Nonlinear Frame Element with Shear–Flexure Interaction for Seismic Analysis of Non-Ductile Reinforced Concrete Columns

Abstract

This paper presents and emphasizes the essence of inclusion of shear response and shear–flexural interaction in the investigation of reinforced concrete (RC) columns characterized by light and inadequately (substandard) detailed transverse reinforcement. This column type commonly exists in old-constructed RC frame buildings before the regulation of modern seismic codes. A stiffness-based RC frame element with shear–flexure interaction is formulated within the framework of Timoshenko beam kinematics assumption. Linked displacement interpolation functions are employed to remedy the problematic shear-locking phenomenon. The axial and flexural actions are interacted via the fiber-section model while shear-strength deterioration with inelastic flexural deformations is accounted for within the framework of the UCSD shear-strength model. The numerical procedure for shear–flexure interaction is modified from the Mergos–Kappos procedure. The proposed element is simple, computationally efficient and able to describe several salient features of RC columns with substandard detailed transverse reinforcement, including gradual spread inelasticity, shear–flexure coupling effects, and shear-strength deterioration with increasing curvature ductility. Three correlation studies are conducted to examine the model accuracy and its capability to predict the rather complex responses of non-ductile RC columns. Comparison with conventional flexural frame element is also presented to emphasize the essence of inclusion of shear response and shear–flexure interaction.

1 Introduction

A large portion of old-constructed reinforced concrete (RC) frame buildings before the regulation of modern seismic codes in the mid-1970s had been severely damaged or even completely collapsed during destructive seismic events. Columns in those RC frame buildings are usually characterized by insufficient and poorly detailed transverse reinforcement (Lynn et al. 1996) and are typically referred to as “substandard” columns. As the most critical structural component in building systems, column shear failure had been observed to be the main source of building damage and collapses, and represented a high risk during destructive seismic events (Moehle et al. 2001; Sezen et al. 2003). Even though some columns may have been initially designed with sectional shear capacity that sufficiently exceeds the sectional shear force induced by flexural yielding, they could eventually fail in shear due to the detrimental action of inelastic flexural deformations on the sectional shear strength (Kim et al. 2013). Several experimental evidences (e.g. Li et al. 1995; Sezen 2002) and postearthquake reconnaissance (e.g. Moehle and Mahin 1991; Sezen et al. 2003) had been observed that substandard RC columns are susceptible to shear collapse after flexural yielding. This observation stems from the fact that widening of flexural–shear cracks associated with inelastic flexural deformations (plastic-hinge formation) impedes the concrete shear transfer mechanism driven by aggregate interlocking, thus reducing the sectional shear capacity. Several shear strength models (e.g. Priestley et al. 1993; Sezen and Moehle 2004) proposed in the research community have taken into account this detrimental effect of inelastic flexural deformations on the sectional shear capacity.

To withstand future earthquakes, seismic enhancement of substandard columns is drastically needed for old-constructed RC frame buildings. Unfortunately, rehabilitating all of these buildings to comply with current seismic regulations is a formidable task since the costs involved are extremely high. Consequently, an efficient and sufficiently accurate frame model with inclusion of shear–flexure coupling effects is required to play an essential role in assessing seismic vulnerability of existing RC frame buildings, and also in scheduling retrofitting priority in seismic rehabilitation programs.

Nowadays, it is widely accepted that nonlinear frame models have become a vital tool in seismic assessment procedures for existing RC frame buildings. However, most nonlinear frame models adopted in the standard seismic evaluation procedures for existing RC frame buildings mainly focus on inelastic flexural behaviors. Thus, development of nonlinear frame models capable of considering inelastic shear responses, detecting shear failure modes, and coupling shear with flexural actions, is still a challenging and open-ended problem.

Up to date, many researchers have proposed a variety of frame models considering inelastic shear responses. The state-of-the-art review on frame models with inclusion of shear responses was excellently presented in Ceresa et al. (2007). These models vary from relatively simple approaches in which a single shear spring is employed to represent the shear response, to frame models with sophisticated multi-dimensional constitutive laws. Pincheira et al. (1999) included shear effects into the frame element using translational springs. Though simple, this approach is only reasonable when variation of bending moment is known beforehand and the inflection point remains fixed during analysis. Filippou et al. (1992) employed a more refined approach using a subelement model in which shear deformations are considered through rotational springs at both element ends. Although this model is more accurate, shear–flexure interaction is not accounted for. Ricles et al. (1998) enhanced the one-component frame model of Giberson (1967) with stress-resultant plasticity-based end hinges. This enhanced one-component frame model could account for shear–flexure interaction and post-shear failure responses but failed to capture the gradual spreading phenomenon associated with inelastic flexural and shear responses. Mergos and Kappos (2008, 2012) proposed a distributed flexibility model with inclusion of shear–flexure interaction. This model employed the spread plasticity and subelement concept of Soleimani (1978) and could account for several salient features of flexure–shear critical columns including gradual spread inelasticity, shear–flexure coupling effects, and shear-strength deterioration with larger curvature ductility. Nevertheless, this model requires an ad-hoc phenomenological hysteretic law for moment–curvature response and does not naturally take into account the axial–flexural interaction. In the last three decades, several finite frame elements based on fiber-section model (e.g. Limkatanyu and Spacone 2002; Panto et al. 2017) have been proposed and shown their ability to accurately predict axial and bending responses of flexure-dominated RC members. Therefore, enhancing a fiber frame model with the ability to account for shear–flexure interaction is a challenging and worth-pursuing research problem. Several researchers have attempted to incorporate shear response into the flexibility-based fiber frame model. For example; Ranzo and Petrangeli (1998) proposed a flexibility-based Timoshenko frame element in which shear and flexural actions were not interacted at the section level but were coupled through the axial-strain dependent shear strength model. Another flexibility-based Timoshenko frame element was also proposed by Marini and Spacone (2006). The flexural and shear actions in this element were not coupled at the section level as well, but were interacted through equilibrium enforcement at the element level. Though very accurate, the flexibility-based formulation usually requires a complicated element state determination procedure (Panto et al. 2017). A number of fiber frame models coupling shear and flexural actions at the section level have recently been proposed by several researchers. For example; Petrangeli et al. (1999) formulated a flexibility-based Timoshenko frame model with the ability to capture shear response and shear–flexure interaction at the section level, accomplished using the microplane model (Ozbolt and Bazant 1992). Ceresa et al. (2009) formulated a stiffness-based Timoshenko frame element in which shear and flexure actions was interacted at the section level using the Modified Compression Field Theory (MCFT) developed by Palermo and Vecchio (2003). Kagermanov and Ceresa (2017) enhanced the flexibility-based frame element of Spacone et al. (1996) with an exact shear strain profile. This novel fiber-section model was developed based on a smeared-crack/fixed crack membrane model by Kagermanov and Ceresa (2016) and allowed for the shear–flexure interaction at the section level. Even though these frame models are very accurate and take into account the sectional shear–flexure interaction in a very precise manner, sophisticated multi-dimensional constitutive laws and high implementation efforts are usually required. These could prohibit the model usage by practicing structural engineers, and the computational costs involved are usually high.

