Theoretical and experimental studies on in-plane stiffness of integrated container structure

Abstract

This article presents analytical, numerical, and experimental studies on the in-plane stiffness of container buildings. First, based on diaphragm theory, parallel corrugated direction stiffness of corrugated sheet has been deduced, and based on energy method, shear modulus of two elastic principal directions of orthotropic plate has been deduced, and through stiffness conversion method, the stiffness relationship between parallel corrugated direction and vertical corrugated direction has been obtained. Combined with container frame, the container stiffness of loading end and non-loading end, as bottom side beam fixed, has been obtained. Second, through the software Abaqus, full-scale container model has been established. The loading–displacement curve of finite element model has been compared with theoretical analysis and has a good agreement. Third, through 20 and 40 ft container, corresponding experimental verification has been done, and by comparison of container stiffness, the theoretical analysis and finite element simulation have been verified. Finally, based on verified finite element model, parametric analysis of corrugated sheet size, corrugated sheeting cross section, elasticity modulus of top side beam, and every plate action for container stiffness have been given. Research result has made feasible in design and construction of container buildings and can provide some references to corresponding specification preparation.

Introduction

As a light steel structure, containers have increasingly been used for building structures as shown in Figure 1. Container buildings have many advantages and applications. First, container steel is recyclable, thus reduces energy consumption and environmental pollution. Second, containers are normally manufactured in factory, transported to and quickly assembled at construction sites, which follows a standardized design and construction process that is easy to manage and construct. Third, containers can be in a variety of forms depending on architectural design requirements. Due to the above advantages, containers have been used not only for general residential buildings but also for public buildings, disaster relief housing, field and military constructions. ²

Figure 1.

The container building. ¹

Although, as a special form of buildings, containers have been widely accepted by the construction industry, there are still demands from the design community for further study on their structural performance. In this aspect, Suzuki and colleagues^3–5 used finite element method to analyze vibration of stacked containers. The numerical model was validated by full-size tests. However, the majority of research has focused on dynamics of container stacking in maritime transport. Børvik and colleagues^6,7 used aluminum board to design envelope structures, and then used finite element method to simulate mechanical properties of the protective structures under impact of explosion. Finally, a full-size container with protective structure was tested to verify the finite element analysis. In these studies, the interactions between the side walls and the frame of container and the integrated container stiffness were not considered. In accordance with the ISO requirements, Sinha and Prakash ⁸ proposed a space frame model for calculating stresses and deflections of containers subjected to some loading conditions when compared with experimental results. It was found that the sidewall simulations provided unsatisfactory results, probably because modeling a container as a space frame for lifting and restraint tests had not represented a good simulation of the actual test conditions. Giriunas and colleagues^9,10 developed finite element models to predict stiffness of a variety of container structures under various loading conditions. Zha and Zuo^11,12 proposed a container optimal model, taking the lightest total weight as the objective function, and the strength and stiffness as the constraint conditions. The optimal cross-sectional size of beam and column was obtained. Zuo and Zha ¹³ used diaphragm theory to give a mechanical analysis and stiffness calculation method of container with holes and then used Abaqus to give the influence law of parameters on the stiffness of container with holes. Finally, five experimental studies of container with holes were verified. Zha and colleagues^14–16 proposed the formula of container stiffness as the single layer and multi-span, multi-layer and single-span, and multi-layer and multi-span based on the principle of displacement coordination and diaphragm theory. Then proposed a new isolation system based on friction energy between containers, and the relationship between equivalent damping ratio and friction coefficient of sliding isolation structure had been obtained.

It can be seen from the above that the majority of research on container buildings were on architectural design for maritime transport, rather than considering containers as building structures. The general code of steel structure^17,18 had often considered a corrugated sheet as building envelope structure without considering its own mechanical property and its interaction with the surrounding frame structure and had no provision of appropriate design and calculation methods. The literature^19–21 introduced that the connection between corrugated sheet and surrounding members was joined by bolt, and this boundary condition was discontinuous. However, the boundary condition of container structure was different, which was joined by welding continuously between corrugated sheet and surrounding members. So, the literature^19–21 failed to reflect the real boundary condition of container structure.

Therefore, for a better and economic design, further research is required to take into account the mechanical property of corrugated sheet, as well as its interaction with container frame.

This article presents a study on the in-plane stiffness of container structures with the considerations of the interactions between corrugated sheet and surroundings members. An analytical solution based on diaphragm theory is proposed. Finite element method is also used to predict the stiffness. Finally, the theoretical analysis and finite element simulations are verified through full-scale test of container structures.

Theoretical analysis

Deformation of corrugated sheets

Assuming that a corrugated sheet is subjected to an in-plane shear, $F\prime$ is applied uniformly on the vertical edges of the corrugated sheet as shown in Figure 2. The corrugated sheet and the container frame are joined by welding. The corrugated sheet of a typical container side wall is as shown in Figure 2.

Figure 2.

The corrugated sheet of container side wall. EBCDEB is a wave band of the corrugated sheet.

In Figure 2, h is the height of the container side wall and $F\prime$ is the unilateral loading which parallel to corrugated direction.

When the sheet is subjected to the in-plane loading,^22,23 a representation part of section is shown in Figure 2(c), lines TT, SS, and BB in the mid-width of the top, side, and bottom plates will be twisted to their respective new positions T′T′, S′S′, and B′B′. The deformations are due to a combined action of bending and shearing as shown in Figure 3.

Figure 3.

The deformation of each plate of corrugated sheet.

In Figure 3, ${\gamma }_{\mathrm{T}}$ , ${\gamma }_{\mathrm{S}}$ , and ${\gamma }_{\mathrm{B}}$ are the in-plane shear strain of center lines at any point of the top, side, and bottom plates, respectively; $u_{\mathrm{T}}$ , $u_{\mathrm{S}}$ , and $u_{\mathrm{B}}$ are the in-plane deflection of center lines at any point of the top, side, and bottom plates, respectively, due to bending deformation; and $b_{\mathrm{T}}$ , $b_{\mathrm{S}}$ , and $b_{\mathrm{B}}$ are the half-length of the top, side, and bottom plates of corrugated sheet, respectively.

Bending deformation

The deformations of the top, side, and bottom plates due to bending are shown in Figure 4. In Figure 4, $u_{\mathrm{T}}$ , $u_{\mathrm{S}}$ , and $u_{\mathrm{B}}$ are decomposed into a linear component plus a Fourier expansion. ²³

Figure 4.

The bending deformation of each plate of corrugated sheet.

In Lawson, ²³ it was shown that sufficient accuracy can be achieved by choosing the first two terms, where for simplicity, only $u_{\mathrm{T}}$ and $u_{\mathrm{S}}$ were considered. In this work, we extend the expression to include $u_{\mathrm{B}}$ , and a full set of new formulation are developed.

Equation (1) is the in-plane displacement of the sheet due to bending

$$\{\begin{array}{c}{u}_{\mathrm{T}}=a_{1}y+a_2\sin \frac{2\pi y}{h}\hfill \\ \multicolumn{1}{c}{u_{\mathrm{S}}=b_{1}y+b_2\sin \frac{2\pi y}{h}}\\ \multicolumn{1}{c}{u_{\mathrm{B}}=c_{1}y+c_2\sin \frac{2\pi y}{h}}\end{array}$$

(1)

where $a_1$ , $a_2$ , $a_3$ , $b_1$ , $b_2$ , $b_3$ , $c_1$ , $c_2$ , and $c_3$ are unknown constant.

