Finite element analysis of creep for plane steel frames in fire

Abstract

Steel is widely used for the construction of bridges, buildings, towers, and other structures because of its great strength, light weight, ductility, and ease of fabrication, but the cost of fireproofing is a major disadvantage. Therefore, the resistance of a steel structure to fire is a significant subject for modern society. In the past, for simplification, creep behavior was not taken into account in research on the resistance of a steel structure to fire. However, it was demonstrated that the effect of creep is considerable at temperatures that commonly reach 600°C and should not be neglected in this context. In this paper, a co-rotational total Lagrangian finite element formulation is derived, and the corresponding numerical model is developed to study the creep behavior of plane steel frames in fire conditions. The geometric nonlinearity, material nonlinearity, high temperature creep, and temperature rate of change are taken into account. To verify the accuracy and efficiency of the numerical model, four prototypical numerical examples are analyzed using this model, and the results show very good agreement with the solutions in the literature. Next, the numerical model is used to analyze the creep behavior of the plane steel frames under decreasing temperatures. The results indicate that the effect of creep is negligible at temperatures lower than 500°C and is considerable at temperatures higher than 500°C. In addition, the heating rate is a critical factor in the failure point of the steel frames. Furthermore, it is demonstrated that the deflection at the midpoint of the steel beam, considering creep behavior, is approximately 13% larger than for the situation in which creep is ignored. At temperatures higher than 500°C, the deformed steel member may recover approximately 20% of the total deflection. The application of the numerical model proposed in this paper is greatly beneficial to the steel industry for creep analysis, and the numerical results make a significant contribution to the understanding of resistance and protection for steel structures against disastrous fires.

Keywords

creep / plane steel frame / fire / finite element method / geometric nonlinearity

Introduction

With the significant advantages of light-weight and efficient construction, steel structures have made a significant contribution to the construction industry. However, the poor fire resistance of steel cannot be ignored, and this limitation of the material becomes an obstacle for further development. This typical problem has also become a concern for experts and scholars worldwide. To ensure rational and economical fire protection for steel structures, the behavior of a steel structure under fire conditions must be calculated through numerical analysis, and the destruction mechanism of the material in the high-temperature environment must be clearly understood. Such a structural fire resistance analysis involves many factors, such as the fire heating curve, temperature distribution of the structural components, thermal expansion caused by heating, and changes in material properties in fire, among others. Therefore, the analysis is a complex and nonlinear procedure. To this end, the finite element method is commonly used to perform analysis for structural response under high temperature and to fully investigate steel performance in fire conditions.

In the 1990s and early 2000s, certain scholars established a few finite element models for investigation of the fire resistance of steel structures. Saab and Nethercot [1] developed a two-dimensional finite element method to simulate the performance of steel-frame structures in fire with consideration of the geometric and material nonlinearities, and a series of equations were derived based on the Ramberg-Osgood nonlinear stress-strain relationship. A nonlinear analysis program named INSTAF was built into this model. The ultimate load and loading conditions specific to the critical temperature could be calculated using the finite element program. The incremental stiffness equation for the displacement was solved using the Newton-Raphson iterative method.

The second-order plastic hinge method was used by Richard [2] to analyze the overall performance of a large steel-framed hall under semi-continuous fire. Wang and Moore [3] investigated the fire resistance of a two-dimensional steel-reinforced concrete structure and steel-concrete composite structures using their own finite element program. The steel and concrete high-temperature stress-strain relationship, non-uniform temperature distribution, strain caused by thermal expansion at high temperature, second-order geometric nonlinearity, residual stress, creep and transient thermal strain of steel and concrete were taken into consideration. The displacement of the steel frame under constant loading in normal and high temperature environments was examined.

Saab and Nethercot [1], Bailey [4,5], Wang, Li and Guo [6], and Lin, Yang and Huang [7] developed finite element code to explore the performance of steel frames under firing in which the interaction of material nonlinearity and geometric nonlinearity, non-uniform temperature distribution, and thermal strain effects were taken into consideration. Huang [8-10] considered the combined effects of the plate structure and analyzed the overall response of the steel frame construction in fire conditions as well as the effect of creep on steel frame structures under a high heating rate.

