Reinforced Concrete¶

CalcpadCE worksheets in this section handle the verification and design of reinforced concrete members at ultimate and serviceability limit states according to Eurocode EN 1992-1-1 — bending, shear, torsion, axial force and their combinations on rectangular and T-cross-sections.

Bending of beams is treated separately for rectangular and T-sections, with a dedicated worksheet for the bending capacity of a T-section given the reinforcement layout. Shear of rectangular sections covers the variable-strut-inclination model and the maximum strut compression check, and the combined bending, shear and torsion sheet adds the space-truss model for the torsional resistance.

Columns are verified for uniaxial and biaxial bending with axial force, with a parallel unit-aware variant and a full N-M interaction diagram generated by sweeping the neutral-axis depth.

Serviceability is addressed by stress, crack-width and deflection checks for rectangular and T-cross-sections, with the static scheme picked from a drop-down list. A small reinforcement-area picker returns the total area \(A_s\) for a chosen bar diameter and count.

Areas of Reinforcement Bars¶

Lookup table that returns the total reinforcement area \(A_s\) for a chosen bar diameter and bar count, with the cells highlighting the picked combination on hover.

Code:

'<style>td:hover {color:red;} table {cursor:default;}</style>
'Required reinforcement area -'A_s,req = ? {7}'cm²
#pre
'<img style="height:104pt; width:122pt;" src="../../Images/structures/rc/design/bars.png" alt="bars.png">
#post
#val
'<table class="bordered">
'<tr><th>Count</th><!--'i = 0''j = 0'-->
#Repeat 10
    '<th id="c'j = j + 1'">'i = i + 1'</th>
    #if i ≡ 6
        '<th id="c'j = j + 1'">'20/3'</th>
    #else if i ≡ 8
        '<th id="c'j = j + 1'">'25/3'</th>
    #end if
#Loop
'</tr><!--'d = 6''j = 0'-->
#Repeat 11
    '<!--'i = 0''j = j + 1'--><tr><th id="r'j'">Ø'd = if(d ≡ 28; 32; if(d ≥ 22; d + 3; d + 2))'</th>
    #Repeat 10
        #hide
        i = i + 1
        A_s = π*d*d/400*i
        #post
        #if A_s > A_s,req
            '<td style="background:#EAFFCA" align="right">'A_s'</td>
        #else
            '<td align="right">'A_s'</td>
        #end if
        #if i ≡ 6
            #hide
            A_s = π*d*d/400*20/3
            #post
            #if A_s > A_s,req
                '<td style="background:#EAFFCA" align="right">'A_s'</td>
            #else
                '<td align="right">'A_s'</td>
            #end if
        #else if i ≡ 8
            #hide
            A_s = π*d*d/400*25/3
            #post
            #if A_s > A_s,req
                '<td style="background:#EAFFCA" align="right">'A_s'</td>
            #else
                '<td align="right">'A_s'</td>
            #end if
        #end if
    #Loop
#Loop
'</table>
'<script>var col=0;var row=0;$("td").hover(function(){$("#r"+row+",#c"+col).css("color","black").css("background-color","#F0F0F0");col=$(this).parent().children().index($(this));row=$(this).parent().parent().children().index($(this).parent());$("#r"+row+",#c"+col).css("color","red").css("background-color","#FFF0A0");;});</script>

Rendered Output:

Required reinforcement area - A_s,req = 7 cm²

Count	1	2	3	4	5	6	6.67	7	8	8.33	9	10
Ø8	0.5	1.01	1.51	2.01	2.51	3.02	3.35	3.52	4.02	4.19	4.52	5.03
Ø10	0.79	1.57	2.36	3.14	3.93	4.71	5.24	5.5	6.28	6.54	7.07	7.85
Ø12	1.13	2.26	3.39	4.52	5.65	6.79	7.54	7.92	9.05	9.42	10.18	11.31
Ø14	1.54	3.08	4.62	6.16	7.7	9.24	10.26	10.78	12.32	12.83	13.85	15.39
Ø16	2.01	4.02	6.03	8.04	10.05	12.06	13.4	14.07	16.08	16.76	18.1	20.11
Ø18	2.54	5.09	7.63	10.18	12.72	15.27	16.96	17.81	20.36	21.21	22.9	25.45
Ø20	3.14	6.28	9.42	12.57	15.71	18.85	20.94	21.99	25.13	26.18	28.27	31.42
Ø22	3.8	7.6	11.4	15.21	19.01	22.81	25.34	26.61	30.41	31.68	34.21	38.01
Ø25	4.91	9.82	14.73	19.63	24.54	29.45	32.72	34.36	39.27	40.91	44.18	49.09
Ø28	6.16	12.32	18.47	24.63	30.79	36.95	41.05	43.1	49.26	51.31	55.42	61.58
Ø32	8.04	16.08	24.13	32.17	40.21	48.25	53.62	56.3	64.34	67.02	72.38	80.42

Bending Capacity of Tee Section¶

Bending capacity \(M_{Rd}\) of a T-cross-section with given reinforcement: rectangular stress block, neutral-axis depth from horizontal equilibrium and lever arm to the resultant of the compressed concrete.

Code:

'<small>According to <strong>Eurocode</strong>: EN 1992-1-1</small>
'<div style="max-width:180mm">
'<img style="width:400pt;" src="../../Images/structures/rc/design/tbeam-bending.png" alt="tbeam-bending.png">
'Design bending moment -'M_Ed = ?'kN·m
'<h4>Cross section dimensions</h4>
'Web -'b_w = ?'mm,'h_w = ?'mm
'Flange -'b_f = ?'mm,'h_f = ?'mm
#post
'Flange area -'A_f = (b_f - b_w)*h_f'mm²
'Section area -'A_c = b_w*h_w + A_f'mm²
#show
'<p><b>Concrete cover to the center of reinforcement</b></p>
'Bottom -'d_1 = ?'mm,&nbsp;&nbsp;&nbsp;&nbsp;Top -'d_2 = ?'mm
#post
'Effective cross section depth -'d = h_w - d_1'mm
#show
'<p><b>Reinforcement</b></p>
'Bottom -'A_s1 = ?'mm²,&nbsp;&nbsp;&nbsp;&nbsp;Top -'A_s2 = ?'mm²
#post
'Reinforcement ratios
'- bottom reinforcement -'ρ_1 = (A_s1)/(b_w*d)
'- top reinforcement -'ρ_2 = (A_s2)/(b_w*d)
#show
'<h4>Material properties</h4>
'<p class="ref" style="float:right">[EN 1992-1-1, Table 3.1]</p>
'<p><b>Concrete</b></p>
'Characteristic compressive cylinder strength -'f_ck = ?'MPa
'Partial safety factor for concrete -'γ_c = 1.5','α_cc = ?
#post
'Design compressive cylinder strength -'f_cd = α_cc*f_ck/γ_c'MPa
'Ultimate compressive strain -'ε_cu2 = 0.0035
'Strain at the end of the parabolic part of the diagram -'ε_c2 = 0.002
'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
'Modulus of elasticity -'E_s = 200'GPa
'<p><b>Strain-stress diagrams</b>, MPa:</p>
'<!--'PlotWidth = 250','PlotHeight = 125'-->
'<table><tr><td>
#hide
n = 2
σ_c(ε) = f_cd*((1 - (1 - ε/ε_c2)^n)*(ε < ε_c2) + (ε ≥ ε_c2))
#post
$Plot{σ_c(ε/1000) @ ε = 0 : ε_cu2*1000}
'</td><td>
#hide
σ_s(ε) = max(-f_yd; min(ε*E_s*1000;f_yd))
#post
$Plot{σ_s(ε/1000) @ ε = -10 : 10}
'</td></tr></table>
'<h4>Design checks</h4>
'<p class="ref">[EN 1992-1-1, § 9.2.1.1]
'Minimum tensile reinforcement ratio
ρ_min = max(0.26*f_ctm/f_yk;0.0013)
#if ρ_1 < ρ_min
    '<p class="err">Tensile reinforcement ratio is lower than minimum:'ρ_1'&lt;'ρ_min'</p>'
#end if
'Maximum tensile reinforcement ratio -'ρ_max = 0.04
#if ρ_1 > ρ_max
    '<p class="err">Tensile reinforcement ratio is greater than maximum:'ρ_1'>'ρ_max'</p>'
#end if
'Internal section forces as a function of the compressive zone depth <i>x</i>.
'Reinforcement strain:
'- in bottom reinforcement -'ε_s1(x) = ε_cu2*(d - x)/x
'- in top reinforcement -'ε_s2(x) = ε_cu2*(d_2 - x)/x
'- at flange bottom edge -'ε_cf(x) = ε_cu2*(x - h_f)/x
'Factors for integration of concrete stress diagram
k = f_cd*ε_cu2
k_1(ε_cu) = $Area{σ_c(ε)/k @ ε = 0 : ε_cu}
k_2(ε_cu) = $Area{σ_c(ε)*ε/(ε_cu2*k) @ ε = 0 : ε_cu}
k_1 = k_1(ε_cu2)
k_2 = k_2(ε_cu2)
'Internal section forces:
'- in concrete -'N_c(x) = -(k_1*x*b_f*(x < h_f) + (k_1*x*b_f - k_1(ε_cf(x))*(x - h_f)*(b_f - b_w))*(x ≥ h_f))*f_cd*10^-3
'- in  bottom reinforcement -'N_s1(x) = σ_s(ε_s1(x))*A_s1*10^-3
'- in  top reinforcement -'N_s2(x) = σ_s(ε_s2(x))*A_s2*10^-3
'Section capacity for axial force
N_Rd(x) = N_c(x) + N_s1(x) + N_s2(x)
$Plot{N_Rd(x) @ x = 0.1 : h_w}
'Compression zone depth is determined from the equilibrium of axial forces
x = $Root{N_Rd(x) @ x = 0.1 : h_w}'mm
'Compression zone depth at reinforcement yield point
x_lim = d* ε_cu2/(ε_cu2 + f_yd/E_s*10^-3)'cm
#if x ≤ x_lim
    'Check:'x'≤'x_lim'- compressive reinforcement is not required.
#else
    'Check:'x'>'x_lim'- compressive reinforcement is required.
#end if
'Distance to the equivalent concrete stress force
'- from the neutral line
#if x < h_f
    z_0 = k_2/k_1*x'cm
#else
    z_0 = (k_2*x^2*b_f - k_2(ε_cf(x))*(x - h_f)^2*(b_f - b_w))/(k_1*x*b_f - k_1(ε_cf(x))*(x - h_f)*(b_f - b_w))'cm
#end if
'- from the bottom edge of the section
z_c = h_w - x + z_0'mm
'Bending moment capacity
M_Rd = (A_s1*σ_s(ε_s1(x))*(z_c - d_1) + A_s2*σ_s(ε_s2(x))*(z_c - h_w + d_2))*10^-6'kN·m
#if M_Ed > M_Rd
    'Design bending moment is greater than bending capacity
    M_Ed'>'M_Rd'kN·m
    '<p class="err">Design checks are NOT satisfied.</p>
#else
    'Design bending moment is lower than bending capacity:
    M_Ed'≤'M_Rd'kN·m
    'Design checks are satisfied.
#end if
#show
'</div>800  300 650 1200    120 50  50  3700    0   20  0.85    500

Bending Design of Rectangular Section¶

Required tension reinforcement \(A_s\) of a rectangular section under a design moment \(M_{Ed}\), with compression reinforcement added when the relative neutral-axis depth exceeds the ductility limit.

Code:

'<small>According to <strong>Eurocode</strong>: EN 1992-1-1</small>
'<div style = "max-width:180mm;">
'<img style="width:330pt;" src="../../Images/structures/rc/design/beam-bending.png" alt="beam-bending.png">
'Design bending moment -'M_Ed = ? {340}'kN·m
'<h4>Cross section dcimensions</h4>
'Width -'b = ? {300}'mm, Height -'h = ? {500}'mm
'Concrete cover to the center of reinforcement -'d_1 = ? {50}'mm
#post
'Effective cross section depth -'d = h - d_1'cm
#show
'<h4>Material properties</h4>
'<p class="ref" style="float:right">[EN 1992-1-1, Table 3.1]</p>
'<p><b>Concrete</b></p>
'Characteristic compressive cylinder strength -'f_ck = ? {20}'MPa
'Partial safety factor for concrete -'γ_c = 1.5','α_cc = ? {0.85}
#post
'Design compressive cylinder strength -'f_cd = α_cc*f_ck/γ_c'MPa
'Factor for effective compression zone depth -'λ = 0.8
'Effective compressive strength factor -'η = 1.0
'Ultimate compressive strain -'ε_cu3 = 0.0035
'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 = ? {500}'MPa
#post
'Partial safety factor for steel -'γ_s = 1.15
'Design yield strength -'f_yd = f_yk/γ_s'MPa
'Modulus of elasticity -'E_s = 200000'MPa
'<h4>Bending design</h4>
'Relative design bending moment -'m_Ed = M_Ed*10^6/(b*d^2*η*f_cd)
'Compression zone depth -'x = d/λ*(1 - sqr(1 - 2*m_Ed))'mm
'Relative compression zone depth -'ξ = x/d'
'Design reinforcement yield strain
ε_yd = f_yd/E_s
'Relative depth of compression zone at yielding of bottom reinforcement
ξ_yd = ε_cu3/(ε_cu3 + ε_yd)
#show
'Limit compression zone depth -'ξ_lim = ? {0.62}
'(enter <i>ξ</i><sub>yd</sub> for elastic or 0.45 for plastic analysis)
#post
#if ξ ≤ ξ_lim
    ξ'≤'ξ_lim'- Compressive reinforcement is NOT required.
    'Lever arm of internal forces -'z = d - 0.5*λ*x'mm
    'Area of required tensile reinforcement -'A_s1 = M_Ed*10^6/(z*f_yd)'mm²
    'Reinforcement ratio
    ρ_1 = A_s1/(b*d)
    ρ_2 = 0
#else
    ξ'&gt;'ξ_lim'- Compressive reinforcement is required.
    'Relative depth is assumed to be'ξ = ξ_lim'and compressive reinforcement is designed
    'Compression zone depth -'x = ξ*d'mm
    'Distance from the center of compressive reinforcement to the concrete surface
    d_2 = h - d'mm
    'Distance between tensile and compressive reinforcement -'z_s = d - d_2'mm
    'Resultant compression force in concrete
    N_c = b*λ*x*η*f_cd*10^-3'kN
    'Required tensile reinforcement area:
    A_s1 = (M_Ed*10^6 + N_c*10^3*(λ*x/2 - d_2))/(f_yd*z_s)'mm²
    'Compressive reinforcement strain
    ε_s2 = (x - d_2)/x*ε_cu3
    'Compressive reinforcement stress
    σ_s2 = min(ε_s2*E_s; f_yd)'MPa
    'Required compressive reinforcement area:
    A_s2 = (M_Ed*10^6 - N_c*10^3*(d - λ*x/2))/(σ_s2*z_s)'mm²
    'Reinforcement ratio:
    '- tensile reinforcement -'ρ_1 = A_s1/(b*d)
    '- compressive reinforcement -'ρ_2 = A_s2/(b*d)
#end if
'<p class="ref">[EN 1992-1-1, § 9.2.1.1]</p>
'Minimum tensile reinforcement ratio
ρ_min = max(0.26*f_ctm/f_yk; 0.0013)
'Minimum area of tensile reinforcement -'A_s_min = ρ_min*b*d'mm²
#if A_s1 < A_s_min
    '<p class="err">Required reinforcement is lower than minimum:'A_s1'mm² &lt;'A_s_min'mm²</p>'
#end if
'<p class="ref">[EN 1992-1-1, § 9.2.1.1]</p>
'Maximum tensile reinforcement ratio -'ρ_max = 0.04
#if ρ_1 > ρ_max
    '<p class="err">Tensile reinforcement ratio is greater than maximum:'ρ_1'>'ρ_max'</p>'
#end if
#if ρ_2 > ρ_max
    '<p class="err">Compressive reinforcement ratio is greater than maximum:'ρ_2'>'ρ_max'</p>'
#end if
#show
'</div>

Bending Design of Tee Section¶

Required tension reinforcement \(A_s\) of a T-section under a design moment \(M_{Ed}\), switching between rectangular flange-only behaviour and T-section behaviour according to the position of the neutral axis.

