1 Background
2 The GSVATM for RC Beams
Plain truss analogy for a RC thin beam under shear:
| |
\( R = \sqrt {C^{2} + T^{2} } \quad (1) \)
\( \upbeta = \arctan \left( {T/C} \right)\quad (2) \)
\( \upgamma = \alpha + \beta \quad (3) \)
\( C = \sigma_{2}^{c} t_{c} d_{v} \cos \upalpha \quad (4) \)
\( T = \sigma_{1}^{c} t_{c} d_{v} \sin \upalpha \quad (5) \)
| |
Space truss analogy for a RC box beam under torsion:
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Equilibrium equations: | |
\( M_{T} = \frac{{2A_{o} R\sin \gamma }}{{d_{v} }}\quad (6) \)
\( t_{c} = \frac{{A_{sl} f_{sl} }}{{\sigma_{2}^{c} p_{o} }}\frac{\cos \beta }{\cos \alpha \cos \upgamma }{\text{ for }}\upgamma = \upalpha + \upbeta \le 90^{ \circ } \quad (7) \)
\( \upalpha = \arctan \left( {\frac{{\sqrt {F^{2} \left( {\tan \upbeta } \right)^{2} + F\left( {\tan \upbeta } \right)^{4} + F + \left( {\tan \upbeta } \right)^{2} } }}{{F\left( {\tan \upbeta } \right)^{2} + 1}}} \right){\text{ with }}F = \frac{{A_{st} f_{st} p_{o} }}{{A_{sl} f_{sl \, } s}}\quad (8) \)
| |
Compatibility equations: | |
\( \varepsilon_{st} = \left( {\frac{{A_{o}^{2} \sigma_{2 \, }^{c} \sin \gamma }}{{p_{o} M_{T \, } \cos \beta \tan \alpha \sin \alpha }} - \frac{1}{2}} \right)\varepsilon_{2s}^{c} \quad (9) \)
\( \varepsilon_{sl} = \left( {\frac{{A_{o}^{2} \sigma_{2 \, }^{c} \sin \upgamma }}{{p_{o} M_{T \, } \cos \upbeta \cot \upalpha \sin \upalpha }} - \frac{1}{2}} \right)\varepsilon_{2s}^{c} \quad (10) \)
\( \theta = \frac{{\varepsilon_{2s}^{c} }}{{2t_{c \, } \sin \upalpha \cos \upalpha }}\quad (11) \)
\( \varepsilon_{1s}^{c} = 2\varepsilon_{1}^{c} = 2\varepsilon_{sl} + 2\varepsilon_{st} + \varepsilon_{2s}^{c} \quad (12) \)
|
\( \sigma_{2}^{c} = \upbeta_{\sigma } f_{c}^{\prime } \left[ {2\left( {\frac{{\upvarepsilon_{2}^{c} }}{{\upbeta_{\upvarepsilon } \upvarepsilon_{o} }}} \right) - \left( {\frac{{\upvarepsilon_{2}^{c} }}{{\upbeta_{\upvarepsilon } \upvarepsilon_{o} }}} \right)^{2} } \right]{\text{ if }}\upvarepsilon_{2}^{c} \le \upbeta_{\upvarepsilon } \upvarepsilon_{o} \quad (13) \)
| |
\( \sigma_{2}^{c} = \upbeta_{\sigma } f_{c}^{\prime } \left[ {1 - \left( {\frac{{\upvarepsilon_{2}^{c} - \upbeta_{\upvarepsilon } \upvarepsilon_{o} }}{{2\upvarepsilon_{o} - \upbeta_{\upvarepsilon } \upvarepsilon_{o} }}} \right)^{2} } \right]{\text{ if }}\upvarepsilon_{2}^{c} > \upbeta_{\upvarepsilon } \upvarepsilon_{o} \quad (14) \)
| |
\( \upbeta_{ * } = \upbeta_{\upsigma } = \upbeta_{\upvarepsilon } = \frac{{R\left( {f_{c}^{\prime } } \right)}}{{\sqrt {1 + \frac{{400\upvarepsilon_{1}^{c} }}{{\upeta^{\prime}}}} }}\quad (15) \)
| |
\( \upeta = \frac{{\rho_{l} f_{ly} }}{{\rho_{t} f_{ty} \, }}\quad (16) \)
