 Original article
 Open Access
 Published:
Fiber Beam Analysis of Reinforced Concrete Members with Cyclic Constitutive and Material Laws
International Journal of Concrete Structures and Materials volume 12, Article number: 61 (2018)
Abstract
This paper presents a nonlinear Timoshenko beam element with axial, bending, and shear force interaction for nonlinear analysis of reinforced concrete structures. The structural material tangent stiffness matrix, which relates the increments of load to corresponding increments of displacement, is properly formulated. Appropriate simplified cyclic uniaxial constitutive laws are developed for cracked concrete in compression and tension. The model also includes the softening effect of the concrete due to lateral tensile strain. To establish the validity of the proposed model, correlation studies with experimentallytested concrete specimens have been conducted.
Background
The response of reinforced concrete (RC) structures is affected by the combined effect of bending, shear, and axial loads. Accurate constitutive model of RC elements for combined loadings is essential for reliably predicting structural behavior. In the past decades constitutive models have markedly improved, thereby improving the accuracy and efficiency of modeling complex RC structures. Efficient constitutive models for concrete and reinforcing bars are typically established from largescale panel testing, and assuming a smeared cracked behavior. Belarbi and Hsu (1995) developed the RotatingAngle SoftenedTruss Model (RASTM). They assumed that shear stresses exist along the crack direction, and proposed a tension stiffening function to account for this effect. This model was further improved by Pang and Hsu (1996), who developed the FixedAngle SoftenedTruss Model (FASTM). In this model, cracks were assumed to be oriented at a fixed angle. Zhu et al. (2001) derived a proper shear modulus compatible with the FASTM model and proposed a robust solution algorithm for analysis of shearcritical concrete elements. Later, the authors evaluated the Hsu/Zhu Poisson’s Ratio (Poisson’s ratio for cracked concrete) (Zhu and Hsu 2002), which resulted in the Softened Membrane Model (SMM). The SMM proved to be able to simulate both the prepeak and postpeak behavior of concrete elements.
Accurate modeling of the complex behavior of RC structures is typically performed with twodimensional membrane elements. However, these elements are computationally very expensive, which renders the analysis timeconsuming. Unlike membrane elements, fiber beam elements proved to provide a good balance between accuracy and numerical efficiency (Belarbi and Hsu 1995). In fiberbased beam elements, the spread of inelasticity along the depth is evaluated through discretization of the section into a large number of fibers with appropriate material models. Typical fiber elements account for the axialflexural interaction effect owing to the assumption that plane sections remain plane after deformation. Lately, Mullapudi and Ayoub (Mullapudi and Ayoub 2010) developed a fiberbased beam element that accounts for shearaxialflexure interaction effects and further improved the element to account for full threedimensional effects, including the combined bending, axial, shear and torsional interaction (Mullapudi and Ayoub 2013).
Research Objective
The main objective of this paper is to derive an appropriate material tangent stiffness matrix for fiber beamcolumn element formulation of shearcritical concrete members (Fig. 1). The developed stiffness matrix does not account for material nonlinearity. Simplified cyclic uniaxial constitutive laws are developed for concrete in both compression and tension. The formulation of the proposed element is based on the flexibility method of analysis. Flexibilitybased formulations (Mullapudi and Ayoub 2009; Labib et al. 2013) are used to overcome most of the locking difficulties associated with the standard displacement model. Shear effects is simulated through a Timoshenkobased approach (Mullapudi and Ayoub 2009). The concrete constitutive law is based on the aforementioned SMM model with Hsu/Zhu ratios. The work also attempts to improve the development of the concrete uniaxial constitutive relations. The model is added to the library of the finite element program FEAPpv (Taylor 2005).
Concrete Constitutive Model
The ACI 318 (American Concrete Institute 2008) building code suggests that the shear strength of an RC member is the combination of concrete strength (V_{C}) and transverse reinforcement strength (V_{S}). The value of V_{C} cannot be calculated in the RASTM model, because the crack angle is assumed to be rotating. However, the FASTM and SMM theories are capable of accounting for the effect of V_{C}; because the cracks are assumed to be oriented at a fixed angle and the proper concrete shear stress term (τ_{c}) is accounted for.
To formulate the SMM model with the inclusion of FRP, three coordinate systems are defined as shown in Fig. 2: the first (x, y) represents the local coordinate of the fiber; the second (1, 2) defines the principal stresses; while the third system (r, d) defines the concrete principal coordinate system in which the concrete shear stress \(\tau_{12}^{c} = 0\). In the figure, α_{1} is the angle between the x and 1axes, and α_{r} is the angle between the x and raxes (Fig. 2).
