Progressive collapse of 2D reinforced concrete structures under sudden column removal

Abstract

Once a column in building is removed due to gas explosion, vehicle impact, terrorist attack, earthquake or any natural disaster, the loading supported by removed column transfers to neighboring structural elements. If these elements are unable to resist the supplementary loading, they continue to fail, which leads to progressive collapse of building. In this paper, an efficient strategy to model and simulate the progressive collapse of multi-story reinforced concrete structure under sudden column removal is presented. The strategy is subdivided into several connected steps including failure mechanism creation, MBS dynamic analysis and dynamic contact simulation, the latter is solved by using conserving/decaying scheme to handle the stiff nonlinear dynamic equations. The effect of gravity loads, structure-ground contact, and structure-structure contact are accounted for as well. The main novelty in this study consists in the introduction of failure function, and the proper manner to control the mechanism creation of a frame until its total failure. Moreover, this contribution pertains to a very thorough investigation of progressive collapse of the structure under sudden column removal. The proposed methodology is applied to a six-story frame, and many different progressive collapse scenarios are investigated. The results illustrate the efficiency of the proposed strategy.

Keywords

failure mechanism / MBS dynamic analysis / gravity loads / structure-ground contact / structure-structure contact / energy conserving/decaying scheme

Introduction

When one or more structural members fail due to an explosion, a natural disaster, vehicle impact or any other extreme action, the structure becomes unstable and no longer stays in a state of equilibrium. To find a new equilibrium state under such extreme action and gravity loads, we ought to relieve the damaged zone with the transfer to the surrounding structural elements. The redistribution of these actions and gravity loads through the structure leads to the propagation of the damage in the remaining structural elements, resulting in progressive collapse of such structure [¹].

Historically, the spur in research in the domain of progressive collapse was the result of the collapse of Ronan Point Apartment in 1968 in Newham, East London, where a gas explosion caused the progressive collapse of the building [²]. However, researches in the area of progressive collapse increased drastically after the September 11 collapse of the World Trade Center twin towers in New York.

Many research works have been carried out on the study of structural collapse. The analysis of the structural failure of reinforced concrete building caused by a blast load is presented in Refs. [³–⁵] which performed numerical analysis in order to reach a prediction of the collapse kinematics of a real building subjected to blast loading. Some other works such as Refs. [⁶–¹¹] used the technique of controlled explosives in the demolition of complex structures. Computational procedures related to progressive collapse simulations based on the Alternate Load Path approach can also be found in Refs. [²,¹²–¹⁴] with proposing static nonlinear calculations by taking into account the dynamic inertial effects via load amplification factors [¹⁵,¹⁶]. In Refs. [¹⁷–²⁰], linear static, linear dynamic, and nonlinear dynamic analyses are used to estimate the progressive collapse resistance of the building subjected to column failure. Most of the recent works available in literature still model structures by using 2D frame elements as in Refs. [¹⁸–²¹]). However, nonlinear 3D dynamic computations with geometrically nonlinear formulations can also be found in the literature, related to steel structures [¹¹,²²,²³].

According to Ref. [²⁴], many design guidance should be undertaken in order to prevent the progressive collapse of such structure, such as: the structural redundancy, reinforcement detailing to provide structural continuity and ductility, capacity for resisting load reversals which can be achieved by using additional reinforcements in some specific zones of structural members, and capacity for preventing shear failure.

In our recent paper [²⁵], we discuss the various aspects problem of progressive collapse, and we give an overall global methodology based on failure mechanism creation (modeling of softening plastic hinges), MBS simulation (beam model, modeling of joint, time integration scheme), and consideration of contact between structure and ground. In this work, we further improve all particular ingredients of the proposed methodology. The first of this improvement concerns the failure mechanism creation. Many works focused on the limit load in structures and used the concept of plastic hinge (see for example [²⁶–³²]). Some works focused on mesh-free methods for modeling discrete cracks in structures [³³–³⁶]. Here we improve on this, by using the localized failure zones that provides full failure in given section, which can be treated in the context of discontinuity introduced in a standard finite element method. The principal idea is the incorporation of localized energy dissipation in finite element formulation, by introducing a strong discontinuity in kinematics and defining a local dissipative mechanism [³⁷–⁴²]. A similar disputation mechanism can be seen in the phase field model [⁴³–⁴⁵]. By constrast, works such as Refs. [⁴⁶,⁴⁷] presented the machine-learning based procedures which can be used for solving very difficult problems.

Further improvement concerns the study of flexible multi-body systems, with respect to a number of research works (see for example, Refs. [⁴⁸–⁵⁰]. Here, we show that the master slave formulation is very suitable for the modeling of flexible mechanisms and can be easily implemented in finite element codes [⁵⁰].

Some improvements are proposed in the context of time integration scheme, where the classical Newmark scheme [⁵¹] becomes unstable when dealing with high frequency problems [⁵²–⁵⁴]. To dissipate the high frequency oscillations and improve the stability of such time integration scheme, some techniques are proposed in Refs. [⁵⁵–⁵⁷] adding numerical dissipation [⁵³,⁵⁴]. The main idea is to modify the computation of the algorithmic stress resultants and velocity updates, so that the energy can either be conserved or decayed in a controllable manner [⁵²–⁵⁴]. The energy conserving/decaying scheme, which is discussed in Ref. [⁵³], provides an unconditional stability in highly nonlinear problems.