EI-Rimawi, Burgess and Plank [11] performed inelastic large-deformation analysis for a semi-rigid framework in high temperatures using the plastic hinge method with consideration of the cooling stage. The proposed plastic hinge method was based on the axial and bending stiffnesses of a nonlinear elastic constitutive model. This group also performed fire-resistant plastic analysis of a simple frame structure to fill the gap between the simplified calculation method and the finite element method. Huang [12] developed a three-dimensional nonlinear finite element program to simulate the steel and composite frame structures under firing.

Using the finite element program FEMFAN, Huang [8-10,12] developed a series of parametric analyses to study the end moment, heating rate and pressure boundary conditions of steel columns with axis constraints and also analyzed the effects of the length-slenderness ratio, axial restraint ratio, rotation constraint ratio, axial load utilization factors and other factors on the calculation results. Huang also proposed a new sub-frame model and an independent component model for fire-resistant performance of the local steel frame. With a boundary constrained by linear combinations of springs and rotating springs, this model was used to simulate the stiffness of the spring of the beam-column connection nodes.

At the current stage, the research on the fire resistance of steel structure contains the following gaps. First, the creep behavior of steel at high temperature has rarely been considered in the previous analyses of steel members. Second, studies on the analysis of steel structures during the cooling stage are rare. In this paper, a co-rotational finite element model is proposed to simulate the entire process of the response of steel frame structures in both the heating and cooling stages. The high-temperature steel constitutive model is explored in detail. The variation in the material properties of a steel-framed structure in a high temperature environment is investigated in-depth. The geometric equations, physical equations, balance equations and the virtual work principle are derived in detail for high-temperature planar steel frames with consideration of geometric and material nonlinearity. Thus, the large-scale impact of the creep and the heating rate on the performance of a steel structure in fire conditions is studied.

Constitutive model of creep in steel at high temperature

At a certain temperature, the phenomenon of slow plastic deformation of material created by continuous loading is called creep. Creep is one of the most significant causes of the collapse of steel structures at high temperatures. After repeated creep tests at different loading conditions, we can obtain a set of creep curves, and these curves can generally be divided into three stages. The first stage corresponds to the creep rate decrease. The second stage is called steady-state creep, in which the basic creep rate is constant. In the third stage, a sharp increase occurs in the creep rate and eventually leads to fracture of the material. In general, whether creep occurs or not, this behavior has a close relationship with the melting point of the metal, Tm. The occurrence of creep can be roughly estimated according to whether the temperature is greater than 0.5 Tm. For most alloys, creep occurs at temperatures between (0.4～0.6) Tm. When the temperature is greater than 0.5 Tm, creep will occur even if the applied stress is less than the yield point of the material. When the temperature is less than 0.5 Tm, the applied stress must be close to or greater than the yield point for creep to occur. Generally, creep at high temperature is explained by two physical mechanisms, the diffusion creep and the dislocation creep. Both of these mechanisms have a relationship to the presence of voids and dislocations. A void appears when a normal atom is transferred as an interstitial atom. A dislocation is a line defect in the crystal structure that glides and climbs.

In the first stage of creep, a large amount of dislocation occurs and slides at the moment of loading; these dislocations result in rapid deformation behavior that eventually slows due to strain hardening. A near-constant creep rate in the second stage could be interpreted as the balance of hardening due to the recovery of dislocations with softening due to the extension of defects. In the third stage, the creep rate increases rapidly. Internal and external damages result in loss of this balance in the second stage. In this paper, the four creep constitutive models are investigated and are explained in the following sections.

The Penny creep model

In this model, three parameters are used to describe the creep behavior: stress, time and temperature.

