Reinforced Concrete: Seismic Detailing¶

CalcpadCE worksheets in this section build the seismic detailing checks of reinforced concrete primary members of medium ductility class (DCM) according to Eurocode 8: the geometry of the critical regions, the minimum and maximum reinforcement ratios, the spacing of the transverse reinforcement and the confinement requirements at the plastic hinges.

The beam detailing sheet covers a rectangular beam with a slab flange and verifies the longitudinal ratio, the spacing of the closed stirrups inside the critical region of length \(l_{cr}\) and the lap-splice requirements for the bottom and top bars. The column detailing sheet checks the dimensions of the cross-section, the longitudinal ratio, the confinement of the boundary regions for the maximum seismic axial load and the minimum mechanical volumetric ratio of the hoops. The shear wall detailing sheet addresses the confined boundary elements at the wall ends, the web horizontal and vertical reinforcement and the curvature ductility verification at the base of the wall.

Beam Detailing for DCM¶

Eurocode 8 detailing of a primary seismic beam for medium ductility class: critical region length \(l_{cr}\), longitudinal ratio limits, spacing of closed stirrups and lap-splice requirements.

Code:

'<small>According to <strong>Eurocode</strong>: EN 1998-1</small>
'<div style="max-width:180mm">
'<img style="width:375pt;" style="max-width:100%;" src="../../Images/structures/rc/detailing/beam.png" alt="beam.png">
'<p><b>Beam dimensions</b></p>
'Cross section - 'b_w = ? {350}'mm, 'h_w = ? {650}'mm
'Clear beam span -'l_cl = ? {5000}'mm
'Slab depth -'h_f = ? {130}'mm
#post
'Cross section area -'A_c = b_w*h_w'mm²
#show
'Dimensions of columns -'b_c = ? {500}'mm,'h_c = ? {500}'mm
'Maximum seismic axial load in beam -'N_Ed = ? {0}'kN

'<p><b>Concrete</b></p>
'Characteristic compressive cylinder strength -'f_ck = ? {25}'MPa
'Partial safety factor -'γ_c = 1.5','α_ct = 1','α_cc = ? {1}
#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