Final improvement is brought in the context of the contact problems, with respect to formulation of the frictionless contact problem [⁵⁸] which only considers static contact between an elastic body and a rigid obstacle. Other similar works have also focused on frictionless contact problems [⁵⁹,⁶⁰]. Here, we focus upon the sudden column removal modeling, which is usually achieved by eliminating one or more columns from the first story as in Ref. [¹⁹]. In the sudden column removal model, the structure must be remodeled without the concerned column, replacing it with its member forces (N, T, and M) which keeps the structure in a state of equilibrium. Then, the member force is suddenly removed in order to initiate progressive collapse [¹⁹]. The subsequent computation is then placed in the framework of dynamic analysis.

In this paper, we thus enrich the methodology proposed in our previous work [²⁵], with the corresponding formulation in the prediction of the softening plastic hinges. In the context of MBS simulation, use the geometrically exact beam first presented in Ref. [⁶¹] for planar problems. For the modeling of joints, we use the master-slave approach, which is very suitable for the modeling of flexible mechanisms [⁵⁰], for time integration scheme we use the energy conserving/decaying scheme [⁵²–⁵⁴] which provides an unconditional stability in highly nonlinear problems.

In this paper, the most important contribution consists in the introduction of the failure function, and the proper manner to control the mechanism creation of a frame until its total failure. Another important contribution pertains to a very thorough investigation of progressive collapse of the structure under sudden column removal. Namely, all the different effects are accounted for such as, the effect of gravity loads, structure-ground contact, and structure-structure contact in the progressive collapse process. We thus can provide a more realistic way of progressive collapse simulation.

The proposed strategy

In this section, we summarize the different aspects needed in the simulation of the progressive collapse of a 2D structure under sudden column removal. Such aspects are explained in the next sections and are applied to a six-story reinforced concrete frame. The study of progressive collapse of such structure is based on two main steps.

1) Failure mechanism creation, which is characterized by appearance of softening plastic hinges in critical zones in the structure. Such plastic hinges lead to full section collapse and can be replaced by joints to get a mechanism for subsequent MBS dynamic analysis.

2) MBS nonlinear dynamic analysis requires: the modeling of flexible beams with Reissner’s beam elements under gravity load, describing large displacements and large rotations; the modeling of joints by using revolute ones and implementing the constraint joints with the master slave approach; the consideration of contact between elements and between structure and ground; the energy conserving/decaying time integration scheme is required to solve nonlinear dynamic equations

Many sources of uncertainties can be associated to progressive collapse process, such as environment loads, structural strength, finite element approximation and limitations in supporting databases [⁶²]. To deal with such uncertainties, many meshes are tested until obtaining a satisfactory convergence, and several progressive collapse scenarios are investigated, in order to visualize the structural collapse in different cases (different loading, contact consideration, etc.).

Failure mechanism creation

The first step in the progressive collapse of such structure is the creation of the failure mechanism. This step is characterized by localization of yielding and plastic strain in a number of critical zones, which are known as plastic hinges.

The classical concept of plastic hinge is characterized by keeping the capacity limit constant during the analysis, while other plastic hinges develop, such a concept cannot describe the post peak structural response. In our case, we are exploiting the softening of the cross section, which leads to the concept of softening plastic hinge. Such a concept can describe the post peak structural response, which is characterized by the decrease of the load capacity along with the increase in deformation [³⁹,⁴²].

A 2D Timoshenko beam is used for building such model, the corresponding kinematics, constitutive laws and equilibrium equations are developed in order to get the beam model that is used in failure analysis and mechanism creation.

Embedded discontinuity

To model the softening plastic hinge, the beam rotation \phi is enhanced with embedded discontinuity (see Fig. 1) rotation jumpα

(1)\psi \left(S\right)=\tilde{\psi }\left(S\right)+\alpha H_{\mathrm{\Gamma }},

where \tilde{\psi }\left(S\right) is the regular part rotation, and \alpha H_{\mathrm{\Gamma }} is the singular part. With

(2)

The expression of total rotation can be written as:

(3)\psi \left(S\right)=\overline{\psi }\left(S\right)+\alpha \left(H_{\mathrm{\Gamma }}-\mathrm{\Omega }\left(S\right)\right),

where

(4)\overline{\psi }\left(S\right)=\tilde{\psi }\left(S\right)+\alpha \mathrm{\Omega }\left(S\right),

and

(5)\mathrm{\Omega }\left(S\right)=\{\begin{array}{ll}\mathrm{0},\hfill & \mathrm{if}\mathrm{\hspace{0.17em}}s=\mathrm{0},\hfill \\ \mathrm{1},\hfill & \mathrm{if}\mathrm{\hspace{0.17em}}s=L.\hfill \end{array}

With this particular form of rotation in Eq. (3), the contribution of discontinuity is cancelled at the element nodes and does not further propagate outside the element domain (for more detail for the choice of \Omega \left(S\right), see Ref. [⁴²]). Such modifications are taken into account in the derivation of the beam deformation and the construction of the finite element model [³⁹,⁴²].

Constitutive laws

Response in axial direction is described by the linear elastic law: N=EA\overline{\epsilon }, where E is the elastic modulus, A is the cross section area, and \overline{\epsilon } is the axial strain. Here, we use the elastoplastic law with bilinear isotropic hardening to describe the bending behavior of the beam (Fig. 2(a)). To describe the softening plastic hinge behavior (after activating the discontinuity), we use the plastic-rigid law characterized by decreasing the moment and increasing the rotation jump (Fig. 2(b)) [³⁹,⁴²].