| |
\( \left\{ {\begin{array}{*{20}l} {\upeta \le 1 \, \Rightarrow \, \upeta^{\prime} = \upeta } \\ {\upeta > 1 \, \Rightarrow \, \upeta^{\prime} = 1/\upeta } \\ \end{array} } \right.\quad (17) \)
| |
\( R\left( {f_{c}^{\prime } } \right) = \frac{5.8}{{\sqrt {f_{c}^{\prime } \, \left( {\text{MPa}} \right)} \, }} \le 0.9\quad (18) \)
| |
\( \upsigma_{1}^{c} = E_{c} \upvarepsilon_{1}^{c} {\text{ if }}\upvarepsilon_{1}^{c} \le \upvarepsilon_{cr} \quad (19) \)
| |
\( \sigma_{1}^{c} = f_{cr} \left( {\frac{{\upvarepsilon_{cr} }}{{\upvarepsilon_{2}^{c} }}} \right)^{0.4} {\text{ if }}\upvarepsilon_{1}^{c} > \upvarepsilon_{cr} \quad (20) \)
| |
\( k_{2}^{c} = \frac{{\upvarepsilon_{2s}^{c} }}{{\upbeta_{\upvarepsilon } \upvarepsilon_{o} }} - \frac{{(\upvarepsilon_{2s}^{c} )^{2} }}{{3(\upbeta_{\upvarepsilon } \upvarepsilon_{o} )^{2} }}\quad (21) \)
| |
\( E_{c} = 3875\;K\sqrt {f_{c}^{\prime } (\text{MPa} )} \quad (22) \)
| |
\( \varepsilon_{cr} = 0.00008\;K\quad K = 1.45{\text{ or }}1.24{\text{ (plain or hollow)}}\quad (23) \)
| |
\( \sigma_{2}^{c} = k_{2}^{c} \upbeta_{\sigma } f_{c}^{\prime } \quad (24) \)
\( k_{2}^{c} = \frac{{\varepsilon_{2s}^{c} }}{{\upbeta_{\varepsilon } \varepsilon_{o} }} - \frac{{(\varepsilon_{2s}^{c} )^{2} }}{{3(\beta_{\varepsilon } \varepsilon_{o} )^{2} }}{\text{ if }}\varepsilon_{2s}^{c} \le \upbeta_{\varepsilon } \varepsilon_{o} \quad (25) \)
\( k_{2}^{c} = 1 - \frac{{\upbeta_{\varepsilon } \varepsilon_{o} }}{{3\varepsilon_{2s}^{c} }} - \frac{{(\varepsilon_{2s}^{c} - \upbeta_{\varepsilon } \varepsilon_{o} )^{3} }}{{3\varepsilon_{2s}^{c} (2\varepsilon_{o} - \upbeta_{\varepsilon } \varepsilon_{o} )^{2} }}{\text{ if }}\varepsilon_{2s}^{c} > \upbeta_{\varepsilon } \varepsilon_{o} \quad (26) \)
| |
\( \sigma_{1}^{c} = k_{1}^{c} f_{cr} \quad (27) \)
\( k_{1}^{c} = \frac{{\varepsilon_{1s}^{c} }}{{2\varepsilon_{cr} }}{\text{ if }}\varepsilon_{1s}^{c} \le \varepsilon_{cr} \quad (28) \)
\( k_{1}^{c} = \frac{{\varepsilon_{cr} }}{{2\varepsilon_{1s}^{c} }} + \frac{{(\varepsilon_{cr} )^{0.4} }}{{0.6\varepsilon_{1s}^{c} }}[(\varepsilon_{1s}^{c} )^{0.6} - (\varepsilon_{cr} )^{0.6} ]{\text{ if }}\varepsilon_{1s}^{c} > \varepsilon_{cr} \quad (29) \)
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\( f_{s} = \frac{{0.975E_{s} \varepsilon_{s} }}{{\left[ {1 + \left( {\frac{{1,1E_{s} \varepsilon_{s} }}{{f_{y} }}} \right)^{m} } \right]^{{\frac{1}{m}}} }} + 0.025E_{s} \varepsilon_{s} \quad (30) \)
| |
\( m = \frac{1}{9B - 0.2} \le 25\quad (31) \)
| |
\( B = \frac{1}{\rho }\left( {\frac{{f_{cr} }}{{f_{y} }}} \right)^{1.5} \quad (32) \)
| |
\( f_{p} = E_{p} \varepsilon_{p} {\text{ if }}\varepsilon_{p} \le \varepsilon_{p0.1\% } = f_{p0.1\% } /E_{p} \quad (33) \)
