - 0.069)*(k - 1.75)/0.25
#else if k ≡ 2
β_xb1 = 0.074
#end if
'- for moments over supports
#if k < 1.1
β_xt = 0.057 + (0.065 - 0.057)*(k - 1)/0.1
#else if k < 1.2
β_xt = 0.065 + (0.071 - 0.065)*(k - 1.1)/0.1
#else if k < 1.3
β_xt = 0.071 + (0.076 - 0.071)*(k - 1.2)/0.1
#else if k < 1.4
β_xt = 0.076 + (0.081 - 0.076)*(k - 1.3)/0.1
#else if k < 1.5
β_xt = 0.081 + (0.084 - 0.081)*(k - 1.4)/0.1
#else if k < 1.75
β_xt = 0.084 + (0.092 - 0.084)*(k - 1.5)/0.25
#else if k < 2
β_xt = 0.092 + (0.098 - 0.092)*(k - 1.75)/0.25
#else if k ≡ 2
β_xt = 0.098
#end if
'For simply supported slab
' - for moments in mid-span
'<table class="bordered centered">
'<tr><th style="width:100mm;" colspan="9">Short span coefficients</th><th style="width:30mm;" rowspan="2">Long span coefficients</th></tr>
'<tr><th><i>k</i> =</th><th>1.0</th><th>1.1</th><th>1.2</th><th>1.3</th><th>1.4</th><th>1.5</th><th>1.75</th><th>2.0</th></tr>
'<tr><td><i>β</i><sub>xb2</sub> = </td><td>0.055</td><td>0.065</td><td>0.074</td><td>0.081</td><td>0.087</td><td>0.092</td><td>0.103</td><td>0.111</td><td>'β_yb2 = 0.056'</td></tr></table>
#if k < 1.1
β_xb2 = 0.055 + (0.065 - 0.055)*(k - 1)/0.1
#else if k < 1.2
β_xb2 = 0.065 + (0.074 - 0.065)*(k - 1.1)/0.1
#else if k < 1.3
β_xb2 = 0.074 + (0.081 - 0.074)*(k - 1.2)/0.1
#else if k < 1.4
β_xb2 = 0.081 + (0.087 - 0.081)*(k - 1.3)/0.1
#else if k < 1.5
β_xb2 = 0.087 + (0.092 - 0.087)*(k - 1.4)/0.1
#else if k < 1.75
β_xb2 = 0.092 + (0.103 - 0.092)*(k - 1.5)/0.25
#else if k < 2
β_xb2 = 0.103 + (0.111 - 0.103)*(k - 1.75)/0.25
#else if k ≡ 2
β_xb2 = 0.111
#end if
#end if
'<p><b>Loads for accounting alternation in adjacent spans</b></p>
p_1 = γ_G*(g_sw + g_k) + γ_Q*q_k/2'kN/m²
p_2 = γ_Q*q_k/2'kN/m²
'<p><b>Bending moments</b></p>
'Span moment (including load alternation)
M_Ed_xb = β_xb1*p_1*l_x^2 + β_xb2*p_2*l_x^2'kN·m/m
#if k ≤ 2
M_Ed_yb = β_yb1*p_1*l_x^2 + β_yb2*p_2*l_x^2'kN·m/m
#end if
'Support moment
M_Ed_xt = β_xt*p*l_x^2'kN·m/m
'<p><b>Average moments at supports from the adjacent spans</b></p>
M_Ed_xt2 = (M_Ed_xt2_*l_x2 + M_Ed_xt*l_x)/(l_x2 + l_x)'kN·m/m
'<p><b>Shear forces</b></p>
#if k ≤ 2
V_Ed_x1 = β_vx1*p*l_x'kN/m
V_Ed_x2 = β_vx2*p*l_x'kN/m
V_Ed_y = β_vy*p*l_x'kN/m
#else
V_Ed_x1 = β_vx1*p*l_x'kN/m
V_Ed_x2 = β_vx2*p*l_x'kN/m
#end if
'<p><b>Edge support moments</b></p>
M_Ed_xte2 = M_Ed_xt2 - V_Ed_x2*min(b_x/200;h/200)'kN·m/m
'</div><div class="fold">
'<h4>Bending design</h4>
'<p><b>Bottom reinforcement parallel to <i>x</i></b></p>
'Effective cross section depth -'d_xb = h - c - d_bxb/20'cm
'Relative design bending moment -'m_Ed = 10*M_Ed_xb/(d_xb^2*η*f_cd)
'Compression zone depth -'x = d_xb/λ*(1 - sqr(1 - 2*m_Ed))'cm
'Relative compression zone depth -'ξ = x/d_xb
'Lever arm of internal forces -'z = d_xb - 0.5*λ*x'cm
'Area of required reinforcement -'A_sxb = M_Ed_xb*10^3/(z*f_yd)'cm²/m
'Reinforcement ratio -'ρ_xb = A_sxb/(d_xb*100)
#if ξ > ξ_lim
'<p class="err">'ξ'>'ξ_lim'- Reinforcement in compression zone is required.</p>
#end if
#hide
#if k ≤ 2
'<p><b>Bottom reinforcement parallel to <i>y</i></b></p>
'Effective cross section depth -'d_yb = h - c - d_bxb/10 - d_byb/20'cm
'Relative design bending moment -'m_Ed = 10*M_Ed_yb/(d_yb^2*η*f_cd)
'Compression zone depth -'x = d_yb/λ*(1 - sqr(1 - 2*m_Ed))'cm
'Relative compression zone depth -'ξ = x/d_yb
'Lever arm of internal forces -'z = d_yb - 0.5*λ*x'cm
'Area of required reinforcement -'A_syb = M_Ed_yb*10^3/(z*f_yd)'cm²/m
'Reinforcement ratio -'ρ_yb = A_syb/(d_yb*100)
#if ξ > ξ_lim
'<p class="err">'ξ'>'ξ_lim'- Reinforcement in compression zone is required.</p>
#end if
#end if
'<p><b>Top reinforcement parallel to <i>x</i> за <i>M</i><sub>Ed_xte2</sub></b></p>
'Effective cross section depth -'d_xt2 = h - c - d_bxt/20'cm
'Relative design bending moment -'m_Ed = 10*M_Ed_xte2/(d_xt2^2*η*f_cd)
'Compression zone depth -'x = d_xt2/λ*(1 - sqr(1 - 2*m_Ed))'cm
'Relative compression zone depth -'ξ = x/d_xt2
'Lever arm of internal forces -'z = d_xt2 - 0.5*λ*x'cm
'Area of required reinforcement -'A_sxt2 = M_Ed_xte2*10^3/(z*f_yd)'cm²/m
'Reinforcement ratio -'ρ_xt2 = A_sxt2/(d_xt2*100)
#if ξ > ξ_lim
'<p class="err">'ξ'>'ξ_lim'- Reinforcement in compression zone is required.</p>
#end if
#post
'<p><b>The rest of the reinforcement is calculated similarly.</b></p>
'</div><div class="fold">
'<h4>Reinforcement detailing</h4>
'<p><b>Bottom reinforcement</b></p>
'<p class="ref">[EN 1992-1-1, Clause 9.2.1.1 (1)]</p>
'Minimum reinforcement ratio
ρ_min = max(0.26*f_ctm/f_yk;0.0013)
'<p class="ref">[EN 1992-1-1, § 9.2.1.1 (3)]</p>
'Maximum reinforcement ratio -'ρ_max = 0.04
#if ρ_xb < ρ_min
'<p><i>ρ</i><sub>xb</sub> < <i>ρ</i><sub>min</sub>. Minimum reinforcement is assumed -'A_sxb = ρ_min*d_xb*100'cm²/m</p>
#else if ρ_xb > ρ_max
'<p class="err">Reinforcement ratio is greater than maximum:'ρ_xb'>'ρ_max'</p>
#end if
#if k > 2
'<p class="ref">[EN 1992-1-1, § 9.3.1.1 (2)]</p>
'Minimum bottom reinforcement parallel to y
A_syb = 0.2*A_sxb'cm²/m
#else
'<!--'
#if ρ_yb < ρ_min
'<p><i>ρ</i><sub>yb</sub> < <i>ρ</i><sub>min</sub>. Minimum reinforcement is assumed -'A_syb = ρ_min*d_yb*100'cm²/m</p>
#else if ρ_yb > ρ_max
'<p class="err">Reinforcement ratio is greater than maximum:'ρ_yb'>'ρ_max'</p>
#end if
'<-->
#end if
'Required bar spacing
s_xb = floor(π*d_bxb^2/(4*A_sxb))'cm
#if k > 2
'<p class="ref">[EN 1992-1-1, § 9.3.1.1 (3)]</p>
'Minimum bottom reinforcement parallel to y
s_yb = min(3*h;40)'cm
'Required bar diameter for bottom reinforcement parallel to y
d_byb = max(ceiling(sqr(4*A_syb*s_yb/π));6)'mm
#else
'<!--'s_yb = floor(π*d_byb^2/(4*A_syb))'cm-->
#end if
'<p class="ref">[EN 1992-1-1, § 9.3.1.1 (3)]</p>
'Maximum bar spacing
s_max = min(2*h;25)'cm
#if s_xb > s_max
'<p><i>s</i><sub>xb</sub> > <i>s</i><sub>max</sub>. Assume's_xb = s_max'cm</p>
#end if
#if (k ≤ 2)*(s_yb > s_max)
'<!--<p><i>s</i><sub>yb</sub> > <i>s</i><sub>max</sub>. Assume's_yb = s_max'cm</p>-->
#end if
#hide
'<p><b>Top reinforcement</b></p>
#if ρ_xt2 < ρ_min
'<p><i>ρ</i><sub>xt2</sub> < <i>ρ</i><sub>min</sub>. Minimum reinforcement is assumed -'A_sxt2 = ρ_min*d_xt2*100'cm²/m</p>
#else if ρ_xt2 > ρ_max
'<p class="err">Reinforcement ratio is greater than maximum:'ρ_xt2'>'ρ_max'</p>
#end if
'Required bar spacing
s_xt2 = floor(π*d_bxt^2/(4*A_sxt2))'cm
'<p class="ref">[EN 1992-1-1, § 9.3.1.1 (3)]</p>
'Maximum bar spacing
s_max = min(2*h;25)'cm
#if s_xt2 > s_max
'<p><i>s</i><sub>xt2</sub> > <i>s</i><sub>max</sub>. It is assumed's_xt2 = s_max'cm</p>
#end if
#post
'<p><b>The rest of the reinforcement is detailed similarly.</b></p>
'</div>
'<h4>Results</h4>
#val
'<table class="bordered centered">
'<tr><th>Reinforcement</th><th>M<sub>Ed</sub><br>kN·m/m</th><th>A<sub>s</sub><br>cm<sup>2</sup>/m</th><th>ρ, %</th><th>Selected</tr>
'<tr><td>Bottom parallel to <i>x</i></td><td>'M_Ed_xb'</td><td>'A_sxb'</td><td>'A_sxb/d_xb'%</td><td>Ø'd_bxb'/'s_xb'</td></tr>
#if k ≤ 2
'<tr><td>Bottom parallel to <i>y</i></td><td>'M_Ed_yb'</td><td>'A_syb'</td><td>'A_syb/d_yb'%</td><td>Ø'd_byb'/'s_yb'</td></tr>
#else
'<tr><td>Bottom parallel to <i>y</i></td><td></td><td></td><td></td><td>Ø'd_byb'/'s_yb'</td></tr>
#end if
'<tr><td>Top parallel to <i>x</i> - right side</td><td>'M_Ed_xte2'</td><td>'A_sxt2'</td><td>'A_sxt2/d_xt2'%</td><td>Ø'd_bxt'/'s_xt2'</td></tr>
'</table>
#end if
#show
'</div> 4 6 25 25 14 2 0 0 2.5 2 25 0.85 500 8 8 6
Clear spans (lx < ly)
lx_cl = 4 m, ly_cl = 6 m
Beam width along x - by = 25 cm
Beam width along y - bx = 25 cm
Slab thickness - h = 14 cm
Concrete cover - c = 2 cm
Design span lengths
lx = lx_cl + min(bx100; h100) = 4 + min(25100; 14100) = 4.14 m
ly = ly_cl + min(by100; h100) = 6 + min(25100; 14100) = 6.14 m
Span ratio - k = lylx = 6.144.14 = 1.48
Design spans lengths and edge moments in adjacent panels
|
Parallel to x - |
lx2 = 0 m; |
MEd_xt2_ = 0 kN·m/m |
(in absence of contiguous span - l = 0)
Characteristic uniform loads
Self weight - gsw = 0.25 · h = 0.25 · 14 = 3.5 kN/m²
Permanent - gk = 2.5 kN/m²
Variable - qk = 2 kN/m²
Partial safety factor - γG = 1.35 ; γQ = 1.5
Design uniform loads
p = ( gsw + gk ) · γG + qk · γQ = ( 3.5 + 2.5 ) · 1.35 + 2 · 1.5 = 11.1 kN/m²
Beam loads
(as uniform loads)
|
Parallel to x - |
gk_x = ( gsw + gk ) · lx_cl4 = ( 3.5 + 2.5 ) · 44 = 6 kN/m |
|
qk_x = qk · lx_cl4 = 2 · 44 = 2 kN/m | |
|
Parallel to y - |
gk_y = ( gsw + gk ) · lx_cl · (ly_cl − lx_cl2)2 · ly_cl = ( 3.5 + 2.5 ) · 4 · (6 − 42)2 · 6 = 8 kN/m |
|
qk_y = qk · lx_cl · (ly_cl − lx_cl2)2 · ly_cl = 2 · 4 · (6 − 42)2 · 6 = 2.67 kN/m |
Concrete [EN 1992-1-1, Table 3.1]
Characteristic compressive cylinder strength - fck = 25 MPa
Partial safety factors for concrete - γc = 1.5 , αcc = 0.85
Design compressive cylinder strength - fcd = αcc · fckγc = 0.85 · 251.5 = 14.17 MPa
Factor defining the effective height of the compression zone - λ = 0.8
Factor defining the effective strength - η = 1
Mean value of axial tensile strength - fctm = 0.3 · fck23 = 0.3 · 2523 = 2.56 MPa
Steel
Characteristic yield strength - fyk = 500 MPa
Partial safety factor for steel - γs = 1.15
Design yield strength - fyd = fykγs = 5001.15 = 434.78 MPa
Limit relative compressive zone depth - ξlim = 0.45
Bar diameters
- Bottom parallel to x - dbxb = 8 mm, parallel to y - dbyb = 8 mm
- Top parallel to x - dbxt = 6 mm
Shear forces and bending moments coefficients
For the actual static scheme
- for shear forces
| Short span coefficients | Long span coefficients | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| k = | 1.0 | 1.1 | 1.2 | 1.3 | 1.4 | 1.5 | 1.75 | 2.0 | ||
| V | βvx1 = | 0.30 | 0.32 | 0.34 | 0.35 | 0.36 | 0.37 | 0.39 | 0.41 | βvy = 0.29 |
| V | βvx2 = | 0.45 | 0.48 | 0.51 | 0.53 | 0.55 | 0.57 | 0.60 | 0.63 | |
βvx1 = 0.36 + ( 0.37 − 0.36 ) · ( k − 1.4 ) 0.1 = 0.36 + ( 0.37 − 0.36 ) · ( 1.48 − 1.4 ) 0.1 = 0.368
βvx2 = 0.55 + ( 0.57 − 0.55 ) · ( k − 1.4 ) 0.1 = 0.55 + ( 0.57 − 0.55 ) · ( 1.48 − 1.4 ) 0.1 = 0.567
- for moments in mid-span
| Short span coefficients | Long span coefficients | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| k = | 1.0 | 1.1 | 1.2 | 1.3 | 1.4 | 1.5 | 1.75 | 2.0 | ||
| +M | βxb1 = | 0.043 | 0.048 | 0.053 | 0.057 | 0.060 | 0.063 | 0.069 | 0.074 | βyb1 = 0.044 |
| -M | βxt = | 0.057 | 0.065 | 0.071 | 0.076 | 0.081 | 0.084 | 0.092 | 0.098 | |
βxb1 = 0.063 + ( 0.06 − 0.063 ) · ( k − 1.4 ) 0.1 = 0.063 + ( 0.06 − 0.063 ) · ( 1.48 − 1.4 ) 0.1 = 0.0605
- for moments over supports
βxt = 0.081 + ( 0.084 − 0.081 ) · ( k − 1.4 ) 0.1 = 0.081 + ( 0.084 − 0.081 ) · ( 1.48 − 1.4 ) 0.1 = 0.0835
For simply supported slab
- for moments in mid-span
| Short span coefficients | Long span coefficients | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| k = | 1.0 | 1.1 | 1.2 | 1.3 | 1.4 | 1.5 | 1.75 | 2.0 | |
| βxb2 = | 0.055 | 0.065 | 0.074 | 0.081 | 0.087 | 0.092 | 0.103 | 0.111 | βyb2 = 0.056 |
βxb2 = 0.087 + ( 0.092 − 0.087 ) · ( k − 1.4 ) 0.1 = 0.087 + ( 0.092 − 0.087 ) · ( 1.48 − 1.4 ) 0.1 = 0.0912
Loads for accounting alternation in adjacent spans
p1 = γG · ( gsw + gk ) + γQ · qk2 = 1.35 · ( 3.5 + 2.5 ) + 1.5 · 22 = 9.6 kN/m²
p2 = γQ · qk2 = 1.5 · 22 = 1.5 kN/m²
Bending moments
Span moment (including load alternation)
MEd_xb = βxb1 · p1 · lx2 + βxb2 · p2 · lx2 = 0.0605 · 9.6 · 4.142 + 0.0912 · 1.5 · 4.142 = 12.3 kN·m/m
MEd_yb = βyb1 · p1 · lx2 + βyb2 · p2 · lx2 = 0.044 · 9.6 · 4.142 + 0.056 · 1.5 · 4.142 = 8.68 kN·m/m
Support moment
MEd_xt = βxt · p · lx2 = 0.0835 · 11.1 · 4.142 = 15.88 kN·m/m
Average moments at supports from the adjacent spans
MEd_xt2 = MEd_xt2_ · lx2 + MEd_xt · lxlx2 + lx = 0 · 0 + 15.88 · 4.140 + 4.14 = 15.88 kN·m/m
Shear forces
VEd_x1 = βvx1 · p · lx = 0.368 · 11.1 · 4.14 = 16.93 kN/m
VEd_x2 = βvx2 · p · lx = 0.567 · 11.1 · 4.14 = 26.04 kN/m
VEd_y = βvy · p · lx = 0.29 · 11.1 · 4.14 = 13.33 kN/m
Edge support moments
MEd_xte2 = MEd_xt2 − VEd_x2 · min(bx200; h200) = 15.88 − 26.04 · min(25200; 14200) = 14.06 kN·m/m
Bottom reinforcement parallel to x
Effective cross section depth - dxb = h − c − dbxb20 = 14 − 2 − 820 = 11.6 cm
Relative design bending moment - mEd = 10 · MEd_xbdxb2 · η · fcd = 10 · 12.311.62 · 1 · 14.17 = 0.0645
Compression zone depth - x = dxbλ · ( 1 − √ 1 − 2 · mEd ) = 11.60.8 · ( 1 − √ 1 − 2 · 0.0645 ) = 0.968 cm
Relative compression zone depth - ξ = xdxb = 0.96811.6 = 0.0834
Lever arm of internal forces - z = dxb − 0.5 · λ · x = 11.6 − 0.5 · 0.8 · 0.968 = 11.21 cm
Area of required reinforcement - Asxb = MEd_xb · 103z · fyd = 12.3 · 10311.21 · 434.78 = 2.52 cm²/m
Reinforcement ratio - ρxb = Asxbdxb · 100 = 2.5211.6 · 100 = 0.00217
The rest of the reinforcement is calculated similarly.