'<p class="ref">[§ 5.1.2 (1)]</p>
'<p><b>Check for normalized axial load</b></p>
ν_d = N_Ed/(A_c*f_cd)*1000
#if ν_d > 0.1
    '<p class="err">'ν_d'&gt; 0.1. The check is NOT satisfied! ❌</p>
#else
    ν_d'≤ 0.1. The check is satisfied! ✓
#end if
'<p class="ref">[§ 5.4.1.2.1 (3)]</p>
'<p><b>Beam width check</b></p>
b_w,max = min(b_c + h_w;2*b_c)'mm
#if b_w ≤ b_w,max
    b_w'mm &le;'b_w,max'mm. The check is satisfied! ✓
#else
    '<p class="err">'b_w'mm &gt;'b_w,max'mm. The check is NOT satisfied! ❌</p>
#end if
'<p class="ref">[§ 5.4.3.1.1 (3)]</p>
'<p><b>Maximum effective flange width</b></p>
'<table class="bordered"><tr><th></th><th>Exterior column</th><th>Interior column</th></tr>
'<tr><th>In the absence of transverse beam</th><td>'b_c'mm</td><td>'b_c + 4*h_f'mm</td></tr>
'<tr><th>With transverse beam</th><td>'b_c + 4*h_f'mm</td><td>'b_c + 8*h_f'mm</td></tr></table>
#show
'<p><b>Longitudinal reinforcement</b></p>
'<p id="C" style="display:none;">'C = ? {0}'</p>
'Characteristic yield strength -'f_yk = ? {500}'MPa
#pre
'<p>Steel class - <select data-target="C">
'<option value="0"> Class B </option>
'<option value="1"> Class C </option>
'</select></p>
#post
#val
#if C ≡ 0
    'Selected steel class <b>B'f_yk'B</b>
#else if C ≡ 1
    'Selected steel class <b>B'f_yk'C</b>
#else
    '<p class="err">Invalid steel class!</p>
#end if
#equ
'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
#show
'<p><b>Bottom reinforcement in critical regions</b></p>
'Bar diameter -'d_bL1 = ? {16}'mm
'Effective cross section depth -'d_1 = ? {590}'mm
'Reinforcement area -'A_s1 = ? {1005}'mm²
#post
'Reinforcement ratio -'ρ_1 = A_s1/(b_w*d_1)
'<p class="ref">[§ 5.4.3.1.2 (5)]</p>
'Minimum reinforcement ratio
ρ_min = 0.5*f_ctm/f_yk
'Maximum reinforcement ratio -'ρ_max = 0.04
#if ρ_1 < ρ_min
    '<p class="err">The reinforcement ratio is less than the minimum:'ρ_1'&lt;'ρ_min'. ❌</p>
#else if ρ_1 > ρ_max
    '<p class="err">The reinforcement ratio is greater than the maximum:'ρ_1'&gt;'ρ_max'. ❌</p>
#else
    'Design check:'ρ_min'≤'ρ_1'≤'ρ_max'. The check is satisfied! ✓
#end if
'<p><b>Design anchorage length</b></p>
η_1 = 1 'when good conditions are obtained
#if d_bL1 ≤ 32
    η_2 = 1'- for'd_bL1'≤ 32 mm
#else
    η_2 = (132 - d_bL1)/100'- for'd_bL1'> 32 mm
#end if
f_ctd = α_ct*f_ctk,005/γ_c'MPa
'<p class="ref">[EN 1992-1-1, § 8.4.2 (2)]</p>
f_bd = 2.25*η_1*η_2*f_ctd'MPa
σ_sd = f_yd'MPa
'<p class="ref">[EN 1992-1-1, § 8.4.3 (2)]</p>
l_b,rqd = d_bL1/4*(σ_sd/f_bd)'mm
'<p class="ref">[EN 1992-1-1, Table 8.2]</p>
α_1 = 1','α_2 = 1','α_3 = 1','α_4 = 1','α_5 = 1
'<p class="ref">[EN 1992-1-1, § 8.4.4 (1)]</p>
l_bd_ = α_1*α_2*α_3*α_4*α_5*l_b,rqd'mm
l_b,min = max(0.6*l_b,rqd;10*d_bL1;100)'mm
l_bd = round(max(l_bd_;l_b,min))'mm
#show
'<p><b>Top reinforcement at supports</b></p>
'Bar diameter -'d_bL2 = ? {20}'mm
'Effective cross section depth -'d_2 = ? {590}'mm
'Reinforcement area -'A_s2 = ? {1571}'mm²
#post
'Reinforcement ratio -'ρ_2 = A_s2/(b_w*d_2)
#show
'Fundamental period of first vibration mode -'T_1 = ? {0.55}'s
'Upper limit period of constant spectral acceleration -'T_C = ? {0.60}'s
'Basic behavior factor value -'q_0 = ? {3.9}
#post
'Curvature ductility factor
'<p class="ref">[§ 5.2.3.4 (3)]</p>
#if T_1 ≥ T_C
    μ_Φ = 1.0*(2*q_0 - 1)'- for T<sub>1</sub> ≥ T<sub>C</sub>
#else
    μ_Φ = 1.0*(1 + 2*(q_0 - 1)*(T_C/T_1))'- for T<sub>1</sub> &lt; T<sub>C</sub>
#end if
#if C ≡ 0
    '<!--'μ_Φ = 1.5*μ_Φ'-->
    '<p class="ref">[§ 5.2.3.4 (4)]</p>
    'For steel class B, ductility factor is increased by 05% -'μ_Φ
#end if
'Design value of steel yield strain -'ε_sy,d = f_yd/E_s
'Maximum reinforcement ratio at critical regions
ρ_max = ρ_1 + 0.0018*f_cd/(μ_Φ*ε_sy,d*f_yd)
#if ρ_2 < ρ_min
    '<p class="err">Reinforcement ratio is less than minimum:'ρ_2'&lt;'ρ_min'. ❌</p>
#else if ρ_2 > ρ_max
    '<p class="err">Reinforcement ratio is greater than maximum:'ρ_2'&gt;'ρ_max'. ❌</p>
#else
    'Design check:'ρ_min'≤'ρ_2'≤'ρ_max'. The check is satisfied! ✓
#end if
'<p><b>Design anchorage length</b></p>
#hide
#if d_bL2 ≤ 32
    η_2 = 1
#else
    η_2 = (132 - d_bL2)/100
#end if
f_bd = 2.25*η_1*η_2*f_ctd
l_b,rqd = d_bL2/4*σ_sd/f_bd
l_bd_ = α_1*α_2*α_3*α_4*α_5*l_b,rqd
l_b,min = max(0.6*l_b,rqd;10*d_bL2;100)
#post
l_bd = round(max(l_bd_;l_b,min))'mm
'<p><b>Limit of bar diameter to column size</b></p>
'Model uncertainty factor -'γ_Rd = 1.0
'Factor reflecting the ductility class -'k_D = 2/3
'Maximum diameter of longitudinal reinforcement 'd_bL = max(d_bL1;d_bL2)'mm
#show
'<p class="ref">[§ 5.6.2.2 (2), a)]</p>
'<p><b>For interior beam-column joints</b></p>
'Minimal normalized design axial force at column -'ν_d = ? {0.14}'(<var>ν</var><sub>d</sub> = <var>N</var><sub>Еd</sub> / <var>f</var><sub>cd</sub><var>A</var><sub>c</sub>)
#post
d_bL,max = h_c*7.5*f_ctm/(γ_Rd*f_yd)*(1 + 0.8*ν_d)/(1 + 0.75*k_D*ρ_1/ρ_max)'mm
#if d_bL > d_bL,max
    '<p class="err">'d_bL'mm &gt;'d_bL,max'mm. The check is NOT satisfied! ❌
#else
    'Design check:'d_bL'mm &le;'d_bL,max'mm. The check is satisfied! ✓
#end if
#show
'<p class="ref">[§ 5.6.2.2 (2), b)]</p>
'<p><b>For exterior beam-column joints</b></p>
'Minimal normalized design axial force at column -'ν_d = ? {0.07}
#post
d_bL,max = h_c*7.5*f_ctm/(γ_Rd*f_yd)*(1 + 0.8*ν_d)'mm
#if d_bL > d_bL,max
    '<p class="err">'d_bL'mm &gt;'d_bL,max'mm. The check is NOT satisfied! ❌'
#else
    'Design check:'d_bL'mm &le;'d_bL,max'mm. The check is satisfied! ✓
#end if
'According to § 5.6.2.2 (4), Top or bottom bars passing through interior joints, shall terminate in the members framing into the joint at a distance not less than <var>l</var><sub>cr</sub>.
#show