Code:

'<small>According to <strong>Eurocode</strong>: EN 1992-1-1</small>
'<div style="max-width:180mm;">
'<img style="width:400pt;" src="../../Images/structures/rc/design/tbeam-bending.png" alt="tbeam-bending.png">
'Design bending moment -'M_Ed = ?'kN·m
'<h4>Cross section dimensions</h4>
'Web -'b_w = ?'mm,'h_w = ?'mm
'Flange -'b_f = ?'mm,'h_f = ?'mm
'Concrete cover to the center of reinforcement -'d_1 = ?'mm
#post
'Effective cross section depth -'d = h_w - d_1'mm
'Flange area -'A_f = (b_f - b_w)*h_f'mm<sup>2</sup>
'Flange first moment of area -'S_f = A_f*(d - h_f/2)'mm<sup>3</sup>
#show
'<h4>Material properties</h4>
'<p class="ref" style="float:right">[EN 1992-1-1, Table 3.1]</p>
'<p><b>Concrete</b></p>
'Characteristic compressive cylinder strength -'f_ck = ?'MPa
'Partial safety factor for concrete -'γ_c = 1.5','α_cc = ?
#post
'Design compressive cylinder strength -'f_cd = α_cc*f_ck/γ_c'MPa
'Factor for effective compression zone depth -'λ = 0.8
'Effective compressive strength factor -'η = 1.0
'Ultimate compressive strain -'ε_cu3 = 0.0035
'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
'Modulus of elasticity -'E_s = 200000'MPa
'<h4>Bending design</h4>
'Check for the location of the neutral line
'Bending moment for neutral line at the bottom edge of the flange
M_f = b_f*h_f*η*f_cd*(d - h_f/2)*10^-6'kN·m
#if M_Ed > M_f
    'The neutral line is below the flange - design for T-section
    'Relative design bending moment -'m_Ed = M_Ed*10^6/(b_w*d^2*η*f_cd)
    'Compression zone depth -'x = d/λ*(1 - sqr(1 - 2*(m_Ed - S_f/(b_w*d^2))))'mm
    'Relative compression zone depth -'ξ = x/d'
    'Design yield strain of reinforcement -'ε_yd = f_yd/E_s
    'Relative depth of compression zone corresponding to design yield strain
    ξ_yd = ε_cu3/(ε_cu3 + ε_yd)
    #show
    'Limit compression zone depth -'ξ_lim = ?
    '(enter <i>ξ</i><sub>yd</sub> for elastic or 0.45 for plastic analysis)
    #post
    #if ξ ≤ ξ_lim
        ξ'≤'ξ_lim'- compressive reinforcement is NOT required.
        'Lever arm of internal forces -'z = d - 0.5*λ*x'mm
        'Required tensile reinforcement area -'A_s1 = M_Ed*10^6/(z*f_yd)'mm<sup>2</sup>
        'Reinforcement ratio -'ρ_1 = A_s1/(b_w*d)
        '<!--'ρ_2 = 0'-->
    #else
        ξ'>'ξ_lim'- compressive reinforcement is required.
        'Relative depth is assumed to be'ξ = ξ_lim'and compressive reinforcement is designed
        'Compression zone depth -'x = ξ*d'mm
        'Distance from the center of compressive reinforcement to the concrete surface
        d_2 = h_w - d'mm
        'Distance between tensile and compressive reinforcement -'z_s = d - d_2'mm
        'Required tensile reinforcement area
        A_s1 = (M_Ed*10^6 + (b_w*λ*x*(λ*x/2 - d_2) + A_f*(h_f/2 - d_2))*η*f_cd)/(f_yd*z_s)'mm<sup>2</sup>
        'Strain is compressive reinforcement
        ε_s2 = (x - d_2)/x*ε_cu3
        'Compressive reinforcement stress
        σ_s2 = min(ε_s2*E_s; f_yd)'MPa
        'Required compressive reinforcement area
        A_s2 = (A_s1*f_yd - (b_w*λ*x + A_f)*η*f_cd)/σ_s2'mm<sup>2</sup>
        'Reinforcement ratios
        '- tensile reinforcement -'ρ_1 = A_s1/(b_w*d)
        '- compressive reinforcement -'ρ_2 = A_s2/(b_w*d)
    #end if
#else
    'The neutral line is within the flange - design of rectangular section with'b_f'mm
    'Relative design bending moment -'m_Ed = M_Ed*10^6/(b_f*d^2*η*f_cd)
    'Compression zone depth -'x = d/λ*(1 - sqr(1 - 2*m_Ed))'mm
    'Lever arm of internal forces -'z = d - 0.5*λ*x'mm
    'Area of required tensile reinforcement -'A_s1 = M_Ed*10^6/(z*f_yd)'mm<sup>2</sup>
    'Reinforcement ratio
    ρ_1 = A_s1/(b_w*d)
    '<!--'ρ_2 = 0'-->
#end if
'<p class="ref">[EN 1992-1-1, § 9.2.1.1]
'Minimum tensile reinforcement ratio
ρ_min = max(0.26*f_ctm/f_yk; 0.0013)
'Minimum reinforcement -'A_s_min = ρ_min*b_w*d'mm<sup>2</sup>
#if A_s1 < A_s_min
    '<p class="err">The reinforcement is then the minimum:'A_s1'mm<sup>2</sup> &lt;'A_s_min'mm<sup>2</sup></p>'
#end if
'Maximum reinforcement ratio -'ρ_max = 0.04
#if '<!--'ρ_1 > ρ_max'-->'
    '<p class="err">Reinforcement ratio is greater than the maximum:'ρ_1'>'ρ_max'mm<sup>2</sup></p>'
#end if
#show
'</div>800  250 600 1200    120 50  20  0.85    500 0.62

Column Design for Bending and Axial Force¶

Verification of a rectangular reinforced concrete column under bending and axial force from the strain compatibility, with symmetric or asymmetric reinforcement and the strut compression check.

Code:

'<small>According to <strong>Eurocode</strong>: EN 1992-1-1</small>
'<div style="max-width:180mm">
'<img class="side" style="width:150pt;"src="../../Images/structures/rc/design/column-design.png" alt="column-design.png">
'<h4>Cross section dimensions</h4>
'Width -'b = ? {300}'mm, Height -'h = ? {700}'mm
#post
'Cross section area -'A_c = b*h'mm²
#show
'<p><b>Reinforcement area</b></p>
'Left -'A_s1 = ? {800}'mm², Right -'A_s2 = A_s1'mm²
#post
'Reinforcement ratio -'ρ = (A_s1 + A_s2)/A_c
#show
'<p><b>Concrete cover to the center of reinforcement</b></p>
'Left -'d_1 = ? {50}'mm, Right -'d_2 = d_1'mm
#post
'Effective cross section depth -'d = h - d_1'mm
#show
'<h4>Section design loads</h4>
'<table>
'<tr><td>№ 1 - </td><td>'M_Ed_1 = ? {100}'kN·m, </td><td>'N_Ed_1 = ? {500}'kN</td></tr>
'<tr><td>№ 2 - </td><td>'M_Ed_2 = ? {100}'kN·m, </td><td>'N_Ed_2 = ? {1000}'kN</td></tr>
'<tr><td>№ 3 - </td><td>'M_Ed_3 = ? {100}'kN·m, </td><td>'N_Ed_3 = ? {1500}'kN</td></tr>
'<tr><td>№ 4 - </td><td>'M_Ed_4 = ? {100}'kN·m, </td><td>'N_Ed_4 = ? {2000}'kN</td></tr>
'<tr><td>№ 5 - </td><td>'M_Ed_5 = ? {100}'kN·m, </td><td>'N_Ed_5 = ? {2500}'kN</td></tr>
'</table>
#pre
'All input values for M<sub>Ed</sub> and N<sub>Ed</sub> must be positive!
#show
'Positive axial force is compressive!
'<h4>Material properties</h4>
'<p class="ref" style="float:right">[EN 1992-1-1, Table 3.1]</p>
'<p><b>Concrete</b></p>
'Characteristic compressive cylinder strength -'f_ck = ? {20}'MPa
'Partial safety factor for concrete -'γ_c = 1.5','α_cc = ? {0.85}
#post
'Mean value of cylinder compressive strength -'f_cm = f_ck + 8'MPa
'Design compressive cylinder strength -'f_cd = α_cc*f_ck/γ_c'MPa
'Ultimate compressive strain -'ε_cu2 = 0.0035'('ε_c2 = 0.002','n = 2')'
'Secant modulus of elasticity -'E_cm = 22*(f_cm/10)^0.3'GPa
#show
'<p><b>Steel</b></p>
'Characteristic yield strength -'f_yk = ? {500}'MPa
#post
'Partial safety factor for steel -'γ_s = 1.15
'Design yield strength -'f_yd = f_yk/γ_s'MPa
'Modulus of elasticity -'E_s = 200'GPa
'<p><b>Strain-stress diagrams</b>, MPa:</p>
'<!--'PlotWidth = 250','PlotHeight = 125'-->
'<table><tr><td>
#hide
σ_c(ε) = f_cd*((1 - (1 - ε/ε_c2)^n)*(ε < ε_c2) + (ε ≥ ε_c2))
#post
$Plot{σ_c(ε/1000) @ ε = 0 : ε_cu2*1000}
'</td><td>
#hide
σ_s(ε) = max(-f_yd; min(ε*E_s*1000; f_yd))
#post
$Plot{σ_s(ε/1000) @ ε = -10 : 10}
'</td></tr></table>
#show
'<h4>Imperfections and second order effects</h4>
'Ratio of quasi-permanent bending moment: M<sub>0Eqp</sub>/M<sub>0Ed</sub> ='K_G = ? {0.75}
#post
'M<sub>0Eqp</sub> is the first order bending moment in quasi-permanent load combination (SLS)
'M<sub>0Ed</sub> is the first order bending moment in design load combination (ULS)
#show
'Long term creep factor - φ(∞,t<sub>0</sub>) ='φ = ? {2.5}
'Column height -'L = ? {4000}'mm
'Effective column  height -'L_o = ? {2}*L'mm
'Number of vertical elements -'m = ? {1}
#hide
α_n_ = 2/sqr(L/1000)
α_n = max(2/3; min(α_n_; 1))
α_m = sqr(0.5*(1 + 1/m))
θ_o = 1/200
θ_i = θ_o*α_n*α_m
e_i_ = θ_i*L_o/2
e_o = max(h/30; 20)
e_i = max(e_i_; e_o)
I_c = (b*h^3)/12
i = sqr(I_c/A_c)
λ = L_o/i
k_1 = sqr(f_ck/20)
φ_ef = φ*K_G
K_s = 1
γ_cE = 1.3
E_cd = E_cm/γ_cE
I_s = A_s1*(h/2 - d_1)^2 + A_s2*(h/2 - d_2)^2
n(N) = N/(A_c*f_cd/1000)
k_2_(N) = n(N)*λ/170
k_2(N) = min(k_2_(N); 0.2)
K_c(N) = k_1*k_2(N)/(1 + φ_ef)
EI_e(N) = (K_c(N)*E_cd*I_c + K_s*E_s*I_s)
N_B(N) = π^2*EI_e(N)/(L_o)^2
#if L > 0
    M_II_1 = (M_Ed_1 + N_Ed_1*e_i/1000)/(1 - N_Ed_1/N_B(N_Ed_1))
    M_II_2 = (M_Ed_2 + N_Ed_2*e_i/1000)/(1 - N_Ed_2/N_B(N_Ed_2))
    M_II_3 = (M_Ed_3 + N_Ed_3*e_i/1000)/(1 - N_Ed_3/N_B(N_Ed_3))
    M_II_4 = (M_Ed_4 + N_Ed_4*e_i/1000)/(1 - N_Ed_4/N_B(N_Ed_4))
    M_II_5 = (M_Ed_5 + N_Ed_5*e_i/1000)/(1 - N_Ed_5/N_B(N_Ed_5))
#else
    M_II_1 = M_Ed_1
    M_II_2 = M_Ed_2
    M_II_3 = M_Ed_3
    M_II_4 = M_Ed_4
    M_II_5 = M_Ed_5
#end if

A_s_min = 0.002*A_c
#if A_s1 < A_s_min
    A_s1 = A_s_min
#end if
#if A_s2 < A_s_min
    A_s2 = A_s_min
#end if

ε_s1,a(x) = ε_cu2*(d - x)/x
ε_s2,a(x) = ε_cu2*(d_2 - x)/x
N_c,a(x) = 17/21*x*b*f_cd/1000
N_s1,a(x) = σ_s(ε_s1,a(x))*A_s1/1000
N_s2,a(x) = σ_s(ε_s2,a(x))*A_s2/1000
N_Rd,a(x) = N_c,a(x) - N_s1,a(x) - N_s2,a(x)
z_c,a(x) = h/2 - 99/238*x
M_Rd,a(x) = (N_c,a(x)*z_c,a(x) + N_s1,a(x)*(h/2 - d_1) - N_s2,a(x)*(h/2 - d_2))/1000

x(ε) = h*ε_cu2/(ε_cu2 + ε)
h_c = h*ε_c2/ε_cu2
ε_s1,b(ε) = ε + (ε_c2 - ε)*d_1/h_c
ε_s2,b(ε) = ε + (ε_c2 - ε)*(h - d_2)/h_c
N_c,b(ε) = (h_c/6*(σ_c(ε) + 4*σ_c((ε + ε_c2)/2) + σ_c(ε_c2)) + (h - h_c)*f_cd)*b/1000
N_s1,b(ε) = σ_s(ε_s1,b(ε))*A_s1/1000
N_s2,b(ε) = σ_s(ε_s2,b(ε))*A_s2/1000
N_Rd,b(ε) = N_c,b(ε) + N_s1,b(ε) + N_s2,b(ε)
M_c0(ε) = (h_c^2/6*(2*σ_c((ε + ε_c2)/2) + σ_c(ε_c2)) + (h^2 - h_c^2)/2*f_cd)*b/1000
z_c,b(ε) = M_c0(ε)/N_c,b(ε) - h/2
M_Rd,b(ε) = (N_c,b(ε)*z_c,b(ε) - N_s1,b(ε)*(h/2 - d_1) + N_s2,b(ε)*(h/2 - d_2))/1000

N_Rd(ε) = N_Rd,a(x(ε))*(ε ≥ 0) + N_Rd,b(abs(ε))*(ε < 0)
M_Rd(ε) = M_Rd,a(x(ε))*(ε ≥ 0) + M_Rd,b(abs(ε))*(ε < 0)

ε_max = $Root{N_Rd(ε) - 1 @ ε = 0 : 2}
#if ε_max > 0.1
    ε_max = 0.1
#end if
N_Rd_max = A_c*f_cd/1000 + σ_s(ε_max)*(A_s1 + A_s2)/1000
#if N_Ed_1 > N_Rd_max
    M_Rd_1 = 0
#else
    ε_1 = $Root{N_Rd(ε) - N_Ed_1 @ ε = -ε_c2 : 2}
    M_Rd_1 = M_Rd(ε_1)
#end if
#if N_Ed_2 > N_Rd_max
    M_Rd_2 = 0
#else
    ε_2 = $Root{N_Rd(ε) - N_Ed_2 @ ε = -ε_c2 : 2}
    M_Rd_2 = M_Rd(ε_2)
#end if
#if N_Ed_3 > N_Rd_max
    M_Rd_3 = 0
#else
    ε_3 = $Root{N_Rd(ε) - N_Ed_3 @ ε = -ε_c2 : 2}
    M_Rd_3 = M_Rd(ε_3)
#end if
#if N_Ed_4 > N_Rd_max
    M_Rd_4 = 0
#else
    ε_4 = $Root{N_Rd(ε) - N_Ed_4 @ ε = -ε_c2 : 2}
    M_Rd_4 = M_Rd(ε_4)
#end if
#if N_Ed_5 > N_Rd_max
    M_Rd_5 = 0
#else
    ε_5 = $Root{N_Rd(ε) - N_Ed_5 @ ε = -ε_c2 : 2}
    M_Rd_5 = M_Rd(ε_5)
#end if