In light of the above discussion, there is still room to propose an additional frame model for the research and practicing-engineer communities. Consequently, the present work is aimed at proposing an improved finite frame element with shear–flexure interaction. This proposed model needs to be simple but sufficiently accurate and capable of being programmed into a general-purpose finite element platform with ease. Furthermore, the model inputs need to be closely related to actual engineering properties to render the proposed model accessible by an even wider range of practicing structural engineers and researchers.

Organization of the present work is as following outlined. Equilibrium, compatibility, and sectional constitutive relations of the Timoshenko frame model are first derived. Stiffness-based framework of the model formulation is presented next. The virtual displacement principle is at the core of the model formulation and the resulting model can be programmed with ease. To remedy the problematic shear locking phenomenon, linked displacement interpolation functions are employed, thus resulting in the locking-free Timoshenko frame element. Subsequently, the shear–flexure interaction procedure is discussed. The present study adopts and modifies the interaction procedure proposed by Mergos and Kappos (2008, 2012) within the framework of the so-called “UCSD Shear-Strength Model” proposed by Priestley et al. (1993). Finally, three correlation studies are conducted to examine the model accuracy and its ability to represent the rather complex responses of non-ductile RC columns. The first two correlation studies focus on flexure–shear critical columns while the third emphasizes on a shear-dominated column. The finite element platform FEAP (Taylor 2000) is used to host the proposed frame element.

2 Governing Equations of Timoshenko Frame Element

2.1 Equilibrium Relations

An infinitesimal segment dx taken from a frame member under transverse load $p_{y} \left( x \right)$ is shown in Fig. 1. Considering axial, moment, and vertical equilibriums of this segment yields the following relations:

$$\frac{dN\left( x \right)}{dx} = 0$$

(1)

$$\frac{dM\left( x \right)}{dx} + V\left( x \right) = 0$$

(2)

$$\frac{dV\left( x \right)}{dx} + p_{y} \left( x \right) = 0$$

(3)

Fig. 1

Infinitesimal segment dx of a frame member under transverse load.

where $N\left( x \right)$ is the frame sectional axial force; $V\left( x \right)$, the frame sectional shear force; and $M\left( x \right)$, the frame sectional bending moment.

The matrix form of Eqs. (1) (2), and (3) can be expressed as:

$${\mathbf{L}}_{TB}^{T} {\mathbf{D}}\left( x \right) - {\mathbf{p}}\left( x \right) = {\mathbf{0}}$$

(4)

where ${\mathbf{D}}\left( x \right) = \left\{ {\begin{array}{*{20}c} {N\left( x \right)} & {M\left( x \right)} & {V\left( x \right)} \\ \end{array} } \right\}^{T}$ represents the element sectional force vector; ${\mathbf{p}}\left( x \right) = \left\{ {\begin{array}{*{20}c} 0 & 0 & {p_{y} \left( x \right)} \\ \end{array} } \right\}^{T}$, the element distributed load vector; and ${\mathbf{L}}_{TB}^{{}}$, the differential operator, given as:

$${\mathbf{L}}_{TB}^{{}} = \left[ {\begin{array}{*{20}c} {\frac{d}{dx}} & 0 & 0 \\ 0 & {\frac{d}{dx}} & 0 \\ 0 & 1 & { - \frac{d}{dx}} \\ \end{array} } \right]$$

(5)

2.2 Compatibility Relations

As work conjugate of the element sectional force vector ${\mathbf{D}}\left( x \right)$, the element sectional deformation vector ${\mathbf{d}}\left( x \right)$ is defined as:

$${\mathbf{d}}\left( x \right) = \left\{ {\begin{array}{*{20}c} {\varepsilon_{0} \left( x \right)} & {\varphi \left( x \right)} & {\gamma \left( x \right)} \\ \end{array} } \right\}^{T}$$

(6)

where $\varepsilon_{0} \left( x \right)$ is the sectional axial strain at reference axis; $\varphi \left( x \right)$, the sectional bending curvature; and $\gamma \left( x \right)$, the sectional shear strain.

At the element level, the following sectional displacements are collected in a vector form as:

$${\mathbf{u}}\left( x \right) = \left\{ {\begin{array}{*{20}c} {u_{0} \left( x \right)} & {\theta \left( x \right)} & {v_{0} \left( x \right)} \\ \end{array} } \right\}^{T}$$

(7)

where $u_{0} \left( x \right)$ is the sectional axial displacement at reference axis; $\theta \left( x \right)$, the sectional rotation; and $v_{0} \left( x \right)$, the sectional transverse displacement.

Figure 2 depicts kinematics description of the frame section following the Timoshenko beam hypothesis.

Fig. 2

Kinematics description of a frame section.