'<p><b>Transverse reinforcement</b></p>
'Hoop diameter -'d_bw = ? {6}'mm
'Characteristic yield strength -'f_ywk = ? {500}'MPa
#post
'Design yield strength -'f_ywd = f_ywk/γ_s'MPa
'<p class="ref">[§ 5.4.3.1.2 (6), a)]</p>
'Minimum diameter -'d_bw,min = 6'mm
#if d_bw < d_bw,min
    '<p class="err">Hoop diameter is smaller than the minimum:</p>
    '<p class="err">'d_bw'&lt;'d_bw,min'mm. ❌</p>
#else
    'Design check:'d_bw'≥'d_bw,min'mm. The check is satisfied! ✓
#end if
'<p class="ref">[§ 5.4.3.1.2 (1)]</p>
'<p><b>Critical region length</b> -'l_cr = h_w'mm</p>
'Minimum longitudinal bars diameter -'d_bL = min(d_bL1; d_bL2)'mm
'Effective cross section depth -'d = min(d_1; d_2)'mm
'<p><b>Hoop spacing</b></p>
'<p class="ref">[§ 5.4.3.1.2 (6), b)]</p>
'Inside critical regions:
s_cr,max = min(h_w/4; 24*d_bw; 8*d_bL; 225)'mm
'<p class="ref">[EN 1992-1-1, § 9.2.2 (6)]</p>
'Outside critical regions:
s_l,max = 0.75*d'mm
'<p class="ref">[§ 5.4.3.1.2 (6), c)]</p>
'Maximum distance from column edge to the first hoop - 50 mm
'Minimum reinforcement ratio for transverse reinforcement
'<p class="ref">[EN 1992-1-1 NA.2.79, § 9.2.2 (5)]</p>
ρ_w,min = 0.10*sqr(f_ck)/f_ywk
'NOTE: All references are according to EN 1998-1, unless noted otherwise.
#show
'</div>

Column Detailing for DCM¶

Eurocode 8 detailing of a primary seismic column for medium ductility class: cross-section dimensions, longitudinal ratio, confinement of the boundary regions and minimum mechanical volumetric ratio of the hoops.

Code:

'<small>According to <strong>Eurocode</strong>: EN 1998-1</small>
'<div style="max-width:180mm">
'<img class="side" style="width:220pt;" src="../../Images/structures/rc/detailing/column-section.png" alt="column-section.png">
'<p><b>Column dimensions</b></p>
'Cross section -'b_c = ? {500}'mm,'h_c = ? {500}'mm
'Clear storey height -'l_cl = ? {2850}'mm
'Cross section area -'A_c = b_c*h_c'mm²
'Maximum seismic axial load -'N_Ed = ? {983.8}'kN

'<p class="ref">[EN 1992-1-1, Table 3.1]</p>
'<p><b>Concrete</b></p>
'Characteristic compressive cylinder strength -'f_ck = ? {25}'MPa
'Partial safety factor -'γ_c = 1.5','α_ct = 1','α_cc = ? {1}
'<img class="side" style="width:130pt;" src="../../Images/structures/rc/detailing/column-view.png" alt="column-view.png">
#post
'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 compressive cylinder strength
f_cd = α_cc*f_ck/γ_c'MPa
#show

'<p><b>Longitudinal reinforcement</b></p>
'<p id="C" style="display:none;">'C = ? {0}'</p>
'Characteristic yield strength -'f_yk = ? {500}'MPa
#pre
'<p>Steel class - <select data-target="C">
'<option value="0"> Class B </option>
'<option value="1"> Class C </option>
'</select></p>
#post
#val
#if C ≡ 0
    'Selected steel class <b>B'f_yk'B</b>
#else if C ≡ 1
    'Selected steel class <b>B'f_yk'C</b>
#else
    '<p class="err">Invalid steel class!</p>
#end if
#equ
'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
#show
'Bar diameter -'d_bL = ? {28}'mm
#post
'<p class="ref">[BS EN 1992-1-1, § 9.5.2 (1)/NA.1]</p>
'Minimum bar diameter -'d_bL,min = 12'mm
#if d_bL < d_bL,min
    '<p class="err">The bar diameter is less than the minimum:'d_bL'mm &lt;'d_bL,min'mm</p>
#end if
#show
'Bar count -'n_b = ? {12}
'Bar count along "<var>h</var><sub>c</sub>" -'n_b1 = ? {4}
#post
'Bar count along "<var>b</var><sub>c</sub>" -'n_b2 = n_b/2 - n_b1 + 2
#if n_b < 4
    '<p class="err">The bar count is'n_b'&lt; 4</p>
#else if (n_b1 < 3) + (n_b2 < 3)
    '<p class="ref">[§ 5.4.3.2.2 (2)]</p>
    '<p class="err">Minimum one intermediate bar at each side is required.</p>
#end if
'Reinforcement area
A_s1 = π*d_bL^2/4'mm²
A_s = n_b*A_s1'mm²
'Reinforcement ratio
ρ_L = A_s/A_c
'<p class="ref">[§ 5.4.3.2.2 (1)]</p>
#if ρ_L < 0.01
    '<p class="err">The reinforcement ratio is less than the minimum:'ρ_L'&lt; 0.01</p>
#else if ρ_L > 0.04
    '<p class="err">The reinforcement ratio is greater than the maximum:'ρ_L'&gt; 0.04</p>
#else
    'Design check: 0.01 ≤'ρ_L'≤ 0.04. The check is satisfied! ✓
#end if

'<p class="ref">[EN 1992-1-1, § 9.5.1 (1)]</p>
'<p><b>Column dimensions check</b></p>
#if h_c/b_c > 4
    '<p class="err">'h_c/b_c'&gt; 4</p>
#else
    h_c/b_c'≤ 4. The check is satisfied! ✓
#end if
'<p class="ref">[§ 5.4.3.2.1 (3)]</p>
'Check for normalized axial load
ν_d = N_Ed/(A_c*f_cd)*1000
#if ν_d > 0.65
    '<p class="err">'ν_d'&gt; 0.65. The check is NOT satisfied!  ❌</p>
#else
    ν_d'≤ 0.65. The check is satisfied! ✓
#end if