The total curvature \mathit{\kappa } decomposes into elastic part {\overline{\mathit{\kappa }}}^{\mathrm{e}}, plastic part {\overline{\mathit{\kappa }}}^{\mathrm{p}}, and singular part \overline{\overline{\mathit{\kappa }}}

(6)\mathit{\kappa }={\overline{\mathit{\kappa }}}^{\mathrm{e}}+{\overline{\mathit{\kappa }}}^{\mathrm{p}}+\overline{\overline{\mathit{\kappa }}}.

Before the activation of the softening plastic hinge; \overline{\overline{\mathit{\kappa }}} is zero. However, after the activation of the softening plastic hinge, \overline{\overline{\mathit{\kappa }}} is none-zero and {\overline{\mathit{\kappa }}}^{\mathrm{p}} remains frozen subsequently.

The expression of the moment in the bulk can be written as

(7)M=EI\left(\overline{\mathit{\kappa }}-{\overline{\mathit{\kappa }}}^{\mathrm{p}}\right),

where I is the inertia moment and E is the elastic modulus.

To control the plastification process in the bulk, we introduce two yield criteria.

The first one is presented by the function {\overline{\phi }}_{\mathrm{c}} as

(8){\overline{\phi }}_{\mathrm{c}}\left(M,{\overline{q}}_{\mathrm{c}}\right)=\left|M\right|-\left(M_{\mathrm{c}}-{\overline{q}}_{\mathrm{c}}\right).

The second one is presented by the function {\overline{\phi }}_{\mathrm{y}} as

(9){\overline{\phi }}_{\mathrm{y}}\left(M,{\overline{q}}_{\mathrm{y}}\right)=\left|M\right|-\left(M_{\mathrm{y}}-{\overline{q}}_{\mathrm{y}}\right).

With these two criteria, we are able to control the plastification of the bulk through two limit points. 1) The cracking of concrete when reaching the elastic limit moment value M_{\mathrm{c}} associated with the hardening parameter K₁ and the stress-like bending hardening variable {\overline{q}}_{\mathrm{c}}=-K_{\mathrm{1}}\xi. 2) The yielding of the steel reinforcement bars when reaching the yielding moment value M_{\mathrm{y}} associated with the hardening parameter K₂ and the stress-like bending hardening variable {\overline{q}}_{\mathrm{y}}=-K_{\mathrm{2}}\xi, where \xi is the strain-like bending hardening variable.

To control the activation of discontinuity, and consequently, the formation of softening plastic hinge at s_{\mathrm{d}}, we introduce the failure function \overline{\overline{\phi }} which can be written as

(10)\overline{\overline{\phi }}\left(\overline{t},\overline{\overline{q}}\right)=\left|\overline{t}\right|-\left(M_{\mathrm{u}}-\overline{\overline{q}}\right).

The expression of moment at discontinuity can be written as

(11)\overline{t}=-{\displaystyle \underset{\mathrm{0}}{\overset{l^{\mathrm{e}}}{\int }}\overline{G}M\mathrm{d}x}.

With the particular form of the failure function in Eq. (10), it is easy to control the formation of the plastic hinge in the beam from reaching the ultimate moment value M_{\mathrm{u}} at one of the integration points until the total failure of the beam when carrying capacity drops to zero. The softening of the section characterized by its modulus K_{\mathrm{s}} is exploited here to conduct the failure of the beam, where \overline{\overline{q}}=-K_{\mathrm{s}}\overline{\overline{\xi }} is the stress-like bending softening variable and \overline{\overline{\xi }} is the strain-like bending softening variable.

Replacing the softening plastic hinges with joints obtain MBS

By applying elastoplastic analysis with hardening/softening laws, we obtain a structure that is completely or partially failed. After the formation of most of the full plastic hinges with softening failure, the structure loses its equilibrium state, with redistribution of actions and gravity loads, resulting with a mechanism that behave in a similar way to a multi-body system. For a subsequent representation of the mechanism behavior, we replace the plastic hinges with revolute joints, which allows a free components rotation in the plan and results in the corresponding MBS (Fig. 3).

Multi-body system dynamic analysis

The second step in the progressive collapse of such structure is the simulation of the mechanism created in the first step, which is composed of flexible beams and columns interconnected in some locations with revolute joints. Such a mechanism behaves like a multi-body system.

To model the multi-body system, a 2D Reissner’s beam undergoing large rotations and displacements, including the master-slave joint constraint is developed and used. Such a model allows for describing the large motions and deformations of the structure during its collapse process.

In the case of revolute joints, the coordinates of the slave node are the same as those of the master node. However, the slave node rotation matrix can be obtained by the multiplication of the master rotation matrix with a relative rotation matrix {\mathit{\Lambda }}_{\mathrm{t}}^{\mathrm{r}} [⁵⁰], which allows to write

(12){\mathit{\phi }}_{\mathrm{t}}^{\mathrm{s}}={\mathit{\phi }}_{\mathrm{t}}^{\mathrm{m}}\mathrm{,}\mathrm{\hspace{0.17em}}{\mathit{\Lambda }}_{\mathrm{t}}^{\mathrm{s}}={\mathit{\Lambda }}_{\mathrm{t}}^{\mathrm{m}}{\mathit{\Lambda }}_{\mathrm{t}}^{\mathrm{r}},

where {\mathit{\Lambda }}_{\mathrm{t}}^{\mathrm{r}} is the relative rotation matrix. In 2D problems, the slave node rotation can be expressed as a sum of master node rotation and a given relative rotation {\psi }^{\mathrm{r}}

(13){\psi }^{\mathrm{s}}={\psi }^{\mathrm{m}}+{\psi }^{\mathrm{r}}.

