ACES - PSC Design Module V{VERSION}: Run date: {DATE}
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Heading: {PROJECT}
Job Name: {JOBNAME}
Designer: {DESIGNER}
Comments: {COMMENT1}
Units: mm, kN, kN.m, MPa
Design Code: {CODE} {DEC 0}
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SECTION: {Sectnum}
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Distance (x) of section from the first node = {x} mm |
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Strand segment number: {SectSSeg} |
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Passive R/F segment number: {SectPSeg} |
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SHEAR CHECK: {ShearDescription} {DEC 1}
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Ultimate design shear force: |
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Calculate the default ultimate design shear force (V*) if the user has not provided one: |
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(V* = Vult = LFsw*Vdsw + LFslab*Vdslab + LFsdl*Vdsdl + LFll*Vdll) |
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Design parameters: |
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{DEC 0} |
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Ultimate design shear force (V* = Vult) |
= |
{Vult} |
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kN |
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Corresponding ultimate design moment (M* = Mult) |
= |
{Mult} |
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kN.m {DEC 2} |
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28 day girder concrete strength (f'cg) |
= |
{f'cg} |
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MPa |
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Allowable principal tensile strength (st = 0.33*f'cg^0.5) |
= |
{st} |
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MPa ({CODE} Clause 8.2.7.2(b)) |
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Characteristic flexural tensile strength (f'cf = 0.6*f'cg^0.5) |
= |
{f'cf} |
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MPa ({CODE} Clause 6.1.1(b)) |
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{DEC 0} |
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Overall depth of composite section (D) |
= |
{D} |
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mm |
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Location of girder centroid (Yb) |
= |
{Yb} |
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mm |
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Location of composite centroid (Yc) |
= |
{Yc} |
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mm |
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Width of a single web (bw) |
= |
{bw} |
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mm |
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Width of bottom flange (Wbf) |
= |
{Wbf} |
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mm {EXP 4} |
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Moment of inertia of girder (Ig) |
= |
{Ig} |
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mm^4 |
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Section modulus of girder at bottom flange (Zb) |
= |
{Zb} |
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mm^3 |
Section modulus of composite girder at bottom flange (Zgb) |
= |
{Zgb} |
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mm^3 |
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Q for composite girder at centroid (Qna) |
= |
{Qna} |
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mm^3 {DEC 0} |
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Modulus of elasticity of prestressing steel (Ep) |
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{Ep} |
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MPa |
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Total area of pretensioned bonded strand (Ap) |
= |
{Ap} |
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mm^2 |
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Area of longitudinal RF in tensile zone (Ast) |
= |
{Ast} |
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mm^2 |
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Area of pretensioned girder (Ag) |
= |
{Ag} |
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mm^2 {DEC 1} |
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Section cracking moment: |
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Final prestress force (P) |
= |
{P} |
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kN |
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Eccentricity CG girder to CG strand group (e) |
= |
{e} |
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Mm {DEC 0} |
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Section cracking moment (Mcr): |
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Mcr = Zgb*10^-6*(f'cf + 1000*P/Ag) + P*e/1000 |
= |
{Mcr} |
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kN.m ({CODE} Eqn 8.1.4.1(1)) |
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Flexure shear cracking: ({CODE} Section 8.2.7.2(a)) |
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Compression stress due to PS at girder bottom (scpf) |
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{DEC 2} |
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scpf = -1000*P/Ag - 1000*P*e*Yb/Ig |
= |
{scpf} |
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MPa {DEC 1} |
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Decompression moment (Mo = - scpf*Zgb/10^6) |
= |
{Mo} |
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kN.m |
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Shear force corresponding to the decompression moment: |
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Vo = Mo*Vult/Mult |
= |
{Vo} |
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kN |
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Dist. of outermost layer of RF from girder bottom (Ybarmin) |
= |
{Ybarmin} |
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mm |
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Distance from extreme compression fibre to centroid of outermost layer of tensile R/F or tendons (do): |
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(Default value for do = D - Ybarmin) |