Bottom reinforcement
[EN 1992-1-1, Clause 9.2.1.1 (1)]
Minimum reinforcement ratio
ρmin = max(0.26 · fctmfyk; 0.0013) = max(0.26 · 2.56500; 0.0013) = 0.00133
[EN 1992-1-1, § 9.2.1.1 (3)]
Maximum reinforcement ratio - ρmax = 0.04
Required bar spacing
sxb = floor(π · dbxb24 · Asxb) = floor(3.14 · 824 · 2.52) = 19 cm
[EN 1992-1-1, § 9.3.1.1 (3)]
Maximum bar spacing
smax = min ( 2 · h; 25 ) = min ( 2 · 14; 25 ) = 25 cm
The rest of the reinforcement is detailed similarly.
| Reinforcement | MEd kN·m/m | As cm2/m | ρ, % | Selected |
|---|---|---|---|---|
| Bottom parallel to x | 12.3 | 2.52 | 0.22% | Ø8/19 |
| Bottom parallel to y | 8.68 | 1.9 | 0.18% | Ø8/25 |
| Top parallel to x - right side | 14.06 | 2.87 | 0.25% | Ø6/9 |
Restrained Panel with Unrestrained Longer Edge¶
Two-way slab panel restrained on three sides with the longer edge left free, typical of a balcony or canopy oriented along the long span.
'<small>According to <strong>Eurocode</strong>: EN 1992-1-1</small>
'<div style="max-width:180mm;">
'<img alt="sfff-span-cl.png" class="side" style="height:180pt;" src="../../Images/structures/rc/spans/sfff-span-cl.png">
'<h4>Dimensions</h4>
'Clear spans (<i>l</i><sub>x</sub> < <i>l</i><sub>y</sub>)
l_x_cl = ?'m,'l_y_cl = ?'m
'Beam width along <i>x</i> -'b_y = ?'cm
'Beam width along <i>y</i> -'b_x = ?'cm
'Slab thickness -'h = ?'cm
'Concrete cover -'c = ?'cm
#post
'Design span lengths
l_x = l_x_cl + min(b_x/100;h/100)'md
l_y = l_y_cl + min(b_y/100;h/100)'m
'Span ratio -'k = l_y/l_x
#if k < 1
'Span length <i>l</i><sub>y</sub> should be greater than <i>l</i><sub>x</sub>
#else
#show
'Design spans lengths and edge moments in adjacent panels
'<img alt="sfff-span.png" class="side" style="height:140pt;" src="../../Images/structures/rc/spans/sfff-span.png">
'<table><tr><td>
'Parallel to <i>x</i> - <br />
'</td><td>
l_x2 = ?'m;
'</td><td>
M_Ed_xt2_ = ?'kN·m/m
'</td></tr></table>
'<!--
#if k ≤ 2
'<-->
'<table><tr><td rowspan="2">
'Parallel to <i>y</i> -
'</td><td>
l_y1 = ?'m;
'</td><td>
M_Ed_yt1_ = ?'kN·m/m
'</td></tr><tr><td>
l_y2 = ?'m;
'</td><td>
M_Ed_yt2_ = ?'kN·m/m
'</td></tr></table>
'<!--
#end if
'<-->
'(in absence of contiguous span - <i>l</i> = 0)
'<h4>Loads</h4>
'Characteristic uniform loads
'Self weight -'g_sw = 0.25*h'kN/m²
'Permanent - 'g_k = ?'kN/m²
'Variable -'q_k = ?'kN/m²
'Partial safety factor -'γ_G = 1.35';'γ_Q = 1.5
#post
'Design uniform loads
p = (g_sw + g_k)*γ_G + q_k*γ_Q'kN/m²
'<div class="fold">
'<p><b>Beam loads</b></p>
'(as uniform loads)
#if k ≤ 2
'<table><tr><td rowspan="2">
'Parallel to <i>x</i> - <br />
'</td><td>
g_k_x = (g_sw + g_k)*l_x_cl/4'kN/m
'</td></tr><tr><td>
q_k_x = q_k*l_x_cl/4'kN/m
'</td></tr><tr><td rowspan="2">
'Parallel to <i>y</i> - <br />
'</td><td>
g_k_y = (g_sw + g_k)*l_x_cl*(l_y_cl - l_x_cl/2)/(2*l_y_cl)'kN/m
'</td></tr><tr><td>
q_k_y = q_k*l_x_cl*(l_y_cl - l_x_cl/2)/(2*l_y_cl)'kN/m
'</td></tr></table>
#else
'Parallel to <i>y</i> -'q_y = p*l_x_cl/2'kN/m
#end if
'</div>
#show
'<h4>Material properties</h4>
'<p><b>Concrete</b> [EN 1992-1-1, Table 3.1]</p>
'Characteristic compressive cylinder strength -'f_ck = ?'MPa
'Partial safety factors for concrete -'γ_c = 1.5','α_cc = ?
#post
'Design compressive cylinder strength -'f_cd = α_cc*f_ck/γ_c'MPa
'Factor defining the effective height of the compression zone -'λ = 0.8'
'Factor defining the effective strength - 'η = 1.0
'Mean value of axial tensile strength -'f_ctm = 0.30*f_ck^(2/3)'MPa
#show
'<p><b>Steel</b></p>
'Characteristic yield strength -'f_yk = ?'MPa
#post
'Partial safety factor for steel - 'γ_s = 1.15
'Design yield strength - 'f_yd = f_yk/γ_s'MPa
'Limit relative compressive zone depth -'ξ_lim = 0.45
#show
'<p><b>Bar diameters</b></p>
'- Bottom parallel to <i>x</i> -'d_bxb = ?'mm, parallel to <i>y</i> -'d_byb = ?'mm
'- Top parallel to <i>x</i> -'d_bxt = ?'mm, parallel to <i>y</i> -'d_byt = ?'mm
#post
'<div class="fold">
'<h4>Static calculations</h4>
'<p><b>Shear forces and bending moments coefficients</b></p>
#if l_y > 2*l_x
l_y/l_x'>'2'- One-way slab
'For the actual static scheme
'- for moments in mid-span -'β_xb1 = 0.0703
'- for moments over supports -'β_xt = 0.125
'- for shear forces -'β_vx1 = 0.375';'β_vx2 = 0.625
'For simply supported slab
'- for span moments -'β_xb2 = 0.125
#else if l_x ≡ l_y
l_y/l_x'- Two-way square slab
'For the actual static scheme
'- for moments in mid-span -'β_xb1 = 0.030';'β_yb1 = 0.028
'- for moments over supports -'β_xt = 0.039';'β_yt = 0.037
'- for shear forces -'β_vx1 = 0.24';'β_vx2 = 0.36';'β_vy = 0.36
'For simply supported slab
'- for span moments -'β_xb2 = 0.055';'β_yb2 = 0.055
#else
'For the actual static scheme
'- for shear forces
'<table class="bordered centered">
'<tr><th rowspan="2"></th><th style="width: 100mm;" colspan="9">Short span coefficients</th><th style="width:30mm;" rowspan="2">Long span coefficients</th></tr>
'<tr><th><i>k</i> =</th><th>1.0</th><th>1.1</th><th>1.2</th><th>1.3</th><th>1.4</th><th>1.5</th><th>1.75</th><th>2.0</th></tr>
'<tr><th>V</th><td><i>β</i><sub>vx1</sub> = </td><td>0.24</td><td>0.27</td><td>0.29</td><td>0.31</td><td>0.32</td><td>0.34</td><td>0.36</td><td>0.38</td><td rowspan="2">'β_vy = 0.36'</td></tr>
'<tr><th>V</th><td><i>β</i><sub>vx2</sub> = </td><td>0.36</td><td>0.40</td><td>0.44</td><td>0.47</td><td>0.49</td><td>0.51</td><td>0.55</td><td>0.59</td></tr></table>
#if k < 1.1
β_vx1 = 0.24 + (0.27 - 0.24)*(k - 1)/0.1
β_vx2 = 0.36 + (0.40 - 0.36)*(k - 1)/0.1
#else if k < 1.2
β_vx1 = 0.27 + (0.29 - 0.27)*(k - 1.1)/0.1
β_vx2 = 0.40 + (0.44 - 0.40)*(k - 1.1)/0.1
#else if k < 1.3
β_vx1 = 0.29 + (0.31 - 0.29)*(k - 1.2)/0.1
β_vx2 = 0.44 + (0.47 - 0.44)*(k - 1.2)/0.1
#else if k < 1.4
β_vx1 = 0.31 + (0.32 - 0.31)*(k - 1.3)/0.1
β_vx2 = 0.47 + (0.49 - 0.47)*(k - 1.3)/0.1
#else if k < 1.5
β_vx1 = 0.32 + (0.34 - 0.32)*(k - 1.4)/0.1
β_vx2 = 0.49 + (0.51 - 0.49)*(k - 1.4)/0.1
#else if k < 1.75
β_vx1 = 0.34 + (0.36 - 0.34)*(k - 1.5)/0.25
β_vx2 = 0.51 + (0.55 - 0.51)*(k - 1.5)/0.25
#else if k < 2
β_vx1 = 0.36 + (0.38 - 0.36)*(k - 1.75)/0.25
β_vx2 = 0.55 + (0.59 - 0.55)*(k - 1.75)/0.25
#else if k ≡ 2
β_vx1 = 0.38
β_vx2 = 0.59
#end if
'- for moments in mid-span
'<table class="bordered centered">
'<tr><th rowspan="2"></th><th style="width:100mm;" colspan="9">Short span coefficients</th><th style="width:30mm;" rowspan="2">Long span coefficients</th></tr>
'<tr><th><i>k</i> =</th><th>1.0</th><th>1.1</th><th>1.2</th><th>1.3</th><th>1.4</th><th>1.5</th><th>1.75</th><th>2.0</th></tr>
'<tr><th>+M</th><td><i>β</i><sub>xb1</sub> = </td><td>0.030</td><td>0.036</td><td>0.042</td><td>0.047</td><td>0.051</td><td>0.055</td><td>0.062</td><td>0.067</td><td>'β_yb1 = 0.028'</td></tr>
'<tr><th>-M</th><td><i>β</i><sub>xt</sub> = </td><td>0.039</td><td>0.049</td><td>0.056</td><td>0.062</td><td>0.068</td><td>0.073</td><td>0.082</td><td>0.089</td><td>'β_yt = 0.037'</td></tr></table>
#if k < 1.1
β_xb1 = 0.030 + (0.036 - 0.030)*(k - 1)/0.1
#else if k < 1.2
β_xb1 = 0.036 + (0.042 - 0.036)*(k - 1.1)/0.1
#else if k < 1.3
β_xb1 = 0.042 + (0.047 - 0.042)*(k - 1.2)/0.1
#else if k < 1.4
β_xb1 = 0.047 + (0.051 - 0.047)*(k - 1.3)/0.1
#else if k < 1.5
β_xb1 = 0.051 + (0.055 - 0.051)*(k - 1.4)/0.1
#else if k < 1.75
β_xb1 = 0.055 + (0.062 - 0.055)*(k - 1.5)/0.25
#else if k < 2
β_xb1 = 0.062 + (0.067 - 0.062)*(k - 1.75)/0.25
#else if k ≡ 2
β_xb1 = 0.067
#end if
'- for moments over supports
#if k < 1.1
β_xt = 0.039 + (0.049 - 0.039)*(k - 1)/0.1
#else if k < 1.2
β_xt = 0.049 + (0.056 - 0.049)*(k - 1.1)/0.1
#else if k < 1.3
β_xt = 0.056 + (0.062 - 0.056)*(k - 1.2)/0.1
#else if k < 1.4
β_xt = 0.062 + (0.068 - 0.062)*(k - 1.3)/0.1
#else if k < 1.5
β_xt = 0.068 + (0.073 - 0.068)*(k - 1.4)/0.1
#else if k < 1.75
β_xt = 0.073 + (0.082 - 0.073)*(k - 1.5)/0.25
#else if k < 2
β_xt = 0.082 + (0.089 - 0.082)*(k - 1.75)/0.25
#else if k ≡ 2
β_xt = 0.089
#end if
'For simply supported slab
' - for moments in mid-span
'<table class="bordered centered">
'<tr><th style="width:100mm;" colspan="9">Short span coefficients</th><th style="width:30mm;" rowspan="2">Long span coefficients</th></tr>
'<tr><th><i>k</i> =</th><th>1.0</th><th>1.1</th><th>1.2</th><th>1.3</th><th>1.4</th><th>1.5</th><th>1.75</th><th>2.0</th></tr>
'<tr><td><i>β</i><sub>xb2</sub> = </td><td>0.055</td><td>0.065</td><td>0.074</td><td>0.081</td><td>0.087</td><td>0.092</td><td>0.103</td><td>0.111</td><td>'β_yb2 = 0.056'</td></tr></table>
#if k < 1.1
β_xb2 = 0.055 + (0.065 - 0.055)*(k - 1)/0.1
#else if k < 1.2
β_xb2 = 0.065 + (0.074 - 0.065)*(k - 1.1)/0.1
#else if k < 1.3
β_xb2 = 0.074 + (0.081 - 0.074)*(k - 1.2)/0.1
#else if k < 1.4
β_xb2 = 0.081 + (0.087 - 0.081)*(k - 1.3)/0.1
#else if k < 1.5
β_xb2 = 0.087 + (0.092 - 0.087)*(k - 1.4)/0.1
#else if k < 1.75
β_xb2 = 0.092 + (0.103 - 0.092)*(k - 1.5)/0.25
#else if k < 2
β_xb2 = 0.103 + (0.111 - 0.103)*(k - 1.75)/0.25
#else if k ≡ 2
β_xb2 = 0.111
#end if
#end if
'<p><b>Loads for accounting alternation in adjacent spans</b></p>
p_1 = γ_G*(g_sw + g_k) + γ_Q*q_k/2'kN/m²
p_2 = γ_Q*q_k/2'kN/m²
'<p><b>Bending moments</b></p>
'Span moment (including load alternation)
M_Ed_xb = β_xb1*p_1*l_x^2 + β_xb2*p_2*l_x^2'kN·m/m
#if k ≤ 2
M_Ed_yb = β_yb1*p_1*l_x^2 + β_yb2*p_2*l_x^2'kN·m/m
#end if
'Support moment
M_Ed_xt = β_xt*p*l_x^2'kN·m/m
#if k ≤ 2
M_Ed_yt = β_yt*p*l_x^2'kN·m/m
#end if
'<p><b>Average moments at supports from the adjacent spans</b></p>
M_Ed_xt2 = (M_Ed_xt2_*l_x2 + M_Ed_xt*l_x)/(l_x2 + l_x)'kN·m/m
#if k ≤ 2
M_Ed_yt1 = (M_Ed_yt1_*l_y1 + M_Ed_yt*l_y)/(l_y1 + l_y)'kN·m/m
M_Ed_yt2 = (M_Ed_yt2_*l_y2 + M_Ed_yt*l_y)/(l_y2 + l_y)'kN·m/m
#end if
'<p><b>Shear forces</b></p>
#if k ≤ 2
V_Ed_x1 = β_vx1*p*l_x'kN/m
V_Ed_x2 = β_vx2*p*l_x'kN/m
V_Ed_y = β_vy*p*l_x'kN/m
#else
V_Ed_x1 = β_vx1*p*l_x'kN/m
V_Ed_x2 = β_vx2*p*l_x'kN/m
#end if
'<p><b>Edge support moments</b></p>
M_Ed_xte2 = M_Ed_xt2 - V_Ed_x2*min(b_x/200;h/200)'kN·m/m
#if k ≤ 2
M_Ed_yte1 = M_Ed_yt1 - V_Ed_y*min(b_y/200;h/200)'kN·m/m
M_Ed_yte2 = M_Ed_yt2 - V_Ed_y*min(b_y/200;h/200)'kN·m/m
#end if
'</div><div class="fold">
'<h4>Bending design</h4>
'<p><b>Bottom reinforcement parallel to <i>x</i></b></p>
'Effective cross section depth -'d_xb = h - c - d_bxb/20'cm
'Relative design bending moment -'m_Ed = 10*M_Ed_xb/(d_xb^2*η*f_cd)
'Compression zone depth -'x = d_xb/λ*(1 - sqr(1 - 2*m_Ed))'cm
'Relative compression zone depth -'ξ = x/d_xb
'Lever arm of internal forces -'z = d_xb - 0.5*λ*x'cm
'Area of required reinforcement -'A_sxb = M_Ed_xb*10^3/(z*f_yd)'cm²/m
'Reinforcement ratio -'ρ_xb = A_sxb/(d_xb*100)
#if ξ > ξ_lim
'<p class="err">'ξ'>'ξ_lim'- Reinforcement in compression zone is required.</p>
#end if
#hide
#if k ≤ 2
'<p><b>Bottom reinforcement parallel to <i>y</i></b></p>
'Effective cross section depth -'d_yb = h - c - d_bxb/10 - d_byb/20'cm
'Relative design bending moment -'m_Ed = 10*M_Ed_yb/(d_yb^2*η*f_cd)
'Compression zone depth -'x = d_yb/λ*(1 - sqr(1 - 2*m_Ed))'cm
'Relative compression zone depth -'ξ = x/d_yb
'Lever arm of internal forces -'z = d_yb - 0.5*λ*x'cm
'Area of required reinforcement -'A_syb = M_Ed_yb*10^3/(z*f_yd)'cm²/m
'Reinforcement ratio -'ρ_yb = A_syb/(d_yb*100)
#if ξ > ξ_lim
'<p class="err">'ξ'>'ξ_lim'- Reinforcement in compression zone is required.</p>
#end if
#end if
'<p><b>Top reinforcement parallel to <i>x</i> for <i>M</i><sub>Ed_xte2</sub></b></p>
'Effective cross section depth -'d_xt2 = h - c - d_bxt/20'cm
'Relative design bending moment -'m_Ed = 10*M_Ed_xte2/(d_xt2^2*η*f_cd)
'Compression zone depth -'x = d_xt2/λ*(1 - sqr(1 - 2*m_Ed))'cm
'Relative compression zone depth -'ξ = x/d_xt2
'Lever arm of internal forces -'z = d_xt2 - 0.5*λ*x'cm