Shear deformation

The in-plane displacement of the top, side, and bottom plates of corrugated sheet due to shearing can be expressed in a similar form as

$$\{\begin{array}{c}{u\prime }_{\mathrm{T}}=d_{1}y+{d\prime }_2\sin \frac{2\pi y}{h}\hfill \\ \multicolumn{1}{c}{{u\prime }_{\mathrm{S}}=e_{1}y+{e\prime }_2\sin \frac{2\pi y}{h}}\\ \multicolumn{1}{c}{{u\prime }_{\mathrm{B}}=f_{1}y+{f\prime }_2\sin \frac{2\pi y}{h}}\end{array}$$

(2)

where ${u\prime }_{\mathrm{T}}$ , ${u\prime }_{\mathrm{S}}$ , and ${u\prime }_{\mathrm{B}}$ are the in-plane deflection of center lines at any point of the top, side, and bottom plates, respectively, due to shear deformation; $d_1$ , ${d\prime }_2$ , $e_1$ , ${e\prime }_2$ , $f_1$ , and ${f\prime }_2$ are unknown constant.

Thus, the in-plane shear strains, ${\gamma }_{\mathrm{T}}$ , ${\gamma }_{\mathrm{S}}$ , and ${\gamma }_{\mathrm{B}}$ by taking derivative of equation (2) with respect to ${u\prime }_{\mathrm{T}}$ , ${u\prime }_{\mathrm{S}}$ , and ${u\prime }_{\mathrm{B}}$ , can be expressed as

$$\{\begin{array}{c}{\gamma }_{\mathrm{T}}=d_1+d_2\cos \frac{2\pi y}{h}\hfill \\ \multicolumn{1}{c}{{\gamma }_{\mathrm{S}}=e_1+e_2\cos \frac{2\pi y}{h}}\\ \multicolumn{1}{c}{{\gamma }_{\mathrm{B}}=f_1+f_2\cos \frac{2\pi y}{h}}\end{array}$$

(3)

where $d_2=(2\pi /h){d\prime }_2$ , $e_2={e\prime }_2(2\pi /h)$ , and $f_2={f\prime }_2(2\mathrm{\pi }/h)$ .

Axial deformation

The axial deformation of the top, side, and bottom plates of the corrugated sheet can be expressed as ²³

$$t_{\mathrm{S}}=\frac{1}{2}((\frac{d{u}_{\mathrm{T}}}{\mathrm{dy}}+{\gamma }_{\mathrm{T}})b_{\mathrm{T}}-(\frac{d{u}_{\mathrm{B}}}{\mathrm{dy}}+{\gamma }_{\mathrm{B}})b_{\mathrm{B}})$$

(4)

where $t_{\mathrm{S}}$ is the in-plane axial displacement of the center line at any point of the side plate.

Introducing equations (1) and (3) into equation (4) yields

$$t_{\mathrm{S}}=h_1+h_2\cos \frac{2\pi y}{h}$$

(5)

where $h_1=(b_{\mathrm{T}}/2)a_1-(b_{\mathrm{B}}/2)b_1+(b_{\mathrm{T}}/2)d_1-(b_{\mathrm{B}}/2)e_1$ and $h_2=(b_{\mathrm{T}}/2)(2\pi /h)a_2-(b_{\mathrm{B}}/2)(2\pi /h)b_2+(b_{\mathrm{T}}/2)d_2-(b_{\mathrm{B}}/2)e_2$ .

Considering deformation compatibility of the top, side, and bottom plates, all the three plates have the same axial deformation, that is

$$t_{\mathrm{T}}=t_{\mathrm{B}}=t_{\mathrm{S}}$$

(6)

where $t_{\mathrm{T}}$ and $t_{\mathrm{B}}$ are the in-plane axial displacement of the top and bottom plates, respectively.

Energy of corrugated sheets

There are two types of bending in the plates, that is, the one causing defection in the plan of the plates and the one causing deflection transverse to the plan of the plates. Therefore, the associated bending energies are calculated separately.

Bending energy due to out-of-plane deformations

Considering the deformation of a wave of the corrugated sheet, this part of energy can be calculated from the deformation shown in Figure 5.

Figure 5.

The deformation of portal frame of corrugated sheet.

In Figure 5, $v_{\mathrm{T}}$ and $v_{\mathrm{S}}$ are the displacements normal to the top and side plates, respectively, at C; $v_{\mathrm{B}}$ is the displacement normal to the bottom plate at B; ${\theta }_{\mathrm{B}}$ and ${\theta }_{\mathrm{C}}$ are the rotation angles at B and C, respectively, of the side plate; $M_{\mathrm{B}}$ and $M_{\mathrm{C}}$ are the bending moments at B and C of the side plate, respectively; $\theta$ is the angle between the side plate and the z axis; and c and d are the length of horizontal and vertical projections of the side plate, respectively.

Since the deformation of a wave of the corrugated sheet appears anti-symmetric, analysis was carried out by Lawson for only half of the wave and the bending moments, and $M_{\mathrm{B}}$ and $M_{\mathrm{C}}$ are calculated as

$$\{\begin{array}{c}{M}_{\mathrm{B}}=\frac{\left(\frac{3b_{\mathrm{S}}+2b_{\mathrm{T}}}{b_{\mathrm{T}}}\right)v_{\mathrm{T}}+\left(\frac{9b_{\mathrm{S}}+3b_{\mathrm{T}}}{2b_{\mathrm{S}}}\right)v_{\mathrm{S}}+\left(\frac{3b_{\mathrm{S}}+2b_{\mathrm{T}}}{b_{\mathrm{B}}}\right)v_{\mathrm{B}}}{3b_{\mathrm{S}}^2+2b_{\mathrm{S}}{b}_{\mathrm{B}}+2b_{\mathrm{T}}{b}_{\mathrm{S}}+b_{\mathrm{T}}{b}_{\mathrm{B}}}D\hfill \\ \multicolumn{1}{c}{M_{\mathrm{C}}=\frac{\left(\frac{3b_{\mathrm{S}}+2b_{\mathrm{B}}}{b_{\mathrm{T}}}\right)v_{\mathrm{T}}-\left(\frac{9b_{\mathrm{S}}+3b_{\mathrm{B}}}{2b_{\mathrm{S}}}\right)v_{\mathrm{S}}+\left(\frac{3b_{\mathrm{S}}+2b_{\mathrm{B}}}{b_{\mathrm{B}}}\right)v_{\mathrm{B}}}{3b_{\mathrm{S}}^2+2b_{\mathrm{S}}{b}_{\mathrm{B}}+2b_{\mathrm{T}}{b}_{\mathrm{S}}+b_{\mathrm{T}}{b}_{\mathrm{B}}}D}\end{array}$$

(7)

where D is the bending stiffness of the sheet; and $v_{\mathrm{T}}$ , $v_{\mathrm{S}}$ , and $v_{\mathrm{B}}$ can be calculated by

$$\{\begin{array}{c}{v}_{\mathrm{T}}=u_{\mathrm{T}}\tan \theta +\frac{u_{\mathrm{S}}}{\cos \theta }\hfill \\ \multicolumn{1}{c}{v_{\mathrm{B}}=u_{\mathrm{B}}\tan \theta +\frac{u_{\mathrm{S}}}{\cos \theta }}\\ \multicolumn{1}{c}{v_{\mathrm{S}}=\frac{(v_{\mathrm{B}}-u_{\mathrm{S}}\cos \theta )}{\sin \theta }-(v_{\mathrm{T}}\tan \theta +u_{\mathrm{T}})\cos \theta }\end{array}$$

(8)

Inserting equation (8) into equation (7) gives

$$\{\begin{array}{c}{M}_{\mathrm{B}}=\frac{B_{1}C+D_{1}E}{A}D\hfill \\ \multicolumn{1}{c}{M_{\mathrm{C}}=\frac{B_{2}C-D_{2}E}{A}D}\end{array}$$

(9)

where $A=(3b_{\mathrm{S}}^2+2b_{\mathrm{S}}{b}_{\mathrm{B}}+2b_{\mathrm{T}}{b}_{\mathrm{S}}+b_{\mathrm{T}}{b}_{\mathrm{B}})d{b}_{\mathrm{T}}{b}_{\mathrm{B}}$ ; $B_1=3b_{\mathrm{S}}+2b_{\mathrm{T}}$ ; $B_2=3b_{\mathrm{S}}+2b_{\mathrm{B}}$ ; $C=2b_{\mathrm{S}}{u}_{\mathrm{S}}(b_{\mathrm{B}}+b_{\mathrm{T}})+c(b_{\mathrm{T}}{u}_{\mathrm{B}}+b_{\mathrm{B}}{u}_{\mathrm{T}})$ ; $D_1=9b_{\mathrm{S}}+3b_{\mathrm{T}}$ ; $D_2=9b_{\mathrm{S}}+3b_{\mathrm{B}}$ ; and $E=(u_{\mathrm{B}}-u_{\mathrm{T}})b_{\mathrm{T}}{b}_{\mathrm{B}}$ .