When constructing the set of global equilibrium equations, the result in Eq. (12) can be used. Such an operation can eliminate completely the slave motion component; which leads to an efficient computational procedure [⁵⁰].

Weak form of the equilibrium equation of 2D Reissner’s beam under gravity load

The kinetic energy of the beam can be cast as a quadratic uncoupled form, which presents an important advantage when constructing the time integration scheme:

(14){\mathrm{\Pi }}_{\mathrm{iner}}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\displaystyle \underset{\mathrm{0}}{\overset{L}{\int }}\left(A_{\rho }\mathrm{}\dot{\mathit{\phi }}\cdot \dot{\mathit{\phi }}\mathrm{}+I_{\rho }{\dot{\mathit{\psi }}}^{\mathrm{2}}\right)\mathrm{d}s}.

The elastic potential energy can be written as

(15){\mathrm{\Pi }}_{\mathrm{int}}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\displaystyle \underset{\mathrm{0}}{\overset{L}{\int }}\left(\mathit{\epsilon }\cdot \mathit{n}+km\right)\mathrm{d}s}.

The external energy is given by:

(16){\mathrm{\pi }}_{\mathrm{ext}}={\mathrm{\Pi }}_{\mathrm{grav}}+{\mathrm{\Pi }}_{\mathrm{ext}},

where {\Pi }_{\mathrm{grav}} is the gravitational potential energy of the beam which can be written as:

(17){\mathbf{\Pi }}_{\mathrm{grav}}={\displaystyle \underset{\mathrm{0}}{\overset{L}{\int }}{A}_{\rho }gy\left(s\right)\mathrm{d}s}.

By using the Hamilton’s principle (see Refs. [⁶³,⁶⁴]), the weak form of the equation of motion of the 2D beam can be written as:

{\displaystyle \underset{\mathrm{0}}{\overset{L}{\int }}\left(\delta \mathit{\epsilon }\cdot \mathit{n+}\delta km\right)\mathrm{d}s}+{\displaystyle \underset{\mathrm{0}}{\overset{L}{\int }}\left(\delta \mathit{\phi }\cdot A_{\rho }\ddot{\mathit{\phi }}\mathrm{+}\delta \psi \cdot I_{\rho }\ddot{\psi }\right)\mathrm{d}s}

(18)+{\displaystyle \underset{\mathrm{0}}{\overset{L}{\int }}{A}_{\rho }g\delta y\left(s\right)\mathrm{d}s}-\delta {\mathrm{\Pi }}_{\mathrm{ext}}=\mathrm{0},

where \mathit{\phi } and \psi are the position vector and rotation, \mathrm{}\dot{\mathit{\phi }} and \dot{\psi } are the velocity components, \ddot{\mathit{\phi }}and \ddot{\psi } are the acceleration components, A_{\rho }={\displaystyle \int \rho \mathrm{d}A} and I_{\rho }={\displaystyle \int \rho {\zeta }^{\mathrm{2}}\mathrm{d}A} are inertia coefficients. g is the gravity acceleration, n and m are the spatial stress resultants, \mathit{\epsilon } and k are the strains and curvature while \delta \mathit{\epsilon } and \delta k are the virtual strains and curvature computed by Lie derivative. \delta {\mathrm{\Pi }}_{\mathrm{ext}} is the virtual work of external forces.

Contact kinematics

Considering complete collapse of a structure, we finally need to describe its contact with the ground. We only consider the frictionless case, which is sufficient to obtain the structure debris dispersion [⁶³].

Furthermore, in order to ensure the impenetrability of structural parts in each other, contact between structural elements is also considered.

Contact constrains can be written in term of the Kuhn-Tucker condition as:

(19)g_{\mathrm{t}}\ge \mathrm{0}\mathrm{;}\mathrm{}{\overline{p}}_{\mathrm{t}}\ge \mathrm{0}\mathrm{;}\mathrm{}{g}_{\mathrm{t}}\cdot {\overline{p}}_{\mathrm{t}}=\mathrm{0.}

The contact problem is handled by the penalty method [⁶³,⁶⁵]. The simple quadratic penalty function can be written in a standard way:

(20)

where k_c is the chosen penalty parameter. The use of this penalty function is equivalent to considering the rigid obstacle as an elastic spring with an elasticity coefficient equal to k_c (see Ref. [⁶³] for more details). The contact force can be written in terms of displacement as:

(21)

Time integration scheme

Since the classical Newmark time integration scheme [⁵¹] becomes unstable in the presence of high frequencies [⁵³,⁵⁴]; it is preferable to turn toward a scheme which can be able to preserve or decay energy [⁶⁶]. To that end, we use an energy conserving/decaying scheme. Such a scheme provides an unconditional stability in highly nonlinear problems [⁵³,⁵⁴].