'Area of required reinforcement -'A_sxt2 = M_Ed_xte2*10^3/(z*f_yd)'cm²/m
'Reinforcement ratio -'ρ_xt2 = A_sxt2/(d_xt2*100)
#if ξ > ξ_lim
'<p class="err">'ξ'>'ξ_lim'- Reinforcement in compression zone is required.</p>
#end if
#if k ≤ 2
'<p><b>Top reinforcement parallel to <i>y</i> for <i>M</i><sub>Ed_yte1</sub></b></p>
'Effective cross section depth -'d_yt1 = h - c - d_bxt/10 - d_byt/20'cm
'Relative design bending moment -'m_Ed = 10*M_Ed_yte1/(d_yt1^2*η*f_cd)
'Compression zone depth -'x = d_yt1/λ*(1 - sqr(1 - 2*m_Ed))'cm
'Relative compression zone depth -'ξ = x/d_yt1
'Lever arm of internal forces -'z = d_yt1 - 0.5*λ*x'cm
'Area of required reinforcement -'A_syt1 = M_Ed_yte1*10^3/(z*f_yd)'cm²/m
'Reinforcement ratio -'ρ_yt1 = A_syt1/(d_yt1*100)
#if ξ > ξ_lim
'<p class="err">'ξ'>'ξ_lim'- Reinforcement in compression zone is required.</p>
#end if
'<p><b>Top reinforcement parallel to <i>y</i> for <i>M</i><sub>Ed_yte2</sub></b></p>
'Effective cross section depth -'d_yt2 = h - c - d_bxt/10 - d_byt/20'cm
'Relative design bending moment -'m_Ed = 10*M_Ed_yte2/(d_yt2^2*η*f_cd)
'Compression zone depth -'x = d_yt2/λ*(1 - sqr(1 - 2*m_Ed))'cm
'Relative compression zone depth -'ξ = x/d_yt2
'Lever arm of internal forces -'z = d_yt2 - 0.5*λ*x'cm
'Area of required reinforcement -'A_syt2 = M_Ed_yte2*10^3/(z*f_yd)'cm²/m
'Reinforcement ratio -'ρ_yt2 = A_syt2/(d_yt2*100)
#if ξ > ξ_lim
'<p class="err">'ξ'>'ξ_lim'- Reinforcement in compression zone is required.</p>
#end if
#end if
#post
'<p><b>The rest of the reinforcement is calculated similarly.</b></p>
'</div><div class="fold">
'<h4>Reinforcement detailing</h4>
'<p><b>Bottom reinforcement</b></p>
'<p class="ref">[EN 1992-1-1, § 9.2.1.1 (1)]</p>
'Minimum reinforcement ratio
ρ_min = max(0.26*f_ctm/f_yk;0.0013)
'<p class="ref">[EN 1992-1-1, § 9.2.1.1 (3)]</p>
'Maximum reinforcement ratio -'ρ_max = 0.04
#if ρ_xb < ρ_min
'<p><i>ρ</i><sub>xb</sub> < <i>ρ</i><sub>min</sub>. Minimum reinforcement is assumed - 'A_sxb = ρ_min*d_xb*100'cm<sup>2</sup>/m</p>
#else if ρ_xb > ρ_max
'<p class="err">Reinforcement ratio is greater than maximum:'ρ_xb'>'ρ_max'</p>
#end if
#if k > 2
'<p class="ref">[EN 1992-1-1, § 9.3.1.1 (2)]</p>
'Minimum bottom reinforcement parallel to y
A_syb = 0.2*A_sxb'cm²/m
#else
'<!--'
#if ρ_yb < ρ_min
'<p><i>ρ</i><sub>yb</sub> < <i>ρ</i><sub>min</sub>. Minimum reinforcement is assumed - 'A_syb = ρ_min*d_yb*100'cm<sup>2</sup>/m</p>
#else if ρ_yb > ρ_max
'<p class="err">Reinforcement ratio is greater than maximum:'ρ_yb'>'ρ_max'</p>
#end if
'<-->
#end if
'Required bar spacing
s_xb = floor(π*d_bxb^2/(4*A_sxb))'cm
#if k > 2
'<p class="ref">[EN 1992-1-1, § 9.3.1.1 (3)]</p>
'Minimum bottom reinforcement parallel to y
s_yb = min(3*h;40)'cm
'Required bar diameter for bottom reinforcement parallel to y
d_byb = max(ceiling(sqr(4*A_syb*s_yb/π));6)'mm
#else
'<!--'s_yb = floor(π*d_byb^2/(4*A_syb))'cm-->
#end if
'<p class="ref">[EN 1992-1-1, § 9.3.1.1 (3)]</p>
'Maximum bar spacing
s_max = min(2*h;25)'cm
#if z> s_max
'<p><i>s</i><sub>xb</sub> > <i>s</i><sub>max</sub>. Assume's_xb = s_max'cm</p>
#end if
#if (k ≤ 2)*(s_yb > s_max)
'<!--<p><i>s</i><sub>yb</sub> > <i>s</i><sub>max</sub>. Assume's_yb = s_max'cm</p>-->
#end if
#hide
'<p><b>Top reinforcement</b></p>
#if ρ_xt2 < ρ_min
'<p><i>ρ</i><sub>xt2</sub> < <i>ρ</i><sub>min</sub>. Minimum reinforcement is assumed -'A_sxt2 = ρ_min*d_xt2*100'cm<sup>2</sup>/m</p>
#else if ρ_xt2 > ρ_max
'<p class="err">Reinforcement ratio is greater than maximum:'ρ_xt2'>'ρ_max'</p>
#end if
#if k ≤ 2
#if ρ_yt1 < ρ_min
'<p><i>ρ</i><sub>yt1</sub> < <i>ρ</i><sub>min</sub>. Minimum reinforcement is assumed -'A_syt1 = ρ_min*d_yt1*100'cm<sup>2</sup>/m</p>
#else if ρ_yt1 > ρ_max
'<p class="err">Reinforcement ratio is greater than maximum:'ρ_yt1'>'ρ_max'</p>
#end if
#if ρ_yt2 < ρ_min
'<p><i>ρ</i><sub>yt2</sub> < <i>ρ</i><sub>min</sub>. Minimum reinforcement is assumed -'A_syt2 = ρ_min*d_yt2*100'cm<sup>2</sup>/m</p>
#else if ρ_yt2 > ρ_max
'<p class="err">Reinforcement ratio is greater than maximum:'ρ_yt2'>'ρ_max'</p>
#end if
#end if
'Required bar spacing
s_xt2 = floor(π*d_bxt^2/(4*A_sxt2))'cm
#if k ≤ 2
s_yt1 = floor(π*d_byt^2/(4*A_syt1))'cm
s_yt2 = floor(π*d_byt^2/4/A_syt2)'cm
#end if
'<p class="ref">[EN 1992-1-1, § 9.3.1.1 (3)]</p>
'Maximum bar spacing
s_max = min(2*h;25)'cm
#if s_xt2 > s_max
'<p><i>s</i><sub>xt2</sub> > <i>s</i><sub>max</sub>. It is assumed's_xt2 = s_max'cm</p>
#end if
#if (k ≤ 2)*(s_yt1 > s_max)
'<p><i>s</i><sub>yt1</sub> > <i>s</i><sub>max</sub>. It is assumed's_yt1 = s_max'cm</p>
#end if
#if (k ≤ 2)*(s_yt2 > s_max)
'<p><i>s</i><sub>yt2</sub> > <i>s</i><sub>max</sub>. It is assumed's_yt2 = s_max'cm</p>
#end if
#post
'<p><b>The rest of the reinforcement is detailed similarly.</b></p>
'</div>
'<h4>Results</h4>
#val
'<table class="bordered centered">
'<tr><th>Reinforcement</th><th>M<sub>Ed</sub><br>kN·m/m</th><th>A<sub>s</sub><br>cm²/m</th><th>ρ, %</th><th>Selected</tr>
'<tr><td>Bottom parallel to <i>x</i></td><td>'M_Ed_xb'</td><td>'A_sxb'</td><td>'A_sxb/d_xb'%</td><td>Ø'd_bxb'/'s_xb'</td></tr>
#if k ≤ 2
'<tr><td>Bottom parallel to <i>y</i></td><td>'M_Ed_yb'</td><td>'A_syb'</td><td>'A_syb/d_yb'%</td><td>Ø'd_byb'/'s_yb'</td></tr>
#else
'<tr><td>Bottom parallel to <i>y</i></td><td></td><td></td><td></td><td>Ø'd_byb'/'s_yb'</td></tr>
#end if
'<tr><td>Top parallel to <i>x</i> - right side</td><td>'M_Ed_xte2'</td><td>'A_sxt2'</td><td>'A_sxt2/d_xt2'%</td><td>Ø'd_bxt'/'s_xt2'</td></tr>
#if k ≤ 2
'<tr><td>Top parallel to <i>y</i> - down side</td><td>'M_Ed_yte1'</td><td>'A_syt1'</td><td>'A_syt1/d_yt1'%</td><td>Ø'd_byt'/'s_yt1'</td></tr>
'<tr><td>Top parallel to <i>y</i> - up side</td><td>'M_Ed_yte2'</td><td>'A_syt2'</td><td>'A_syt2/d_yt2'%</td><td>Ø'd_byt'/'s_yt2'</td></tr>
#end if
'</table>
#end if
#show
'</div> 4 6 25 25 14 2 0 0 0 0 0 0 2.5 2 25 0.85 500 8 8 6 6
Clear spans (lx < ly)
lx_cl = 4 m, ly_cl = 6 m
Beam width along x - by = 25 cm
Beam width along y - bx = 25 cm
Slab thickness - h = 14 cm
Concrete cover - c = 2 cm
Design span lengths
lx = lx_cl + min(bx100; h100) = 4 + min(25100; 14100) = 4.14 md
ly = ly_cl + min(by100; h100) = 6 + min(25100; 14100) = 6.14 m
Span ratio - k = lylx = 6.144.14 = 1.48
Design spans lengths and edge moments in adjacent panels
|
Parallel to x - |
lx2 = 0 m; |
MEd_xt2_ = 0 kN·m/m |
|
Parallel to y - |
ly1 = 0 m; |
MEd_yt1_ = 0 kN·m/m |
|
ly2 = 0 m; |
MEd_yt2_ = 0 kN·m/m |
(in absence of contiguous span - l = 0)
Characteristic uniform loads
Self weight - gsw = 0.25 · h = 0.25 · 14 = 3.5 kN/m²
Permanent - gk = 2.5 kN/m²
Variable - qk = 2 kN/m²
Partial safety factor - γG = 1.35 ; γQ = 1.5
Design uniform loads
p = ( gsw + gk ) · γG + qk · γQ = ( 3.5 + 2.5 ) · 1.35 + 2 · 1.5 = 11.1 kN/m²
Beam loads
(as uniform loads)
|
Parallel to x - |
gk_x = ( gsw + gk ) · lx_cl4 = ( 3.5 + 2.5 ) · 44 = 6 kN/m |
|
qk_x = qk · lx_cl4 = 2 · 44 = 2 kN/m | |
|
Parallel to y - |
gk_y = ( gsw + gk ) · lx_cl · (ly_cl − lx_cl2)2 · ly_cl = ( 3.5 + 2.5 ) · 4 · (6 − 42)2 · 6 = 8 kN/m |
|
qk_y = qk · lx_cl · (ly_cl − lx_cl2)2 · ly_cl = 2 · 4 · (6 − 42)2 · 6 = 2.67 kN/m |
Concrete [EN 1992-1-1, Table 3.1]
Characteristic compressive cylinder strength - fck = 25 MPa
Partial safety factors for concrete - γc = 1.5 , αcc = 0.85
Design compressive cylinder strength - fcd = αcc · fckγc = 0.85 · 251.5 = 14.17 MPa
Factor defining the effective height of the compression zone - λ = 0.8
Factor defining the effective strength - η = 1
Mean value of axial tensile strength - fctm = 0.3 · fck23 = 0.3 · 2523 = 2.56 MPa
Steel
Characteristic yield strength - fyk = 500 MPa
Partial safety factor for steel - γs = 1.15
Design yield strength - fyd = fykγs = 5001.15 = 434.78 MPa
Limit relative compressive zone depth - ξlim = 0.45
Bar diameters
- Bottom parallel to x - dbxb = 8 mm, parallel to y - dbyb = 8 mm
- Top parallel to x - dbxt = 6 mm, parallel to y - dbyt = 6 mm
Shear forces and bending moments coefficients
For the actual static scheme
- for shear forces
| Short span coefficients | Long span coefficients | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| k = | 1.0 | 1.1 | 1.2 | 1.3 | 1.4 | 1.5 | 1.75 | 2.0 | ||
| V | βvx1 = | 0.24 | 0.27 | 0.29 | 0.31 | 0.32 | 0.34 | 0.36 | 0.38 | βvy = 0.36 |
| V | βvx2 = | 0.36 | 0.40 | 0.44 | 0.47 | 0.49 | 0.51 | 0.55 | 0.59 | |
βvx1 = 0.32 + ( 0.34 − 0.32 ) · ( k − 1.4 ) 0.1 = 0.32 + ( 0.34 − 0.32 ) · ( 1.48 − 1.4 ) 0.1 = 0.337
βvx2 = 0.49 + ( 0.51 − 0.49 ) · ( k − 1.4 ) 0.1 = 0.49 + ( 0.51 − 0.49 ) · ( 1.48 − 1.4 ) 0.1 = 0.507
- for moments in mid-span
| Short span coefficients | Long span coefficients | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| k = | 1.0 | 1.1 | 1.2 | 1.3 | 1.4 | 1.5 | 1.75 | 2.0 | ||
| +M | βxb1 = | 0.030 | 0.036 | 0.042 | 0.047 | 0.051 | 0.055 | 0.062 | 0.067 | βyb1 = 0.028 |
| -M | βxt = | 0.039 | 0.049 | 0.056 | 0.062 | 0.068 | 0.073 | 0.082 | 0.089 | βyt = 0.037 |
βxb1 = 0.051 + ( 0.055 − 0.051 ) · ( k − 1.4 ) 0.1 = 0.051 + ( 0.055 − 0.051 ) · ( 1.48 − 1.4 ) 0.1 = 0.0543
- for moments over supports
βxt = 0.068 + ( 0.073 − 0.068 ) · ( k − 1.4 ) 0.1 = 0.068 + ( 0.073 − 0.068 ) · ( 1.48 − 1.4 ) 0.1 = 0.0722
For simply supported slab
- for moments in mid-span
| Short span coefficients | Long span coefficients | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| k = | 1.0 | 1.1 | 1.2 | 1.3 | 1.4 | 1.5 | 1.75 | 2.0 | |
| βxb2 = | 0.055 | 0.065 | 0.074 | 0.081 | 0.087 | 0.092 | 0.103 | 0.111 | βyb2 = 0.056 |
βxb2 = 0.087 + ( 0.092 − 0.087 ) · ( k − 1.4 ) 0.1 = 0.087 + ( 0.092 − 0.087 ) · ( 1.48 − 1.4 ) 0.1 = 0.0912
Loads for accounting alternation in adjacent spans
p1 = γG · ( gsw + gk ) + γQ · qk2 = 1.35 · ( 3.5 + 2.5 ) + 1.5 · 22 = 9.6 kN/m²
p2 = γQ · qk2 = 1.5 · 22 = 1.5 kN/m²
Bending moments
Span moment (including load alternation)
MEd_xb = βxb1 · p1 · lx2 + βxb2 · p2 · lx2 = 0.0543 · 9.6 · 4.142 + 0.0912 · 1.5 · 4.142 = 11.28 kN·m/m
MEd_yb = βyb1 · p1 · lx2 + βyb2 · p2 · lx2 = 0.028 · 9.6 · 4.142 + 0.056 · 1.5 · 4.142 = 6.05 kN·m/m
Support moment
MEd_xt = βxt · p · lx2 = 0.0722 · 11.1 · 4.142 = 13.73 kN·m/m
MEd_yt = βyt · p · lx2 = 0.037 · 11.1 · 4.142 = 7.04 kN·m/m
Average moments at supports from the adjacent spans
MEd_xt2 = MEd_xt2_ · lx2 + MEd_xt · lxlx2 + lx = 0 · 0 + 13.73 · 4.140 + 4.14 = 13.73 kN·m/m
MEd_yt1 = MEd_yt1_ · ly1 + MEd_yt · lyly1 + ly = 0 · 0 + 7.04 · 6.140 + 6.14 = 7.04 kN·m/m
MEd_yt2 = MEd_yt2_ · ly2 + MEd_yt · lyly2 + ly = 0 · 0 + 7.04 · 6.140 + 6.14 = 7.04 kN·m/m
Shear forces
VEd_x1 = βvx1 · p · lx = 0.337 · 11.1 · 4.14 = 15.47 kN/m
VEd_x2 = βvx2 · p · lx = 0.507 · 11.1 · 4.14 = 23.28 kN/m
VEd_y = βvy · p · lx = 0.36 · 11.1 · 4.14 = 16.54 kN/m
Edge support moments
MEd_xte2 = MEd_xt2 − VEd_x2 · min(bx200; h200) = 13.73 − 23.28 · min(25200; 14200) = 12.1 kN·m/m
MEd_yte1 = MEd_yt1 − VEd_y · min(by200; h200) = 7.04 − 16.54 · min(25200; 14200) = 5.88 kN·m/m
MEd_yte2 = MEd_yt2 − VEd_y · min(by200; h200) = 7.04 − 16.54 · min(25200; 14200) = 5.88 kN·m/m
Bottom reinforcement parallel to x
Effective cross section depth - dxb = h − c − dbxb20 = 14 − 2 − 820 = 11.6 cm
Relative design bending moment - mEd = 10 · MEd_xbdxb2 · η · fcd = 10 · 11.2811.62 · 1 · 14.17 = 0.0592
Compression zone depth - x = dxbλ · ( 1 − √ 1 − 2 · mEd ) = 11.60.8 · ( 1 − √ 1 − 2 · 0.0592 ) = 0.885 cm
Relative compression zone depth - ξ = xdxb = 0.88511.6 = 0.0763
Lever arm of internal forces - z = dxb − 0.5 · λ · x = 11.6 − 0.5 · 0.8 · 0.885 = 11.25 cm
Area of required reinforcement - Asxb = MEd_xb · 103z · fyd = 11.28 · 10311.25 · 434.78 = 2.31 cm²/m
Reinforcement ratio - ρxb = Asxbdxb · 100 = 2.3111.6 · 100 = 0.00199
The rest of the reinforcement is calculated similarly.
