Theoretical Study on Local Buckling of Steel Plate in Concrete-filled Tube Column under Axial Compression

Abstract

Concrete filled tube (CFT) columns have been increasingly used as the load-bearing systems in engineering applications. The fact that the CFT column has steel plate restrained by infilled concete and the availability of high strength structural steel leads to the application of thin steel plates in CFT columns. However, this gives rise to the local instability problem of thin steel plates under axial loadings. Furthermore, most of the studies on the local buckling of steel plates in contact with concrete reported in the literature were concerned with the cases where the four edges of the steel plate were assumed to be clamped or simply supported. This cannot reflect the real state of steel plate in CFT columns, where the unloaded edges of steel plate is more appropriate to be regarded as elastically restrained. This paper presents an analytical study of local buckling of steel plates in CFT columns. The steel plate, subjected to uniform axial compression, is assumed to be elastically rotationally restrained along loaded and unloaded edges. The approximate solution is obtained by Rayleigh-Rize method. The solution of rotationally restrained plates is verified by comparing with available experimental and theoretical data in the literature. Good agreement is found between them for local buckling stress of the steel plate. This research provides basis for capacity design of CFT columns under axial compression.

1. Introduction

CFT columns have been increasingly used in North America and Asia. A typical CFT column consists of four steel plates infilled with concrete. The composite action of the steel and concrete provides excellent strength and stiffness for the column. In a CFT column, steel plates under compression can only buckle locally outward owning to the restraints of the infilled concrete. This buckling mode results in a considerable increase in the critical local buckling strength of the steel plates as well as the load-bearing capacity of the column. Meanwhile, the steel plates offer confinement to the concrete, which enhances the concrete performance and thus, increasing the stability and strength of the column as a system.

The steel plate in CFT column can be modeled as an isotropic plate which is rotationally restrained along the two loaded and two unloaded edges and the rotational stiffness of the elastic restraint at the four edges is introduced to account for the flexibility of the conjunctions. As a result of different loading states and rotational restraints provided by concrete, the loaded and unloaded edges of the steel plate can be assumed as simply supported or clamped.

The fact that the CFT column has steel plates restrained by concrete and the availability of high strength structural steel leads to the application of thin steel plates in CFT columns. However, this gives rise to the local buckling issues of thin steel plates under axial loadings, which is common in CFT columns.

A number of research have been undertaken on the local buckling of plates with various edge restraints and under different loadings. It is well recognized that numerical methods such as the finite element method,^1,2) finite difference method,³⁾ and finite strip method⁴⁾ could provide explicit analytical solutions for instability analysis of plates. However, it is computationally complicated, time consuming and tedious for broad use in design as several iterations may be necessary to ensure the reasonable solution.

As an alternative to the numerical approach, the classical Rayleigh-Ritz method has been adopted by many researchers to obtain the local buckling solution of plates. In applying the variational formulation of this approach, Qiao and his research group^5,6,7,8) conducted extensive analyses to develop the explicit local buckling of composite plates with rotational restraint under uniaxial compression, uniform shear, and biaxial loading, and fully restrained under combined linearly varying axial and in-plane shear loadings. Kang and Leissa⁹⁾ proposed an exact buckling solution of plates subjected to linearly varying normal stresses and with two opposite edges simply supported, and the other two edges may be clamped, simply supported, free, or elastically supported. Zhong and Gu¹⁰⁾ also presented an exact buckling solution for simply supported symmetrical cross-ply rectangular plates subjected to linearly varying loadings. Mittelstedt¹¹⁾ presented an energy-based approach to obtain the local buckling loads of blade-stringer-stiffened and compressively loaded orthotropic composite plates. Seif and Kabir¹²⁾ developed an analytical approach for both symmetric and anti-symmetric local buckling of thin-plate in finite sizes and with a center crack under tension, the solution of which was based on the principle of minimum total potential energy. Recently, Galerkin method, which is a powerful numerical solution approach in solving differential equations, was used by Jaberzadeh and Azhari¹³⁾ to analyze the elastic and inelastic local buckling of flat rectangular plates with centerline boundary conditions under non-uniform in-plane compression and shear stress.