To following matrix R(θ) is used to rotate the stress and strain vectors from one coordinate system to another:
where θ is the angle between the two coordinate systems.
The compatibility equations in the x–y system are:
Similar to the strain transformations, stress transformation equations in the x–y system are:
The transformation equations are graphically represented by Mohr’s stress and strain circles in Fig. 3.
For a fiber Timoshenkotype beam element formulation, the state determination at the fiber level uses the strain state {ɛ_{x}, ɛ_{y}, γ_{xy}} to evaluate the fiber stresses {σ_{x}, σ_{y}, τ_{xy}}. In this case, the values of ɛ_{x}, γ_{xy} are given, but the lateral strain ɛ_{y} value is unknown and has to be calculated from the equilibrium equations as described next.
Process to Evaluate Lateral Strains
To evaluate the lateral strain, the equilibrium equations between concrete and steel are given below:
where f_{sx} and f_{sy} are the reinforcing bar stresses in the x and y directions respectively, and ρ_{sx}, ρ_{sy} are the smeared steel ratios in the x and y directions respectively.
The lateral strain ɛ_{y} in fiber i is calculated from the second of Equations in (4), knowing that the value of σ_{y} equals zero. In order to evaluate the value of the lateral strain ɛ_{y}, an iterative procedure is needed owing to the nonlinear behavior of the materials used (Mullapudi and Ayoub 2010).
UniAxial Constitutive Relationships of the Materials
The biaxial strains in the x–y direction \(\left\{ {\begin{array}{*{20}c} {\varepsilon_{x} } & {\varepsilon_{y} } & {\gamma_{xy} } \\ \end{array} } \right\}^{T}\) need to be converted to equivalent uniaxial strains in the principal 1–2 direction \(\left\{ {\begin{array}{*{20}c} {\bar{\varepsilon }_{1} } & {\bar{\varepsilon }_{2} } & {\gamma_{12} } \\ \end{array} } \right\}^{T}\) in order to evaluate the concrete stresses as explained in (Mullapudi and Ayoub 2010).
To evaluate the rotating crack angle, α_{r} corresponding to a concrete shear stress \(\tau_{12}^{c} = 0\) (Fig. 3) the following expression is used:
The calculation of the rotating angle, α_{r} is dependent on the strain state. The calculation of the trial rotating angle, \(\alpha_{r}^{*}\) from the Mohr circle (Fig. 3) is based on the strain values
If the value of the difference between the axial and transverse strains ɛ_{x} − ɛ_{y} = 0, then the value of the rotating angle α_{r} depends upon the value of the shear strain γ_{xy} as follows.
Rotating angle α_{r} = 45° when the value of γ_{xy} > 0, and α_{r} = 135° when the value of the γ_{xy} < 0.
If the value of the shear strain γ_{xy} = 0 then the rotating angle depends upon the value of the ɛ_{x} and ɛ_{y} as follows.
The rotating angle α_{r} = 0° when the value of ɛ_{x} > ɛ_{y} and, α_{r} = 90° when the value of ɛ_{x} < ɛ_{y}.
If both the shear strain and the difference of the axial and transverse strains are nonzero numbers, then the following laws will be applied.

If the value of ɛ_{x} > ɛ_{y} and the shear strain γ_{xy} > 0, then the value of the rotating angle α_{r} becomes the value of the \(\alpha_{r}^{*}\).

If the value of ɛ_{x} > ɛ_{y} and the shear strain γ_{xy} < 0, then the value of the rotating angle α_{r} = 180° − \(\alpha_{r}^{*}\).

If the value of ɛ_{x} < ɛ_{y} and the shear strain γ_{xy} > 0, then the value of the rotating angle α_{r} = 90° − \(\alpha_{r}^{*}\).

If the value of ɛ_{x} < ɛ_{y} and the shear strain γ_{xy} < 0, then the value of the rotating angle α_{r} = 90° + \(\alpha_{r}^{*}\).