(1){ϵ}_{cr}=f\left(\sigma ,t,T\right),

(2){ϵ}_{cr}=f_{\mathrm{1}}(\sigma )f_{\mathrm{2}}(t)f_{\mathrm{3}}(T),

where

(3)f_{\mathrm{1}}(\sigma )=A_{\mathrm{1}}{\sigma }^m,

(4)f_{\mathrm{1}}(\sigma )=A_{\mathrm{2}}\sinh (\sigma /{\sigma }_{\mathrm{0}}),

(5)f_{\mathrm{1}}(\sigma )=A_{\mathrm{3}}\exp (\sigma /{\sigma }_{\mathrm{0}}),

(6)f_{\mathrm{1}}(\sigma )=A_{\mathrm{4}}{[\sinh (\sigma /{\sigma }_{\mathrm{0}})]}^m,

(7)f_{\mathrm{2}}(t)=(\mathrm{1}+b{t}^{\mathrm{1}/\mathrm{3}})\exp (kt)-\mathrm{1},

(8)f_{\mathrm{2}}(t)=B_{\mathrm{1}}{t}^n,

(9)f_{\mathrm{2}}(t)=B_{\mathrm{2}}[\mathrm{1}-\exp (-qt)]+B_{\mathrm{3}}t,

(10)f_{\mathrm{2}}(t)={\sum a_{i}t}^{n_i},

(11)f_{\mathrm{3}}(T)=\exp \left(-{\displaystyle \frac{Q}{RT\prime }}\right),

Here, A_j, B_j, a, b, k, m, n, and q are constants, σ₀ is the initial stress, Q is the creep potential energy, R is the gas constant, and T’ is the absolute temperature.

The Dorn creep model

In Dorn’s opinion, metal creep at high temperature has no relationship to stress and strain. The Dorn creep model concentrates on combining temperature and time as the variable θ. The creep strain can be given as

(12){ϵ}_{cr}=f(\theta ),

where \theta ={\displaystyle {\int }_{\mathrm{0}}^t\exp (-\mathrm{\Delta }H/RT)dt} ,and ∆H is the creep potential energy. In the Dorn creep model, the strain rate can be expressed as

(13){\stackrel{·}{ϵ}}_{cr}=\exp (-\mathrm{\Delta }H/RT)S\phi (\sigma ),

where S is the resistance of creep deformation, as determined by stress history; in the area of high stress, \phi (\sigma )=\exp (B\sigma ), and in the area of low stress, \phi (\sigma )={\sigma }^N, and B and N are constants.

Upon simplification, the relationship of creep stress ϵ_cr to the variable θ can be divided into two parts at θ = θ₀ . The creep strain can be given as

(14){ϵ}_{cr}={ϵ}_{cr,\mathrm{0}}\cdot \mathrm{2}{(z\cdot \theta /{ϵ}_{cr,\mathrm{0}})}^{\frac{\mathrm{1}}{\mathrm{2}}}\mathrm{0}\le \theta \le {\theta }_{\mathrm{0}},

(15){ϵ}_{cr}={ϵ}_{cr,\mathrm{0}}[\mathrm{1}+z\cdot \theta /{ϵ}_{cr,\mathrm{0}}]\theta \ge {\theta }_{\mathrm{0}},

where {\theta }_{\mathrm{0}}={ϵ}_{cr,\mathrm{0}}/z. In selected situations, a straight-line model is used as follows:

(16){ϵ}_{cr}={ϵ}_{cr,\mathrm{0}}+z\theta when\theta \ge \mathrm{0},

where z is Zemer-Hollomom constant and ϵ_cr,0 is the initial creep strain.

The Fujimoto-Furumura creep model

Based on the large number of experiments, Fujimoto and Furumura proposed an expression of the initial creep for SS41 steel:

(17){ϵ}_{cr}={\mathrm{10}}^{\left(\frac{H_{\mathrm{1}}}{T_s}+H_{\mathrm{2}}\right)}\cdot {(g\cdot \sigma )}^{\left(\frac{H_{\mathrm{3}}}{T}+H_{\mathrm{4}}\right)\cdot }{t}^{H_{\mathrm{5}}\cdot T_s+H_{\mathrm{6}}},

where T_s is the absolute temperature, g is the acceleration of gravity and H_j are material constants.