| |
\( f_{p} = \frac{{E_{p} \varepsilon_{p} }}{{\left[ {1 + \left( {\frac{{E_{p} \varepsilon_{p} }}{{f_{pt} }}} \right)^{4.38} } \right]^{{\frac{1}{4.38}}} }}{\text{ if }}\varepsilon_{p} > \varepsilon_{p0.1\% } \quad (34) \)
|
3 The GSVATM for PC Beams
3.1 Assumptions to Incorporate Prestress
-
The calculation of the \( M_{T}{-}\theta \) curve for the pre-decompression stage is not relevant because the small associated part of the \( M_{T}{-}\theta \) curve is perfectly linear. Then, it is assumed that GSVATM will only start the calculation procedure after the concrete decompression. This procedure is similar to the same one assumed by Hsu and Mo (1985b) to extend the VATM to PC beams under torsion. This assumption allows to simplificate the solution procedure because the strain and stress gradients in the concrete strut and tie do not need to include the initial compressive stress state in concrete due to prestress. The modelling of this initial stress state would complicate needlessly the calculation procedure for the very low loading stages (Jeng et al. 2010);
-
Related with the previous assumption, it should be referred that the influence of long term response of the PC beams was neglected in this study, as also assumed by Jeng et al. (2010). In such study, the pre-decompression response of the PC beams was computed and the stress variation in the concrete was only due to the incremental torsional loading. However, it should be referred that stress variations in the concrete decompression stage in fact exists due to viscous effects in the concrete, in accordance with the long term response of structural concrete structures (Price and Anderson 1992; Johnson 1994; Ascione et al. 2011; Berardi and Mancusi 2012, 2013);
-
From the referred previously, despite the initial compressive stress state due to prestress is not directly modelled to compute the strain and stress states in concrete before decompression, it must be considered to compute the initial strain in the longitudinal prestress and non-prestress reinforcement, in order to compute the strain at concrete decompression. In addition, the torsional moment corresponding to the tensile strength of concrete, that is the cracking torque, must be corrected to include the favourable effect due to prestress. This correction is performed by using a simple prestress factor;
-
After the concrete decompression, and as for the non-prestress reinforcement, longitudinal prestress reinforcement participates for the longitudinal equilibrium of the beam. Then, equilibrium equations must incorporate the force in the prestress reinforcement;