= |
{do} |
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mm |
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Minimum allowable value of 'do' (doMin = 0.8D) |
= |
{doMin} |
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mm |
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Effective width of both webs for shear (bv = 2*bw) |
= |
{bv} |
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mm |
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Shear coefficients: |
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{DEC 2} |
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ß1 = 1.1*(1.6 - do/1000): If ß1<1.1 then ß1 = 1.1 |
= |
{Beta1} |
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Shear coefficient ß2: |
= |
{Beta2} |
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(For Super-Ts: {CODE} Sectn 8.2.7.1) |
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Shear coefficient ß3: (where X = x or Span – x) |
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ß3=2*do*1000/X: If ß3<1 then ß3=1: If ß3>2 then ß3=2 |
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{Beta3} |
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Flexural shear capacity (Vuct): |
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{DEC 0} |
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Vuct = ß1*ß2*ß3*bv*do*((Ast+Ap)*f'cg/(bv*do))^.333/1000 |
= |
{Vuct} |
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kN ({CODE} Section 8.2.7.2(1)) |
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Ultimate flexural shear capacity of concrete alone (Vucs): |
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Vucs = Vuct + Vo |
= |
{Vucs} |
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kN |
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Web shear cracking at the centroid of the composite section: ({CODE} 8.2.7.2(b)) |
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Since there is no bending stress at the centroid, the ultimate shear capacity of |
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the concrete can be determined directly. |
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{DEC 2} |
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Flexural stress at composite centroid, uncracked sectn (sc): |
= |
{sc} |
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MPa {EXP 4} |
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sc = -P*1000/Ag + P*e*1000*(Yc - Yb)/Ig |
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First moment of area of composite section above composite centroid & about composite centroid: |
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Qlbwc = Qna/(Ic*2*bw) |
= |
{Qlbwc} |
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mm^-2 {DEC 2} |
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Shear stress (tau = tc = (st^2 - st*sc)^0.5 = 0 if sc < st) |
= |
{tc} |
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MPa |
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Allowable principal tensile strength (st = 0.33*f'cg^0.5) |
= |
{st} |
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MPa |
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Principal tensile stress due to design prestress (s1c): |
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s1c = ((0.5*sc)^2 + tc*tc)^0.5 + 0.5*sc |
= |
{s1c} |
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MPa {DEC 0} |
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Concrete ultimate shear capacity assuming web-cracking (Vtc) |
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Vtc = tc/(Qlbwc*1000) |
= |
{Vtc} |
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kN |
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Moment corresponding to Vtc (Mtc = Mult*Vtc /Vult) |
= |
{Mtc} |
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kN.m |
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Section cracking moment (Mcr) |
= |
{Mcr} |
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kN.m |
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If Mtc > Mcr then the section is cracked; otherwise it is --------------> |
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{VuccNte$} |
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and the ultimate shear capacity of conc alone (Vucc = Vtc or 0) |
= |
{Vucc} |
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kN {DEC 1} |
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Web shear cracking at the bottom of the web-flange interface: |
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The solution for the ultimate concrete shear capacity assuming web-cracking (Vtf) is based on |
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the method proposed by RF Warner, BV Rangan, AS Hall and KA Faulkes in their text book |
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"Concrete Structures", (Addison Wesley Longman, Australia, page 331, ISBN 0582 802474) |
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Given that: |
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e = Eccentricity (CG girder - CG strand group) |
= |
{e} |
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mm {DEC 0} |
Dwf = Dist from bottom of girder to web/flange interface |
= |
{Dwf} |
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mm |
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y2 = Dist from web/flange joint to comp centroid (Yc-Dwf) |
= |
{y2} |
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mm |
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y3 = Dist from web/flange join to centroid of gird (Yb-Dwf) |
= |
{y3} |
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mm |
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Bw2 = Sum of widths of both webs |
= |
{Bw2} |
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mm |
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Ag = Area of precast girder |
= |
{Ag} |
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mm^2 {EXP 4} |
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Ig = Moment of Inertia of precast girder |
= |
{Ig} |
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mm^4 |
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Ic = Composite moment of inertia |
= |
