Implementation of total Lagrangian formulation for the elasto-plastic analysis of plane steel frames exposed to fire

Abstract

In this paper, the co-rotational total Lagrangian forms of finite element formulations are derived to perform elasto-plastic analysis for plane steel frames that either experience increasing external loading at ambient temperature or constant external loading at elevated temperatures. Geometric nonlinearities and thermal-expansion effects are considered. A series of programs were developed based on these formulations. To verify the accuracy and efficiency of the nonlinear finite element programs, numerical benchmark tests were performed, and the results from these tests are in a good agreement with the literature. The effects of the nonlinear terms of the stiffness matrices on the computational results were investigated in detail. It was also demonstrated that the influence of geometric nonlinearities on the incremental steps of the finite element analysis for plane steel frames in the presence of fire is limited.

Keywords

co-rotational / total Lagrangian / geometrical nonlinearity / fire / elasto-plastic

Introduction

For the finite element model, there are two different approaches (Bathe [1], Belytschko and Liu [2], and Wang [3]): the total Lagrangian formation and the updated Lagrangian formation. In the total Lagrangian formulation, the displacement from the initial configuration is used as an independent variable. In contrast, discrete equations are formulated for the current configuration of the updated Lagrangian formulation. Mitchell and Owen [4] adopted both approaches to examine geometric and material nonlinearities. In the past two decades, studies have extended these two approaches to study the nonlinear behavior of plane steel structures in fire. Zhao [5] reported a direct iteration method for the nonlinear analysis of steel frames. The Rankine formulation, which was originally derived by Rankine and then modified by Merchant, was extended by Tang et al. [6] to study the behavior of steel frames when exposed to fire. At the same time, Toh et al. [7] extended the three classical plastic theorems, i.e., the lower-bound, the upper-bound, and the uniqueness theorems, into the elevated temperature regime.

For numerical simulations, a finite element model for the analysis of two-dimensional steel frames at either ambient or elevated temperature was proposed by Tan et al. [8]. An elasto-plastic analysis of semirigid steel frames that were subjected to elevated temperatures was conducted by Vimonsatit et al. [9]. A series of numerical studies of thermally restrained steel columns that were predominantly subjected to axial loads was presented by Huang et al. [10]. Based on a simple design approach, a new sub-frame model and an isolated member model for determining the fire resistance of beams and columns subjected to compartment fires were proposed by Huang and Tan [11]. A method that uses the axis arc-length and the section rotation of the deformed beam as basic variables was proposed by Li and Guo [12]. Huang et al. [13] proposed a total Lagrangian approach to the finite element model for the analysis of two-dimensional steel frames, either at ambient or elevated temperatures. However, studies that consider the effects of geometric nonlinearities in the two approaches to the incremental finite element method for elasto-plastic analyses of plane steel frames in fire are rare.

In this paper, we investigate the influence of the geometric nonlinear terms of tangent stiffness matrices on the total Lagrangian finite element elasto-plastic analysis of two-dimensional steel frames that are subjected to monotonically increasing loadings at ambient temperature or are subjected to a rising temperature under a constant load. A series of programs was developed based on these formulations. To verify the accuracy and efficiency of the total Lagrangian formulation, the force-deflection relations of the numerical experiments were tested. These results show that the predictions agree well with those in the literature. The effects of geometric nonlinearities on the finite element analysis of steel frames in a fire are discussed in detail. It is demonstrated that the influence of geometric nonlinearities on the incremental step finite element analysis for plane steel frames in fire is limited.

Governing equations

Assumptions

The total Lagrangian formulation is derived for the analysis of plane steel frames that are subjected to nodal loads at ambient and elevated temperatures. The assumptions adopted in the analysis are as follows:

1) The plane cross section remains in the initial plane after deformation (Plane cross-section assumption).

2) Local and lateral-torsional buckling are neglected.

3) The temperature-spatial distribution varies linearly or bi-linearly in the direction of one of the principal axes of the cross section. Additionally, in the direction perpendicular to this axis, the temperature remains constant.

4) Temperature profiles are identical along each element length. Temperature changes only induce thermal longitudinal strain; the thermal effect in the transverse direction is neglected.

5) Only rectangular and I sections are considered.

6) The element is straight and prismatic.

7) Beam and bar elements are used in the finite element model.

8) Shear deformations are negligible.

The displacement equations

The displacement components u and v on an arbitrary point A can be described by

(1)u=u_{\mathrm{0}}-y{\displaystyle \frac{\mathrm{d}{v}_{\mathrm{0}}}{\mathrm{d}x}},

where x and y are the coordinates of A.

The displacement-strain relation

For the arbitrary point A, the normal strain can be described by

(2){ϵ}_x={\displaystyle \frac{\mathrm{d}u}{\mathrm{d}x}}+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\left({\left({\displaystyle \frac{\mathrm{d}u}{\mathrm{d}x}}\right)}^{\mathrm{2}}+{\left({\displaystyle \frac{\mathrm{d}v}{\mathrm{d}x}}\right)}^{\mathrm{2}}\right).