Through the compatibility conditions, element sectional deformations are related to the element sectional displacements as:

$$\varepsilon_{0} \left( x \right) = \frac{{du_{0} \left( x \right)}}{dx}$$

(8)

$$\varphi \left( x \right) = \frac{d\theta \left( x \right)}{dx}$$

(9)

$$\gamma \left( x \right) = \theta \left( x \right) - \frac{{dv_{0} \left( x \right)}}{dx}$$

(10)

and can be written together in the matrix form as:

$${\mathbf{d}}\left( x \right) = {\mathbf{L}}_{TB}^{{}} {\mathbf{u}}\left( x \right)$$

(11)

2.3 Sectional Force–Deformation Relations

In this work, the nonlinear feature of RC responses is entirely based on a nonlinear deformation-based constitutive model, defined as:

$${\mathbf{D}}\left( x \right) = {\varvec{\Psi}}\left[ {{\mathbf{d}}\left( x \right)} \right]$$

(12)

Equation (13) expresses the linearized incremental form of Eq. (12) as:

$${\mathbf{D}}\left( x \right) = {\mathbf{D}}^{0} \left( x \right) + {\mathbf{k}}^{0} \left( x \right)\Delta {\mathbf{d}}\left( x \right)$$

(13)

where ${\mathbf{D}}^{0} \left( x \right)$ and ${\mathbf{k}}^{0} \left( x \right)$ are the sectional force vector and the stiffness matrix at the initial point, respectively. Superscript 0 indicates the value of a vector or matrix at the initial point of the incremental-iterative solution procedure.

In this study, the fiber-section model is employed to derive the incremental relation of Eq. (13). The fiber-section model can automatically account for the axial–flexural interaction (Spacone and Limkatanyu 2000). Figure 3 shows the fiber-section discretization. The sectional axial force $N\left( x \right)$ and sectional moment $M\left( x \right)$ are defined as:

$$N\left( x \right) = \sum\limits_{m = 1}^{nfib} {\sigma_{m} A_{m} } \;\;\;{\text{and}}\;\;\;M\left( x \right) = - \sum\limits_{m = 1}^{nfib} {y_{m} \sigma_{m} A_{m} }$$

(14)

where m defines the generic fiber, and nfib represents the number of fibers in the section; $y_{m}$, $\sigma_{m}$, and $A_{m}$ represent the distance from the reference axis x (Fig. 3), the normal stress, and the area, respectively, of the mth fiber in the section.

Fig. 3

Fiber discretization of reinforced concrete section.

With Eq. (14), the element sectional force vector ${\mathbf{D}}\left( x \right)$ can be rewritten as:

$${\mathbf{D}}\left( x \right) = \left\{ {\begin{array}{*{20}c} {\sum\limits_{m = 1}^{nfib} {\sigma_{m} A_{m} } } & { - \sum\limits_{m = 1}^{nfib} {y_{m} \sigma_{m} A_{m} } } & {V\left( x \right)} \\ \end{array} } \right\}^{T}$$

(15)

It is worth observing from Eq. (15) that nfib fibers are required in section discretization to represent the axial and bending actions due to variation of the normal stress $\sigma_{m}$ along the section depth, while there is no need for fiber-section discretization to represent the shear action due to the constant shear stress along the section depth following the Timoshenko beam theory (Onate 2013). In other words, the fiber-section discretization for shear action is analogous to the one-fiber discretization.

The sectional stiffness matrix ${\mathbf{k}}\left( x \right)$ can be expressed via the fiber-section model as:

$${\mathbf{k}}\left( x \right) = \left[ {\begin{array}{*{20}c} {\sum\limits_{m = 1}^{nfib} {E_{m} A_{m} } } & { - \sum\limits_{m = 1}^{nfib} {y_{m} E_{m} A_{m} } } & 0 \\ { - \sum\limits_{m = 1}^{nfib} {y_{m} E_{m} A_{m} } } & {\sum\limits_{m = 1}^{nfib} {y_{m}^{2} E_{m} A_{m} } } & 0 \\ 0 & 0 & {GA_{s} \left( x \right)} \\ \end{array} } \right]$$

(16)

where $E_{m}$ is the modulus of the mth fiber in the section; and $GA_{s} \left( x \right)$ is the sectional shear stiffness.

From Eq. (16), there is no coupling between shear and flexural actions. However, interaction between these two effects can be triggered via the shear strength model employed in this study, and will be discussed later in the paper.

Uniaxial cyclic constitutive laws employed in this study for concrete, reinforcing steel, and shear responses are schematically shown in Fig. 4 to ease model implementation. Kent and Park (1971) law is used for concrete response; Menegotto and Pinto (1973) law is used for reinforcing steel response; and the tri-linear envelope curve proposed by Mergos and Kappos (2012) is used for shear response.

Fig. 4

Uniaxial constitutive laws of material models.

3 Stiffness-Based Formulation of Timoshenko Frame Element

3.1 Formulation

In this work, the proposed element is constructed with the stiffness-based finite element formulation. Through appropriate interpolation functions, element sectional displacements ${\mathbf{u}}\left( x \right)$ can be expressed in terms of the element nodal displacements U. Element deformations ${\mathbf{d}}\left( x \right)$ are obtained by enforcing the section compatibility of Eq. (11).

Equation (17) defines the weighted residual expression of Eq. (4) as:

$$\int\limits_{L} {\delta {\mathbf{u}}^{T} \left( x \right)\left[ {{\mathbf{L}}_{TB}^{T} {\mathbf{D}}\left( x \right) - {\mathbf{p}}\left( x \right)} \right]} \,dx = 0$$

(17)

with $\delta {\mathbf{u}}\left( x \right)$ being a vector containing compatible virtual displacement fields.