Similar to the local buckling problems of plates, a special case about the local buckling of steel plates in steel-concrete composite members has been studied by a few researchers. Many of the studies were based on the numerical approach. Uy and Bradford¹⁴⁾ proposed a modified semi-analytical finite strip method to study the elastic local buckling behavior of steel plates in composite steel-concrete members. Liang and Uy¹⁵⁾ studied the local and post-local buckling of steel plates in concrete-filled tubular columns under axial compression by finite element method and proposed effective width formulas for the steel plates. Furthermore, Liang et al.^16,17) studied the local buckling strength of steel plates in double skin composite panels and concrete-filled thin-walled steel tubular beam-columns by using finite element method. On the other hand, some were based on energy method. Wright¹⁸⁾ described an analytical model for evaluating local buckling of a variety of sections where the steel plates was in contact with a rigid medium. Recently, Cai and Long¹⁹⁾ and Long et al.²⁰⁾ applied Rayleigh-Ritz method to obtain the solution of elastic buckling of steel plates in rectangular concrete-filled steel tubular columns with binding bars under axial and eccentric compressions. Arabzade et al.²¹⁾ used theoretical models to obtain the elastic buckling coefficients of steel plates in composite steel plate walls with various aspect ratios under shear loading based on Rayleigh-Ritz method.

However, most of the studies on the local buckling of steel plates in contact with concrete reported in the literature were concerned with the cases where the four edges of the steel plate were assumed to be clamped or simply supported. It should be acknowledged that the four edges of the steel plate in CFT columns were restrained to some extent by the surrounded components, but the restraints are normally not stiff enough to be regarded as clamped. The partial restraint has a pronounced impact on the local buckling behavior of CFT columns under out-of-plane compression. Therefore, the analytical solution of the CFT columns elastically restrained along four edges is needed in order to predict accurately the local buckling response of steel plates in CFT columns.

This paper presents a theoretical study on the local buckling response of steel plates in CFT columns based on the Rayleigh-Ritz method. Two opposite edges are subjected to uniform axial compression and all four edges (both loaded and unloaded) are assumed to be rotationally-restrained. The solution of rotationally restrained plates is verified by comparing with available experimental and theoretical data in the literature.

2. Analytical Formulation

2.1. Yield Line Model for Orthotropic Problem

The Rayleigh-Ritz method was first adopted by Timoshenko and Gere²²⁾ to obtain the solution of stability problems of the plate. The analysis in this paper is based on this method, which also resembles the method used by several previous researchers such as Mittelstedt¹¹⁾ and Cai and Long.¹⁹⁾ Figure 1 shows a steel plate in CFT column subjected to axial compression N_x along the x-direction.

Fig. 1.

Analytical model of elastically rotationally restrained steel plate.

According to classical theory of elastic stability,²²⁾ in the calculation of critical values of forces applied in the middle plane of a plate at which the flat form of equilibrium becomes unstable, the governing differential equation of the deflection surface for the buckled plate is given by the following equation   

$$\frac{{\partial }^{4}w}{\partial x^4}+2\frac{{\partial }^{4}w}{\partial x^2\partial y^2}+\frac{{\partial }^{4}w}{\partial y^4}=\frac{1}{D}\left(N_x\frac{{\partial }^{2}w}{\partial x^2}+N_y\frac{{\partial }^{2}w}{\partial y^2}+2N_{xy}\frac{{\partial }^{2}w}{\partial x\partial y}\right)$$(1)

where D is called the flexural rigidity of the plate and is given as,   

$$D=\frac{E{t}^3}{12\left(1-{\nu }^2\right)}$$(2)

Using the energy method to investigate the local buckling of the elastically restrained steel plate subjected to one axial in-plane compression, the total potential energy of the steel plate (Π) is the summation of the strain energy stored in the plate associate with the thin plate deforming (U), the strain energy associate with the elastic restraint along the rotationally restrained boundary of the plate (V), and the energy associate with the external loads (U_Γ). It can be mathematically expressed by,   

$$\mathrm{\Pi }=U+V+U_{\mathrm{\Gamma }}$$(3)

The strain energy stored in the steel plate U is given as   

$$\begin{array}{l}U=\\ \frac{D}{2}{\iint }_{\mathrm{\Omega }}\left\{{\left(\frac{{\partial }^{2}w}{\partial x^2}+\frac{{\partial }^{2}w}{\partial y^2}\right)}^2-2\left(1-\nu \right)\left[\frac{{\partial }^{2}w}{\partial x^2}\frac{{\partial }^{2}w}{\partial y^2}-{\left(\frac{{\partial }^{2}w}{\partial x\partial y}\right)}^2\right]\right\}dxdy\end{array}$$(4)

where Ω is the area of the steel plate.

