Finite element modelling of reinforced concrete framed structures including catenary action

doi:10.1016/j.compstruc.2010.01.002

Computers & Structures

Volume 88, Issues 9–10, May 2010, Pages 529-538

https://doi.org/10.1016/j.compstruc.2010.01.002 Get rights and content

Abstract

In this paper, a 1D discrete element is formulated for analysis of reinforced concrete frames with catenary action. A force-based formulation is developed based on the total secant stiffness approach and an associated direct iterative solution scheme is derived. The effect of material nonlinearity as well as softening of concrete under compression is taken into account and a nonlocal averaging technique is employed to maintain the objectivity of displacement and force responses. Concerning geometrical nonlinearities, the fibre strains are assumed to be small; however, the effect of transverse displacement on the axial strain is large and it is taken into account as well as the effect of shear on the axial force. Using a Simpson integration scheme, together with a piecewise interpolation of curvature, the deformed shape of the element is consistently updated. The formulation is verified with numerical examples.

Introduction

Catenary action of frame elements when a structure is subjected to extreme loading scenarios, such as blast and fire, can assist the elements to withstand gravity loads and prevent progressive collapse. Accordingly, an accurate progressive collapse assessment procedure has to take account of the catenary effects. The catenary action of steel and composite elements has been studied extensively [1], [2], [3], [4], [5], whereas the effect of catenary action within reinforced concrete elements has not been studied that much, except for a few simplified cases [6], [7].

In a progressive collapse assessment of frames based on an alternate load path, one or two of the load-bearing elements within different scenarios are removed and the ensuing structural response is obtained [8], [9]. In this case, the beams bridging over the removed columns experience large vertical displacements and develop catenary action. In unrestrained reinforced concrete beams, after cracking the neutral axis moves within the section such that at the initial reference axis (i.e. neutral axis of the uncracked section) the strains are tensile. That is the average strain in the member is tensile. However, in a framed structure, due to in-plane axial restraints provided by columns, the member cannot extend freely and these restraints set up axial compressive forces in the beams that must be considered in the analytical model [7].

In 1D frame elements, softening of concrete under compression after peak is associated with a lack of objectivity of the post-peak results and localisation of failure that is observed at either the element or section level, depending on the formulation type [10]. In such cases, the objectivity of response can be maintained by using lumped nonlinearity models with a softening hinge [11], scaling of the constitutive law [10], using a plastic-hinge integration method [12], adopting a nonlocal averaging technique [13], [14], [15], or by using a gradient-dependent plasticity model within the distributed nonlinearity context [16]. In this study the nonlocal averaging method is adopted as it can maintain the objectivity of displacement and moment-curvature responses without additional provisions and the position of plastic hinges are not required to be determined a priori.

In the finite element context, the displacement and force-based formulations, more or less, have the same degree of approximations. For frame elements, however, force-based formulations leads to superior accuracy compared with displacement-based formulations due to exact fulfillment of equilibrium equations [17], [18], [19]. Consequently, in this paper the force interpolation concept within a framework of a total secant stiffness approach is adopted to formulate the element.

Adopting the concept of the “natural approach”, the geometrical nonlinearity is taken into account by decomposing element displacements into a rigid body rotation and deformations [20], [21], [22]. The effect of transverse displacement on the axial strain is taken into account; however, the strains are taken to be small. Further, the formulation takes account of varying axial force along the element caused by geometrical nonlinearity, because the definition of axial load adopted in the formulation includes the second order effects as well as the effect of shear (although small) on the axial force. A composite Simpson integration scheme, accompanied with a parabolic piecewise interpolation of curvature function is used to establish the displacements and rotation of the longitudinal sections required for the geometrical nonlinearity analysis. While the formulation is developed for 2D cases, the requirements for extension to 3D are given in [23].

Section snippets

Compatibility equations

Adopting perfect bond and the Navier–Bernoulli assumptions, section compatibility requirements yield $ε_{x} = ε_{r} - y κ$ where ε_x denotes the total strain at the integration point in local x–x direction (along the element axis, see Fig. 1a), ε_r denotes the section axial strain, κ denotes the total curvature of section and y is the distance of the integration point (fibre) from the mid-plane of the element and parallel with x–z plane (see Fig. 1a).

Fig. 1a shows a two-node plane frame element AB with three

Displacement interpolation along the element axis

It is observed that incorporating the geometrical nonlinearity into the formulation necessitates the deformed shape of the element to be available. In the displacement-based formulation, the deformed shape of the element is obtained based on the nodal displacement values and adopted shape functions. In the force-based element, however, there is no displacement shape function to be used and a different technique is required. Carol and Murcia [24] have used combination of Simpson integration and

Implementation of the nonlocal integral formulation

The transverse deflection of reinforced concrete beams in a framed structure could be associated with compressive force as explained in Section 1. The stresses caused by this compressive axial force combined with compressive stresses caused by bending moment could lead to compressive softening of concrete fibres, which is associated with a lack of objectivity and with spurious mesh sensitivity.