M_Rd_ei(ε) = N_Rd(ε)*e_i/1000
M_Rd_II(ε) = M_Rd_ei(ε)/(1 - N_Rd(ε)/N_B(N_Rd(ε)))/(N_Rd(ε) < 0.95*N_B(N_Rd(ε)))
ε_ei = $Root{M_Rd_ei(ε) - M_Rd(ε) @ ε = -ε_c2 : ε_max}
M_Rd_ei = M_Rd(ε_ei)
N_Rd_ei = N_Rd(ε_ei)
#if N_Rd_max < N_B(N_Rd_max)
    ε_NB = ε_ei
#else
    ε_NB = $Root{N_Rd(ε) - 0.95*N_B(N_Rd(ε)) @ ε = -ε_c2 : ε_max}
#end if
ε_II = ε_NB
ε_II = $Root{M_Rd_ei(ε)/(1 - N_Rd(ε)/N_B(N_Rd(ε))) - M_Rd(ε) @ ε = ε_NB : ε_max}
M_Rd_II = M_Rd(ε_II)
N_Rd_II = N_Rd(ε_II)
PlotWidth = 600
PlotHeight = 600
#post
'<h4>Interaction diagram</h4>
'<p><i>N</i><sub>Rd</sub></p>
$Plot{M_Rd(ε)|N_Rd(ε) & M_Rd_ei(ε)|N_Rd(ε) & M_Rd_II(ε)|N_Rd(ε) & M_Rd_ei|N_Rd_ei & M_Rd_II|N_Rd_II & M_II_1|N_Ed_1 & M_II_2|N_Ed_2 & M_II_3|N_Ed_3 & M_II_4|N_Ed_4 & M_II_5|N_Ed_5 @ ε = ε_max : -ε_c2}
'<i>M</i><sub>Rd</sub>
'<p><b>Legend:</b></p>
'<p><b style="color:Tomato">━━</b> Interaction diagram</p>
'<p><b style="color:YellowGreen">━━</b> Line of geometric imperfections</p>
'<p><b style="color:CornflowerBlue">━━</b> Buckling curve</p>
'<p><b style="color:Gold">●</b> Design resistance including geometric imperfections and accidental eccentricity - <i>N</i><sub>Rd_ei</sub></p>
'<p><b style="color:MediumVioletRed">●</b> Design buckling resistance - <i>N</i><sub>Rd_II</sub></p>
'<p><b style="color:MediumSpringGreen">●</b> Design load № 1, <b style="color:BlueViolet">●</b> Design load № 2, <b style="color:LightSalmon">●</b> Design load № 3,</p>
'<p><b style="color:DeepPink">●</b> Design load № 4, <b style="color:DarkTurquoise">●</b> Design load № 5</p>
'<h4>Design checks</h4>
#val
'<table class="bordered centered">
'<tr><th>№</th><th>N<sub>Ed</sub><br/>kN</th><th></th><th>N<sub>B</sub><br/>kN</th><th>M<sub>Ed</sub><br/>kN·m</th><th>M<sub>II</sub><br/>kN·m</th><th> </th><th>M<sub>Rd</sub><br/>kN·m</th></tr>
#if N_Ed_1 > N_B(N_Ed_1)
    '<tr class="err"><td>1</td><td>'N_Ed_1'</td><td>&gt;</td><td>'N_B(N_Ed_1)'</td><td></td><td></td><td></td><td></td></tr>
#else if M_Rd_1 > M_II_1
    '<tr><td>1</td><td>'N_Ed_1'</td><td>&lt;</td><td>'N_B(N_Ed_1)'</td><td>'M_Ed_1'</td><td>'M_II_1'</td><td>&le;</td><td>'M_Rd_1'</td></tr>
#else
    '<tr class="err"><td>1</td><td>'N_Ed_1'</td><td>&lt;</td><td>'N_B(N_Ed_1)'</td><td>'M_Ed_1'</td><td>'M_II_1'</td><td>&gt;</td><td>'M_Rd_1'</td></tr>
#end if
#if N_Ed_2 > N_B(N_Ed_2)
    '<tr class="err"><td>2</td><td>'N_Ed_2'</td><td>&gt;</td><td>'N_B(N_Ed_2)'</td><td></td><td></td><td></td><td></td></tr>
#else if M_Rd_2 > M_II_2
    '<tr><td>2</td><td>'N_Ed_2'</td><td>&lt;</td><td>'N_B(N_Ed_2)'</td><td>'M_Ed_2'</td><td>'M_II_2'</td><td>&le;</td><td>'M_Rd_2'</td></tr>
#else
    '<tr class="err"><td>2</td><td>'N_Ed_2'</td><td>&lt;</td><td>'N_B(N_Ed_2)'</td><td>'M_Ed_2'</td><td>'M_II_2'</td><td>&gt;</td><td>'M_Rd_2'</td></tr>
#end if
#if N_Ed_3 > N_B(N_Ed_3)
    '<tr class="err"><td>3</td><td>'N_Ed_3'</td><td>&gt;</td><td>'N_B(N_Ed_3)'</td><td></td><td></td><td></td><td></td></tr>
#else if M_Rd_3 > M_II_3
    '<tr><td>3</td><td>'N_Ed_3'</td><td>&lt;</td><td>'N_B(N_Ed_3)'</td><td>'M_Ed_3'</td><td>'M_II_3'</td><td>&le;</td><td>'M_Rd_3'</td></tr>
#else
    '<tr class="err"><td>3</td><td>'N_Ed_3'</td><td>&lt;</td><td>'N_B(N_Ed_3)'</td><td>'M_Ed_3'</td><td>'M_II_3'</td><td>&gt;</td><td>'M_Rd_3'</td></tr>
#end if
#if N_Ed_4 > N_B(N_Ed_4)
    '<tr class="err"><td>4</td><td>'N_Ed_4'</td><td>&gt;</td><td>'N_B(N_Ed_4)'</td><td></td><td></td><td></td><td></td></tr>
#else if M_Rd_4 > M_II_4
    '<tr><td>4</td><td>'N_Ed_4'</td><td>&lt;</td><td>'N_B(N_Ed_4)'</td><td>'M_Ed_4'</td><td>'M_II_4'</td><td>&le;</td><td>'M_Rd_4'</td></tr>
#else
    '<tr class="err"><td>4</td><td>'N_Ed_4'</td><td>&lt;</td><td>'N_B(N_Ed_4)'</td><td>'M_Ed_4'</td><td>'M_II_4'</td><td>&gt;</td><td>'M_Rd_4'</td></tr>
#end if
#if N_Ed_5 > N_B(N_Ed_5)
    '<tr class="err"><td>5</td><td>'N_Ed_5'</td><td>&gt;</td><td>'N_B(N_Ed_5)'</td><td></td><td></td><td></td><td></td></tr>
#else if M_Rd_5 > M_II_5
    '<tr><td>5</td><td>'N_Ed_5'</td><td>&lt;</td><td>'N_B(N_Ed_5)'</td><td>'M_Ed_5'</td><td>'M_II_5'</td><td>&le;</td><td>'M_Rd_5'</td></tr>
#else
    '<tr class="err"><td>5</td><td>'N_Ed_5'</td><td>&lt;</td><td>'N_B(N_Ed_5)'</td><td>'M_Ed_5'</td><td>'M_II_5'</td><td>&gt;</td><td>'M_Rd_5'</td></tr>
#end if
'</table>
#equ
'<p><b>Ultimate capacity for axial force:</b></p>
'- cross section only -'N_Rd_max'kN
'- column with geometric imperfections -'N_Rd_ei'kN
'- column with imperfections and II order effects -'N_Rd_II'kN
'Buckling factor -'φ = N_Rd_II/N_Rd_max
#show
'</div>

Column Design for Biaxial Bending and Axial Force¶

Verification of a rectangular reinforced concrete column under axial force and bending about both axes by superposition of the two uniaxial interaction surfaces with the EN 1992-1-1 power-law combination.

Code:

'<small>According to <strong>Eurocode</strong>: EN 1992-1-1</small>
'<div style="max-width:180mm">
'<img class="side" style="width:150pt;" src="../../Images/structures/rc/design/column-design.png" alt="column-design.png">
'<h4>Cross section dimensions</h4>
'Width -'b = ?'mm, Height -'h = ?'mm
#post
'Cross section area -'A_c = b*h'mm²
#show
'<p><b>Reinforcement</b></p>
'Left -'A_s1 = ?'mm², Right -'A_s2 = ?'mm²
#post
'Reinforcement ratio -'ρ = (A_s1 + A_s2)/A_c
#show
'<p><b>Concrete cover to the center of reinforcement</b></p>
'Left -'d_1 = ?'mm, Right -'d_2 = ?'mm
#post
'Effective cross section depth -'d = h - d_1'mm
#show
'<p><b>Section design loads</b></p>
'Axial force -'N_Ed = ?'kN (positive for compression)
'Bending moment -'M_Ed = ?'kN·m
'<h4>Material properties</h4>
'<p class="ref" style="float:right">[EN 1992-1-1, Table 3.1]</p>
'<p><b>Concrete</b></p>
'Characteristic compressive cylinder strength -'f_ck = ?'MPa
'Partial safety factor for concrete -'γ_c = 1.5','α_cc = ?
#post
'Design compressive cylinder strength -'f_cd = α_cc*f_ck/γ_c'MPa
'Mean value of cylinder compressive strength -'f_cm = f_ck + 8'MPa
'Secant modulus of elasticity -'E_cm = 22*(f_cm/10)^0.3'GPa
'Ultimate compressive strain -'ε_cu2 = 0.0035
'Strain  at the end of parabolic part of the diagram -'ε_c2 = 0.002
#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
'Modulus of elasticity -'E_s = 200'GPa
'<p><b>Strain-stress diagrams</b>, MPa:</p>
'<!--'PlotWidth = 250','PlotHeight = 125'-->
'<table><tr><td>
#hide
n = 2
σ_c(ε) = f_cd*((1 - (1 - ε/ε_c2)^n)*(ε < ε_c2) + (ε ≥ ε_c2))
#post
$Plot{σ_c(ε/1000) @ ε = 0 : ε_cu2*10^3}
'</td><td>
#hide
σ_s(ε) = max(-f_yd; min(ε*E_s*10^3;f_yd))
#post
$Plot{σ_s(ε/1000) @ ε = -10 : 10}
'</td></tr></table>
#show
'<h4>Imperfections and second order effects</h4>
'Ratio of quasi-permanent bending moment: M<sub>0Eqp</sub>/M<sub>0Ed</sub> ='K_G = ?
#post
'M<sub>0Eqp</sub> is the first order bending moment in quasi-permanent load combination (SLS)
'M<sub>0Ed</sub> is the first order bending moment in design load combination (ULS)
#show
'Long term creep factor - φ(∞,t<sub>0</sub>) ='φ = ?
'Column height -'L = ?'mm
'Effective column height -'L_o = ?*L'mm
#post
'<div class="fold">
#show
'<p><b>Geometric imperfections and accidental eccentricity</b></p>
'Number of vertical members contributing to the total effect -'m = ?
#post
'Reduction factor for height:
α_h_ = 2/sqr(L*10^-3)
'This value is limited within the interval: 2/3 ≤ α<sub>n</sub> ≤ 1
α_h = max(2/3;min(α_h_;1))
'Reduction factor for number of members
α_m = sqr(0.5*(1 + 1/m))
'Basic inclination value -'θ_o = 1/200
'<p class="ref">[EN 1992-1-1 (5.1)]</p>
'Inclination -'θ_i = θ_o*α_h*α_m
'<p class="ref">[EN 1992-1-1 (5.2)]</p>
'Eccentricity -'e_i_ = θ_i*L_o/2'mm
'<p class="ref">[EN 1992-1-1, § 6.1(4)]</p>
'Minimum eccentricity -'e_o_ = h/30'mm or 20 mm
e_o = max(e_o_;20)'mm
e_i = max(e_i_;e_o)'mm
'</div>
M_0Ed = M_Ed + N_Ed*e_i*10^-3'kN.m
'<div class="fold">
'<p><b>Second order effects based on the nominal stiffness method</b></p>
'Moment of inertia of the concrete cross section
I_c = (b*h^3)/12'mm<sup>4</sup>
'Radius of gyration
i = sqr(I_c/A_c)'mm
'Slenderness ratio
λ = L_o/i
'Relative axial force
n = N_Ed/(A_c*f_cd*10^-3)
'<p class="ref">[EN 1992-1-1 (5.23)]</p>
'Factor depending on the concrete strength class
k_1 = sqr(f_ck/20)
'<p class="ref">[EN 1992-1-1 (5.24)]</p>
'Factor depending on axial force and slenderness
k_2_ = n*λ/170
'Maximum value - k<sub>2</sub> &lt; 0.20
k_2 = min(k_2_;0.20)
'Effective creep ratio
'φ<sub>ef</sub> = φ(∞,t0)·M<sub>0Eqp</sub>/M<sub>0Ed</sub>
φ_ef = φ*K_G
'<p class="ref">[EN 1992-1-1 (5.22)]</p>
K_c = k_1*k_2/(1 + φ_ef)'– factor for effects of cracking, creep etc.
K_s = 1'– factor for contribution of reinforcement
'Second moment of area of the reinforcement, about the centroid
I_s = A_s1*(h/2 - d_1)^2 + A_s2*(h/2 - d_2)^2'mm<sup>4</sup>
'<p class="ref">[BS EN 1992-1-1 (5.20), NA.1]</p>
γ_cE = 1.2
'<p class="ref">[EN 1992-1-1 (5.20)]</p>
'Design value of the modulus of elasticity of concrete
E_cd = E_cm/γ_cE'GPa
'<p class="ref">[EN 1992-1-1 (5.21)]</p>
'Nominal stiffness
EI = (K_c*E_cd*I_c + K_s*E_s*I_s)'kN·mm²
'<p class="ref">[EN 1992-1-1 (5.17)]</p>
'Buckling load
N_B = π^2*EI/(L_o)^2'kN
'</div>
'<p class="ref">[EN 1992-1-1 (5.30)]</p>
'Total design moment (including second order moment)
M_Ed = M_0Ed/(1 - N_Ed/N_B)'kN·m
'<h4>Design checks</h4>
'Design axial resistance of section when strain'ε = ε_c2
N_Rd_max = A_c*f_cd*10^-3 + σ_s(ε)*(A_s1 + A_s2)*10^-3'kN
#if N_Ed > N_Rd_max
    'Design axial load is greater than the ultimate axial force capacity
    N_Ed'>'N_Rd_max'kN
    '<p class="err">The design check is NOT satisfied.</p>
#else
    'Internal forces in the section are expressed as functions of the compression zone depth<i>x</i>.
    'Reinforcement strain:
    '- bottom reinforcement -'ε_s1(x) = ε_cu2*(d - x)/x
    '- top reinforcement -'ε_s2(x) = ε_cu2*(d_2 - x)/x
    'Internal forces:
    '- concrete -'N_c(x) = 17/21*x*b*f_cd*10^-3
    '- bottom reinforcement -'N_s1(x) = σ_s(ε_s1(x))*A_s1*10^-3
    '- top reinforcement -'N_s2(x) = σ_s(ε_s2(x))*A_s2*10^-3
    'Section resistance for axial force
    N_Rd(x) = N_c(x) - N_s1(x) - N_s2(x)
    'Axial force, corresponding to triangular strain distribution
    N_Rd_h = N_Rd(h)'kN
    #if N_Ed < N_Rd_h
        'Section is partially in tension -'N_Ed'kN &lt;'N_Rd_h'kN
        'Compression zone depth is determined from the equilibrium of the axial forces in the section
        x = $Root{N_Rd(x) - N_Ed @ x = d_2 : h}'mm
        'Lever arm of concrete stress to the cross section centroid
        z_c = h/2 - 99/238*x'mm
        'Bending resistance at <i>N</i><sub>Ed</sub> = <i>N</i><sub>Rd</sub>
        M_Rd = (N_c(x)*z_c + N_s1(x)*(h/2 - d_1) - N_s2(x)*(h/2 - d_2))*10^-3'kN·m
    #else
        'Section is entirely in compression -'N_Ed'≥'N_Rd_h'kN
        'Distance from the point with constant strain to the bottom edge of the section
        h_c = h*ε_c2/ε_cu2
        'Internal forces are expressed as functions of the strain <i>ε</i> at the bottom edge.
        'Reinforcement strain:
        '- bottom reinforcement -'ε_s1(ε) = ε + (ε_c2 - ε)*d_1/h_c
        '- top reinforcement -'ε_s2(ε) = ε + (ε_c2 - ε)*(h - d_2)/h_c
        'Internal forces:
        '- concrete -'N_c(ε) = (h_c/6*(σ_c(ε) + 4*σ_c((ε + ε_c2)/2) + σ_c(ε_c2)) + (h - h_c)*f_cd)*b*10^-3
        '- bottom reinforcement -'N_s1(ε) = σ_s(ε_s1(ε))*A_s1*10^-3
        '- top reinforcement -'N_s2(ε) = σ_s(ε_s2(ε))*A_s2*10^-3
        'Section resistance for axial force
        N_Rd(ε) = N_c(ε) + N_s1(ε) + N_s2(ε)
        'Strain at bottom edge is determined from the equilibrium of the axial forces in the section
        ε_c = $Root{N_Rd(ε) - N_Ed @ ε = 0 : ε_c2}
        'Equivalent bending moment about the lower section edge, due to concrete stress
        M_c0 = (h_c^2/6*(2*σ_c((ε_c + ε_c2)/2) + σ_c(ε_c2)) + (h^2 - h_c^2)/2*f_cd)*b*10^-3'kNm
        'Lever arm of concrete stress equivalent force about the section centroid
        z_c = M_c0/N_c(ε_c) - h/2'mm
        'Bending resistance at <i>N</i><sub>Ed</sub> = <i>N</i><sub>Rd</sub>
        M_Rd = (N_c(ε_c)*z_c - N_s1(ε_c)*(h/2 - d_1) + N_s2(ε_c)*(h/2 - d_2))*10^-3'kN·m
    #end if
    #if M_Ed > M_Rd
        'Design bending is greater than the bending resistance:
        M_Ed'>'M_Rd'kN·m
        '<p class="err">The design check is NOT satisfied.</p>
    #else
        'Bending moment is less than moment resistance:
        M_Ed'≤'M_Rd'kN·m
        'The design check is satisfied.
        #pre
        '<img class="side" style="width:190pt;" src="../../Images/structures/rc/design/column-biaxial.png" alt="column-biaxial.png">
        '<h4>Biaxial bending design</h4>
        'For the other direction:
        'Design moment -'M_Edy = ?'kN·m
        'Bending resistance -'M_Rdy = ?'kN·m
        'Effective length -'L_oy = ?*L'mm
        #post
        #if M_Edy > 0
            '<img class="side" style="width:190pt;" src="../../Images/structures/rc/design/column-biaxial.png" alt="column-biaxial.png">
            '<h4>Biaxial bending design</h4>
            'For the other direction:
            'Design moment -'M_Edy'kN.m
            'Bending resistance -'M_Rdy'kN.m
            'Effective length -'L_oy'mm
            'Radius of inertia -'i_y = b/sqr(12)'mm
            'Slenderness ratio -'λ_y = L_oy/i_y
            'Eccentricity -'e_z = M_Edy/N_Ed*10^3'mm
            'For the current direction:
            'Design moment -'M_Edz = M_Ed'kN·m
            'Bending resistance -'M_Rdz = M_Rd'kN·m
            'Effective length -'L_oz = L_o'mm
            'Radius of inertia -'i_z = h/sqr(12)'mm
            'Slenderness ratio -'λ_z = L_oz/i_z
            'Eccentricity -'e_y = M_Edz/N_Ed*10^3'mm
            '<p class="ref">[EN 1992-1-1 (5.38)]</p>
            '<p><b>Check if biaxial bending design is required</b></p>
            #if (λ_y/λ_z ≤ 2)*(λ_z/λ_y ≤ 2)*((e_y/h*(b/e_z) ≤ 0.2) + (e_z/b*(h/e_y) ≤ 0.2))
                λ_y/λ_z'≤ 2 and'λ_z/λ_y'≤ 2 and ('e_y/h*(b/e_z)'≤ 0.2 or'e_z/b*(h/e_y)'≤ 0.2)'
                'The condition is satisfied. Biaxial bending design is NOT required. Separate checks may be performed for each direction
            #else
                #if λ_y/λ_z > 2
                    λ_y/λ_z'&gt; 2
                #end if
                #if λ_z/λ_y > 2
                    λ_z/λ_y'&gt; 2
                #end if
                #if (e_y/h*b/e_z > 0.2)*(e_z/b*(h/e_y) > 0.2)
                    e_y/h*(b/e_z)'&gt; 0.2 and'e_z/b*(h/e_y)'&gt; 0.2
                #end if
                'The condition is NOT satisfied. Biaxial bending design is required.
                'Ultimate axial force resistance-'N_Rd = N_Rd_max'kN
                N_Ed/N_Rd
                'Calculation of the exponent factor <i>a</i>
                #if N_Ed/N_Rd ≤ 0.1
                    a = 1
                #else if N_Ed/N_Rd ≤ 0.7
                    a = 1 + (N_Ed/N_Rd - 0.1)*0.8333
                #else
                    a = 1.5 + (N_Ed/N_Rd - 0.7)*1.6667
                #end if
                '<p class="ref">[EN 1992-1-1 (5.39)]</p>
                '<p><b>Biaxial bending check</b></p>
                k = (M_Edz/M_Rdz)^a + (M_Edy/M_Rdy)^a
                #if k ≤ 1
                    '(<i>M</i><sub>Edz</sub>/<i>M</i><sub>Rdz</sub>)<sup>a</sup> + (<i>M</i><sub>Edy</sub>/<i>M</i><sub>Rdy</sub>)<sup>a</sup> ≤ 1. The design check is satisfied.
                #else
                    '(<i>M</i><sub>Edz</sub>/<i>M</i><sub>Rdz</sub>)<sup>a</sup> + (<i>M</i><sub>Edy</sub>/<i>M</i><sub>Rdy</sub>)<sup>a</sup> > 1. <span class="err">The design check is NOT satisfied.</span>
                #end if
            #end if
        #end if
    #end if
#end if
#show
'</div>300  700 800 800 50  50  1200    200 20  0.85    500 0.75    2.5 4000    1   1   50  100 1