The first square term is usually neglected because the square of an infinitesimal number is much less than the infinitesimal number. However, the second square term is considerable because it represents the effect of bending on the normal strain. Differentiation of the displacement components with respect to x yields

(3){\displaystyle \frac{\mathrm{d}u}{\mathrm{d}x}}={\displaystyle \frac{\mathrm{d}{u}_{\mathrm{0}}}{\mathrm{d}x}}-y{\displaystyle \frac{{\mathrm{d}}^{\mathrm{2}}{v}_{\mathrm{0}}}{\mathrm{d}{x}^{\mathrm{2}}}}.

Ignoring the effects of the rotational deflection and substituting Eq. (3) into Eq. (2), the normal strain can be described by

(4){ϵ}_x={\displaystyle \frac{\mathrm{d}{u}_{\mathrm{0}}}{\mathrm{d}x}}+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\left({\displaystyle \frac{\mathrm{d}{v}_{\mathrm{0}}}{\mathrm{d}x}}\right)}^{\mathrm{2}}-y{\displaystyle \frac{{\mathrm{d}}^{\mathrm{2}}{v}_{\mathrm{0}}}{\mathrm{d}{x}^{\mathrm{2}}}}.

The constitutive models

There are various elevated-temperature models for steel, such as the ECCS, modified ECCS, and CTICM models. Details of these constitutive models are included in Huang [14].

The total Lagrangian formulation

The normal deflection is a function of the position, given as u=u\left(x\right). The displacement at a time t in the initial configuration can be described as {\mathrm{}}_{\mathrm{0}}^{t}u={\mathrm{}}_{\mathrm{0}}^{t}u\left(x\right). The displacement at the next time increment is the sum of the displacement at time t and the increase in the displacement, described as follows:

(5){\mathrm{}}_{\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{0}}^{t+\mathrm{\Delta }t}u={\mathrm{}}_{\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{0}}^{t+\mathrm{\Delta }t}u\left(x\right)={\mathrm{}}_{\mathrm{0}}^{t}u+u.

The strain at the current time is described by

(6){\mathrm{}}_{\mathrm{0}}^t{ϵ}_x={\mathrm{}}_{\mathrm{0}}^t{\displaystyle \frac{\mathrm{d}u}{\mathrm{d}x}}+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\left({\mathrm{}}_{\mathrm{0}}^t{\displaystyle \frac{\mathrm{d}v}{\mathrm{d}x}}\right)}^{\mathrm{2}}+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\left({\mathrm{}}_{\mathrm{0}}^t{\displaystyle \frac{\mathrm{d}u}{\mathrm{d}x}}\right)}^{\mathrm{2}}.

The strain at the next time increment can be derived from

(7){\mathrm{}}_{\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{0}}^{t+\mathrm{\Delta }t}{ϵ}_x={\mathrm{}}_{\mathrm{0}}^t{ϵ}_x+{\mathrm{}}_{\mathrm{0}}{e}_x+{\mathrm{}}_{\mathrm{0}}{\eta }_x,

where

{\mathrm{}}_{\mathrm{0}}{e}_x={\mathrm{}}_{\mathrm{0}}{\displaystyle \frac{\mathrm{d}u}{\mathrm{d}x}}{+}_{\mathrm{0}}^t{{\displaystyle \frac{\mathrm{d}v}{\mathrm{d}x}}}_{\mathrm{0}}{\displaystyle \frac{\mathrm{d}v}{\mathrm{d}x}}{+}_{\mathrm{0}}^t{{\displaystyle \frac{\mathrm{d}u}{\mathrm{d}x}}}_{\mathrm{0}}{\displaystyle \frac{\mathrm{d}u}{\mathrm{d}x}},

{\mathrm{}}_{\mathrm{0}}{\eta }_x={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\left({\mathrm{}}_{\mathrm{0}}{{\displaystyle \frac{\mathrm{d}v}{\mathrm{d}x}}}_{\mathrm{0}}{\displaystyle \frac{\mathrm{d}v}{\mathrm{d}x}}+{\mathrm{}}_{\mathrm{0}}{{\displaystyle \frac{\mathrm{d}u}{\mathrm{d}x}}}_{\mathrm{0}}{\displaystyle \frac{\mathrm{d}u}{\mathrm{d}x}}\right).

The bending deflection is a function of the position, which is given by v=v\left(x\right). The displacement at the current time in the initial configuration can be described by {\mathrm{}}_{\mathrm{0}}^{t}v={\mathrm{}}_{\mathrm{0}}^{t}v\left(x\right). The displacement at the next time increment is the sum of the displacement at time t and the increase of the displacement, described as follows:

(8){\mathrm{}}_{\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{0}}^{t+\mathrm{\Delta }t}v={\mathrm{}}_{\mathrm{0}}^{t+\mathrm{\Delta }t}v\left(x\right)={\mathrm{}}_{\mathrm{0}}^{t}v+v.