In this study, the concept of a nonlocal integral model is adopted to maintain the objectivity of the

Rigid body motion and corresponding transformation

The flexibility formulation presented in the previous section was derived in the element reference system without rigid body motion. Thus, a transformation is required to relate the deformation and corresponding force vectors in the system without rigid body modes to the system with rigid body modes.

Satisfying the equilibrium equations, the following transformation between the force vectors of the systems with and without rigid body modes can be established (Fig. 6), $Q = T^{T} \bar{Q}$ where $Q = [\begin{matrix} Q_{1} & Q_{2} & Q_{3} & Q_{4} & Q_{5} & Q_{6} \end{matrix}]^{T}$

Direct iteration solution scheme

The solution procedure consists of a single loop that carries out the iteration at the structural level and, in which, element equilibrium as well as compatibility is satisfied. If the element is subjected to element load, an inner loop is required to restore the section (element) equilibrium. In the algorithm presented below, the right superscript i denotes the iteration number and right subscript k indicates the load step. If the results at the end of the kth load step are available, then

Linear elastic analysis of a pin supported indeterminate beam

This first example is used to demonstrate the performance of the developed formulation, for a linear-elastic beam and the results are compared with more demanding displacement-based formulations. A statically indeterminate simple beam with pin supports at both ends and subjected to point load at mid-span is analysed (Fig. 7). The elastic modulus is E = 25,000 MPa and as the material behaviour is linear-elastic, there is no need to use the nonlocal formulation. Thus, one half of the beam is

Conclusions

A novel flexibility-based formulation in the framework of the total secant stiffness matrix is derived for 1D frame elements and a direct iterative solution scheme, consistent with the formulation, is presented. In adopting a total secant solution strategy, the positive definiteness of the stiffness matrix is preserved and improves the stability of the solution. Furthermore, the inelastic deformation of the elements at each stage of the loading can be obtained directly without resorting to an

References (29)

Z. Ma et al.
Structural behaviour of composite slim floor frames in fire conditions
J Construct Steel Res
(2006)
Y.Z. Yin et al.
Analysis of catenary action in steel beams using a simplified hand calculation method. Part 1: Theory and validation for uniform temperature distribution
J Construct Steel Res
(2005)
G. Kaewkulchai et al.
Beam element formulation and solution procedure for dynamic progressive collapse analysis
Comput Struct
(2004)
G. Cocchetti et al.
Elastic–plastic and limit-state analyses of frames with softening plastic-hinge models by mathematical programming
Int J Solid Struct
(2003)
I. Carol et al.
Nonlinear time-dependent analysis of planar frames using an ‘exact’ formulation. I. Theory
Comput Struct
(1989)
C. Oran et al.
Large deformation of framed structures under static and dynamic loads
Comput Struct
(1976)
I. Carol et al.
Nonlinear time-dependent analysis of planar frames using an ‘exact’ formulation. II. Computer implementation for R.C. structures and examples
Comput Struct
(1989)
M. Jirasek
Nonlocal models for damage and fracture: comparison of approaches
Int J Solid Struct
(1998)
M. Byfield et al.
Catenary action in steel-framed buildings
Proc Inst Civil Eng: Struct Build
(2007)
Khandelwal K, El-Tawil S. Catenary action during collapse of steel MRF buildings, St. Louis, MO, United States; 2006....

Y.C. Wang et al.

A simplified analysis of catenary action in steel beams in fire and implications on fire resistant design

Steel Compos Struct

(2006)

B.A. Izzuddin et al.

Failure of lightly reinforced concrete members under fire. I: Analytical modeling.

J Struct Eng

(2004)

M. Sasani et al.

Progressive collapse analysis of an RC structure

Struct Des Tall Special Build

(2007)

M. Sasani et al.

Progressive collapse of reinforced concrete structures: a multihazard perspective

ACI Struct J

(2008)

Cited by (38)