Since the bending moments across the width of the plate are constant and can be calculated from the moments at the intersections, the bending energy due to out-of-plane bending can be computed as

$$\begin{array}{c}\multicolumn{1}{c}{U_{\mathrm{F}}=\frac{2}{D}\underset{0}{\overset{\frac{h}{2}}{\int }}(\underset{0}{\overset{b_{\mathrm{B}}}{\int }}{\left(\frac{M_{\mathrm{B}}}{b_{\mathrm{B}}}s\right)}^2\mathrm{ds}}\\ \multicolumn{1}{c}{+\underset{0}{\overset{2b_{\mathrm{S}}}{\int }}{[\left(\frac{M_{\mathrm{C}}-M_{\mathrm{B}}}{2b_{\mathrm{S}}}\right)(s-2b_{\mathrm{S}})+M_{\mathrm{C}}]}^2\mathrm{ds}+\underset{0}{\overset{b_{\mathrm{T}}}{\int }}{\left(\frac{M_{\mathrm{C}}}{b_{\mathrm{T}}}s\right)}^2\mathrm{ds})\mathrm{dy}}\end{array}$$

(10)

where s is the coordinate along the cross-sectional length of a wave band.

Introducing equation (9) into equation (10), the bending energy is obtained as follows

$$\begin{array}{c}\multicolumn{1}{c}{U_{\mathrm{F}}=F_{\mathrm{C}}(c^2{b}_{\mathrm{B}}^2(\frac{a_1{h}^3}{24}+\frac{a_1{a}_2{h}^2}{\pi }+\frac{a_2^{2}h}{4})+b_{\mathrm{S}}^2{(b_{\mathrm{B}}+b_{\mathrm{T}})}^2(\frac{b_1{h}^3}{6}+\frac{4b_1{b}_2{h}^2}{\pi }+b_2^{2}h)+c^2{b}_{\mathrm{T}}^2(\frac{c_1{h}^3}{24}+\frac{c_1{c}_2{h}^2}{\pi }+\frac{c_2^{2}h}{4})}\\ \multicolumn{1}{c}{+c{b}_{\mathrm{S}}(b_{\mathrm{B}}+b_{\mathrm{T}})(\left(\frac{b_{\mathrm{T}}{c}_1+b_{\mathrm{B}}{a}_1}{6}\right)b_1{h}^3+\frac{(b_1{c}_2+b_2{c}_1)b_{\mathrm{T}}+(a_1{b}_2+a_2{b}_1)b_{\mathrm{B}}}{\pi }2h^2+(c_2{b}_{\mathrm{T}}+a_2{b}_{\mathrm{B}})b_{2}h)}\\ \multicolumn{1}{c}{+c^2{b}_{\mathrm{T}}{b}_{\mathrm{B}}(\frac{a_1{c}_1{h}^3}{12}+\frac{(a_1{c}_2+a_2{c}_1)h^2}{\pi }+\frac{a_2{c}_{2}h}{2}))}\\ \multicolumn{1}{c}{+F_{\mathrm{E}}\left(b_{\mathrm{T}}^2{b}_{\mathrm{B}}^2(\frac{c_1+a_1-2a_1{c}_1}{24}{h}^3+\frac{c_1{c}_2+a_1{a}_2-a_1{c}_2-a_2{c}_1}{\pi }{h}^2+\frac{{(a_2-c_2)}^2}{4}h)\right)}\\ \multicolumn{1}{c}{+F_{\mathrm{CE}}((b_{\mathrm{T}}{b}_{\mathrm{S}}{b}_{\mathrm{B}}^2+b_{\mathrm{T}}^2{b}_{\mathrm{S}}{b}_{\mathrm{B}})(\frac{c_1-a_1}{12}{b}_1{h}^3+\frac{b_1{c}_2+b_2{c}_1-a_1{b}_2-a_2{b}_1}{\pi }{h}^2+\frac{c_2-a_2}{2}{b}_{2}h)}\\ \multicolumn{1}{c}{+c{b}_{\mathrm{T}}{b}_{\mathrm{B}}^2(\frac{c_1-1}{24}{a}_1{h}^3+\frac{a_1{c}_2+a_2{c}_1-2a_1{a}_2}{2\pi }{h}^2+\frac{c_2-a_2}{4}{a}_{2}h)}\\ \multicolumn{1}{c}{+{\mathrm{cb}}_{\mathrm{T}}^2{b}_{\mathrm{B}}(\frac{1-a_1}{24}{c}_1{h}^3+\frac{2c_1{c}_2-a_1{c}_2-a_2{c}_1}{2\pi }{h}^2+\frac{c_2-a_2}{4}{c}_{2}h))}\end{array}$$

(11)

where $U_{\mathrm{F}}$ is the bending energy of a wave band

$$\begin{array}{c}\multicolumn{1}{c}{F_{\mathrm{CE}}=\frac{4D}{3A^2}}\\ \multicolumn{1}{c}{(-B_2{D}_2{b}_{\mathrm{T}}+(2B_1{D}_1-B_1{D}_2+B_2{D}_1-2B_2{D}_2)b_{\mathrm{S}}+B_1{D}_1{b}_{\mathrm{B}})}\\ \multicolumn{1}{c}{F_{\mathrm{C}}=\frac{2D}{3A^2}(B_2^2{b}_{\mathrm{T}}+2(B_1^2+B_2^2+B_1{B}_2)b_{\mathrm{S}}+B_1^2{b}_{\mathrm{B}})}\\ \multicolumn{1}{c}{F_{\mathrm{E}}=\frac{2D}{3A^2}(D_2^2{b}_{\mathrm{T}}+2(D_1^2+D_2^2-D_1{D}_2)b_{\mathrm{S}}+D_1^2{b}_{\mathrm{B}})}\end{array}$$

Bending energy due to in-plane deformations

The bending within the plane of plates can be formulated as

$$\{\begin{array}{c}\frac{d^2{u}_{\mathrm{T}}}{d{y}^2}=\frac{M\left(y\right)}{E{I}_{\mathrm{T}}}\hfill \\ \multicolumn{1}{c}{\frac{d^2{u}_{\mathrm{S}}}{d{y}^2}=\frac{M\left(y\right)}{E{I}_{\mathrm{S}}}}\\ \multicolumn{1}{c}{\frac{d^2{u}_{\mathrm{B}}}{d{y}^2}=\frac{M\left(y\right)}{E{I}_{\mathrm{B}}}}\end{array}$$

(12)

where $M(y)$ is the moment along the y direction in mid-width line of each plates; E is the elastic modulus of container steel; and $I_{\mathrm{T}}$ , $I_{\mathrm{S}}$ , and $I_{\mathrm{B}}$ are the moment of areas of the top, side, and bottom plates, respectively.