-
An additional \( \sigma{-}\varepsilon \) relationship for the prestress steel reinforcement in tension must be implemented to model the behavior of this material and compute the stresses.
3.2 Changes in the GSVATM
3.3 Solution Procedure
4 Comparative Analysis with Experimental Results and Codes of Practice
Beam | Sectiona | x (cm) | y (cm) | t (cm) |
\( A_{sl} \)
(cm2) |
\( A_{st} /s \)
(cm2/m) |
\( A_{p} \)
(cm2) |
\( \rho_{l} \)
( %) |
\( \rho_{t} \)
( %) |
\( \rho_{p} \)
( %) |
\( f_{c} \)
(MPa) |
\( f_{sly} \)
(MPa) |
\( f_{sty} \)
(MPa) |
\( f_{p0.1\% } \)
(MPa) |
\( f_{p,i} \)
(MPa) |
\( f_{cp} \)
(MPa) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
P2 (Mitchell and Collins 1974) | H | 35.6 | 43.2 | 8.9 | 5.6 | 7.4 | 4.6 | 0.37 | 0.61 | 0.29 | 32.9 | 407 | 407 | 1476 | 1145 | 4.9 |
P3 (Mitchell and Collins 1974) | H | 35.6 | 43.2 | – | 4.3 | 7.4 | 1.2 | 0.28 | 0.63 | 0.07 | 34.0 | 328 | 328 | 1476 | 1145 | 0.9 |
P8 (Hsu and Mo 1985b) | H | 25.4 | 38.1 | – | 5.2 | 22.6 | 9.6 | 0.53 | 2.90 | 1.02 | 31.0 | 334 | 336 | 959 | 690 | 6.8 |
D1 (Bernardo et al. 2013) | H | 60.0 | 60.0 | 11.4 | 23.8 | 11.2 | 4.2 | 0.66 | 0.68 | 0.11 | 80.8 | 724 | 715 | 1670 | 640 | 1.8 |
D2 (Bernardo et al. 2013) | H | 60.0 | 60.0 | 11.5 | 23.8 | 11.2 | 5.6 | 0.66 | 0.69 | 0.15 | 58.8 | 724 | 715 | 1670 | 1100 | 3.1 |
PA1R (El-Degwy and McMullen 1985) | P | 25.4 | 25.4 | – | 2.8 | 4.5 | 0.9 | 0.40 | 0.80 | 0.20 | 43.6 | 435 | 310 | 1638 | 1207 | 1.7 |
PA2 (El-Degwy and McMullen 1985) | P | 25.4 | 25.4 | – | 5.2 | 7.7 | 1.5 | 0.80 | 1.40 | 0.20 | 45.6 | 483 | 310 | 1663 | 1207 | 2.8 |
PA3 (El-Degwy and McMullen 1985) | P | 25.4 | 25.4 | – | 8.0 | 7.9 | 2.1 | 1.20 | 1.40 | 0.40 | 41.8 | 389 | 435 | 1744 | 1303 | 4.2 |
PA4 (El-Degwy and McMullen 1985) | P | 25.4 | 25.4 | – | 11.4 | 11.0 | 3.0 | 1.80 | 2.00 | 0.40 | 42.2 | 419 | 435 | 1709 | 1303 | 6.0 |
PB1 (El-Degwy and McMullen 1985) | P | 17.8 | 35.6 | – | 2.8 | 4.5 | 0.9 | 0.40 | 0.80 | 0.20 | 45.8 | 435 | 310 | 1638 | 1207 | 1.8 |
PB2 (El-Degwy and McMullen 1985) | P | 17.8 | 35.6 | – | 5.2 | 7.7 | 1.5 | 0.80 | 1.50 | 0.30 | 45.8 | 483 | 310 | 1663 | 1207 | 2.9 |
PB3 (El-Degwy and McMullen 1985) | P | 17.8 | 35.6 | – | 8.0 | 7.5 | 2.1 | 1.30 | 1.40 | 0.30 | 45.5 | 389 | 435 | 1744 | 1303 | 4.2 |
PB4 (El-Degwy and McMullen 1985) | P | 17.8 | 35.6 | – | 11.4 | 10.2 | 3.0 | 1.80 | 2.00 | 0.50 | 45.5 | 419 | 435 | 1709 | 1303 | 6.1 |
PC1 (El-Degwy and McMullen 1985) | P | 14.6 | 43.8 | – | 2.8 | 3.9 | 0.9 | 0.40 | 0.80 | 0.20 | 42.2 | 435 | 310 | 1638 | 1207 | 1.8 |
PC2 (El-Degwy and McMullen 1985) | P | 14.6 | 43.8 | – | 5.2 | 6.9 | 1.5 | 0.80 | 1.50 | 0.20 | 45.1 | 483 | 310 | 1663 | 1207 | 2.8 |