After evaluating theɛ_{y}term that satisfies the equilibrium condition (Eq. 4), the principal angle α_{1} is evaluated as:
Similar to Eq. (2), the biaxial principal strains are calculated as follow:
The biaxial strains in the x–y direction \(\left\{ {\begin{array}{*{20}c} {\varepsilon_{x} } & {\varepsilon_{y} } & {\gamma_{xy} } \\ \end{array} } \right\}^{T}\) need to be converted to equivalent uniaxial strains in the principal 1–2 direction \(\left\{ {\begin{array}{*{20}c} {\bar{\varepsilon }_{1} } & {\bar{\varepsilon }_{2} } & {\gamma_{12} } \\ \end{array} } \right\}^{T}\) in order to evaluate the concrete stresses as explained in (Mullapudi and Ayoub 2010). This is done using the Hsu/Zhu ratios (μ_{12}, μ_{21}) (Zhu and Hsu 2002). μ_{12} is the ratio of the tensile strain increment in direction 1 to the compressive strain increment in direction 2, and μ_{21} is the ratio of the compressive strain increment in direction 2 to the tensile strain increment in direction 1. Based on test data the following expressions are proposed by Zhu and Hsu (2002).
where ɛ_{sf} is the strain in the steel bar that yields first and ɛ_{y} is the yield strain.
After cracking, the value of the Hsu/Zhu ratio μ_{12} is larger than maximum value of 0.5 for Poisson ratios of continuous materials. Before cracking, the Hsu/Zhu ratio μ_{21} = 0.2 and, after cracking, μ_{21} = 0, meaning the tensile strain does not affect the compressive strain.
The Hsu/Zhu ratio is used to elate the uniaxial strains to the biaxial principal strains:
where
The longitudinal and transverse reinforcement uniaxial principal strains are then:
The equivalent uniaxial longitudinal and transverse steel stresses, f_{sx} and f_{sy} respectively are evaluated from the corresponding steel strains \(\bar{\varepsilon }_{sx}\) and \(\bar{\varepsilon }_{sy}\) through a proper steel constitutive model. The equivalent uniaxial strains \(\bar{\varepsilon }_{1}\) and \(\bar{\varepsilon }_{2}\) are also used to evaluate the concrete stresses \(\sigma_{1}^{c}\) and \(\sigma_{2}^{c}\).
Concrete Model
The uniaxial concrete material model adopted follows the wellestablished modified Kent and Park model (Park et al. 1982). However, the model was modified to account for the following effects:

First, the softening effect for both, the stresses and strains, is accounted for.

Second, the cyclic stiffness degradation for both, the unloading and reloading branches, is introduced.

Third, the tensionstiffening effect is accounted for (Belarbi and Hsu 1994).
According to Kent and Park (Park et al. 1982), the monotonic stress–strain envelope of concrete follows a parabolic curve (Fig. 4):
It was observed from experimental tests of concrete panels that the compressive stress–strain curve is reduced due to the effect of perpendicular tensile stresses. This effect is accounted for through a softening coefficient \(\varsigma\). When the softened stress–strain curve is developed, it is assumed that the lines that connect the origin to the peak stress of the softened and nonsoftened curve have the same slope as shown in Fig. 4. Similarly, the prepeak and postpeak curves of the softened member are assumed to follow a parabolic shape. The descending branch of the softened envelope is gently sloped until the stress reaches a value that equals 20% of the maximum stress \(\varsigma f_{c}^{\prime }\) at a strain of ɛ_{20}. The residual concrete compressive strength is assumed to be 20% of the softened concrete compressive strength (\(\varsigma f_{c}^{\prime }\)). The sse of this value in the model is very common and has accurately predicted the experimental results (Mullapudi and Ayoub 2013).
For the softened behavior, the following relationships are adopted:
The cyclic model (Fig. 5) considers the concrete damage and accounts for the crack opening and closing. The envelope for the cyclic stress–strain curves of concrete adheres to the monotonic stress–strain curve. The unloading and reloading path of the compressive side is simplified, as all the loading paths start from a common point R, which determines the degradation stiffness (i.e. the ratio between the slope of any given loading path and that of the monotonic envelope at the origin), which limit is provided by the slope of the path RB.
The unloading modulus E_{20} at point B of the monotonic envelope curve is \(\frac{{0.2\varsigma f_{c}^{'}  f_{r} }}{{\varepsilon_{20}  \varepsilon_{r} }}\). E_{20} and must be determined experimentally. The stress and strain at the intersection of point R and the origin are given by the following expressions:
in which E_{c} is the initial tangent modulus at origin in compression; in the current model, it is assumed to equal \(\frac{{2f_{c}^{'} }}{{\varepsilon_{0} }}\). The unloading stress \(f_{m}^{1}\) and strain \(\varepsilon_{m}^{1}\) values at point D on the compressive monotonic envelope are used to calculate the reloading modulus and strain \(\varepsilon_{t}^{1}\) at zero stress point H from the following expressions:
From any unloading point D, the stress will reach the zero stress axis at point H after completing two smaller cycles that are defined by these expressions:
The loading and unloading cycles are carried out with the assumption that the model follows a linear behavior with modulus E_{c}. The trial stress \(f_{c}^{T}\) and tangent modulus E_{t} are based on a linear elastic behavior with initial tangent modulus E_{c}; later this assumption is corrected to fall under the line HD and line HE.