#if l_cl/h_c < 3
    '<p class="ref">[§ 5.4.3.2.2 (5)]</p>
    '<p><b>Critical region length</b></p>
    l_cr = l_cl'mm - when'l_cl/h_c'&lt; 3, the entire height is a critical region.
#else
    '<p class="ref">[§ 5.4.3.2.2 (4)]</p>
    '<p><b>Critical region length</b></p>
    l_cr = max(h_c;l_cl/6;450)'mm
#end if
'Design anchorage length
η_1 = 1'- when good conditions are provided
#if d_bL ≤ 32
    η_2 = 1'- for'd_bL'≤ 32 mm
#else
    η_2 = (132 - d_bL)/100'- for'd_bL'&gt; 32 mm
#end if
f_ctd = α_ct*f_ctk,005/γ_c'MPa
'<p class="ref">[EN 1992-1-1, § 8.4.2 (2)]</p>
f_bd = 2.25*η_1*η_2*f_ctd'MPa
σ_sd = f_yd'MPa
'<p class="ref">[EN 1992-1-1, § 8.4.3 (2)]</p>
l_b,rqd = d_bL/4*(σ_sd/f_bd)'mm
'<p class="ref">[EN 1992-1-1, Table 8.2]</p>
α_1 = 1','α_2 = 1','α_3 = 1','α_5 = 1','α_6 = 1.5
'<p class="ref">[EN 1992-1-1, § 8.7.3 (1)]</p>
l_0_ = α_1*α_2*α_3*α_5*α_6*l_b,rqd'mm
l_0,min = max(0.3*α_6*l_b,rqd;15*d_bL;200)'mm
l_0 = round(max(l_0_;l_0,min))'mm
'Mid zone length
l_1 = max(0;l_cl - l_cr - l_0)'mm
#show

'<p><b>Transverse reinforcement</b></p>
'Hoop diameter -'d_bw = ? {10}'mm
'Characteristic yield strength -'f_ywk = ? {500}'MPa
#post
'Design yield strength -'f_ywd = f_ywk/γ_s'MPa
'Concrete cover to hoops -'c = 40'mm
'<p class="ref">[EN 1992-1-1, § 9.5.3 (1)]</p>
'Minimum diameter -'d_bw,min = max(6;0.25*d_bL)'mm
#if d_bw < d_bw,min
    '<p class="err">The hoop diameter is less than minimum:</p>
    '<p class="err">'d_bw'&lt;'d_bw,min'mm</p>
#else
    'Hoop diameter check:
    d_bw'≥'d_bw,min'mm. The check is satisfied! ✓
#end if

'Confined core dimensions (between centerlines of links)
b_0 = b_c - (d_bw + 2*c)'mm
h_0 = h_c - (d_bw + 2*c)'mm
'Maximum bar spacing
d_b1 = (h_c - 2*(d_bw + c) - d_bL)/(n_b1 - 1)'mm
d_b2 = (b_c - 2*(d_bw + c) - d_bL)/(n_b2 - 1)'mm
'Maximum distance between consecutive longitudinal bars engaged by hoops
'<p class="ref">[§ 5.4.3.2.2 (11), b)]</p>
d_h,max = 200'mm
'Distance between bars engaged by links
n_h1 = max(floor(d_h,max/d_b1);1)
n_h2 = max(floor(d_h,max/d_b2);1)
'Distance between bars engaged by links
d_h1 = n_h1*d_b1
d_h2 = n_h2*d_b2
#if d_h1 > d_h,max
    '<p class="err">The distance is greater than the maximum:'d_h1'mm >'d_h,max'mm</p>
#else if d_h2 > d_h,max
    '<p class="err">The distance is greater than the maximum:'d_h2'mm >'d_h,max'mm</p>
#end if
'Distance between bars engaged by links
n_h1 = round((n_b1 - 1)*d_b1/d_h1)
n_h2 = round((n_b2 - 1)*d_b2/d_h2)

'Hoop spacing in critical regions
'<p class="ref">[§ 5.4.3.2.2 (11), a)]</p>
s_cr = min(b_0/2;8*d_bL;175)'mm
'Hoop spacing in mid zone
'<p class="ref">[EN 1992-1-1, § 9.5.3 (3)]</p>
s = min(b_c;20*d_bL;400)'mm
'Hoop spacing in lap zone
'<p class="ref">[§ 5.6.3 (3), c)]</p>
s_l = min(100;b_c/4)'mm

'<b>Transverse reinforcement in lap zones</b>
'Required area of one leg
'<p class="ref">[§ 5.6.3 (4)]</p>
A_st = s_l*(d_bL/50)*(f_yd/f_ywd)'mm²
'Provided area of one leg
A_sw1 = π*d_bw^2/4'mm²
#if A_sw1 < A_st
    '<p class="err">Insufficient area of one leg in the lap zone:'A_sw1'mm² &lt;'A_st'mm²</p>
#else
    'Design check:'A_sw1'mm² ≥'A_st'mm². The check is satisfied! ✓
#end if
#if d_bL > 20
    'Check for bar diameters > 20 mm:
    'Number of legs in the outer 1/3 of lap zone
    n_w = round(2*l_0/(3*s_l))
    'Total area of legs in the outer 1/3 of lap zone
    ΣA_sw = A_sw1*n_w
    '<p class="ref">[EN 1992-1-1, § 8.7.4.1 (3)]</p>
    #if ΣA_sw < A_sw1*n_w
        '<p class="err">Insufficient transverse reinforcement area in the lap zone:'ΣA_sw'mm² &lt;'A_s1'mm²</p>
    #else
        'Design check:'ΣA_sw'mm² ≥'A_s1'mm²
    #end if
    'Additional hoop is required for compressed bars
    '<p class="ref">[EN 1992-1-1, § 8.7.4.2 (1)]</p>
    'at'4*d_bL'mm from the end of the lap zone
#end if