Assuming that the steel plate is elastically restrained along four edges with the elastic rotational restraint stiffness coefficients k_x along the two opposite loaded edges and k_y along the two opposite unloaded edges, the analytical model of the steel plate in CFT column in indicated in Fig. 1. For steel plate with rotational restraints distributed along the four edges, the strain energy U_Γ stored in the equivalent elastic rotational springs is expressed as   

$$\begin{array}{c}{U}_{\mathrm{\Gamma }}=\frac{1}{2}{\int }_{\mathrm{\Gamma }x}{k}_y{\left({\frac{\partial w}{\partial y}|}_{y=0}\right)}^{2}dx+\frac{1}{2}{\int }_{\mathrm{\Gamma }x}{k}_y{\left({\frac{\partial w}{\partial y}|}_{y=b}\right)}^{2}dx\\ \hspace{1em}+\frac{1}{2}{\int }_{\mathrm{\Gamma }y}{k}_x{\left({\frac{\partial w}{\partial x}|}_{x=0}\right)}^{2}dy+\frac{1}{2}{\int }_{\mathrm{\Gamma }y}{k}_x{\left({\frac{\partial w}{\partial x}|}_{x=a}\right)}^{2}dy\end{array}$$(5)

where k_x is the rotational stiffness of the elastic restraint at the loaded edges of x = 0 and a (see Fig. 1) and Γ_y is along the width of the plate (Γ_y = 0 to b); k_y is the rotational stiffness of the elastic restraint at the unloaded edges of y = 0 and b (see Fig. 1) and Γ_x is along the length of the plate (Γ_x = 0 to a).

The energy associated with the external loads U_Γ can be written as   

$$V=\frac{1}{2}{\iint }_{\mathrm{\Omega }}\left[N_x{\left(\frac{\partial w}{\partial x}\right)}^2+N_y{\left(\frac{\partial w}{\partial y}\right)}^2+2N_{xy}\frac{\partial w}{\partial x}\frac{\partial w}{\partial y}\right]dxdy$$(6)

where N_x is the axial force per unit length at the boundary of x = 0 and a; N_y is the axial force per unit length at the boundary of y = 0 and b; and N_xy is the force per unit in the transverse direction.

For the CFT columns subjected to axial compression, the boundary stress of the steel plate is σ_y = τ_xy = 0, and thus   

$$N_x=-{\sigma }_{x}t$$(7)

  

$$N_y=-{\sigma }_{y}t=0$$(8)

  

$$N_{xy}=-{\tau }_{xy}t=0$$(9)

2.2. Displacement Shape Function

It is of significant importance to choose the suitable form of the plate out-of-plane buckling displacement function w in order to solve the eigenvalue problem. In this research, a unique plate buckled displacement shape was adopted to obtain the explicit analytical solution for local buckling of the steel plate in CFT column under uniform in-plane compression in the x direction, as shown in Fig. 1.

The steel plate in composite wall only buckles outwards due to the restraint of the infilled concrete which prevents the plate buckling inwards. Assuming that the displacement shape in the x direction is a combine sine and cosine function and the deflection in the y direction is a biquadratic function, the displacement shape function can be expressed as   

$$\begin{array}{l}w\left(x,y\right)=C\left[\frac{y}{b}+{\varphi }_1{\left(\frac{y}{b}\right)}^2+{\varphi }_2{\left(\frac{y}{b}\right)}^3+{\varphi }_3{\left(\frac{y}{b}\right)}^4\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\times \left[\left(1-\omega \right)sin\frac{\pi x}{a}+\omega \left(1-cos\frac{2\pi x}{a}\right)\right]\end{array}$$(10)

where C is a constant, and ϕ₁, ϕ₂, ϕ₃ and ω are the constants to be determined which should satisfy both the boundary conditions and the requirement of compatibility.

