Parametric Study of the Behavior of Longitudinally and Transversally Prestressed Concrete under Pure Torsion
Abstract
:1. Introduction
2. Modified Variable Angle Truss Model (MVATM)
2.1. Non-Cracked State
2.2. Cracked State
2.3. Zones 2.b and 3 (Cracked and Ultimate States)
3. Extension of the MVATM for TPC Beams
3.1. Non-Cracked State
- ρpt = transversal prestress reinforcement ratio;
- fpt0.1% = conventional proportional stress of the transversal prestress reinforcement.
- -
- 1st step: Select εds. Calculate εdsi (Equation (86)) and Equation (55));
- -
- …
- -
- 4th step: Calculate T (Equation (1)), εst (Equation (6)), (Equation (63)), (Equation (83)), εpl (Equation (67), εpt (Equation (81)), fst and fsl (Equations (58)), fpl and fpt (Equations (65) and (66));
- -
- …
- -
- 7th step: Calculate α′ (Equation (78));
- -
- …
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- 12th step: T > Tcr,ef ? (Equation (74));
- -
- …
3.2. Zone 2a (Cracked State)
3.3. Zones 2.b and 3 (Cracked and Ultimate States)
4. Theoretical Parametric Study
4.1. Reference Beams and Study Variables
4.2. Comparative Analysis
5. Conclusions
Author Contributions
Conflicts of Interest
References
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where: = concrete compressive strength (uniaxial); Ao = area limited by the center line of the flow of shear stresses; Ac = area limited by the outer perimeter of the section (includes hollow area): , with x and y the width and height of the cross section; Alh; Ath = homogenized steel areas in the longitudinal and transversal direction, respectively; Acl,eq; Act,eq = equivalent area of effective concrete in tension for the longitudinal and transversal direction, pc = perimeter of area Ac; po = perimeter of area Ao; t = thickness of the wall; u = perimeter of the centerlines of the closed stirrups: , with x1 and y1 the width and height of the centerlines of legs of the closed stirrups; s = spacing of the transversal reinforcement; fd = stress in the diagonal concrete strut; Asl; Apl = total area of the longitudinal ordinary and prestress reinforcement, respectively; Ast = area of one unit of the transversal reinforcement; εds = maximum compressive strain in the outer fiber in the strut direction; fsl; fpl = stress in the longitudinal ordinary and prestress reinforcement, respectively; fst = stress in the transversal reinforcement; fcp = stress in the concrete due to prestress; nc = Ec/Es, with Ec and Es the Young’s Modulus for concrete and ordinary reinforcement, respectively; np = Ep/Es, with Ep and Es the Young’s Modulus for prestress and ordinary reinforcement, respectively; ρl; ρt = longitudinal and transversal reinforcement ratio, respectively (see Section 3.1); Kt,eq = equivalent secant torsional stiffness; Kt,eq,tot = equivalent total torsional stiffness of the full section (including concrete core); Kt,c = torsional stiffness of the concrete core; β = St. Venant’s coefficient. |
|
where: xc; yc = width and height of the cross section core, respectively; Δθmax = correction of the twists. |
where: εo = strain corresponding to ; εc1 = tensile strain in the perpendicular direction to the strut; εdsi = initial compressive strain in the outer fiber of the concrete strut due to prestress; = effective compressive strain in the outer fiber of the concrete strut; ρl; ρt = longitudinal and transversal reinforcement ratio, respectively (see Section 3.1); ρp = longitudinal prestress reinforcement ratio (see Section 3.1); fsly; fsty = yielding stress in the longitudinal and transversal reinforcement, respectively; fpl0.1% = conventional proportional stress of the longitudinal prestress reinforcement. |
where: Es; Ec = Young’s modulus of the ordinary reinforcement and concrete, respectively; Ac = area limited by the external perimeter of the section; Ah = hollow area of the section (for plain sections: Ah = 0); fcr = concrete cracking stress; ρ = reinforcement ratio (see Section 3.1); εsli = initial compressive strain in the longitudinal ordinary reinforcement; = effective strain in the longitudinal ordinary reinforcement; fsy = reinforcement yielding stress. |