The Harmathy creep model

Experimental results show that creep is a shear deformation, and the volume change is so small that it can be negligible. Therefore, when the creep constitutive model is established, the spherical stress can be neglected. In other words, creep behavior can be mainly described using the creep strain rate. Harmathy proposed the expression of creep strain rate {\dot{ϵ}}_{\mathrm{c}\mathrm{r}}for time hardening as:

(18){\stackrel{·}{ϵ}}_{cr}=Z\exp \left({\displaystyle \frac{-Q}{R{T}_R}}\right){\coth }^{\mathrm{2}}{\left[{\displaystyle \frac{\mathrm{3}tZ\exp \left({\displaystyle \frac{-Q}{R{T}_R}}\right)}{{ϵ}_{cr,\mathrm{0}}}}\right]}^{\mathrm{1}/\mathrm{3}},

and for strain hardening as:

(19){\stackrel{·}{ϵ}}_{cr}=Z\exp \left({\displaystyle \frac{-Q}{R{T}_R}}\right){\coth }^{\mathrm{2}}\left[{\displaystyle \frac{{ϵ}_{cr}^{\ast }}{{ϵ}_{cr,\mathrm{0}}}}\right],

where Z = 0.026S^4.7 when S\le \mathrm{15000} (psi), Z = 1.23×10¹⁶exp(0.0003S) when (psi), {ϵ}_{\mathrm{c}\mathrm{r},\mathrm{0}}=\mathrm{1.7}\times {\mathrm{10}}^{-\mathrm{10}}{S}^{\mathrm{1.75}} , T_R = 491.67+ 1.8T (T is in Celsius degrees), Q/R is 700000°Rankine, and {ϵ}_{\mathrm{c}\mathrm{r}}^{*}=\mathrm{\Sigma }\left|{\dot{ϵ}}_{\mathrm{c}\mathrm{r}}\right|\mathrm{\Delta }t.

The co-rotational finite element method

In this section, a co-rotational finite element model is proposed in detail. The displacement expression, the strain-displacement equation, the stress resultants, and the potential energy equation are derived based on the co-rotational coordinate system. Finally, all equations are written in matrix form, and the numerical model is completed.

The governing equations

The displacement components u and v on an arbitrary point A can be given as

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

where x and y are coordinates. On the arbitrary point A, the normal strain can be given as

(21){ϵ}_x={\displaystyle \frac{du}{dx}}+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\left({\left({\displaystyle \frac{du}{dx}}\right)}^{\mathrm{2}}+\left({\frac{dv}{dx}}^{\mathrm{2}}\right)\right),

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

(22){\displaystyle \frac{du}{dx}}={\displaystyle \frac{d{u}_{\mathrm{0}}}{dx}}-y{\displaystyle \frac{d^{\mathrm{2}}{v}_{\mathrm{0}}}{d{x}^{\mathrm{2}}}},

Ignoring the effect of the rotational deflection and substituting Eq. (20) into Eq. (21), the normal strain can be given as

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

With the developed displacement-strain relationship, the total and incremental equilibrium equations are derived using the virtual work principle. The virtual work principle of the element is written as

(24)\delta W={\displaystyle {\int }_V{\sigma }_x\delta {ϵ}_{x}dV}-\lfloor Q\rfloor \{\delta q\}=\mathrm{0},

Substitution of Eq. (21-23) into Eq. (24) yields

(25)\delta W={\displaystyle {\int }_l{\displaystyle {\int }_A{\sigma }_x}\left({\displaystyle \frac{d\delta u_{\mathrm{0}}}{dx}}+{\displaystyle \frac{d{v}_{\mathrm{0}}}{dx}}{\displaystyle \frac{d{\delta }_{\mathrm{0}}}{dx}}-y{\displaystyle \frac{d^{\mathrm{2}}\delta v_{\mathrm{0}}}{d{x}^{\mathrm{2}}}}\right)}dAdx-\lfloor Q\rfloor \{\delta q\}=\mathrm{0},

The stress resultants are defined as

(26)N={\displaystyle {\int }_A{\sigma }_{X}dA,}

(27)M_z={\displaystyle {\int }_A{\sigma }_X}ydA,

where N is the normal force vector and Mz is the moment with respect to the z axis. Substitution of Eqs. (26-27) into Eq. (25) yields