The corresponding strain at the current time is described by

(9){\mathrm{}}_{\mathrm{0}}^t{ϵ}_x=-y_{\mathrm{0}}^t{\displaystyle \frac{{\mathrm{d}}^{\mathrm{2}}v}{\mathrm{d}{x}^{\mathrm{2}}}}.

The corresponding strain at the next time increment is described by

(10){\mathrm{}}_{\mathrm{0}}^{t+\mathrm{\Delta }t}{ϵ}_x=-y_{\mathrm{0}}^{t+\mathrm{\Delta }t}{\displaystyle \frac{{\mathrm{d}}^{\mathrm{2}}v}{\mathrm{d}{x}^{\mathrm{2}}}}=-y\left({\mathrm{}}_{\mathrm{0}}^t{\displaystyle \frac{{\mathrm{d}}^{\mathrm{2}}v}{\mathrm{d}{x}^{\mathrm{2}}}}+{\mathrm{}}_{\mathrm{0}}{\displaystyle \frac{{\mathrm{d}}^{\mathrm{2}}v}{\mathrm{d}{x}^{\mathrm{2}}}}\right).

The weak form of the structural mechanics in the total Lagrangian formulation can be described by

(11)\int {\mathrm{}}_{{\mathrm{0}}_V}{\mathrm{}}_{\mathrm{0}}^{t+\mathrm{\Delta }t}{S}_x{\delta }_{\mathrm{0}}^{t+\mathrm{\Delta }t}{ϵ}_x^{\mathrm{0}}\mathrm{d}V=\int {\mathrm{}}_{{\mathrm{0}}_S}{\mathrm{}}_{\mathrm{0}}^{t+\mathrm{\Delta }t}{t}_k\delta u_k^{\mathrm{0}}\mathrm{d}S+\int {\mathrm{}}_{{\mathrm{0}}_V}^{\mathrm{0}}{\rho }_{\mathrm{0}}^{t+\mathrm{\Delta }t}{f}_k\delta u_k^{\mathrm{0}}\mathrm{d}V.

Substitution of the local decomposition into the previous weak form yields

(12)\int {\mathrm{}}_{{\mathrm{0}}_V}{\mathrm{}}_{\mathrm{0}}{S}_x{\delta }_{\mathrm{0}}{ϵ}_x^{\mathrm{0}}\mathrm{d}V+\int {\mathrm{}}_{{\mathrm{0}}_V}{\mathrm{}}_{\mathrm{0}}^t{S}_x{\delta }_{\mathrm{0}}{\eta }_x^{\mathrm{0}}\mathrm{d}V+\int {\mathrm{}}_{{\mathrm{0}}_V}{\mathrm{}}_{\mathrm{0}}^t{S}_x{\delta }_{\mathrm{0}}{e}_x^{\mathrm{0}}\mathrm{d}V=\int {\mathrm{}}_{{\mathrm{0}}_S}\left({\mathrm{}}_{\mathrm{0}}^t{t}_k+{\mathrm{}}_{\mathrm{0}}{t}_k\right)\delta u_k^{\mathrm{0}}\mathrm{d}S+\int {\mathrm{}}_{{\mathrm{0}}_V}^{\mathrm{0}}\rho \left({\mathrm{}}_{\mathrm{0}}^t{f}_k+{\mathrm{}}_{\mathrm{0}}{f}_k\right)\delta u_k^{\mathrm{0}}\mathrm{d}V.

Substitution of the corresponding stress and strain vectors for the current problem, which involves steel frames in fire, into the weak form yields

(13)$$\begin{gathered} \int {\mathrm{}}_{{\mathrm{0}}_V}{\delta }_{\mathrm{0}}{e}_x{E}_{\mathrm{0}}{e}_x^{\mathrm{0}}\mathrm{d}V+\int {\mathrm{}}_{{\mathrm{0}}_V}{\mathrm{}}_{\mathrm{0}}^t{S}_x{\delta }_{\mathrm{0}}{\eta }_x^{\mathrm{0}}\mathrm{d}V+\int {\mathrm{}}_{{\mathrm{0}}_V}{\mathrm{}}_{\mathrm{0}}^t{S}_x{\delta }_{\mathrm{0}}{e}_x^{\mathrm{0}}\mathrm{d}V \\ ={\left[{\mathrm{}}_{\mathrm{0}}^{t+\mathrm{\Delta }t}F\delta u\right]}_{\mathrm{0}}^l+\int {\mathrm{}}_{\mathrm{0}}^l{\mathrm{}}_{\mathrm{0}}^{t+\mathrm{\Delta }t}q\delta v\mathrm{d}x \\ +{\left[{\mathrm{}}_{\mathrm{0}}^{t+\mathrm{\Delta }t}{F}_Q\delta v\right]}_{\mathrm{0}}^l-{\left[{\mathrm{}}_{\mathrm{0}}^{t+\mathrm{\Delta }t}M{\delta }_{\mathrm{0}}{\displaystyle \frac{\mathrm{d}v}{\mathrm{d}x}}\right]}_{\mathrm{0}}^l. \end{gathered}$$

Numerical model

In this section, the second Piola-Kirchhoff stress that corresponds to the Green strains in the matrix notation may be used. Hermitian functions are used as the shape functions.