Here, \(f_{c}^{P}\) is the previous stress and Δɛ_{c} is the strain increment.
The actual stress f_{c} and tangent modulus E_{t} are calculated based on the trial stress state
When the unloading begins from points D to E, the reloading will follow the same path back to D. When the unloading reaches point F, then reloading will result in the loop DEFGD. If unloading reaches point H, then reloading will follow loop DEHD. The reloading path will always rejoin the compression envelope at the point of initial unloading, D. If unloading continues below point H, then reloading begins in tension. After the start of the reloading in compression, the model will reenter the compression branch at point H. Subsequent loading in the tension branch will not affect the behavior once the model returns back to the compression branch.
Steel Model
The smeared stress–strain relationship of steel embedded in concrete under uniaxial loading has been developed by Belarbi and Hsu (1994, 1995). The steel strain at cracked sections typically increases rapidly compared to adjacent regions because part of the stress is resisted by the concrete. Steel stresses are averaged along the reinforcing bar crossing several cracks, and the resulting smeared steel stress at first yield is reduced compared to the local yield stress of a bare bar. The smeared stress–strain relationship of embedded steel bars (solid curves), as well as that of bare bar (dotted curve) are plotted in Fig. 6.
The difference between the bare bar yield stress f_{y} and smeared steel bar yield stress \(f_{y}^{\prime }\) depends on the parameter B defined by Belarbi and Hsu (1994, 1995). The parameter Bis derived to be a function of three variables such as percentage of steel (ρ), cracking strength of concrete (f_{cr}), and yield stress of the bare bar (f_{y}). The parameter B depends on the crack width, crack spacing and interface bond slip behavior of steel bar and concrete. It can be seen that when the steel ratio ρ is decreased or when f_{cr} is increased, then the smeared yield stress \(f_{y}^{\prime }\) decreases and the smeared stress–strain plot moves downward.
As shown in Fig. 6, the shape of both, the bare bar and the smeared steel bar, curves of mild steel follow two straight lines. These two straight lines have slopes of E_{s} before yielding and \(E_{p}^{\prime }\) after yielding for smeared steel, and slope of E_{p} after yielding for bare steel bar as illustrated in Fig. 6. The slope of the strainhardening branch of the bare steel bar is assumed to equal 0.025 E_{s}. The stress value at which the two straight lines of smeared steel intersect is the smeared yield stress \(f_{y}^{\prime }\) and the corresponding strain is the smeared yield strain \(\varepsilon_{y}^{\prime }\).
The equations of the pre yield and post yield lines are given as:
The vertical intercept of the post yield line \(f_{{_{o} }}^{\prime }\) is evaluated as:
Belarbi and Hsu (1994, 1995) defined the parameter Bas:
where f_{cr} = 0.31 \(\sqrt {f_{c}^{'} (MPa)}\) and ρ ≥ 0.15%.
The smeared yield stress of steel bar is calculated as:
The smeared steel bar yield strain is calculated as:
The slope of the strain hardening branch of the smeared steel bar is calculated as
The smeared steel stress before yielding can be calculated as:
When the steel stress f_{s} reaches a value of f_{p} and begins to unload, then the unloading branch follows a straight line with a slope of E_{s}. The unloading stress is expressed as:
The cyclic response of the smeared steel bar is formulated according to the Filippou et al. (1983) model which accounts for isotropic strain hardening and Bauschinger effect.
Tangent Material Constitutive Relations
The concrete constitutive law is simplified as an orthotropic material with two perpendicular planes of elastic symmetry. Directions one and two are the local principal material axes that are normal to the planes of symmetry. With the equivalent uniaxial strains, the stiffness values \(\bar{E}_{1}^{c}\) and \(\bar{E}_{2}^{c}\) are determined from a material uniaxial stress–strain relationship. The material behavior is expressed as:
In this equation, \(\left\{ {\sigma_{12}^{c} } \right\}\) is the concrete stress vector, {ɛ_{12}} is the principal strain vector, and [D_{lo}]^{c} is the principal local uniaxial concrete material tangent stiffness matrix.
The tangent stiffness matrix of an RC element is defined as:
where ρ_{si} is the reinforcement ratio in the ith direction; and [R(− α_{1})] and [R(− α_{si})] are the transformation matrices from the local 1–2 coordinate and the x_{si }− y_{si} reinforcement coordinate system, to the x–y coordinate system, respectively (Fig. 7).