(28){\displaystyle {\int }_l\left(a_{\mathrm{1}}{\displaystyle \frac{d\delta u}{dx}}+a_{\mathrm{2}}{\displaystyle \frac{d\delta v_{\mathrm{0}}}{dx}}+a_{\mathrm{3}}{\displaystyle \frac{d^{\mathrm{2}}\delta v_{\mathrm{0}}}{d{x}^{\mathrm{2}}}}\right)}dx-\lfloor Q\rfloor \{\delta q\}=\mathrm{0},

where a_{\mathrm{1}}=N , a_{\mathrm{2}}=N{\displaystyle \frac{d{v}_{\mathrm{0}}}{dx}} and a_{\mathrm{3}}=-M_z . If displacements u₀ and v₀ are written in the form of a discrete set of displacement coordinates q_i, Eq. (21) can be written as

(29){\psi }_i={\displaystyle {\int }_l{a}_{\mathrm{1}}{\displaystyle \frac{\partial }{\partial \delta q_i}}\left({\displaystyle \frac{d\delta u_{\mathrm{0}}}{dx}}\right)}+a_{\mathrm{2}}{\displaystyle \frac{\partial }{\partial \delta q_i}}\left({\displaystyle \frac{d\delta v_{\mathrm{0}}}{dx}}\right)+a_{\mathrm{3}}{\displaystyle \frac{\partial }{\partial \delta q_i}}\left({\displaystyle \frac{d^{\mathrm{2}}\delta v_{\mathrm{0}}}{d{x}^{\mathrm{2}}}}\right)dx-Q_i=\mathrm{0},

In other words, the internal resultant force and the external resultant force must be equal, which represents a total equilibrium. Because Eq. (12) is nonlinear, an iterative approach to the solution is necessary. The Newton-Raphson method offers a first-order approximation to Eq. (12) that can be rewritten as

(30){\psi }_i^{(n+\mathrm{1})}={\psi }_i^{(n)}+{\left({\displaystyle \frac{\partial {\psi }_i}{\partial q_j}}\right)}^{\left(n\right)}\mathrm{\Delta }{q}_j^{\left(n\right)}=\mathrm{0},

where n is the iterative counter and q_j is also a set of displacement coordinates. Substitution of Eq. (30) into Eq. (29) yields

(31)\begin{array}{l}\left({\displaystyle {\int }_l{\displaystyle \frac{\partial a_{\mathrm{1}}}{\partial q_j}}{\displaystyle \frac{\partial }{\partial \delta q_i}}\left({\displaystyle \frac{d\delta u_{\mathrm{0}}}{dx}}\right)+{\displaystyle \frac{\partial a_{\mathrm{2}}}{\partial q_j}}{\displaystyle \frac{\partial }{\partial \delta q_i}}\left({\displaystyle \frac{d\delta v_{\mathrm{0}}}{dx}}\right)+{\displaystyle \frac{\partial a_{\mathrm{3}}}{\partial q_j}}{\displaystyle \frac{\partial }{\partial \delta q_i}}\left({\displaystyle \frac{d^{\mathrm{2}}\delta v_{\mathrm{0}}}{d{x}^{\mathrm{2}}}}\right)dx}\right)\mathrm{\Delta }{q}_j\hfill \\ =Q_i-{\displaystyle {\int }_l{a}_{\mathrm{1}}{\displaystyle \frac{\partial }{\partial \delta q_i}}\left({\displaystyle \frac{d\delta u_{\mathrm{0}}}{dx}}\right)+a_{\mathrm{2}}{\displaystyle \frac{\partial }{\partial \delta q_i}}\left({\displaystyle \frac{d\delta v_{\mathrm{0}}}{dx}}\right)}+a_{\mathrm{3}}{\displaystyle \frac{\partial }{\partial \delta q_i}}\left({\displaystyle \frac{d^{\mathrm{2}}\delta v_{\mathrm{0}}}{d{x}^{\mathrm{2}}}}\right)dx\hfill \end{array},

The previous equation can be written in matrix form as

(32)[K_T]\{\mathrm{\Delta }q\}=\{\mathrm{\Delta }Q\},

where [K_T] is called the Jacobian matrix in mathematics and the element tangential stiffness matrix in finite element analysis; it is a geometrically and materially nonlinear equation. Additionally, {∆Q} is the unbalanced force vector, and {∆q} is the incremental displacement vector.