The co-rotational coordinate system

In the co-rotational coordinate system, the nodal displacement and force components, can be described by

(14)\left\{\mathrm{d}\right\}=\left\{\begin{array}{l}\hfill {\alpha }_{\mathrm{1}}\hfill \\ \hfill {\alpha }_{\mathrm{2}}\hfill \\ \hfill \mathrm{\Delta }\hfill \end{array}\right\},

(15)\left\{f\right\}=\left\{\begin{array}{l}\hfill \overline{M_{\mathrm{1}}}\hfill \\ \hfill \overline{M_{\mathrm{2}}}\hfill \\ \hfill \overline{N}\hfill \end{array}\right\},

where {\alpha }_{\mathrm{1}} and {\alpha }_{\mathrm{2}} are the nodal angular deflections at nodes 1 and 2, respectively, \mathrm{\Delta } is the elongation of the element, {\overline{M}}_{\mathrm{1}} and {\overline{M}}_{\mathrm{2}} are the bending moments and N is the axial force, as shown in Fig. 1.

The displacement equations in matrix form

The displacement equations in matrix form can be described by

(16)u=\lfloor N_u\rfloor \left\{d\right\},

(17)v=\lfloor N_v\rfloor \left\{d\right\},

(18)d=\lfloor N_u\rfloor \left\{d\right\}-y\lfloor {\displaystyle \frac{\mathrm{d}{N}_v}{\mathrm{d}x}}\rfloor \left\{d\right\},

where u and v are the displacements in the x and y directions, respectively, and \lfloor N_u\rfloor and \lfloor N_v\rfloor are the shape functions of the finite element model.

The displacement-strain relationship in matrix form

The displacement-strain relationship in matrix form can be described by

(19)ϵ=\lfloor B\rfloor \left\{d\right\},

where

\lfloor B\rfloor =\lfloor B_L\rfloor +\lfloor B_{NL}\rfloor ,

\lfloor B_L\rfloor =\lfloor {\displaystyle \frac{\mathrm{d}{N}_u}{\mathrm{d}x}}\rfloor -y\lfloor {\displaystyle \frac{{\mathrm{d}}^{\mathrm{2}}{N}_v}{\mathrm{d}{x}^{\mathrm{2}}}}\rfloor ,

\lfloor B_{NL}\rfloor ={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\left(\lfloor d\rfloor \left\{{\displaystyle \frac{\mathrm{d}{N}_u}{\mathrm{d}x}}\right\}\lfloor {\displaystyle \frac{\mathrm{d}{N}_u}{\mathrm{d}x}}\rfloor +\lfloor d\rfloor \left\{{\displaystyle \frac{\mathrm{d}{N}_v}{\mathrm{d}x}}\right\}\lfloor {\displaystyle \frac{\mathrm{d}{N}_v}{\mathrm{d}x}}\rfloor \right).

Let us define \left[C_u\right], \left[C_v\right] and \left[C\right] as

(20)\left[C_u\right]=\left\{{\displaystyle \frac{\mathrm{d}{N}_u}{\mathrm{d}x}}\right\}\lfloor {\displaystyle \frac{\mathrm{d}{N}_u}{\mathrm{d}x}}\rfloor ,

(21)\left[C_v\right]=\left\{{\displaystyle \frac{\mathrm{d}{N}_v}{\mathrm{d}x}}\right\}\lfloor {\displaystyle \frac{\mathrm{d}{N}_v}{\mathrm{d}x}}\rfloor ,

(22)\left[C\right]=\left[C_u\right]+\left[C_v\right].

Thus,\lfloor B_{NL}\rfloorcan be described by

(23)\lfloor B_{NL}\rfloor ={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\lfloor d\rfloor \left(\left[C_u\right]+\left[C_v\right]\right)={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\lfloor d\rfloor \left[C\right].

The tangent stiffness matrix

Equation (13) can be described in matrix form by

(24)\lfloor f\rfloor \left\{\delta d\right\}=\int {\mathrm{}}_V\sigma \delta ϵ\mathrm{d}V.

The nodal force vector can be described by

(25)\lfloor f\rfloor =\int {\mathrm{}}_V\sigma \lfloor B\rfloor \mathrm{d}V.

