3.1 Assumptions to Incorporate Prestress
Longitudinal prestress is favourable for the torsional behavior of RC beams under torsion (Jeng et al. 2010; Bernardo et al. 2013). The combined action of longitudinal prestress and torsion induces a biaxial stress state (initial longitudinal compressive stress + shear stress) which increases the cracking torque of the beam. In addition, after concrete decompression the PC beam behaves like a current RC beam and longitudinal prestress reinforcement behaves as ordinary reinforcement, contributing to increase the resistance torque. Concrete decompression occurs when the strain in the nonprestress longitudinal reinforcement, initially in compression due to prestress, becomes zero due to the increasing torque.
To include the effect of longitudinal prestress, the following assumptions are made to extend the GSVATM for PC beams under torsion:

The calculation of the \( M_{T}{}\theta \) curve for the predecompression 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 predecompression 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 nonprestress 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 nonprestress 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
To consider indirectly the influence in the noncracked stage (for \( \varepsilon_{1}^{c} \le \varepsilon_{cr} \)) of the initial compressive stress state in the concrete due to prestress, Eq. (6) (Table 1) to compute the torsional moment, \( M_{T} \), is multiplied by a prestress factor \( \gamma_{p} \):
$$ M_{T} = \gamma_{p} \frac{{2A_{o} R\sin \gamma }}{{d_{v} }} $$
(35)
$$\gamma _{p} = \left\{ \begin{aligned} \sqrt {1 + 10\frac{{f_{{cp,i}} }}{{f^{\prime}_{c} }}}\quad {\text{if}}\ \varepsilon _{1}^{c} \le \varepsilon _{{cr}} \\ 1\quad{\text{ if }}\varepsilon _{1}^{c} > \varepsilon _{{cr}} \\ \end{aligned} \right. $$
(36)
In Eq. (36), \( f_{cp,i} \) is the initial compressive stress in concrete due to the longitudinal prestress. This prestress factor was proposed by Hsu (1984), based on Cowan’s failure criterion. This prestress factor proved to be a simple parameter to correct the cracking torque due to prestress and has been used in previous analytical models for PC beams under torsion (Hsu 1984; Lopes and Bernardo 2014; Andrade and Bernardo 2013). From Eqs. (35) and (36), the maximum torsional moment for which the prestress factor is higher than unity corresponds to the cracking torque \( M_{Tcr} \) (which occurs when \( \varepsilon_{1}^{c} = \varepsilon_{cr} \)). This simplified procedure to correct the torsional moment is acceptable because in the noncracked stage the \( M_{T}{}\theta \) curve is linear. In this stage, the most important key point of the \( M_{T} {} \theta \) curve is the upper limit, with coordinates (\( \theta_{cr} \);\( M_{Tcr} \)).
For consistency, in the noncracked stage, the twists \( \theta \) must also be multiplied by \( \gamma_{p} \), in order to maintain unchanged the torsional stiffness in this stage. Then, Eq. (11) (Table 1) must also be corrected:
$$ \theta = \gamma_{p} \frac{{\varepsilon_{2s}^{c} }}{{2t_{c \, } \sin \alpha \cos \alpha }} $$
(37)
To account for the initial deformation due to prestress, after the concrete decompression, the strain in the longitudinal prestress reinforcement, \( \varepsilon_{p} \), can be calculated as follows:
$$ \varepsilon_{p} = \varepsilon_{dec} + \varepsilon_{sl} $$
(38)
where \( \varepsilon_{dec} \) = decompression strain; \( \varepsilon_{sl} \) = strain in the longitudinal nonprestress reinforcement due to the external torque.
The decompression strain \( \varepsilon_{dec} \) is the sum of the initial tensile strain in the prestress reinforcement due to prestress, \( \varepsilon_{p,i} \), with the strain in the longitudinal nonprestress reinforcement necessary to reach the decompression (Eq. 39). This last one is equal, in modulus, to the initial compressive strain in the longitudinal nonprestress reinforcement due to prestress, \( \varepsilon_{sl,i} \). It should be referred that when concrete decompression occurs, \( \varepsilon_{sl} = 0 \)
$$ \varepsilon_{dec} = \varepsilon_{p,i} + \varepsilon_{sl,i} $$
(39)
By using Hooke’s law and based on the homogenized cross section, the initial strains are computed as follows:
$$ \varepsilon_{p,i} = \frac{{f_{p,i} }}{{E_{p} }} $$
(40)
$$ \varepsilon_{sl,i} = \frac{{A_{p} f_{p,i} }}{{A_{sl} \left( {E_{s}  E_{c} } \right) + \left( {A_{c}  A_{h}  A_{p} } \right)E_{c} }} $$
(41)
where \( f_{p,i} \) = initial stress in the prestress reinforcement, due to prestress; \( E_{p} \) = Young’s Modulus for prestress steel; \( A_{p} \) = total area of longitudinal prestress reinforcement; \( A_{h} \) = area of the hollow part for hollow sections (for plain sections, \( A_{h} = 0 \)).