PC3 (El-Degwy and McMullen 1985) | P | 14.6 | 43.8 | – | 8.0 | 6.8 | 2.1 | 1.30 | 1.40 | 0.30 | 41.3 | 389 | 435 | 1744 | 1303 | 4.2 |
PC4 (El-Degwy and McMullen 1985) | P | 14.6 | 43.8 | – | 11.4 | 9.5 | 3.0 | 1.80 | 2.00 | 0.40 | 42.1 | 419 | 435 | 1709 | 1303 | 6.1 |
H3AR (Wafa et al. 1995) | P | 14.0 | 42.0 | – | 8.0 | 10.3 | 4.0 | 1.40 | 2.00 | 0.60 | 92.2 | 487 | 390 | 1816 | 1366 | 8.3 |
H2A (Wafa et al. 1995) | P | 17.0 | 34.0 | – | 8.0 | 12.6 | 4.0 | 1.40 | 2.20 | 0.70 | 91.9 | 487 | 390 | 1816 | 1338 | 8.7 |
H1AR (Wafa et al. 1995) | P | 24.0 | 24.0 | – | 8.0 | 12.6 | 4.0 | 1.40 | 2.10 | 0.70 | 94.7 | 487 | 390 | 1816 | 1374 | 6.4 |
H3B (Wafa et al. 1995) | P | 14.0 | 42.0 | – | 6.2 | 5.6 | 2.1 | 1.00 | 1.10 | 0.40 | 91.5 | 374 | 387 | 1841 | 1309 | 4.0 |
H2B (Wafa et al. 1995) | P | 17.0 | 34.0 | – | 6.2 | 6.0 | 2.1 | 1.10 | 1.10 | 0.30 | 95.6 | 374 | 387 | 1841 | 1344 | 4.3 |
H1B (Wafa et al. 1995) | P | 24.0 | 24.0 | – | 6.2 | 6.5 | 2.1 | 1.10 | 1.10 | 0.30 | 89.8 | 374 | 387 | 1841 | 1258 | 4.3 |
M3A (Wafa et al. 1995) | P | 14.0 | 42.0 | – | 8.0 | 10.3 | 4.0 | 1.40 | 2.00 | 0.60 | 69.9 | 487 | 390 | 1816 | 1036 | 6.5 |
M2A (Wafa et al. 1995) | P | 17.0 | 34.0 | – | 8.0 | 11.3 | 4.0 | 1.40 | 2.00 | 0.70 | 70.1 | 487 | 390 | 1816 | 1021 | 6.2 |
M1A (Wafa et al. 1995) | P | 24.0 | 24.0 | – | 8.0 | 12.6 | 4.0 | 1.40 | 2.10 | 0.70 | 72.5 | 487 | 390 | 1816 | 1053 | 6.4 |
M3B (Wafa et al. 1995) | P | 14.0 | 42.0 | – | 6.2 | 5.6 | 2.1 | 1.00 | 1.10 | 0.40 | 69.3 | 374 | 387 | 1841 | 991 | 3.2 |
M2B (Wafa et al. 1995) | P | 17.0 | 34.0 | – | 6.2 | 6.0 | 2.1 | 1.10 | 1.10 | 0.30 | 69.7 | 374 | 387 | 1841 | 979 | 3.3 |
M1B (Wafa et al. 1995) | P | 24.0 | 24.0 | – | 6.2 | 6.5 | 2.1 | 1.10 | 1.10 | 0.30 | 72.0 | 374 | 387 | 1841 | 1008 | 3.2 |
Beam |
\( M_{Tcr,\exp } \)
(kN m) |
\( M_{{Tcr,\text{th} }} \)
(kN m) |
\( \frac{{M_{Tcr,\exp } }}{{M_{{Tcr,\text{th} }} }} \)
|
\( \theta_{Tcr,\exp } \)
(º/m) | \( \theta_{{Tcr,\text{th} }} \) º/m |
\( \frac{{\theta_{Tcr,\exp } }}{{\theta_{{Tcr,\text{th} }} }} \)
|
\( M_{Tu,\exp } \)
(kN m) |
\( M_{{Tu,\text{th} }} \)
(kN m) |
\( \frac{{M_{Tu,\exp } }}{{M_{{Tu,\text{th} }} }} \)
|
\( \theta_{Tu,\exp } \)
(º/m) |
\( \theta_{{Tu,\text{th} }} \)
(º/m) |
\( \frac{{\theta_{Tu,\exp } }}{{\theta_{{Tu,\text{th} }} }} \)
|
---|---|---|---|---|---|---|---|---|---|---|---|---|
P2 (Mitchell and Collins 1974) | 58.0 | 55.4 | 1.05 | 0.14 | 0.09 | 1.55 | 87.1 | 84.5 | 1.03 | 2.80 | 1.86 | 1.50 |
P3 (Mitchell and Collins 1974) | 41.2 | 44.4 | 0.93 | 0.21 | 0.10 | 2.06 | 55.8 | 71.4 | 0.78 | 3.14 | 2.60 | 1.21 |