With the tangent stiffness matrix, the nodal force vector is described by

(26)\lfloor f\rfloor =\left[K_T\right]\left\{\mathrm{\Delta }d\right\},

where K_{Tij}={\displaystyle \frac{\partial f_i}{\partial f_j}}=\int {\mathrm{}}_V\left({\displaystyle \frac{\partial \sigma }{\partial ϵ}}{\displaystyle \frac{\partial ϵ}{\partial d_j}}{B}_i+\sigma {\displaystyle \frac{\partial B_i}{\partial d_j}}\right)\mathrm{d}Vi,\mathrm{}\mathrm{}j=\mathrm{1},\mathrm{2},...,\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{o}\mathrm{f}\mathrm{deg}\mathrm{r}\mathrm{e}\mathrm{e}\mathrm{s}\mathrm{o}\mathrm{f}\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}\mathrm{d}\mathrm{o}\mathrm{m}

For E\equiv {\displaystyle \frac{\partial \sigma }{\partial ϵ}}, the tangent stiffness components can be described by

(27)K_{Tij}=\int {\mathrm{}}_V\left(E{B}_j{B}_i+\sigma C_{ij}\right)\mathrm{d}V,

In matrix form, the tangent stiffness matrix is described by

(28)\left[K_T\right]=\left[K_L\right]+\left[K_E\right]+\left[K_{\sigma }\right],

where

\left[K_L\right]=\int {\mathrm{}}_V\left\{B_L\right\}E\lfloor B_L\rfloor \mathrm{d}V,

\left[K_E\right]=\int {\mathrm{}}_V\left(\left\{B_L\right\}E\lfloor B_{NL}\rfloor +\left\{B_{NL}\right\}+\left\{B_{NL}\right\}E\lfloor B_L\rfloor \right)\mathrm{d}V,

\left[K_{\sigma }\right]=\int {\mathrm{}}_V\sigma \left[C\right]\mathrm{d}V,

\left[K_L\right] is the linear tangent stiffness matrix, \left[K_E\right] is the nonlinear tangent stiffness matrix as a function of displacement that incorporates the elastic modulus, and \left[K_{\sigma }\right] is the nonlinear constant tangent stiffness matrix that incorporates the current stress.

Transformation from local to global coordinates

The transformation matrix from global to chord displacements can be described by

(29)\left[T\right]=\left[\begin{array}{ccc}\hfill -{\displaystyle \frac{s^{\prime }}{L^{\prime }}}\hfill & \hfill -{\displaystyle \frac{s^{\prime }}{L^{\prime }}}\hfill & \hfill -c^{\prime }\hfill \\ \hfill {\displaystyle \frac{c^{\prime }}{L^{\prime }}}\hfill & \hfill {\displaystyle \frac{c^{\prime }}{L^{\prime }}}\hfill & \hfill -s^{\prime }\hfill \\ \hfill \mathrm{1}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill \\ \hfill {\displaystyle \frac{s^{\prime }}{L^{\prime }}}\hfill & \hfill {\displaystyle \frac{s^{\prime }}{L^{\prime }}}\hfill & \hfill c^{\prime }\hfill \\ \hfill -{\displaystyle \frac{c^{\prime }}{L^{\prime }}}\hfill & \hfill -{\displaystyle \frac{c^{\prime }}{L^{\prime }}}\hfill & \hfill s^{\prime }\hfill \\ \hfill \mathrm{0}\hfill & \hfill \mathrm{1}\hfill & \hfill \mathrm{0}\hfill \end{array}\right],

where c^{\prime }={\displaystyle \frac{X^{\prime }}{L^{\prime }}}, s^{\prime }={\displaystyle \frac{Y^{\prime }}{L^{\prime }}}, and L’, X’, Y’, L, ρ, ρ’, ∆, α₁, and α₂ are defined as shown in Fig. 1. For the total Lagrangian formulations, L’, X’ and Y’ are the magnitudes at the initial time, but they are at the current time step in the updated Lagrangian forms.

The transformation from the chord to the global tangent stiffness matrix is described by

(30)\left[K_G\right]=\left[T\right]\left[K_c\right]{\left[T\right]}^{\mathrm{T}}+\left[\left({\overline{M}}_{\mathrm{1}}+{\overline{M}}_{\mathrm{2}}\right)\left[G_{\mathrm{1}}\right]+N\left[G_{\mathrm{2}}\right]\right],

where

\left[G_{\mathrm{1}}\right]=\left[\begin{array}{cccccc}\hfill -k_{\mathrm{1}}\hfill & \hfill k_{\mathrm{2}}\hfill & \hfill \mathrm{0}\hfill & \hfill k_{\mathrm{1}}\hfill & \hfill -k_{\mathrm{2}}\hfill & \hfill \mathrm{0}\hfill \\ \hfill k_{\mathrm{2}}\hfill & \hfill k_{\mathrm{1}}\hfill & \hfill \mathrm{0}\hfill & \hfill -k_{\mathrm{2}}\hfill & \hfill -k_{\mathrm{1}}\hfill & \hfill \mathrm{0}\hfill \\ \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill \\ \hfill k_{\mathrm{1}}\hfill & \hfill -k_{\mathrm{2}}\hfill & \hfill \mathrm{0}\hfill & \hfill -k_{\mathrm{1}}\hfill & \hfill k_{\mathrm{2}}\hfill & \hfill \mathrm{0}\hfill \\ \hfill -k_{\mathrm{2}}\hfill & \hfill -k_{\mathrm{1}}\hfill & \hfill \mathrm{0}\hfill & \hfill k_{\mathrm{2}}\hfill & \hfill k_{\mathrm{1}}\hfill & \hfill \mathrm{0}\hfill \\ \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill \end{array}\right],

