5.2 Design for Shear (Part I)

Transcription

1 5. Design or Shear (Par I) This seion overs he ollowing opis. General Commens Limi Sae o Collapse or Shear 5..1 General Commens Calulaion o Shear Demand The objeive o design is o provide ulimae resisane or shear (V ur ) greaer han he shear demand under ulimae loads (V u ). For simply suppored presressed beams, he maximum shear near he suppor is given by he beam heory. For oninuous presressed beams, a rigorous analysis an be done by he momen disribuion mehod. Else, he shear oeiiens in Table 13 o IS:456 - an be used under ondiions o uniorm ross-seion o he beams, uniorm loads and similar lenghs o span. Design o Sirrups The design is done or he riial seion. The riial seion is deined in Clause.6. o IS: In general ases, he ae o he suppor is onsidered as he riial seion. When he reaion a he suppor inrodues ompression a he end o he beam, he riial seion an be seleed a a disane eeive deph rom he ae o he suppor. The eeive deph is seleed as he greaer o d p or d s. d p = deph o CGS rom he exreme ompression iber d s = deph o enroid o non-presressed seel. Sine he CGS is a a higher loaion near he suppor, he eeive deph will be equal o d s. To vary he spaing o sirrups along he span, oher seions may be seleed or design. Usually he ollowing sheme is seleed or beams under uniorm load. 1) Close spaing or quarer o he span adjaen o he suppors. ) Wide spaing or hal o he span a he middle.

2 For large beams, more variaion o spaing may be seleed. The ollowing skeh shows he ypial variaion o spaing o sirrups. The span is represened by L. L / 4 L / L / 4 Figure 5-.1 Typial variaion o spaing o sirrups 5.. Limi Sae o Collapse or Shear The shear is sudied based on he apaiy o a seion whih is he limi sae o ollapse. The apaiy (or ulimae resisane) o a seion (V ur ) onsiss o a onree onribuion (V ) and he sirrup onribuion (V S ). V ur = V C + V S (5-.1) V inludes V z (onribuion rom unraked onree), V a (aggregae inerlok) and V d (dowel aion). The value o V depends on wheher he seion is raked due o lexure. Seion.4 o IS: gives wo expressions o V, one or raked seion and he oher or unraked seion. Usually, he expression or he unraked seion will govern near he suppor. The expression or he raked seion will govern near he mid span. O ourse, boh he expressions need o be evaluaed a a pariular seion. The lower value obained rom he wo expressions is seleed. For unraked seions, V =V o V =.67bD +.8 (5-.) p V o is he shear ausing web shear raking a CGC. In he above expression, b = breadh o he seion = b w, breadh o he web or langed seions

3 D = oal deph o he seion (h) = ensile srengh o onree =.4 k p = ompressive sress in onree a CGC due o he presress = P e /A. The value o p is aken as posiive (numeri value). Noe ha, a redued eeive presress needs o be onsidered in he ransmission lengh (explained in Seion 7.1) region o a pre-ensioned beam. The previous equaion an be derived based on he expression o he prinipal ensile sress (σ 1 ) a CGC. p v p σ 1 σ ( p,v) σ σ 1 Sae o sress a CGC Prinipal sresses Mohr s irle Figure 5-. Sae o sresses a a poin on he neural axis or a presressed beam The prinipal ensile sress is equaed o he dire ensile srengh o onree ( ). p p σ 1 =- + +v p p V Q = Ib = 4 (5-.3) In he previous equaion, I = gross momen o ineria Q = A y A = area o seion above CGC y = verial disane o enroid o A rom CGC.

4 A CGC + y Figure 5-.3 Cross-seion o a beam showing he variables or alulaing shear sress in he web Transposing he erms, Ib V = + Q p.67bd +.8 (5-.4) p The erm.67bd represens Ib/Q or he seion. I is exa or a reangular seion and onservaive or oher seions. To be onservaive, only 8% o he presressing ore is onsidered in he erm.8 p. For a langed seion, when he CGC is in he lange, he inerseion o web and lange is onsidered o be he riial loaion. The expression o V is modiied by subsiuing.8 p wih.8 (he sress in onree a he level o he inerseion o web and lange). In presene o inlined endons or verial presress, he verial omponen o he presressing ore (V p ) an be added o V. V V +V p =.67bD +.8 +V p p (5-.5) For raked seions, V =V r V = bd k pe pk τbd + M V M u u (5-.6)

5 V r is he shear orresponding o lexure shear raking. The erm (1.55 pe / pk )τ bd is he addiional shear ha hanges a lexural rak o a lexure shear rak. The noaions in he previous equaion are as ollows. pe = eeive presress in he endon aer all losses.6 pk pk = haraerisi srengh o presressing seel τ = ulimae shear sress apaiy o onree, obained rom Table 6 o IS: I is given or values o A p / bd, where d is he deph o CGS. The values are ploed in he nex igure. b = breadh o he seion = b w, breadh o he web or langed seions d = disane rom he exreme ompression ibre o he enroid o he endons a he seion onsidered M = momen iniiaing a lexural rak M u = momen due o ulimae loads a he design seion V u = shear due o ulimae loads a he design seion. 1. τ (N/mm ) A p /bd x 1 4 M3 M4 Figure 5-.4 Variaion o shear srengh o onree The erm (M /M u )V u is he shear orresponding o he momen M, ha deompresses (nulliies he ee o presress) he ension ae and iniiaes a lexural rak. The expression o M is given below. M = I y.8 p (5-.7)

6 In he above expression, p = magniude o he ompressive sress in onree a he level o CGS due o presress only. An equal amoun o ensile sress is required o deompress he onree a he level o CGS. The orresponding momen is p I / y. In he expression o M, I = gross momen o ineria y = deph o he CGS rom CGC. The aor.8 implies ha M is esimaed o be 8% o he momen ha deompresses he onree a he level o CGS. Sine he onree is raked and he inlinaion o endon is small away rom he suppors, any verial omponen o he presressing ore is no added o V r. Maximum Permissible Shear Sress To hek he rushing o onree in shear ompression ailure, he shear sress is limied o a maximum value (τ,max ). The value o τ,max depends on he grade o onree and is given in Table 7 o IS: Vu bd τ,max (5-.8) In he previous expression, d = greaer o d p or d s d p = deph o CGS rom he exreme ompression iber d s = deph o enroid o regular seel V u = shear ore a a seion due o ulimae loads.

7 6 τ, max (N/mm ) k (N/mm ) Figure 5-.5 Variaion o maximum permissible shear sress in onree