Bottom reinforcement
[EN 1992-1-1, § 9.2.1.1 (1)]
Minimum reinforcement ratio
ρmin = max(0.26 · fctmfyk; 0.0013) = max(0.26 · 2.56500; 0.0013) = 0.00133
[EN 1992-1-1, § 9.2.1.1 (3)]
Maximum reinforcement ratio - ρmax = 0.04
Required bar spacing
sxb = floor(π · dbxb24 · Asxb) = floor(3.14 · 824 · 2.31) = 21 cm
[EN 1992-1-1, § 9.3.1.1 (3)]
Maximum bar spacing
smax = min ( 2 · h; 25 ) = min ( 2 · 14; 25 ) = 25 cm
The rest of the reinforcement is detailed similarly.
| Reinforcement | MEd kN·m/m | As cm²/m | ρ, % | Selected |
|---|---|---|---|---|
| Bottom parallel to x | 11.28 | 2.31 | 0.2% | Ø8/21 |
| Bottom parallel to y | 6.05 | 1.44 | 0.13% | Ø8/25 |
| Top parallel to x - right side | 12.1 | 2.46 | 0.21% | Ø6/11 |
| Top parallel to y - down side | 5.88 | 1.48 | 0.13% | Ø6/19 |
| Top parallel to y - up side | 5.88 | 1.48 | 0.13% | Ø6/19 |
Restrained Panel with Unrestrained Shorter Edge¶
Two-way slab panel restrained on three sides with the shorter edge left free, typical of a balcony or canopy oriented across the short span.
'<small>According to <strong>Eurocode</strong>: EN 1992-1-1</small>
'<div style="max-width:180mm;">
'<img alt="fsff-span-cl.png" class="side" style="height:180pt;" src="../../Images/structures/rc/spans/fsff-span-cl.png">
'<h4>Dimensions</h4>
'Clear spans (<i>l</i><sub>x</sub> < <i>l</i><sub>y</sub>)
l_x_cl = ?'m,'l_y_cl = ?'m
'Beam width along <i>x</i> -'b_y = ?'cm
'Beam width along <i>y</i> -'b_x = ?'cm
'Slab thickness -'h = ?'cm
'Concrete cover -'c = ?'cm
#post
'Design span lengths
l_x = l_x_cl + min(b_x/100;h/100)'m
l_y = l_y_cl + min(b_y/100;h/100)'m
'Span ratio -'k = l_y/l_x
#if k < 1
'Span length <i>l</i><sub>y</sub> should be greater than <i>l</i><sub>x</sub>
#else
#show
'Design spans lengths and edge moments in adjacent panels
'<img alt="fsff-span.png" class="side" style="height:140pt;" src="../../Images/structures/rc/spans/fsff-span.png">
'<table><tr><td rowspan="2">
'Parallel to <i>x</i> - <br />
'</td><td>
l_x1 = ?'m;
'</td><td>
M_Ed_xt1_ = ?'kN·m/m
'</td></tr><tr><td>
l_x2 = ?'m;
'</td><td>
M_Ed_xt2_ = ?'kN·m/m
'</td></tr></table>
'<!--
#if k ≤ 2
'<-->
'<table><tr><td>
'Parallel to <i>y</i> -
'</td><td>
l_y1 = ?'m;
'</td><td>
M_Ed_yt1_ = ?'kN·m/m
'</td></tr></table>
'<!--
#end if
'<-->
'(in absence of contiguous span - <i>l</i> = 0)
'<h4>Loads</h4>
'Characteristic uniform loads
'Self weight -'g_sw = 0.25*h'kN/m<sup>2</sup>
'Permanent -'g_k = ?'kN/m<sup>2</sup>
'Variable -'q_k = ?'kN/m<sup>2</sup>
'Partial safety factor -'γ_G = 1.35';'γ_Q = 1.5
#post
'Design uniform loads
p = (g_sw + g_k)*γ_G + q_k*γ_Q'kN/m²
'<div class="fold">
'<p><b>Beam loads</b></p>
'(as uniform loads)
#if k ≤ 2
'<table><tr><td rowspan="2">
'Parallel to <i>x</i> - <br />
'</td><td>
g_k_x = (g_sw + g_k)*l_x_cl/4'kN/m
'</td></tr><tr><td>
q_k_x = q_k*l_x_cl/4'kN/m
'</td></tr><tr><td rowspan="2">
'Parallel to <i>y</i> - <br />
'</td><td>
g_k_y = (g_sw + g_k)*l_x_cl*(l_y_cl - l_x_cl/2)/(2*l_y_cl)'kN/m
'</td></tr><tr><td>
q_k_y = q_k*l_x_cl*(l_y_cl - l_x_cl/2)/(2*l_y_cl)'kN/m
'</td></tr></table>
#else
'Parallel to <i>y</i> -'q_y = p*l_x_cl/2'kN/m
#end if
'</div>
#show
'<h4>Material properties</h4>
'<p><b>Concrete</b> [EN 1992-1-1, Table 3.1]</p>
'Characteristic compressive cylinder strength -'f_ck = ?'MPa
'Partial safety factors for concrete -'γ_c = 1.5','α_cc = ?
#post
'Design compressive cylinder strength -'f_cd = α_cc*f_ck/γ_c'MPa
'Factor defining the effective height of the compression zone -'λ = 0.8'
'Factor defining the effective strength - 'η = 1.0
'Mean value of axial tensile strength -'f_ctm = 0.30*f_ck^(2/3)'MPa
#show
'<p><b>Steel</b></p>
'Characteristic yield strength -'f_yk = ?'MPa
#post
'Partial safety factor for steel - 'γ_s = 1.15
'Design yield strength - 'f_yd = f_yk/γ_s'MPa
'Limit relative compressive zone depth -'ξ_lim = 0.45
#show
'<p><b>Bar diameters</b></p>
'- Bottom parallel to <i>x</i> -'d_bxb = ?'mm, parallel to <i>y</i> -'d_byb = ?'mm
'- Top parallel to <i>x</i> -'d_bxt = ?'mm, parallel to <i>y</i> -'d_byt = ?'mm
#post
'<div class="fold">
'<h4>Static calculations</h4>
'<p><b>Shear forces and bending moments coefficients</b></p>
#if l_y > 2*l_x
l_y/l_x'>'2'- One-way slab
'For the actual static scheme
'- for moments in mid-span -'β_xb1 = 0.04167
'- for moments over supports -'β_xt = 0.08333
'- for shear forces -'β_vx = 0.5
'For simply supported slab
'- for span moments -'β_xb2 = 0.125
#else if l_x ≡ l_y
l_y/l_x'- Two-way square slab
'For the actual static scheme
'- for moments in mid-span -'β_xb1 = 0.029';'β_yb1 = 0.028
'- for moments over supports -'β_xt = 0.039';'β_yt = 0.037
'- for shear forces -'β_vx = 0.36';'β_vy1 = 0.36';'β_vy2 = 0.24
'For simply supported slab
'- for span moments -'β_xb2 = 0.055';'β_yb2 = 0.055
#else
'For the actual static scheme
'- for shear forces
'<table class="bordered centered">
'<tr><th rowspan="2"></th><th style="width:100mm;" colspan="9">Short span coefficients</th><th style="width:30mm;" rowspan="2">Long span coefficients</th></tr>
'<tr><th><i>k</i> =</th><th>1.0</th><th>1.1</th><th>1.2</th><th>1.3</th><th>1.4</th><th>1.5</th><th>1.75</th><th>2.0</th></tr>
'<tr><th>V</th><td><i>β</i><sub>vx</sub> = </td><td>0.36</td><td>0.39</td><td>0.42</td><td>0.44</td><td>0.45</td><td>0.47</td><td>0.50</td><td>0.52</td><td>'β_vy1 = 0.36'<BR>'β_vy2 = 0.24'</td></tr></table>
#if k < 1.1
β_vx = 0.36 + (0.39 - 0.36)*(k - 1)/0.1
#else if k < 1.2
β_vx = 0.39 + (0.42 - 0.39)*(k - 1.1)/0.1
#else if k < 1.3
β_vx = 0.42 + (0.44 - 0.42)*(k - 1.2)/0.1
#else if k < 1.4
β_vx = 0.44 + (0.45 - 0.44)*(k - 1.3)/0.1
#else if k < 1.5
β_vx = 0.45 + (0.47 - 0.45)*(k - 1.4)/0.1
#else if k < 1.75
β_vx = 0.47 + (0.50 - 0.47)*(k - 1.5)/0.25
#else if k < 2
β_vx = 0.50 + (0.52 - 0.50)*(k - 1.75)/0.25
#else if k ≡ 2
β_vx = 0.52
#end if
'- for moments in mid-span
'<table class="bordered centered">
'<tr><th rowspan="2"></th><th style="width:100mm;" colspan="9">Short span coefficients</th><th style="width:30mm;" rowspan="2">Long span coefficients</th></tr>
'<tr><th><i>k</i> =</th><th>1.0</th><th>1.1</th><th>1.2</th><th>1.3</th><th>1.4</th><th>1.5</th><th>1.75</th><th>2.0</th></tr>
'<tr><th>+M</th><td><i>β</i><sub>xb1</sub> = </td><td>0.029</td><td>0.033</td><td>0.036</td><td>0.039</td><td>0.041</td><td>0.043</td><td>0.047</td><td>0.050</td><td>'β_yb1 = 0.028'</td></tr>
'<tr><th>-M</th><td><i>β</i><sub>xt</sub> = </td><td>0.039</td><td>0.044</td><td>0.048</td><td>0.052</td><td>0.055</td><td>0.058</td><td>0.063</td><td>0.067</td><td>'β_yt = 0.037'</td></tr></table>
#if k < 1.1
β_xb1 = 0.029 + (0.033 - 0.029)*(k - 1)/0.1
#else if k < 1.2
β_xb1 = 0.033 + (0.036 - 0.033)*(k - 1.1)/0.1
#else if k < 1.3
β_xb1 = 0.036 + (0.039 - 0.036)*(k - 1.2)/0.1
#else if k < 1.4
β_xb1 = 0.039 + (0.041 - 0.039)*(k - 1.3)/0.1
#else if k < 1.5
β_xb1 = 0.041 + (0.043 - 0.041)*(k - 1.4)/0.1
#else if k < 1.75
β_xb1 = 0.043 + (0.047 - 0.043)*(k - 1.5)/0.25
#else if k < 2
β_xb1 = 0.047 + (0.050 - 0.047)*(k - 1.75)/0.25
#else if k ≡ 2
β_xb1 = 0.050
#end if
'- for moments over supports
#if k < 1.1
β_xt = 0.039 + (0.044 - 0.039)*(k - 1)/0.1
#else if k < 1.2
β_xt = 0.044 + (0.048 - 0.044)*(k - 1.1)/0.1
#else if k < 1.3
β_xt = 0.048 + (0.052 - 0.048)*(k - 1.2)/0.1
#else if k < 1.4
β_xt = 0.052 + (0.055 - 0.052)*(k - 1.3)/0.1
#else if k < 1.5
β_xt = 0.055 + (0.058 - 0.055)*(k - 1.4)/0.1
#else if k < 1.75
β_xt = 0.058 + (0.063 - 0.058)*(k - 1.5)/0.25
#else if k < 2
β_xt = 0.063 + (0.067 - 0.063)*(k - 1.75)/0.25
#else if k ≡ 2
β_xt = 0.067
#end if
'For simply supported slab
' - for moments in mid-span
'<table class="bordered centered">
'<tr><th style="width:100mm;" colspan="9">Short span coefficients</th><th style="width:30mm;" rowspan="2">Long span coefficients</th></tr>
'<tr><th><i>k</i> =</th><th>1.0</th><th>1.1</th><th>1.2</th><th>1.3</th><th>1.4</th><th>1.5</th><th>1.75</th><th>2.0</th></tr>
'<tr><td><i>β</i><sub>xb2</sub> = </td><td>0.055</td><td>0.065</td><td>0.074</td><td>0.081</td><td>0.087</td><td>0.092</td><td>0.103</td><td>0.111</td><td>'β_yb2 = 0.056'</td></tr></table>
#if k < 1.1
β_xb2 = 0.055 + (0.065 - 0.055)*(k - 1)/0.1
#else if k < 1.2
β_xb2 = 0.065 + (0.074 - 0.065)*(k - 1.1)/0.1
#else if k < 1.3
β_xb2 = 0.074 + (0.081 - 0.074)*(k - 1.2)/0.1
#else if k < 1.4
β_xb2 = 0.081 + (0.087 - 0.081)*(k - 1.3)/0.1
#else if k < 1.5
β_xb2 = 0.087 + (0.092 - 0.087)*(k - 1.4)/0.1
#else if k < 1.75
β_xb2 = 0.092 + (0.103 - 0.092)*(k - 1.5)/0.25
#else if k < 2
β_xb2 = 0.103 + (0.111 - 0.103)*(k - 1.75)/0.25
#else if k ≡ 2
β_xb2 = 0.111
#end if
#end if
'<p><b>Loads for accounting alternation in adjacent spans</b></p>
p_1 = γ_G*(g_sw + g_k) + γ_Q*q_k/2'kN/m²
p_2 = γ_Q*q_k/2'kN/m²
'<p><b>Bending moments</b></p>
'Span moment (including load alternation)
M_Ed_xb = β_xb1*p_1*l_x^2 + β_xb2*p_2*l_x^2'kN·m/m
#if k ≤ 2
M_Ed_yb = β_yb1*p_1*l_x^2 + β_yb2*p_2*l_x^2'kN·m/m
#end if
'Support moment
M_Ed_xt = β_xt*p*l_x^2'kN·m/m
#if k ≤ 2
M_Ed_yt = β_yt*p*l_x^2'kN·m/m
#end if
'<p><b>Average moments at supports from the adjacent spans</b></p>
M_Ed_xt1 = (M_Ed_xt1_*l_x1 + M_Ed_xt*l_x)/(l_x1 + l_x)'kN·m/m
M_Ed_xt2 = (M_Ed_xt2_*l_x2 + M_Ed_xt*l_x)/(l_x2 + l_x)'kN·m/m
#if k ≤ 2
M_Ed_yt1 = (M_Ed_yt1_*l_y1 + M_Ed_yt*l_y)/(l_y1 + l_y)'kN·m/m
#end if
'<p><b>Shear forces</b></p>
#if k ≤ 2
V_Ed_x = β_vx*p*l_x'kN/m
V_Ed_y1 = β_vy1*p*l_x'kN/m
V_Ed_y2 = β_vy2*p*l_x'kN/m
#else
V_Ed_x = β_vx*p*l_x'kN/m
#end if
'<p><b>Edge support moments</b></p>
M_Ed_xte1 = M_Ed_xt1 - V_Ed_x*min(b_x/200;h/200)'kN·m/m
M_Ed_xte2 = M_Ed_xt2 - V_Ed_x*min(b_x/200;h/200)'kN·m/m
#if k ≤ 2
M_Ed_yte1 = M_Ed_yt1 - V_Ed_y1*min(b_y/200;h/200)'kN·m/m
#end if
'</div><div class="fold">
'<h4>Bending design</h4>
'<p><b>Bottom reinforcement parallel to <i>x</i></b></p>
'Effective cross section depth -'d_xb = h - c - d_bxb/20'cm
'Relative design bending moment -'m_Ed = 10*M_Ed_xb/(d_xb^2*η*f_cd)
'Compression zone depth -'x = d_xb/λ*(1 - sqr(1 - 2*m_Ed))'cm
'Relative compression zone depth -'ξ = x/d_xb
'Lever arm of internal forces -'z = d_xb - 0.5*λ*x'cm
'Area of required reinforcement -'A_sxb = M_Ed_xb*10^3/(z*f_yd)'cm<sup>2</sup>/m
'Reinforcement ratio -'ρ_xb = A_sxb/(d_xb*100)
#if ξ > ξ_lim
'<p class="err">'ξ'>'ξ_lim'- Reinforcement in compression zone is required.</p>
#end if
#hide
#if k ≤ 2
'<p><b>Bottom reinforcement parallel to <i>y</i></b></p>
'Effective cross section depth -'d_yb = h - c - d_bxb/10 - d_byb/20'cm
'Relative design bending moment -'m_Ed = 10*M_Ed_yb/(d_yb^2*η*f_cd)
'Compression zone depth -'x = d_yb/λ*(1 - sqr(1 - 2*m_Ed))'cm
'Relative compression zone depth -'ξ = x/d_yb
'Lever arm of internal forces -'z = d_yb - 0.5*λ*x'cm
'Area of required reinforcement -'A_syb = M_Ed_yb*10^3/(z*f_yd)'cm<sup>2</sup>/m
'Reinforcement ratio -'ρ_yb = A_syb/(d_yb*100)
#if ξ > ξ_lim
'<p class="err">'ξ'>'ξ_lim'- Reinforcement in compression zone is required.</p>
#end if
#end if
'<p><b>Top reinforcement parallel to <i>x</i> за <i>M</i><sub>Ed_xtel</sub></b></p>
'Effective cross section depth -'d_xt1 = h - c - d_bxt/20'cm
'Relative design bending moment -'m_Ed = 10*M_Ed_xte1/(d_xt1^2*η*f_cd)
'Compression zone depth -'x = d_xt1/λ*(1 - sqr(1 - 2*m_Ed))'cm
'Relative compression zone depth -'ξ = x/d_xt1
'Lever arm of internal forces -'z = d_xt1 - 0.5*λ*x'cm
'Area of required reinforcement -'A_sxt1 = M_Ed_xte1*10^3/(z*f_yd)'cm<sup>2</sup>/m
'Reinforcement ratio -'ρ_xt1 = A_sxt1/(d_xt1*100)