Thus, the bending energy of the top, side, and bottom plates of corrugated sheet can be expressed as

$$\{\begin{array}{c}{U}_{\mathrm{BT}}=\underset{-\frac{h}{2}}{\overset{\frac{h}{2}}{\int }}\frac{M^2\left(y\right)\mathrm{dy}}{2E{I}_{\mathrm{T}}}\hfill \\ \multicolumn{1}{c}{U_{\mathrm{BS}}=2\underset{-\frac{h}{2}}{\overset{\frac{h}{2}}{\int }}\frac{M^2\left(y\right)\mathrm{dy}}{2E{I}_{\mathrm{S}}}}\\ \multicolumn{1}{c}{U_{\mathrm{BB}}=\underset{-\frac{h}{2}}{\overset{\frac{h}{2}}{\int }}\frac{M^2\left(y\right)\mathrm{dy}}{2E{I}_{\mathrm{B}}}}\end{array}$$

(13)

where $U_{\mathrm{BT}}$ , $U_{\mathrm{BS}}$ , and $U_{\mathrm{BB}}$ are the bending energy of the top, side, and bottom plates, respectively.

Considering equations (1), (12), and (13), the bending energy of the top, side, and bottom plates of corrugated sheet is calculated as

$$\{\begin{array}{c}{U}_{\mathrm{BT}}=\frac{8{\pi }^4{\mathrm{Etb}}_{\mathrm{T}}^3{a}_2}{3h^3}\hfill \\ \multicolumn{1}{c}{U_{\mathrm{BS}}=\frac{16{\pi }^4{\mathrm{Etb}}_{\mathrm{S}}^3{b}_2}{3h^3}}\\ \multicolumn{1}{c}{U_{\mathrm{BB}}=\frac{8{\pi }^4{\mathrm{Etb}}_{\mathrm{B}}^3{c}_2}{3h^3}}\end{array}$$

(14)

where t is the cross-sectional thickness of the top, side, and bottom plates.

Shear energy due to in-plane deformations

Shear strain energy density of the top, side, and bottom plates of corrugated sheet can be expressed as

$$\{\begin{array}{c}{\upsilon }_{\mathrm{T}}=\underset{0}{\overset{{\gamma }_{\mathrm{T}}}{\int }}{\tau }_{\mathrm{T}}d\gamma =\frac{1}{2}{\tau }_{\mathrm{T}}{\gamma }_{\mathrm{T}}\hfill \\ \multicolumn{1}{c}{{\upsilon }_{\mathrm{S}}=\underset{0}{\overset{{\gamma }_{\mathrm{S}}}{\int }}{\tau }_{\mathrm{S}}d\gamma =\frac{1}{2}{\tau }_{\mathrm{S}}{\gamma }_{\mathrm{S}}}\\ \multicolumn{1}{c}{{\upsilon }_{\mathrm{B}}=\underset{0}{\overset{{\gamma }_{\mathrm{B}}}{\int }}{\tau }_{\mathrm{B}}d\gamma =\frac{1}{2}{\tau }_{\mathrm{B}}{\gamma }_{\mathrm{B}}}\end{array}$$

(15)

where ${\upsilon }_{\mathrm{T}}$ , ${\upsilon }_{\mathrm{S}}$ , and ${\upsilon }_{\mathrm{B}}$ are the shear strain energy density of the top, side, and bottom plates, respectively; and ${\tau }_{\mathrm{T}}$ , ${\tau }_{\mathrm{S}}$ , and ${\tau }_{\mathrm{B}}$ are the in-plane shear stresses of the top, side, and bottom plates, respectively.

Thus, the respective shear energies are

$$\{\begin{array}{c}{U}_{\mathrm{ST}}=\underset{-\frac{h}{2}}{\overset{\frac{h}{2}}{\int }}2b_{\mathrm{T}}t\frac{1}{2}{\tau }_{\mathrm{T}}{\gamma }_{\mathrm{T}}\mathrm{dy}\hfill \\ \multicolumn{1}{c}{U_{\mathrm{SS}}=2\underset{-\frac{h}{2}}{\overset{\frac{h}{2}}{\int }}2b_{\mathrm{S}}t\frac{1}{2}{\tau }_{\mathrm{S}}{\gamma }_{\mathrm{S}}\mathrm{dy}}\\ \multicolumn{1}{c}{U_{\mathrm{SB}}=\underset{-\frac{h}{2}}{\overset{\frac{h}{2}}{\int }}2b_{\mathrm{B}}t\frac{1}{2}{\tau }_{\mathrm{B}}{\gamma }_{\mathrm{B}}\mathrm{dy}}\end{array}$$

(16)

where $U_{\mathrm{ST}}$ , $U_{\mathrm{SS}}$ , and $U_{\mathrm{SB}}$ are the shear energy of the top, side, and bottom plates, respectively.

Introducing equation (3) into equation (16), the shear energies of the top, side, and bottom plates are obtained as

$$\{\begin{array}{c}{U}_{\mathrm{ST}}=\mathrm{Gth}{b}_{\mathrm{T}}(d_1+\frac{d_2^2}{2})\hfill \\ \multicolumn{1}{c}{U_{\mathrm{SS}}=2\mathrm{Gth}{b}_{\mathrm{S}}(e_1+\frac{e_2^2}{2})}\\ \multicolumn{1}{c}{U_{\mathrm{SB}}=\mathrm{Gth}{b}_{\mathrm{B}}(f_1+\frac{f_2^2}{2})}\end{array}$$

(17)

where G is the shear modulus of container steel.

Tension energy due to in-plane deformations

The tension energy of the side plate can be expressed as

$$U_{\mathrm{TS}}=2\underset{-\frac{h}{2}}{\overset{\frac{h}{2}}{\int }}\frac{N_{\mathrm{S}}^2}{2E{A}_{\mathrm{S}}}\mathrm{dy}=2\mathrm{Et}{b}_{\mathrm{S}}\underset{-\frac{h}{2}}{\overset{\frac{h}{2}}{\int }}{\left(\frac{d{t}_{\mathrm{S}}}{\mathrm{dy}}\right)}^2\mathrm{dy}$$

(18)

where $U_{\mathrm{TS}}$ is the tension energy of the side plate; $A_{\mathrm{S}}$ is the cross-sectional area of the side plate; and $N_{\mathrm{S}}$ is the axial force of the side plate.

Since the top and bottom plates have the same axial deformation as the side plate does, tension energy of the top and bottom plates can be written as

$$\{\begin{array}{c}{U}_{\mathrm{TT}}=\mathrm{Et}{b}_{\mathrm{T}}\underset{-\frac{h}{2}}{\overset{\frac{h}{2}}{\int }}{\left(\frac{d{t}_{\mathrm{S}}}{\mathrm{dy}}\right)}^2\mathrm{dy}\hfill \\ \multicolumn{1}{c}{U_{\mathrm{TB}}=\mathrm{Et}{b}_{\mathrm{B}}\underset{-\frac{h}{2}}{\overset{\frac{h}{2}}{\int }}{\left(\frac{d{t}_{\mathrm{S}}}{\mathrm{dy}}\right)}^2\mathrm{dy}}\end{array}$$

(19)

where $U_{\mathrm{TT}}$ and $U_{\mathrm{TB}}$ are the tension energies of the top and bottom plates, respectively.