The energy conserving/decaying scheme is designed in a controllable manner to overcome the loss of accuracy, especially for the computed internal forces in presence of high frequency contributions. This scheme is based on the mid-point rule approximation. Such a concept is used for time derivation of weak form [⁵³], which should be written in the middle of time step increment. For the energy decaying scheme, the constitutive equations and updates of velocities are constructed/modified in a way that ensures that energy will be dissipated by filtering out the high frequency contributions over each time step where needed. The latter is achieved by using two control parameters for internal and kinetic energy {\alpha }_{\mathrm{1}} and {\beta }_{\mathrm{1}}. With such parameters, the scheme is able to dissipate energy with filtering out the high frequency contributions. With the proposed choice of parameters {\alpha }_{\mathrm{1}}= {\beta }_{\mathrm{1}}=0, the dissipation terms are equal to zero, which leads to an energy conserving scheme preserving total energy. For more details about the energy conserving/decaying scheme, we refer to Refs. [⁵³,⁵⁴].

Numerical simulation

We present in the following: a numerical example to validate the failure analysis part, after that, we present an application of our methodology to simulate the progressive collapse of a six story frame.

Failure analysis validation

We consider a two-story reinforced concrete frame with height H = 2 m and span L = 3.5 m (Fig. 4). Columns and beams have the same rectangular cross section b\times h= \mathrm{0.3}\mathrm{\hspace{0.17em}}\mathrm{m}\times \mathrm{0.4}\mathrm{\hspace{0.17em}}\mathrm{m}. The material proprieties needed for computation (see Ref. [⁴¹]) are given as follow:

for columns:

M_{\mathrm{c}}=\mathrm{100}\mathrm{\hspace{0.17em}}\mathrm{k}\mathrm{N}·\mathrm{m}\mathrm{;}\mathrm{\hspace{0.17em}}{M}_{\mathrm{y}}=\mathrm{245}\mathrm{\hspace{0.17em}}\mathrm{k}\mathrm{N}·\mathrm{m}\mathrm{;}\mathrm{\hspace{0.17em}}{M}_{\mathrm{u}}=\mathrm{265}\mathrm{\hspace{0.17em}}\mathrm{k}\mathrm{N}·\mathrm{m}\mathrm{;}

K_{\mathrm{h}\mathrm{1}}=\mathrm{12450}\mathrm{\hspace{0.17em}}\mathrm{k}\mathrm{N}·{\mathrm{m}}^{\mathrm{2}}\mathrm{;}\mathrm{\hspace{0.17em}}{K}_{\mathrm{h}\mathrm{2}}=\mathrm{195}\mathrm{\hspace{0.17em}}\mathrm{k}\mathrm{N}·{\mathrm{m}}^{\mathrm{2}}\mathrm{;}\mathrm{\hspace{0.17em}}

K_{\mathrm{S}}=-\mathrm{2410}\mathrm{\hspace{0.17em}}\mathrm{k}\mathrm{N}·{\mathrm{m}}^{\mathrm{2}}\mathrm{;}

for beams:

M_{\mathrm{c}}=\mathrm{30}\mathrm{\hspace{0.17em}}\mathrm{k}\mathrm{N}·\mathrm{m}\mathrm{;}\mathrm{\hspace{0.17em}}{M}_{\mathrm{y}}=\mathrm{150}\mathrm{\hspace{0.17em}}\mathrm{k}\mathrm{N}·\mathrm{m}\mathrm{;}\mathrm{\hspace{0.17em}}{M}_{\mathrm{u}}=\mathrm{170}\mathrm{\hspace{0.17em}}\mathrm{k}\mathrm{N}·\mathrm{m}\mathrm{;}

K_{\mathrm{h}\mathrm{1}}=\mathrm{11190}\mathrm{\hspace{0.17em}}\mathrm{k}\mathrm{N}·{\mathrm{m}}^{\mathrm{2}}\mathrm{;}\mathrm{\hspace{0.17em}}{K}_{\mathrm{h}\mathrm{2}}=\mathrm{137}\mathrm{\hspace{0.17em}}\mathrm{k}\mathrm{N}·{\mathrm{m}}^{\mathrm{2}}\mathrm{;}\mathrm{\hspace{0.17em}}

K_{\mathrm{S}}=-\mathrm{1310}\mathrm{\hspace{0.17em}}\mathrm{k}\mathrm{N}·{\mathrm{m}}^{\mathrm{2}}\mathrm{.}

Columns are clamped at the bottom. The load is applied in two steps. First, constant vertical force act at the top of both columns. Second, we impose a horizontal displacement u at the left top of the frame.

The structure is modeled by 120 finite elements. Figure 5 shows that our results are in good agreement with experimental results found by Ref. [²⁶] and the previous numerical results given by Refs. [³⁹,⁶⁷]. The results illustrate that our model is able to predict the complete failure after creating six plastic hinges located in beam-column joints and two bottom clamped supports (Fig. 6).

Application to a six-story reinforced concrete frame

The progressive collapse of the multi-story reinforced concrete framed structure is investigated here with considering sudden column removal at the periphery of the structure. After removing the column, the structure becomes unstable in the given equilibrium state. The gravity loads and actions are redistributed into the surrounding structural elements, and the structure continues to collapse, until finding a new equilibrium state, where the structural elements support the abnormal loadings. In the opposite case, the structure continues to collapse until it reaches the ground [¹⁹].

To apply the proposed strategy for the simulation of the progressive collapse of a 2D structure under sudden column removal, we consider a six-story frame with floor height H = 3 m and span L = 4 m (Fig. 7). Columns and beams have the same rectangular cross section b\times h= \mathrm{0.3}\mathrm{\hspace{0.17em}}\mathrm{m}\times \mathrm{0.4}\mathrm{\hspace{0.17em}}\mathrm{m}.

