5.1 Undamaged Primary Curve
For sectional shear response, the envelope curve proposed by Mergos and Kappos (2008, 2012) is adopted to define the undamaged primary curve. There are four linear portions with three different slopes defined on the undamaged primary curve of Fig. 7.
The first linear portion oa with uncracked slope \(\left( {GA_{0} } \right)\) represents the elastic behavior of uncracked section in shear, and connects the origin point o to the cracking point a (\(V_{cr} ,\;\gamma_{cr}\)) at which the nominal principal tensile stress becomes larger than the nominal tensile strength of concrete. The cracking shear force \(V_{cr}\) and the uncracked slope \(\left( {GA_{0} } \right)\) are suggested by Sezen and Moehle (2004) as:
$$V_{cr} = \left( {\frac{{f_{t}^{'} }}{{\left( {L_{a} /h} \right)}}\sqrt {\left( {1 + \frac{N}{{f_{t}^{'} A_{g} }}} \right)} } \right)0.80A_{g} \;\;\;\;{\text{and}}\;\;\;\;\left( {GA} \right)_{0} = 0.80\,GA_{g}$$
(35)
where \(f_{t}^{{\prime }}\) is the nominal tensile strength of concrete; G, the concrete shear modulus; and \(L_{a} /h\), the shear span ratio. The cracking shear strain is simply defined as:
$$\gamma_{cr} = \frac{{V_{cr} }}{{\left( {GA} \right)_{0} }}$$
(36)
The second linear ab and the third linear bc portions have the same slope as \(\left( {GA} \right)_{1}\). The second linear portion ab connects the cracking point a (\(V_{cr} ,\;\gamma_{cr}\)) to the flexuralyielding point b (\(V_{y} ,\;\gamma_{y}\)) at which the longitudinal reinforcement experiences a yielding state for the first time. The yielding information of the longitudinal reinforcement is provided by the fibersection model. The third linear portion bc links the flexuralyielding point b (\(V_{y} ,\;\gamma_{y}\)) to the point c (\(V_{u0} ,\;\gamma_{st}\)) at which the shear force reaches its ultimate value \(V_{u0}\) while the shear strain \(\gamma_{st}\) corresponds to the verge of transverse reinforcement yielding. Traditionally, the value of shear strain \(\gamma_{st}\) can simply be computed based on the truss analogy (Park and Paulay 1975). However, Mergos and Kappos (2008, 2012) recognized that values of shear strain \(\gamma_{st}\) based on the truss analogy did not correspond well with experimental results since the effects of axial load and member aspect ratio on the shear strain \(\gamma_{st}\) were not considered in the truss analogy approach. To account for these two effects, Mergos and Kappos (2012) recommended the following two correction parameters \(\kappa\) and \(\gamma\) based on regression analyses and proposed the new expression for the shear strain \(\gamma_{st}\) as:
$$\gamma_{st} = \kappa \lambda \gamma_{truss}$$
(37)
where \(\kappa = 1  1.07\left( {\frac{N}{{f_{c}^{'} A_{g} }}} \right)\) is the axialforce correction parameter; \(\lambda = 5.37  1.59\;\hbox{min} \left( {2.5,\frac{{L_{a} }}{h}} \right)\), the memberaspectratio correction parameter; and \(\gamma_{truss}\), the shear strain associated with the yielding of transverse reinforcement based on the truss analogy approach (Park and Paulay 1975), and can be expressed as:
$$\gamma_{truss} = \frac{{V_{cr} }}{{\left( {GA} \right)_{0} }} + \frac{{A_{v} f_{yv}^{{}} \left( {\sin^{4} \phi + \frac{{E_{s} }}{{E_{c} }}\rho_{w} } \right)}}{{sE_{s} b\rho_{w} \sin^{4} \phi \cot \phi }}$$
(38)
where \(E_{s}\) is the steel modulus of elasticity; \(E_{c}\), the concrete modulus of elasticity; b, the section width; \(\rho_{w}\), the volumetric ratio of transverse reinforcement; and \(\phi\), the angle characterized by the member axis and the direction of diagonal struts. Mergos and Kappos (2012) performed regression analyses between experimental and analytical results and recommended that angle \(\phi\) of \(45{^\circ }\) is the optimal value.
The flattop portion cd characterizing plastic behavior of shear response connects the shearyielding point c (\(V_{u0} ,\;\gamma_{st}\)) to the ultimate point d (\(V_{u0} ,\;\gamma_{u}\)) at which the shear strain reaches its ultimate value \(\gamma_{u}\). This flattop portion corresponds to the experimental observation that shearcritical reinforced concrete members can experience additional shear deformation under sustained shear force before the onset of shear failure (Ma et al. 1976; Aboutaha et al. 1999). Consequently, the shear strain \(\gamma_{u}\) associated with the onset of shear failure (significant strength deterioration) could be considerably larger than the shear strain \(\gamma_{st}\) (Gerin and Adebar 2004; Sezen 2008; Mergos and Kappos 2012). Based on regression analyses of experimental results for 25 RC members eventually failing in shear, Mergos and Kappos (2012) proposed the following expression for the ultimate shear strain \(\gamma_{u}\):