Since the calculation procedure of the GSVATM starts at concrete decompression (\( \varepsilon_{sl} = 0 \)), Eqs. (9, 10) (Table 1) remain valid to compute the strains in the longitudinal and transverse nonprestress reinforcement.
To start the calculation procedure (Sect. 3.3), the input value is the strain at the outer fiber of the concrete strut, \( \varepsilon_{2s}^{c} = 2\varepsilon_{2}^{c} \) (see Table 2). Since the calculation procedure starts at concrete decompression, \( \varepsilon_{2s}^{c} \) must starts from zero.
After the concrete decompression, PC beams behaves as RC beams. Then, the participation of the longitudinal prestress reinforcement must be considered for the longitudinal equilibrium. For this, Eq. (7) (Table 1), to compute the effective width of the concrete strut and tie, must include the force in the longitudinal prestress reinforcement, \( A_{p} f_{p} \):
$$ t_{c} = \frac{{A_{sl} f_{sl} + A_{p} f_{p} }}{{\sigma_{2}^{c} p_{o} }}\frac{\cos \beta }{\cos \alpha \cos \gamma }{\text{ for }}\gamma = \alpha + \beta \le 90^{ \circ } $$
(42)
Parameter \( F \) in Eq. (8) (Table 1), to compute the angle of the concrete strut to the longitudinal axis, must also include the force in the longitudinal prestress reinforcement, \( A_{p} f_{p} \):
$$ \alpha = \arctan \left( {\frac{{\sqrt {F^{2} \left( {\tan \beta } \right)^{2} + F\left( {\tan \beta } \right)^{4} + F + \left( {\tan \beta } \right)^{2} } }}{{F\left( {\tan \beta } \right)^{2} + 1}}} \right){\text{ with }}F = \frac{{A_{st} f_{st} p_{o} }}{{\left( {A_{sl} f_{sl \, } + A_{p} f_{p} } \right)s}} $$
(43)
To characterize the behavior of the materials, the same \( \sigma{}\varepsilon \) relationships assumed in the GSVATM for RC beams (Bernardo et al. 2015a) are also used here. However, for concrete in compression, parameter \( \eta \) (Eq. (16), Table 2), which accounts for the ratio between the longitudinal and transverse resistance forces in the reinforcements, must be rewritten to account for the additional resistance force of the longitudinal prestress reinforcement:
$$ \eta = \frac{{\rho_{l} f_{sly} + \rho_{p} f_{p0.1\% } }}{{\rho_{t} f_{sty} }} $$
(44)
where \( \rho_{p} \) = longitudinal prestress reinforcement ratio (\( \rho_{p} = A_{p} /A_{c} \)); \( f_{p0.1\% } \) = proportional conventional limit stress to 0.1% for the longitudinal prestressing steel.
To model the behavior of the prestress steel reinforcement in tension, a \( \sigma{}\varepsilon \) relationship must be adopted. Bernardo and Andrade (2017) found that the σ − ε relationship for prestress reinforcement in tension proposed by Ramberg–Osgood (Eqs. 33, 34, Table 2), which was also used by Hsu and Mo (1985b) for the VATM, provides good results. This \( \sigma{}\varepsilon \) relationship is also adopted for this study. In Table 2, \( f_{p} \) and \( \varepsilon_{p} \) are the stress and strain in the prestress steel reinforcement, respectively, \( E_{p} \) is the Young’s Modulus for prestressing steel, \( \varepsilon_{p0.1\% } \) is the strain corresponding to \( f_{p0.1\% } \) and \( f_{pt} \) is the tensile strength of prestress steel reinforcement.
3.3 Solution Procedure
As for the GSVATM for RC beams (Bernardo et al. 2015a), the solution procedure for the GSVATM for PC beams to compute the theoretical \( M_{T} {} \theta \) curve is based on a trialanderror technique. This is because some unknown and interdependent variables must be assumed or estimated at the starting of the calculations. Figure 1 shows the flowchart for the iterative calculation algorithm used in this study. The first input value to starts the calculation procedure, which is incremented for each new cycle, is the strain at the outer fiber of the concrete struts \( \varepsilon_{2s}^{c} \). Each cycle of the solution procedure corresponds to a solution point of the \( M_{T}{} \theta \) curve, with coordinates (\( \theta \); \( M_{T} \)).
The end point of the theoretical \( M_{T}{} \theta \) curve corresponds to the theoretical failure of the PC beam under torsion. This one occurs when the maximum compressive strain at the outer fiber of the concrete strut, \( \varepsilon_{2s}^{c} \), reaches its conventional ultimate value. \( \varepsilon_{cu} \), or when the tensile strain for the reinforcements, \( \varepsilon_{s} \) or \( \varepsilon_{p} \), reaches its conventional ultimate value, \( \varepsilon_{su} \) or \( \varepsilon_{pu} \). In this study, these conventional values are defined from Eurocode 2 procedures (NP EN 199211 1992).
In this study, the solution procedure was implemented with computing language Delphi. The resulting computer program was used to compute the \( M_{T}{} \theta \) curve of several reference PC beams under torsion, as presented in Sect. 4.