The material properties needed for computation are given by (see Ref. [⁶⁸] for the material properties identification):

for columns:

M_{\mathrm{c}}=\mathrm{100}\mathrm{\hspace{0.17em}}\mathrm{k}\mathrm{N}·\mathrm{m}\mathrm{;}\mathrm{\hspace{0.17em}}{M}_{\mathrm{y}}=\mathrm{245}\mathrm{\hspace{0.17em}}\mathrm{k}\mathrm{N}·\mathrm{m}\mathrm{;}\mathrm{\hspace{0.17em}}{M}_{\mathrm{u}}=\mathrm{265}\mathrm{\hspace{0.17em}}\mathrm{k}\mathrm{N}·\mathrm{m}\mathrm{;}

K_{\mathrm{h}\mathrm{1}}=\mathrm{12450}\mathrm{\hspace{0.17em}}\mathrm{k}\mathrm{N}·{\mathrm{m}}^{\mathrm{2}}\mathrm{;}\mathrm{\hspace{0.17em}}{K}_{\mathrm{h}\mathrm{2}}=\mathrm{195}\mathrm{\hspace{0.17em}}\mathrm{k}\mathrm{N}·{\mathrm{m}}^{\mathrm{2}}\mathrm{;}

for beams:

M_{\mathrm{c}}=\mathrm{30}\mathrm{\hspace{0.17em}}\mathrm{k}\mathrm{N}·\mathrm{m}\mathrm{;}\mathrm{\hspace{0.17em}}{M}_{\mathrm{y}}=\mathrm{150}\mathrm{\hspace{0.17em}}\mathrm{k}\mathrm{N}·\mathrm{m}\mathrm{;}\mathrm{\hspace{0.17em}}{M}_{\mathrm{u}}=\mathrm{170}\mathrm{\hspace{0.17em}}\mathrm{k}\mathrm{N}·\mathrm{m}\mathrm{;}

K_{\mathrm{h}\mathrm{1}}=\mathrm{11190}\mathrm{\hspace{0.17em}}\mathrm{k}\mathrm{N}·{\mathrm{m}}^{\mathrm{2}}\mathrm{;}\mathrm{\hspace{0.17em}}{K}_{\mathrm{h}\mathrm{2}}=\mathrm{137}\mathrm{\hspace{0.17em}}\mathrm{k}\mathrm{N}·{\mathrm{m}}^{\mathrm{2}}\mathrm{.}

Columns are clamped at the bottom.

The progressive collapse analysis is performed in two steps: 1) creation of softening plastic hinges; 2) progressive collapse simulation.

Creation of softening plastic hinges

The frame is modeled with 84, 144, and 288 elements. As a first step, a static analysis is performed by using an elastoplastic law with bi-linear isotropic hardening to describe the bending behavior of the reinforced concrete framed structure in the bulk, and a softening plastic-rigid law to describe the softening plastic hinge behavior.

The load is applied in two steps. First, a constant vertical force acts on the top columns. Second, we impose horizontal displacement with function u(t), which increases linearly with pseudo-time, at the top left node of the frame.

Figure 8 presents the response of the frame using different meshes. It is important to note that, many meshes are tested until obtaining a satisfactory convergence with 288 elements.

The finite element model used in the remaining simulations is constructed with 288 elements, each one having 2-nodes with one integration point. To predict the creation of most softening plastic hinges, many values of softening modulus are tested, and it is found that the best choice for that is the following values for the softening modulus: K_{\mathrm{S}}=-\mathrm{393}\mathrm{\hspace{0.17em}}\mathrm{k}\mathrm{N}·\mathrm{m} for beams and K_{\mathrm{S}}=-\mathrm{723} \mathrm{k}\mathrm{N}·\mathrm{m} for columns.

By using these adequate parameters, a second analysis has been performed by applying constant vertical forces acting at the top columns, and an imposed horizontal displacement u_{\mathrm{0}} on the top left node of the frame, where the reaction of the same node is monitored.

Figure 9 shows the response of the reinforced concrete framed structure up to total failure. Figure 10 shows the location of the softening plastic hinges in the six-story framed structure. The softening plastic hinges are designed with colored circles and indicated in Fig. 9 in order to clarify their appearance in a chronological way.

The first six plastic hinges (group (1)) are created when the horizontal displacement of top left node reaches u = 1.213 m and the reaction of the same node starts to be constant. The creation of plastic hinges of group (2) take place at u = 1.254 m and that of group (3) at u = 1.264 m. The capacity load is still constant while other plastic hinges of group (4) and (5) are created at u = 1.295 m and u = 1.304 m, respectively. At u = 1.331 m, the structure starts to fail where the plastic hinges of group (6) are created. The structure continues to fail with creating the plastic hinges of group (7) at u = 1.338 m where the softening behavior is observed with little negative slope. This later can be observed clearly with decreasing the capacity load and increasing the displacement where the rest of softening plastic hinges are created, as described below:

1) group (8): two plastic hinges are created at u = 1.434 m, a third at u = 1.501 m, and a fourth at u = 1.527 m;

2) group (9): two plastic hinges are created at u = 1.536 m, a third at u = 1.596 m, a fourth and a fifth at u = 1.6 m, and a sixth at u = 1.605 m;

3) group (10): first plastic hinge is created at u = 1.63 m and a second at u = 1.636 m.