\left[G_{\mathrm{2}}\right]=\left[\begin{array}{cccccc}\hfill k_{\mathrm{5}}\hfill & \hfill -k_{\mathrm{4}}\hfill & \hfill \mathrm{0}\hfill & \hfill -k_{\mathrm{5}}\hfill & \hfill k_{\mathrm{4}}\hfill & \hfill \mathrm{0}\hfill \\ \hfill -k_{\mathrm{4}}\hfill & \hfill k_{\mathrm{3}}\hfill & \hfill \mathrm{0}\hfill & \hfill k_{\mathrm{4}}\hfill & \hfill -k_{\mathrm{3}}\hfill & \hfill \mathrm{0}\hfill \\ \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill \\ \hfill -k_{\mathrm{5}}\hfill & \hfill k_{\mathrm{4}}\hfill & \hfill \mathrm{0}\hfill & \hfill k_{\mathrm{5}}\hfill & \hfill -k_{\mathrm{4}}\hfill & \hfill \mathrm{0}\hfill \\ \hfill k_{\mathrm{4}}\hfill & \hfill k_{\mathrm{3}}\hfill & \hfill \mathrm{0}\hfill & \hfill -k_{\mathrm{4}}\hfill & \hfill k_{\mathrm{3}}\hfill & \hfill \mathrm{0}\hfill \\ \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill \end{array}\right],

where

k_1={\displaystyle \frac{2c^{\prime }{s}^{\prime }}{{L^{\prime }}^2}},k_2={\displaystyle \frac{{c^{\prime }}^2-{s^{\prime }}^2}{{L^{\prime }}^2}},k_3={\displaystyle \frac{{c^{\prime }}^2}{{L^{\prime }}^2}},

k_4={\displaystyle \frac{c^{\prime }{s}^{\prime }}{{L^{\prime }}^2}},k_5={\displaystyle \frac{{s^{\prime }}^2}{L^{\prime }}}.

The temperature gradient

To define the temperature gradient across a section and allow for gradual yielding, the cross section is discretised into horizontal slices with an appropriate thickness. Throughout this study, each flange is divided into four layers, and the web is divided into ten layers considering the bending of the major axis. The temperature variations over the cross section are incorporated throughout the distribution of slice temperatures, which are assumed to be uniform in each slice.

Verification

Based on the finite element formulations that were derived in the previous section, a computer program (Co-Rotational Total Lagrangian Plane Frame Element (CTPFE)) was developed for the nonlinear elasto-plastic analysis of plane steel beams, columns and frames. The validity of the CTPFE is confirmed in this section using two numerical tests. The first test is an analysis of a simply supported column that is subjected to a cyclic concentrated moment. The second test is an analysis of a single-floor double-span frame that is subjected to concentrated forces at elevated temperatures.

A simply supported column that is subjected to a cyclic concentrated moment

As shown in Fig. 2, a simple supported column is subjected to a cyclic tensional force. The elastic modulus E of the steel is 2.1×10⁵ MPa, and the yield strength f_y is 310 MPa. The two constitutive models that are used here are MECCS and EC3. Harmathy’s thermal constitutive law for steel is used. The tested section of the column is a 100 cm² square, as shown in Fig. 2(a). The cyclic loading includes loading, unloading, reverse loading, and reloading, as shown in Fig.€2(b). The time increment ∆t in the incremental step simulation is 0.5 min. The tested temperature is 400°C. The results from the numerical test are provided in Fig. 3. These results are in good agreement with the global bilinear stress-strain model MECCS and the global bilinear-elliptic model EC3 at 400°C.

A single-floor double-span frame that is subjected to concentrated forces at elevated temperatures

Tests for the single-floor double-span frame that is subjected to concentrated forces at elevated temperatures were performed. The standard I10 section is adopted for the beams, while a 100 cm² square section is used for the columns (Fig. 4(a)). The elastic modulus is 203 GPa for the beams and 218 GPa for the columns. The yield stress is 294.4 MPa for the beams and 334.2 MPa for the columns. The failure stress is 524.9 MPa for the beams and 461.41 MPa for the columns. The thermal elongation parameter is 0.10 for the beams and 0.12 for the columns. Figure 4 shows that the frame is subjected to four concentrated forces under a rising temperature. The laboratory experiment was performed by Li et al. [15]. The temperature-time relationship at select measurement points within Span A is shown in Fig. 4(b). Additional experimental details are provided in Li et al. [15]. A comparison of the laboratory results, the numerical solutions from select software, and computational results from this study are provided in Figs.5–6. The difference between these values is less than 10%. Thus, the accuracy of the CTPFE program was verified with the previous two numerical tests regarding the elasto-plastic analysis for plane steel structures in fire.