Substituting Eq. (44) into Eq. (43) gives:
Equation (45) is split into a concrete stiffness [D_{gl}]^{c} and a reinforcement stiffness [D_{gl}]^{s} as:
and
Thus the total stiffness becomes:
where [D_{gl}]^{c} is the tangent material constitutive matrix of concrete; and [D_{gl}]^{s} is the tangent material constitutive matrix of the steel bars.
The equivalent uniaxial strains can be calculated from the global strains using:
where [μ] is the Hsu/Zhu ratio matrix as shown in Eq. 12.
After substituting Eq. (49) into Eq. (46):
From Eq. (50), the concrete local tangential stiffness can be written as:
After substituting Eq. (51) into Eq. (50):
The diagonal terms in Eq. (64) matrix can be found directly from the uniaxial stresses and strains in the respective directions. The first diagonal term \(\frac{{\partial \sigma_{1}^{c} }}{{\partial \bar{\varepsilon }_{1} }} = \bar{E}_{1}^{c}\) is the tangential uniaxial modulus of concrete in the 1direction, the second diagonal term \(\frac{{\partial \sigma_{2}^{c} }}{{\partial \bar{\varepsilon }_{2} }} = \bar{E}_{2}^{c}\) is the tangential uniaxial modulus of concrete in the 2direction, and the third diagonal term \(\frac{{\partial \tau_{12}^{c} }}{{\partial \left( {\frac{1}{2}\gamma_{12} } \right)}} = \frac{{\sigma_{1}^{c}  \sigma_{2}^{c} }}{{\varepsilon_{1}  \varepsilon_{2} }} = G_{12}^{c}\) is the shear modulus. The offdiagonal terms \(\frac{{\partial \sigma_{1}^{c} }}{{\partial \bar{\varepsilon }_{2} }}\) and \(\frac{{\partial \sigma_{2}^{c} }}{{\partial \bar{\varepsilon }_{1} }}\) are obtained using the uniaxial stresses and the uniaxial strains in the orthogonal direction. These offdiagonal terms are not zero because the stress and strains of the concrete in compression is softened by the perpendicular tensile strains.
Therefore, [D_{lo}]^{c} can be written as:
The offdiagonal terms can be determined with three cases as described below.
Equivalent uniaxial strains \(\bar{\varepsilon }_{1} > 0\) and \(\bar{\varepsilon }_{2} > 0\)
When \(\bar{\varepsilon }_{1} > 0\) and \(\bar{\varepsilon }_{2} > 0\) then the uniaxial concrete stresses \(\sigma_{1}^{c}\) and \(\sigma_{2}^{c}\) are calculated only from uniaxial strains \(\bar{\varepsilon }_{1}\) and \(\bar{\varepsilon }_{2}\), respectively.
Therefore,
Equivalent uniaxial strains \(\bar{\varepsilon }_{1} > 0\) and \(\bar{\varepsilon }_{2} < 0\)
When the uniaxial strain is \(\bar{\varepsilon }_{1} > 0\) then the uniaxial compressive stress \(\sigma_{1}^{c}\) is calculated directly from the \(\bar{\varepsilon }_{1}\), and \(\sigma_{1}^{c}\) is not a function of the orthogonal concrete strain \(\bar{\varepsilon }_{2}\). Therefore, \(\frac{{\partial \sigma_{1}^{C} }}{{\partial \bar{\varepsilon }_{2} }} = 0\).
To obtain \(\frac{{\partial \sigma_{2}^{C} }}{{\partial \bar{\varepsilon }_{1} }}\), the constitutive law of the concrete strut in compression is needed.
In order to obtain \(\frac{{\partial \sigma_{2}^{C} }}{{\partial \overline{\varepsilon }_{1} }}\), the constitutive law of concrete in compression is needed.
where the softening coefficient ζ (Hsu and Zhu 2002)
In Eq. (57), α_{r1} is in degrees. If the value of the α_{r1} is in radians then the value of the 24° should be converted into radians. From panel tests up to 100 MPa of concrete compressive strength, the deviation angle α_{r1} is equal to or less than 24° (Wang 2006)
After making the required changes, Eq. (57) becomes
where ɛ_{1} and ɛ_{2} are the biaxial strains in the 1–2 coordinate system and \(\bar{\varepsilon }_{1}\) and \(\bar{\varepsilon }_{2}\) are the equivalent uniaxial strains in the 1–2 coordinate system.