'<b>Calculation of hoop count</b>
'In the lap zone -'n_w,l = round(l_0/s_l)
'In the middle zone -'n_w,1 = round(l_1/s)
#if n_w,1 ≡ 0
    l_cr,top = max(l_cl - l_0;0)
    'In critical region -'n_w,cr = round(l_cr,top/s_cr)
#else
    'In critical region -'n_w,cr = round(l_cr/s_cr)
#end if
'Total number of hoops -'n_w = n_w,l + n_w,1 + n_w,cr

'<p><b>Detailing for local ductility in the critical region at column base</b></p>
'Total length of confining hoops
Σl_i = (n_h1 + 1)*b_0 + (n_h2 + 1)*h_0
'Mechanical volumetric ratio of confining hoops within the critical region
ω_d = (A_sw1*Σl_i)/(b_0*h_0*min(s_cr;s_l))*(f_ywd/f_cd)
'<p class="ref">[§ 5.4.3.2.2 (8)]</p>
'The minimum value is 0.08.
#if ω_d < 0.08
    '<p class="err">Design check:'ω_d'&lt;'0.08'.Mechanical volumetric ratio is less than minimum.</p>
#else
    'Design check:'ω_d'&ge;'0.08'. The condition is satisfied!
#end if
'Sum of the squares of the spacing between consecutive engaged bars
Σb2_i = 2*(n_h1*d_h1^2 + n_h2*d_h2^2)
'Confinement effectiveness factors for bars and hoops
α_n = 1 - Σb2_i/(6*b_0*h_0)
α_s = (1 - s_cr/(2*b_0))*(1 - s_cr/(2*h_0))
α = α_n*α_s
#show
'<p><b>Analysis results</b></p>
'Fundamental period of first vibration mode -'T_1 = ? {0.55}'s
'Upper limit period of constant spectral acceleration -'T_C = ? {0.6}'s
'Basic behavior factor value -'q_0 = ? {3.9}
#post
'<p><b>Curvature ductility factor</b></p>
'<p class="ref">[§ 5.2.3.4 (3)]</p>
#if T_1 ≥ T_C
    μ_Φ = 2*q_0 - 1'- for T<sub>1</sub> ≥ T<sub>C</sub>
#else
    μ_Φ = 1 + 2*(q_0 - 1)*(T_C/T_1)'- for T<sub>1</sub> &lt; T<sub>C</sub>
#end if
#if C ≡ 0
    '<!--'μ_Φ = 1.5*μ_Φ'-->
    '<p class="ref">[§ 5.2.3.4 (4)]</p>
    'For steel class B, ductility factor is increased by 05% -'μ_Φ
#end if
'Design value of steel yield strain - 'ε_sy_d = f_yd/E_s
'<p class="ref">[§ 5.4.3.2.2 (8)]</p>
'Design check: <var>αω</var><sub>d</sub> &ge; <var>αω</var><sub>d,min</sub> = 30·<var>μ</var><sub>Φ</sub>·<var>ν</var><sub>d</sub>·<var>ε</var><sub>sy_d</sub>·<var>b</var><sub>c</sub>/<var>b</var><sub>0</sub> – 0.035
αω_d = α*ω_d
αω_d,min = 30*μ_Φ*ν_d*ε_sy_d*(b_c/b_0) - 0.035
#if αω_d < αω_d,min
    '<p class="err">The required curvature ductility is NOT provided:'αω_d'&lt;'αω_d,min'</p>
#else
    'The required curvature ductility is provided:'αω_d'≥'αω_d,min
#end if
'NOTE: All references are according to EN 1998-1, unless noted otherwise.
#show
'</div>

Shear Wall Detailing for DCM¶

Eurocode 8 detailing of a primary seismic shear wall for medium ductility class: confined boundary elements at the wall ends, web horizontal and vertical reinforcement and curvature ductility check at the wall base.

Code:

'<small>According to <strong>Eurocode</strong>: EN 1998-1</small>
'<div style="max-width:180mm">
'<img class="side" src="../../Images/structures/rc/detailing/shear-wall-section.png" alt="shear-wall-section.png" style="width:260pt;">
'<p><b>Shear wall dimensions</b></p>
'Length -'l_w = ? {4000}'mm
'Web thickness -'b_wo = ? {300}'mm
'Total height -'h_w = ? {19000}'mm
'Clear storey height -'h_s = ? {3820}'mm
'Number of storeys -'n_s = ? {6}
'Confined zone dimensions
b_c = ? {300}'mm,'h_c = ? {875}'mm
#post
'Cross section area
'Area of confined boundary element
A_f = b_c*h_c'mm²
'Web area
A_w = (l_w - 2*h_c)*b_wo'mm²
'Total area
A_c = A_w + 2*A_f'mm²
#show
'Maximum seismic axial load -'N_Ed = ? {2254}'kN

'<p><b>Concrete</b> &nbsp;&nbsp;[EN 1992-1-1, Table 3.1]</p>
'Characteristic compressive cylinder strength
f_ck = ? {25}'MPa
'Partial safety factor -'γ_c = 1.5','α_ct = 1','α_cc = ? {1}'
#post
'Mean value of axial tensile strength
f_ctm = 0.30*f_ck^(2/3)'MPa
'<img class="side" src="../../Images/structures/rc/detailing/shear-wall-view.png" alt="shear-wall-view.png" style="width:100pt;">
'Characteristic axial tensile strength
f_ctk,005 = 0.7*f_ctm'MPa
'Design compressive cylinder strength
f_cd = α_cc*f_ck/γ_c'MPa
'Unconfined concrete ultimate strain
ε_cu2 = 0.0035
'Ultimate compressive strain -'ε_c2 = 0.002
#show