#if ξ > ξ_lim
'<p class="err">'ξ'>'ξ_lim'- Reinforcement in compression zone is required.</p>
#end if
'<p><b>Top reinforcement parallel to <i>x</i> за <i>M</i><sub>Ed_xte2</sub></b></p>
'Effective cross section depth -'d_xt2 = h - c - d_bxt/20'cm
'Relative design bending moment -'m_Ed = 10*M_Ed_xte2/(d_xt2^2*η*f_cd)
'Compression zone depth -'x = d_xt2/λ*(1 - sqr(1 - 2*m_Ed))'cm
'Relative compression zone depth -'ξ = x/d_xt2
'Lever arm of internal forces -'z = d_xt2 - 0.5*λ*x'cm
'Area of required reinforcement -'A_sxt2 = M_Ed_xte2*10^3/(z*f_yd)'cm²/m
'Reinforcement ratio -'ρ_xt2 = A_sxt2/(d_xt2*100)
#if ξ > ξ_lim
'<p class="err">'ξ'>'ξ_lim'- Reinforcement in compression zone is required.</p>
#end if
#if k ≤ 2
'<p><b>Top reinforcement parallel to <i>y</i> за <i>M</i><sub>Ed_yte1</sub></b></p>
'Effective cross section depth -'d_yt1 = h - c - d_bxt/10 - d_byt/20'cm
'Relative design bending moment -'m_Ed = 10*M_Ed_yte1/(d_xt1^2*η*f_cd)
'Compression zone depth -'x = d_yt1/λ*(1 - sqr(1 - 2*m_Ed))'cm
'Relative compression zone depth -'ξ = x/d_yt1
'Lever arm of internal forces -'z = d_yt1 - 0.5*λ*x'cm
'Area of required reinforcement -'A_syt1 = M_Ed_yte1*10^3/(z*f_yd)'cm²/m
'Reinforcement ratio -'ρ_yt1 = A_syt1/(d_yt1*100)
#if ξ > ξ_lim
'<p class="err">'ξ'>'ξ_lim'- Reinforcement in compression zone is required.</p>
#end if
#end if
#post
'<p><b>The rest of the reinforcement is calculated similarly.</b></p>
'</div><div class="fold">
'<h4>Reinforcement detailing</h4>
'<p><b>Bottom reinforcement</b></p>
'<p class="ref">[EN 1992-1-1, § 9.2.1.1 (1)]</p>
'Minimum reinforcement ratio
ρ_min = max(0.26*f_ctm/f_yk;0.0013)
'<p class="ref">[EN 1992-1-1, § 9.2.1.1 (3)]</p>
'Maximum reinforcement ratio -'ρ_max = 0.04
#if ρ_xb < ρ_min
'<p><i>ρ</i><sub>xb</sub> < <i>ρ</i><sub>min</sub>. Minimum reinforcement is assumed -'A_sxb = ρ_min*d_xb*100'cm²/m</p>
#else if ρ_xb > ρ_max
'<p class="err">Reinforcement ratio is greater than maximum:'ρ_xb'>'ρ_max'</p>
#end if
#if k > 2
'<p class="ref">[EN 1992-1-1, § 9.3.1.1 (2)]</p>
'Minimum bottom reinforcement parallel to y
A_syb = 0.2*A_sxb'cm<sup>2</sup>/m
#else
'<!--'
#if ρ_yb < ρ_min
'<p><i>ρ</i><sub>yb</sub> < <i>ρ</i><sub>min</sub>. Minimum reinforcement is assumed -'A_syb = ρ_min*d_yb*100'cm²/m</p>
#else if ρ_yb > ρ_max
'<p class="err">Reinforcement ratio is greater than maximum:'ρ_yb'>'ρ_max'</p>
#end if
'<-->
#end if
'Required bar spacing
s_xb = floor(π*d_bxb^2/(4*A_sxb))'cm
#if k > 2
'<p class="ref">[EN 1992-1-1, § 9.3.1.1 (3)]</p>
'Minimum bottom reinforcement parallel to y
s_yb = min(3*h;40)'cm
'Required bar diameter for bottom reinforcement parallel to y
d_byb = max(ceiling(sqr(4*A_syb*s_yb/π));6)'mm
#else
'<!--'s_yb = floor(π*d_byb^2/(4*A_syb))'cm-->
#end if
'<p class="ref">[EN 1992-1-1, § 9.3.1.1 (3)]</p>
'Maximum bar spacing
s_max = min(2*h;25)'cm
#if s_xb > s_max
'<p><i>s</i><sub>xb</sub> > <i>s</i><sub>max</sub>. Assume's_xb = s_max'cm</p>
#end if
#if (k ≤ 2)*(s_yb > s_max)
'<!--<p><i>s</i><sub>yb</sub> > <i>s</i><sub>max</sub>. Assume's_yb = s_max'cm</p>-->
#end if
#hide
'<p><b>Top reinforcement</b></p>
#if ρ_xt1 < ρ_min
'<p><i>ρ</i><sub>xt1</sub> < <i>ρ</i><sub>min</sub>. Minimum reinforcement is assumed -'A_sxt1 = ρ_min*d_xt1*100'cm<sup>2</sup>/m</p>
#else if ρ_xt1 > ρ_max
'<p class="err">Reinforcement ratio is greater than maximum:'ρ_xt1'>'ρ_max'</p>
#end if
#if ρ_xt2 < ρ_min
'<p><i>ρ</i><sub>xt2</sub> < <i>ρ</i><sub>min</sub>. Minimum reinforcement is assumed -'A_sxt2 = ρ_min*d_xt2*100'cm<sup>2</sup>/m</p>
#else if ρ_xt2 > ρ_max
'<p class="err">Reinforcement ratio is greater than maximum:'ρ_xt2'>'ρ_max'</p>
#end if
#if k ≤ 2
#if ρ_yt1 < ρ_min
'<p><i>ρ</i><sub>yt1</sub> < <i>ρ</i><sub>min</sub>. Minimum reinforcement is assumed -'A_syt1 = ρ_min*d_yt1*100'cm<sup>2</sup>/m</p>
#else if ρ_yt1 > ρ_max
'<p class="err">Reinforcement ratio is greater than maximum:'ρ_yt1'>'ρ_max'</p>
#end if
#end if
'Required bar spacing
s_xt1 = floor(π*d_bxt^2/(4*A_sxt1))'cm
s_xt2 = floor(π*d_bxt^2/(4*A_sxt2))'cm
#if k ≤ 2
s_yt1 = floor(π*d_byt^2/(4*A_syt1))'cm
#end if
'<p class="ref">[EN 1992-1-1, § 9.3.1.1 (3)]</p>
'Maximum bar spacing
s_max = min(2*h;25)'cm
#if s_xt1 > s_max
'<p><i>s</i><sub>xt1</sub> > <i>s</i><sub>max</sub>. It is assumed's_xt1 = s_max'cm</p>
#end if
#if s_xt2 > s_max
'<p><i>s</i><sub>xt2</sub> > <i>s</i><sub>max</sub>. It is assumed's_xt2 = s_max'cm</p>
#end if
#if (k ≤ 2)*(s_yt1 > s_max)
'<p><i>s</i><sub>yt1</sub> > <i>s</i><sub>max</sub>. It is assumed's_yt1 = s_max'cm</p>
#end if
#post
'<p><b>The rest of the reinforcement is detailed similarly</b></p>
'</div>
'<h4>Results</h4>
#val
'<table class="bordered centered">
'<tr><th>Reinforcement</th><th>M<sub>Ed</sub><br>kN·m/m</th><th>A<sub>s</sub><br>cm²/m</th><th>ρ, %</th><th>Selected</tr>
'<tr><td>Bottom parallel to <i>x</i></td><td>'M_Ed_xb'</td><td>'A_sxb'</td><td>'A_sxb/d_xb'%</td><td>Ø'd_bxb'/'s_xb'</td></tr>
#if k ≤ 2
'<tr><td>Bottom parallel to <i>y</i></td><td>'M_Ed_yb'</td><td>'A_syb'</td><td>'A_syb/d_yb'%</td><td>Ø'd_byb'/'s_yb'</td></tr>
#else
'<tr><td>Bottom parallel to <i>y</i></td><td></td><td></td><td></td><td>Ø'd_byb'/'s_yb'</td></tr>
#end if
'<tr><td>Top parallel to <i>x</i> - left side</td><td>'M_Ed_xte1'</td><td>'A_sxt1'</td><td>'A_sxt1/d_xt1'%</td><td>Ø'd_bxt'/'s_xt1'</td></tr>
'<tr><td>Top parallel to <i>x</i> - right side</td><td>'M_Ed_xte2'</td><td>'A_sxt2'</td><td>'A_sxt2/d_xt2'%</td><td>Ø'd_bxt'/'s_xt2'</td></tr>
#if k ≤ 2
'<tr><td>Top parallel to <i>y</i> - down side</td><td>'M_Ed_yte1'</td><td>'A_syt1'</td><td>'A_syt1/d_yt1'%</td><td>Ø'd_byt'/'s_yt1'</td></tr>
#end if
'</table>
#end if
#show
'</div>
4 6 25 25 14 2 0 0 0 0 0 0 2.5 2 25 0.85 500 8 8 6 6
Clear spans (lx < ly)
lx_cl = 4 m, ly_cl = 6 m
Beam width along x - by = 25 cm
Beam width along y - bx = 25 cm
Slab thickness - h = 14 cm
Concrete cover - c = 2 cm
Design span lengths
lx = lx_cl + min(bx100; h100) = 4 + min(25100; 14100) = 4.14 m
ly = ly_cl + min(by100; h100) = 6 + min(25100; 14100) = 6.14 m
Span ratio - k = lylx = 6.144.14 = 1.48
Design spans lengths and edge moments in adjacent panels
|
Parallel to x - |
lx1 = 0 m; |
MEd_xt1_ = 0 kN·m/m |
|
lx2 = 0 m; |
MEd_xt2_ = 0 kN·m/m |
|
Parallel to y - |
ly1 = 0 m; |
MEd_yt1_ = 0 kN·m/m |
(in absence of contiguous span - l = 0)
Characteristic uniform loads
Self weight - gsw = 0.25 · h = 0.25 · 14 = 3.5 kN/m2
Permanent - gk = 2.5 kN/m2
Variable - qk = 2 kN/m2
Partial safety factor - γG = 1.35 ; γQ = 1.5
Design uniform loads
p = ( gsw + gk ) · γG + qk · γQ = ( 3.5 + 2.5 ) · 1.35 + 2 · 1.5 = 11.1 kN/m²
Beam loads
(as uniform loads)
|
Parallel to x - |
gk_x = ( gsw + gk ) · lx_cl4 = ( 3.5 + 2.5 ) · 44 = 6 kN/m |
|
qk_x = qk · lx_cl4 = 2 · 44 = 2 kN/m | |
|
Parallel to y - |
gk_y = ( gsw + gk ) · lx_cl · (ly_cl − lx_cl2)2 · ly_cl = ( 3.5 + 2.5 ) · 4 · (6 − 42)2 · 6 = 8 kN/m |
|
qk_y = qk · lx_cl · (ly_cl − lx_cl2)2 · ly_cl = 2 · 4 · (6 − 42)2 · 6 = 2.67 kN/m |
Concrete [EN 1992-1-1, Table 3.1]
Characteristic compressive cylinder strength - fck = 25 MPa
Partial safety factors for concrete - γc = 1.5 , αcc = 0.85
Design compressive cylinder strength - fcd = αcc · fckγc = 0.85 · 251.5 = 14.17 MPa
Factor defining the effective height of the compression zone - λ = 0.8
Factor defining the effective strength - η = 1
Mean value of axial tensile strength - fctm = 0.3 · fck23 = 0.3 · 2523 = 2.56 MPa
Steel
Characteristic yield strength - fyk = 500 MPa
Partial safety factor for steel - γs = 1.15
Design yield strength - fyd = fykγs = 5001.15 = 434.78 MPa
Limit relative compressive zone depth - ξlim = 0.45
Bar diameters
- Bottom parallel to x - dbxb = 8 mm, parallel to y - dbyb = 8 mm
- Top parallel to x - dbxt = 6 mm, parallel to y - dbyt = 6 mm
Shear forces and bending moments coefficients
For the actual static scheme
- for shear forces
| Short span coefficients | Long span coefficients | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| k = | 1.0 | 1.1 | 1.2 | 1.3 | 1.4 | 1.5 | 1.75 | 2.0 | ||
| V | βvx = | 0.36 | 0.39 | 0.42 | 0.44 | 0.45 | 0.47 | 0.50 | 0.52 | βvy1 = 0.36 βvy2 = 0.24 |
βvx = 0.45 + ( 0.47 − 0.45 ) · ( k − 1.4 ) 0.1 = 0.45 + ( 0.47 − 0.45 ) · ( 1.48 − 1.4 ) 0.1 = 0.467
- for moments in mid-span
| Short span coefficients | Long span coefficients | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| k = | 1.0 | 1.1 | 1.2 | 1.3 | 1.4 | 1.5 | 1.75 | 2.0 | ||
| +M | βxb1 = | 0.029 | 0.033 | 0.036 | 0.039 | 0.041 | 0.043 | 0.047 | 0.050 | βyb1 = 0.028 |
| -M | βxt = | 0.039 | 0.044 | 0.048 | 0.052 | 0.055 | 0.058 | 0.063 | 0.067 | βyt = 0.037 |
βxb1 = 0.041 + ( 0.043 − 0.041 ) · ( k − 1.4 ) 0.1 = 0.041 + ( 0.043 − 0.041 ) · ( 1.48 − 1.4 ) 0.1 = 0.0427
- for moments over supports
βxt = 0.055 + ( 0.058 − 0.055 ) · ( k − 1.4 ) 0.1 = 0.055 + ( 0.058 − 0.055 ) · ( 1.48 − 1.4 ) 0.1 = 0.0575
For simply supported slab
- for moments in mid-span
| Short span coefficients | Long span coefficients | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| k = | 1.0 | 1.1 | 1.2 | 1.3 | 1.4 | 1.5 | 1.75 | 2.0 | |
| βxb2 = | 0.055 | 0.065 | 0.074 | 0.081 | 0.087 | 0.092 | 0.103 | 0.111 | βyb2 = 0.056 |
βxb2 = 0.087 + ( 0.092 − 0.087 ) · ( k − 1.4 ) 0.1 = 0.087 + ( 0.092 − 0.087 ) · ( 1.48 − 1.4 ) 0.1 = 0.0912
Loads for accounting alternation in adjacent spans
p1 = γG · ( gsw + gk ) + γQ · qk2 = 1.35 · ( 3.5 + 2.5 ) + 1.5 · 22 = 9.6 kN/m²
p2 = γQ · qk2 = 1.5 · 22 = 1.5 kN/m²
Bending moments
Span moment (including load alternation)
MEd_xb = βxb1 · p1 · lx2 + βxb2 · p2 · lx2 = 0.0427 · 9.6 · 4.142 + 0.0912 · 1.5 · 4.142 = 9.36 kN·m/m
MEd_yb = βyb1 · p1 · lx2 + βyb2 · p2 · lx2 = 0.028 · 9.6 · 4.142 + 0.056 · 1.5 · 4.142 = 6.05 kN·m/m
Support moment
MEd_xt = βxt · p · lx2 = 0.0575 · 11.1 · 4.142 = 10.94 kN·m/m
MEd_yt = βyt · p · lx2 = 0.037 · 11.1 · 4.142 = 7.04 kN·m/m
Average moments at supports from the adjacent spans
MEd_xt1 = MEd_xt1_ · lx1 + MEd_xt · lxlx1 + lx = 0 · 0 + 10.94 · 4.140 + 4.14 = 10.94 kN·m/m
MEd_xt2 = MEd_xt2_ · lx2 + MEd_xt · lxlx2 + lx = 0 · 0 + 10.94 · 4.140 + 4.14 = 10.94 kN·m/m
MEd_yt1 = MEd_yt1_ · ly1 + MEd_yt · lyly1 + ly = 0 · 0 + 7.04 · 6.140 + 6.14 = 7.04 kN·m/m
Shear forces
VEd_x = βvx · p · lx = 0.467 · 11.1 · 4.14 = 21.44 kN/m
VEd_y1 = βvy1 · p · lx = 0.36 · 11.1 · 4.14 = 16.54 kN/m
VEd_y2 = βvy2 · p · lx = 0.24 · 11.1 · 4.14 = 11.03 kN/m
Edge support moments
MEd_xte1 = MEd_xt1 − VEd_x · min(bx200; h200) = 10.94 − 21.44 · min(25200; 14200) = 9.44 kN·m/m
MEd_xte2 = MEd_xt2 − VEd_x · min(bx200; h200) = 10.94 − 21.44 · min(25200; 14200) = 9.44 kN·m/m
MEd_yte1 = MEd_yt1 − VEd_y1 · min(by200; h200) = 7.04 − 16.54 · min(25200; 14200) = 5.88 kN·m/m
Bottom reinforcement parallel to x
Effective cross section depth - dxb = h − c − dbxb20 = 14 − 2 − 820 = 11.6 cm
Relative design bending moment - mEd = 10 · MEd_xbdxb2 · η · fcd = 10 · 9.3611.62 · 1 · 14.17 = 0.0491
Compression zone depth - x = dxbλ · ( 1 − √ 1 − 2 · mEd ) = 11.60.8 · ( 1 − √ 1 − 2 · 0.0491 ) = 0.731 cm
Relative compression zone depth - ξ = xdxb = 0.73111.6 = 0.063
Lever arm of internal forces - z = dxb − 0.5 · λ · x = 11.6 − 0.5 · 0.8 · 0.731 = 11.31 cm
Area of required reinforcement - Asxb = MEd_xb · 103z · fyd = 9.36 · 10311.31 · 434.78 = 1.9 cm2/m
Reinforcement ratio - ρxb = Asxbdxb · 100 = 1.911.6 · 100 = 0.00164
The rest of the reinforcement is calculated similarly.