Inserting equation (5) into equations (18) and (19) yields the tension energies of the top, side, and bottom plates

$$\{\begin{array}{c}{U}_{\mathrm{TT}}=\frac{2{\pi }^2{\mathrm{Eth}}_2^2{b}_{\mathrm{T}}}{h}\hfill \\ \multicolumn{1}{c}{U_{\mathrm{TS}}=\frac{4{\pi }^2{\mathrm{Eth}}_2^2{b}_{\mathrm{S}}}{h}}\\ \multicolumn{1}{c}{U_{\mathrm{TB}}=\frac{2{\pi }^2{\mathrm{Eth}}_2^2{b}_{\mathrm{B}}}{h}}\end{array}$$

(20)

Deformation compatibility of corrugated sheets

The literature ¹⁸ derived the following formulas for calculating $a_1$ and $a_2$

$$\{\begin{array}{c}{a}_1=\frac{{\mathrm{\Delta }″}_{y=\frac{h}{2}}}{2b_{\mathrm{T}}}-\frac{2b_{\mathrm{S}}}{b_{\mathrm{T}}}{b}_1-\frac{b_{\mathrm{B}}}{b_{\mathrm{T}}}{c}_1-d_1-\frac{2b_{\mathrm{S}}}{b_{\mathrm{T}}}{e}_1-\frac{b_{\mathrm{B}}}{b_{\mathrm{T}}}{f}_1\hfill \\ \multicolumn{1}{c}{a_2=-\frac{2b_{\mathrm{S}}}{b_{\mathrm{T}}}{b}_2-\frac{b_{\mathrm{B}}}{b_{\mathrm{T}}}{c}_2-\frac{h}{2\pi }{d}_2-\frac{b_{\mathrm{S}}h}{b_{\mathrm{T}}\pi }{e}_2-\frac{b_{\mathrm{B}}h}{2\pi b_{\mathrm{T}}}{f}_2}\end{array}$$

(21)

where ${\mathrm{\Delta }″}_{y=\frac{h}{2}}=2b_{\mathrm{T}}(a_1+d_1)+4b_{\mathrm{S}}(b_1+e_1)+2b_{\mathrm{B}}(c_1+f_1)$

Longitudinal in-plane stiffness of corrugated sheets

The total energy of a wave of corrugated sheet can be expressed as

$$U=U_{\mathrm{F}}+U_{\mathrm{BT}}+U_{\mathrm{BS}}+U_{\mathrm{BB}}+U_{\mathrm{ST}}+U_{\mathrm{SS}}+U_{\mathrm{SB}}+U_{\mathrm{TT}}+U_{\mathrm{TS}}+U_{\mathrm{TB}}$$

(22)

where U is the total energy of a wave band.

Introducing equations (11), (14), (17), and (20) into equation (22), the total energy is

$$\begin{array}{c}\multicolumn{1}{c}{U=U_{\mathrm{F}}+\frac{8{\pi }^4{\mathrm{Etb}}_{\mathrm{T}}^3{a}_2}{3h^3}+\frac{16{\pi }^4{\mathrm{Etb}}_{\mathrm{S}}^3{b}_2}{3h^3}+\frac{8{\pi }^4{\mathrm{Etb}}_{\mathrm{B}}^3{c}_2}{3h^3}}\\ \multicolumn{1}{c}{+\mathrm{Gth}{b}_{\mathrm{T}}(d_1+\frac{d_2^2}{2})+2\mathrm{Gth}{b}_{\mathrm{S}}(e_1+\frac{e_2^2}{2})}\\ \multicolumn{1}{c}{+\mathrm{Gth}{b}_{\mathrm{B}}(f_1+\frac{f_2^2}{2})}\\ \multicolumn{1}{c}{+\frac{2{\pi }^2{\mathrm{Eth}}_2^2{b}_{\mathrm{T}}}{h}+\frac{4{\pi }^2{\mathrm{Eth}}_2^2{b}_{\mathrm{S}}}{h}+\frac{2{\pi }^2{\mathrm{Eth}}_2^2{b}_{\mathrm{B}}}{h}}\end{array}$$

(23)

It can be seen from equation (23), the total energy of a wave of corrugated sheet is a function of 14 unknown coefficients, that is, $a_1$ , $a_2$ , $b_1$ , $b_2$ , $c_1$ , $c_2$ , $d_1$ , $d_2$ , $e_1$ , $e_2$ , $f_1$ , $f_2$ , $h_1$ , and $h_2$ . Considering equation (21), the number of unknown coefficients is reduced to 12, that is, $b_1$ , $b_2$ , $c_1$ , $c_2$ , $d_1$ , $d_2$ , $e_1$ , $e_2$ , $f_1$ , $f_2$ , $h_1$ , and $h_2$ . From the principle of energy conservation and by taking partial derivatives of equation (23) with respect to the above unknown parameters, we have

$$\{\begin{array}{c}{b}_1^{a_1}{a}_1+b_1^{a_2}{a}_2+b_1^{b_1}{b}_1+b_1^{b_2}{b}_2+b_1^{c_1}{c}_1+b_1^{c_2}{c}_2+b_1^C=0\hfill \\ \multicolumn{1}{c}{b_2^{a_1}{a}_1+b_2^{a_2}{a}_2+b_2^{b_1}{b}_1+b_2^{b_2}{b}_2+b_2^{c_1}{c}_1+b_2^{c_2}{c}_2+b_2^C=0}\\ \multicolumn{1}{c}{c_1^{a_1}{a}_1+c_1^{a_2}{a}_2+c_1^{b_1}{b}_1+c_1^{b_2}{b}_2+c_1^{c_1}{c}_1+c_1^{c_2}{c}_2+c_1^C=0}\\ \multicolumn{1}{c}{c_2^{a_1}{a}_1+c_2^{a_2}{a}_2+c_2^{b_1}{b}_1+c_2^{b_2}{b}_2+c_2^{c_1}{c}_1+c_2^{c_2}{c}_2+c_2^C=0}\\ \multicolumn{1}{c}{d_1^{a_2}{a}_2+d_1^{b_1}{b}_1+d_1^{b_2}{b}_2+d_1^{c_1}{c}_1+d_1^{c_2}{c}_2+d_1^C=0}\\ \multicolumn{1}{c}{d_2^{a_1}{a}_1+d_2^{a_2}{a}_2+d_2^{b_1}{b}_1+d_2^{b_2}{b}_2+d_2^{c_1}{c}_1+d_2^{c_2}{c}_2+d_2^{d_2}{d}_2+d_2^C=0}\\ \multicolumn{1}{c}{e_1^{a_2}{a}_2+e_1^{b_1}{b}_1+e_1^{b_2}{b}_2+e_1^{c_1}{c}_1+e_1^{c_2}{c}_2+e_1^C=0}\\ \multicolumn{1}{c}{e_2^{a_1}{a}_1+e_2^{a_2}{a}_2+e_2^{b_1}{b}_1+e_2^{b_2}{b}_2+e_2^{c_1}{c}_1+e_2^{c_2}{c}_2+e_2^{e_2}{e}_2+e_2^C=0}\\ \multicolumn{1}{c}{f_1^{a_2}{a}_2+f_1^{b_1}{b}_1+f_1^{b_2}{b}_2+f_1^{c_1}{c}_1+f_1^{c_2}{c}_2+f_1^C=0}\\ \multicolumn{1}{c}{f_2^{a_1}{a}_1+f_2^{a_2}{a}_2+f_2^{b_1}{b}_1+f_2^{b_2}{b}_2+f_2^{c_1}{c}_1+f_2^{c_2}{c}_2+f_2^{f_2}{f}_2+f_2^C=0}\\ \multicolumn{1}{c}{h_1^{a_1}{a}_1+h_1^{a_2}{a}_2+h_1^{b_1}{b}_1+h_1^{b_2}{b}_2+h_1^{h_1}{h}_1=0}\\ \multicolumn{1}{c}{h_2^{a_1}{a}_1+h_2^{a_2}{a}_2+h_2^{c_1}{c}_1+h_2^{c_2}{c}_2+h_2^{h_2}{h}_2=0}\\ \multicolumn{1}{c}{2b_{\mathrm{T}}{a}_1+4b_{\mathrm{S}}{b}_1+2b_{\mathrm{B}}{c}_1+4b_{\mathrm{S}}{e}_1+2b_{\mathrm{T}}{d}_1+2b_{\mathrm{B}}{f}_1={\mathrm{\Delta }″}_{y=\frac{h}{2}}}\\ \multicolumn{1}{c}{2\pi b_{\mathrm{T}}{a}_2+4\pi b_{\mathrm{S}}{b}_2+2\pi b_{\mathrm{B}}{c}_2+2b_{\mathrm{S}}h{e}_2+h{b}_{\mathrm{T}}{d}_2+b_{\mathrm{B}}h{f}_2=0}\end{array}$$

(24)

where coefficients before the unknown parameters can be found in Appendix 1.