As shown in Fig. 1, the boundary conditions of the plate elastically restrained against rotation at loaded and unloaded edges are given by   

$$w\left(0,y\right)=w\left(a,y\right)=0$$(11a)

  

$$M_x\left(0,y\right)=-D{\left(\frac{{\partial }^{2}w}{\partial x^2}\right)}_{x=0}=-k_x{\left(\frac{\partial w}{\partial x}\right)}_{x=0}$$(11b)

  

$$M_x\left(a,y\right)=-D{\left(\frac{{\partial }^{2}w}{\partial x^2}\right)}_{x=a}=+k_x{\left(\frac{\partial w}{\partial x}\right)}_{x=a}$$(11c)

  

$$w\left(x,0\right)=w\left(x,b\right)=0$$(12a)

  

$$M_y\left(x,0\right)=-D{\left(\frac{{\partial }^{2}w}{\partial y^2}\right)}_{y=0}=-k_y{\left(\frac{\partial w}{\partial y}\right)}_{y=0}$$(12b)

  

$$M_y\left(x,b\right)=-D{\left(\frac{{\partial }^{2}w}{\partial y^2}\right)}_{y=b}=+k_y{\left(\frac{\partial w}{\partial y}\right)}_{y=b}$$(12c)

Substituting the derivative function of Eq. (10) with respect to y into Eq. (11) gives the weight constant ω in terms of the rotational stiffness of the elastic restraint (k_x) at the loaded edges:   

$$\omega =\frac{k_{x}a}{k_{x}a+4\pi D}$$(13)

Substituting the first-order partial derivative and the second-order partial derivative of Eq. (10) with respective to y into Eq. (12), the unknown constants ϕ₁, ϕ₂, ϕ₃ can be solved in terms of the elastic rotational restraint stiffness (k_y) along the two opposite unloaded edges as   

$${\varphi }_1=\frac{k_{y}b}{2D}$$(14a)

  

$${\varphi }_2=-\frac{2D+k_{y}b}{D}$$(14b)

  

$${\varphi }_3=\frac{2D+k_{y}b}{2D}$$(14c)

By substituting Eq. (10) into Eqs. (4), (5), and (6), integrating and summing them according to Eq. (3), the total potential energy of the steel plate is given as   

$$\begin{array}{l}\mathrm{\Pi }=\frac{D}{2}[\frac{b{\pi }^3{C}^2}{a^3}{A}_1{B}_1+\frac{a{C}^2}{b^3}{A}_2{B}_2\\ +2\left(1-\nu \right)\frac{\pi C^2}{ab}{A}_3{B}_3-2\nu \frac{\pi C^2}{ab}{A}_4{B}_4]\\ -\frac{{\sigma }_{x}tb\pi C^2}{2a}{A}_1{B}_4+\frac{k_{y}a{C}^2}{2b^2}\left(1+A_5^2\right)B_5+\frac{k_{x}b{\pi }^2{C}^2{\left(1-\omega \right)}^2}{a^2}{A}_6\end{array}$$(15)

where A₁, A₂, A₃, A₄, A₅, A₆, B₁, B₂, B₃, B₄, and B₅ are defined as   

$$A_1=\frac{1}{3}+\frac{{\varphi }_1}{2}+\frac{{\varphi }_3+{\varphi }_1{\varphi }_2}{3}+\frac{{\varphi }_2{\varphi }_3}{4}+\frac{{\varphi }_1^2+2{\varphi }_2}{5}+\frac{{\varphi }_2^2+2{\varphi }_1{\varphi }_3}{7}+\frac{{\varphi }_3^2}{9}$$(16a)

  

$$A_2=4{\varphi }_1^2+12{\varphi }_2^2+\frac{144}{5}{\varphi }_3^2+12{\varphi }_1{\varphi }_2+16{\varphi }_1{\varphi }_3+36{\varphi }_2{\varphi }_3$$(16b)

  

$$\begin{array}{l}{A}_3=1+2{\varphi }_1+2{\varphi }_2+2{\varphi }_3+\frac{4}{3}{\varphi }_1^2+\frac{9}{5}{\varphi }_2^2+\frac{16}{7}{\varphi }_3^2\\ \hspace{1em}\hspace{0.5em}+3{\varphi }_1{\varphi }_2+\frac{16}{5}{\varphi }_1{\varphi }_3+4{\varphi }_2{\varphi }_3\end{array}$$(16c)