Bottom reinforcement
[EN 1992-1-1, § 9.2.1.1 (1)]
Minimum reinforcement ratio
ρmin = max(0.26 · fctmfyk; 0.0013) = max(0.26 · 2.56500; 0.0013) = 0.00133
[EN 1992-1-1, § 9.2.1.1 (3)]
Maximum reinforcement ratio - ρmax = 0.04
Required bar spacing
sxb = floor(π · dbxb24 · Asxb) = floor(3.14 · 824 · 1.9) = 26 cm
[EN 1992-1-1, § 9.3.1.1 (3)]
Maximum bar spacing
smax = min ( 2 · h; 25 ) = min ( 2 · 14; 25 ) = 25 cm
sxb > smax. Assume sxb = smax = 25 cm
The rest of the reinforcement is detailed similarly
| Reinforcement | MEd kN·m/m | As cm²/m | ρ, % | Selected |
|---|---|---|---|---|
| Bottom parallel to x | 9.36 | 1.9 | 0.16% | Ø8/25 |
| Bottom parallel to y | 6.05 | 1.44 | 0.13% | Ø8/25 |
| Top parallel to x - left side | 9.44 | 1.9 | 0.16% | Ø6/14 |
| Top parallel to x - right side | 9.44 | 1.9 | 0.16% | Ø6/14 |
| Top parallel to y - down side | 5.88 | 1.48 | 0.13% | Ø6/19 |
Restrained Shorter Edge¶
Two-way slab panel with one shorter edge restrained and the other three simply supported, typical of an edge panel oriented across the short span.
'<small>According to <strong>Eurocode</strong>: EN 1992-1-1</small>
'<div style="max-width:180mm;">
'<img alt="sssf-span-cl.png" class="side" style="height:180pt;" src="../../Images/structures/rc/spans/sssf-span-cl.png">
'<h4>Dimensions</h4>
'Clear spans (<i>l</i><sub>x</sub> < <i>l</i><sub>y</sub>)
l_x_cl = ?'m,'l_y_cl = ?'m
'Beam width along <i>x</i> -'b_y = ?'cm
'Beam width along <i>y</i> -'b_x = ?'cm
'Slab thickness -'h = ?'cm
'Concrete cover -'c = ?'cm
#post
'Design span lengths
l_x = l_x_cl + min(b_x/100;h/100)'m
l_y = l_y_cl + min(b_y/100;h/100)'m
'Span ratio -'k = l_y/l_x
#if k < 1
'Span length <i>l</i><sub>y</sub> should be greater than <i>l</i><sub>x</sub>
#else
#show
'Design spans lengths and edge moments in adjacent panels
'<img alt="sssf-span.png" class="side" style="height:140pt;" src="../../Images/structures/rc/spans/sssf-span.png">
'<!--
#if k ≤ 2
'<-->
'<table><tr><td>
'Along <i>y</i> -
'</td><td>
l_y1 = ?'m;
'</td><td>
M_Ed_yt1_ = ?'kN·m/m
'</td></tr></table>
'<!--
#end if
'<-->
'(in absence of contiguous span - <i>l</i> = 0)
'<h4>Loads</h4>
'Characteristic uniform loads
'Self weight -'g_sw = 0.25*h'kN/m²
'Permanent -'g_k = ?'kN/m²
'Variable -'q_k = ?'kN/m²
'Partial safety factor -'γ_G = 1.35';'γ_Q = 1.5
#post
'Design uniform loads
p = (g_sw + g_k)*γ_G + q_k*γ_Q'kN/m²
'<div class="fold">
'<p><b>Beam loads</b></p>
'(as uniform loads)
#if k ≤ 2
'<table><tr><td rowspan="2">
'Parallel to <i>x</i> - <br />
'</td><td>
g_k_x = (g_sw + g_k)*l_x_cl/4'kN/m
'</td></tr><tr><td>
q_k_x = q_k*l_x_cl/4'kN/m
'</td></tr><tr><td rowspan="2">
'Parallel to <i>y</i> - <br />
'</td><td>
g_k_y = (g_sw + g_k)*l_x_cl*(l_y_cl - l_x_cl/2)/(2*l_y_cl)'kN/m
'</td></tr><tr><td>
q_k_y = q_k*l_x_cl*(l_y_cl - l_x_cl/2)/(2*l_y_cl)'kN/m
'</td></tr></table>
#else
'Parallel to <i>y</i> -'q_y = p*l_x_cl/2'kN/m
#end if
'</div>
#show
'<h4>Material properties</h4>
'<p><b>Concrete</b> [EN 1992-1-1, EN 1992-1-1, Table 3.1]</p>
'Characteristic compressive cylinder strength -'f_ck = ?'MPa
'Partial safety factors for concrete -'γ_c = 1.5','α_cc = ?
#post
'Design compressive cylinder strength -'f_cd = α_cc*f_ck/γ_c'MPa
'Factor defining the effective height of the compression zone -'λ = 0.8'
'Factor defining the effective strength - 'η = 1.0
'Mean value of axial tensile strength -'f_ctm = 0.30*f_ck^(2/3)'MPa
#show
'<p><b>Steel</b></p>
'Characteristic yield strength -'f_yk = ?'MPa
#post
'Partial safety factor for steel - 'γ_s = 1.15
'Design yield strength - 'f_yd = f_yk/γ_s'MPa
'Limit relative compressive zone depth -'ξ_lim = 0.45
#show
'<p><b>Bar diameters</b></p>
'- Bottom parallel to <i>x</i> -'d_bxb = ?'mm, parallel to <i>y</i> -'d_byb = ?'mm
'- Top parallel to <i>y</i> -'d_byt = ?'mm
#post
'<div class="fold">
'<h4>Static calculations</h4>
'<p><b>Shear forces and bending moments coefficients</b></p>
#if l_y > 2*l_x
l_y/l_x'>'2'- One-way slab
'For the actual static scheme
'- for moments in mid-span -'β_xb1 = 0.125
'- for moments over supports -'β_vx = 0.5
'For simply supported slab
'- for span moments -'β_xb2 = 0.125
#else if l_x ≡ l_y
l_y/l_x'- Two-way square slab
'For the actual static scheme
'- for moments in mid-span -'β_xb1 = 0.042';'β_yb1 = 0.044
'- for moments over supports -'β_yt = 0.058
'- for shear forces -'β_vx = 0.29';'β_vy1 = 0.45';'β_vy2 = 0.30
'For simply supported slab
'- for span moments -'β_xb2 = 0.055';'β_yb2 = 0.055
#else
'For the actual static scheme
'- for shear forces
'<table class="bordered centered">
'<tr><th rowspan="2"></th><th style="width:100mm;" colspan="9">Short span coefficients</th><th style="width:30mm;" rowspan="2">Long span coefficients</th></tr>
'<tr><th><i>k</i> =</th><th>1.0</th><th>1.1</th><th>1.2</th><th>1.3</th><th>1.4</th><th>1.5</th><th>1.75</th><th>2.0</th></tr>
'<tr><th>V</th><td><i>β</i><sub>vx</sub> = </td><td>0.29</td><td>0.33</td><td>0.36</td><td>0.38</td><td>0.40</td><td>0.42</td><td>0.45</td><td>0.480</td><td>'β_vy1 = 0.45'<BR>'β_vy2 = 0.30'</td></tr></table>
#if k < 1.1
β_vx = 0.29 + (0.33 - 0.29)*(k - 1)/0.1
#else if k < 1.2
β_vx = 0.33 + (0.36 - 0.33)*(k - 1.1)/0.1
#else if k < 1.3
β_vx = 0.36 + (0.38 - 0.36)*(k - 1.2)/0.1
#else if k < 1.4
β_vx = 0.38 + (0.40 - 0.38)*(k - 1.3)/0.1
#else if k < 1.5
β_vx = 0.40 + (0.42 - 0.40)*(k - 1.4)/0.1
#else if k < 1.75
β_vx = 0.42 + (0.45 - 0.42)*(k - 1.5)/0.25
#else if k < 2
β_vx = 0.45 + (0.48 - 0.45)*(k - 1.75)/0.25
#else if k ≡ 2
β_vx = 0.48
#end if
'- for moments in mid-span
'<table class="bordered centered">
'<tr><th rowspan="2"></th><th style="width:100mm;" colspan="9">Short span coefficients</th><th style="width:30mm;" rowspan="2">Long span coefficients</th></tr>
'<tr><th><i>k</i> =</th><th>1.0</th><th>1.1</th><th>1.2</th><th>1.3</th><th>1.4</th><th>1.5</th><th>1.75</th><th>2.0</th></tr>
'<tr><th>+M</th><td><i>β</i><sub>xb1</sub> = </td><td>0.042</td><td>0.054</td><td>0.063</td><td>0.071</td><td>0.078</td><td>0.084</td><td>0.096</td><td>0.105</td><td>'β_yb1 = 0.044'<BR>'β_yt = 0.058'</td></tr></table>
#if k < 1.1
β_xb1 = 0.042 + (0.054 - 0.042)*(k - 1)/0.1
#else if k < 1.2
β_xb1 = 0.054 + (0.063 - 0.054)*(k - 1.1)/0.1
#else if k < 1.3
β_xb1 = 0.063 + (0.071 - 0.063)*(k - 1.2)/0.1
#else if k < 1.4
β_xb1 = 0.071 + (0.078 - 0.071)*(k - 1.3)/0.1
#else if k < 1.5
β_xb1 = 0.078 + (0.084 - 0.078)*(k - 1.4)/0.1
#else if k < 1.75
β_xb1 = 0.084 + (0.096 - 0.084)*(k - 1.5)/0.25
#else if k < 2
β_xb1 = 0.096 + (0.105 - 0.096)*(k - 1.75)/0.25
#else if k ≡ 2
β_xb1 = 0.105
#end if
'For simply supported slab
'- for moments in mid-span
'<table class="bordered centered">
'<tr><th style="width:100mm;" colspan="9">Short span coefficients</th><th style="width:30mm;" rowspan="2">Long span coefficients</th></tr>
'<tr><th><i>k</i> =</th><th>1.0</th><th>1.1</th><th>1.2</th><th>1.3</th><th>1.4</th><th>1.5</th><th>1.75</th><th>2.0</th></tr>
'<tr><td><i>β</i><sub>xb2</sub> = </td><td>0.055</td><td>0.065</td><td>0.074</td><td>0.081</td><td>0.087</td><td>0.092</td><td>0.103</td><td>0.111</td><td>'β_yb2 = 0.056'</td></tr></table>
#if k < 1.1
β_xb2 = 0.055 + (0.065 - 0.055)*(k - 1)/0.1
#else if k < 1.2
β_xb2 = 0.065 + (0.074 - 0.065)*(k - 1.1)/0.1
#else if k < 1.3
β_xb2 = 0.074 + (0.081 - 0.074)*(k - 1.2)/0.1
#else if k < 1.4
β_xb2 = 0.081 + (0.087 - 0.081)*(k - 1.3)/0.1
#else if k < 1.5
β_xb2 = 0.087 + (0.092 - 0.087)*(k - 1.4)/0.1
#else if k < 1.75
β_xb2 = 0.092 + (0.103 - 0.092)*(k - 1.5)/0.25
#else if k < 2
β_xb2 = 0.103 + (0.111 - 0.103)*(k - 1.75)/0.25
#else if k ≡ 2
β_xb2 = 0.111
#end if
#end if
'<p><b>Loads for accounting alternation in adjacent spans</b></p>
p_1 = γ_G*(g_sw + g_k) + γ_Q*q_k/2'kN/m²
p_2 = γ_Q*q_k/2'kN/m²
'<p><b>Bending moments</b></p>
'Span moment (including load alternation)
M_Ed_xb = β_xb1*p_1*l_x^2 + β_xb2*p_2*l_x^2'kN·m/m
#if k ≤ 2
M_Ed_yb = β_yb1*p_1*l_x^2 + β_yb2*p_2*l_x^2'kN·m/m
'Support moment
M_Ed_yt = β_yt*p*l_x^2'kN·m/m
#end if
#if k ≤ 2
'<p><b>Average moments at supports from the adjacent spans</b></p>
M_Ed_yt1 = (M_Ed_yt1_*l_y1 + M_Ed_yt*l_y)/(l_y1 + l_y)'kN·m/m
#end if
'<p><b>Shear forces</b></p>
#if k ≤ 2
V_Ed_x = β_vx*p*l_x'kN/m
V_Ed_y1 = β_vy1*p*l_x'kN/m
V_Ed_y2 = β_vy2*p*l_x'kN/m
#else
V_Ed_x = β_vx*p*l_x'kN/m
#end if
#if k ≤ 2
'<p><b>Edge support moments</b></p>
M_Ed_yte1 = M_Ed_yt1 - V_Ed_y1*min(b_y/200;h/200)'kN·m/m
#end if
'</div><div class="fold">
'<h4>Bending design</h4>
'<p><b>Bottom reinforcement parallel to <i>x</i></b></p>
'Effective cross section depth -'d_xb = h - c - d_bxb/20'cm
'Relative design bending moment -'m_Ed = 10*M_Ed_xb/(d_xb^2*η*f_cd)
'Compression zone depth -'x = d_xb/λ*(1 - sqr(1 - 2*m_Ed))'cm
'Relative compression zone depth -'ξ = x/d_xb
'Lever arm of internal forces -'z = d_xb - 0.5*λ*x'cm
'Area of required reinforcement -'A_sxb = M_Ed_xb*10^3/(z*f_yd)'cm²/m
'Reinforcement ratio -'ρ_xb = A_sxb/(d_xb*100)
#if ξ > ξ_lim
'<p class="err">'ξ'>'ξ_lim'- Reinforcement in compression zone is required.</p>
#end if
#hide
#if k ≤ 2
'<p><b>Bottom reinforcement parallel to <i>y</i></b></p>
'Effective cross section depth -'d_yb = h - c - d_bxb/10 - d_byb/20'cm
'Relative design bending moment -'m_Ed = 10*M_Ed_yb/(d_yb^2*η*f_cd)
'Compression zone depth -'x = d_yb/λ*(1 - sqr(1 - 2*m_Ed))'cm
'Relative compression zone depth -'ξ = x/d_yb
'Lever arm of internal forces -'z = d_yb - 0.5*λ*x'cm
'Area of required reinforcement -'A_syb = M_Ed_yb*10^3/(z*f_yd)'cm²/m
'Reinforcement ratio -'ρ_yb = A_syb/(d_yb*100)
#if ξ > ξ_lim
'<p class="err">'ξ'>'ξ_lim'- Reinforcement in compression zone is required.</p>
#end if
'<p><b>Горна армировка Along <i>y</i> за <i>M</i><sub>Ed_yte1</sub></b></p>
'Effective cross section depth -'d_yt1 = h - c - d_byt/10'cm
'Relative design bending moment -'m_Ed = 10*M_Ed_yte1/(d_yt1^2*η*f_cd)
'Compression zone depth -'x = d_yt1/λ*(1 - sqr(1 - 2*m_Ed))'cm
'Relative compression zone depth -'ξ = x/d_yt1
'Lever arm of internal forces -'z = d_yt1 - 0.5*λ*x'cm
'Area of required reinforcement -'A_syt1 = M_Ed_yte1*10^3/(z*f_yd)'cm²/m
'Reinforcement ratio -'ρ_yt1 = A_syt1/(d_yt1*100)
#if ξ > ξ_lim
'<p class="err">'ξ'>'ξ_lim'- Reinforcement in compression zone is required.</p>
#end if
#end if
#post
'<p><b>The rest of the reinforcement is calculated similarly.</b></p>
'</div><div class="fold">
'<h4>Reinforcement detailing</h4>
'<p><b>Bottom reinforcement</b></p>
'<p class="ref">[EN 1992-1-1, § 9.2.1.1 (1)]</p>
'Minimum reinforcement ratio
ρ_min = max(0.26*f_ctm/f_yk;0.0013)
'<p class="ref">[EN 1992-1-1, § 9.2.1.1 (3)]</p>