P8 (Hsu and Mo 1985b) | 45.2 | 47.6 | 0.95 | 0.24 | 0.21 | 1.13 | 61.8 | 67.4 | 0.92 | 1.89 | 1.66 | 1.14 |
D1 (Bernardo et al. 2013) | 172.9 | 184.4 | 0.94 | 0.07 | 0.05 | 1.41 | 396.0 | 450.6 | 0.88 | 1.73 | 1.98 | 0.87 |
D2 (Bernardo et al. 2013) | 184.7 | 178.7 | 1.03 | 0.07 | 0.08 | 0.89 | 447.7 | 434.2 | 1.03 | 1.93 | 1.68 | 1.15 |
PA1R (El-Degwy and McMullen 1985) | 18.6 | 16.9 | 1.10 | 0.14 | 0.18 | 0.78 | 21.8 | 21.4 | 1.02 | 2.98 | 4.34 | 0.69 |
PA2 (El-Degwy and McMullen 1985) | 22.8 | 23.1 | 0.99 | 0.61 | 0.23 | 2.71 | 29.3 | 31.5 | 0.93 | 2.87 | 2.85 | 1.01 |
PA3 (El-Degwy and McMullen 1985) | 25.1 | 26.7 | 0.94 | 0.38 | 0.29 | 1.31 | 34.0 | 35.2 | 0.96 | 2.68 | 2.72 | 0.98 |
PA4 (El-Degwy and McMullen 1985) | 26.3 | 28.6 | 0.92 | 0.29 | 0.26 | 1.09 | 37.4 | 40.0 | 0.94 | 2.95 | 2.37 | 1.24 |
PB1 (El-Degwy and McMullen 1985) | 16.4 | 14.9 | 1.10 | 0.51 | 0.23 | 2.23 | 22.2 | 20.5 | 1.08 | 4.47 | 4.50 | 1.00 |
PB2 (El-Degwy and McMullen 1985) | 18.9 | 17.0 | 1.11 | 0.43 | 0.23 | 1.87 | 27.5 | 29.7 | 0.93 | 2.75 | 3.24 | 0.85 |
PB3 (El-Degwy and McMullen 1985) | 21.8 | 18.6 | 1.17 | 0.56 | 0.31 | 1.81 | 32.6 | 32.9 | 0.99 | 3.08 | 2.98 | 1.03 |
PB4 (El-Degwy and McMullen 1985) | 24.1 | 22.3 | 1.08 | 0.44 | 0.26 | 1.68 | 37.6 | 36.9 | 1.02 | 2.63 | 2.55 | 1.03 |
PC1 (El-Degwy and McMullen 1985) | 13.9 | 13.5 | 1.03 | 0.71 | 0.39 | 1.83 | 19.7 | 18.0 | 1.10 | 5.23 | 4.48 | 1.17 |
PC2 (El-Degwy and McMullen 1985) | 17.2 | 19.3 | 0.89 | 0.33 | 0.22 | 1.54 | 28.6 | 26.5 | 1.08 | 3.38 | 3.46 | 0.98 |
PC3 (El-Degwy and McMullen 1985) | 18.5 | 15.9 | 1.16 | 0.55 | 0.30 | 1.80 | 32.8 | 28.8 | 1.14 | 4.74 | 3.22 | 1.47 |
PC4 (El-Degwy and McMullen 1985) | 21.6 | 22.7 | 0.95 | 0.50 | 0.32 | 1.55 | 38.7 | 33.5 | 1.16 | 3.84 | 2.71 | 1.42 |
H3AR (Wafa et al. 1995) | 22.7 | 20.4 | 1.11 | – | 0.29 | – | 33.7 | 36.8 | 0.91 | – | 2.84 | – |
H2A (Wafa et al. 1995) | 15.1 | 20.8 | 0.73 | – | 0.27 | – | 35.8 | 41.6 | 0.86 | – | 2.60 | – |
H1AR (Wafa et al. 1995) | 31.6 | 27.7 | 1.14 | – | 0.25 | – | 38.4 | 44.0 | 0.87 | – | 2.34 | – |
H3B (Wafa et al. 1995) | 20.7 | 22.6 | 0.91 | – | 0.37 | – | 26.4 | 26.8 | 0.98 | – | 3.31 | – |
H2B (Wafa et al. 1995) | 27.1 | 29.2 | 0.93 | – | 0.37 | – | 29.5 | 29.1 | 1.01 | – | 2.90 | – |
H1B (Wafa et al. 1995) | 29.9 | 27.8 | 1.07 | – | 0.37 | – | 31.3 | 30.7 | 1.02 | – | 2.85 | – |
M3A (Wafa et al. 1995) | 19.6 | 17.9 | 1.09 | – | 0.36 | – | 30.0 | 32.6 | 0.92 | – | 2.87 | – |
M2A (Wafa et al. 1995) | 27.0 | 28.8 | 0.94 | – | 0.44 | – | 31.9 | 35.3 | 0.91 | – | 2.58 | – |
M1A (Wafa et al. 1995) | 31.5 | 33.1 | 0.95 | – | 0.33 | – | 35.4 | 39.1 | 0.91 | – | 2.37 | – |
M3B (Wafa et al. 1995) | 18.3 | 15.5 | 1.18 | – | 0.38 | – | 24.7 | 24.6 | 1.01 | – | 3.48 | – |