After creating all the softening plastic hinges, the structure is considered completely failed and is ready to start the collapse simulation.

Progressive collapse simulation of a six-story framed structure under sudden column removal

Having obtained the completely failed state, when the structure becomes a mechanism, we must replace all softening plastic hinges with revolute joints. We thus obtain the mechanism composed of beams and columns interconnected in a specific location with joints. In this paper, we are interested in the progressive collapse of a structure with removed column. To that end, we remove the right bottom column of the reinforced framed structure as shown in Fig. 11.

To investigate the progressive collapse of the new reinforced concrete framed structure, a multi-body system nonlinear dynamic analysis is performed on the six-story framed mechanism. The behavior of the structural mechanism is assumed elastic and contact between the structure and the ground is considered. The energy conserving/decaying scheme is used with dissipation parameters {\alpha }_{\mathrm{1}}= {\beta }_{\mathrm{1}}=0.25, to eliminate the high frequency contributions and capture a realistic response of the structure.

First, the dynamic analysis is performed with many time step values: \mathrm{\Delta }t=0.01 s, 0.05 s, 0.1 s, 0.25 s (see Fig. 12). Figure 13 shows the difference in the dynamic response of the six-story frame when using different time step values.

It is clear from Fig. 13 that the use of a smaller time step produces more high frequency oscillations than the larger one. In other words, the used time integration scheme seems stable even when using a larger time step. In addition to its accuracy and stability, the ability of the energy conserving/decaying scheme to take larger time step presents more advantage leading to an increase in the computational cost.

Effect of gravity load and structure-structure contact on the progressive collapse of structure

To investigate the effect of gravity load and contact between elements, the progressive collapse simulation has been done in four cases.

1) First case: no gravity load and no contact between elements.

In this case, the gravity load is not considered in the dynamic analysis, and contact between elements of the structure is not considered as well. An imposed horizontal displacement is applied at the top left node of the structure, and the reaction of the same node is monitored.

Figure 14 illustrates the progressive collapse of the reinforced concrete framed structure. It is clear from Fig. 14 that the structure collapses progressively until it touches the ground at time t=\mathrm{155}\mathrm{\hspace{0.17em}}\mathrm{s}. The structure continues to collapse keeping contact with ground until t=\mathrm{165}\mathrm{\hspace{0.17em}}\mathrm{s}, where the contact force is enough to push-up the structure after t=\mathrm{165}\mathrm{\hspace{0.17em}}\mathrm{s}. After that, the structure is tearing-down again slowly, which requires larger force at the final stage of the collapse.

In the pushing-up/tearing-down stage of the collapse, it can be observed that the structural elements (beams, columns) arbitrarily penetrated in each other. This observation can be explained by the high contact reaction of the ground on the one hand, and the lack of consideration of contact kinematics between structural elements on the other hand.

Figure 15 shows the response of the structure up to complete collapse to the ground. Different penalty parameters are used in computing the contact force of the ground: {\mathit{\kappa }}_{\mathrm{c}}= \mathrm{3}\times {\mathrm{10}}^{\mathrm{3}}, {\mathit{\kappa }}_{\mathrm{c}}= \mathrm{5}\times {\mathrm{10}}^{\mathrm{3}}, and {\mathit{\kappa }}_{\mathrm{c}}= \mathrm{8}\times {\mathrm{10}}^{\mathrm{3}}. It is important to note that the use of k_c inferior to 3×10³ is not enough to satisfy the impenetrability condition. As a result, the structure can penetrate into the ground, which presents an unrealistic behavior.

Figure 16 (a zoom from Fig. 15) illustrates the effect of penalty parameters. With the three values of penalty: {\mathit{\kappa }}_{\mathrm{c}}= \mathrm{3}\times {\mathrm{10}}^{\mathrm{3}}, {\mathit{\kappa }}_{\mathrm{c}}= \mathrm{5}\times {\mathrm{10}}^{\mathrm{3}}, and {\mathit{\kappa }}_{\mathrm{c}}= \mathrm{8}\times {\mathrm{10}}^{\mathrm{3}}, the response of the structure presents an equal jump in force which takes place when the structure is pushed-up. For precision, we always need a much larger penalty parameter to converge to the realistic solution of the structure.

2) Second case: with gravity load and no contact between elements.

The same analysis presented in the first case is performed here, with the only difference that the gravity load is now considered, with gravitational acceleration g = 9.81 m/s².

Figure 17 illustrates the progressive collapse of the reinforced concrete framed structure. Figure 18 shows the response of the structure until complete collapse. Through these two Figs., it has been observed that we have the same results as the previous case (without gravity load) with a little difference in the deformed shape of structural elements (when the gravity load is considered, it seems that the deformed shape is nonlinear). Moreover, in presence of the controlled displacement, the gravity load is not dominating and its effect cannot be investigated (Fig. 19).

3) Third case: with gravity load and contact between elements.

In the third case, the same analysis is performed as the previous ones, with considering the gravity load and the structure-structure contact.