where: Ep = Young’s modulus of the prestress reinforcement; fpi = initial stress due to prestress in the prestress reinforcement; fpt = ultimate stress of the prestress reinforcement; fp0.1% = conventional proportional stress of the longitudinal prestress reinforcement; εpl = strain in the prestress reinforcement; εdec = strain in the prestress reinforcement at concrete’s decompression; εpli = initial tensile strain in the prestress reinforcement; εsli = initial compressive strain in the longitudinal ordinary reinforcement; = effective strain in the longitudinal ordinary reinforcement; εp0.1% = strain corresponding to fp0.1%. |
Beam | x; y cm | T cm | x1 cm | y1 cm | Asl cm2 | Ast/s cm2/m | ρsl % | ρst % | fcm MPa | fctm MPa | flym MPa | ftym MPa | Ec GPa | % | εcu % |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
A2 | 60 | 10.7 | 53.8 | 53.1 | 14.0 | 6.3 | 0.39 | 0.37 | 47.3 | 3.5 | 672 | 696 | 36.1 | 0.20 | 0.35 |
A3 | 60 | 10.9 | 54.0 | 53.5 | 18.1 | 8.3 | 0.50 | 0.49 | 46.2 | 3.4 | 672 | 715 | 35.8 | 0.20 | 0.35 |
A5 | 60 | 10.4 | 52.8 | 52.8 | 30.7 | 14.1 | 0.85 | 0.83 | 53.1 | 3.8 | 724 | 672 | 37.5 | 0.20 | 0.35 |
B2 | 60 | 10.8 | 53.3 | 53.4 | 14.6 | 6.7 | 0.41 | 0.40 | 69.8 | 4.1 | 672 | 696 | 39.4 | 0.21 | 0.33 |
B4 | 60 | 11.2 | 52.3 | 53.6 | 32.2 | 15.1 | 0.89 | 0.89 | 79.8 | 4.4 | 724 | 672 | 41.0 | 0.21 | 0.31 |
C2 | 60 | 10.0 | 53.2 | 53.3 | 14.0 | 6.3 | 0.39 | 0.37 | 94.8 | 4.9 | 672 | 696 | 43.2 | 0.22 | 0.28 |
C3 | 60 | 10.3 | 54.5 | 54.0 | 23.8 | 10.5 | 0.66 | 0.63 | 91.6 | 4.8 | 724 | 715 | 42.8 | 0.22 | 0.28 |
C4 | 60 | 10.3 | 54.6 | 54.5 | 30.7 | 14.1 | 0.85 | 0.86 | 91.4 | 4.8 | 724 | 672 | 42.7 | 0.22 | 0.28 |
C6 | 60 | 10.4 | 53.3 | 52.9 | 48.3 | 22.6 | 1.34 | 1.34 | 87.5 | 4.7 | 724 | 724 | 42.2 | 0.22 | 0.29 |
Type | LPC | TPC | LTPC | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Beam | Apl cm2 | ρpl % | fpli MPa | fcpl MPa | Apt/sp cm2/m | ρpt % | fpti MPa | fcpt MPa | Apl cm2 | Apt/sp cm2/m | ρpl % | ρpt % | fpli MPa | fcpl MPa | fpti MPa | fcpt MPa |
A2 | 16.63 | 0.46 | 1350 | 10.64 | 8.44 | 0.46 | 1350 | 10.64 | 8.32 | 4.22 | 0.23 | 0.23 | 1350 | 5.32 | 1350 | 5.32 |
A3 | 16.48 | 0.46 | 1350 | 10.40 | 8.39 | 0.46 | 1350 | 10.40 | 8.24 | 4.20 | 0.23 | 0.23 | 1350 | 5.20 | 1350 | 5.20 |
A5 | 18.26 | 0.51 | 1350 | 11.95 | 9.20 | 0.51 | 1350 | 11.95 | 9.13 | 4.60 | 0.25 | 0.25 | 1350 | 5.97 | 1350 | 5.97 |
B2 | 24.73 | 0.69 | 1350 | 15.71 | 12.56 | 0.69 | 1350 | 15.71 | 12.36 | 6.28 | 0.34 | 0.34 | 1350 | 7.85 | 1350 | 7.85 |
B4 | 29.08 | 0.81 | 1350 | 17.96 | 14.90 | 0.81 | 1350 | 17.96 | 14.54 | 7.45 | 0.40 | 0.40 | 1350 | 8.98 | 1350 | 8.98 |
C2 | 31.60 | 0.88 | 1350 | 21.33 | 15.80 | 0.88 | 1350 | 21.33 | 15.80 | 7.90 | 0.44 | 0.44 | 1350 | 10.67 | 1350 | 10.67 |
C3 | 31.26 | 0.87 | 1350 | 20.61 | 15.72 | 0.87 | 1350 | 20.61 | 15.63 | 7.86 | 0.43 | 0.43 | 1350 | 10.31 | 1350 | 10.31 |
C4 | 31.19 | 0.87 | 1350 | 20.57 | 15.69 | 0.87 | 1350 | 20.57 | 15.60 | 7.85 | 0.43 | 0.43 | 1350 | 10.28 | 1350 | 10.28 |
C6 | 30.09 | 0.84 | 1350 | 19.69 | 15.17 | 0.84 | 1350 | 19.69 | 15.05 | 7.58 | 0.42 | 0.42 | 1350 | 9.84 | 1350 | 9.84 |
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Bernardo, L.; Andrade, J.; Teixeira, M. Parametric Study of the Behavior of Longitudinally and Transversally Prestressed Concrete under Pure Torsion. Designs 2018, 2, 12. https://doi.org/10.3390/designs2020012
Bernardo L, Andrade J, Teixeira M. Parametric Study of the Behavior of Longitudinally and Transversally Prestressed Concrete under Pure Torsion. Designs. 2018; 2(2):12. https://doi.org/10.3390/designs2020012
Chicago/Turabian StyleBernardo, Luís, Jorge Andrade, and Mafalda Teixeira. 2018. "Parametric Study of the Behavior of Longitudinally and Transversally Prestressed Concrete under Pure Torsion" Designs 2, no. 2: 12. https://doi.org/10.3390/designs2020012