M2B (Wafa et al. 1995) | 26.2 | 24.5 | 1.07 | – | 0.41 | – | 26.2 | 26.3 | 1.00 | – | 3.11 | – |
M1B (Wafa et al. 1995) | 26.8 | 27.5 | 0.97 | – | 0.43 | – | 28.9 | 28.6 | 1.01 | – | 2.88 | – |
\( \bar{x} \) = 1.02 | \( \bar{x} \) = 1.60 | \( \bar{x} \) = 0.98 | \( \bar{x} \) = 1.10 | |||||||||
s = 0.10 | s = 0.49 | s = 0.09 | s = 0.22 | |||||||||
cv = 10.3% | cv = 30.6% | cv = 8.8% | cv = 20.3% |
Beam | ACI 318R-14 (ACI Committee 318 2014) | MC 10 (CEB-FIP MODEL 2010) | EC 2 (NP EN 1992-1-1 1992) | CAN3-A23.3-14 (CSA Standard 2014) | ||||||
---|---|---|---|---|---|---|---|---|---|---|
\( M_{Tcr,n} \)
(kN m) |
\( \frac{{M_{Tcr,\exp } }}{{M_{Tcr,n} }} \)
|
\( M_{Tu,n} \)
(kN m) |
\( \frac{{M_{Tu,\exp } }}{{M_{Tu,n} }} \)
|
\( M_{Tu,n} \)
(kN m) |
\( \frac{{M_{Tu,\exp } }}{{M_{Tu,n} }} \)
|
\( M_{Tu,n} \)
(kN m) |
\( \frac{{M_{Tu,\exp } }}{{M_{Tu,n} }} \)
|
\( M_{Tu,n} \)
(kN m) |
\( \frac{{M_{Tu,\exp } }}{{M_{Tu,n} }} \)
| |
P2 (Jeng et al. 2010) | 137.18 | 0.42 | 68.11 | 1.28 | 111.28 | 0.78 | 112.13 | 0.78 | 142.08 | 0.613 |
P3 (Jeng et al. 2010) | 88.74 | 0.46 | 46.26 | 1.21 | 100.27 | 0.56 | 54.41 | 1.03 | 68.95 | 0.809 |
P8 (Hsu and Mo 1985b) | 75.08 | 0.60 | 30.86 | 2.00 | 36.60 | 1.69 | 116.31 | 0.53 | 64.67 | 0.956 |
D1 (Bernardo et al. 2015c) | 502.16 | 0.34 | 505.64 | 0.78 | 730.46 | 0.54 | 594.78 | 0.67 | 753.65 | 0.525 |
D2 (Bernardo et al. 2015c) | 515.56 | 0.36 | 548.28 | 0.82 | 612.36 | 0.73 | 644.94 | 0.69 | 817.21 | 0.548 |
PA1R (Bernardo et al. 2013) | 30.55 | 0.61 | 18.23 | 1.20 | 20.28 | 1.07 | 21.45 | 1.02 | 27.18 | 0.802 |
PA2 (Bernardo et al. 2013) | 35.00 | 0.65 | 19.22 | 1.52 | 23.41 | 1.25 | 37.72 | 0.78 | 47.79 | 0.613 |
PA3 (Bernardo et al. 2013) | 38.33 | 0.65 | 19.18 | 1.77 | 21.26 | 1.60 | 52.17 | 0.65 | 46.65 | 0.729 |
PA4 (Bernardo et al. 2013) | 43.65 | 0.60 | 19.27 | 1.94 | 21.32 | 1.75 | 71.68 | 0.52 | 47.09 | 0.794 |
PB1 (Bernardo et al. 2013) | 28.70 | 0.57 | 17.01 | 1.31 | 23.05 | 0.96 | 20.01 | 1.11 | 25.36 | 0.876 |
PB2 (Bernardo et al. 2013) | 32.33 | 0.58 | 16.54 | 1.66 | 25.84 | 1.06 | 34.95 | 0.79 | 42.12 | 0.653 |
PB3 (Bernardo et al. 2013) | 36.41 | 0.60 | 17.30 | 1.88 | 24.10 | 1.35 | 47.06 | 0.69 | 43.91 | 0.742 |
PB4 (Bernardo et al. 2013) | 41.30 | 0.58 | 17.30 | 2.17 | 23.82 | 1.58 | 67.25 | 0.56 | 43.91 | 0.856 |
PC1 (Bernardo et al. 2013) | 25.83 | 0.54 | 14.73 | 1.34 | 22.10 | 0.89 | 17.33 | 1.14 | 21.96 | 0.897 |
PC2 (Bernardo et al. 2013) | 29.87 | 0.58 | 13.93 | 2.05 | 25.67 | 1.11 | 30.12 | 0.95 | 35.21 | 0.812 |
PC3 (Bernardo et al. 2013) | 32.72 | 0.57 | 14.13 | 2.32 | 23.09 | 1.42 | 41.22 | 0.80 | 34.17 | 0.960 |