Column Design for Biaxial Bending and Axial Force (with Units)¶

Biaxial bending check of a rectangular column with the same strain-compatibility model as the uniaxial worksheet, expressed entirely with explicit CalcpadCE units so that the input fields use cm or mm directly.

Code:

'<small>According to <strong>Eurocode</strong>: EN 1992-1-1</small>
'<div style="max-width:180mm">
'<img class="side" style="width:150pt;" src="../../Images/structures/rc/design/column-design.png" alt="Column_Design.png">
'<h4>Cross section dimensions</h4>
'Width -'b = ?mm', Height - 'h = ?mm
#post
'Cross section area -'A_c = b*h|cm^2
#show
'<p><b>Reinforcement</b></p>
'Left -'A_s1 = ?*cm^2', Right -'A_s2 = ?*cm^2
#post
'Reinforcement ratio -'ρ = (A_s1 + A_s2)/A_c
#show
'<p><b>Concrete cover to the center of reinforcement</b></p>
'Left -'d_1 = ?mm', Right -'d_2 = ?mm
#post
'Effective cross section depth -'d = h - d_1
#show
'<p><b>Section design loads</b></p>
'Axial force -'N_Ed = ?kN' (positive for compression)
'Bending moment -'M_Ed = ?kNm
'<h4>Material properties</h4>
'<p class="ref" style="float:right">[EN 1992-1-1, Table 3.1]</p>
'<p><b>Concrete</b></p>
'Characteristic compressive cylinder strength -'f_ck = ?MPa
'Partial safety factor for concrete -'γ_c = 1.5','α_cc = ?
#post
'Design compressive cylinder strength -'f_cd = α_cc*f_ck/γ_c
'Mean value of cylinder compressive strength -'f_cm = f_ck + 8MPa
'Secant modulus of elasticity -'E_cm = 22GPa*(f_cm/10MPa)^0.3
'Ultimate compressive strain -'ε_cu2 = 0.0035
'Strain  at the end of parabolic part of the diagram -'ε_c2 = 0.002
#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
'Modulus of elasticity -'E_s = 200 GPa
'<p><b>Strain-stress diagrams</b>, MPa:</p>
'<!--'PlotWidth = 250','PlotHeight = 125'-->
'<table><tr><td>
#hide
n = 2
σ_c(ε) = f_cd*((1 - (1 - ε/ε_c2)^n)*(ε < ε_c2) + (ε ≥ ε_c2))
#post
$Plot{σ_c(ε/1000) @ ε = 0 : ε_cu2*1000}
'</td><td>
#hide
σ_s(ε) = max(-f_yd;min(ε*E_s;f_yd))
#post
$Plot{σ_s(ε/1000) @ ε = -10 : 10}
'</td></tr></table>
#show
max(-f_yd;min(0.01*E_s;f_yd))
'<h4>Imperfections and second order effects</h4>
'Ratio of quasi-permanent bending moment: M<sub>0Eqp</sub>/M<sub>0Ed</sub> ='K_G = ?
#post
'M<sub>0Eqp</sub> is the first order bending moment in quasi-permanent load combination (SLS)
'M<sub>0Ed</sub> is the first order bending moment in design load combination (ULS)
#show
'Long term creep factor - φ(∞,t<sub>0</sub>) ='φ = ?
'Column height -'L = ?m
'Effective column height -'L_o = ?*L
#post
'<div class="fold">
#show
'<p><b>Geometric imperfections and accidental eccentricity</b></p>
'Number of vertical members contributing to the total effect -'m = ?
#post
'Reduction factor for height:
α_h_ = 2/sqr(L/1m)
'This value is limited within the interval: 2/3 ≤ α<sub>n</sub> ≤ 1
α_h = max(2/3;min(α_h_;1))
'Reduction factor for number of members
α_m = sqr(0.5*(1 + 1/m))
'Basic inclination value -'θ_o = 1/200
'<p class="ref">[EN 1992-1-1 (5.1)]</p>
'Inclination -'θ_i = θ_o*α_h*α_m
'<p class="ref">[EN 1992-1-1 (5.2)]</p>
'Eccentricity -'e_i = θ_i*L_o/2|mm
'</div>
M_0Ed = M_Ed + N_Ed*e_i
'<div class="fold">
'<p><b>Second order effects based on the nominal stiffness method</b></p>
'Moment of inertia of the concrete cross section
I_c = (b*h^3)/12|cm^4
'Radius of gyration
i = sqr(I_c/A_c)|mm
'Slenderness ratio
λ = L_o/i
'Relative axial force
n = N_Ed/(A_c*f_cd)
'<p class="ref">[EN 1992-1-1 (5.23)]</p>
'Factor depending on the concrete strength class
k_1 = sqr(f_ck/20MPa)
'<p class="ref">[EN 1992-1-1 (5.24)]</p>
'Factor depending on axial force and slenderness
k_2_ = n*λ/170
'Maximum value - k<sub>2</sub> &lt; 0.20
k_2 = min(k_2_;0.20)
'Effective creep ratio
'φ<sub>ef</sub> = φ(∞,t0)·M<sub>0Eqp</sub>/M<sub>0Ed</sub>
φ_ef = φ*K_G
'<p class="ref">[EN 1992-1-1 (5.22)]</p>
K_c = k_1*k_2/(1 + φ_ef)'– factor for effects of cracking, creep etc.
K_s = 1'– factor for contribution of reinforcement
'Second moment of area of the reinforcement, about the centroid
I_s = A_s1*(h/2 - d_1)^2 + A_s2*(h/2 - d_2)^2|cm^4
'<p class="ref">[BS EN 1992-1-1 (5.20), NA.1]</p>
γ_cE = 1.2
'<p class="ref">[EN 1992-1-1 (5.20)]</p>
'Design value of the modulus of elasticity of concrete
E_cd = E_cm/γ_cE
'<p class="ref">[EN 1992-1-1 (5.21)]</p>
'Nominal stiffness
EI = (K_c*E_cd*I_c + K_s*E_s*I_s)|kN*cm^2
'<p class="ref">[EN 1992-1-1 (5.17)]</p>
'Buckling load
N_B = π^2*EI/(L_o)^2|kN
'</div>
'<p class="ref">[EN 1992-1-1 (5.30)]</p>
'Bending moment with second order effects
M_Ed_ = M_0Ed/(1 - N_Ed/N_B)
'<p class="ref">[EN 1992-1-1, Clause 6.1(4)]</p>
'Minimum eccentricity
e_o = max(h/30;20mm)|mm
'Design bending moment
M_Ed = max(M_Ed_;N_Ed*e_o)
'<h4>Design checks</h4>
'Design axial resistance of section when strain'ε = ε_c2
N_Rd_max = A_c*f_cd + σ_s(ε)*(A_s1 + A_s2)|kN
#if N_Ed > N_Rd_max
    'Design axial load is greater than the ultimate axial force capacity
    N_Ed'&gt;'N_Rd_max
    '<p class="err">The design check is NOT satisfied.</p>
#else
    'Internal forces in the section are expressed as functions of the compression zone depth <var>x</var>.
    'Reinforcement strain:
    '- bottom reinforcement -'ε_s1(x) = ε_cu2*(d - x)/x
    '- top reinforcement -'ε_s2(x) = ε_cu2*(d_2 - x)/x
    'Internal forces:
    '- concrete -'N_c(x) = 17/21*x*b*f_cd
    '- bottom reinforcement -'N_s1(x) = σ_s(ε_s1(x))*A_s1
    '- top reinforcement -'N_s2(x) = σ_s(ε_s2(x))*A_s2
    'Section resistance for axial force
    N_Rd(x) = N_c(x) - N_s1(x) - N_s2(x)
    'Axial force, corresponding to triangular strain distribution
    N_Rd_h = N_Rd(h)|kN
    #if N_Ed < N_Rd_h
        'Section is partially in tension -'N_Ed' &lt;'N_Rd_h
        'Compression zone depth is determined from the equilibrium of the axial forces in the section
        x = $Root{N_Rd(x) = N_Ed @ x = d_2 : h}
        'Lever arm of concrete stress to the cross section centroid
        z_c = h/2 - 99/238*x
        'Bending resistance at <i>N</i><sub>Ed</sub> = <i>N</i><sub>Rd</sub>
        M_Rd = (N_c(x)*z_c + N_s1(x)*(h/2 - d_1) - N_s2(x)*(h/2 - d_2))|kNm
    #else
        'Section is entirely in compression -'N_Ed'≥'N_Rd_h
        'Distance from the point with constant strain to the bottom edge of the section
        h_c = h*ε_c2/ε_cu2
        'Internal forces are expressed as functions of the strain <i>ε</i> at the bottom edge.
        'Reinforcement strain:
        '- bottom reinforcement -'ε_s1(ε) = ε + (ε_c2 - ε)*d_1/h_c
        '- top reinforcement -'ε_s2(ε) = ε + (ε_c2 - ε)*(h - d_2)/h_c
        'Internal forces:
        '- concrete -'N_c(ε) = (h_c/6*(σ_c(ε) + 4*σ_c((ε + ε_c2)/2) + σ_c(ε_c2)) + (h - h_c)*f_cd)*b
        '- bottom reinforcement -'N_s1(ε) = σ_s(ε_s1(ε))*A_s1
        '- top reinforcement -'N_s2(ε) = σ_s(ε_s2(ε))*A_s2
        'Section resistance for axial force
        N_Rd(ε) = N_c(ε) + N_s1(ε) + N_s2(ε)
        'Strain at bottom edge is determined from the equilibrium of the axial forces in the section
        ε_c = $Root{N_Rd(ε) = N_Ed @ ε = 0 : ε_c2}
        'Equivalent bending moment about the lower section edge, due to concrete stress
        M_c0 = (h_c^2/6*(2*σ_c((ε_c + ε_c2)/2) + σ_c(ε_c2)) + (h^2 - h_c^2)/2*f_cd)*b|kNm
        'Lever arm of concrete stress equivalent force about the section centroid
        z_c = M_c0/N_c(ε_c) - h/2
        'Bending resistance at <i>N</i><sub>Ed</sub> = <i>N</i><sub>Rd</sub>
        M_Rd = (N_c(ε_c)*z_c - N_s1(ε_c)*(h/2 - d_1) + N_s2(ε_c)*(h/2 - d_2))|kNm
    #end if
    #if M_Ed > M_Rd
        'Design bending is greater than the bending resistance:
        M_Ed'>'M_Rd
        '<p class="err">The design check is NOT satisfied.</p>
    #else
        'Bending moment is less than moment resistance:
        M_Ed'≤'M_Rd
        'The design check is satisfied.
        #pre
        '<img class="side" style="width:190pt;" src="../../Images/structures/rc/design/column-biaxial.png" alt="Column_Biaxial.png">
        '<h4>Biaxial bending design</h4>
        'For the other direction:
        'Design moment -'M_Edy = ?kNm
        'Bending resistance -'M_Rdy = ?kNm
        'Effective length -'L_oy = ?*L
        #post
        #if M_Edy > 0kNm
            '<img class="side" style="width:190pt;" src="../../Images/structures/rc/design/column-biaxial.png" alt="Column_Biaxial.png">
            '<h4>Biaxial bending design</h4>
            'For the other direction:
            'Design moment -'M_Edy
            'Bending resistance -'M_Rdy
            'Effective length -'L_oy
            'Radius of inertia -'i_y = b/sqr(12)|mm
            'Slenderness ratio -'λ_y = L_oy/i_y
            'Eccentricity -'e_z = M_Edy/N_Ed|mm
            'For the current direction:
            'Design moment -'M_Edz = M_Ed
            'Bending resistance -'M_Rdz = M_Rd
            'Effective length -'L_oz = L_o
            'Radius of inertia -'i_z = h/sqr(12)|mm
            'Slenderness ratio -'λ_z = L_oz/i_z
            'Eccentricity -'e_y = M_Edz/N_Ed|mm
            '<p class="ref">[EN 1992-1-1 (5.38)]</p>
            '<p><b>Check if biaxial bending design is required</b></p>
                #if (λ_y/λ_z ≤ 2)*(λ_z/λ_y ≤ 2)*((e_y/h*b/e_z ≤ 0.2) + (e_z/b*h/e_y ≤ 0.2))
                λ_y/λ_z'≤ 2 and'λ_z/λ_y'≤ 2 and ('e_y/h*(b/e_z)'≤ 0.2 or'e_z/b*(h/e_y)'≤ 0.2)'
                'The condition is satisfied. Biaxial bending design is NOT required. Separate checks may be performed for each direction
            #else
                #if λ_y/λ_z > 2
                    λ_y/λ_z'&gt; 2
                #end if
                #if λ_z/λ_y > 2
                    λ_z/λ_y'&gt; 2
                #end if
                #if (e_y/h*b/e_z > 0.2)*(e_z/b*h/e_y > 0.2)
                    e_y/h*(b/e_z)'&gt; 0.2 and'e_z/b*(h/e_y)'&gt; 0.2
                #end if
                'The condition is NOT satisfied. Biaxial bending design is required.
                'Ultimate axial force resistance-'N_Rd = N_Rd_max'kN
                N_Ed/N_Rd
                'Calculation of the exponent factor <i>a</i>
                #if N_Ed/N_Rd ≤ 0.1
                    a = 1
                #else if N_Ed/N_Rd ≤ 0.7
                    a = 1 + (N_Ed/N_Rd - 0.1)*0.8333
                #else
                    a = 1.5 + (N_Ed/N_Rd - 0.7)*1.6667
                #end if
                '<p class="ref">[EN 1992-1-1 (5.39)]</p>
                '<p><b>Biaxial bending check</b></p>
                k = (M_Edz/M_Rdz)^a + (M_Edy/M_Rdy)^a
                #if k ≤ 1
                    '(<var>M</var><sub>Edz</sub>/<var>M</var><sub>Rdz</sub>)<sup>a</sup> + (<var>M</var><sub>Edy</sub>/<var>M</var><sub>Rdy</sub>)<sup>a</sup> ≤ 1. The design check is satisfied.
                #else
                    '(<var>M</var><sub>Edz</sub>/<var>M</var><sub>Rdz</sub>)<sup>a</sup> + (<var>M</var><sub>Edy</sub>/<var>M</var><sub>Rdy</sub>)<sup>a</sup> > 1. <span class="err">The design check is NOT satisfied.</span>
                #end if
            #end if
        #end if
    #end if
#end if
#show
'</div>300  700 8   8   50  50  1200    200 20  0.85    500 0.75    2.5 4   1   1   50  100 1

Combined Bending, Shear and Torsion¶

Combined bending, shear and torsion design of a T-beam: variable-strut-inclination model for the shear contribution, space-truss model for the torsional resistance and superposition for the longitudinal reinforcement.