$$\gamma_{u} = \lambda_{1} \lambda_{2} \lambda_{3} \gamma_{st} \ge \gamma_{st}$$
(39)
where \(\lambda_{1} = 1  2.5\;\text{min} \left( {0.4,\frac{N}{{f_{c}^{'} A_{g} }}} \right)\) is the parameter accounting for the axial load; \(\lambda_{2} = \text{min} \left( {2.5,\frac{{L_{a} }}{h}} \right)^{2}\) is the parameter accounting for the member aspect ratio; and \(\lambda_{3} = 0.31 + 17.8\,\hbox{min} \left( {\frac{{A_{v} f_{yv} }}{{bsf_{c}^{'} }},0.08} \right)\) is the parameter accounting for the amount of transverse reinforcement.
5.2 Modified Mergos–Kappos Shear–Flexure Interaction Procedure
Adverse influences of inelastic flexural deformation on shear resistance have long been recognized in the research community. Several researchers have noticed and demonstrated that shear strength of an RC section in the plastic hinge region decreases with increasing inelastic flexural deformation (Ghee et al. 1989; Priestley et al. 1993; Sezen 2002). This shearstrength deterioration is caused by concrete disintegration associated with inelastic flexural deformation (plastichinge formation). Moreover, several experimental results (e.g. Lynn 2001; Sezen 2002) indicate that sectional shear strain in the plastic hinge region increases drastically following formation of the plastic hinge despite approximately constant shear force confined by the flexural yielding. These two phenomena result from interaction between the shear and flexural actions and can be considered together by integrating the UCSD shearstrength model with the truss analogy approach as suggested by Mergos and Kappos (2008, 2012).
This study adopts and modifies the shear–flexure interaction procedure suggested by Mergos and Kappos (2008, 2012). Figure 8 shows the general scheme for the shear–flexure interaction procedure and the evolution of the reduced shear envelope curve with increasing curvature ductility. Degradation in the shear strength is associated with reduction in the concrete shearstrength contribution \(V_{c}\) as dictated by the UCSD shearstrength model, and is accounted for by reducing the ordinate of the undamaged shear envelope. The sectional shear response starts to deviate from the undamaged envelope curve when there is degradation in the shear strength. The damaged (reduced) shear envelope curve keeps on updating with evolution of the reduced shear strength and the resulting envelope curve is along the path \(o  a  b  e  f^{{\prime }}  g^{{\prime }}  h^{{\prime }}  c^{g}  d^{g}\). Figure 8 shows that there are three cases encountered when the shear–flexure interaction is triggered once yielding of flexural reinforcement takes place at point b. General representation of the shear–flexure interaction procedure adopted herein is depicted in Fig. 9 for all cases.
In Case I, shown in Fig. 9a, the sectional curvature ductility does not attain the value of 3. Consequently, there is no strength degradation in shear following the UCSD shearstrength model (Fig. 6). In this case, sectional shear response points at the start and the end of the load increment step both lie on the undamaged shear envelop with the cracked sectional shear stiffness \(\left( {GA} \right)_{1}.\)
In Case II, shown in Fig. 9b, the sectional curvature ductility exceeds the value of 3 for the first time. As a result, there is strength degradation in shear based on the UCSD shearstrength model (Fig. 6). In this case, the sectional shear response point at the start of the load increment step lies on the undamaged shear envelop while the sectional shear response point at the end of the load increment step is on the damaged (reduced) shear envelop with the effective sectional shear stiffness \(\left( {GA} \right)_{eff}\).
In Case III, shown in Fig. 9c, the sectional shear response points at the start and the end of the load increment step both lie on the damaged shear envelop with the effective sectional shear stiffness \(\left( {GA} \right)_{eff}\).