Numerical and simplified analytical investigation on RC frame behaviours under progressive collapse scenarios
2022, Structures
Citation Excerpt :
In conjunction with experimental research, numerical investigations have also been extensively carried out to study aspects of the structural behaviour of the RC sub-structures which are difficult to observe experimentally due to cost and safety constraints. The applications of physics-based solid modelling using the finite element (FE) method [16–19] have shown good agreement with actual test data. When considering static conditions, simplified numerical analyses using fibre elements or beam-column joint models can be used [4,20].
Although progressive collapse research has been extensively carried out for more than two decades after from the tragic events of 9/11, not many actual dynamic experiments have been conducted due to the constraint in laboratory facilities and expense. Instead, numerical and simplified analytical studies were preferred to achieve a comprehensive understanding of the structural behaviour against these rare dynamic events. In this research, a numerical and analytical investigation programme is carried out to study the mobilisation of alternate-load-path mechanisms, such as compressive arch or catenary action, in RC beam-column substructures undergoing missing column scenarios. First, a physics-based finite element modelling is developed and validated with two published test series conducted under quasi-static and free-fall dynamic conditions. These tests concentrated on 2D reinforced concrete sub-frame structures, investigating two column-loss scenarios, i.e. internal and penultimate columns. After being validated, the model is then employed to further investigate important parameters which can significantly affect the behaviour of a 2D frame structure under both static and dynamic environments, such as concrete grade, release time, column pre-loaded force, etc. Thereafter, a simplified analysis using an equivalent SDOF system is developed to provide a convenient tool for assessing dynamic capacities of RC frame structures. The simplified method produces conservative predictions compared to actual tests and detailed numerical analyses, however it requires very little computing time.
Reliability of fully assembled precast concrete frame structures against progressive collapse
2022, Journal of Building Engineering
Citation Excerpt :
In addition, Yu et al. [12] and Feng et al. [13,14] also applied macro model-based FE analysis to investigate the progressive collapse behavior of concrete beam-column sub-assemblages, and parametric studies were conducted. Due to the convenience of macro model, it can be used to estimate the structural performance of frame structures [15–19]. Pushdown analysis [20] is very efficient to determine the collapse modes and ultimate load-carrying capacity of frame structures with lost or damaged critical members.
Although fully assembled precast concrete (PC) frame structures using dry connections have been widely promoted, limited studies have been carried out on anti-progressive collapse performance. This study presented a methodology to analyze and compare the reliabilities of entire structural systems between RC and fully assembled PC frame structures, which evaluates the risk and vulnerability of a structure when column failure occurs under accidental loads. First, based on the existing test results of three half-scale moment sub-structures, nonlinear finite element (FE) macro-models were established in OpenSEES software. The verified FE models were applied to three seven-story frame structures (one RC frame and two PC frames). To evaluate the progressive collapse resistance, nonlinear pushdown analysis was conducted on the frame structures using the alternate load path (ALP) approach. Combining the Nataf transformation-based point estimate method (i.e. the improved point estimate method (IPEM)) with the nonlinear analysis results, various load amplification factors ( $α_{m a x}$ ) were estimated. Finally, the reliability indices of the intact and damaged frame structures were calculated using the Zhao-Ono high-order moment method (HOMM). The results revealed that the fully assembled PC frames had higher failure probability (2.1–6.6%) than the RC frame (0.2%) in a catastrophic event.
Robustness quantification of reinforced concrete structures subjected to progressive collapse via the probability density evolution method
2020, Engineering Structures
Robustness is the most comprehensive and acceptable index that describes the ability of structures to withstand progressive collapse induced by accidental extreme events such as impact, explosion and terrorist attacks. As found in the literature, several approaches have been proposed to quantify the robustness of a structure, e.g., approaches based on deterministic structural performance, failure probabilities (or the collapse reliabilities), and collapse risks. In this paper, the reliability-based approach is adopted to quantify the structural robustness of reinforced concrete (RC) structures subjected to progressive collapse since it is a trade-off between comprehensiveness and operability. An efficient calculation framework is developed based on the probability density evolution method (PDEM). Emphasis is placed on two aspects, i.e., the progressive collapse behavior modeling and the evaluation of the structural reliability. The static nonlinear pushdown method is employed to represent the progressive collapse capacity of the structures, and the force-based frame element is used to generate the finite element model. Then, the PDEM incorporated with the equivalent extreme value event is used to capture the reliability indices before and after progressive collapse. With the reliability indices, the robustness index can be easily computed. The developed framework is applied to two prototype RC frames designed in accordance with the Chinese design code. The reliability and robustness indices of the frames under different initial local damage scenarios (namely, removal of columns in typical pushdown method) are obtained, and the influences of the position of the initial damage scenarios on the robustness are also discussed.
Simplified analysis method for catenary action of restrained cellular steel beams at elevated temperature considering strain reversal
2018, Fire Safety Journal
Citation Excerpt :