Code:

'<small>According to <strong>Eurocode</strong>: EN 1992-1-1</small>
'<div style="width:180mm">
'<img style="width:400pt;" src="../../Images/structures/rc/design/torsion.png" alt="torsion.png">
'<h4>Cross section dimensions</h4>
b_w = ? {250}'mm,'h_w = ? {500}'mm
b_f = ? {1200}'mm,'h_f = ? {120}'mm
'Shear link legs count -'n_w = ? {2}
'Bottom bars diameter:'Φ_b = ? {20}'mm
'Top and middle bars diameter:'Φ_t = ? {12}'mm
'Links diameter (closed and anchored):'Φ_s = ? {8}'mm
'Concrete cover -'c_nom = ? {25}'mm
#post
'Effective cross section depth
d = h_w - c_nom - Φ_s - Φ_b/2'mm
#show
'Number of middle rows of bars -'n_ef = ? {2}
#hide
a_l = (2*d - h_w)/(n_ef + 1)'mm
#post
#val
'<p class="ref">[EN 1992-1-1, § 9.2.3 (4)]</p>
#if a_l > 350'mm
    'Distance between rows of bars is 'a_l'mm &gt; 350 mm
    'The number of middle rows is not sufficient.
#else
    'Distance between rows of bars is 'a_l'mm &le; 350 mm
    'The number of middle rows of bars is sufficient.
#end if
#equ
#show
'<h4>Section design loads</h4>
'Torsional moment -'T_Ed = ? {20}'kN·m
'Bending moment -'M_Ed = ? {200}'kN·m
'Shear force -'V_Ed = ? {120}'kN
'<h4>Material properties</h4>
'<p><b>Concrete</b> [EN 1992-1-1, Table 3.1]</p>
'Characteristic compressive cylinder strength -'f_ck = ? {20}'MPa,
'Partial safety factors for concrete -'γ_c = 1.5','α_cc = ? {0.85}
#post
'Design compressive cylinder strength -'f_cd = α_cc*f_ck/γ_c'MPa
'Mean value of axial tensile strength -'f_ctm = 0.30*f_ck^(2/3)'MPa
'Characteristic axial tensile strength -'f_ctk,005 = 0.7*f_ctm'MPa
'Design axial tensile strength -'f_ctd = f_ctk,005/γ_c'MPa
#show
'<p><b>Steel</b></p>
'Characteristic yield strength -'f_yk = ? {500}'MPa
#post
'Partial safety factor for steel -'γ_s = 1.15
'Design yield strength -'f_yd = f_yk/γ_s'MPa
'Modulus of elasticity -'E_s = 200000'MPa
'<div class="fold">
'<h4>Bending design</h4>
'Factor for effective compression zone depth -'λ = 0.8
'Effective strength factor -'η = 1.0
'Ultimate compressive strain in concrete -'ε_cu3 = 0.0035
'Flange area -'A_f = (b_f - b_w)*h_f'mm²
'Flange first moment of area -'S_f = A_f*(d - h_f/2)'mm³
'Bending moment for neutral line at bottom edge of flange
M_f = b_f*h_f*η*f_cd*(d - h_f/2)*10^-6'kN·m
#if M_Ed > M_f
    M_Ed'&gt;'M_f'- the neutral line is below the flange - design for T-section
    'Relative design bending moment -'m_Ed = M_Ed*10^6/(b_w*d^2*η*f_cd)
    'Compression zone depth -'x = d/λ*(1 - sqr(1 - 2*(m_Ed - S_f/(b_w*d^2))))'mm
    'Relative compression zone depth -'ξ = x/d
    'Design yield strain of reinforcement -'ε_yd = f_yd/E_s
    'Relative depth of compression zone corresponding to design yield strain
    ξ_yd = ε_cu3/(ε_cu3 + ε_yd)
    #show
    'Limit compression zone depth -'ξ_lim = ? {0.62}
    '(enter <i>ξ</i><sub>yd</sub> for elastic or 0.45 for plastic analysis)
    #post
    #if ξ ≤ ξ_lim
        ξ'≤'ξ_lim'- compressive reinforcement is NOT required.
        'Lever arm of internal forces -'z = d - 0.5*λ*x'mm
        'Required tensile reinforcement area -'A_s1 = M_Ed*10^6/(z*f_yd)'mm²
        'Reinforcement ratio -'ρ_1 = A_s1/(b_w*d)
        '<!--'A_s2 = 0','ρ_2 = 0'-->
    #else
        ξ'&gt;'ξ_lim'- compressive reinforcement is required.
        'Relative depth is assumed to be'ξ = ξ_lim'and compressive reinforcement is designed
        'Compression zone depth -'x = ξ*d'mm
        'Distance from the center of compressive reinforcement to the concrete surface
        d_2 = h_w - d'mm
        'Distance between tensile and compressive reinforcement -'z_s = d - d_2'mm
        'Required tensile reinforcement area
        A_s1 = (M_Ed*10^6 + (b_w*λ*x*(λ*x/2 - d_2) + A_f*(h_f/2 - d_2))*η*f_cd)/(f_yd*z_s)'mm²
        'Strain is compressive reinforcement
        ε_s2 = (x - d_2)/x*ε_cu3
        'Compressive reinforcement stress
        σ_s2 = min(ε_s2*E_s; f_yd)'MPa
        'Required compressive reinforcement area
        A_s2 = (A_s1*f_yd - (b_w*λ*x + A_f)*η*f_cd)/σ_s2'mm²
        'Reinforcement ratios
        '- tensile reinforcement -'ρ_1 = A_s1/(b_w*d)
        '- compressive reinforcement -'ρ_2 = A_s2/(b_w*d)
    #end if
#else
    'The neutral line is within the flange - design of rectangular section with'b_f'mm
    'Relative design bending moment -'m_Ed = M_Ed*10^6/(b_f*d^2*η*f_cd)
    'Compression zone depth -'x = d/λ*(1 - sqr(1 - 2*m_Ed))'mm
    'Lever arm of internal forces -'z = d - 0.5*λ*x'mm
    'Required tensile reinforcement area -'A_s1 = M_Ed*10^6/(z*f_yd)'mm²
    'Reinforcement ratio -'ρ_1 = A_s1/(b_w*d)
    '<!--'A_s2 = 0','ρ_2 = 0'-->
#end if
'<p class="ref">[EN 1992-1-1, § 9.2.1.1]</p>
'Minimum tensile reinforcement ratio
ρ_min = max(0.26*f_ctm/f_yk; 0.0013)
'Minimum area of tensile reinforcement -'A_s,min = ρ_min*(b_w*d)'mm²
'</div>
'<div class="fold">
'<h4>Shear (V<sub>Ed</sub>) and torsion (Т<sub>Ed</sub>) design</h4>
'<p class="ref">[EN 1992-1-1, § 6.2.2 and 6.3.2]</p>
'<p><b>Bearing capacity without shear reinforcement</b></p>
k = min(1 + sqr(200/d); 2)
C_Rd_c = 0.18/γ_c
ρ_L = min(max(ρ_min; ρ_1); 0.02)
ν_min = 0.035*k^(3/2)*sqr(f_ck)
V_Rd,c_ = C_Rd_c*k*(100*ρ_L*f_ck)^(1/3)*b_w*d*10^-3'kN
V_Rd,c,min = ν_min*b_w*d*10^-3'kN
V_Rd,c = max(V_Rd,c,min; V_Rd,c_)'kN
'Effective wall thickness
t_ef = max(b_w*h_w/(2*(b_w + h_w)); 2*c_nom)'mm
'Total area of the cross section within the outer perimeter
A_k = (b_w - t_ef)*(h_w - t_ef)'mm²
'Outer perimeter of the cross section
u_k = 2*(b_w + h_w - 2*t_ef)'mm
'Section subjected to pure torsion
T_Rd,c = 2*A_k*t_ef*f_ctd*10^-6'kN·m
#if T_Ed/T_Rd,c + V_Ed/V_Rd,c ≤ 1
    '</div>
    '<p class="ref">[EN 1992-1-1, § 6.3.2 (5)]</p>
    T_Ed/T_Rd,c + V_Ed/V_Rd,c'≤'1
    'Shear reinforcement is not required by calculation!
    'Minimum area of shear reinforcement
    '<p class="ref">[EN 1992-1-1, § 9.2.2 (5)]</p>
    A_sw,min = 0.08*sqr(f_ck)/f_yk*b_w'mm²/mm
    '<!--'θ = 21.8'-->
#else
    '<p class="ref">[EN 1992-1-1, § 6.3.2 (5)]</p>
    T_Ed/T_Rd,c + V_Ed/V_Rd,c'>'1
    'Shear reinforcement is required!
    '<p><b>Bearing capacity with shear reinforcement</b></p>
    'Lever arm of internal forces -'z = 0.9*d'mm
    α_cw = 1'(when axial force N<sub>Ed</sub> = 0)
    '<p class="ref">[EN 1992-1-1, §6.2.2 (5)]</p>
    ν_1 = 0.6*(1 - f_ck/250)
    '<p class="ref">[(6.9) and (6.30) from EN 1992-1-1]</p>
    'Maximum capacity for shear and torsion (θ = 45°)
    V_Rd,max,45 = α_cw*b_w*z*ν_1*f_cd/2*10^-3'kN
    T_Rd,max,45 = ν_1*α_cw*f_cd*A_k*t_ef*10^-6'kN·m
    '<p class="ref">[(6.29) from EN 1992-1-1]</p>
    'Maximum bearing capacity check
    #if T_Ed/T_Rd,max,45 + V_Ed/V_Rd,max,45 ≤ 1
        T_Ed/T_Rd,max,45 + V_Ed/V_Rd,max,45'≤'1.'The design check is satisfied!
        'Angle between the concrete struts and the longitudinal reinforcement
        θ = 0.5*asin(T_Ed/T_Rd,max,45 + V_Ed/V_Rd,max,45)'°
        'The angle is limited within 21,8° ≤ θ ≤ 45°
        θ = max(21.8; min(θ; 45))'°
        '<p class="ref">[(6.9) and (6.30) from EN 1992-1-1]</p>
        'Bearing capacity check at'θ'°
        V_Rd,max = α_cw*b_w*z*ν_1*f_cd*cot(θ)/(1 + cot(θ)^2)*10^-3'kN
        T_Rd,max = 2*ν_1*α_cw*f_cd*A_k*t_ef*sin(θ)*cos(θ)*10^-6'kN·m
        T_Ed/T_Rd,max + V_Ed/V_Rd,max
        '<p class="ref">(6.8) from EN 1992-1-1</p>
        'Shear reinforcement area
        A_sw,V = V_Ed*10^3/(z*f_yd*cot(θ))'mm²/mm
        'Torsion reinforcement area
        A_sw,T = T_Ed*10^6*tan(θ)/(2*A_k*f_yd)'mm²/mm
        '<p class="ref">[EN 1992-1-1, § 9.2.2 (5)]</p>
        'Minimum area of shear reinforcement
        A_sw,min = 0.08*sqr(f_ck)/f_yk*b_w'mm²/mm
        '<p><b>Total reinforcement required for shear and torsion</b></p>
        A_sw = max(A_sw,V + A_sw,T; A_sw,min)'mm²/mm
        'Area of one leg -'A_sw,1 = π/4*(Φ_s)^2'mm²
        'Calculation of shear link spacing
        s_w = n_w*A_sw,1/A_sw'mm
        'Maximum shear link spacing
        '<p class="ref">[EN 1992-1-1, §9.2.2]</p>
        s_w,max,V = min(0.75*d; 300)'mm
        '<p class="ref">[EN 1992-1-1, § 9.2.3]</p>
        s_w,max,T = min(u_k/8; b_w)'mm
        #hide
        s_w,max = min(s_w,max,V; s_w,max,T)
        s_w = floor(min(s_w; s_w,max)/10)*10
        #post
        '</div>
        #val
        'Provide <b>doubly closed links</b>: Ø'Φ_s'/'s_w'mm with total area A<sub>s,w</sub> = 'n_w*A_sw,1/s_w' mm²/mm
        #equ
    #else
        '</div>
        T_Ed/T_Rd,max,45 + V_Ed/V_Rd,max,45'>'1
        '<p class="err">The condition is NOT satisfied!. Change concrete grade or cross section dimensions.</p>
        '<!--'θ = 45'-->
    #end if
#end if
'Required longitudinal reinforcement area to resist torsion<span class="ref">(6.28) from EN 1992-1-1</span>
A_s,T = T_Ed*10^6*u_k*cot(θ)/(2*A_k*f_yd)'mm²
#hide
A_s,m = A_s,T/(n_ef + 2)
A_s,b = max(A_s1 + A_s,m; A_s,min)
A_s,t = max(A_s2 + A_s,m; A_s,min)
A_s1,b = π*Φ_b^2/4
A_s1,t = π*Φ_t^2/4
n_b = max(ceiling(A_s,b/A_s1,b); 2)
n_t = max(ceiling(A_s,t/A_s1,t); 2)
n_m = max(ceiling(A_s,m/A_s1,t); 2)
#post
#val
'<p><b>Total longitudinal reinforcement for bending and torsion</b></p>
'Bottom reinforcement: A<sub>s,b</sub> ='A_s,b'mm² ('n_b'Ø'Φ_b' bars)
'Top reinforcement: A<sub>s,t</sub> ='A_s,t'mm² ('n_t'Ø'Φ_t' bars)
'Middle reinforcement: A<sub>s,m</sub> ='A_s,m*n_ef'mm² ('n_m'Ø'Φ_t' bars in 'n_ef' rows @ spacing 'a_l' mm)
#show
'</div>

Interaction Diagram of RC Section¶

N-M interaction diagram of a symmetrically reinforced rectangular section, generated by sweeping the neutral-axis depth from pure compression to pure tension and plotting the resulting axial force and bending moment.