The co-rotational coordinate system

In the co-rotational coordinate system, the nodal displacement and force components can be written as

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

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

where {\alpha }_{\mathrm{1}} and {\alpha }_{\mathrm{2}} are the nodal angular deflections in node 1 and node 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. Please refer to Huang et al. [4-6] for details.

The displacement equations in matrix form

The displacement equations in matrix form can be written as

(35)u=\lfloor N_u\rfloor \{d\},

(36)v=\lfloor N_v\rfloor \{d\},

(37)d=\lfloor N_u\rfloor \{d\}-y\lfloor {\displaystyle \frac{d{N}_v}{dx}}\rfloor \{d\},

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 relation in matrix form

The displacement-strain relation can be written in matrix form as

(38)ϵ=\lfloor B\rfloor \{d\},

where \lfloor B\rfloor =\lfloor B_L\rfloor +\lfloor B_{NL}\rfloor ; and

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

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

Let us define [C_u] , [C_v] and [C] as

(41)[C_u]=\left\{{\displaystyle \frac{d{N}_u}{dx}}\right\}\lfloor {\displaystyle \frac{d{N}_u}{dx}}\rfloor ,

(42)[C_v]=\left\{{\displaystyle \frac{d{N}_v}{dx}}\right\}\lfloor {\displaystyle \frac{d{N}_v}{dx}}\rfloor ,

(43)[C]=[C_u]+[C_v]\mathrm{.}

Hence, \lfloor B_{NL}\rfloor can be written as

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

The tangential stiffness matrix

Equation (25) can be written in matrix form as

(45)\lfloor f\rfloor \{\delta d\}={\displaystyle {\int }_V\sigma \delta ϵdV}\mathrm{.}

The nodal force vector can be written as

(46)\lfloor f\rfloor ={\displaystyle {\int }_V\sigma }\lfloor B\rfloor dV\mathrm{.}

With the tangential stiffness matrix, the nodal force vector is given as

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

where K_{Tij}={\displaystyle \frac{\partial f_i}{\partial d_j}}={\displaystyle {\int }_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)}dV and i, j = 1... are the number of degrees of freedom. If we define {\displaystyle \frac{\partial \sigma }{\partial ϵ}}=E , then the components of the tangential stiffness can be given as

(48)K_{Tij}={\displaystyle {\int }_V(E{B}_j{B}_i+\sigma C_{ij})dV,}

In matrix form, the tangential stiffness matrix is given as

(50)[K_T]=[K_L]+[K_E]+[K_{\mathit{\sigma }}],

where

[K_L]={\displaystyle {\int }_V\{B_L\}E}\lfloor B_L\rfloor dV,

[K_E]={\displaystyle {\int }_V(\{B_L\}E\lfloor B_{NL}\rfloor +\{B_{NL}\}E\lfloor B_L\rfloor +\{B_{NL}\}E\lfloor B_{NL}\rfloor )dV,}

[K_{\sigma }]={\displaystyle {\int }_V\sigma }[C]dV,

It is indicated that [K_L] is the linear tangential stiffness matrix, [K_E] is the nonlinear tangential stiffness matrix function of the displacement incorporating the elastic modulus, and [K_{\sigma }] is the nonlinear constant tangential stiffness matrix incorporating the current stress.

Verification

Based on the basic theory of the nonlinear finite element method, a program is developed to analyze the creep of plane steel frames under firing. In this section, comparisons are made among the present results and solutions from the literature to verify the efficiency and accuracy of the proposed numerical model.