{Ic} |
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mm^4 |
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Qfw = Shear flow constant at flange/web junction |
= |
{Qfw} |
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mm^3 {DEC 0} |
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P = Final prestress force |
= |
{P} |
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kN {DEC 2} |
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Mcf = Corresponding ult moment factor (= Mult/Vult) |
= |
{Mcf} |
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and if: |
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sf = Flexural stress web-flange interface (uncracked section) |
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st = Allowable principal tensile strength |
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tf = Ultimate shear stress capacity of concrete on its own (tau) |
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The relationships in the solution process are: |
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sf = (-P/Ag) - (P*e*y3/Ig) + Vtf*Mcf*y2/Ic ..................... (1) Flexural stress at web-flange interface |
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st = 0.33SQRT(f'cg) = SQRT((sf/2)^2 + tf^2) + sf/2 ........ (2) Tensile stress capacity |
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Vtf = tf*Ic*Bw2/Qfw ......................................................... (3) Shear force capacity |
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Assuming: |
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{DEC 3} |
Mcf = Mult/Vult |
= |
{Mcf} |
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st = 0.33SQRT(f'cg) |
= |
{st} |
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MPa |
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s1 = - P*1000/Ag - P*e*y3*1000/Ig |
= |
{s1} |
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MPa {EXP 4} |
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s2 = y2*10^6/Ic |
= |
{s2} |
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Qlbwf = Qfw/(Ic*Bw2) |
= |
{Qlbwf} |
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mm^-2 |
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Combine equns 1-3 to form a quadratic equation where the only unknown is Vtf viz: |
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(Qlbwf*Vtf)^2 + (st*s2*Mcf)*Vtf + (st*s1 - st^2) = 0 ........ (4) |
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Setting: |
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a = Qlbwf*Qlbwf*10^6 |
= |
{a} |
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b = st*s2*Mcf |
= |
{b} |
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{DEC 2} |
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c = st*s1 - st^2 |
= |
{c} |
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Expression (4) reduces to: |
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a*Vtf^2 + b*Vtf + c = 0 and |
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Vtf = (-b +- SQRT(b*b - 4*a*c))/2*a ................... (5) |
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The determinant of equation (5) is: |
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{EXP 4} |
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f = b*b - 4*a*c = (st*s2*Mcf)^2 - 4*Qlbwf^2*(st*s1 - st^2) |
= |
{f} |
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If f > 0 then a solution exists and: |
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{VucfNte$} |
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{DEC 1} |
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Vtf = (-b + SQRT(f))/2*a |
= |
{Vtf} |
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kN |
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Moment corresponding to Vtf (Mtf = Mult*Vtf/Vult) |
= |
{Mtf} |
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kN.m |
Section cracking moment (Mcr) |
= |
{Mcr} |
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kN.m {DEC 2} |
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NOTE: If f < 0 then Mtf is set to 999999 and Vtf to 0.0 |
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If f > 0 flexural stress at web-flange interface, uncracked sectn: |
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sf = -P*1000/Ag + P*e*1000*y3/Ig + Vtf*Mcf*y2*10^6/Ic |
= |
{sf} |
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MPa |
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If f > 0 ultimate shear stress capacity of concrete on its own (tf): |
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tf = 1000*Vtf*Qlbwf |
= |
{tf} |
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MPa |
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If f > 0 principal tension stress due to design prestress (s1f): |
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s1f = ((0.5*sf)^2 + tf*tf)^0.5 + 0.5*sf |
= |
{s1f} |
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MPa |
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Allowable principal tensile strength (st = 0.33*f'cg^0.5) |
= |
{st} |
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MPa {DEC 0} |
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If Mtf > Mcr then the section is cracked; otherwise it is ---------> |
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{VucfNte$} |
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and the ultimate shear capacity of concrete alone (Vucf = Vtf) |
= |
{Vucf} |
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kN |
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Ultimate Shear Capacity of Concrete Alone: |
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Ultimate design moment corresponding to V* (Mult) |
= |
{Mult} |
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kN.m |
Section cracking moment (Mcr) |
= |
{Mcr} |
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kN.m |
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If Mult > Mcr the section is cracked in flexure |
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Ultimate shear capacity of concrete alone (ignoring shear R/F) is given by: |
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Vuc = Min of Vucs, Vucc, Vucf or Vucs if section is cracked in flexure |