where μ_{12} and μ_{21} are the Hsu/Zhu ratios (Zhu and Hsu 2002)
\(\frac{\partial \zeta }{{\partial \bar{\varepsilon }_{1} }}\) can be derived after substituting Eq. (61) into Eq. (58):
where \(\frac{{\partial \sigma_{2}^{C} }}{\partial \zeta }\) can be derived as follow when \(\frac{{\bar{\varepsilon }_{2} }}{{\zeta \varepsilon_{0} }} \le 1\):
when \(\frac{{\bar{\varepsilon }_{2} }}{{\zeta \varepsilon_{0} }} > 1\):
Let
Equation (67) can be written as:
where
Let
Equation (71) can be written as:
where
Substituting Eq. (74) into Eq. (73) gives:
Substituting Eq. (75) into Eq. (70) gives:
Substituting Eqs. (66) and (64) into Eq. (62) gives:When
Substituting Eqs. (76) and (64) into Eq. (62) gives:When
When the equivalent uniaxial strains \(\bar{\varepsilon }_{1} < 0\) and \(\bar{\varepsilon }_{2} > 0\), then the same procedure should be followed.
Equivalent uniaxial strains \(\bar{\varepsilon }_{1} < 0\) and \(\bar{\varepsilon }_{2} < 0\)
When \(\bar{\varepsilon }_{1} <\) 0 and \(\bar{\varepsilon }_{2} < 0\) then concrete will not soften; instead it increases its compressive strength in one direction depending on the confining stress in the orthogonal direction. Because of this reason, the value of ζ should be greater than or equal to 1. The current research uses the Vecchio’s (Vecchio 1992) simplified version of Kupfer et al. (1969) biaxial compression strength envelope. These equations are strengthbased; strainbased equations are not available in the literature and need to be investigated in future. Accordingly:
The global tangential constitutive matrix of steel [D_{gl}]^{s}can be derived From Fig. 7 as:
where \(\bar{\varepsilon }_{si}\) is the equivalent uniaxial longitudinal strain of the reinforcement, \(\bar{\varepsilon }_{{si^{*} }}\) is the equivalent uniaxial transverse or dowel strain of the reinforcement and γ_{si} is the equivalent uniaxial shear strain of the reinforcement.
Substituting Eq. (82) into Eq. (47) gives:
Let
where \(\frac{{\partial f_{si} }}{{\partial \bar{\varepsilon }_{si} }} = \bar{E}_{si}\), which is the equivalent uniaxial tangential modulus in the longitudinal direction of the reinforcement. The dowel action of the reinforcement is neglected, thus \(\frac{{\partial f_{si} }}{{\bar{\varepsilon }_{{si^{*} }} }} = 0\), and also the shear deformation in the reinforcing bar is neglected, thus \(\frac{{\partial f_{si} }}{{\partial \left( {\frac{1}{2}\gamma_{si} } \right)}}\) = 0.
With these simplifications Eq. (86) can be written as:
Equation (85) can be written as:
After substituting Eqs. (88) and (51) into Eq. (48) the global tangent material constitutive matrix [D_{gl}]^{c+s} can be evaluated as:
The total section stiffness is evaluated from the sum of concrete and steel stiffness:
The total section force is evaluated from the sum of concrete and steel forces in their respective directions:
where n is the number of concrete and steel fibers in a section.
The element stiffness and forces are calculated with numerical integration of section stiffness and section forces at different section points along the length of the element.
Numerical Correlations with Experiments
The simplified uniaxial material and constitutive laws are implemented in the developed fiber beam element and analyzed the structural members under cyclic loading. Model predicted the experimental results throughout the loading history and could be used to simulate the behavior of RC structures under seismic loading.
Comparison of Concrete Model with the experimental 1D Cyclic Stress–Strain Curves
The uniaxial material models developed in this paper are compared with the experimental results of Mansour (2001).
Mansour (2001) tested three panels of the CVE3—series. The steel grids in these panels are set parallel to the applied principal stresses in horizontal and vertical directions. The three panels of this series CVE31, CVE32 and CVE33 are subjected to 1D cyclic loading in the horizontal direction, while maintaining a constant lateral tensile strain (ε_{t}) of 0.0044, 0.012 and 0.030.
Each panel has the following dimensions: length 1397 mm, height 1397 mm, and thickness 178 mm. The panels are reinforced in each direction with No. 6 bars at 267 mm spacing. The concrete compressive strengths of CVE31, CVE32, and CVE33 are 48, 41, and 43 MPa. The yield stress of longitudinal and transverse steel of panels CVE31, CVE32, and CVE33 are 425.4 MPa respectively.