In the present work, the shear–flexure interaction procedure originally proposed by Mergos and Kappos (2008, 2012) is modified to compute the incremental sectional shear force \(\Delta V\) and the effective sectional shear stiffness \(\left( {GA} \right)_{eff}\) for a given incremental sectional shear strain \(\Delta \gamma\) for all abovementioned cases. Consequently, the authors will refer to the employed procedure as the “modified Mergos–Kappos” shear–flexure interaction procedure.
In this procedure, the socalled reference shear stiffness \(\left( {GA_{ref} } \right)_{i}^{k}\) is defined as:
$$\left( {GA_{ref} } \right)_{i}^{k} = \frac{{V_{0,\,\,i}^{k + 1}  V_{{}}^{k} }}{{\Delta \gamma_{i}^{k} }}$$
(40)
where \(V_{0,\,\,i}^{k + 1}\) is the nondegraded sectional shear force corresponding to the sectional shear strain \(\gamma_{i}^{k + 1} = \gamma_{{}}^{k} + \Delta \gamma_{i}^{k}\) and can be defined as:
$$V_{0,\,\,i}^{k + 1} = V_{cr} + \left( {GA} \right)_{1} \left( {\gamma_{i}^{k + 1}  \gamma_{cr} } \right)$$
(41)
It is noted that in Cases I and II, the reference sectional shear stiffness \(\left( {GA_{ref} } \right)_{i}^{k}\) is simply equal to the cracked sectional shear stiffness \(\left( {GA} \right)_{1}\) as shown in Fig. 9a, b. Considering the geometric relation in Fig. 9 leads to the following expression:
$$\Delta \gamma_{i}^{k} = \frac{{\Delta V_{i}^{k} }}{{\left( {GA_{eff} } \right)_{i}^{k} }} = \frac{{\Delta V_{i}^{k} + \left( {\Delta V_{c}^{\deg } } \right)_{i}^{k} }}{{\left( {GA_{ref} } \right)_{i}^{k} }}$$
(42)
where \(\Delta V_{i}^{k}\) is the incremental sectional shear force, and \(\left( {\Delta V_{c}^{\deg } } \right)_{i}^{k}\) is the reduction in sectional shear force associated with the concrete shear strength degradation and can be defined as:
$$\left( {\Delta V_{c}^{\deg } } \right)_{i}^{k} = \left( {GA_{ref} } \right)_{i}^{k} \Delta \gamma_{i}^{k}  \left( {\frac{{V_{ui}^{k}  V_{{}}^{k} }}{{\gamma_{st}  \gamma^{k} }}} \right)\Delta \gamma_{i}^{k}$$
(43)
where \(V_{ui}^{k}\) is the reduced shear strength dictated by variation of the concretecontribution coefficient \(k_{\varphi }\) with sectional curvature ductility \(\mu_{\varphi }\) (Fig. 6).
Based on Eq. (42), relation between the effective sectional shear stiffness \(\left( {GA_{eff} } \right)_{i}^{k}\) and the reference sectional shear stiffness \(\left( {GA_{ref} } \right)_{i}^{k}\) can be established as:
$$\left( {GA_{eff} } \right)_{i}^{k} = \frac{{\Delta V_{i}^{k} }}{{\Delta V_{i}^{k} + \left( {\Delta V_{c}^{\deg } } \right)_{i}^{k} }}\left( {GA_{ref} } \right)_{i}^{k}$$
(44)
It is observed from Eqs. (42) and (44) that the effective sectional shear stiffness \(\left( {GA_{eff} } \right)_{i}^{k}\) and the incremental sectional shear force \(\Delta V_{i}^{k}\) are mutually dependent. To obtain these two quantities, an additional iterative procedure is required within the element iterative step \(i\) of the load increment \(k\). It is worth remarking that the following quantities \(\left( {GA_{ref} } \right)_{i}^{k}\), \(\left( {\Delta V_{c}^{\deg } } \right)_{i}^{k}\), and \(\Delta \gamma_{i}^{k}\) are unchanged during this additional iterative process. An additional subscript “j” is appended to \(\left( {GA_{eff} } \right)_{i}^{k}\) and \(\Delta V_{i}^{k}\) to indicate the iterative step within the shear–flexure interaction procedure.
Based on Eqs. (42) and (44), the residual function \(\varPhi \left( {\left( {GA_{eff} } \right)_{i,j}^{k} } \right)\) can be defined as:
$$\varPhi \left( {\left( {GA_{eff} } \right)_{i,j}^{k} } \right) = \left( {GA_{eff} } \right)_{i,j}^{k} \Delta \gamma_{i}^{k}  \frac{{\left( {GA_{eff} } \right)_{i,j}^{k} \left( {\Delta V_{c}^{\deg } } \right)_{i}^{k} }}{{\left( {GA_{ref} } \right)_{i}^{k}  \left( {GA_{eff} } \right)_{i,j}^{k} }}$$
(45)
The Newton–Raphson method is to be employed to solve for the solution to Eq. (45). The derivative of Eq. (45) with respect to \(\left( {GA_{eff} } \right)_{i,j}^{k}\) is:
$$\begin{aligned}\frac{{d\varPhi \left( {\left( {GA_{eff} } \right)_{i,j}^{k} } \right)}}{{d\left( {GA_{eff} } \right)_{i,j}^{k} }} & = \Delta \gamma_{i}^{k}  \frac{{\left( {\Delta V_{c}^{\deg } } \right)_{i}^{k} }}{{\left( {GA_{ref} } \right)_{i}^{k}  \left( {GA_{eff} } \right)_{i,j}^{k} }} \\ & \quad  \frac{{\left( {GA_{eff} } \right)_{i,j}^{k} \left( {\Delta V_{c}^{\deg } } \right)_{i}^{k} }}{{\left( {\left( {GA_{ref} } \right)_{i}^{k}  \left( {GA_{eff} } \right)_{i,j}^{k} } \right)^{2} }} \end{aligned}$$
(46)
The stepbystep algorithm shown in Fig. 10 for the Newton–Raphson iterative procedure within the shear–flexure interaction procedure is as follows:

1.
Compute the reference sectional shear stiffness \(\left( {GA_{ref} } \right)_{i}^{k}\) from Eq. (40), and the reduction in sectional shear force associated with the concrete shear strength degradation \(\left( {\Delta V_{c}^{\deg } } \right)_{i}^{k}\) from Eq. (43).

2.
Assume an initial value of \(\left( {GA_{eff} } \right)_{i,j = 1}^{k}\). In this study, it is suggested to set \(\left( {GA_{eff} } \right)_{i,j = 1}^{k} = \left( {GA_{eff} } \right)_{i  1}^{k}\).

3.
Start the iterative procedure (\(j \ge 1\)) for the shear–flexure interaction within the element iterative step \(i\) of the load increment \(k\),

a.
Compute the residual function \(\varPhi \left( {\left( {GA_{eff} } \right)_{i,j}^{k} } \right)\) based on Eq. (45):
$$\varPhi \left( {\left( {GA_{eff} } \right)_{i,j}^{k} } \right) = \left( {GA_{eff} } \right)_{i,j}^{k} \Delta \gamma_{i}^{k}  \frac{{\left( {GA_{eff} } \right)_{i,j}^{k} \left( {\Delta V_{c}^{\deg } } \right)_{i}^{k} }}{{\left( {GA_{ref} } \right)_{i}^{k}  \left( {GA_{eff} } \right)_{i,j}^{k} }}$$