  

$$\begin{array}{l}{A}_4={\varphi }_1+2{\varphi }_2+3{\varphi }_3+\frac{2}{3}{\varphi }_1^2+\frac{6}{5}{\varphi }_2^2+\frac{12}{7}{\varphi }_3^2\\ \hspace{1em}\hspace{0.5em}+2{\varphi }_1{\varphi }_2+\frac{14}{5}{\varphi }_1{\varphi }_3+3{\varphi }_2{\varphi }_3\end{array}$$(16d)

  

$$A_5=1+2{\varphi }_1+3{\varphi }_2+4{\varphi }_3$$(16e)

  

$$\begin{array}{l}{A}_6=\frac{1}{3}+\frac{1}{2}{\varphi }_1+\frac{2}{5}{\varphi }_2+\frac{1}{3}{\varphi }_3+\frac{1}{3}{\varphi }_1{\varphi }_2+\frac{2}{7}{\varphi }_1{\varphi }_3\\ \hspace{1em}\hspace{0.5em}+\frac{1}{4}{\varphi }_2{\varphi }_3+\frac{1}{5}{\varphi }_1^2+\frac{1}{7}{\varphi }_2^2+\frac{1}{9}{\varphi }_3^2\end{array}$$(16f)

  

$$B_1=\frac{\pi {\left(1-\omega \right)}^2}{2}+8\pi {\omega }^2-\frac{16\omega \left(1-\omega \right)}{3}$$(16g)

  

$$B_2=\frac{{\left(1-\omega \right)}^2}{2}+\frac{3{\omega }^2}{2}+\frac{14\omega \left(1-\omega \right)}{3}$$(16h)

  

$$B_3=\frac{\pi {\left(1-\omega \right)}^2}{2}+2\pi {\omega }^2+\frac{8\omega \left(1-\omega \right)}{3}$$(16i)

  

$$B_4=\frac{\pi {\left(1-\omega \right)}^2}{2}+2\pi {\omega }^2+\frac{16\omega \left(1-\omega \right)}{3}$$(16j)

  

$$B_5=\frac{{\left(1-\omega \right)}^2}{2}+\frac{3{\omega }^2}{2}+\frac{4\omega \left(1-\omega \right)}{\pi }$$(16k)

The first polynomial in Eq. (15) represents the change in strain energy due to the deforming of the plate, the second represents the work done by the applied loads, and the third and the fourth represent the energy associated with the elastic rotational restraint at the unloaded and loaded edges, respectively. Note that k_x = 0 or k_y = 0 stands for the loaded or unloaded edges being simply supported; while k_x → ∞ or k_y → ∞ represents loaded or unloaded edges being clamped. Any values of k_x or k_y between the two extreme conditions corresponds to the boundary edges being elastically restrained.

2.3. Explicit Solution

According to the principle of minimum potential energy, the value of σ_xt can be obtained by the condition of minimizing the value of Π by taking a partial derivative of Eq. (15) with respect to C and setting the derivative equal to zero:   

$$\frac{\partial \mathrm{\Pi }}{\partial C}=0$$(17)

Substituting Eq. (15) into (17) and Rearranging gives   

$$\begin{array}{l}{\sigma }_{x}t=\frac{{\pi }^{2}D}{b^2}\left[\begin{array}{l}\frac{B_1}{{\gamma }^2{B}_4}+\frac{{\gamma }^2{A}_2{B}_2}{{\pi }^3{A}_1{B}_4}+\frac{2\left(1-\nu \right)A_3{B}_3-2\nu A_4{B}_4}{{\pi }^2{A}_1{B}_4}\\ +\frac{2{\lambda }_y{\gamma }^2\left(1+A_5^2\right)B_5}{{\pi }^3{A}_1{B}_4}+\frac{4{\lambda }_x{\left(1-\omega \right)}^2{A}_6}{{\gamma }^2\pi A_1{B}_4}\end{array}\right]\\ \hspace{1em}\hspace{0.5em}=\frac{k{\pi }^{2}D}{b^2}\end{array}$$(18)

where γ is the half-wave parameter or so called aspect ratio and is defined as γ = a/b; k is elastic local buckling coefficient of the steel plate as given in Eq. (19); λ_x and λ_y are elastically restraining factors of loaded and unloaded edges, respectively, as defined in Eqs. (20)–(21).   