Substituting Eq. (13) into (17) and enforcing compatibility of Eq. (11) lead to:

$$\int\limits_{L} {\delta {\mathbf{u}}^{T} \left( x \right)\left[ {{\mathbf{L}}_{TB}^{T} \left( {{\mathbf{D}}^{0} \left( x \right) + {\mathbf{k}}^{0} \left( x \right){\mathbf{L}}_{TB}^{{}} \Delta {\mathbf{u}}\left( x \right)} \right) - {\mathbf{p}}\left( x \right)} \right]} \,dx = 0$$

(18)

Integration by parts is applied to Eq. (18), thus resulting in:

$$\begin{aligned} & \int\limits_{L} {\left( {{\mathbf{L}}_{TB}^{{}} \delta {\mathbf{u}}^{{}} \left( x \right)} \right)^{T} {\mathbf{k}}^{0} \left( x \right)} \,\left( {{\mathbf{L}}_{TB}^{{}} \Delta {\mathbf{u}}\left( x \right)} \right)dx \hfill \\ & = \delta {\mathbf{U}}^{T} {\mathbf{P}} + \int\limits_{L} {\delta {\mathbf{u}}^{T} \left( x \right){\mathbf{p}}\left( x \right)} \,dx - \int\limits_{L} {\left( {{\mathbf{L}}_{TB}^{{}} \delta {\mathbf{u}}^{{}} \left( x \right)} \right)^{T} {\mathbf{D}}^{0} \left( x \right)} \,dx \hfill \\ \end{aligned}$$

(19)

where $\delta {\mathbf{U}}^{T} {\mathbf{P}}$ defines the external virtual work done and is obtained from the boundary terms during integration by parts.

Through the displacement interpolation function matrix ${\mathbf{N}}_{TB} \left( x \right)$, the element sectional displacements ${\mathbf{u}}\left( x \right)$ are expressed in terms of the element nodal displacements U as:

$${\mathbf{u}}\left( x \right) = {\mathbf{N}}_{TB} \left( x \right){\mathbf{U}}$$

(20)

For the Timoshenko frame element, the selection of displacement interpolation functions must be made with care and is to be discussed in the subsequent section. Substituting Eq. (20) into (19) and accounting for the arbitrariness of $\delta {\mathbf{U}}$ lead to:

$$\left[ {\int\limits_{L} {{\mathbf{B}}_{TB} \left( x \right)^{T} {\mathbf{k}}^{0} \left( x \right)} \,{\mathbf{B}}_{TB} \left( x \right)dx} \right]\,\Delta {\mathbf{U}} = {\mathbf{P}} + \int\limits_{L} {{\mathbf{N}}_{TB} \left( x \right)^{T} {\mathbf{p}}\left( x \right)} \,dx - \int\limits_{L} {{\mathbf{B}}_{TB} \left( x \right)^{T} {\mathbf{D}}^{0} \left( x \right)} \,dx$$

(21)

where ${\mathbf{B}}_{TB} \left( x \right) = {\mathbf{L}}_{TB}^{{}} {\mathbf{N}}_{TB} \left( x \right)$ is the sectional deformation-displacement matrix.

The matrix form of Eq. (21) is:

$${\mathbf{K}}^{0} \Delta {\mathbf{U}} = \left( {{\mathbf{P}} + {\mathbf{P}}_{{\mathbf{p}}} } \right) - {\mathbf{Q}}^{0}$$

(22)

where ${\mathbf{K}}^{0} = \int\nolimits_{L} {{\mathbf{B}}_{TB} \left( x \right)^{T} {\mathbf{k}}^{0} \left( x \right)} \,{\mathbf{B}}_{TB} \left( x \right)dx$ is the frame element stiffness matrix; ${\mathbf{P}}_{{\mathbf{p}}} = \int\nolimits_{L} {{\mathbf{N}}_{TB} \left( x \right)^{T} {\mathbf{p}}\left( x \right)} \,dx$ is the equivalent load vector due to the distributed load ${\mathbf{p}}\left( x \right)$; ${\mathbf{Q}}^{0} = \int\nolimits_{L} {{\mathbf{B}}_{TB} \left( x \right)^{T} {\mathbf{D}}^{0} \left( x \right)} \,dx$ is the element resisting force vector.

Equation (22) represents the incremental form of the element stiffness equation and the term on its right-hand side defines the residual force vector associated with the weak statement of equilibrium equations. Once equilibrium configuration is reached during the incremental-iterative solution procedure, this residual force vector vanishes.

3.2 Linked Displacement Interpolation Functions

In Timoshenko beam theory, the sectional rotation field $\theta \left( x \right)$ is independent of the sectional transverse displacement field $v_{0} \left( x \right)$. Consequently, the sectional rotation $\theta \left( x \right)$ and sectional transverse displacement $v_{0} \left( x \right)$ can be interpolated independently to their nodal values in the stiffness-based formulation of Timoshenko frame model. Nevertheless, inattentive selection of independent interpolation functions for the sectional transverse displacement $v_{0} \left( x \right)$ and sectional rotation $\theta \left( x \right)$ fields may cause the resulting Timoshenko frame model to suffer from the problematic phenomenon known as “shear locking”. Diagnosis of the shear-locking phenomenon is thoroughly discussed in Onate (2013).

In this study, the consistent interpolation approach is employed to overcome the shear-locking problem. This approach requires that the choice of interpolation functions for the sectional transverse displacement $v_{0} \left( x \right)$ and the sectional rotation $\theta \left( x \right)$ fields has to be such that $dv_{0} \left( x \right)/dx$ and $\theta \left( x \right)$ are polynomials of the same order. In other words, the interpolation function for the sectional transverse displacement field $v_{0} \left( x \right)$ must be one-degree higher than that for the sectional rotation field $\theta \left( x \right)$. Consequently, a standard linear Timoshenko frame element, shown in Fig. 5, is to be enhanced with a quadratic (bubble) term for the sectional transverse displacement field $v_{0} \left( x \right)$. The resulting interpolation functions are:

$$u_{0} \left( x \right) = \left( {1 - \frac{x}{L}} \right)U_{1} + \frac{x}{L}U_{4}$$

(23)

$$v_{0} \left( x \right) = \left( {1 - \frac{x}{L}} \right)U_{2} + \frac{x}{L}U_{5} + \underbrace {{\alpha x\left( {x - L} \right)}}_{Enhanced\;Bubble\;Function}$$

(24)

$$\theta \left( x \right) = \left( {1 - \frac{x}{L}} \right)U_{3} + \frac{x}{L}U_{6}$$

(25)

with ${\mathbf{U}} = \left\{ {\begin{array}{*{20}c} {U_{1} } & {U_{2} } & {U_{3} } & {U_{4} } & {U_{5} } & {U_{6} } \\ \end{array} } \right\}^{T}$ being a vector collecting element nodal displacements, and $\alpha$ is a parameter to be determined from the limit Euler–Bernoulli condition of vanishing shear strain ($\gamma = 0$) for slender beams.