b.
Compute the slope \(d\varPhi \left( {\left( {GA_{eff} } \right)_{i,j}^{k} } \right)/d\left( {GA_{eff} } \right)_{i,j}^{k}\) based on Eq. (46):
$$\begin{aligned}\frac{{d\varPhi \left( {\left( {GA_{eff} } \right)_{i,j}^{k} } \right)}}{{d\left( {GA_{eff} } \right)_{i,j}^{k} }} & = \Delta \gamma_{i}^{k}  \frac{{\left( {\Delta V_{c}^{\deg } } \right)_{i}^{k} }}{{\left( {GA_{ref} } \right)_{i}^{k}  \left( {GA_{eff} } \right)_{i,j}^{k} }} \\ & \quad  \frac{{\left( {GA_{eff} } \right)_{i,j}^{k} \left( {\Delta V_{c}^{\deg } } \right)_{i}^{k} }}{{\left( {\left( {GA_{ref} } \right)_{i}^{k}  \left( {GA_{eff} } \right)_{i,j}^{k} } \right)^{2} }} \end{aligned}$$

c.
Update the effective sectional shear stiffness \(\left( {GA_{eff} } \right)_{i,j + 1}^{k}\):
$$\left( {GA_{eff} } \right)_{i,j + 1}^{k} \, = \;\,\left( {GA_{eff} } \right)_{i,j}^{k} \,  \left( {\frac{{\varPhi \left( {\left( {GA_{eff} } \right)_{i,j}^{k} } \right)}}{{\left( {{{d\varPhi \left( {\left( {GA_{eff} } \right)_{i,j}^{k} } \right)} \mathord{\left/ {\vphantom {{d\varPhi \left( {\left( {GA_{eff} } \right)_{i,j}^{k} } \right)} {d\left( {GA_{eff} } \right)_{i,j}^{k} }}} \right. \kern0pt} {d\left( {GA_{eff} } \right)_{i,j}^{k} }}} \right)}}} \right)$$

d.
Compute the updated residual value \(\varPhi \left( {\left( {GA_{eff} } \right)_{i,j + 1}^{k} } \right)\)

4.
Check if the updated residual value in step 3(d) is less than the convergence tolerance \(\varepsilon_{tol}\):

i.
If no, set \(j = j + 1\) and return to step 3(a).

ii.
If yes, return the current sectional shear force \(V_{i}^{k + 1} = V_{i}^{k} + \left( {GA_{eff} } \right)_{i,j}^{k} \Delta \gamma_{i}^{k}\) and the current effective sectional shear stiffness \(\left( {GA_{eff} } \right)_{i,j}^{k}\).
5.3 Hysteretic Shear Force–Shear Strain Response
To describe the sectional shear response under cyclic loading, a hysteretic model is required. In this work, a general hysteretic model presented by Filippou et al. (1992) and later enhanced by Martino (1999) is adopted and modified to describe the sectional shear force–deformation response under cyclic load reversals. This hysteretic model is attractive since it can represent both the damage and the pinching effects associated with shear crack closing and opening. The general shape of the modified hysteretic shear model is shown in Fig. 11 and its general feature of the hysteresis law can be briefly described as follows:
The section is loaded first along the monotonic branch 012 and then is unloaded along the path 2–3 with the initial stiffness \(\left( {GA} \right)_{0}\) until it reaches the abscissa at point 3. As unloading is reloading, and reloading continues in the opposite direction along the path 345, the section experiences a crack closing process until it reaches the monotonic envelope on the opposite side at point 4 and continues loading along the path 4–5. At point 5, the section is unloaded with the initial stiffness \(\left( {GA} \right)_{0}\) along the path 5–6 and starts to reload in the opposite direction along the path 678 on which the section experiences the crack closing process, thus resulting in the pinching response. Along the path 8910, the section response travels along the reduced envelope. More details on the hysteresis law can be found in Filippou et al. (1992) and Martino (1999).