Equation (24) can be solved by letting ${\mathrm{\Delta }″}_{y=\frac{h}{2}}=1$ , from which the strain energy of a wave of the corrugated sheet can be calculated.

From Zha, ²⁴ the stiffness of the n-wave plates shown in Figure 6 can be calculated by

$${k\prime }_{\mathrm{CS}}=\frac{2U}{n}$$

(25)

where n is the number of wave band of corrugated sheet and ${k\prime }_{\mathrm{CS}}$ is the in-plane shear stiffness of the corrugated sheet against shearing in x-direction.

Figure 6.

The deformation of corrugated sheet with n wave bands.

Stiffness conversion of corrugated sheets

Equation (25) provides a solution for calculating in-plane shear stiffness of the n-wave corrugated sheet in the flute direction. In order to find the in-plane shear stiffness in the transverse direction, the conversion process proposed by the literature^19,25 is adopted in the following calculation.

The shear modulus of corrugated sheet relative to the x- and y-directions

The deformation of a corrugated sheet can be treated as the deformation of an orthotropic plate as proposed in the literature^19,25 and shown in Figure 7.

Figure 7.

The deformation of corrugated sheet: (a) parallel to corrugated direction and (b) vertical to corrugated direction.

In Figure 7, ${\gamma }_{\mathrm{B}}$ and ${\gamma }_{\mathrm{S}}$ are the bending and shear strain angles, respectively, due to f; ${\gamma \prime }_{\mathrm{B}}$ and ${\gamma \prime }_{\mathrm{S}}$ are the bending and shear strain angles, respectively, due to $f\prime$ ; ${\delta }_{\mathrm{B}}$ and ${\delta }_{\mathrm{S}}$ are the displacements normal to corrugated direction due to bending and shear, respectively; ${\delta \prime }_{\mathrm{B}}$ and ${\delta \prime }_{\mathrm{S}}$ are the displacements parallel to corrugated direction due to bending and shear, respectively; f and $f\prime$ are the loads normal and parallel to the corrugated direction, respectively; $\delta$ and $\delta \prime$ are the displacements normal and parallel to corrugated direction, respectively; and $z_1$ and $z_2$ are the lengths of corrugated sheet in the y- and x-direction, respectively.

From the literature^19,25 and Figure 7, the shear modulus relative to the y-direction and x-direction can be expressed as

$$G_y=\frac{z_2}{{z\prime }_2}G$$

(26)

$$G_x=\frac{{z\prime }_2}{z_2}G$$

(27)

where ${z\prime }_2$ is the length of corrugated sheet in the x-direction after developed; $G_y$ is the shear modulus relative to y-direction; and $G_x$ is the shear modulus relative to x-direction.

Stiffness conversion

From Figure 7, the following stiffness can be calculated

$${k\prime }_{\mathrm{CS}}=\frac{f\prime }{\delta \prime }=\frac{f\prime }{{\delta \prime }_{\mathrm{S}}+{\delta \prime }_{\mathrm{B}}}=\frac{f\prime }{{\gamma \prime }_{\mathrm{S}}{z}_2+{\gamma \prime }_{\mathrm{B}}{z}_2}$$

(28)

$$k_{\mathrm{CS}}=\frac{f}{\delta }=\frac{f}{{\delta }_{\mathrm{S}}+{\delta }_{\mathrm{B}}}=\frac{f}{{\gamma }_{\mathrm{S}}{z}_2+{\gamma }_{\mathrm{B}}{z}_2}$$

(29)

where $k_{\mathrm{CS}}$ is in-plane shear stiffness of the corrugated sheet in the y-direction.

Considering that the shear strain of Figure 7 can be expressed as

$${\gamma \prime }_{\mathrm{S}}=\frac{{\gamma \prime }_{\mathrm{S}}{z}_2{k\prime }_{\mathrm{CS}}}{z_{1}t{G}_{\mathrm{y}}}$$

(30)

$${\gamma }_{\mathrm{S}}=\frac{{\gamma }_{\mathrm{S}}{z}_2{k}_{\mathrm{CS}}}{{z\prime }_{2}t{G}_{\mathrm{x}}}$$

(31)

Hence, from equations (30) and (31)

$$\frac{k_{\mathrm{CS}}}{{k\prime }_{\mathrm{CS}}}=\frac{{z\prime }_2{G}_{\mathrm{x}}}{z_1{G}_{\mathrm{y}}}$$

(32)

Introducing equations (26) and (27) into equation (32) yields

$$k_{\mathrm{CS}}=\frac{n{(2b_{\mathrm{T}}+4b_{\mathrm{S}}+2b_{\mathrm{B}})}^3}{z_1{e}^2}{k\prime }_{\mathrm{CS}}$$

(33)

or

$$k_{\mathrm{CS}}=\lambda {k\prime }_{\mathrm{CS}}$$

(34)

where e is the horizontal projection length of a wave band of corrugated sheet; $\lambda$ is the stiffness conversion coefficient of corrugated sheet, and $\lambda =(n(2b_{\mathrm{T}}+4b_{\mathrm{S}}+2b_{\mathrm{B}}{)}^3)/z_1{e}^2$ .

In-plane shear stiffness of container side wall

Under the external loading as shown in Figure 8, the wall of container should deform coordinately. The axial stiffness of top side beam is rarely and only part of F can be transmit to the non-loading end (NLE). In this case, it can be assumed that axial force of the top side beam is distributed trapezoidal, and the maximum axial force $N_{\mathrm{FMAX}}$ acts at the LE, and the minimum axial force $N_{\mathrm{FMIN}}$ acts at the NLE.

Figure 8.

The axial force of top side beam.

In Figure 8, ${\mathrm{\Delta }}_{\mathrm{FL}}$ and ${\mathrm{\Delta }}_{\mathrm{FNL}}$ are the displacement of LE and NLE, respectively.

The internal force in the top beam consist two components, that is, an axial force $\alpha F$ that is uniformly distributed along the beam and a triangularly distributed axial force with its maximum $\beta F$ at the LE.

where

$$\alpha +\beta =1$$

(35)

Consequently, $\alpha F$ causes a uniform displacement, ${\mathrm{\Delta }}_{\mathrm{FNL}}$ , along the top beam that can be calculated by

$${\mathrm{\Delta }}_{\mathrm{FNL}}=\frac{{\alpha }_{\mathrm{C}}\alpha F}{k_{\mathrm{C}}}=\frac{{\alpha }_{\mathrm{CS}}\alpha F}{k_{\mathrm{CS}}}$$

(36)

where $k_{\mathrm{C}}$ is the bending stiffness of a single column; ${\alpha }_{\mathrm{C}}$ is the percentage of the $\alpha F$ resisted by a column; and ${\alpha }_{\mathrm{CS}}$ is the percentage of the $\alpha F$ carried over to the top beam and thus acting as a shear force on the top edge of the side wall. Therefore, ${\alpha }_{\mathrm{C}}$ and ${\alpha }_{\mathrm{CS}}$ satisfy

$$2{\alpha }_{\mathrm{C}}+{\alpha }_{\mathrm{CS}}=1$$

(37)

From equations (36) and (37), we have

$${\mathrm{\Delta }}_{\mathrm{FNL}}=\frac{\alpha F}{2k_{\mathrm{C}}+k_{\mathrm{CS}}}$$

(38)

As shown in Figure 9(b), container stiffness is mainly born by top side beam. As shown in Figure 10, and based on the equilibrium condition of top side beam, the following can be obtained as shown in equation (39).

Figure 9.

The decomposition of axial force of top side beam. (a) Part of the axial force of rectangular distribution αF acting on the corner column and corrugated sheeting and (b) Part of the axial force of triangular distribution βF acting on the top side beam.

Figure 10.

The triangular distribution of axial force of top side beam.