Fig. 5

Two-node displacement-based Timoshenko frame element.

Based on Eqs. (10), (24), and (25), the sectional shear strain can be expressed as:

$$\begin{aligned} \gamma \left( x \right) & = \left( {\frac{{U_{2} - U_{5} }}{L}} \right) + U_{3} + \alpha L \\ & \quad + x\left( {\frac{{U_{6} - U_{3} }}{L} - 2\alpha } \right) \end{aligned}$$

(26)

To ensure the condition of vanishing shear strain ($\gamma = 0$) for slender beams within the limit, Eq. (26) must be independent of the coordinate x, thus leading to the following relation:

$$\alpha = \frac{{U_{6} - U_{3} }}{2L}$$

(27)

Combining Eq. (24) and Eq. (27) yields the expression for the enhanced sectional transverse displacement field $v_{0} \left( x \right)$ as:

$$\begin{aligned} v_{0} \left( x \right) & = \left( {1 - \frac{x}{L}} \right)U_{2} + \frac{x}{L}U_{5} \\ & \quad + \left( {\frac{x}{2} - \frac{{x^{2} }}{2L}} \right)U_{3} + \left( { - \frac{x}{2} + \frac{{x^{2} }}{2L}} \right)U_{6} \end{aligned}$$

(28)

Equation (28) clearly indicates that the sectional transverse displacement field $v_{0} \left( x \right)$ is interpolated not only in terms of nodal displacements ($U_{2}$ and $U_{5}$) but also in terms of nodal rotations ($U_{3}$ and $U_{6}$), thus leading to the so-called “linked” displacement interpolation functions (Fraeijs de Veubeke 1965).

Based on Eqs. (23), (25), and (28), the displacement interpolation function matrix ${\mathbf{N}}_{TB} \left( x \right)$ is:

$${\mathbf{N}}_{TB} \left( x \right) = \left[ {\begin{array}{*{20}c} {1 - \frac{x}{L}} & 0 & 0 & {\frac{x}{L}} & 0 & 0 \\ 0 & 0 & {1 - \frac{x}{L}} & 0 & 0 & {\frac{x}{L}} \\ 0 & {1 - \frac{x}{L}} & {\frac{x}{2} - \frac{{x^{2} }}{2L}} & 0 & {\frac{x}{L}} & { - \frac{x}{2} + \frac{{x^{2} }}{2L}} \\ \end{array} } \right]$$

(29)

The sectional deformation-displacement matrix ${\mathbf{B}}_{TB} \left( x \right)$ is:

$${\mathbf{B}}_{TB} \left( x \right) = {\mathbf{L}}_{TB}^{{}} {\mathbf{N}}_{TB} \left( x \right) = \left[ {\begin{array}{*{20}c} { - \frac{1}{L}} & 0 & 0 & {\frac{1}{L}} & 0 & 0 \\ 0 & 0 & { - \frac{1}{L}} & 0 & 0 & {\frac{1}{L}} \\ 0 & {\frac{1}{L}} & {\frac{1}{2}} & 0 & { - \frac{1}{L}} & {\frac{1}{2}} \\ \end{array} } \right]$$

(30)

4 UCSD Shear Strength Model

Among several shear-strength models proposed in the research community, the so-called “UCSD Shear-Strength Model” presented by Priestley et al. (1993) is adopted herein because of its validity and capability. The UCSD shear-strength model has an ability to consider the shear strength deterioration with increasing curvature ductility demand. This feature is very attractive and is suited to the development of RC frame element considering shear–flexure interaction effects in this work.

In the UCSD shear-strength model, there are three resisting mechanisms contributing to the shear strength $V_{u}$, namely: concrete mechanism $V_{c}$; truss mechanism (transverse reinforcement) $V_{s}$; and arch mechanism (axial force) $V_{a}$.

$$V_{u} = V_{c} + V_{s} + V_{a}$$

(31)

The concrete shear strength $V_{c}$ is defined as:

$$V_{c} = k_{\varphi } \sqrt {f_{c}^{{\prime }} } \left( {0.8A_{g} } \right)$$

(32)

where $f_{c}^{{\prime }}$ is the concrete compressive cylinder strength (in MPa); $A_{g}$, the gross cross sectional area; and $k_{\varphi }$, a coefficient considering the influence of curvature ductility $\mu_{\varphi }$ on concrete shear strength $V_{c}$ as shown in Fig. 6.

Fig. 6

Relationship between imposed curvature ductility and concrete shear strength contribution.

For a rectangular section, the transverse-reinforcement shear strength $V_{s}$ associated with the truss mechanism (Park and Paulay 1975) is defined as:

$$V_{s} = \frac{{A_{v} f_{yv} D^{{\prime }} }}{s}\cot 30{^\circ }$$

(33)

where s is the spacing of transverse reinforcement measured parallel to the longitudinal reinforcement; $A_{v}$, the transverse reinforcement area with spacing s; $f_{yv}$, the yield strength of transverse reinforcement; and $D^{{\prime }}$, the distance measured parallel to the applied shear between centers of the longitudinal reinforcement.

The axial-force shear strength $V_{a}$ associated with the arch mechanism is expressed as:

$$V_{a} = N\tan \beta$$

(34)

where $N$ is the compressive axial force, and $\beta$ is the angle between the column axis and the line connecting the centers of the flexural compression zones at the top and the bottom of the column ends. The axial-force shear strength $V_{a}$ is not degraded with increasing ductility as pointed out by Priestley et al. (1993).