The analytical result of the three panels CVE31, CVE32, and CVE33 with the current model and corresponding experimental results are presented in Figs. 8, 9, and 10. In these figures, the horizontal axes represent the smeared concrete strain in the longitudinal direction, and the vertical axes represent the smeared concrete stress in the longitudinal direction.
Comparison of the current model results for the three panels in the CVE3 series showed an increase in lateral tensile strain; the ultimate value of the horizontal compression stress decreases because of the softening behavior of the concrete. Figures 8, 9, 10, 11, 12, 13 show that the current model predicts fairly well the experimental behavior at both the compression and tension regions.
Simulation of Columns
Three hollow, rectangular prototype bridge piers PI1, PI2, and PS1 are tested under reverse cyclical loading at the National Center for Research on Earthquake Engineering in Taiwan (Yeh and Mo 1999). These prototypes are analyzed using the fiber beam element. The specimens dimensions along with the material properties of concrete and reinforcement as given in Table 1. Figure 11 shows the details of the specimens’ cross section dimensions and reinforcement details. The columns are tested under displacement control with cyclicallyreversed horizontal load.
The specimens are modeled with only one force based element and five Gauss–Lobatto integration points. The cross section is divided into 28 fiber sections. The boundary condition at the bottom is assumed to be fixed while the lateral load is applied to the top of the column. The longitudinal and transverse steel ratios are calculated based on the dimensions and spacing of the reinforcements. Concrete enveloped by the stirrups is modeled as confined concrete, while the remaining concrete (mostly in the cover) is considered as unconfined. The horizontal forces are increased based on the cyclic displacement control scheme.
The analytical shear force versus displacement relationships of the specimens are predicted with the 2D fiber beam element as shown in Figs. 12, 13, 14; the results are then compared to the experimental data.
The moment arms for specimens PS1, PI1, and PI2 are 6.5, 4.5, and 3.5 m respectively. The reinforcement of the columns is designed such that column PS1 is dominated by a flexural failure, column PI1 is dominated by a flexureshear failure, and column PI2 is dominated by shear failure. The experimental results (Yeh and Mo 1999) showed that specimens PS1 and PI1 failed in a flexure mode with the formation of plastic hinges at the bottom of the column and specimen PI2 failed under shear failure mode without rupturing the longitudinal reinforcement. The failure modes and ductility levels are reflected in the shape of the load displacement plots. The rebar yielded significantly prior to crushing of the concrete in specimens PS1 and PI1 (Figs. 12 and 13), which resulted in a long yield plateau and higher energy dissipation. Specimen PI2 (Fig. 14) showed a much shorter yield plateau and pinching with less energy dissipation than specimens PS1 and PI1. Specimen PS1 developed a displacement ductility of 10.8, specimen PI1 developed a displacement ductility of 7.8, and specimen PI2 developed a displacement ductility of only 3.7.
The analytical load–displacement relationships with the fiber beam element accurately captured the different behaviors of each of the three specimens. The fiber beam element predicted the initial stiffness, yield point, ultimate strength, and ductility of specimens PS1 and PI1 very well. The predicted cyclic load–displacement curve of symmetric specimen PI1 (Fig. 13) showed less ultimate strength in the negative cycles compared to experimental results. The fiber beam element also predicted the behavior of specimen PI2 (Fig. 14) to a degree in the positive direction, including the ultimate strength and strength degradation in the descending branch. The predicted ultimate strength of specimen PI2 in the negative direction is slightly higher than the test result, while the ultimate strength is slightly less with the plane stress element analysis. The predicted hysteresis loops of specimen PI2 with fiber beam element showed much less energy dissipation and predicted the experimental results. Table 2 summarizes the specimens’ strength values for both the experimental results and analytical predictions.
Conclusions
This paper represents a new element for cyclic analysis of reinforced concrete structures. A fiberbased beam element is developed to analyze reinforced concrete structures with the incorporation of mechanisms of shear deformation and strength. Simplified cyclic uniaxial constitutive relations are developed and checked with the 1D cyclic test panels of Mansour (2001). The tangent stiffness is formulated with the inclusion of the softening and dilatation effects. The reverse cyclic analyses of different columns with rectangular crosssections are analyzed with the 2D fiber beam element. The cyclic analysis of columns tested by Yeh and Mo (1999) are analyzed with the 2D fiber beam element. The columns with different aspect ratio exhibit different behaviors. The numerical results concerning the columns agree with the experimental data throughout the entire loading history. The finite element analysis using the generalized Softened Membrane Model predicted the experimental results throughout the loading history, including the initial stiffness, yield point, ultimate strength, and failure mode; and could therefore be used to simulate the behavior of RC structures under seismic loading.