= |
{Vuc} |
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kN |
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Shear Rating Check: |
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{DEC 1} |
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Shear due to self-weight (Vsw) |
= |
{Vsw} |
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kN |
Shear due to DL of the slab (Vslab) |
= |
{Vslab} |
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kN |
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Shear due to the superimposed dead load (Vsdl) |
= |
{Vsdl} |
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kN |
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Load Factor on self-weight (LFsw) |
= |
{LFsw} |
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Load Factor on the DL of slab (LFslab) |
= |
{LFslab} |
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Load Factor on the DL of superimposed DL (LFsdl) |
= |
{LFsdl} |
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Load Factor on Live Load |
= |
{LFll} |
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Live load shear: |
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Vucll = Vuc - (LFsw*Vsw + LFslab*Vslab + LFsdl*Vsdl) |
= |
{Vucll} |
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kN |
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Shear capacity rating (limited to 4.0 if it exceeds this value): |
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Vuratll = Vucll / (LFll*Vll) |
= |
{Vuratll} |
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Design of web shear reinforcement: ({CODE} Section 8.2.8 - 8.2.10) |
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Ultimate design shear force (V* = Vult) |
= |
{Vult} |
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kN {DEC 1} |
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Dist. from extreme compn fibre to outer tensile RF or tendon (do) |
= |
{Dsr} |
mm {DEC 2} |
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Capacity reduction factor - shear (Ø = Os) |
= |
{Os} |
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{DEC 0} |
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Ultimate shear strength of girder having minimum amount of shear R/F (ØVumin): |
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ØVumin = Ø*(Vuc + 0.6*2*bw*do/1000) |
= |
{OVumin} |
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kN |
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Ultimate shear strength of girder limited by web crushing failure (ØVumax): |
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ØVumax = Ø*0.2*f'cg*2*bw*do/1000 + Pv |
= |
{OVumax} |
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kN |
where Pv = Post-tensioning force |
= |
{Pv} |
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kN |
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Yield strength of shear reinforcement (fsysr) |
= |
{fsysr} |
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MPa {DEC 2} |
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Diameter of shear reinforcement (Dsr) |
= |
{Dsr} |
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mm {DEC 1} |
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Area of both legs of shear stirrup (Asv = 2*3.14*Dsr^2/4) |
= |
{Asv} |
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mm^2 {DEC 2} |
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OVumin > Vult: Concrete web shear capacity is sufficient |
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Shear steel required |
= |
{AsNote1$} |
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Angle between inclined conc comp strut & long. axis (Øv) |
= |
{Ov} |
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Degrees (prescribed minimum) |
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Required area of shear reinforcmt (Asvsr1 = 0.35*2*bw/fsysr) |
= |
{Asvsr1} |
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mm^2 per mm {DEC 1} |
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Actual shear reinforcement spacing (Svr = max allowed = Svmax) |
= |
{Svmax} |
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mm |
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OVumin < Vult: Concrete web shear capacity is insufficient: |
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{AsNote2$} |
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Required contribution of steel R/F to shear capacity of the section (Vus): {DEC 0} |
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Vus = (Vult - 0.7*Vuc)/0.7 |
= |
{Vus} |
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kN {DEC 1} |
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Angle between inclined conc comp strut & long. axis (Øv1) |
= |
{Ov1} |
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Degrees |
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Øv2 = 30 + 15*(Vult - ØVumin)/ (ØVumax - ØVumin) |
= |
{Ov2} |
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If Øv1 < 30 then Øv1 = 30 |
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If Øv1 > 45 then Øv1 = 45 otherwise Øv1 = Øv2 |
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Required area of shear R/F per unit length along beam (Asvsr2) {DEC 2}: |
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Asvsr2 = ABS(Vus*1000*TAN(Øv1)/(do*fsysr))) |
= |
{Asvsr2} |
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mm^2 per mm {DEC 0} |
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Required shear reinforcement spacing (Svr2 = Asv/Asvsr2) |
= |
{Svr2} |
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mm |
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Maximum allowable shear reinforcement spacing (Svmax) |
= |
{Svmax} |
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mm |
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Shear Reinforcement: |
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Actual shear stirrup spacing (Sv) |
= |
{Sv} |
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mm {DEC 2} |
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Actual steel area per unit length (Asvs) |
= |
{Asvs} |
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mm^2 per mm |
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