Analysis for the plane steel frame at elevated temperature

The steel frame was the subject of laboratory experiments by Rubert and Schaumann. The steel structure, shown in Fig. 1, is 1170 mm in height and 1220 mm in span. The beam-column junction was represented by a rigid connection, and two columns are hinge-connected with the ground. To prevent lateral instability from occurring in the two columns of the plane steel frame, the positions h/4, h/2, 3h/4, and h in height (with × shown in Fig. 1) are constrained in the direction of out-of-plane displacement and torsion. The elastic constant is set at 2.1 × 105 MPa, and the yield point at normal temperatures is 382 N/mm². The cross-section of the beam and the columns is the same as IPE80 steel, as shown in Fig. 1, and the entire frame is heated evenly. The two columns are under a concentrated vertical loading of 65 kN applied to the top, and the right column is also under a concentrated horizontal loading of 2.5 kN also applied to the top. The horizontal displacements of the two beam-column connection points are computed.

In the numerical model, the ECCS model is used as the constitutive model. The beam of the frame is discretized into four elements. The flange is set as one single slice, and the web is divided into four slices of the same thickness. Four Gaussian points are set in the thickness and three in the length. As shown in Fig. 2, the results from the present study at different temperatures are in good agreement with the experimental data and other analytical solutions, with errors of approximately 5%. This comparison indicates that the proposed numerical model is accurate.

Creep analysis of the simple beam at elevated temperature

The total strain and stress are related to the heating process. It has been demonstrated in the previous studies that creep has little effect on the global structural behavior when the temperature is lower than 600°C. In other words, at temperatures lower than 600°C, creep strain can be neglected. In this section, we focus on the effect of creep on steel components at temperatures higher than 600°C.

As shown in Fig. 3, the span of the simple beam is 2902 mm, with a cross-section UBS356 × 171 × 45. The beam is under a distributed load of 10 kN/m at elevated temperatures, and the temperature is distributed evenly in the length and in the cross-section. The elastic modulus is set at 2.1 × 105 MPa, and the yield point is set as 305 Mpa. The Harmathy creep model is used in this analysis with consideration of three different heating rates: 2°C/min, 5°C/min, and 20°C/min. The results are compared with the solutions from Huang’s FEMFAN program, as shown in Fig. 4.

In the numerical model, the Donus-Golrang model is used for the stress-strain relationship, and the beam is discretized into 20 elements. The flange is divided into two slices, and the web is divided as six slices. Three Gaussian points are set both in the length and thickness. Before 25 min, the time increment is set to 0.05 min and then to 0.02 min. When the displacement reaches l/20 of the span of the simple beam, the structure is deemed as ‘failed’, and the temperature at that time is regarded as the critical temperature of the beam. The present results are in good agreement with solutions from the analytical model.

Creep analysis of the plane steel frame at elevated temperature

In this section, the results from the plane steel structure analysis are compared with the experimental data. The plane steel frame, shown in Fig. 0.5, consists of one story and two spans and is 1400 mm in height with a span of 1620 mm in width. The structure is fixedly connected with the ground. The cross-sections of beams and columns are given in Table 1.

In this section, the material properties of steel are given in Table 2.

The lateral supports are set on the top of the structure at a position 550 mm below the top (marked with ‘x’, as shown in Fig. 5) to prevent out-of-plane instability for the columns. The frame is under a concentrated static loading of 30 kN at the position shown in Fig. 5. Then, the temperature is elevated while the loading is held constant.

Figure 6 shows the temperature-time relationship of the measurement points. It is noted that the temperature is not constant on the cross-section. EC3 is the constitutive model used in the calculation. The initial bending of the component and the residual stress are neglected. The coefficient of thermal expansion is 1.4 × 10^-5m /(m · °C).

In the finite element model, each beam of the plane steel frame is divided into six elements and each column into seven elements. In the cross-section, the flange is regarded as one slice and the web as four slices, and there are four Gaussian points in the thickness and three in the length. The relationship between the displacements of the midpoints of the two spans and time are shown in Fig. 7. It is indicated that the solutions from the present study are in a good agreement with the experimental results.

Numerical experiments

Creep analysis of a steel simple beam under heating and cooling

In this section, the results of creep analysis of a simple steel beam are compared with results from the literature. The cross-section of the simple beam is 356 × 171 × 51 UB, as shown in Fig. 8, and its span is 6 m in length. The elastic modulus is 2.1 × 105 MPa, the yield point is 308 MPa, and the distributed loading is 30.6 kN/m.