'Maximum reinforcement ratio -'ρ_max = 0.04
#if ρ_xb < ρ_min
'<p><i>ρ</i><sub>xb</sub> < <i>ρ</i><sub>min</sub>. Minimum reinforcement is assumed -'A_sxb = ρ_min*d_xb*100'cm<sup>2</sup>/m</p>
#else if ρ_xb > ρ_max
'<p class="err">Reinforcement ratio is greater than maximum:'ρ_xb'>'ρ_max'</p>
#end if
#if k > 2
'<p class="ref">[EN 1992-1-1, § 9.3.1.1 (2)]</p>
'Minimum bottom reinforcement parallel to y
A_syb = 0.2*A_sxb'cm²/m
#else
'<!--'
#if ρ_yb < ρ_min
'<p><i>ρ</i><sub>yb</sub> < <i>ρ</i><sub>min</sub>. Minimum reinforcement is assumed -'A_syb = ρ_min*d_yb*100'cm²/m</p>
#else if ρ_yb > ρ_max
'<p class="err">Reinforcement ratio is greater than maximum:'ρ_yb'>'ρ_max'</p>
#end if
'<-->
#end if
'Required bar spacing
s_xb = floor(π*d_bxb^2/(4*A_sxb))'cm
#if k > 2
'<p class="ref">[EN 1992-1-1, § 9.3.1.1 (3)]</p>
'Minimum bottom reinforcement parallel to y
s_yb = min(3*h;40)'cm
'Required bar diameter for bottom reinforcement parallel to y
d_byb = max(ceiling(sqr(4*A_syb*s_yb/π));6)'mm
#else
'<!--'s_yb = floor(π*d_byb^2/(4*A_syb))'cm-->
#end if
'<p class="ref">[EN 1992-1-1, § 9.3.1.1 (3)]</p>
'Maximum bar spacing
s_max = min(2*h;25)'cm
#if s_xb > s_max
'<p><i>s</i><sub>xb</sub> > <i>s</i><sub>max</sub>. Assume's_xb = s_max'cm</p>
#end if
#if (k ≤ 2)*(s_yb > s_max)
'<!--<p><i>s</i><sub>yb</sub> > <i>s</i><sub>max</sub>. Assume's_yb = s_max'cm</p>-->
#end if
#hide
'<p><b>Top reinforcement</b></p>
#if k ≤ 2
#if ρ_yt1 < ρ_min
'<p><i>ρ</i><sub>yt1</sub> < <i>ρ</i><sub>min</sub>. Minimum reinforcement is assumed -'A_syt1 = ρ_min*d_yt1*100'cm²/m</p>
#else if ρ_yt1 > ρ_max
'<p class="err">Reinforcement ratio is greater than maximum:'ρ_yt1'>'ρ_max'</p>
#end if
#end if
'Required bar spacing
#if k ≤ 2
s_yt1 = floor(π*d_byt^2/(4*A_syt1))'cm
#end if
'<p class="ref">[EN 1992-1-1, § 9.3.1.1 (3)]</p>
'Maximum bar spacing
s_max = min(2*h;25)'cm
#if (k ≤ 2)*(s_yt1 > s_max)
'<p><i>s</i><sub>yt1</sub> > <i>s</i><sub>max</sub>. It is assumed's_yt1 = s_max'cm</p>
#end if
#post
'<p><b>The rest of the reinforcement is detailed similarly.</b></p>
'</div>
'<h4>Results</h4>
#val
'<table class="bordered centered">
'<tr><th>Reinforcement</th><th>M<sub>Ed</sub><br>kN·m/m</th><th>A<sub>s</sub><br>cm<sup>2</sup>/m</th><th>ρ, %</th><th>Selected</tr>
'<tr><td>Bottom parallel to <i>x</i></td><td>'M_Ed_xb'</td><td>'A_sxb'</td><td>'A_sxb/d_xb'%</td><td>Ø'd_bxb'/'s_xb'</td></tr>
#if k ≤ 2
'<tr><td>Bottom parallel to <i>y</i></td><td>'M_Ed_yb'</td><td>'A_syb'</td><td>'A_syb/d_yb'%</td><td>Ø'd_byb'/'s_yb'</td></tr>
#else
'<tr><td>Bottom parallel to <i>y</i></td><td></td><td></td><td></td><td>Ø'd_byb'/'s_yb'</td></tr>
#end if
#if k ≤ 2
'<tr><td>Top parallel to <i>y</i> - down side</td><td>'M_Ed_yte1'</td><td>'A_syt1'</td><td>'A_syt1/d_yt1'%</td><td>Ø'd_byt'/'s_yt1'</td></tr>
#end if
'</table>
#end if
#show
'</div> 4 6 25 25 14 2 0 0 2.5 2 25 0.85 500 8 8 6 6
Clear spans (lx < ly)
lx_cl = 4 m, ly_cl = 6 m
Beam width along x - by = 25 cm
Beam width along y - bx = 25 cm
Slab thickness - h = 14 cm
Concrete cover - c = 2 cm
Design span lengths
lx = lx_cl + min(bx100; h100) = 4 + min(25100; 14100) = 4.14 m
ly = ly_cl + min(by100; h100) = 6 + min(25100; 14100) = 6.14 m
Span ratio - k = lylx = 6.144.14 = 1.48
Design spans lengths and edge moments in adjacent panels
|
Along y - |
ly1 = 0 m; |
MEd_yt1_ = 0 kN·m/m |
(in absence of contiguous span - l = 0)
Characteristic uniform loads
Self weight - gsw = 0.25 · h = 0.25 · 14 = 3.5 kN/m²
Permanent - gk = 2.5 kN/m²
Variable - qk = 2 kN/m²
Partial safety factor - γG = 1.35 ; γQ = 1.5
Design uniform loads
p = ( gsw + gk ) · γG + qk · γQ = ( 3.5 + 2.5 ) · 1.35 + 2 · 1.5 = 11.1 kN/m²
Beam loads
(as uniform loads)
|
Parallel to x - |
gk_x = ( gsw + gk ) · lx_cl4 = ( 3.5 + 2.5 ) · 44 = 6 kN/m |
|
qk_x = qk · lx_cl4 = 2 · 44 = 2 kN/m | |
|
Parallel to y - |
gk_y = ( gsw + gk ) · lx_cl · (ly_cl − lx_cl2)2 · ly_cl = ( 3.5 + 2.5 ) · 4 · (6 − 42)2 · 6 = 8 kN/m |
|
qk_y = qk · lx_cl · (ly_cl − lx_cl2)2 · ly_cl = 2 · 4 · (6 − 42)2 · 6 = 2.67 kN/m |
Concrete [EN 1992-1-1, EN 1992-1-1, Table 3.1]
Characteristic compressive cylinder strength - fck = 25 MPa
Partial safety factors for concrete - γc = 1.5 , αcc = 0.85
Design compressive cylinder strength - fcd = αcc · fckγc = 0.85 · 251.5 = 14.17 MPa
Factor defining the effective height of the compression zone - λ = 0.8
Factor defining the effective strength - η = 1
Mean value of axial tensile strength - fctm = 0.3 · fck23 = 0.3 · 2523 = 2.56 MPa
Steel
Characteristic yield strength - fyk = 500 MPa
Partial safety factor for steel - γs = 1.15
Design yield strength - fyd = fykγs = 5001.15 = 434.78 MPa
Limit relative compressive zone depth - ξlim = 0.45
Bar diameters
- Bottom parallel to x - dbxb = 8 mm, parallel to y - dbyb = 8 mm
- Top parallel to y - dbyt = 6 mm
Shear forces and bending moments coefficients
For the actual static scheme
- for shear forces
| Short span coefficients | Long span coefficients | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| k = | 1.0 | 1.1 | 1.2 | 1.3 | 1.4 | 1.5 | 1.75 | 2.0 | ||
| V | βvx = | 0.29 | 0.33 | 0.36 | 0.38 | 0.40 | 0.42 | 0.45 | 0.480 | βvy1 = 0.45 βvy2 = 0.3 |
βvx = 0.4 + ( 0.42 − 0.4 ) · ( k − 1.4 ) 0.1 = 0.4 + ( 0.42 − 0.4 ) · ( 1.48 − 1.4 ) 0.1 = 0.417
- for moments in mid-span
| Short span coefficients | Long span coefficients | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| k = | 1.0 | 1.1 | 1.2 | 1.3 | 1.4 | 1.5 | 1.75 | 2.0 | ||
| +M | βxb1 = | 0.042 | 0.054 | 0.063 | 0.071 | 0.078 | 0.084 | 0.096 | 0.105 | βyb1 = 0.044 βyt = 0.058 |
βxb1 = 0.078 + ( 0.084 − 0.078 ) · ( k − 1.4 ) 0.1 = 0.078 + ( 0.084 − 0.078 ) · ( 1.48 − 1.4 ) 0.1 = 0.083
For simply supported slab
- for moments in mid-span
| Short span coefficients | Long span coefficients | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| k = | 1.0 | 1.1 | 1.2 | 1.3 | 1.4 | 1.5 | 1.75 | 2.0 | |
| βxb2 = | 0.055 | 0.065 | 0.074 | 0.081 | 0.087 | 0.092 | 0.103 | 0.111 | βyb2 = 0.056 |
βxb2 = 0.087 + ( 0.092 − 0.087 ) · ( k − 1.4 ) 0.1 = 0.087 + ( 0.092 − 0.087 ) · ( 1.48 − 1.4 ) 0.1 = 0.0912
Loads for accounting alternation in adjacent spans
p1 = γG · ( gsw + gk ) + γQ · qk2 = 1.35 · ( 3.5 + 2.5 ) + 1.5 · 22 = 9.6 kN/m²
p2 = γQ · qk2 = 1.5 · 22 = 1.5 kN/m²
Bending moments
Span moment (including load alternation)
MEd_xb = βxb1 · p1 · lx2 + βxb2 · p2 · lx2 = 0.083 · 9.6 · 4.142 + 0.0912 · 1.5 · 4.142 = 16 kN·m/m
MEd_yb = βyb1 · p1 · lx2 + βyb2 · p2 · lx2 = 0.044 · 9.6 · 4.142 + 0.056 · 1.5 · 4.142 = 8.68 kN·m/m
Support moment
MEd_yt = βyt · p · lx2 = 0.058 · 11.1 · 4.142 = 11.03 kN·m/m
Average moments at supports from the adjacent spans
MEd_yt1 = MEd_yt1_ · ly1 + MEd_yt · lyly1 + ly = 0 · 0 + 11.03 · 6.140 + 6.14 = 11.03 kN·m/m
Shear forces
VEd_x = βvx · p · lx = 0.417 · 11.1 · 4.14 = 19.15 kN/m
VEd_y1 = βvy1 · p · lx = 0.45 · 11.1 · 4.14 = 20.68 kN/m
VEd_y2 = βvy2 · p · lx = 0.3 · 11.1 · 4.14 = 13.79 kN/m
Edge support moments
MEd_yte1 = MEd_yt1 − VEd_y1 · min(by200; h200) = 11.03 − 20.68 · min(25200; 14200) = 9.59 kN·m/m
Bottom reinforcement parallel to x
Effective cross section depth - dxb = h − c − dbxb20 = 14 − 2 − 820 = 11.6 cm
Relative design bending moment - mEd = 10 · MEd_xbdxb2 · η · fcd = 10 · 1611.62 · 1 · 14.17 = 0.0839
Compression zone depth - x = dxbλ · ( 1 − √ 1 − 2 · mEd ) = 11.60.8 · ( 1 − √ 1 − 2 · 0.0839 ) = 1.27 cm
Relative compression zone depth - ξ = xdxb = 1.2711.6 = 0.11
Lever arm of internal forces - z = dxb − 0.5 · λ · x = 11.6 − 0.5 · 0.8 · 1.27 = 11.09 cm
Area of required reinforcement - Asxb = MEd_xb · 103z · fyd = 16 · 10311.09 · 434.78 = 3.32 cm²/m
Reinforcement ratio - ρxb = Asxbdxb · 100 = 3.3211.6 · 100 = 0.00286
The rest of the reinforcement is calculated similarly.
Bottom reinforcement
[EN 1992-1-1, § 9.2.1.1 (1)]
Minimum reinforcement ratio
ρmin = max(0.26 · fctmfyk; 0.0013) = max(0.26 · 2.56500; 0.0013) = 0.00133
[EN 1992-1-1, § 9.2.1.1 (3)]
Maximum reinforcement ratio - ρmax = 0.04
Required bar spacing
sxb = floor(π · dbxb24 · Asxb) = floor(3.14 · 824 · 3.32) = 15 cm
[EN 1992-1-1, § 9.3.1.1 (3)]
Maximum bar spacing
smax = min ( 2 · h; 25 ) = min ( 2 · 14; 25 ) = 25 cm
The rest of the reinforcement is detailed similarly.
| Reinforcement | MEd kN·m/m | As cm2/m | ρ, % | Selected |
|---|---|---|---|---|
| Bottom parallel to x | 16 | 3.32 | 0.29% | Ø8/15 |
| Bottom parallel to y | 8.68 | 1.9 | 0.18% | Ø8/25 |
| Top parallel to y - down side | 9.59 | 1.99 | 0.17% | Ø6/14 |
Simply Supported Slab Panel¶
Two-way slab panel simply supported on all four edges, the limiting case of the family with the highest span moments and no support reinforcement requirement.
'<small>According to <strong>Eurocode</strong>: EN 1992-1-1</small>
'<div style="max-width:180mm;">
'<img alt="ss-span-cl.png" style="height:170pt;" class="side" src="../../Images/structures/rc/spans/ss-span-cl.png">
'<h4>Dimensions</h4>
'Clear spans (<i>l</i><sub>x</sub> < <i>l</i><sub>y</sub>)
l_x_cl = ?'m,'l_y_cl = ?'m
'Beam width along <i>x</i> -'b_y = ?'cm
'Beam width along <i>y</i> -'b_x = ?'cm
'Slab thickness -'h = ?'cm
'Concrete cover -'c = ?'cm
#post
'Design span lengths
l_x = l_x_cl + min(b_x/100;h/100)'m
l_y = l_y_cl + min(b_y/100;h/100)'m
'Span ratio -'k = l_y/l_x
#if k < 1
#show
'Span length <i>l</i><sub>y</sub> should be greater than <i>l</i><sub>x</sub>
#hide
#else
#show
'<img alt="ss-span.png" class="side" style="height:140pt;" src="../../Images/structures/rc/spans/ss-span.png">
'<h4>Span length</h4>
'Design spans lengths and edge moments in adjacent panels
'Self weight -'g_sw = 0.25*h'kN/m²
'Permanent-'g_k = ?'kN/m²
'Variable -'q_k = ?'kN/m²
'Partial safety factor -'γ_G = 1.35';'γ_Q = 1.5
#post
'Design uniform loads
p = (g_sw + g_k)*γ_G + q_k*γ_Q'kN/m²
'<div class="fold">
'<p><b>Beam loads</b></p>
'(as uniform loads)
#if k ≤ 2
'<table><tr><td rowspan="2">
'Parallel to <i>x</i> - <br /><br />
'</td><td>
g_k_x = (g_sw + g_k)*l_x_cl/4'kN/m
'</td></tr><tr><td>
q_k_x = q_k*l_x_cl/4'kN/m
'</td></tr><tr><td rowspan="2">
'Parallel to <i>y</i> - <br />
'</td><td>
g_k_y = (g_sw + g_k)*l_x_cl*(l_y_cl - l_x_cl/2)/(2*l_y_cl)'kN/m
'</td></tr><tr><td>
q_k_y = q_k*l_x_cl*(l_y_cl - l_x_cl/2)/(2*l_y_cl)'kN/m
'</td></tr></table>
#else
'Parallel to <i>y</i> -'q_y = p*l_x_cl/2'kN/m
#end if
'</div>
#show
'<h4>Material properties</h4>
'<p><b>Concrete</b> [EN 1992-1-1, Table 3.1]</p>
'Characteristic compressive cylinder strength -'f_ck = ?'MPa
'Partial safety factors for concrete -'γ_c = 1.5','α_cc = ?