5 Sectional Shear Constitutive Model

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.

Fig. 7

Undamaged envelope curve for sectional shear response.

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 flexural-yielding 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 fiber-section model. The third linear portion bc links the flexural-yielding 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 axial-force correction parameter; $\lambda = 5.37 - 1.59\;\hbox{min} \left( {2.5,\frac{{L_{a} }}{h}} \right)$, the member-aspect-ratio 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 flat-top portion cd characterizing plastic behavior of shear response connects the shear-yielding 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 flat-top portion corresponds to the experimental observation that shear-critical 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 shear-strength deterioration is caused by concrete disintegration associated with inelastic flexural deformation (plastic-hinge 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 shear-strength 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 shear-strength contribution $V_{c}$ as dictated by the UCSD shear-strength 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.

Fig. 8

Evolution of damaged (reduced) envelope curve with increasing curvature ductility.

Fig. 9

Incremental sectional shear force–shear strain relation.

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 shear-strength 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 shear-strength 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 above-mentioned cases. Consequently, the authors will refer to the employed procedure as the “modified Mergos–Kappos” shear–flexure interaction procedure.

In this procedure, the so-called 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 non-degraded 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 concrete-contribution 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 step-by-step algorithm shown in Fig. 10 for the Newton–Raphson iterative procedure within the shear–flexure interaction procedure is as follows:

Fig. 10

Iterative algorithm for determining effective sectional shear stiffness $\left( {GA_{eff} } \right)_{i}^{k}$.

• 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 $\left( {GA_{eff} } \right)_{i,j + 1}^{k}$:

    $$\left( {GA_{eff} } \right)_{i,j + 1}^{k} \, = \;\,\left( {GA_{eff} } \right)_{i,j}^{k} \, - \left( {\frac{{\varPhi \left( {\left( {GA_{eff} } \right)_{i,j}^{k} } \right)}}{{\left( {{{d\varPhi \left( {\left( {GA_{eff} } \right)_{i,j}^{k} } \right)} \mathord{\left/ {\vphantom {{d\varPhi \left( {\left( {GA_{eff} } \right)_{i,j}^{k} } \right)} {d\left( {GA_{eff} } \right)_{i,j}^{k} }}} \right. \kern-0pt} {d\left( {GA_{eff} } \right)_{i,j}^{k} }}} \right)}}} \right)$$

  • 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:

Fig. 11

Hysteretic shear force–shear strain response.

The section is loaded first along the monotonic branch 0-1-2 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 3-4-5, 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 6-7-8 on which the section experiences the crack closing process, thus resulting in the pinching response. Along the path 8-9-10, 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).

6 Model Validation Against Experimental Evidence

To examine the validity of the proposed RC frame element and to discuss the importance of including shear–flexure interaction effects, three correlation investigations are conducted. The first set considers cyclic tests of two RC columns which eventually failed in shear following flexural yielding and the effects of shear–flexure interaction on both global and local responses are discussed. This type of member is referred to as a “flexure–shear” critical member. The second set studies the cyclic response of an RC column failed in shear before reaching its flexural strength. This type of member is referred to as a “shear” critical member. In all numerical models, 16 proposed elements are used to discretize the column; seven Gauss–Lobatto integration points are used for all elements; and forty fibers (layers) are used to discretize the column cross section. The flexural model proposed by Spacone et al. (1996) is also used to obtain numerical results.

6.1 Flexure–Shear Critical Members

6.1.1 Sezen (2002): Column 2CLD12

Sezen (2002) conducted a series of tests on RC square columns representing older existing columns with substandard seismic reinforcement details. One of these columns, labeled column 2CLD12, is modeled by the proposed RC frame element. Mergos and Kappos (2008) used the same column to validate their distributed flexibility frame element with shear–flexure interaction. The geometry, material properties, and reinforcement detail of column 2CLD12 as provided by Sezen (2002) are shown in Fig. 12.

Fig. 12

Geometry and loads of column 2CLD12 (Sezen 2002).

In Fig. 13a the tip load–displacement response from the experimental test is superimposed on the numerical results obtained using both the proposed and the Mergos–Kappos models, while in Fig. 13b the experimental result is compared with the numerical result simulated by the flexural model. Clearly, Fig. 13a shows that the proposed element can represent reasonably well the salient characteristics of the experimental load–displacement behavior. However, the proposed model cannot accurately represent initial stiffness of the experimental load–displacement response since the concrete tensile strength is neglected herein. Compared to the Mergos–Kappos model, the proposed model represents a smoother load–displacement response, thus resulting in a more accurate yielding process in flexure. This benefit is due to the fiber-section model employed to characterize the column section response. It is worth remarking that the Mergos–Kappos model employs the bilinear yield-oriented moment–curvature curve to characterize the column section response, thus failing to capture the smooth yielding process of column cross section in flexure. From the numerical simulation, the proposed model predicts that the lateral displacement $\Delta_{y}$ and the sectional shear strain $\gamma_{y}$ associated with the plastic-hinge formation are approximately 13.80 mm and 1.58 × 10⁻³, respectively. Based on the experimental observation (Sezen 2002), the lateral displacement $\Delta_{u}$ at the onset of drastic shear strength deterioration is 28.0 mm. This value is close to the one given by the numerical model ($\Delta_{u} = 2 8. 2 1\;\;{\text{mm}}$). The essence of considering the shear–flexure interaction effect is emphasized in Fig. 13b in which the experimental result is compared with the numerical result simulated by the flexural model. As expected, the flexural model can predict well the column strength. However, shear failure following the plastic-hinge formation cannot be predicted by the flexural model. Thus, the frame element with inclusion of shear–flexure interaction effect is necessary and essential in simulating the response of RC columns prone to flexure–shear failure (ductile shear failure).

Fig. 13

Experimental and numerical responses of column 2CLD12: a proposed model; b flexural model.