$$\begin{array}{l}k=\frac{B_1}{{\gamma }^2{B}_4}+\frac{{\gamma }^2{A}_2{B}_2}{{\pi }^3{A}_1{B}_4}+\frac{2\left(1-\nu \right)A_3{B}_3-2\nu A_4{B}_4}{{\pi }^2{A}_1{B}_4}\\ \hspace{1em}+\frac{2{\lambda }_y{\gamma }^2\left(1+A_5^2\right)B_5}{{\pi }^3{A}_1{B}_4}+\frac{4{\lambda }_x{\left(1-\omega \right)}^2{A}_6}{{\gamma }^2\pi A_1{B}_4}\end{array}$$(19)

  

$${\lambda }_x=\frac{k_{x}a}{2D}$$(20)

  

$${\lambda }_y=\frac{k_{y}b}{2D}$$(21)

By taking a partial derivative of Eq. (19) with respect to γ, the minimum of Eq. (19) can be found when the critical aspect ratio γ_cr is given by   

$${\gamma }_{cr}={\left[\frac{{\pi }^3{A}_1{B}_1+4{\lambda }_x{\pi }^2{\left(1-\omega \right)}^2{A}_6}{A_2{B}_2+2{\lambda }_y\left(1+A_5^2\right)B_5}\right]}^{\frac{1}{4}}$$(22)

Substituting Eq. (22) into Eq. (19), the critical elastic local buckling coefficient k_cr can be obtained. When the steel plate is relatively long (e.g., γ ≥ γ_cr), which is generally the case for steel plate in CFT column, the critical local buckling stress of the steel plate (σ_cr) elastically restrained against rotation at loaded and unloaded edges can be determined by substituting k_cr and D (Eq. (2)) into Eq. (18) as   

$${\sigma }_{cr}=\frac{k_{cr}{\pi }^{2}E}{12\left(1-{\nu }^2\right){\left(b/t\right)}^2}$$(23)

While for some rare case with relatively short-span plates (e.g., γ < γ_cr), the critical local buckling stress should be calculated directly by employing Eq. (18).

3. Validity of the Explicit Solution

In this section, the explicit formulas for two special cases which are commonly used in the practical CFT column design and analysis are first discussed and compared with previous theory and data. A more general case was then presented and the accuracy of the explicit local buckling solution based on the energy method was validated against experimental results.

3.1. Case 1: Clamped at Loaded Edge and Simply-supported at Unloaded Edges

When the elastic rotational restraint stiffness k_y = 0, and k_x → ∞, it results in ω = 1 from Eq. (13), λ_x = 0 from Eq. (20), λ_y = 0 from Eq. (21), and γ_cr = 1.519 from Eq. (22). This case is equivalent to the condition that the steel plate is clamped at the loaded edges and simply supported at the unloaded edges. Substituting γ_cr = 1.519 into Eq. (19) gives the value k_cr = 5.467. Then the critical local buckling stress is   

$${\sigma }_{cr}=\frac{5.467{\pi }^{2}E}{12\left(1-{\nu }^2\right){\left(b/t\right)}^2}$$(24)

This result is close to k_cr = 5.6 suggested and used by Uy and Bradford¹⁴⁾ based on finite strip method.

3.2. Case 2: Clamped at All Edges

When the elastic rotational restraint stiffness k_y → ∞, and k_x → ∞, it results in ω = 1, λ_x → ∞, λ_y → ∞, and γ_cr = 1.008. This case is equivalent to the condition that the steel plate is clamped at the loaded and unloaded edges. Substituting γ_cr = 1.008 into Eq. (19) gives the value k_cr = 10.311. Then the critical local buckling stress is   

$${\sigma }_{cr}=\frac{10.311{\pi }^{2}E}{12\left(1-{\nu }^2\right){\left(b/t\right)}^2}$$(25)

This result is close to k_cr = 10.31 which was proposed by Uy and Bradford,¹⁴⁾ k_cr = 9.99 recommended by Bridge and O’Shea²³⁾ based on finite strip analysis, and k_cr = 9.81 obtained by Liang and Uy¹⁵⁾ and Liang et al.¹⁷⁾ by a linear finite element buckling analysis of plates based on the bifurcation buckling theory.

