Chapter 5: Flexure o f Two-way Slabs Strengthened with Prestressed CFRP Sheets
6.7. Prediction o f Punching Shear Load
In order to p red ict the p unching shear capacity o f the tested slabs, four different approaches w ere studied, such as the application o f yield lines, the flexure/punching interaction, code equations, and FEA .
6.7.1. Yield line analysis
As a tw o-w ay slab experiences h ig h er loads, the crack lines start to form yield lines, as clearly show n in Fig. 6.9; therefore, the slab m ay be divided into several elastic parts by boundaries o f the yield lines. E ach divided section m ay be assum ed to rem ain as an elastic body. T he principal o f virtual w o rk is applied to calculate the load-carrying capacity. N ote that the elastic d efo n n a tio n o f a slab, in the case o f the y ield line analysis, is usually neglected. M ore details o f the yield line analysis are pro v id ed in reference (P ark and G am ble 2000). A m ong m any available m ethods, the m ethods p roposed by E lstner and H ognestad (1956) and G esund (1980) are considered for this study (L ongw orth 2004). It w as assum ed in this yield line application fo r the strengthened slabs that the C F R P sheets w ere evenly d istributed o ver the tension side o f the slab and an average am ount o f C FR P p e r w id th w as used. E lstn er and H ognestad (1956) assum ed that the yield lines form ed around the colum n and spread tow ards the co rn er o f a tw o-w ay slab, including neg ativ e yield lines w ith u p lift o f the slab at the com ers. T hey prop o sed an equation to p red ict the ultim ate flexural load o r colum n reaction (V//ex):
182 Y ail J. K im , P.E ng., P h.D . T hesis
C h ap ter 6: T w o-w ay S lab -C o lu m n C onnections
w here m c and mo are the radial m om ent resistan ces in the colum n strip and in the outer strips, respectively, and L and c are the w idth o f a slab and a colum n, respectively.
G esund (1980) stated that the yield lines for the above equation w ere o v ersim plified; then proposed the y ie ld fa n s n ear the corner o f a slab. U sing the p rincipal o f virtual w ork, the follow ing eq uation w as proposed:
w here m u is the ultim ate m om ent capacity. N o te that m u and the reinforcem ent ratio w ere assum ed to be constant across the entire slab w idth.
6.7.2. The flexure/punching interaction
The pu n ch in g shear capacity m ay be affected by flexural b eh aviour o f a slab b ecause the m axim um b en d in g m om ent usually occurs n ea r the location w here the pu n ch in g shear occurs. T herefore, the in teraction b etw een the flexure and the shear m ay decrease the ultim ate shear strength o f a tw o-w ay slab. M oe (1961) included this in teraction concept in the p ro p o sed em pirical equation (L ongw orth 2004):
183 Y ail J. K im , P .E ng., P h.D . T hesis
and d are the w idth o f a colum n, p erim e te r o f the colum n, and effective depth o f the slab, re s p e c tiv e ly ,/’c is the specified concrete strength, and Vfjex is the flexural load capacity.
6.7.3. The code equations and the FEA
The p u n ch in g shear capacity w as calcu lated as p e r C S A A 23.3-94 (1995). Since the code exhibits em pirical form ulas, including the aggregate interaction and dow el action o f rebars, the contribution o f the C F R P s could no t be included into the code equations, in contrast to the above tw o analytical m ethods. N o te that C SA A 23.3-04 (2006) indicated the sam e results. T he ultim ate p u n ch in g shear capacity in the F E A w as obtained as the highest load at w hich num erical converg en ce w as achieved.
6.7.4. Correlation o f the predictions with the experim ent
Fig. 6.11 show s a general com p ariso n o f each p red ictiv e m ethod, explained above, w ith the experim ent. T he approach em ploying the yield line m eth o d o verestim ated the p unching shear strength. T he possible reason is attributed to the fact that a tw o-w ay slab usually exhibits very ductile flexural response; thus the lim it o f the p u n ch in g shear failure m ight be exceeded w h en the flexural capacity w as obtained. In addition, the initial assum ption o f an evenly spread strengthening effect m ight influence the overestim ation.
T he flexure/punching shear in teraction m eth o d pred icted the experim ental failure loads relatively w ell w ith an average erro r o f 6.5 %. A s the C FR P strengthening effect increased, the difference b etw een the experim ent and the predictio n (Eq. 6.3) decreased.
T he code eq uation indicated relativ ely good agreem ent in the unstren g th en ed slab, including an average error o f 12.3 % ; how ever, it underestim ated the capacity o f the 184 Yail J. K im , P.E ng., Ph.D . T hesis
C hapter 6: T w o-w ay S lab -C o lu m n C o nnections
strengthened slabs due to a lack o f the strengthening contribution. T he FEA p ro v id ed the best p re d ic tio n o f the fo u r different m ethods, including an average error o f 5.1 % , as show n in Fig. 6.11, and the difference w as due to the initial assum ption o f the m odels (i.e., p erfect bond).