Figure 14 shows hysteretic shear responses at various monitoring sections (sections I, II, III, and IV) along the column height. Plastic hinges can merely take place at the column ends (section I) due to restrained conditions at column ends. Thus, the hysteretic shear response at section I located within the plastic-hinge region is different from those at other sections (sections II, III, and IV) located outside the plastic-hinge zone due to the shear–flexure interaction. Figure 14a shows that once the sectional curvature ductility attains its threshold value of 3, the sectional shear response starts to deviate from the undamaged envelope curve as dictated by the shear–flexure interaction law of the UCSD shear-strength model. Figure 14b–d show that the hysteretic shear responses at sections II–IV are almost identical. This relies on the fact that the sectional shear–flexure interaction is not activated at sections II, III, and IV since they are located outside the plastic-hinge region. Furthermore, it is observed from Fig. 14 that even though all sections experience the same magnitude of shear force as governed by the equilibrium, the sectional shear strain at section I is much larger than the sectional shear strain at sections II, III, and IV. Eventually, the shear failure is detected at section I when the sectional shear strain attains its ultimate value of $\gamma_{u} = 12 \times 10^{ - 3}$. It is worth mentioning that this value is close to the experimental result of $10 \times 10^{ - 3}$ as given by Mergos and Kappos (2012).

Fig. 14

Sectional shear responses at various monitoring sections along the height of column 2CLD12: a x = – L/2 and L/2; b x = – L/3 and L/3; c x = – L/6 and L/6; d x = – L/6 and L/6.

Variations of shear force (demand) and shear strength (capacity) with the curvature ductility at section I are presented in Fig. 15. The shear-strength curve can be considered as the failure envelope and is constructed following the UCSD shear-strength model. The initial shear strength $V_{u0} = 515.6\;{\text{kN}}$ calculated from Eq. (31) is higher than the peak shear force $V_{{\text{max}} } = 316.1\;{\text{kN}}$. If the flexural and shear resisting mechanisms had been independent, column 2CLD12 would not have failed in shear. However, the shear–flexure interaction results in the shear-strength deterioration with increasing curvature ductility ($\mu_{\varphi } \ge 3$). Column 2CLD12 will eventually fail in shear when the shear demand meets the shear failure envelope with the associated value of the curvature ductility $\mu_{\varphi } = 14.83$ as shown in Fig. 15

Fig. 15

Variations of shear demand and shear strength with imposed curvature ductility at end section: column 2CLD12.

.

6.1.2 Lynn (2001): Column 2CMH18

A series of eight RC square columns containing a variety of reinforcement details and subjecting to two different levels of axial loading were conducted by Lynn (2001) under cyclic lateral displacements in double bending. These column specimens were constructed to represent existing RC columns built before the 1970s. One of these columns, labeled column 2CMH18, is modeled by the proposed RC frame element.

The geometry, material properties and reinforcement detail of column 2CMH18 as provided by Lynn (2001) are shown in Fig. 16. The amount and reinforcing detail of the transverse reinforcement is intended for characterizing the substandard seismic reinforcement details employed before the 1970s.

Fig. 16

Geometry and loads of column 2CMH18 (Lynn 2001).

In Fig. 17a the tip load–displacement response from the experimental test is compared with the numerical result obtained using the proposed model, while in Fig. 17b the experimental result is superimposed on the numerical result obtained using the flexural model. Clearly, both models are capable of characterizing the salient features of the experimental load–displacement response. From Fig. 17a, the proposed model predicts that the lateral displacement $\Delta_{y}$ and the sectional shear strain $\gamma_{y}$ associated with the plastic-hinge formation approximately equal to 8.30 mm and $1.21 \times 10^{ - 3}$, respectively. The lateral displacement $\Delta_{u}$ associated with the ultimate shear strain $\gamma_{u}$ is equal to 15.30 mm as obtained by the proposed model. This value of lateral displacement corresponds well with the experimental observation that the onset of rapid shear strength degradation occurred at lateral displacement of approximately 15.30 mm. Due to the loss of its axial load carrying capacity, the column specimen 2CMH18 eventually collapsed. The importance of including shear–flexure interaction effect is shown in Fig. 17b in which the experimental result is compared with the numerical result simulated with the flexural model. Although the flexural model can predict reasonably well the general features of the experimental load–displacement response, shear failure following the plastic-hinge formation cannot be predicted by the flexural model. Consequently, the frame element considering shear–flexure interaction is necessary and essential in the seismic evaluation of RC columns prone to flexure–shear failure (ductile shear failure).

Fig. 17

Experimental and numerical responses of column 2CMH18: a proposed model; b flexural model.

Hysteretic shear responses at various monitoring sections (sections I, II, III, and IV) along the column height are presented in Fig. 18. The hysteretic shear response at section I resting inside the plastic-hinge region is different from those at other sections (sections II, III, and IV) resting outside the plastic-hinge zone due to the shear–flexure interaction. It is worth mentioning that plastic hinges are located only at column ends due to their restrained conditions. Figure 18a indicates that the sectional shear response begins to deviate from the undamaged envelope curve when the sectional curvature ductility reaches its threshold value of 3 as dictated by the shear–flexure interaction law of the UCSD shear-strength model. Figure 18b–d indicate that the hysteretic shear responses at sections II-IV are almost identical. This relies on the fact that the sectional flexure–shear interaction is not activated at sections II, III, and IV since they are located outside the plastic-hinge zone. Comparison between the sectional shear responses in Fig. 18a and Fig. 18b–d indicates that although all sections are subjected to the same magnitude of shear force as governed by the equilibrium, the sectional shear strain at section I is much larger than the sectional shear strains at sections II, III, and IV. Eventually, the shear failure occurs at section I when the sectional shear strain reaches its ultimate value of $\gamma_{u} = 4.6 \times 10^{ - 3}$. It is worth remarking that for the transverse reinforcement detail of the column specimen 2CMH18, the shear strain $\gamma_{st}$ corresponding to the verge of transverse reinforcement yielding and the shear strain $\gamma_{u}$ corresponding to the onset of shear failure (significant strength deterioration) are equal based on the undamaged primary curve proposed by Mergos and Kappos (2012).