The effects of geometric nonlinearities

The nonlinear co-rotational total Lagrangian finite element method is an effective method for the elasto-plastic analysis of a plane frame in fire. However, this method is time consuming for analyses of large scale steel structures. Therefore, the overall structural analysis is usually simplified as a partial analysis of a sub-structure model. The connection with the other structural components can be simplified as an end restraint with a specified rigidity. Huang and Tan [11] adopted this method to analyze the plane steel frame in an indoor fire accident. This study isolated the room with the fire into a sub-structure model, which was further simplified to a single restrained beam and column model. Comparisons of the numerical predictions of the internal forces and displacements that were based on the different simplified models and the full structural model were performed. However, the quantification of the restraint at the beam-to-column connections from the adjoining cool structures has not been fully addressed due to the many types of structures and beam-to-column connections. In this section, we focus on the effects of the geometric nonlinear terms in the tangent stiffness matrices for the selected connection rigidity. There are two numerical studies that are performed in this section. The first study involves a portal frame that is subjected to monotonically increasing concentrated forces until the structural fails. The second study involves a restrained steel beam that is subjected to vertical concentrated forces with a rising temperature.

A portal frame subjected to vertical and horizontal concentrated forces

Figure 7(a) shows the dimensions and loading of the portal frame. Both the beam and the column of the frame have a rectangular section with a width of 116 mm and height of 173 mm. The elastic modulus of the steel is 2.1×10⁵ MPa. The force increment in the finite element simulation is 10 kNm. The objective of this numerical experiment is to investigate the effects of each geometric nonlinear term from the tangent stiffness matrices on the final computational results. Based on Eq. (13) and Eq. (30), the elemental stiffness matrix in the co-rotational coordinates can be expressed as

(31)\left[K_T\right]=\left[K_{\mathrm{1}}\right]+\left[K_{\mathrm{2}}\right]+\left[K_{\mathrm{3}}\right],

where \left[K_{\mathrm{1}}\right]=\left[T\right]\left[K_c\right]\left[T\right], \left[K_{\mathrm{2}}\right]=\left({\overline{M}}_{\mathrm{1}}+{\overline{M}}_{\mathrm{2}}\right)\left[G_{\mathrm{1}}\right], and \left[K_{\mathrm{3}}\right]=N\left[G_{\mathrm{2}}\right]. All of these matrices are functions of the element length L, x coordinate X, and y coordinate Y. To investigate the effects of each geometric nonlinear term from the tangent stiffness matrices in detail, four different stiffness matrices are chosen for the simulation. Figure 7(b) shows the horizontal displacement curve of the dimensionless load on the top of the left column.

A restrained steel beam subjected to vertical concentrated forces at elevated temperatures

When a structural component is considered an element for finite element analysis, the role of the structural component with respect to other components can be simplified as a rigidity matrix. The geometric nonlinearity of a structure is mainly caused by the relative horizontal deflection at both ends of the column’s structural components. The nonlinear restrained stiffness at the end of the structural component model described in this section aims to include the side sway effect, which is related to the overall horizontal deflection of the structure, into the restrained stiffness of the partial structural model. This technique is essentially the restrained stiffness calculation method for the end of the structural part model that is provided in the section.

As shown in Fig. 8(a), the frame that is subjected to several vertical concentrated loads is simplified as a restrained beam with nonlinear restrained stiffness at both end connections. The nonlinear restrained stiffness at the end of the restrained beam is provided by the column (for example, the restraint in End A is jointly provided by columns AE and AC). Considering column AE, the nonlinear restraint stiffness of the connection provided by the column is described by

{\left\{K_H^A\right\}}^{AE}=\left\{\begin{array}{c}\hfill {\displaystyle \frac{\mathrm{1}}{L^{\mathrm{2}}}}\left(C_{\mathrm{1}}+C_{\mathrm{2}}+\mathrm{2}S\right)\hfill \\ \hfill -{\displaystyle \frac{\mathrm{\Delta }}{L^{\mathrm{3}}}}\left(C_{\mathrm{1}}+C_{\mathrm{2}}+\mathrm{2}S\right)\hfill \\ \hfill {\displaystyle \frac{\mathrm{1}}{L}}\left(C_{\mathrm{2}}+S\right)\hfill \end{array}\right\}{\displaystyle \frac{EI}{L}}