PC4 (Bernardo et al. 2013) | 37.40 | 0.58 | 14.27 | 2.71 | 23.33 | 1.66 | 57.54 | 0.67 | 34.83 | 1.111 |
H3AR (Jeng 2015) | 47.48 | 0.48 | 43.09 | 0.78 | 33.03 | 1.02 | 50.69 | 0.66 | 64.23 | 0.525 |
H2A (Jeng 2015) | 51.12 | 0.30 | 51.86 | 0.69 | 33.45 | 1.07 | 61.00 | 0.59 | 77.29 | 0.463 |
H1AR (Jeng 2015) | 48.96 | 0.65 | 55.28 | 0.69 | 29.72 | 1.29 | 65.03 | 0.59 | 82.39 | 0.466 |
H3B (Jeng 2015) | 37.37 | 0.55 | 24.04 | 1.10 | 32.13 | 0.82 | 28.28 | 0.93 | 35.84 | 0.737 |
H2B (Jeng 2015) | 41.14 | 0.66 | 27.06 | 1.09 | 31.83 | 0.93 | 31.83 | 0.93 | 40.33 | 0.732 |
H1B (Jeng 2015) | 42.42 | 0.70 | 29.95 | 1.05 | 27.45 | 1.14 | 35.23 | 0.89 | 44.64 | 0.701 |
M3A (Jeng 2015) | 39.85 | 0.49 | 43.09 | 0.70 | 27.75 | 1.08 | 50.69 | 0.59 | 57.79 | 0.519 |
M2A (Jeng 2015) | 41.60 | 0.65 | 49.19 | 0.65 | 27.08 | 1.18 | 57.86 | 0.55 | 64.37 | 0.496 |
M1A (Jeng 2015) | 44.96 | 0.70 | 55.27 | 0.64 | 25.13 | 1.41 | 65.02 | 0.54 | 72.52 | 0.488 |
M3B (Jeng 2015) | 31.94 | 0.57 | 24.05 | 1.03 | 26.98 | 0.92 | 28.28 | 0.87 | 35.84 | 0.689 |
M2B (Jeng 2015) | 34.11 | 0.77 | 27.06 | 0.97 | 26.08 | 1.00 | 31.83 | 0.82 | 40.33 | 0.650 |
M1B (Jeng 2015) | 36.07 | 0.74 | 29.95 | 0.96 | 23.88 | 1.21 | 35.23 | 0.82 | 44.64 | 0.647 |
\( \bar{x} \) = 0.59 | \( \bar{x} \) = 1.17 | \( \bar{x} \) = 1.13 | \( \bar{x} \) = 0.77 | \( \bar{x} \) = 0.68 | ||||||
s = 0.12 | s = 0.63 | s = 0.22 | s = 0.18 | s = 0.19 | ||||||
cv = 20% | cv = 54% | cv = 20% | cv = 23% | cv = 28% |
5 Conclusions
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It was shown that the predictions from the GSVATM compare very well with the cracking torque and reasonably with the corresponding twist. In general, the transition from the non-cracked stage to the cracked stage is well predicted. These observations show that the assumptions of GSVATM to model the influence of the initial stress state due to prestress are valid, despite pre-decompression concrete stage was not computed neither the influence of long term response was considered;
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It was also shown that the predictions from the GSVATM compare very well with the ultimate (resistance) torque and, again, reasonably with the corresponding twist. This observation shows that the assumptions of GSVATM to model the contribution of the prestress reinforcement for the torsional strength, namely the used tensile σ–ε relationship for prestress steel, are also valid;
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In addition, good agreement were also observed for the torsional stiffness, for the non-cracked stage and also the cracked stage for high loading levels.