Fig. 18

Sectional shear responses at various monitoring sections along the height of column 2CMH18: a x = – L/2 and L/2; b x = – L/3 and L/3; c x = – L/6 and L/6; d x = 0.

Figure 19 shows the variations of shear force (demand) and shear strength (capacity) with the curvature ductility at section I. The shear-strength curve based on the UCSD shear-strength model can be regarded as the failure envelope. The initial shear strength $V_{u0} = 477.2\; {\text{kN}}$ calculated from Eq. (31) is higher than the peak shear force $V_{{\text{max}} } = 316.0\; {\text{kN}}$. If the shear–flexure interaction had not been considered, the column 2CMH18 would not have failed in shear. However, the shear–flexure interaction results in the shear-strength deterioration with increasing curvature ductility ($\mu_{\varphi } \ge 3$). Figure 19 shows that column 2CMH18 will eventually fail in shear when the shear demand meets the shear failure envelope with the associated value of the curvature ductility $\mu_{\varphi } = 5.86$. When compared to column 2CMH18 (Lynn 2001) presently studied, column 2CLD12 (Sezen 2002) discussed previously experiences shear failure at a much larger value of curvature ductility ($\mu_{\varphi } = 14.83$). This observation relies on the fact that column 2CMH18 was subjected to a higher axial loading and was on a more ductile sectional shear envelope of the column 2CLD12.

Fig. 19

Variations of shear demand and shear strength with imposed curvature ductility at end section: column 2CMH18.

6.2 Shear Critical Member

6.2.1 Bett et al. (1985): Column Specimen No. 1-1

A series of two-thirds scale RC columns with square cross section was tested by Bett et al. (1985) to assess the efficiency of strengthening and repairing techniques for short columns designed in seismic regions of the US in the 1950s and 1960s. One of these columns, labeled Specimen No. 1-1, is used herein to examine the validity of the proposed RC frame element in modeling a shear-dominated (shear critical) column. Figure 20 shows the geometry, material properties, and reinforcement detail of column specimen No. 1-1 as provided by Bett et al. (1985).

Fig. 20

Geometry and loads of column specimen No. 1-1 (Bett et al. 1985).

In Fig. 21a the tip load–displacement response from the experimental test is superimposed on the numerical result obtained using the proposed model, while in Fig. 21b the experimental result is compared with the numerical result simulated by the flexural model. Figure 21a shows that the proposed element can predict well the salient characteristics of the experimental load–displacement response. Unlike the load–displacement responses of flexural–shear critical members (column 2CLD12 and column 2CMH18) previously investigated, pinching of the hysteretic loops can be clearly noticed in the load–displacement response of column No. 1-1. This is expected since column No. 1-1 is considered a shear-dominated member. From Fig. 21a, the proposed model predicts that lateral displacement $\Delta_{u}$ and sectional shear strain $\gamma_{u}$ corresponding to the onset of drastic shear strength deterioration equal to 7.6 mm and $7.10 \times 10^{ - 3}$, respectively, and there is no plastic-hinge formation in column No. 1-1. This prediction corresponds well with the experimental observation. Figure 21b shows the adverse consequence of considering only flexural response in the numerical simulation. Clearly, the flexural model fails to simulate the load–displacement response of column No. 1-1. With the flexural model, both the member strength and the amount of dissipated hysteretic energy are drastically overestimated. Consequently, frame element with inclusion of sectional shear response is necessary and essential in assessing seismic performance of the shear-dominated RC columns.

Fig. 21

Experimental and numerical responses of column specimen No. 1-1: a Proposed Model; b Flexural Model.

Hysteretic shear responses at various monitoring sections (sections I, II, III, and IV) along the column length are presented in Fig. 22. Unlike the sectional shear responses of column 2CLD12 and column 2CMH18, the hysteretic shear responses at all monitoring sections of column No. 1-1 are identical. This relies on the fact that all column sections have not experienced yielding in flexure. Consequently, sectional shear–flexure interaction is not triggered in column No. 1-1 and the hysteretic shear responses follow the undamaged primary curve. The pinching characteristic is noticeable in the sectional shear response. Eventually, shear failure takes place once the sectional shear strain attains its ultimate value of $\gamma_{u} = 7.10 \times 10^{ - 3}$.

Fig. 22

Sectional shear responses at various monitoring sections along the column height: column specimen No. 1-1.

7 Summary and Conclusions

The present work proposes a fiber frame element for an inelastic analysis of reinforced concrete (RC) members prone to flexure–shear failure as well as shear failure. The proposed frame element is constructed within the framework of stiffness-based model and relies on the Timoshenko beam kinematics assumption, thus leading to comparatively simple and readily implemented equations. The choice of displacement interpolation functions is selected with care to obtain the locking-free Timoshenko frame element. Material nonlinearities are considered in the uniaxial hysteretic laws for concrete, steel, and sectional shear. Adverse influences of inelastic flexural deformation (plastic-hinge formation) on shear capacity are accounted for within the framework of the UCSD shear-strength model. The numerical procedure for shear–flexure interaction is modified following the Mergos–Kappos procedure. The model inputs are based on general engineering properties of RC members. The proposed element is simple, computationally efficient and able to describe several distinct characteristics of non-ductile RC column responses. Accuracy of the proposed model is validated through correlation studies on non-ductile RC columns under cyclic loadings.

For flexure–shear critical columns, the proposed model can predict reasonably well the member strength, stiffness deterioration with larger displacement magnitude, the amount of dissipated hysteretic energy, the general shape of hysteretic response, and the failure mode. The proposed model can represent well the shear-strength deterioration and the drastic increase of shear deformations after plastic-hinge formation. The essence of including the shear–flexure interaction is further emphasized when the flexural model is used to simulate the experimental response.

For the shear critical column, the proposed model can represent reasonably well the member strength, stiffness deterioration with larger displacement magnitude, the amount of dissipated hysteretic energy, the general pinched shape of the hysteretic response, and the failure mode. The essence of including shear response is confirmed when the flexural model is replaced by the proposed model.