In Figure 10, ${\tau }_{\mathrm{TSB}}(x)$ it is the shear between the sheet and the bottom of top side beam, and l is the length of top side beam of container

$${\tau }_{\mathrm{TSB}}\left(0\right)=\frac{2\beta F}{w_{\mathrm{TSB}}l}$$

(39)

where $w_{\mathrm{TSB}}$ is the bottom width of top side beam.

As shown in Figure 10, the shear of any cross section can be expressed as

$${\tau }_{\mathrm{TSB}}\left(x\right)=\frac{{\tau }_{\mathrm{TSB}}\left(0\right)(l-x)}{l}$$

(40)

And then, the axial deformation of top side beam can be expressed as

$${\mathrm{\Delta }\prime }_{\mathrm{FL}}=\underset{0}{\overset{l}{\int }}\frac{1}{E{A}_{\mathrm{TSB}}}(\beta F-\frac{{\tau }_{\mathrm{TSB}}\left(x\right)w_{\mathrm{TSB}}+{\tau }_{\mathrm{TSB}}\left(0\right)w_{\mathrm{TSB}}}{2}x)\mathrm{dx}$$

(41)

where $A_{\mathrm{TSB}}$ is the cross-sectional area of top side beam.

As shown in Figure 10, the energy of top side beam can be expressed as

$$U_{\mathrm{TSB}}=\underset{0}{\overset{l}{\int }}\frac{{(\beta F-\frac{{\tau }_{\mathrm{TSB}}\left(x\right)w_{\mathrm{TSB}}+{\tau }_{\mathrm{TSB}}\left(0\right)w_{\mathrm{TSB}}}{2}x)}^2}{2E{A}_{\mathrm{TSB}}}\mathrm{dx}$$

(42)

where $U_{\mathrm{TSB}}$ is the energy of top side beam.

From Castigliano’s theorem, the relationship between ${\mathrm{\Delta }\prime }_{\mathrm{FL}}$ and $U_{\mathrm{TSB}}$ can be expressed as

$$\frac{\partial U_{\mathrm{TSB}}}{\partial F}={\mathrm{\Delta }\prime }_{\mathrm{FL}}$$

(43)

And then, the $\alpha$ and $\beta$ can be calculated as

$$\alpha =\frac{4-l^2{w}_{\mathrm{TSB}}^2}{4}$$

(44)

$$\beta =\frac{l^2{w}_{\mathrm{TSB}}^2}{4}$$

(45)

Substituting equations (44) and (45) into equations (38) and (41), we obtain

$${\mathrm{\Delta }}_{\mathrm{FNL}}=\frac{F(4-l^2{w}_{\mathrm{TSB}}^2)}{4(2k_{\mathrm{C}}+k_{\mathrm{CS}})}$$

(46)

$${\mathrm{\Delta }\prime }_{\mathrm{FL}}=\frac{F{l}^3{w}_{\mathrm{TSB}}^2}{12E{A}_{\mathrm{TSB}}}$$

(47)

As shown in Figure 9, superimposed Figure 9(a) and (b), the displacement of LE and NLE of container can be expressed as

$$\{\begin{array}{c}{\mathrm{\Delta }}_{\mathrm{FL}}={\mathrm{\Delta }}_{\mathrm{FNL}}+{\mathrm{\Delta }\prime }_{\mathrm{FL}}\hfill \\ \multicolumn{1}{c}{{\mathrm{\Delta }}_{\mathrm{FNL}}={\mathrm{\Delta }}_{\mathrm{FNL}}+0}\end{array}$$

(48)

And then, the container stiffness of LE and NLE can be expressed as follows. The container component size can be referred to the literature ²⁶

$$\{\begin{array}{c}{K}_{\mathrm{I}}^{\mathrm{FL}}=\frac{F}{{\mathrm{\Delta }}_{\mathrm{FL}}}=\frac{24E{A}_{\mathrm{TSB}}{\omega }_2}{3E{A}_{\mathrm{TSB}}{\omega }_1+l^3{w}_{\mathrm{TSB}}^2{\omega }_2}\hfill \\ \multicolumn{1}{c}{K_{\mathrm{I}}^{\mathrm{FNL}}=\frac{F}{{\mathrm{\Delta }}_{\mathrm{FNL}}}=\frac{4{\omega }_2}{{\omega }_1}}\end{array}$$

(49)

where $K_{\mathrm{I}}^{\mathrm{FL}}$ and $K_{\mathrm{I}}^{\mathrm{FNL}}$ are the unilateral container stiffness of LE and NLE, respectively, as bottom side beam fixed

$$\begin{array}{c}\multicolumn{1}{c}{{\omega }_1=4-l^2{w}_{\mathrm{TSB}}^2}\\ \multicolumn{1}{c}{{\omega }_2=2k_{\mathrm{C}}+k_{\mathrm{CS}}}\end{array}.$$

According to the container size in the literature, ²⁶ the unilateral container stiffness of LE and NLE of 20 and 40 ft, respectively, can be obtained as shown in Table 1.

Table 1.

The unilateral container stiffness of theoretical value (kN/mm).

	20 ft container	40 ft container
LE	80.13	85.13
NLE	3002.45	5987.3

LE: loading end; NLE: non-loading end.

Finite element simulation

Container model establishment

Nonlinear finite element software Abaqus^13,14,27 is used as primary means of finite element analysis. The member and size of container are shown in Tables 2 and 3, and more extensive information of container can be referred to the relevant literature. ²⁶

Table 2.

The main container members.

	Legend	Color	Container member
		Red	Corner fitting
		Orange	Bottom side beam
		Yellow	Top side beam
		Blue	Front corner column
		Green	Rear corner column
		White	Side wall
		Black	Top wall
		Gray	End wall

Table 3.

The main container size (mm).

	Length	Width	Height
20-ft container	6058	2438	2591
40-ft container	12192	2438	2591

Material parameter and constitutive relationship of container model are based on the material experiment as shown in Table 4 and Figure 11. The density of container steel is 7.85 × 10⁻⁹ t/mm³.

Table 4.

The material parameters of container model.

Specimen	E (MPa)	$\mu$	${\sigma }_{\mathrm{s}}$ (MPa)	${\sigma }_{\mathrm{b}}$ (MPa)
Corner column	2.03 × 105	0.3	320.83	461.87
U-steel	2.04 × 105	0.28	325.89	476.21
Top side beam	1.99 × 105	0.31	314.33	449.12
Bottom side beam	2.01 × 105	0.3	318.44	458.07
Side wall: 1.6 mm	1.97 × 105	0.3	318.67	448.2
Side wall: 2 mm	1.98 × 105	0.29	324.17	452.95

$\mu$ , ${\sigma }_{\mathrm{s}}$ , and ${\sigma }_{\mathrm{b}}$ are Poisson’s ratio, yielding strength, and ultimate strength of container steel, respectively.

Figure 11.

The constitutive relationship of container model.

Element size, element shape, technique of mesh control, and element type of container model are shown in Tables 5 and 6.

Table 5.

The elements of container model.

Member name	Top side beam	Bottom side beam	Front corner column	Rear corner column	Corner fitting
Member form
Size (mm)	15	15	15	15	10
Shape	Quad	Quad	Quad	Quad	Hex
Technique	Structured	Structured	Structured	Structured	Structured
Type	S4R	S4R	S4R	S4R	C3D8R

Table 6.

The elements of container model.

Member name	Top end beam	Bottom end beam	Side wall	Top wall	End wall
Member form
Size (mm)	15	15	10	10	10
Shape	Quad	Quad	Quad	Tri	Quad
Technique	Structured	Structured	Structured	Free	Structured
Type	S4R	S4R	S4R	S3	S4R

Through the above analysis, the container model is as shown in Figure 12, and the finite element model has a good agreement with the real container as shown in Figure 12.

Figure 12.

The container model: (a) the 20-ft container model and (b) the 40-ft container model.