(32)+\left\{\begin{array}{c}\hfill {\left({\displaystyle \frac{\mathrm{\Delta }}{L}}\right)}^{\mathrm{2}}\hfill \\ \hfill {\displaystyle \frac{\mathrm{\Delta }}{L}}\hfill \\ \hfill \mathrm{0}\hfill \end{array}\right\}{\displaystyle \frac{EA}{L}},

(33){\left\{K_V^A\right\}}^{AE}=\left\{\begin{array}{c}\hfill {\displaystyle \frac{\mathrm{1}}{L^{\mathrm{3}}}}\left(C_{\mathrm{1}}+C_{\mathrm{2}}+\mathrm{2}S\right)\hfill \\ \hfill -{\displaystyle \frac{{\mathrm{\Delta }}^{\mathrm{2}}}{L^{\mathrm{4}}}}\left(C_{\mathrm{1}}+C_{\mathrm{2}}+\mathrm{2}S\right)\hfill \\ \hfill -{\displaystyle \frac{\mathrm{\Delta }}{L^{\mathrm{2}}}}\left(C_{\mathrm{2}}+S\right)\hfill \end{array}\right\}{\displaystyle \frac{EI}{L}}+\left\{\begin{array}{c}\hfill {\displaystyle \frac{\mathrm{\Delta }}{L}}\hfill \\ \hfill \mathrm{1}\hfill \\ \hfill \mathrm{0}\hfill \end{array}\right\}{\displaystyle \frac{EA}{L}},

(34){\left\{K_R^A\right\}}^{AE}=\left\{\begin{array}{c}\hfill {\displaystyle \frac{\mathrm{1}}{L}}\left(C_{\mathrm{2}}+\mathrm{2}S\right)\hfill \\ \hfill -{\displaystyle \frac{\mathrm{\Delta }}{L^{\mathrm{2}}}}\left(C_{\mathrm{2}}+\mathrm{2}S\right)\hfill \\ \hfill C_{\mathrm{2}}\hfill \end{array}\right\}{\displaystyle \frac{EI}{L}},

where ∆ refers to the longitudinal elongation or contraction at the column top. The stiffness matrices \left\{K_H^A\right\}, \left\{K_V^A\right\} and \left\{K_R^A\right\} are functions of the side sway displacement. The previous two formulas consider the side sway effect for the determination of the restrained stiffness. If the side sway effect is neglected, the two mentioned formulas can be revised as follows:

(35){\left\{K_H^A\right\}}^{AE}=\left\{\begin{array}{c}\hfill {\displaystyle \frac{\mathrm{1}}{L^{\mathrm{2}}}}\left(C_{\mathrm{1}}+C_{\mathrm{2}}+\mathrm{2}S\right)\hfill \\ \hfill \mathrm{0}\hfill \\ \hfill {\displaystyle \frac{\mathrm{1}}{L}}\left(C_{\mathrm{2}}+S\right)\hfill \end{array}\right\}{\displaystyle \frac{EI}{L}},

(36){\left\{K_R^A\right\}}^{AE}=\left\{\begin{array}{c}\hfill {\displaystyle \frac{\mathrm{1}}{L}}\left(C_{\mathrm{2}}+\mathrm{2}S\right)\hfill \\ \hfill \mathrm{0}\hfill \\ \hfill C_{\mathrm{2}}\hfill \end{array}\right\}{\displaystyle \frac{EI}{L}},

The calculation model for nonlinear restrained stiffness is presented at the end of this study for use with real applications. It is assumed that the axial deformation of all of the structural components and the relative vertical deflection of all of the beam ends are neglected, and all of the provided restraints have fixed values. The geometric nonlinearity results only from the side sway effect for the beam. The nonlinear restrained stiffness at both ends of the restrained beam refers only to the restraint function of the beam horizontal side displacement. Figure 9 shows the deflection-time curve at the mid-point of the beam span. Figure 10 shows the internal axial force-time curve at the mid-point of the beam span. Figure 11 shows the moment-time curve at the right end of the beam.

From the calculated results of the co-rotational column element stiffness matrix, the leading elements of the geometric nonlinearities of the steel frame are located in the relative horizontal deflections at both ends of the column structural components. It is demonstrated that the effect of side sway on the column stiffness is limited. The calculations for the restrained steel beam that experiences a fire accident reveal that the calculated results from the structural parts model with nonlinear restrained stiffness are, at most, 10% different from the results that neglect the side sway effect. Although all of the geometric nonlinear elements of the beam are neglected, the calculated results are still similar to the precise results.

Conclusions

In this study, the total Lagrangian form of the finite element model is developed to analyze two-dimensional steel frames that experience an increasing external load at ambient temperature or a constant external load at elevated temperatures. Geometric nonlinearities and thermal expansion are considered. To verify the accuracy and efficiency of the finite element model, analytical patch tests were performed, and the results from these tests are in a good agreement with those from the literature. The effects of the nonlinear terms of the stiffness matrices on the computational results are investigated in detail. The influence of geometric nonlinearities on the incremental finite element analysis for plane steel frames in fire is demonstrated to be limited.

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