#post
'Design compressive cylinder strength -'f_cd = α_cc*f_ck/γ_c'MPa
'Factor defining the effective height of the compression zone -'λ = 0.8'
'Factor defining the effective strength - 'η = 1.0
'Mean value of axial tensile strength -'f_ctm = 0.30*f_ck^(2/3)'MPa
#show
'<p><b>Steel</b></p>
'Characteristic yield strength -'f_yk = ?'MPa
#post
'Partial safety factor for steel - 'γ_s = 1.15
'Design yield strength - 'f_yd = f_yk/γ_s'MPa
'Limit relative compressive zone depth -'ξ_lim = 0.45
#show
'Bar diameters
'- Bottom parallel to <i>x</i> -'d_bxb = ?'mm
'- Top parallel to <i>y</i> -'d_byb = ?'mm
#post
'<div class="fold">
'<h4>Static calculations</h4>
'<p><b>Shear forces and bending moments coefficients</b></p>
#if l_y > 2*l_x
l_y/l_x'>'2'- One-way slab
'- for moments in mid-span-'β_x = 0.125
'- for shear forces -'β_vx = 0.5
#else if l_x ≡ l_y
l_y/l_x'- Two-way square slab
'- for moments in mid-span -'β_x = 0.055';'β_y = 0.055
'- for shear forces-'β_vx = 0.33';'β_vy = 0.33
#else
'- for shear forces
'<table class="bordered centered">
'<tr><th rowspan="2"></th><th style="width:100mm;" colspan="9">Short span coefficients</th><th style="width:30mm;" rowspan="2">Long span coefficients</th></tr>
'<tr><th><i>k</i> =</th><th>1.0</th><th>1.1</th><th>1.2</th><th>1.3</th><th>1.4</th><th>1.5</th><th>1.75</th><th>2.0</th></tr>
'<tr><th>V</th><td><i>β</i><sub>vx</sub> = </td><td>0.33</td><td>0.36</td><td>0.39</td><td>0.41</td><td>0.43</td><td>0.45</td><td>0.48</td><td>0.50</td><td>'β_vy = 0.33'</td></tr></table>
#if k < 1.1
β_vx = 0.33 + (0.36 - 0.33)*(k - 1)/0.1
#else if k < 1.2
β_vx = 0.36 + (0.39 - 0.36)*(k - 1.1)/0.1
#else if k < 1.3
β_vx = 0.39 + (0.41 - 0.39)*(k - 1.2)/0.1
#else if k < 1.4
β_vx = 0.41 + (0.43 - 0.41)*(k - 1.3)/0.1
#else if k < 1.5
β_vx = 0.43 + (0.45 - 0.43)*(k - 1.4)/0.1
#else if k < 1.75
β_vx = 0.45 + (0.48 - 0.45)*(k - 1.5)/0.25
#else if k < 2
β_vx = 0.48 + (0.50 - 0.48)*(k - 1.75)/0.25
#else if k ≡ 2
β_vx = 0.50
#end if
'- for moments in mid-span
'<table class="bordered centered">
'<tr><th style="width:100mm;" colspan="9">Short span coefficients</th><th style="width:30mm;" rowspan="2">Long span coefficients</th></tr>
'<tr><th><i>k</i> =</th><th>1.0</th><th>1.1</th><th>1.2</th><th>1.3</th><th>1.4</th><th>1.5</th><th>1.75</th><th>2.0</th></tr>
'<tr><td><i>β</i><sub>x</sub> = </td><td>0.055</td><td>0.065</td><td>0.074</td><td>0.081</td><td>0.087</td><td>0.092</td><td>0.103</td><td>0.111</td><td>'β_y = 0.056'</td></tr></table>
#if k < 1.1
β_x = 0.055 + (0.065 - 0.055)*(k - 1)/0.1
#else if k < 1.2
β_x = 0.065 + (0.074 - 0.065)*(k - 1.1)/0.1
#else if k < 1.3
β_x = 0.074 + (0.081 - 0.074)*(k - 1.2)/0.1
#else if k < 1.4
β_x = 0.081 + (0.087 - 0.081)*(k - 1.3)/0.1
#else if k < 1.5
β_x = 0.087 + (0.092 - 0.087)*(k - 1.4)/0.1
#else if k < 1.75
β_x = 0.092 + (0.103 - 0.092)*(k - 1.5)/0.25
#else if k < 2
β_x = 0.103 + (0.111 - 0.103)*(k - 1.75)/0.25
#else if k ≡ 2
β_x = 0.111
#end if
#end if
'<p><b>Bending moments</b></p>
M_Ed_xb = β_x*p*l_x^2'kN·m/m
#if k ≤ 2
M_Ed_yb = β_y*p*l_x^2'kN·m/m
#end if
'<p><b>Shear forces</b></p>
#if k ≤ 2
V_Ed_x = β_vx*p*l_x'kN/m
V_Ed_y = β_vy*p*l_x'kN/m
#else
V_Ed_x = β_vx*p*l_x'kN/m
#end if
'</div><div class="fold">
'<h4>Bending design</h4>
'<p><b>Bottom reinforcement parallel to <i>x</i></b></p>
'Effective cross section depth -'d_xb = h - c - d_bxb/20'cm
'Relative design bending moment -'m_Ed = 10*M_Ed_xb/(d_xb^2*η*f_cd)
'Compression zone depth -'x = d_xb/λ*(1 - sqr(1 - 2*m_Ed))'cm
'Relative compression zone depth -'ξ = x/d_xb
'Lever arm of internal forces -'z = d_xb - 0.5*λ*x'cm
'Area of required reinforcement -'A_sxb = M_Ed_xb*10^3/(z*f_yd)'cm²/m
'Reinforcement ratio -'ρ_xb = A_sxb/(d_xb*100)
#if ξ > ξ_lim
'<p class="err">'ξ'>'ξ_lim'- Reinforcement in compression zone is required.</p>
#end if
#if k ≤ 2
#hide
'<p><b>Bottom reinforcement parallel to <i>y</i></b></p>
'Effective cross section depth -'d_yb = h - c - d_bxb/10 - d_byb/20'cm
'Relative design bending moment-'m_Ed = 10*M_Ed_yb/(d_yb^2*η*f_cd)
'Compression zone depth -'x = d_yb/λ*(1 - sqr(1 - 2*m_Ed))'cm
'Relative compression zone depth -'ξ = x/d_yb
'Lever arm of internal forces -'z = d_yb - 0.5*λ*x'cm
'Area of required reinforcement -'A_syb = M_Ed_yb*10^3/(z*f_yd)'cm²/m
'Reinforcement ratio -'ρ_yb = A_syb/(d_yb*100)
#if ξ > ξ_lim
'<p class="err">'ξ'>'ξ_lim'- Reinforcement in compression zone is required. </p>
#end if
#post
'<p><b>The rest of the reinforcement is calculated similarly.</b></p>
#end if
'</div><div class="fold">
'<h4>Reinforcement detailing</h4>
'<p><b>Bottom reinforcement</b></p>
'<p class="ref">[EN 1992-1-1, §. 9.2.1.1 (1)]</p>
'Minimum reinforcement ratio
ρ_min = max(0.26*f_ctm/f_yk;0.0013)
'<p class="ref">[EN 1992-1-1, §. 9.2.1.1 (3)]</p>
'Maximum reinforcement ratio -'ρ_max = 0.04
#if ρ_xb < ρ_min
'<p><i>ρ</i><sub>xb</sub> < <i>ρ</i><sub>min</sub>. Minimum reinforcement is assumed -'A_sxb = ρ_min*d_xb*100'cm<sup>2</sup>/m</p>
#else if ρ_xb > ρ_max
'<p class="err">Reinforcement ratio is greater than maximum:'ρ_xb'>'ρ_max'</p>
#end if
#if k > 2
'<p class="ref">[ EN 1992-1-1, §. 9.3.1.1 (2)]</p>
'Minimum bottom reinforcement parallel to y
A_syb = 0.2*A_sxb'cm²/m
#else
#hide
#if ρ_yb < ρ_min
'<p><i>ρ</i><sub>yb</sub> < <i>ρ</i><sub>min</sub>. Minimum reinforcement is assumed -'A_syb = ρ_min*d_yb*100'cm<sup>2</sup>/m</p>
#else if ρ_yb > ρ_max
'<p class="err">Reinforcement ratio is greater than maximum:'ρ_yb'>'ρ_max'</p>
#end if
#post
#end if
'Required bar spacing
s_xb = floor(π*d_bxb^2/(4*A_sxb))'cm
#if k > 2
'<p class="ref">[EN 1992-1-1, §. 9.3.1.1 (3)]</p>
'Minimum bottom reinforcement parallel to y
s_yb = min(3*h;40)'cm
'Required bar diameter for bottom reinforcement parallel to y
d_byb = max(ceiling(sqr(4*A_syb*s_yb/π));6)'mm
#else
#hide
s_yb = floor(π*d_byb^2/(4*A_syb))'cm
#post
#end if
'<p class="ref">[EN 1992-1-1, §. 9.3.1.1 (3)]</p>
'Maximum bar spacing
s_max = min(2*h;25)'cm
#if s_xb > s_max
'<p><i>s</i><sub>xb</sub> > <i>s</i><sub>max</sub>. Assume's_xb = s_max'cm</p>
#end if
#if k ≤ 2
#hide
#if s_yb > s_max
'<p><i>s</i><sub>yb</sub> > <i>s</i><sub>max</sub>. Assume's_yb = s_max'cm</p>
#end if
#post
'<p><b>The rest of the reinforcement is detailed similarly.</b></p>
#end if
'</div>
'<h4>Results</h4>
#val
'<table class="bordered centered">
'<tr><th>Reinforcement</th><th>M<sub>Ed</sub><br>kN·m/m</th><th>A<sub>s</sub><br>cm<sup>2</sup>/m</th><th>ρ, %</th><th>Selected</tr>
'<tr><td>Bottom parallel to <i>x</i></td><td>'M_Ed_xb'</td><td>'A_sxb'</td><td>'A_sxb/d_xb'%</td><td>Ø'd_bxb'/'s_xb'</td></tr>
#if k ≤ 2
'<tr><td>Bottom parallel to <i>y</i></td><td>'M_Ed_yb'</td><td>'A_syb'</td><td>'A_syb/d_yb'%</td><td>Ø'd_byb'/'s_yb'</td></tr>
#else
'<tr><td>Top parallel to <i>y</i></td><td></td><td></td><td></td><td>Ø'd_byb'/'s_yb'</td></tr>
#end if
'</table>
#end if
#show
'</div> 4 6 25 25 14 2 2.5 2 25 0.85 500 8 6
Clear spans (lx < ly)
lx_cl = 4 m, ly_cl = 6 m
Beam width along x - by = 25 cm
Beam width along y - bx = 25 cm
Slab thickness - h = 14 cm
Concrete cover - c = 2 cm
Design span lengths
lx = lx_cl + min(bx100; h100) = 4 + min(25100; 14100) = 4.14 m
ly = ly_cl + min(by100; h100) = 6 + min(25100; 14100) = 6.14 m
Span ratio - k = lylx = 6.144.14 = 1.48
Design spans lengths and edge moments in adjacent panels
Self weight - gsw = 0.25 · h = 0.25 · 14 = 3.5 kN/m²
Permanent- gk = 2.5 kN/m²
Variable - qk = 2 kN/m²
Partial safety factor - γG = 1.35 ; γQ = 1.5
Design uniform loads
p = ( gsw + gk ) · γG + qk · γQ = ( 3.5 + 2.5 ) · 1.35 + 2 · 1.5 = 11.1 kN/m²
Beam loads
(as uniform loads)
|
Parallel to x - |
gk_x = ( gsw + gk ) · lx_cl4 = ( 3.5 + 2.5 ) · 44 = 6 kN/m |
|
qk_x = qk · lx_cl4 = 2 · 44 = 2 kN/m | |
|
Parallel to y - |
gk_y = ( gsw + gk ) · lx_cl · (ly_cl − lx_cl2)2 · ly_cl = ( 3.5 + 2.5 ) · 4 · (6 − 42)2 · 6 = 8 kN/m |
|
qk_y = qk · lx_cl · (ly_cl − lx_cl2)2 · ly_cl = 2 · 4 · (6 − 42)2 · 6 = 2.67 kN/m |
Concrete [EN 1992-1-1, Table 3.1]
Characteristic compressive cylinder strength - fck = 25 MPa
Partial safety factors for concrete - γc = 1.5 , αcc = 0.85
Design compressive cylinder strength - fcd = αcc · fckγc = 0.85 · 251.5 = 14.17 MPa
Factor defining the effective height of the compression zone - λ = 0.8
Factor defining the effective strength - η = 1
Mean value of axial tensile strength - fctm = 0.3 · fck23 = 0.3 · 2523 = 2.56 MPa
Steel
Characteristic yield strength - fyk = 500 MPa
Partial safety factor for steel - γs = 1.15
Design yield strength - fyd = fykγs = 5001.15 = 434.78 MPa
Limit relative compressive zone depth - ξlim = 0.45
Bar diameters
- Bottom parallel to x - dbxb = 8 mm
- Top parallel to y - dbyb = 6 mm
Shear forces and bending moments coefficients
- for shear forces
| Short span coefficients | Long span coefficients | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| k = | 1.0 | 1.1 | 1.2 | 1.3 | 1.4 | 1.5 | 1.75 | 2.0 | ||
| V | βvx = | 0.33 | 0.36 | 0.39 | 0.41 | 0.43 | 0.45 | 0.48 | 0.50 | βvy = 0.33 |
βvx = 0.43 + ( 0.45 − 0.43 ) · ( k − 1.4 ) 0.1 = 0.43 + ( 0.45 − 0.43 ) · ( 1.48 − 1.4 ) 0.1 = 0.447
- for moments in mid-span
| Short span coefficients | Long span coefficients | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| k = | 1.0 | 1.1 | 1.2 | 1.3 | 1.4 | 1.5 | 1.75 | 2.0 | |
| βx = | 0.055 | 0.065 | 0.074 | 0.081 | 0.087 | 0.092 | 0.103 | 0.111 | βy = 0.056 |
βx = 0.087 + ( 0.092 − 0.087 ) · ( k − 1.4 ) 0.1 = 0.087 + ( 0.092 − 0.087 ) · ( 1.48 − 1.4 ) 0.1 = 0.0912
Bending moments
MEd_xb = βx · p · lx2 = 0.0912 · 11.1 · 4.142 = 17.34 kN·m/m
MEd_yb = βy · p · lx2 = 0.056 · 11.1 · 4.142 = 10.65 kN·m/m
Shear forces
VEd_x = βvx · p · lx = 0.447 · 11.1 · 4.14 = 20.52 kN/m
VEd_y = βvy · p · lx = 0.33 · 11.1 · 4.14 = 15.16 kN/m
Bottom reinforcement parallel to x
Effective cross section depth - dxb = h − c − dbxb20 = 14 − 2 − 820 = 11.6 cm
Relative design bending moment - mEd = 10 · MEd_xbdxb2 · η · fcd = 10 · 17.3411.62 · 1 · 14.17 = 0.091
Compression zone depth - x = dxbλ · ( 1 − √ 1 − 2 · mEd ) = 11.60.8 · ( 1 − √ 1 − 2 · 0.091 ) = 1.39 cm
Relative compression zone depth - ξ = xdxb = 1.3911.6 = 0.119
Lever arm of internal forces - z = dxb − 0.5 · λ · x = 11.6 − 0.5 · 0.8 · 1.39 = 11.05 cm
Area of required reinforcement - Asxb = MEd_xb · 103z · fyd = 17.34 · 10311.05 · 434.78 = 3.61 cm²/m
Reinforcement ratio - ρxb = Asxbdxb · 100 = 3.6111.6 · 100 = 0.00311
The rest of the reinforcement is calculated similarly.
Bottom reinforcement
[EN 1992-1-1, §. 9.2.1.1 (1)]
Minimum reinforcement ratio
ρmin = max(0.26 · fctmfyk; 0.0013) = max(0.26 · 2.56500; 0.0013) = 0.00133
[EN 1992-1-1, §. 9.2.1.1 (3)]
Maximum reinforcement ratio - ρmax = 0.04
Required bar spacing
sxb = floor(π · dbxb24 · Asxb) = floor(3.14 · 824 · 3.61) = 13 cm
[EN 1992-1-1, §. 9.3.1.1 (3)]
Maximum bar spacing
smax = min ( 2 · h; 25 ) = min ( 2 · 14; 25 ) = 25 cm
The rest of the reinforcement is detailed similarly.
| Reinforcement | MEd kN·m/m | As cm2/m | ρ, % | Selected |
|---|---|---|---|---|
| Bottom parallel to x | 17.34 | 3.61 | 0.31% | Ø8/13 |
| Bottom parallel to y | 10.65 | 2.32 | 0.21% | Ø6/12 |
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