In the finite element model, each beam of the plane steel frame is divided into 20 elements. In the cross-section, the flange is set as two slices and the web as six slices. There are three Gaussian points in the thickness and in the length. To verify the accuracy of the numerical method, the displacement of the midpoint of the simple beam, without consideration of the creep, is calculated and compared with results from the literature. The comparison is presented in Fig. 9, and the error among the three results is less than 10%. This comparison indicates that the numerical method is reliable.

After verification, the displacement of the midpoint of the simple beam is calculated without consideration of creep. Figure 10 shows the comparison between the displacements of the midpoint of the beam with and without consideration of creep. These results indicate that the displacement with consideration of creep is approximately 25% larger than that without consideration of creep during the cooling period. Therefore, the effect of creep on the behavior of the steel structure in fire conditions is significant.

Creep analysis of the plane steel frame under heating and cooling

In this section, creep analysis is performed for a plane steel frame under heating and cooling. As shown in Fig. 11, the cross-section of the simple beam is the same as that of standard I10 steel, and the cross-section of the columns is 100 mm × 100 mm × 4.2 mm. The concentrated loading P is 30.6 kN/m, and ECCS is used as the constitutive law for the steel structure.

Figure 12 shows the comparison between the displacements of the midpoint of the beam with and without consideration of creep. These results indicate that the displacement with consideration of creep is approximately 10% larger than that without consideration of creep in the cooling period. Therefore, the effect of creep on the behavior of steel structure in fire conditions is significant.

Results and discussion

The results show that the impact of creep is higher in the cooling period than in the heating period. When the temperature begins to decline, the deformation does not recover, but continues to slowly increase. This observation indicates that the structure might collapse in the cooling period due to the increase in deformation.

The results of the second numerical experiment demonstrate that, in the initial heating stage, the thermal expansion of the steel beam at high temperature is constrained so that the axial force of the beam increases with temperature. However, up to a certain temperature, the beam deflection will increase dramatically, and the axial force decreases rapidly. If the beam has been constrained at the end, the beam will generate a very large axial tension. The effect of the axial tension force might be significant for the global response of the steel structure under cooling.

Comparing the overall response of the simple beam with that of the plane frame under heating and cooling, there is less growth in deflection for the plane frame than for the simple beam, and this is due to the axial tension that exists in the frame beam. From this perspective, the frame with constraints at both ends offers better fire resistance than the simple beam. In general, the beam deformation recovery is relatively small compared with the total deformation.

Conclusions and prospects

In this paper, a co-rotational total Lagrangian finite element formulation is derived, and the corresponding numerical model is developed to study the creep behavior of plane steel frames in fire conditions. The geometric nonlinearity, material nonlinearity, high temperature creep, and rate of temperature increase are taken into account. To verify the accuracy and efficiency of the numerical model, four prototypical numerical examples are analyzed using this model. The results are in very good agreement with the solutions found in the literature. Additionally, the numerical model is used to analyze the creep behavior of the plane steel frames when temperatures are decreased. The results indicate that the effect of creep is negligible at temperatures lower than 500°C and are considerable at temperatures higher than 500°C. In addition, the heating rate is a critical factor in the failure point of the steel frame. Furthermore, it is demonstrated that the deflection at the midpoint of the steel beam with consideration of creep is approximately 13% larger than the situation that ignores creep. At temperatures higher than 500°C, the deformation of the steel member might recover approximately 20% of the total deflection. The application of the numerical model proposed in this paper is tremendously significant to the steel industry for creep analysis, and the numerical results make a significant contribution to the understanding of resistance and protection for steel structures against disastrous fires.

This work represents a very preliminary attempt to study the effect of creep on plane steel structures under heating and cooling. We have used only a few simple constitutive models for creep to explore a few very basic questions. Therefore, many issues remain which are worthy of further study, such as the effect of cooling rate and the relationship between the creep and time, among others. We look forward to future research for exploration in these directions.

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