'<p><b>Longitudinal reinforcement</b></p>
'<p id="C" style="display:none;">'C = ? {0}'</p>
'Characteristic yield strength -'f_yk = ? {500}'MPa
#pre
'<p>Steel class - <select data-target="C">
'<option value="0"> Class B </option>
'<option value="1"> Class C </option>
'</select></p>
#post
#val
#if C ≡ 0
    'Selected steel class <b>B'f_yk'B</b>
#else if C ≡ 1
    'Selected steel class <b>B'f_yk'C</b>
#else
    '<p class="err">Invalid steel class!</p>
#end if
#equ
'Partial safety factor -'γ_s = 1.15
'Design yield strength -'f_yd = f_yk/γ_s'MPa
'Modulus of elasticity -'E_s = 200000'MPa
#show
'<p><b>Reinforcement for each confined boundary element</b></p>
'Bar diameter -'d_bL = ? {25}'mm
#post
'<p class="ref">[BS EN 1992-1-1, § 9.5.2 (1)/NA.1]</p>
'Minimum bar diameter -'d_bL,min = 12'mm
#if d_bL < d_bL,min
    '<p class="err">The bar diameter is less than the minimum:'d_bL'mm &lt;'d_bL,min'mm. ❌</p>
#end if
#show
'Bar count -'n_b = ? {13}
'Bar count along "<var>h</var><sub>0</sub>" -'n_b1 = ? {6}
#post
'Bar count along "<var>b</var><sub>0</sub>" -'n_b2 = ceiling(n_b/2 - n_b1 + 2)
#if n_b < 4
    '<p class="err">The bar count is'n_b'&lt; 4</p>
#end if
'Reinforcement area
A_s1 = π*d_bL^2/4'mm²
A_s = n_b*A_s1'mm²
'Reinforcement ratio
ρ_L = A_s/A_f
'<p class="ref">[§ 5.4.3.4.2 (8)]</p>
#if ρ_L < 0.005
    '<p class="err">The reinforcement ratio is less than the minimum:'ρ_L'&lt; 0.005. ❌</p>
#else if ρ_L > 0.04
    '<p class="err">The reinforcement ratio is greater than the maximum:'ρ_L'&gt; 0.04. ❌</p>
#else
    'Design check: 0.005 ≤'ρ_L'≤ 0.04. The check is satisfied! ✓
#end if
#show
'<p><b>Vertical web reinforcement</b></p>
'Bar diameter -'d_bv = ? {10}'mm
'Bar spacing -'s_v = ? {250}'mm
#post
'<p class="ref">[EN 1992-1-1, § 9.6.2 (3)]</p>
'Maximum bar spacing
s_v,max = min(3*b_wo; 400)'mm
#if s_v > s_v,max
    '<p class="err">Bar spacing's_v'mm &gt;'s_v,max'mm. ❌</p>
#end if
'Single bar area -'A_sv1 = π*d_bv^2/4'mm²
'Reinforcement ratio -'ρ_v = 2*A_sv1/(s_v*b_wo)
'<p class="ref">[EN 1992-1-1, § 9.6.2 (1)]</p>
'Minimum reinforcement ratio -'ρ_v,min = 0.002
#if ρ_v < ρ_v,min
    '<p class="err">The reinforcement ratio is less than the minimum -'ρ_v'&lt;'ρ_v,min'. ❌</p>
#end if
'<p class="ref">[§ 5.4.3.4.2 (11)]</p>
'Minimum reinforcement ratio for zones with compressive strain > 0.002
ρ_v,min = 0.005
#show
'<p><b>Horizontal web reinforcement</b></p>
'Bar diameter -'d_bh = ? {12}'mm
'Bar spacing -'s_h = ? {150}'mm
#post
'<p class="ref">[EN 1992-1-1, § 9.6.3 (2)]</p>
'Maximum bar spacing -'s_h,max = 400'mm
#if s_h > s_h,max
    '<p class="err">Bar spacing's_h'mm &gt;'s_h,max'mm. ❌</p>
#end if
'Single bar area -'A_sh1 = π*d_bh^2/4'mm²
'Reinforcement ratio -'ρ_h = 2*A_sh1/(s_h*b_wo)
'<p class="ref">[EN 1992-1-1, § 9.6.3 (1)]</p>
'Minimum reinforcement ratio
ρ_h,min = max(0.25*ρ_v; 0.001)
#if ρ_h < ρ_h,min
    '<p class="err">The reinforcement ratio is less than the minimum -'ρ_h'&lt;'ρ_h,min'. ❌</p>
#end if
#show

'<p><b>Transverse reinforcement in confined boundary elements</b></p>
'Characteristic yield strength -'f_ywk = ? {500}'MPa
'Design yield strength -'f_ywd = f_ywk/γ_s'MPa
'Concrete cover to hoops -'c = ? {42}'mm
'Hoop diameter -'d_bw = ? {8}'mm
#post
'<p class="ref">[EN 1992-1-1, § 9.5.3 (1)]</p>
'Minimum diameter
d_bw,min = max(6; 0.25*d_bL)'mm
#if d_bw < d_bw,min
    '<p class="err">The hoop diameter is less than the minimum:</p>
    '<p class="err">'d_bw'&lt;'d_bw,min'mm ❌</p>
#else
    'Hoop diameter check:
    d_bw'≥'d_bw,min'mm. The check is satisfied! ✓
#end if

'<p class="ref">[§ 5.4.3.4.2 (1)]</p>
'<p><b>Critical region height</b></p>
h_cr_ = max(l_w; h_w/6)'mm
'Must not be greater than
#if n_s ≤ 6
    h_cr,max = min(2*l_w; h_s)'mm, for number of storeys'n_s'&le; 6
#else
    h_cr,max = min(2*l_w; 2*h_s)'mm, for number of storeys'n_s'&gt; 6
#end if
h_cr = min(h_cr_; h_cr,max)'mm

