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}

 

Distance (x) of section from the first node = {x}  mm

 

   

 

 

 

 

 

 

 

Strand segment number:  {SectSSeg}

 

 

 

 

Passive R/F segment number:  {SectPSeg}

 

 

 

SHEAR CHECK:    {ShearDescription} {DEC 1}

 

Ultimate design shear force:

 

 

 

 

  

 

 

Calculate the default ultimate design shear force (V*) if the user has not provided one:

 

   (V* = Vult = LFsw*Vdsw + LFslab*Vdslab + LFsdl*Vdsdl + LFll*Vdll)

 

 

 

 

 

 

 

Design parameters:

 

 

 

{DEC 0}

 

 

 

 

 

 

 

Ultimate design shear force (V* = Vult)

=

{Vult}

 

kN

 

Corresponding ultimate design moment (M* = Mult)

=

{Mult}

 

kN.m {DEC 2}

 

 

 

 

 

 

 

28 day girder concrete strength (f'cg)

=

{f'cg}

 

MPa

 

Allowable principal tensile strength (st = 0.33*f'cg^0.5)

=

{st}

 

MPa ({CODE} Clause 8.2.7.2(b))

 

Characteristic flexural tensile strength (f'cf = 0.6*f'cg^0.5)   

=

{f'cf}

 

MPa ({CODE} Clause 6.1.1(b))

 

 

 

 

 

{DEC 0} 

 

Overall depth of composite section (D)

=

{D}

 

mm

 

Location of girder centroid (Yb)

=

{Yb}

 

mm

 

Location of composite centroid (Yc)

=

{Yc}

 

mm

 

Width of a single web (bw)

=

{bw}

 

mm

 

Width of bottom flange (Wbf)

=

{Wbf}

 

mm {EXP 4}

 

 

 

Moment of inertia of girder (Ig)

=

  {Ig}

 

mm^4

 

Section modulus of girder at bottom flange (Zb)

=

{Zb}

 

mm^3

Section modulus of composite girder at bottom flange (Zgb)

=

{Zgb}

 

mm^3

 

Q for composite girder at centroid (Qna)

=

{Qna}

 

mm^3 {DEC 0}

 

 

 

 

 

 

 

Modulus of elasticity of prestressing steel (Ep)

=

{Ep}

 

MPa

 

Total area of pretensioned bonded strand (Ap)

=

{Ap}

 

mm^2

 

Area of longitudinal RF in tensile zone (Ast)

=

{Ast}

 

mm^2

 

Area of pretensioned girder (Ag)

=

{Ag}

 

mm^2 {DEC 1}

 

 

 

 

 

 

 

Section cracking moment:

 

 

 

 

 

 

 

 

 

 

 

Final prestress force (P)

=

{P}

 

kN 

 

Eccentricity CG girder to CG strand group (e)

=

{e}

 

Mm {DEC 0}

 

 

 

 

 

 

 

Section cracking moment (Mcr):

 

 

 

 

     Mcr = Zgb*10^-6*(f'cf  + 1000*P/Ag) + P*e/1000

=

{Mcr}

 

kN.m  ({CODE} Eqn 8.1.4.1(1))

 

 

Flexure shear cracking:   ({CODE} Section 8.2.7.2(a))

 

 

 

 

 

 

 

Compression stress due to PS at girder bottom (scpf)

 

 

 

{DEC 2}  

 

     scpf = -1000*P/Ag - 1000*P*e*Yb/Ig

=

{scpf}

 

MPa {DEC 1}

 

Decompression moment (Mo = - scpf*Zgb/10^6)

=

{Mo}

 

kN.m

 

 

 

 

 

 

 

Shear force corresponding to the decompression moment:

 

 

 

 

 

     Vo = Mo*Vult/Mult

=

{Vo}

 

kN

 

 

Dist. of outermost layer of RF from girder bottom (Ybarmin)

=

{Ybarmin}

 

mm

 

Distance from extreme compression fibre to centroid of outermost layer of tensile R/F or tendons (do):

 

     (Default value for do = D - Ybarmin)

=

{do}

 

mm

 

Minimum allowable value of 'do' (doMin = 0.8D)

=

{doMin}

 

mm

 

Effective width of both webs for shear (bv = 2*bw)

=

{bv}

 

mm

 

 

 

 

 

 

 

Shear coefficients:

 

 

 

{DEC 2}

 

     1 = 1.1*(1.6 - do/1000):   If 1<1.1 then 1 = 1.1

=

{Beta1}

 

 

 

Shear coefficient 2:

=

{Beta2}

 

(For Super-Ts: {CODE} Sectn 8.2.7.1) 