References
American Concrete Institute. (2008). ACI 318, Building code requirements for structural concrete and commentary. Farmington Hills, MI, USA: American Concrete Institute.
Belarbi, A., & Hsu, T. T. C. (1994). Constitutive laws of concrete in tension and reinforcing bars stiffened by concrete. Structural Journal, American Concrete Institute, 91, 465–474.
Belarbi, A., & Hsu, T. T. C. (1995). Constitutive laws of softened concrete in biaxial tensioncompression. Structural Journal of the American Concrete Institute, 92(5), 562–573.
Filippou, F. C., Popov, E. P., Bertero, V. V. (1983). Effects of bond deterioration on hysteretic behavior of reinforced concrete joints. Report SESM 771, Division of Structural Engineering and Structural Mechanics. Berkeley: University of California.
Hsu, T. T. C., & Zhu, R. R. H. (2002). Softened membrane model for reinforced concrete elements in shear. Structural Journal of the American Concrete Institute, 99(4), 460–469.
Kupfer, H. B., Hildorf, H. K., & Rusch, H. (1969). Behavior of concrete under biaxial stresses. Structural Journal, American Concrete Institute, 66(8), 656–666.
Labib, M., Mullapudi, T. R. S., & Ayoub, A. S. (2013). Analysis of RC structures subjected to multidirectional shear loads. Journal of Advanced Concrete Technology, 11, 22–34.
Mansour, M. (2001). Behavior of reinforced concrete membrane elements under cyclic shear experiments to theory. Ph.D. dissertation. Department of Civil and Environmental Engineering, University of Houston.
Mullapudi, T. R. S., & Ayoub, A. S. (2009). Fiber beam element formulation using the softened membrane model. American concrete institute special publication (pp. 283–308). Farmington Hills: SP265, ACI.
Mullapudi, T. R. S., & Ayoub, A. S. (2010). Modeling of the seismic behavior of shearcritical reinforced concrete columns. Journal of Engineering Structures, 32(11), 3601–3615.
Mullapudi, T. R. S., & Ayoub, A. S. (2013). Analysis of reinforced concrete columns subjected to combined axial, flexure, shear and torsional loads. Journal of Structural Engineering ASCE, 139(4), 561–573.
Pang, X. B., & Hsu, T. T. C. (1996). FixedAngle softenedtruss model for reinforced concrete. Structural Journal of the American Concrete Institute, 93(2), 197–207.
Park, R., Priestley, M. J. N., & Gill, W. D. (1982). Ductility of square confined concrete columns. Journal of Structural Davison, ASCE, 108(4), 929–950.
Taylor, R. L. (2005). FEAP User Manual v2.0. Berkeley: Department of Civil and Environmental Engineering, University of California. http://www.ce.berkeley.edu/~rlt/feap/.
Vecchio, F. J. (1992). Finite element modeling of concrete expansion and confinement. Journal of Structural Engineering, ASCE, 118(9), 2390–2405.
Wang, J. (2006). Constitutive relationships of prestressed concrete membrane elements. Ph.D. dissertation. Houston: University of Houston.
Yeh, Y. K., Mo, Y. L. (1999). Full scale tests on ductility, shear strength and retrofit of reinforced concrete hollow columns (I). Report, No. NCREE99024. Taipei: National Center for Research on Earthquake Engineering. (in Chinese).
Zhu, R. H., & Hsu, T. T. C. (2002). Poisson effect of reinforced concrete membrane elements. Structural Journal, American Concrete Institute, 99(5), 631–640.
Zhu, R. H., Hsu, T. T. C., & Lee, J. Y. (2001). Rational shear modulus for smeared crack analysis of reinforced concrete. Structural Journal, American Concrete Institute, 98(4), 443–450.
Authors' contributions
TRM and AA developed the model presented. TRM conducted the computer coding and drafted the first version of the manuscript. Both authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Author information
Authors and Affiliations
Corresponding author
Additional information
Journal information: ISSN 19760485 / eISSN 22341315
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Ravi Mullapudi, T., Ayoub, A. Fiber Beam Analysis of Reinforced Concrete Members with Cyclic Constitutive and Material Laws. Int J Concr Struct Mater 12, 61 (2018). https://doi.org/10.1186/s400690180286z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s400690180286z
Keywords
 combined loading
 Timoshenko beam
 R/C beams
 uniaxial constitutive relations
 tangent stiffness