'<p class="ref">[§ 5.1.2 (1)]</p>
'<p><b>Shear wall dimensions check</b></p>
#if l_w/b_wo < 4
    '<p class="err">'l_w/b_wo'&lt; 4. The check is NOT satisfied! ❌</p>
#else
    l_w/b_wo'&ge; 4. The check is satisfied! ✓
#end if
'<p class="ref">[§ 5.4.1.2.3 (1)]</p>
'Minimum thickness -'b_w,min = max(150; h_s/20)'mm
#if b_wo < b_w,min
    '<p class="err">'b_wo'mm &lt;'b_w,min'mm. The check is NOT satisfied! ❌</p>
#else
    b_wo'mm &ge;'b_w,min'mm. The check is satisfied! ✓
#end if
'<p class="ref">[ § 5.4.3.4.2 (6)]</p>
'Confined boundary element length
l_c = h_c - (d_bw + 2*c)'mm
'Minimum confined boundary element length
l_c,min = max(0.15*l_w; 1.5*b_c)'mm
#if l_c < l_c,min
    '<p class="err">'l_c'mm &lt;'l_c,min'mm. The check is NOT satisfied! ❌</p>
#else
    l_c'mm &ge;'l_c,min'mm. The check is satisfied! ✓
#end if
'<p class="ref">[§ 5.4.3.4.2 (10)]</p>
'Minimum confined boundary element thickness
#if l_c ≤ max(2*b_c; 0.2*l_w)
    'For'l_c'mm &le;'max(2*b_c; 0.2*l_w)'mm:
    b_c,min = max(h_s/15; 200)'mm
#else
    'For'l_c'mm &gt;'max(2*b_c; 0.2*l_w)'mm:
    b_c,min = max(h_s/10; 200)'mm
#end if
#if b_c < b_c,min
    '<p class="err">'b_c'mm &lt;'b_c,min'mm. The check is NOT satisfied! ❌</p>
#else
    b_c'mm &ge;'b_c,min'mm. The check is satisfied! ✓
#end if

'<p class="ref">[§ 5.4.3.4.1 (2)]</p>
'<p><b>Check for normalized axial load</b></p>
ν_d = N_Ed/(A_c*f_cd)*10^3
#if ν_d > 0.4
    '<p class="err">'ν_d'&gt; 0.4. The check is NOT satisfied! ❌</p>
#else
    ν_d'≤ 0.4. The check is satisfied! ✓
#end if

'<p><b>Design anchorage length</b></p>
η_1 = 1'- when good conditions are provided
#if d_bL ≤ 32
    η_2 = 1'- for'd_bL'≤ 32 mm
#else
    η_2 = (132 - d_bL)/100'- for'd_bL'> 32 mm
#end if
f_ctd = α_ct*f_ctk,005/γ_c'MPa
'<p class="ref">[EN 1992-1-1, § 8.4.2 (2)]</p>
f_bd = 2.25*η_1*η_2*f_ctd'MPa
σ_sd = f_yd'MPa
'<p class="ref">[EN 1992-1-1, § 8.4.3 (2)]</p>
l_b,rqd = d_bL/4*(σ_sd/f_bd)'mm
'<p class="ref">[EN 1992-1-1, Table 8.2]</p>
α_1 = 1','α_2 = 1','α_3 = 1','α_5 = 1','α_6 = 1.5
'<p class="ref">[EN 1992-1-1, § 8.7.3 (1)]</p>
l_0_ = α_1*α_2*α_3*α_5*α_6*l_b,rqd'mm
l_0,min = max(0.3*α_6*l_b,rqd; 15*d_bL; 200)'mm
l_0 = round(max(l_0_; l_0,min))'mm

'Confined core dimensions (between centerlines of hoops)
b_0 = b_c - (d_bw + 2*c)'mm
h_0 = h_c - (d_bw + 2*c)'mm
'Maximum bar spacing
d_b1 = (h_c - 2*(d_bw + c) - d_bL)/(n_b1 - 1)'mm
d_b2 = (b_c - 2*(d_bw + c) - d_bL)/(n_b2 - 1)'mm
'Maximum distance between consecutive longitudinal bars engaged by hoops
'<p class="ref">[§ 5.4.3.4.2 (9)]</p>
d_h,max = 200'mm
'Distance between bars engaged by hoops
n_h1 = max(floor(d_h,max/d_b1); 1)
n_h2 = max(floor(d_h,max/d_b2); 1)
'Distance between bars engaged by hoops
d_h1 = n_h1*d_b1
d_h2 = n_h2*d_b2
#if d_h1 > d_h,max
    '<p class="err">The distance is greater than the maximum:'d_h1'mm >'d_h,max'mm ❌</p>
#else if d_h2 > d_h,max
    '<p class="err">The distance is greater than the maximum:'d_h2'mm >'d_h,max'mm ❌</p>
#end if
'Distance between bars engaged by hoops
n_h1 = round((n_b1 - 1)*d_b1/d_h1)
n_h2 = round((n_b2 - 1)*d_b2/d_h2)

'Hoop spacing in the critical region
'<p class="ref">[§ 5.4.3.4.2 (9)]</p>
s_cr = min(b_0/2; 8*d_bL; 175)'mm
'Hoop spacing in lap zone
'<p class="ref">[§ 5.6.3 (3), c)]</p>
s_l = min(100; b_c/4)'mm
'Hoop spacing outside lap zone
'<p class="ref">[EN 1992-1-1, § 9.5.3 (3)]</p>
s = min(b_c; 20*d_bL; 400)'mm