Figure 20 illustrates the progressive collapse of the reinforced concrete frame structure. The structure collapses progressively until it reaches the ground with its whole right side at t=\mathrm{157.5}\mathrm{\hspace{0.17em}}\mathrm{s}. The structure continues to collapse without penetration between structural elements until its complete collapse at t=\mathrm{180}\mathrm{\hspace{0.17em}}\mathrm{s}. It is important to note that, in this case, i.e., where the structure-structure contact is considered, the phase of pushing-up/tearing-down is not clearly observed. This later can be the results of structural elements pushing each other and not allowing the penetration between them, which presents a more realistic behavior of the progressive collapse of the reinforced concrete framed structure.

Figure 21 shows the response of the structure up to the total collapse with the use of three penalty parameters for both structure-ground contact and structure-structure contact: {\mathit{\kappa }}_{\mathrm{c}}= \mathrm{3}\times {\mathrm{10}}^{\mathrm{3}}, {\mathit{\kappa }}_{\mathrm{c}}= \mathrm{5}\times {\mathrm{10}}^{\mathrm{3}}, and {\mathit{\kappa }}_{\mathrm{c}}= \mathrm{8}\times {\mathrm{10}}^{\mathrm{3}}.

As shown in Fig. 22, with the three values of penalty parameters, we have a small equal jump in force, which takes place when the structure drops to the ground and starts to push-up again. Such a small jump compared to the previous cases (see Fig. 23) shows that, the pushing-up of the structure after contact is not considerably large due to the presence of the contact forces between structural elements, which reduce the contact force of the ground and allow the collapsed structure to stabilize when in contact with the ground.

4) Fourth case: under only the gravity load and considering contact between elements.

To investigate the effect of the gravity load in the progressive collapse of reinforced concrete framed structure, we consider the same six-story frame with a removed column studied before, under only the gravity load. A nonlinear dynamic analysis with energy conserving scheme is performed to the mechanism and the arc-length method is used to solve the nonlinear equations problem.

We consider a distributed mass of \mathrm{300}\mathrm{\hspace{0.17em}}\mathrm{kg/m} for both columns and beams (only the structural masses are considered with the reinforced concrete density is \rho= \mathrm{2500}\mathrm{\hspace{0.17em}}{\mathrm{kg/m}}^{\mathrm{3}}). The considered gravity acceleration is g=\mathrm{9.81}\mathrm{\hspace{0.17em}}{\mathrm{m/s}}^{\mathrm{3}}.

When we remove the bottom right column, the gravity loads and actions carried by this column are transferred to adjacent columns. However, these later are not capable of supporting the abnormal loads transferred, which leads to a partial collapse and the structure is stabilized after the top of the removed column is displaced downward with 1.292 m (see Fig. 24).

To investigate the effect of the additional gravity loads, we consider supplementary static loading masses of 900 and \mathrm{1200}\mathrm{\hspace{0.17em}}\mathrm{kg/m} for loading 1 and loading 2, respectively. The additional masses are added to the structural beam masses.

The same analysis is performed, considering the additional loading described above. In the case where additional loading is \mathrm{900}\mathrm{\hspace{0.17em}}\mathrm{kg/m}, the loads carried by the removed column are again transferred to the adjacent columns. the structure is partially collapsed and stabilized after the displacement of the top of the removed column downward about 2.5 m (Fig. 25).

In the case of an additional loading of \mathrm{1200}\mathrm{\hspace{0.17em}}\mathrm{kg/m}, the rest of the structural elements cannot support the abnormal loading transferred from the location of the removed column, and the structure is completely collapsed; the top of the removed column touches the ground at t=\mathrm{4}\mathrm{\hspace{0.17em}}\mathrm{s}, and the whole of the right side reaches the ground at t=\mathrm{4.5}\mathrm{\hspace{0.17em}}\mathrm{s}. It is clear that the structure is pushed-up again at t=\mathrm{5}\mathrm{\hspace{0.17em}}\mathrm{s}, because of the ground contact reaction, and stabilized in contact with the ground after that. No penetration is observed either between structural elements, or between structure and ground (Fig. 26), which proves the efficiency of kinematic contact to be considered (Fig. 27).

Conclusions

The progressive collapse of multi-story reinforced concrete frame structure under sudden column removal is presented. Different aspects of problem are discussed in order to create the most appropriate methodology capable of simulating the progressive collapse process.

Many concluding remarks can be drawn.

1) It must create enough number of softening plastic hinges which lead to predict the complete failure in the structure. The number of the created softening plastic hinges, its location, and the prediction of complete failure are related to the used softening modulus in the failure analysis.

2) The conserving/decaying energy scheme introduces a desirable numerical dissipation of high frequency oscillations that can be observed in dynamic structure response. In addition to the scheme’s accuracy and stability, its ability to take larger time step presents an important advantage leading to the increase of the computational cost.

3) The contact considered between structure and ground is needed to satisfy the impenetrability condition with a sufficient penalty parameter.

4) Contact kinematics should be considered between elements to overcome such arbitrary penetration between the structural elements and minimize the contact force of the ground.

5) When using the structural element masses only, the progressive collapse process cannot reach the final stage and the structure stabilizes when it finds a new equilibrium state after the redistribution of gravity loads. By using enough additional gravity loads (which could be masses of slabs and other non-structural elements in the construction), the progressive collapse process is investigated until the total demolition of the structure with its dropping to the ground.

Our methodology for modeling and simulating the progressive collapse of reinforced concrete framed structure is capable of describing the structure demolition process taking into account the gravity load, structure-ground contact and structure-structure contact in a realistic way where the efficiency and robustness of the strategy followed is illustrated through a 2D six-story reinforced concrete frame.

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