3.3. Case 3: Clamped at Loaded Edges and Elastically-restrained at Unloaded Edges

For a more general case where the CFT column is under axial compression in practical engineering application, the loaded edges of the steel plate can be regarded as be clamped while the unloaded edges is more exactly to consider elastically rotationally restraint, i.e., it can be assumed that k_x → ∞. In this case and referring back to Eq. (19), the most important issue to deal with is to find out the most suitable solutions for λ_y. Bleich²⁴⁾ proposed an equation to predict the elastically restraining factor for steel plate in box steel tube without infilled concrete as shown in Eqs. (26), (27), (28).   

$${\lambda }_y={\left(\frac{t_w}{t_f}\right)}^3\frac{r}{\rho }$$(26)

  

$$r=1-{\left(\frac{t_f{b}_w}{t_w{b}_f}\right)}^2$$(27)

  

$$\rho =\frac{1}{\pi }tanh\left(\frac{\pi b_w}{4b_f}\right)\left[1+\frac{\pi b_w/2b_f}{sinh\left(\pi b_w/2b_f\right)}\right]$$(28)

where b_f and t_f are the width and thickness of the calculated steel plate, respectively; b_w and t_w are the width and thickness of the adjacent steel plate, respectively. Bleich²⁴⁾ suggested values of k_cr = 4.0 and k_cr = 6.97 for steel plate with loaded edges clamped and unloaded edges simply supported and clamped, respectively. Note that in Bleich’s research, the steel plate was not restrained by the concrete. Comparing these results with the critical elastic local buckling coefficient for steel plate infilled by concrete (case 1 and case 2 in this paper), it can be observed that the critical local buckling stress increases by around 37% to 50% due to the presence of concrete. Therefore, in order to account for this effect, a reduction factor of 0.5 is considered to reduce r. It should be mentioned that a larger value (0.5 rather than 0.37) was selected due to the fact that this leads to the predicted value of σ_cr on the conservative side as can be seen from Eqs. (19), (23) and (26). Consequently, Eq. (27) is modified as   

$$r^{\prime }=1-{\beta }_r{\left(\frac{t_f{b}_w}{t_w{b}_f}\right)}^2$$(29)

where β_r = 0.5 is the reduction factor used for considering the beneficial restraining effects offered by concrete.

In order to validate the accuracy of the proposed formulas, the theoretical results from the equations proposed above are compared with the experimental data obtained by Uy^25,26) and Mo et al.,²⁷⁾ as shown in Table 1. The theoretical value of σ_cr was obtained according to Eq. (23), where k_cr was calculated from Eq. (19) and λ_y was determined from Bleich’s formula of Eq. (26) including modified r′. It should be note that the data of cases where the yielding occurs before the local buckling of steel plate have been excluded in Table 1.

Table 1. Comparison of experiments by other researchers and theoretical results.

Specimen No.	b (mm)	t1 (mm)	t2 (mm)	σcr,e (MPa)	σcr,p (MPa)	σcr,p/σcr,e	Reference
LB7	240	3	3	200	197	0.985	Uy (1998)
LB9	300	3	3	120	126	1.050
FB1	360	3	3	93.4	88	0.942	Uy (2001)
FB2	420	3	3	79.9	64	0.801
FB3	480	3	3	43.7	49	1.121
FB4	540	3	3	38.8	39	1.005
SCC2	200	3	3	246	283	1.150	Mo et al. (2004)
SCC3	200	3	2	191	221	1.157
SCC4	200	2	5	166	183	1.102
SCC6	200	2	2	118	126	1.068
Average						1.038
Standard						0.104
deviation

*Note: t₁ and t₂ = thickness of steel plate at the loaded and unloaded sides, respectively; σ_cr,e = local buckling strength from experimental results; σ_cr,p = local buckling strength determined from analytical model.

Uy²⁵⁾ conducted an extensive set of experiments for the local buckling response of steel plates restrained by infilled concrete. The tests included five specimens (with concrete) which are identified in Table 1. The experiments were undertaken in a 5000 kN capacity compression testing machine. The composite section was designed to bear the load uniformly. The local buckling half-wavelengths were measured throughout the tests. The results demonstrated that the measured half wavelengths for all the tested specimens were initially equal to the width of the steel plate as expected from the theoretical assumptions. The half-wavelength reduced in all the specimens due to the yielding and shortening. The calculated local buckling stress σ_cr,e was based on the relation σ_cr,e = Eε_e for the specimens that buckled in the elastic region.