'<b>Transverse reinforcement in the lap zone</b>
'Required area of one leg
'<p class="ref">[§ 5.6.3 (4)]</p>
A_st = s_l*(d_bL/50)*(f_yd/f_ywd)'mm²
'Provided area of one leg
A_sw1 = π*d_bw^2/4'mm²
#if A_sw1 < A_st
    '<p class="err">Insufficient area of one leg in the lap zone:'A_sw1'mm² &lt;'A_st'mm² ❌</p>
#else
    'Design check:'A_sw1'mm² ≥'A_st'mm². The check is satisfied! ✓
#end if
#if d_bL > 20
    'Check for bar diameters > 20 mm:
    'Number of legs in the outer 1/3 of lap zone
    n_w = round(2*l_0/(3*s_l))
    'Total area of legs in the outer 1/3 of lap zone
    ΣA_sw = A_sw1*n_w
    '<p class="ref">[EN 1992-1-1 § 8.7.4.1 (3)]</p>
    #if ΣA_sw < A_sw1*n_w
        '<p class="err">Insufficient transverse reinforcement area in the lap zone:'ΣA_sw'mm² &lt;'A_s1'mm² ❌</p>
    #else
        'Design check:'ΣA_sw'mm² ≥'A_s1'mm²
    #end if
    'An additional hoop is required for compressed bars
    '<p class="ref">[EN 1992-1-1 § 8.7.4.2 (1)]</p>
    'at'4*d_bL'mm from the end of the lap zone.
#end if

'<p><b>Detailing for local ductility in the critical region</b></p>
'Total length of confining links
Σl_i = (n_h1 + 1)*b_0 + (n_h2 + 1)*h_0
'Mechanical volumetric ratio of confining hoops within the critical region
ω_d = (A_sw1*Σl_i)/(b_0*h_0*s_cr)*(f_ywd/f_cd)
'<p class="ref">[§ 5.4.3.2.2 (8)]</p>
'The minimum value is 0.08.
#if ω_d < 0.08
    '<p class="err">Design check:'ω_d'&lt;'0.08'. Mechanical volumetric ratio is less than minimum. ❌</p>
#else
    'Design check:'ω_d'&ge;'0.08'. The check is satisfied! ✓
#end if
'Sum of the squares of the distances between consecutive engaged bars
Σb2_i = 2*(n_h1*d_h1^2 + n_h2*d_h2^2)
'Confinement effectiveness factors for bars and links
α_n = 1 - Σb2_i/(6*b_0*h_0)
α_s = (1 - s_cr/(2*b_0))*(1 - s_cr/(2*h_0))
α = α_n*α_s
#show
'<p><b>Analysis results</b></p>
'Fundamental period of first vibration mode -'T_1 = ? {0.6795}'s
'Upper limit period of constant spectral acceleration -'T_C = ? {0.4}'s
'Basic behavior factor value -'q_0 = ? {3}
'Design bending moment -'M_Ed = ? {9591}'kNm
'Bending moment capacity -'M_Rd = ? {13268}'kNm
'(The above values refer to the section above the base)
#post
'<p><b>Curvature ductility factor</b></p>
'<p class="ref">[§ 5.2.3.4 (3)]</p>
#if T_1 ≥ T_C
    μ_Φ = 2*q_0*(M_Ed/M_Rd) - 1'- for T<sub>1</sub> ≥ T<sub>C</sub>
#else
    μ_Φ = 1 + 2*(q_0*(M_Ed/M_Rd) - 1)*(T_C/T_1)'- for T<sub>1</sub> &lt; T<sub>C</sub>
#end if
#if C ≡ 0
    '<!--'μ_Φ = 1.5*μ_Φ'-->
    '<p class="ref">[§ 5.2.3.4 (4)]</p>
    'For steel class B, ductility factor is increased by 50% -'μ_Φ
#end if
'Design value of steel yield strain - 'ε_sy,d = f_yd/E_s
'Mechanical ratio of vertical web reinforcement
ω_v = ρ_v*(f_yd/f_cd)
'<p class="ref">[§ 5.4.3.4.2 (4)]</p>
'Design check: <var>αω</var><sub>d</sub> &ge; <var>αω</var><sub>d_min</sub> = 30·<var>μ</var><sub>Φ</sub>·(<var>ν</var><sub>d</sub> + <var>ω</var><sub>v</sub> )·<var>ε</var><sub>sy_d</sub>·<var>b</var><sub>c</sub>/<var>b</var><sub>0</sub> – 0.035
αω_d = α*ω_d
αω_d,min = 30*μ_Φ*(ν_d + ω_v)*ε_sy,d*(b_c/b_0) - 0.035
#if αω_d < αω_d,min
    '<p class="err">The required curvature ductility is NOT provided:'αω_d'&lt;'αω_d,min'. ❌</p>
#else
    'The required curvature ductility is provided:'αω_d'≥'αω_d,min'. ✓
#end if
'Ultimate strain of confined concrete
ε_cu2,c = 0.0035 + 0.1*αω_d
'Neutral axis depth at ultimate curvature
x_u = (ν_d + ω_v)*l_w*(b_c/b_0)'mm
'Confined boundary element length
l_c,req = x_u*(1 - ε_cu2/ε_cu2,c)'mm
#if l_c < l_c,req
    '<p class="err">'l_c'mm &lt;'l_c,req'mm. The check is NOT satisfied! ❌</p>
#else
    'Design check:'l_c'mm &ge;'l_c,req'mm. The check is satisfied! ✓
#end if
'NOTE: All references are according to EN 1998-1, unless noted otherwise.
#show
'</div>

Spotted an error? Edit these examples.

	Exterior column	Interior column
In the absence of transverse beam	`b`_c = 500 mm	`b`_c + 4 · `h`_f = 500 + 4 · 130 = 1020 mm
With transverse beam	`b`_c + 4 · `h`_f = 500 + 4 · 130 = 1020 mm	`b`_c + 8 · `h`_f = 500 + 8 · 130 = 1540 mm