Code:

'<small>According to <strong>Eurocode</strong>: EN 1992-1-1</small>
'<div style="max-width:180mm">
'<img class="side" style="width:150pt;" src="../../Images/structures/rc/design/column-design.png" alt="column-design.png">
'<h4>Cross section dimensions</h4>
'Width -'b = ?'mm, Height -'h = ?'mm
#post
'Cross section area -'A_c = b*h'mm²
#show
'<p><b>Reinforcement</b></p>
'Left -'A_s1 = ?'mm², Right -'A_s2 = A_s1'mm²
#post
'Reinforcement ratio -'ρ = (A_s1 + A_s2)/A_c
#show
'<p><b>Concrete cover</b> (to the center of the reinforcement)</p>
'Left -'d_1 = ?'mm, Right -'d_2 = d_1'mm
#post
'Effective cross section depth -'d = h - d_1'mm
#show
'<h4>Material properties</h4>
'<!--'PlotWidth = 250','PlotHeight = 125'-->
'<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
'Mean value of cylinder compressive strength -'f_cm = f_ck + 8'MPa
'Design compressive cylinder strength -'f_cd = α_cc*f_ck/γ_c'MPa
'Ultimate compressive strain -'ε_cu2 = 0.0035'('ε_c2 = 0.002','n = 2')'
'Secant modulus of elasticity -'E_cm = 22*(f_cm/10)^0.3'GPa
#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
'Modulus of elasticity -'E_s = 200'GPa
'<p><b>Strain-stress diagrams</b>, MPa:</p>
'<!--'PlotWidth = 250','PlotHeight = 125'-->
'<table><tr><td>
#hide
σ_c(ε) = f_cd*((1 - (1 - ε/ε_c2)^n)*(ε < ε_c2) + (ε ≥ ε_c2))
#post
$Plot{σ_c(ε/1000) @ ε = 0 : ε_cu2*1000}
'</td><td>
#hide
σ_s(ε) = max(-f_yd; min(ε*E_s*1000;f_yd))
#post
$Plot{σ_s(ε/1000) @ ε = -10 : 10}
'</td></tr></table>
'<h4>Interaction diagram</h4>
#hide
A_s_min = 0.00002*A_c
#if A_s1 < A_s_min
    A_s1 = A_s_min
#end if
#if A_s2 < A_s_min
    A_s2 = A_s_min
#end if
ε_s1,a(x) = ε_cu2*(d - x)/x
ε_s2,a(x) = ε_cu2*(d_2 - x)/x
N_c,a(x) = 17/21*x*b*f_cd/1000
N_s1,a(x) = σ_s(ε_s1,a(x))*A_s1/1000
N_s2,a(x) = σ_s(ε_s2,a(x))*A_s2/1000
N_Rd,a(x) = N_c,a(x) - N_s1,a(x) - N_s2,a(x)
z_c,a(x) = h/2 - 99/238*x
M_Rd,a(x) = (N_c,a(x)*z_c,a(x) + N_s1,a(x)*(h/2 - d_1) - N_s2,a(x)*(h/2 - d_2))/1000
x(ε) = h*ε_cu2/(ε_cu2 + ε)
h_c = h*ε_c2/ε_cu2
ε_s1,b(ε) = ε + (ε_c2 - ε)*d_1/h_c
ε_s2,b(ε) = ε + (ε_c2 - ε)*(h - d_2)/h_c
N_c,b(ε) = (h_c/6*(σ_c(ε) + 4*σ_c((ε + ε_c2)/2) + σ_c(ε_c2)) + (h - h_c)*f_cd)*b/1000
N_s1,b(ε) = σ_s(ε_s1,b(ε))*A_s1/1000
N_s2,b(ε) = σ_s(ε_s2,b(ε))*A_s2/1000
N_Rd,b(ε) = N_c,b(ε) + N_s1,b(ε) + N_s2,b(ε)
M_c0(ε) = (h_c^2/6*(2*σ_c((ε + ε_c2)/2) + σ_c(ε_c2)) + (h^2 - h_c^2)/2*f_cd)*b/1000
z_c,b(ε) = M_c0(ε)/N_c,b(ε) - h/2
M_Rd,b(ε) = (N_c,b(ε)*z_c,b(ε) - N_s1,b(ε)*(h/2 - d_1) + N_s2,b(ε)*(h/2 - d_2))/1000
N_Rd(ε) = N_Rd,a(x(ε))*(ε ≥ 0) + N_Rd,b(abs(ε))*(ε < 0)
M_Rd(ε) = M_Rd,a(x(ε))*(ε ≥ 0) + M_Rd,b(abs(ε))*(ε < 0)
ε_max = $Root{N_Rd(ε) - 1 @ ε = 0 : 2}
#if ε_max > 0.1
    ε_max = 0.1
#end if
PlotWidth = 600
PlotHeight = 600
#post
'<p><i>N</i><sub>Rd</sub></p>
$Plot{M_Rd(ε)|N_Rd(ε) @ ε = ε_max : -ε_c2}
'<p style=float:right><i>M</i><sub>Rd</sub></p>
#show
'</div>300  700 800 80  20  0.85    500

SLS Design of Beam with Rectangular Section¶

Serviceability checks for a rectangular beam: stress in the concrete and reinforcement, crack-width and long-term deflection, with the static scheme selected from a drop-down list (only in the Desktop version) of cantilevers and simply supported beams.

Code:

'<small>According to <strong>Eurocode</strong>: EN 1992-1-1</small>
'<div style="max-width:180mm">
'<h4>Static scheme</h4>
'<p id="type" class="flip_base" style="display:none">'type = ?'</p>
#pre
'<p><select data-target="type" class="flip_trigger">
'<option value="1">Cantilever with concentrated load</option>
'<option value="2">Cantilever with uniformly distributed load</option>
'<option value="3">Simply supported beam with concentrated load</option>
'<option value="0">Simply supported beam with uniformly distributed load</option>
'</select></p>
#hide
#if type ≡ 1
    #show
    '<img class="side flip flip_1" src="../../Images/structures/rc/design/deflection-cantilever-force.png" alt="deflection-cantilever-force.png" style="width:150pt;">
    #post
    'Cantilever with concentrated load
    #hide
#else if type ≡ 2
    #show
    '<img class="side flip flip_2" src="../../Images/structures/rc/design/deflection-cantilever-uniform.png" alt="deflection-cantilever-uniform.png" style="width:150pt;">
    #post
    'Cantilever with uniformly distributed load
    #hide
#else if type ≡ 3
    #show
    '<img class="side flip flip_3" src="../../Images/structures/rc/design/deflection-beam-force.png" alt="deflection-beam-force.png" style="width:150pt;">
    #post
    'Simply supported beam with concentrated load
    #hide
#else if type ≡ 0
    #show
    '<img class="side flip flip_0" src="../../Images/structures/rc/design/deflection-beam-uniform.png" alt="deflection-beam-uniform.png" style="width:150pt;">
    #post
    'Simply supported beam with uniformly distributed load
    #hide
#end if
#show
'Beam length -'L = ?'m
'Exposure class:
'<p id="w_max" style="display:none">'w_max = ?'</p>
#pre
'<p><select data-target="w_max">
'<option value="0.4">X0 or XC1</option>
'<option value="0.3">XC2, XC3, XC4, XD1, XD2, XS2 or XS3</option>
'</select></p>
#post
#if w_max ≡ 0.4
    'X0 or XC1
#else
    'XC2, XC3, XC4, XD1, XD2, XS2 or XS3
#end if
#show
''
'<h4>Bending moments</h4>
'Characteristic combination (<i>g</i> + <i>q</i>) -'M_k = ?'kNm
'Quasi-permanent combination (<i>g</i> + <i>ψ</i><sub>2</sub><i>q</i>) -'M_qp = ?'kNm
'
'<img class="side" src="../../Images/structures/rc/design/beam-section.png" alt="beam-section.png" style="width:120pt;">
'<h4>Cross section dimensions</h4>
'Width -'b = ?'mm, Height -'h = ?'mm
'Concrete cover -'c = ?'mm (to bars surface)
'<p><b>Tension reinforcement</b></p>
'Bar count -'n_1 = ?'with diameter -'Φ_1 = ?'mm
#post
'Reinforcement area -'A_s1 = n_1*π*Φ_1^2/4'mm²
#show
'Concrete cover to the center of reinforcement -'d_1 = ?'mm
#post
'Effective cross section depth -'d = h - d_1'mm
#show
'Spacing between bar centers -'s = ?'mm
'<p><b>Compression reinforcement</b></p>
'Bar count -'n_2 = ?'with diameter -'Φ_2 = ?'mm
#post
'Reinforcement area -'A_s2 = n_2*π*Φ_2^2/4'mm²
#show
'Concrete cover to the center of reinforcement -'d_2 = ?'mm
'
'<h4>Material properties</h4>
'<p><b>Concrete</b> [EN 1992-1-1, Table 3.1]</p>
'Characteristic compressive cylinder strength -'f_ck = ?'MPa
#post
'Mean value of cylinder compressive strength -'f_cm = f_ck + 8'MPa
'Mean value of axial tensile strength -'f_ctm = 0.30*f_ck^(2/3)'MPa
'Secant modulus of elasticity -'E_cm = 22*(f_cm/10)^0.3'GPa
#show
'<p><b>Steel</b></p>
'Characteristic yield strength -'f_yk = ?'MPa
#post
'Modulus of elasticity -'E_s = 200'GPa
'Ratio of steel to conrete moduli of elasticity -'α = E_s/E_cm
'
'<div class="fold">
#show
'<h4>Concrete creep and shrinkage</h4>
'<p class = "ref">[EN 1992-1-1 B.1(1)]</p>
'<p><b>Creep</b></p>
'Relative humidity of the environment -'RH = ?'%
#post
'Perimeter of cross section in contact with the atmosphere
u = 2*(b + h)'mm
'Cross section area -'A_c = b*h'mm²
'Notional size of the cross section
'<p class = "ref">(eq. B.6)</p>
h_0 = 2*A_c/u'mm
'Relative humidity factor
#if f_cm ≤ 35
    '<p class = "ref">(eq. B.3a)</p>
    φ_RH = (1 + (1 - RH/100)/(0.1*h_0^0.333))
#else
    '<p class = "ref">(eq. B.8c)</p>
    α_1 = (35/f_cm)^0.7','α_2 = (35/f_cm)^0.2
    '<p class = "ref">(eq. B.3b)</p>
    φ_RH = (1 + (1 - RH/100)/(0.1*h_0^0.333)*α_1)*α_2
#end if
'Concrete strength factor
'<p class = "ref">(eq. B.4)</p>
β_f_cm = 16.8/sqr(f_cm)
'Concrete age at the moment of loading -'t_0 = 28'дни
'Concrete age factor
'<p class = "ref">(eq. B.5)</p>
β_t0 = 1/(0.1 + t_0^0.20)
'Notional creep coefficient
'<p class = "ref">(eq. B.2)</p>
φ_8_t0 = φ_RH*β_f_cm*β_t0
'Effective concrete modulus of elasticity
'<p class = "ref">[EN 1992-1-1 §7.4.3(5)]</p>
E_c_eff = E_cm/(1 + φ_8_t0)'GPa
'Effective ratio of modules of elasticity
'<p class = "ref">[EN 1992-1-1 §7.4.3(6)]</p>
α_e = E_s/E_c_eff
'<p class = "ref">[EN 1992-1-1 B.2(1)]</p>
'<p><b>Shrinkage</b></p>
'Coefficient depending on the notional size
#if h_0 ≤ 100
    k_h = 1.00
#else if h_0 ≤ 200
    k_h = 1.00 - 0.0015*(h_0 - 100)
#else if h_0 ≤ 300
    k_h = 0.85 - 0.001*(h_0 - 200)
#else if h_0 ≤ 500
    k_h = 0.75 - 0.00025*(h_0 - 300)
#else
    k_h = 0.70
#end if
α_ds1 = 4'- for cement Class N
α_ds2 = 0.12'- for cement Class N
'<p class = "ref">(eq. B.12)</p>
β_RH = 1.55*(1 - (RH/100)^3)
'Basic drying shrinkage strain
'<p class = "ref">(eq. B.11)</p>
ε_cd0 = 0.85*(220 + 110*α_ds1)*e^(-α_ds2*f_cm/10)*10^-6*β_RH
'<p class = "ref">[EN 1992-1-1 §3.1.4(6)]</p>
'Drying shrinkage strain in time
'<p class = "ref">(eq. 3.9)</p>
ε_cd = k_h*ε_cd0
'Autogenous shrinkage strain
'<p class = "ref">(eq. 3.11)</p>
ε_ca = 2.5*(f_ck - 10)*10^(-6)
'Total shrinkage strain
'<p class = "ref">(eq. 3.8)</p>
ε_cs = ε_cd + ε_ca
'</div>
'
'<div class="fold">
'<h4>Cross section properties</h4>
'Total reinforcement area -'A_s = A_s1 + A_s2'mm²
'Concrete area -'A_c = b*h'mm²
'Effective section area
A_red = A_c + α_e*A_s'mm²
'Section modulus about the bottom edge
S_c = b*h^2/2'mm<sup>3</sup>
'Effective section modulus about the bottom edge
S_red = S_c + α_e*(A_s1*d_1 + A_s2*(h - d_2))'mm<sup>3</sup>
'Effective depth of cross section center
z_c = S_red/A_red'mm
'Second moment of area of the concrete section
I_c = b*h^3/12 + A_c*(z_c - S_c/A_c)^2'mm<sup>4</sup>
'Second moment of area for uncracked section
I_red = I_c + α_e*(A_s1*(z_c - d_1)^2 + A_s2*(h - z_c - d_2)^2)'mm<sup>4</sup>
'Depth of neutral axis for cracked section
x = (sqr(A_s^2 + 2*b*(d*A_s1 + d_2*A_s2)/α_e) - A_s)*α_e/b'mm
'Second moment of area for cracked section
I_red_II = b*x^3/3 + α_e*(A_s1*(d - x)^2 + A_s2*(x - d_2)^2)'mm<sup>4</sup>
'</div>
'
'<div class="fold">
'<h4>Control of concrete stress</h4>
'<p><b>Characteristic combination</b></p>
'Stress at the bottom edge of the section
σ_ct = M_k*10^6/I_red*z_c'MPa
'Check for crack opening
#if σ_ct ≤ f_ctm
    σ_ct'MPa &lе;'f_ctm'MPa - section is uncracked
    'Concrete stress (uncracked section)
    σ_c = M_k*10^6/I_red*(h - z_c)'MPa
    'Reinforcement stress (uncracked section)
    σ_s1 = α_e*M_k*10^6/I_red*(z_c - d_1)'MPa
    σ_s2 = -α_e*M_k*10^6/I_red*(h - z_c - d_2)'MPa
#else
    σ_ct'MPa &gt;'f_ctm'MPa - section is cracked
    'Concrete stress (cracked section)
    σ_c = M_k*10^6/I_red_II*x'MPa
    'Reinforcement stress (cracked section)
    σ_s1 = α_e*M_k*10^6/I_red_II*(d - x)'MPa
    σ_s2 = -α_e*M_k*10^6/I_red_II*(x - d_2)'MPa
#end if
'<p><b>Stress limitation</b></p>
'<p class = "ref">[EN 1992-1-1 §7.2(2)]</p>
'Concrete stress,'k_1 = 0.60
#if σ_c > k_1*f_ck
    σ_c'MPa &gt;'k_1*f_ck'MPa - <span class="err">The condition is NOT satisfied!</span>
#else
    σ_c'MPa &le;'k_1*f_ck'MPa - The condition is satisfied!
#end if
'<p class = "ref">[EN 1992-1-1 §7.2(5)]</p>
'Reinforcement stress,'k_3 = 0.80
#if σ_s1 > k_3*f_yk
    σ_s1'MPa &gt;'k_3*f_yk'MPa - <span class="err">The condition is NOT satisfied!</span>
#else
    σ_s1'MPa &le;'k_3*f_yk'MPa - The condition is satisfied!
#end if
'<p><b>Quasi-permanent combination</b></p>
#if σ_ct ≤ f_ctm
    'Concrete stress for uncracked section
    σ_c = M_qp*10^6/I_red*(h - z_c)'MPa
#else
    'Concrete stress for cracked section
    σ_c = M_qp*10^6/I_red_II*x'MPa
#end if
'<p class = "ref">[EN 1992-1-1 §7.2(3)]</p>
'Stress limitation,'k_2 = 0.45
#if σ_c > k_2*f_ck
    σ_c'MPa &gt;'k_2*f_ck'MPa - <span class="err">The condition is NOT satisfied!</span>
    '<p class = "ref">[EN 1992-1-1 §3.1.4(4)]</p>
    'Non-linear creep must be considered!
    k_σ = σ_c/f_ck
    'Non-linear notional creep coefficient
    φ_k_8_t0 = φ_8_t0*e^(1.5*(k_σ - 0.45))
    'Effective concrete modulus of elasticity
    E_c_eff = E_cm/(1 + φ_k_8_t0)'MPa
#else
    σ_c'MPa &le;'k_2*f_ck'MPa - The condition is satisfied!
    'Non-linear creep is NOT calculated!
#end if
'</div>
'
'<div class="fold">
'<h4>Control of cracks</h4>
'<p class = "ref">[EN 1992-1-1 §7.3.2(2)]</p>
'<p><b>Minimum reinforcement area</b></p>
'Allowable reinforcement stress -'σ_s = f_yk'MPa
'Approximate value of axial tensile strength -'f_ct_eff = f_ctm'MPa
'Area of tensile zone before opening of first crack
A_ct = b*z_c'mm²
#if h ≤ 300
    k = 1'- for h ≤ 300 mm
#else if h ≥ 800
    k = 0.65'- for h ≥ 800 mm
#else
    k = 1 - 0.35*(h - 300)/500'- for 300 mm &lt; h &lt; 800 mm
#end if
k_c = 0.4
#if h ≤ 1000
    h_1 = h'- for h ≤ 1000 mm
#else
    h_1 = 1000'- for h &gt; 1000 mm
#end if
'Minimum reinforcement
A_s_min = k_c*k*f_ct_eff*A_ct/σ_s'mm²
#if A_s1 < A_s_min
    A_s1'mm² &lt;'A_s_min'mm² - <span class="err">The condition is NOT satisfied!</span>
#else
    A_s1'mm² &ge;'A_s_min'mm² - The condition is satisfied!
#end if
'<p><b>Calculation of crack widths</b></p>
'<p class = "ref">[EN 1992-1-1 §7.3.2(3)]</p>
'Depth of the effective area
h_c_eff = min(2.5*(h - d);min(h - x/3;h/2))'mm
'Concrete effective area within tensile zone
A_c_eff = b*h_c_eff'mm²
ρ_p_eff = A_s1/A_c_eff
'<p class = "ref">[EN 1992-1-1 §7.3.4(3)]</p>
#if s > 5*(c + Φ_1/2)
    'For's'mm &gt;'5*(c + Φ_1/2)'mm:
    'Maximum final crack spacing -'s_r_max = 1.3*(h - x)'mm
#else
    'For's'mm &le;'5*(c + Φ_1/2)'mm:
    k_1 = 0.8'- for high bond bars
    k_2 = 0.5'- for bending
    k_3 = 3.4','k_4 = 0.425
    'Maximum final crack spacing
    s_r_max = k_3*c + k_1*k_2*k_4*Φ_1/ρ_p_eff'mm
#end if
'The check is performed for quasi-permanent combination
'Reinforcement stress
σ_s = α_e*M_qp*10^6/I_red_II*(d - x)'MPa
'Long-term load factor -'k_t = 0.4
'<p class = "ref">[EN 1992-1-1 §7.3.4(2)]</p>
'Difference between mean values of concrete and reinforcement strains
'<p><i>ε</i><sub>sm</sub> - <i>ε</i><sub>cm</sub> ='Δε = max((σ_s - k_t*f_ct_eff/ρ_p_eff*(1 + α*ρ_p_eff));0.6*σ_s)/(E_s*10^3)'</p>
'<p class = "ref">[EN 1992-1-1 §7.3.4(1)]</p>
'Crack widths
w_k = s_r_max*Δε'mm
'<p class = "ref">[EN 1992-1-1 §7.3.1(5)]</p>
'Limiting crack width value
'<p class = "ref">[EN 1992-1-1, Table NA.4]</p>
#if w_max ≡ 0.4
    w_max'mm - for exposure classes X0, XC1
#else
    w_max'mm - for exposure classes XC2, XC3, XC4, XD1, XD2, XS2, XS3
#end if
'Crack width limitation
'</div>
#if w_k > w_max
    w_k'mm &gt;'w_max'mm - <span class="err">The condition is NOT satisfied!</span>
#else
    w_k'mm &le;'w_max'mm - The condition is satisfied!
#end if
'
'<div class="fold">
'<h4>Control of deflections</h4>
'Crack opening moment
#if σ_ct ≤ f_ctm
    'Distribution coefficient
    '<p class = "ref">[EN 1992-1-1 §7.4.3(3)]</p>
    ζ = 0'- for uncracked section
#else
    M_cr = f_ctm*I_red/z_c*10^-6'kNm
    'Coefficient for duration of loads
    β = 0.5'- for long-term loads
    '<p class = "ref">[EN 1992-1-1 §7.4.3(3)]</p>
    'Distribution coefficient
    ζ = 1 - β*(M_cr/M_qp)^2'- for cracked section
#end if
'<p><b>Calculation of curvature</b></p>
'First moment of reinforcement area about centroid of uncracked section
S = A_s1*(z_c - d_1) - A_s2*(h - z_c - d_2)'mm<sup>3</sup>
'Curvature due to shrinkage of uncracked section
'<p class = "ref">[EN 1992-1-1 §7.4.3(6)]</p>
θ_cs_I = ε_cs*α_e*S/I_red
'Curvature of uncracked section
θ_I = M_qp/(E_c_eff*I_red)*10^3 + θ_cs_I
'First moment of reinforcement area about centroid of cracked section
S = A_s1*(d - x) - A_s2*(x - d_2)'mm<sup>3</sup>
'Curvature due to shrinkage of cracked section
'<p class = "ref">[EN 1992-1-1 §7.4.3(6)]</p>
θ_cs_II = ε_cs*α_e*S/I_red_II
'Curvature of cracked section
θ_II = M_qp/(E_c_eff*I_red_II)*10^3 + θ_cs_II
'Total curvature
'<p class = "ref">[EN 1992-1-1 §7.4.3(3)]</p>
θ = ζ*θ_II + (1 - ζ)*θ_I
'Factor considering the static scheme
#if type ≡ 1
    k = 1/3'- for cantilever with concentrated load
#else if type ≡ 2
    k = 1/4'- for cantilever with uniformly distributed load
#else if type ≡ 3
    k = 1/12'- for simply supported beam with concentrated load
#else if type ≡ 0
    k = 5/48'- for simply supported beam with uniformly distributed load
#end if
'Deflection due to quasi-permanent combination -'δ = k*θ*L^2*10^6'mm
'<p class = "ref">[EN 1992-1-1 §7.4.1(4)]</p>
#if (type ≡ 1) + (type ≡ 2)
    'Maximum deflection -'δ_max = L*10^3/125'mm
#else
    'Maximum deflection -'δ_max = L*10^3/250'mm
#end if
'Deflection check:
'</div>
#if δ > δ_max
    δ'mm &gt;'δ_max'mm - <span class="err">The condition is NOT satisfied!</span>
#else
    δ'mm &le;'δ_max'mm - The condition is satisfied!
#end if
#show
'</div>
#pre
'<script>if(window.jQuery){$(".flip_trigger").change(function(){$(".flip").hide();$(".flip_"+$(this).val()).show();});$(".flip").hide();$(".flip_"+$(".flip_base input").val()).show();}</script>0    7   0.4 200 150 300 650 30  4   20  45  74  2   14  45  20  500 50