Wong [10] presented a simple technique to model the axial restraints imposed on a steel beam by adjacent members and discussed the catenary action due to large deflection effect for beams in a fire. Valipour and Foster [11] formulated a 1D discrete element for the analysis of reinforced concrete frames with catenary action. Comparison with experimental results and available analytical results showed that the formulation was capable to accurately capture the catenary action of the element.
A Simplified Analysis Method (SAM) for predicting catenary action of restrained Cellular Steel Beams (CSB) at elevated temperature, as an alternative of finite element method, was proposed in this paper. A finite element model was developed and validated by existing test results and numerical results first. The temperature distribution along the section height was assumed to be linear based on available research results. Then the simplified analysis method was developed based on equilibrium equations, constitutive equations and geometric equations. The approach considered non-uniform temperature, geometric non-linearity, material non-linearity, as well as strain reversal when calculating section stresses. A series of parametric study using verified numerical model were then carried out to assess the accuracy of the proposed method. It was found that the simplified method yielded results which agreed well with those obtained from Finite Element Analysis (FEA). A lower compressive force and catenary tension were predicted when the web at perforated section was not considered. And the ignorance of strain reversal in the SAM would lead to a larger deflection prediction. The simple approach could also be used to predict the behavior of the beam subjected to non-uniform temperature distribution across the section in the latter stage of a fire, which may not always be obtained from FEA due to convergence problem.
Progressive collapse fragility models of European reinforced concrete framed buildings based on pushdown analysis
2017, Engineering Structures
Citation Excerpt :
Other parameters were considered as deterministic variables or deterministic transformations of RVs. Fiber-based modeling of structures was selected because it is currently used in earthquake engineering [58,61,63–65] and progressive collapse simulation [11,12,17,19,25,29–32,35,37–39,50]. The downward load on beams, which is here denoted as Qb, was assumed to be the IM.
Structural safety for extreme loads that may cause local damage to single primary components or even the progressive collapse of the structure has been probabilistically assessed in a few studies, hence neglecting uncertainties in loads and system capacity. As such, this paper moves from a deterministic to a probabilistic framework, proposing new progressive collapse fragility models based on pushdown analysis of low-rise, reinforced concrete framed bare structures. Two building classes representative of structures designed for either gravity loads or earthquake resistance in accordance with current European codes were investigated. Monte Carlo simulation was used to generate random realizations of 2D and 3D structural models. Fiber-based finite element models were developed within an open source platform. The primary output consisted of fragility functions for each damage state of interest, given the loss of corner column at the ground floor. The fragility models were compared to those derived through incremental dynamic analysis (IDA) to assess the inaccuracy of progressive collapse fragility functions derived through pushdown analysis. Load capacity predictions provided by those analysis methods were used to develop regression models for a quick estimation of dynamic amplification factor at a given displacement/drift level. The analysis results show a significant influence of both seismic design and secondary beams on robustness of the case-study building classes.
Numerical simulation of bond-slip interface and tension stiffening in GFRP RC tensile elements
2016, Composite Structures
Bond between reinforcement and concrete highly affects the structural behaviour of reinforced concrete (RC). Introduction of bond-slip models into numerical simulation allows taking into account the bond interaction and analyse its effect on local and global behaviour. Unlike conventional steel reinforcement, no standard bond-slip law exists for FRP reinforcement, as bond is the result of a combination of parameters such as reinforcing material and bar surface configuration, among others. Therefore, there is a need in developing a methodology to easily implement bond-slip response from experimental bond tests.
In this work, a methodology to implement bond-slip behaviour between concrete and reinforcement into a FEM model is presented. The inverse analysis is used to characterise the bond mechanisms active in a pull-out test. The obtained constitutive behaviours are thereafter implemented into a FE program by using connector elements and surface-to-surface contacts, and GFRP RC tensile elements are modelled. Numerical results are compared to experimental ones available in the literature, showing good accuracy in terms of load-deformation, crack width and crack spacing, as well as strains and bond stress and slip distributions along the reinforcing bar.

View all citing articles on Scopus

View full text

Finite element modelling of reinforced concrete framed structures including catenary action

Abstract

Introduction

Section snippets

Compatibility equations

Displacement interpolation along the element axis

Implementation of the nonlocal integral formulation

Rigid body motion and corresponding transformation

Direct iteration solution scheme

Linear elastic analysis of a pin supported indeterminate beam

Conclusions

J Construct Steel Res

J Construct Steel Res

Comput Struct

Int J Solid Struct

Comput Struct

Comput Struct

Comput Struct

Int J Solid Struct

Catenary action in steel-framed buildings

Proc Inst Civil Eng: Struct Build

A simplified analysis of catenary action in steel beams in fire and implications on fire resistant design

Steel Compos Struct

Failure of lightly reinforced concrete members under fire. I: Analytical modeling.

J Struct Eng

Progressive collapse analysis of an RC structure

Struct Des Tall Special Build

Progressive collapse of reinforced concrete structures: a multihazard perspective

ACI Struct J