The shear strength of the PHCS is typically determined by the web-shear capacity at the end regions. The web-shear capacity of prestressed concrete (PSC) members without shear reinforcement can be defined as the shear force acting on the cross-section when the principal tensile stress of the web concrete (
\(\sigma_{1}\)) reaches the cracking strength of concrete (
\(f_{cr}\)), assuming that the web-shear capacity is close enough to the web-shear cracking strength. Thus, the web-shear capacity (
\(V_{cw}\)) of PHCS can be expressed, based on theory of elasticity (Ugural and Fenster
2003), as follows:
$$V_{cw} = \frac{{I_{g} b_{w} }}{Q}\sqrt {f_{ct}^{2} + \alpha f_{se} f_{ct} \frac{{A_{ps} }}{{A_{g} }}}$$
(1)
where
\(Q\) is the first moment about the centroidal axis of the part of the cross-sectional area lying farther from the centroidal axis than the point where the shear stresses are being calculated,
\(I_{g}\) is the moment of inertia of the gross section,
\(b_{w}\) is the sum of the total web widths of the PHCS,
\(f_{ct}\) is the tensile strength of the concrete,
\(\alpha\) is the coefficient for the reduced effective prestress at the critical section,
\(f_{se}\) is the effective prestress, and
\(A_{ps}\) and
\(A_{g}\) are the cross-sectional area of tendon and concrete, respectively.
The design codes in North America, such as ACI318-08 (ACI Committee 318
2008) and AASHTO-LRFD (AASHTO
2007) assume an average shear stress distribution in the cross-section, and thus the web-shear capacity (
\(V_{cw}\)) of PSC members can be expressed, as follows:
$$V_{cw} = \left( {0.29\lambda \sqrt {f_{c} '} + 0.3f_{pc} } \right)b_{w} d_{p} + V_{p}$$
(2)
$$V_{cw} = \left( {0.16\lambda \sqrt {f_{c} '} + 0.3f_{pc} } \right)b_{w} d_{v} + V_{p}$$
(3)
where
\(f_{c} '\) is the specified compressive strength of concrete,
\(\lambda\) is the light-weight concrete coefficient,
\(f_{pc}\) is the compressive stress in concrete at centroid of the cross-section resisting externally applied loads or at junction of web and flange when the centroid lies within the flange,
\(d_{p}\) is the distance from extreme compression fiber to centroid of prestressing steel,
\(d_{v}\) is the effective depth, and
\(V_{p}\) is the vertical component of the effective prestress force. In ACI318-08 (ACI Committee 318
2008) and AASHTO-LRFD (AASHTO
2007), it is also specified that the effective prestress (
\(f_{se}\)) at the critical section should be reduced by accounting for its linear change within the transfer length that is
\(50d_{b}\) or
\(60d_{b}\), respectively. ACI318-08 (ACI Committee 318
2008) also specifies that the critical sections of PSC members are located at the distance
\(h/2\) from the member ends, while, in AASHTO-LRFD (AASHTO
2007), the critical section is assumed to be located at the effective shear depth (
\(d_{v}\)) or
\(0.5d_{v} \cot \theta\) from the member ends, where
\(\theta\) is the diagonal crack angle. In addition, as afore-mentioned, the current ACI318 code (ACI Committee 318
2014) specifies that the minimum shear reinforcement should be provided, if the factored shear (
\(V_{u}\)) exceeds
\(0.5\phi V_{cw}\) for the hollowed-section members with the untopped height exceeding 315 mm. In other words, the web-shear capacity of the PHCS with the net member height over 315 mm, produced by the extrusion method, should be reduced by half, as follows:
$$V_{cw} = \left( {0.29\lambda \sqrt {f_{c} '} + 0.3f_{pc} } \right)b_{w} d_{p} /2 + V_{p} /2$$
(4)