SLS Design of Beam with Tee Section¶

Serviceability checks for a T-beam: same stress, crack-width and deflection verifications as the rectangular variant, with the cracked-section second moment of area derived for both flange-in-compression and web-in-compression cases.

Code:

'<small>According to <strong>Eurocode</strong>: EN 1992-1-1</small>
'<div style="max-width:180mm">
'<h4>Static scheme</h4>
'<p id="type" class="flip_base" style="display:none">'type = ?'</p>
#pre
'<p><select data-target="type" class="flip_trigger">
'<option value="1">Cantilever with concentrated load</option>
'<option value="2">Cantilever with uniformly distributed load</option>
'<option value="3">Simply supported beam with concentrated load</option>
'<option value="0">Simply supported beam with uniformly distributed load</option>
'</select></p>
#hide
#if type ≡ 1
    #show
    '<img class="side flip flip_1" src="../../Images/structures/rc/design/deflection-cantilever-force.png" alt="deflection-cantilever-force.png" style="width:150pt;">
    #post
    'Cantilever with concentrated load
    #hide
#else if type ≡ 2
    #show
    '<img class="side flip flip_2" src="../../Images/structures/rc/design/deflection-cantilever-uniform.png" alt="deflection-cantilever-uniform.png" style="width:150pt;">
    #post
    'Cantilever with uniformly distributed load
    #hide
#else if type ≡ 3
    #show
    '<img class="side flip flip_3" src="../../Images/structures/rc/design/deflection-beam-force.png" alt="deflection-beam-force.png" style="width:150pt;">
    #post
    'Simply supported beam with concentrated load
    #hide
#else if type ≡ 0
    #show
    '<img class="side flip flip_0" src="../../Images/structures/rc/design/deflection-beam-uniform.png" alt="deflection-beam-uniform.png" style="width:150pt;">
    #post
    'Simply supported beam with uniformly distributed load
    #hide
#end if
#show
'Beam length -'L = ?'m
'Exposure class:
'<p id="w_max" style="display:none">'w_max = ?'</p>
#pre
'<p><select data-target="w_max">
'<option value="0.4">X0 or XC1</option>
'<option value="0.3">XC2, XC3, XC4, XD1, XD2, XS2 или XS3</option>
'</select></p>
#post
#if w_max ≡ 0.4
    'X0 or XC1
#else
    'XC2, XC3, XC4, XD1, XD2, XS2 или XS3
#end if
#show
'
'<h4>Bending moments</h4>
'Characteristic combination (<i>g</i> + <i>q</i>) -'M_k = ?'kNm
'Quasi-permanent combination (<i>g</i> + <i>ψ</i><sub>2</sub><i>q</i>) -'M_qp = ?'kNm
'
'<img class="side" src="../../Images/structures/rc/design/beam-tsection.png" alt="beam-tsection.png" style="width:150pt;">
'<h4>Cross section dimensions</h4>
'Stem:'b = ?'mm,'h = ?'mm
'Flange:'b_f = ?'mm,'h_f = ?'mm
'Concrete cover -'c = ?'mm (to bars surface)
'<p><b>Tension reinforcement</b></p>
'Bar count -'n_1 = ?'with diameter -'Φ_1 = ?'mm
#post
'Reinforcement area -'A_s1 = n_1*π*Φ_1^2/4'mm²
#show
'Concrete cover to the center of reinforcement -'d_1 = ?'mm
#post
'Effective cross section depth -'d = h - d_1'mm
#show
'Spacing between bar centers -'s = ?'mm
'<p><b>Compression reinforcement</b></p>
'Bar count -'n_2 = ?'with diameter -'Φ_2 = ?'mm
#post
'Reinforcement area -'A_s2 = n_2*π*Φ_2^2/4'mm²
#show
'Concrete cover to the center of reinforcement -'d_2 = ?'mm
'
'<h4>Material properties</h4>
'<p><b>Concrete</b> [EN 1992-1-1, Table 3.1]</p>
'Characteristic compressive cylinder strength -'f_ck = ?'MPa
#post
'Mean value of cylinder compressive strength -'f_cm = f_ck + 8'MPa
'Mean value of axial tensile strength -'f_ctm = 0.30*f_ck^(2/3)'MPa
'Secant modulus of elasticity -'E_cm = 22*(f_cm/10)^0.3'GPa
#show
'<p><b>Steel</b></p>
'Characteristic yield strength -'f_yk = ?'MPa
#post
'Modulus of elasticity -'E_s = 200'GPa
'Ratio of steel to concrete moduli of elasticity -'α = E_s/E_cm
'
'<div class="fold">
#show
'<h4>Concrete creep and shrinkage</h4>
'<p class = "ref">[EN 1992-1-1 B.1(1)]</p>
'<p><b>Creep</b></p>
'Relative humidity of the environment -'RH = ?'%
#post
'Perimeter of cross section in contact with the atmosphere
u = 2*(b + h)'mm
'Cross section area -'A_c = b*h'mm²
'Notional size of the cross section
'<p class = "ref">(eq. B.6)</p>
h_0 = 2*A_c/u'mm
'Relative humidity factor
#if f_cm ≤ 35
    '<p class = "ref">(eq. B.3a)</p>
    φ_RH = (1 + (1 - RH/100)/(0.1*h_0^0.333))
#else
    '<p class = "ref">(eq. B.8c)</p>
    α_1 = (35/f_cm)^0.7','α_2 = (35/f_cm)^0.2
    '<p class = "ref">(eq. B.3b)</p>
    φ_RH = (1 + (1 - RH/100)/(0.1*h_0^0.333)*α_1)*α_2
#end if
'Concrete strength factor
'<p class = "ref">(eq. B.4)</p>
β_f_cm = 16.8/sqr(f_cm)
'Concrete age at the moment of loading -'t_0 = 28'дни
'Concrete age factor
'<p class = "ref">(eq. B.5)</p>
β_t0 = 1/(0.1 + t_0^0.20)
'Notional creep coefficient
'<p class = "ref">(eq. B.2)</p>
φ_8_t0 = φ_RH*β_f_cm*β_t0
'Effective concrete modulus of elasticity
'<p class = "ref">[EN 1992-1-1 §7.4.3(5)]</p>
E_c_eff = E_cm/(1 + φ_8_t0)'GPa
'Effective ratio of modules of elasticity
'<p class = "ref">[EN 1992-1-1 §7.4.3(6)]</p>
α_e = E_s/E_c_eff
'<p class = "ref">[EN 1992-1-1 B.2(1)]</p>
'<p><b>Shrinkage</b></p>
'Coefficient depending on the notional size
#if h_0 ≤ 100
    k_h = 1.00
#else if h_0 ≤ 200
    k_h = 1.00 - 0.0015*(h_0 - 100)
#else if h_0 ≤ 300
    k_h = 0.85 - 0.001*(h_0 - 200)
#else if h_0 ≤ 500
    k_h = 0.75 - 0.00025*(h_0 - 300)
#else
    k_h = 0.70
#end if
α_ds1 = 4'- for cement Class N
α_ds2 = 0.12'- for cement Class N
'<p class = "ref">(eq. B.12)</p>
β_RH = 1.55*(1 - (RH/100)^3)
'Basic drying shrinkage strain
'<p class = "ref">(eq. B.11)</p>
ε_cd0 = 0.85*(220 + 110*α_ds1)*e^(-α_ds2*f_cm/10)*10^-6*β_RH
'<p class = "ref">[EN 1992-1-1 §3.1.4(6)]</p>
'Drying shrinkage strain in time
'<p class = "ref">(eq. 3.9)</p>
ε_cd = k_h*ε_cd0
'Autogenous shrinkage strain
'<p class = "ref">(eq. 3.11)</p>
ε_ca = 2.5*(f_ck - 10)*10^(-6)
'Total shrinkage strain
'<p class = "ref">(eq. 3.8)</p>
ε_cs = ε_cd + ε_ca
'</div>
'
'<div class="fold">
'<h4>Cross section properties</h4>
'Total reinforcement area -'A_s = A_s1 + A_s2'mm²
'Flange area -'A_f = (b_f - b)*h_f'mm²
'T-section area -'A_c = b*h + A_f'mm²
'Effective section area
A_red = A_c + α_e*A_s'mm²
'Section modulus about the bottom edge
S_c = b*h^2/2 + A_f*(h - h_f/2)'mm<sup>3</sup>
'Effective section modulus about the bottom edge
S_red = S_c + α_e*(A_s1*d_1 + A_s2*(h - d_2))'mm<sup>3</sup>
'Effective depth of cross section center
z_c = S_red/A_red'mm
'Second moment of area of the concrete section
I_c = (b*h^3 + (b_f - b)*h_f^3)/12 + b*h*(z_c - h/2)^2 + A_f*((h - z_c - h_f/2)^2)'mm<sup>4</sup>
'Effective second moment of area for uncracked section
I_red = I_c + α_e*(A_s1*(z_c - d_1)^2 + A_s2*(h - z_c - d_2)^2)'mm<sup>4</sup>
'Uncracked section properties for SLS
'Equivalent bending moment and axial force due to concrete stress about the neutral axis
N_c(x) = b_f*x^2/2 - (x - h_f)^2*(b_f - b)/2*(x > h_f)
'Equivalent axial force due to internal stress
N_bal(x) = N_c(x) + α_e*(A_s2*(x - d_2) - A_s1*(d - x))
'Depth of neutral axis
x = $Root{N_bal(x) @ x = 0 : h}'mm
'Second moment of area for cracked section
#if x > h_f
    I_red_II = b_f*x^3/3 - (x - h_f)^3*(b_f - b)/3 + α_e*(A_s1*(d - x)^2 + A_s2*(x - d_2)^2)'mm<sup>4</sup>
#else
    I_red_II = b_f*x^3/3 + α_e*(A_s1*(d - x)^2 + A_s2*(x - d_2)^2)'mm<sup>4</sup>
#end if
'</div>
'
'<div class="fold">
'<h4>Control of concrete stress</h4>
'<p><b>Characteristic combination</b></p>
'Stress at the bottom edge of the section
σ_ct = M_k*10^6/I_red*z_c'MPa
'Check for crack opening
#if σ_ct ≤ f_ctm
    σ_ct'MPa &lе;'f_ctm'MPa - section is uncracked
    'Concrete stress (uncracked section)
    σ_c = M_k*10^6/I_red*(h - z_c)'MPa
    'Reinforcement stress (uncracked section)
    σ_s1 = α_e*M_k*10^6/I_red*(z_c - d_1)'MPa
    σ_s2 = -α_e*M_k*10^6/I_red*(h - z_c - d_2)'MPa
#else
    σ_ct'MPa &gt;'f_ctm'MPa - section is cracked
    'Concrete stress (cracked section)
    σ_c = M_k*10^6/I_red_II*x'MPa
    'Reinforcement stress (cracked section)
    σ_s1 = α_e*M_k*10^6/I_red_II*(d - x)'MPa
    σ_s2 = -α_e*M_k*10^6/I_red_II*(x - d_2)'MPa
#end if
'<p><b>Stress limitation</b></p>
'<p class = "ref">[EN 1992-1-1 §7.2(2)]</p>
'Concrete stress,'k_1 = 0.60
#if σ_c > k_1*f_ck
    σ_c'MPa &gt;'k_1*f_ck'MPa - <span class="err">The condition is NOT satisfied!</span>
#else
    σ_c'MPa &le;'k_1*f_ck'MPa - The condition is satisfied!
#end if
'<p class = "ref">[EN 1992-1-1 §7.2(5)]</p>
'Reinforcement stress,'k_3 = 0.80
#if σ_s1 > k_3*f_yk
    σ_s1'MPa &gt;'k_3*f_yk'MPa - <span class="err">The condition is NOT satisfied!</span>
#else
    σ_s1'MPa &le;'k_3*f_yk'MPa - The condition is satisfied
#end if
'<p><b>Quasi-permanent combination</b></p>
#if σ_ct ≤ f_ctm
    'Concrete stress for uncracked section
    σ_c = M_qp*10^6/I_red*(h - z_c)'MPa
#else
    'Concrete stress for cracked section
    σ_c = M_qp*10^6/I_red_II*x'MPa
#end if
'<p class = "ref">[EN 1992-1-1 §7.2(3)]</p>
'Stress limitation,'k_2 = 0.45
#if σ_c > k_2*f_ck
    σ_c'MPa &gt;'k_2*f_ck'MPa - <span class="err">The condition is NOT satisfied!</span>
    '<p class = "ref">[EN 1992-1-1 §3.1.4(4)]</p>
    'Non-linear creep must be considered!
    k_σ = σ_c/f_ck
    'Non-linear notional creep coefficient
    φ_k_8_t0 = φ_8_t0*e^(1.5*(k_σ - 0.45))
    'Effective concrete modulus of elasticity
    E_c_eff = E_cm/(1 + φ_k_8_t0)'MPa
#else
    σ_c'MPa &le;'k_2*f_ck'MPa - The condition is satisfied!
    'Non-linear creep is NOT calculated!
#end if
'</div>
'
'<div class="fold">
'<h4>Control of cracks</h4>
'<p class = "ref">[EN 1992-1-1 §7.3.2(2)]</p>
'<p><b>Minimum reinforcement area</b></p>
'Allowable reinforcement stress -'σ_s = f_yk'MPa
'Approximate value of axial tensile strength -'f_ct_eff = f_ctm'MPa
'Area of tensile zone before opening of first crack
A_ct = b*z_c'mm²
#if h ≤ 300
    k = 1'- for h ≤ 300
#else if h ≥ 800
    k = 0.65'- for h ≥ 800
#else
    k = 1 - 0.35*(h - 300)/500'- for 300 &lt; h &lt; 800
#end if
    k_c = 0.4
#if h ≤ 1000
    h_1 = h'- for h ≤ 1000
#else
    h_1 = 1000'- for h &gt; 1000
#end if
'Minimum reinforcement
A_s_min = k_c*k*f_ct_eff*A_ct/σ_s'mm²
#if A_s1 < A_s_min
    A_s1'mm² &lt;'A_s_min'mm² - <span class="err">The condition is NOT satisfied!</span>
#else
    A_s1'mm² &ge;'A_s_min'mm² - The condition is satisfied!
#end if
'<p><b>Calculation of crack widths</b></p>
'<p class = "ref">[EN 1992-1-1 §7.3.2(3)]</p>
'Depth of the effective area
h_c_eff = min(2.5*(h - d);min(h - x/3;h/2))'mm
'Concrete effective area within tensile zone
A_c_eff = b*h_c_eff'mm²
ρ_p_eff = A_s1/A_c_eff
'<p class = "ref">[EN 1992-1-1 §7.3.4(3)]</p>
#if s > 5*(c + Φ_1/2)
    'For's'mm &gt;'5*(c + Φ_1/2)'mm:
    'Maximum final crack spacing -'s_r_max = 1.3*(h - x)'mm
#else
    'For's'mm &le;'5*(c + Φ_1/2)'mm:
    k_1 = 0.8'- for high bond bars
    k_2 = 0.5'- for bending
    k_3 = 3.4','k_4 = 0.425
    'Maximum final crack spacing
    s_r_max = k_3*c + k_1*k_2*k_4*Φ_1/ρ_p_eff'mm
#end if
'The check is performed for quasi-permanent combination
'Reinforcement stress
σ_s = α_e*M_qp*10^6/I_red_II*(d - x)'MPa
'Long-term load factor -'k_t = 0.4
'<p class = "ref">[EN 1992-1-1 §7.3.4(2)]</p>
'Difference between mean values of concrete and reinforcement strains
'<p><i>ε</i><sub>sm</sub> - <i>ε</i><sub>cm</sub> ='Δε = max((σ_s - k_t*f_ct_eff/ρ_p_eff*(1 + α*ρ_p_eff));0.6*σ_s)/(E_s*10^3)'</p>
'<p class = "ref">[EN 1992-1-1 §7.3.4(1)]</p>
'Crack widths
w_k = s_r_max*Δε'mm
'<p class = "ref">[EN 1992-1-1 §7.3.1(5)]</p>
'Limiting crack width value
'<p class = "ref">[EN 1992-1-1, Table NA.4]</p>
#if w_max ≡ 0.4
    w_max'mm - for exposure classes X0, XC1
#else
    w_max'mm - for exposure classes XC2, XC3, XC4, XD1, XD2, XS2, XS3
#end if
'Crack width limitation
'</div>
#if w_k > w_max
    w_k'mm &gt;'w_max'mm - <span class="err">The condition is NOT satisfied!</span>
#else
    w_k'mm &le;'w_max'mm - The condition is satisfied!
#end if
'
'<div class="fold">
'<h4>Control of deflections</h4>
'Crack opening moment
#if σ_ct ≤ f_ctm
    'Distribution coefficient
    '<p class = "ref">[EN 1992-1-1 §7.4.3(3)]</p>
    ζ = 0'- for uncracked section
#else
    M_cr = f_ctm*I_red/z_c*10^-6'kNm
    'Coefficient for duration of loads
    β = 0.5'- for long-term loads
    '<p class = "ref">[EN 1992-1-1 §7.4.3(3)]</p>
    'Distribution coefficient
    ζ = 1 - β*(M_cr/M_qp)^2'- for cracked section
#end if
'<p><b>Calculation of curvature</b></p>
'First moment of reinforcement area about centroid of uncracked section
S = A_s1*(z_c - d_1) - A_s2*(h - z_c - d_2)'mm<sup>3</sup>
'Curvature due to shrinkage of uncracked section
'<p class = "ref">[EN 1992-1-1 §7.4.3(6)]</p>
θ_cs_I = ε_cs*α_e*S/I_red
'Curvature of uncracked section
θ_I = M_qp/(E_c_eff*I_red)*10^3 + θ_cs_I
'First moment of reinforcement area about centroid of cracked section
S = A_s1*(d - x) - A_s2*(x - d_2)'mm<sup>3</sup>
'Curvature due to shrinkage of cracked section
'<p class = "ref">[EN 1992-1-1 §7.4.3(6)]</p>
θ_cs_II = ε_cs*α_e*S/I_red_II
'Curvature of cracked section
θ_II = M_qp/(E_c_eff*I_red_II)*10^3 + θ_cs_II
'Total curvature
'<p class = "ref">[EN 1992-1-1 §7.4.3(3)]</p>
θ = ζ*θ_II + (1 - ζ)*θ_I
'Factor considering the static scheme
#if type ≡ 1
    k = 1/3'- for cantilever with concentrated load
#else if type ≡ 2
    k = 1/4'- for cantilever with uniformly distributed load
#else if type ≡ 3
    k = 1/12'- for simply supported beam with concentrated load
#else if type ≡ 0
    k = 5/48'- for simply supported beam with uniformly distributed load
#end if
'Deflection due to quasi-permanent combination -'δ = k*θ*L^2*10^6'mm
'<p class = "ref">[EN 1992-1-1 §7.4.1(4)]</p>
#if (type ≡ 1) + (type ≡ 2)
    'Maximum deflection -'δ_max = L*10^3/125'mm
#else
    'Maximum deflection -'δ_max = L*10^3/250'mm
#end if
'Deflection check:
'</div>
#if δ > δ_max
    δ'mm &gt;'δ_max'mm - <span class="err">The condition is NOT satisfied!</span>
#else
    δ'mm &le;'δ_max'mm - The condition is satisfied!
#end if
#show
'</div>
#pre
'<script>if(window.jQuery){$(".flip_trigger").change(function(){$(".flip").hide();$(".flip_"+$(this).val()).show();});$(".flip").hide();$(".flip_"+$(".flip_base input").val()).show();}</script>0    5.75    0.4 200 157 250 550 2400    140 30  4   20  45  50  2   8   45  25  500 60