 

Shear coefficient 3:   (where X = x or Span x)

 

 

 

 

 

     3=2*do*1000/X:  If 3<1 then 3=1:   If 3>2 then 3=2   

=

{Beta3}

 

 

 

 

 

 

 

 

 

Flexural shear capacity (Vuct):

 

 

 

{DEC 0} 

 

   Vuct = 1*2*3*bv*do*((Ast+Ap)*f'cg/(bv*do))^.333/1000   

=

{Vuct}

 

kN  ({CODE} Section 8.2.7.2(1))

 

Ultimate flexural shear capacity of concrete alone (Vucs):

 

 

 

 

 

   Vucs = Vuct + Vo

=

{Vucs}

 

kN

 

 

 

 

 

 

 

Web shear cracking at the centroid of the composite section:   ({CODE} 8.2.7.2(b))

 

 

Since there is no bending stress at the centroid, the ultimate shear capacity of

 

the concrete can be determined directly.

 

 

 

 

 

{DEC 2} 

 

Flexural stress at composite centroid, uncracked sectn (sc):   

=

{sc}

 

MPa {EXP 4}

 

  sc = -P*1000/Ag + P*e*1000*(Yc - Yb)/Ig

 

 

 

 

 

 

 

 

 

 

 

First moment of area of composite section above composite centroid & about composite centroid:

 

     Qlbwc = Qna/(Ic*2*bw)

=

{Qlbwc}

 

mm^-2 {DEC 2}

 

Shear stress  (tau = tc = (st^2 - st*sc)^0.5 = 0 if sc < st)

=

{tc}

 

MPa

 

 

 

 

 

 

Allowable principal tensile strength (st = 0.33*f'cg^0.5)

=

{st}

 

MPa

 

Principal tensile stress due to design prestress (s1c):

 

 

 

 

 

     s1c = ((0.5*sc)^2 + tc*tc)^0.5 + 0.5*sc

=

{s1c}

 

MPa {DEC 0}

 

 

 

 

 

 

 

Concrete ultimate shear capacity assuming web-cracking (Vtc)

 

 

 

 

 

     Vtc = tc/(Qlbwc*1000)

=

{Vtc}

 

kN

 

 

 

 

 

 

 

Moment corresponding to Vtc (Mtc = Mult*Vtc /Vult)

=

{Mtc}

 

kN.m

 

Section cracking moment (Mcr)

=

{Mcr}

 

kN.m

 

 

 

 

 

 

 

If Mtc > Mcr then the section is cracked; otherwise it is -------------->

 

{VuccNte$}

 

 

 

  and the ultimate shear capacity of conc alone (Vucc = Vtc or 0)   

=

{Vucc}

 

kN {DEC 1}

 

 

 

 

 

 

 

Web shear cracking at the bottom of the web-flange interface:

 

 

 

 

The solution for the ultimate concrete shear capacity assuming web-cracking (Vtf) is based on

 

the method proposed by RF Warner, BV Rangan, AS Hall and KA Faulkes in their text book

 

"Concrete Structures", (Addison Wesley Longman, Australia, page 331, ISBN 0582 802474)

 

 

 

 

 

 

 

Given that:

 

 

 

 

 

 

 

 

 

 

 

     e = Eccentricity (CG girder - CG strand group)

=

{e}

 

mm {DEC 0}

     Dwf = Dist from bottom of girder to web/flange interface

=

{Dwf}

 

mm

 

     y2 = Dist from web/flange joint to comp centroid (Yc-Dwf)   

=

{y2}

 

mm

 

     y3 = Dist from web/flange join to centroid of gird (Yb-Dwf)  

=

{y3}

 

mm

 

     Bw2 = Sum of widths of both webs

=

{Bw2}

 

mm

 

     Ag = Area of precast girder

=

{Ag}

 

mm^2 {EXP 4}

 

     Ig = Moment of Inertia of precast girder

=

{Ig}

 

mm^4

 

     Ic = Composite moment of inertia

=

{Ic}

 

mm^4

 

     Qfw = Shear flow constant at flange/web junction

=

{Qfw}

 

mm^3 {DEC 0}

 

 

 

 

 

 

 

     P = Final prestress force

=

{P}

 

kN {DEC 2}

 

     Mcf = Corresponding ult moment factor (= Mult/Vult)

=

{Mcf}

 

 

 

 

 

 

 

 

 

and if:

 

 

 

 

 

     sf = Flexural stress web-flange interface (uncracked section)  

 

 

 

 

 

     st = Allowable principal tensile strength

 

 

 

 

 

     tf = Ultimate shear stress capacity of concrete on its own (tau)