Uy²⁶⁾ tested five concrete-filled steel sections. The load was applied only on the steel by virtue of a 20 mm recess produced at each end. Meanwhile, the tested specimens were pretreated with grease between the concrete and steel to ensure that no bond or load transfer to the concrete would be developed. The range of slenderness varied from 120 to 180.

Mo et al.²⁷⁾ experimentally investigated the local buckling of steel plates in concrete-filled steel tube column. Six specimens with the cross-sectional area of 200×200 mm was tested under axial compression. Both steel plate and concrete were designed to bear the axial load. Different thicknesses ranging from 2 mm to 5 mm were selected as the variables. The result of Specimen SCC5 was excluded from Table 1 as the recorded experimental data was not valid.

It can be found that, in general, the formulas proposed in the present study has good accuracy with the experimental values. The ratio of the predicted values (σ_cr,p) to the experimental ones (σ_cr,e) range from 0.801 to 1.157 with a mean of 1.027 and a standard deviation of 0.089. These values closely corresponded. Meanwhile, it is obvious that an increase of the b/t values leads to a decrease in the critical local buckling stress.

4. Summary and Conclusions

In this paper, an approximate classical solution, based on Rayleigh-Rize method, was provided for local buckling problem of steel plate subjected to uniform axial compression and with elastically-rotational restraint at loaded and unloaded edges. The resulting solution is obtained based on the principle of minimum total potential energy. The proposed solution was first discussed based on two extreme cases where the steel plate was clamped at the loaded edges and simply supported or clamped at the unloaded edges. The predicted values of elastic local buckling coefficient agrees well with the numerical results by various researchers. A more general case was then presented and comparisons made between the proposed formulas and extensive experimental data show that the model provides remarkably accurate predictions of the local buckling stresses that are suitable for use. Furthermore, the increase in the width to thickness ratio results in a reduction in the critical buckling stress. The impact of thickness to width ratio of the steel plate becomes significant for the cases where the local buckling occurs before the yielding. The work in this paper provide basis for the capacity design of CFT column under axial compression

Acknowledgments

This work is sponsored by Jiangsu Provincial Natural Science Foundation of China (Grant No. BK20170685), A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD). The authors also appreciate the idea on development of displacement function by Dr. Luyang Shan and Prof. Pizhong Qiao at Washington State University, which provides solid basis for the development of this paper.

Notation

a: a half buckling wave length

b: width of steel plate

b_f, b_w: width of the calculated steel plate and adjacent steel plate, respectively

C: A constant

D: flexural rigidity of the plate

E: elastic modulus of steel plate

k: elastic local buckling coefficient of the steel plate

k_cr: critical elastic local buckling coefficient of the steel plate

k_x: rotational stiffness of the elastic restraint at the loaded edges of x = 0 and a

k_y: rotational stiffness of the elastic restraint at the unloaded edges of y = 0 and b

N_x: surface force in the normal direction along x-axis

N_y: surface force in the normal direction along y-axis

N_xy: surface force in the transverse direction

t: thickness of steel plate

t₁, t₂: thickness of steel plate at the loaded and unloaded sides, respectively

t_f, t_w: thickness of the calculated steel plate and adjacent steel plate, respectively

U: strain energy stored in the plate associate with the thin plate deforming

U_Γ: work done by the external loads

V: strain energy stored in elastic restraint edges

σ_cr: critical local buckling stress of the steel plate

σ_cr,e: measured local buckling stress from the experiments

σ_cr,p: predicted local buckling stress based on the proposed formulas

σ_x, σ_y: normal stress in the x and y direction, respectively

τ_xy: shear stress in the steel plate

w: deflection function

ν: Poison’s ratio of steel (ν = 0.3)

Π: total potential energy of the steel plate

Ω: area of the plate

Γ_x: along the length of the plate (Γ_x = 0 to a)

Γ_y: along the width of the plate (Γ_y to b)

ϕ₁, ϕ₂, ϕ₃: unkown constants representing the deflection along the y direction

ω: weight constant representing the deflection shape along the x direction

ε_e: measured local buckling strain from the tests

β_r: reduction factor used for considering the beneficial restraining effects offered by concrete

γ: aspect ratio (γ = a/b)

γ_cr: critical aspect ratio

λ_x, λ_y: elastically restraining factors of loaded and unloaded edges, respectively

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