Shear Design of Rectangular Section¶

Shear design of a rectangular section under EN 1992-1-1: shear strength without links \(V_{Rd,c}\), required link area for a strut angle \(\theta\) and check of the maximum strut compression.

Code:

'<small>According to <strong>Eurocode</strong>: EN 1992-1-1</small>
'<div style="max-width:180mm;">
'<img class="side" style="width:270pt;" src="../../Images/structures/rc/design/beam-shear.png" alt="beam-shear.png">
'<h4>Cross section dimensions</h4>
'Width -'b = ?'mm, Height'h = ?'mm
#post
'Cross section area -'A_c = b*h'mm²
#show
'Tensile reinforcement area -'A_sl = ?'mm²
'Effective depth -'d = h - ?'mm
#post
'Reinforcement ratio -'ρ_l = min(A_sl/(b*d); 0.02)
#show
'Shear links diameter -'d_w = ?'mm
'Number of legs for one link -'n_w = ?
'<p><b>Section design loads</b></p>
'Shear force -'V_Ed = ?'kN&nbsp;&emsp;Axial force -'N_Ed = ?'kN
'<h4>Material properties</h4>
'<p class="ref">[EN 1992-1-1, Table 3.1]</p>
'<p><b>Concrete</b></p>
'Characteristic compressive cylinder strength -'f_ck = ?'MPa
'Partial safety factor for concrete -'γ_c = 1.5','α_cc = ?
#post
'Design compressive cylinder strength -'f_cd = α_cc*f_ck/γ_c'MPa
#show
'<p><b>Steel for shear reinforcement</b></p>
'Characteristic yield strength -'f_yk = ?'MPa
#post
'Partial safety factor for steel -'γ_s = 1.15
'Design yield strength -'f_ywd = f_yk/γ_s'MPa
'
'<div class="fold">
'<h4>Shear capacity without reinforcement<small class="ref">[EN 1992-1-1, § 6.2.2]</small></h4>
k = min(1 + sqr(200/d); 2)
'Stress due to axial force (N<sub>Ed</sub> > 0 for compression)
σ_cp_ = N_Ed*10^3/A_c'MPa
'Maximum stress value:'0.2*f_cd'MPa
σ_cp = min(σ_cp_;0.2*f_cd)'MPa
C_Rd_c = 0.18/γ_c
v_min = 0.035*k^(3/2)*sqr(f_ck)
k_1 = 0.15
'<p class="ref">(6.2a)</p>
'Shear resistance
V_Rd_c_ = (C_Rd_c*k*(100*ρ_l*f_ck)^(1/3) + k_1*σ_cp)*b*d*10^-3'kN
'Minimum shear resistance
'<p class="ref">(6.2b)</p>
V_Rd_c_min = (v_min + k_1*σ_cp)*b*d*10^-3'kN
V_Rd_c = max(V_Rd_c_min;V_Rd_c_)'kN
'Design check
'</div>
#if V_Ed ≤ V_Rd_c
    V_Ed'kN ≤'V_Rd_c'kN. Shear reinforcement is NOT required by calculation!
#else
    V_Ed'kN &gt;'V_Rd_c'kN. Shear reinforcement is required!
    '<div class="fold">
    '<h4>Shear capacity with reinforcement <small class="ref">[EN 1992-1-1, §6.2.3]</small></h4>
    'Lever arm of internal forces -'z = 0.9*d'mm
    '<p class="ref">(6.6N)</p>
    ν_1 = 0.6*(1 - f_ck/250)
    k_σ = σ_cp_/f_cd
    'Factor α<sub>cw</sub> depends on the value of k<sub>σ</sub>
    #if k_σ < 0.25
        α_cw = 1 + k_σ'for 0 &lt; <i>k</i><sub>σ</sub> &lt; 0.25
    #else if k_σ < 0.5
        α_cw = 1.25'for 0.25 &lt; <i>k</i><sub>σ</sub> &lt; 0.5
    #else if k_σ < 1
        α_cw = 2.5*(1 - k_σ)'for 0.5 &lt; <i>k</i><sub>σ</sub> &lt; 1
    #else
        '<p class="err">The value for σ<sub>cp</sub>is NOT admissible</p>
    #end if
    '<p class="ref">(6.9)</p>
    'Maximum shear capacity at θ = 45°
    V_Rd_max_45 = α_cw*b*z*ν_1*f_cd/2*10^-3'kN
    #if V_Ed ≤ V_Rd_max_45
        V_Ed'kN ≤'V_Rd_max_45'kN. The check is satisfied!
        'Angle of concrete struts
        θ_ = 0.5*asin(V_Ed/V_Rd_max_45)'°
        '<p class="ref">(6.7N)</p>
        'The angle is limited between 21,8° ≤ θ ≤ 45°(1 ≤ cot(θ) ≤ 2.5)
        θ = max(21.8;min(θ_;45))'°
        '<p class="ref">(6.9)</p>
        'Bearing capacity check at'θ'°
        V_Rd_max = α_cw*b*z*ν_1*f_cd/(cot(θ) + tan(θ))*10^-3'kN
        '</div>
        'Design check:'V_Ed'kN ≤'V_Rd_max'kN
        '<p class="ref">(6.8)</p>
        'Required shear reinforcement area
        A_sw = V_Ed*10^6/(z*f_ywd*cot(θ))'mm²/m'
        'Shear links with'n_w'legs and diameter -'d_w'mm are provided
        'Area of one leg -'A_s1 = π*(d_w/2)^2'mm²
        'Required links spacing -'s = 1000*n_w*A_s1/A_sw'mm
        '<p class="ref">[EN 1992-1-1, §9.2.2]</p>
        'Maximum spacing -'s_max = min(0.75*d;300)'mm
        #if s > s_max
            s'mm &gt;'s_max'mm. Spacing is greater than the maximum.
            'The relevant spacing is accepted:'s = s_max'mm
        #end if
        '<p class="ref">(9.4)</p>
        'Reinforcement ratio
        ρ_w = A_sw/(s*b)
        '<p class="ref">(9.5N)</p>
        ρ_w_min = 0.08*sqr(f_ck)/f_yk
        #if ρ_w < ρ_w_min
            ρ_w'&lt;'ρ_w_min'. Reinforcement ratio is lower than the minimum. Minimum reinforcement ratio is accepted.
        #end if
        '<p class="ref">(6.18)</p>
        'Additional force in tensile reinforcement
        ΔF_td = 0.5*V_Ed*cot(θ)'kN
        '<p class="ref">(9.2)</p>
        'Bending moment diagram shifting distance
        a_l = z*cot(θ)/2'mm
    #else
        '</div>
        V_Ed'kN &gt;'V_Rd_max_45'kN
        '<p class="err">The check is NOT satisfied!. Increase cross section dimensions or concrete grade.</p>
    #end if
#end if
#show
'</div>300  800 1200    50  8   2   600 0   20  1   500

Spotted an error? Edit these examples.

№ 1 -	`M`_{Ed_1} = 100 kN·m,	`N`_{Ed_1} = 500 kN
№ 2 -	`M`_{Ed_2} = 100 kN·m,	`N`_{Ed_2} = 1000 kN
№ 3 -	`M`_{Ed_3} = 100 kN·m,	`N`_{Ed_3} = 1500 kN
№ 4 -	`M`_{Ed_4} = 100 kN·m,	`N`_{Ed_4} = 2000 kN
№ 5 -	`M`_{Ed_5} = 100 kN·m,	`N`_{Ed_5} = 2500 kN

№	N_Ed kN		N_B kN	M_Ed kN·m	M_II kN·m		M_Rd kN·m
1	500	<	4959.97	100	124.19	≤	345.91
2	1000	<	5478.61	100	150.87	≤	407.57
3	1500	<	5997.25	100	180.03	≤	358.3
4	2000	<	6515.9	100	211.62	≤	275.04
5	2500	<	6561.5	100	255.79	>	156.97