 

 

 

 

 

 

 

 

 

 

 

The relationships in the solution process are:

 

 

 

 

 

 

 

     sf = (-P/Ag) - (P*e*y3/Ig) + Vtf*Mcf*y2/Ic ..................... (1) Flexural stress at web-flange interface

 

     st = 0.33SQRT(f'cg) = SQRT((sf/2)^2 + tf^2) + sf/2 ........ (2) Tensile stress capacity

 

     Vtf = tf*Ic*Bw2/Qfw ......................................................... (3) Shear force capacity

 

 

 

 

 

 

Assuming:

 

 

 

{DEC 3}

     Mcf = Mult/Vult 

=

{Mcf}

     st = 0.33SQRT(f'cg)

=

{st}

 

MPa

 

     s1 = - P*1000/Ag - P*e*y3*1000/Ig

=

{s1}

 

MPa  {EXP 4}

    

 

     s2 = y2*10^6/Ic

=

{s2}

 

 

     Qlbwf = Qfw/(Ic*Bw2)

=

{Qlbwf}

 

mm^-2

 

 

 

 

 

 

 

Combine equns 1-3 to form a quadratic equation where the only unknown is Vtf viz:

 

     (Qlbwf*Vtf)^2 + (st*s2*Mcf)*Vtf + (st*s1 - st^2) = 0 ........ (4)

 

 

 

 

 

 

 

Setting:

 

 

 

 

 

     a = Qlbwf*Qlbwf*10^6

=

{a}

 

 

 

     b = st*s2*Mcf

=

{b}

 

{DEC 2}

 

     c = st*s1 - st^2

=

{c}

 

 

 

 

 

 

 

 

 

Expression (4) reduces to:

 

 

 

 

 

     a*Vtf^2 + b*Vtf + c = 0     and

 

 

 

 

 

     Vtf = (-b +- SQRT(b*b - 4*a*c))/2*a ................... (5)

 

 

 

 

 

 

 

 

 

 

 

The determinant of equation (5) is:

 

 

 

{EXP 4}

 

     f = b*b - 4*a*c = (st*s2*Mcf)^2 - 4*Qlbwf^2*(st*s1 - st^2)  

=

{f}

 

 

 

 

 

If f > 0 then a solution exists and:

 

{VucfNte$}

 

{DEC 1}

 

     Vtf = (-b + SQRT(f))/2*a

=

{Vtf}

 

kN

 

 

     Moment corresponding to Vtf  (Mtf = Mult*Vtf/Vult)

=

{Mtf}

 

kN.m

     Section cracking moment  (Mcr)

=

{Mcr}

 

kN.m {DEC 2}

 

     NOTE:   If f < 0 then Mtf is set to 999999 and Vtf to 0.0  

 

 

 

 

 

 

 

 

 

 

 

If f > 0 flexural stress at web-flange interface, uncracked sectn:

 

 

 

 

 

     sf = -P*1000/Ag + P*e*1000*y3/Ig + Vtf*Mcf*y2*10^6/Ic

=

{sf}

 

MPa

 

 

 

 

 

 

 

If f > 0 ultimate shear stress capacity of concrete on its own (tf):

 

 

 

 

 

     tf = 1000*Vtf*Qlbwf

=

{tf}

 

MPa

 

 

 

 

 

 

 

If f > 0 principal tension stress due to design prestress (s1f):

 

 

 

 

 

     s1f = ((0.5*sf)^2 + tf*tf)^0.5 + 0.5*sf

=

{s1f}

 

MPa

 

Allowable principal tensile strength (st = 0.33*f'cg^0.5)

=

{st}

 

MPa {DEC 0}

 

 

 

 

 

 

 

If Mtf > Mcr then the section is cracked; otherwise it is --------->

 

{VucfNte$}

 

 

 

    and the ultimate shear capacity of concrete alone (Vucf = Vtf)

=

{Vucf}

 

kN

 

 

 

 

 

 

 

Ultimate Shear Capacity of Concrete Alone:

 

 

 

 

 

 

 

 

 

     Ultimate design moment corresponding to V* (Mult)

=

{Mult}

 

kN.m

     Section cracking moment  (Mcr)

=

{Mcr}

 

kN.m

     If Mult > Mcr the section is cracked in flexure

 

     

 

 

 

 

 

Ultimate shear capacity of concrete alone (ignoring shear R/F) is given by:

 

    Vuc = Min of Vucs, Vucc, Vucf  or Vucs if section is cracked in flexure

=

{Vuc}

 

kN

 

 

 

 

 

 

 

Shear Rating Check:

 

{DEC 1} 

 

 

 

 

 

 

 

     Shear due to self-weight  (Vsw)

=

{Vsw}

 

kN

     Shear due to DL of the slab  (Vslab)

=

{Vslab}

 

kN

 

     Shear due to the superimposed dead load (Vsdl)   

=

{Vsdl}

 

kN

 

     Load Factor on self-weight (LFsw)  

=

{LFsw}

 

 

     Load Factor on the DL of slab  (LFslab)

=

{LFslab}

 

 

     Load Factor on the DL of superimposed DL (LFsdl)

=

{LFsdl}

 

 

     Load Factor on Live Load

=

{LFll}

 

 

 

 

 

 

 

 

Live load shear:

 

 

 

 

 

     Vucll = Vuc - (LFsw*Vsw + LFslab*Vslab + LFsdl*Vsdl)

=

{Vucll}

 

kN

 

 

 

 

 

 

 

Shear capacity rating (limited to 4.0 if it exceeds this value):

 

 

 

 

 

     Vuratll = Vucll / (LFll*Vll)

=

{Vuratll}

 

 

 

 

 

 

 

 

Design of web shear reinforcement:   ({CODE} Section 8.2.8 - 8.2.10)

 

 

 

 

 

 

 

Ultimate design shear force (V* = Vult)

=

{Vult}

 

kN {DEC 1}

 

Dist. from extreme compn fibre to outer tensile RF or tendon (do)

=

{Dsr}

mm  {DEC 2}

 

Capacity reduction factor - shear ( = Os) 

=

{Os}

 

{DEC 0}

 

 

 

Ultimate shear strength of girder having minimum amount of shear R/F (Vumin):

 

    Vumin = *(Vuc + 0.6*2*bw*do/1000)

=

{OVumin}

 

kN

 

 

 

 

 

 

 

Ultimate shear strength of girder limited by web crushing failure (Vumax):

 

    Vumax = *0.2*f'cg*2*bw*do/1000 + Pv

=

{OVumax}

 

kN

                       where Pv = Post-tensioning force

=

{Pv}

 

kN

 

 

 

 

 

 

 

Yield strength of shear reinforcement (fsysr)

=

{fsysr}

 

MPa {DEC 2}

 

Diameter of shear reinforcement (Dsr)

=

{Dsr}

 

mm {DEC 1}

 

Area of both legs of shear stirrup (Asv = 2*3.14*Dsr^2/4)

=

{Asv}

 

mm^2 {DEC 2}

 

 

 

 

 

 

 

OVumin > Vult: Concrete web shear capacity is sufficient

 

 

 

 

 

  Shear steel required

=

{AsNote1$}

 

 

 

  Angle between inclined conc comp strut & long. axis (v)

=

{Ov}

 

Degrees (prescribed minimum)

 

  Required area of shear reinforcmt (Asvsr1 = 0.35*2*bw/fsysr)

=

{Asvsr1}

 

mm^2 per mm {DEC 1}

 

  Actual shear reinforcement spacing (Svr = max allowed = Svmax)   

=

{Svmax}

 

mm

 

 

 

 

 

 

 

OVumin < Vult: Concrete web shear capacity is insufficient:

 

 {AsNote2$}

 

 

 

  Required contribution of steel R/F to shear capacity of the section (Vus): {DEC 0}

 

      Vus = (Vult - 0.7*Vuc)/0.7

=

{Vus}

 

kN {DEC 1}

 

  Angle between inclined conc comp strut & long. axis (v1)   

=

{Ov1}

 

Degrees

 

      v2 = 30 + 15*(Vult - Vumin)/ (Vumax - Vumin)

=

{Ov2}

 

 

 

      If v1 < 30 then v1 = 30

 

 

 

 

 

      If v1 > 45 then v1 = 45 otherwise v1 = v2

 

 

 

 

 

  Required area of shear R/F per unit length along beam (Asvsr2) {DEC 2}:

 

      Asvsr2 = ABS(Vus*1000*TAN(v1)/(do*fsysr)))

=

{Asvsr2}

 

mm^2 per mm {DEC 0}

 

  Required shear reinforcement spacing (Svr2 = Asv/Asvsr2)  

=

{Svr2}

 

mm

 

  Maximum allowable shear reinforcement spacing  (Svmax)

=

{Svmax}

 

mm

 

 

 

 

 

 

 

Shear Reinforcement:

 

 

 

 

 

  Actual shear stirrup spacing (Sv)

=

{Sv}

 

mm {DEC 2}

 

  Actual steel area per unit length (Asvs)

=

{Asvs}

 

mm^2 per mm