Longitudinal concentrated loading is symmetrically applied to two front top corner fittings of container. The major research is on the mechanical property of elastic stage of container; each item is loaded until one component on container began to yield. The constraint condition mainly uses corner fitting and bottom side beam fixed as shown in Table 7.

Table 7.

The loading and constraint conditions of container model.

Loading condition	Constraint condition

Result of finite element simulation

Through finite element simulation, the Mises stress of each component on container is in elastic range as shown in Figure 13, and the unilateral container stiffness of each item is shown in Figure 14. Through comparison, the container stiffness of LE and NLE of finite element simulation (F) and theoretical analysis (T) have a good agreement as shown in Figure 14.

Figure 13.

The Mises stress contour of container model: (a) the 20-ft container model and (b) the 40-ft container model.

Figure 14.

The comparison of unilateral container stiffness (T and F): (a) 20-ft LE, (b) 20-ft NLE, (c) 40-ft LE, and (d) 40-ft NLE.

Experimental verification

Taking into account, the container steel is special weathering steel (Corten Steel), the performance of which is different from general building structural steel. So, in order to fully understand its performance, container material experiment is carried out as shown in Figure 15 and relevant experimental data are shown in Table 4. Experimental method is in accordance with the relevant literature. ²⁸

Figure 15.

Tensile experiment: (a) the tensile specimen size (mm) and (b) the experimental phenomenon.

Experimental preparation

Experimental equipment and arrangement

Experimental equipment mainly includes the reaction frame, as shown in Figure 16(a); the DH3816 static strain measurement system, as shown in Figure 16(b); force sensor (range 0–22 t, sensitivity coefficient 0.03%FS); dial gauge (range 0–50 mm, sensitivity coefficient 0.01 mm); and electric resistance strain gauge (sensitivity coefficient 2.06–2.12). Containers are standard 20 and 40-ft container, as shown in Figure 16(c) and (d).

Figure 16.

Experimental equipment: (a) the reaction frame, (b) the DH3816 static strain measurement system, (c) the 20-ft container, and (d) the 40-ft container.

The arrangement of experimental equipment and container is shown in Figure 17.

Figure 17.

The arrangement of experimental equipment and container.

Loading program

The constraint and loading are similar with finite element simulation. According to finite element simulation and material tests, the loading program adopts loading control method. The corresponding loading program is shown in Table 8.

Table 8.

The loading program of container experiment (kN).

Experimental item	Unilateral maximum loading	Unilateral each level loading	Loading series
20-ft container	150	7.5	20
40-ft container	120	6	20

Experimental result

The experiment of 20 and 40 ft is shown in Figure 18.

Figure 18.

The container experiment: (a) the 20-ft container and (b) the 40-ft container.

From unilateral loading–displacement curve of LE and NLE, each unilateral loading–displacement curve is relatively symmetrical and synchronous and indicates that every experimental item is in the elastic stage and no large torsion, as shown in Figure 19.

Figure 19.

The unilateral loading–displacement curve of each experimental item: (a) 20-ft LE, (b) 20-ft NLE, (c) 40-ft LE, and (d) 40-ft NLE.

As can be seen from experiment, container stiffness of each experiment is shown in Table 9.

Table 9.

The unilateral container stiffness of each experimental item (kN/mm).

	20-ft container	40-ft container
LE1	85.46	88.45
LE2	83.51	90.17
NLE1	3131.61	6194.8
NLE2	3350.12	6157

LE: loading end; NLE: non-loading end.

As the bottom side beam is being fixed, the container side wall is constraint, the corrugated sheet at LE bulges outward slightly, and with the container length increasing, the bulge deformation only occurs in the LE of corrugated sheet. So, the LE stiffness of 20-ft container is slightly lesser than 40-ft container. As the bottom side beam is being fixed, the increase in the container length has no significant effect on container stiffness.

Through comparison, theoretical analysis (T) and finite element simulation (F) have a good agreement with experiment (E) and can be verified from the above conclusion, as shown in Tables 10 and 11.

Table 10.

The verification of 20-ft container stiffness (kN/mm).

	20-ft container
	T	F	E		Error (TE), %	Error (FE), %
LE	80.13	87.33	LE1	85.46	6.24	2.19
			LE2	83.51	4.05	4.57
NLE	3002.45	3632.11	NLE1	3131.61	4.12	12.79
			NLE2	3350.12	10.38	5.43

LE: loading end; NLE: non-loading end.

Table 11.

The verification of 40-ft container stiffness (kN/mm).

	40-ft container
	T	F	E		Error (TE), %	Error (FE), %
LE	85.13	92.54	LE1	88.45	3.75	4.62
			LE2	90.17	5.59	2.63
NLE	5987.3	6200.32	NLE1	6194.8	3.35	0.09
			NLE2	6157	2.76	0.70

LE: loading end; NLE: non-loading end.

Parametric analysis and corresponding proposals

Parametric analysis of finite element model

In order to fully understand the mechanical property of the container and its member, the parametric analysis of finite element model is analyzed.

Size of corrugated sheet

The influence law of container stiffness by the corrugated sheet size is shown in Figure 20. As can be seen from the figure, the container stiffness reduces gradually as the corrugated sheet height increases, and the container stiffness initially increases gradually then unchanged basically as the number of wave band of corrugated sheet n increases.

Figure 20.

The influence law of container stiffness (corrugated sheet size).

Section of corrugated sheet

The cross section of corrugated sheet is shown in Figure 21.

Figure 21.

The cross section of corrugated sheet.

The influence law of container stiffness by t and E is shown in Figure 22. As can be seen from the figure, the container stiffness increases gradually as the t increases, and the container stiffness increases gradually as the E increases.

Figure 22.

The influence law of container stiffness (t and E).

The influence law of container stiffness by e and d is shown in Figure 23. As can be seen from the figure, the container stiffness increases gradually as the e increases, and the container stiffness reduces gradually as the d increases.

Figure 23.

The influence law of container stiffness (e and d).

The influence law of container stiffness by $b_{\mathrm{B}}$ and φ is shown in Figure 24. As can be seen from the figure, the container stiffness reduces gradually as the $b_{\mathrm{B}}$ increases, and the container stiffness reduces gradually as the φ increases.

Figure 24.

The influence law of container stiffness ( $b_{\mathrm{B}}$ and φ).

Elasticity modulus of top side beams

The influence law of container stiffness by the elasticity modulus of top side beams is shown in Figure 25. As can be seen from the figure, the container stiffness increases gradually as the elasticity modulus of top side beams increases. As elasticity modulus of top side beams increases, 40-ft container stiffness increases greater than 20-ft container.

Figure 25.

The influence law of container stiffness (the elasticity modulus of the top side beams).

Side wall, front wall, and roof

The influence law of container stiffness by side wall, front wall, and roof is shown in Figure 26. As can be seen from the figure, the front wall has no effect on container stiffness basically, and the side wall has a great effect on container stiffness, and the roof has a small effect on container stiffness. FW, SW, and R represent the side wall, front wall, and roof, respectively; the 20-ft SW represents the 20-ft container with the side wall removed; and the other symbols have the same meaning, as shown in Figure 26.

Figure 26.

The influence law of container stiffness (the side wall, front wall, and roof).

Conclusion

1. Based on diaphragm theory, the stiffness of corrugated sheet which is parallel to corrugated direction had been deduced. Based on energy method, the shear modulus of two elastic principal directions of orthotropic plate had been deduced. Based on stiffness conversion method, the stiffness of corrugated sheet which is vertical to corrugated direction had been deduced.

2. Based on the stiffness of corrugated sheet and decomposition of axial force of top side beam, the container stiffness formula of LE and NLE as bottom side beam fixed had been deduced.

3. Through 20 and 40-ft container experiment, theoretical analysis and finite element simulation had been verified, and the loading–displacement curve of each item had been obtained.

4. Based on verified finite element model, the influence law of container stiffness—such as corrugated sheet size and cross section—elasticity modulus of top side beam, and every plate action had been given.