2.1 Displacement field

The Cartesian coordinate system (x, y, z) is taken as global coordinate, the coordinate origin is located in the cross-section centroid; the curvilinear coordinate system (z, s, n) is used as local coordinate, and the origin is located in the midline of cross section. n, s are the normal and the tangential direction of the middle line respectively, as shown in Figure 1. The composite material is a typical anisotropic material, and the shear modulus is a small amount compared with the elastic modulus. Therefore, the shear deformation under the lateral load cannot be neglected. The flat section assumption in the classical beam theory is also no longer applicable. The analytic model of the composite laminated box beam adopts the first-order shear-deformable beam theory, considering the transverse shear deformation, the primary and secondary warping effects and the non-uniform torsional effect. The assumptions made are as follows:

Figure 1

Geometry and coordinate system of composite laminated box beam

1. The cross section does not undergo in-plane deformation;

2. The transverse shear strains γ_xy and γ_xz are evenly distributed in the cross section;

3. The torsion angle φ is a function of the longitudinal coordinate;

4. In addition to considering the main warping displacement along the contour of the centerline, the secondary warpingdisplacement deviating from the contour of the centerline is also considered;

5. Only small deformations in the linear elastic range occur.

According to the above assumptions, the displacement field of an arbitrary point on the cross-section is assumed to be [13, 14]

where

u, v,w are the displacements along the x, y, and z directions of the coordinate axis. θ_x(z), θ_x(z) and φ(z) are the rotation angles around the coordinate axis x, y, and z, respectively. u₀(z), v₀(z) and w₀(z) are rigid body displacements in three directions. φ is the angle between the n and the x direction (see Figure 2). γ_xy(z) and γ_xz(z) are two transverse shear strains. If the transverse shear and the out-of-plane warping are neglected, γ_xy = γ_xz = 0, the model is simplified to the Euler-Bernoulli beam model without considering shear deformation. a(s) is the height of the right triangle in the geometric relationship. F_w(s) is the generalized fan coordinate. Ψ(s) is the warping function. For box section with a wall thickness of T, height of h, and width of b:

Figure 2

Displacement field of composite laminated box beam

2.2 Geometric equation

Based on the assumptions (1), (5) and the displacement field, there are only three strains existing, namely ϵ_z, γ_sz, γ_nz. The axial strain, ϵ_z, can be obtained as:

where εz0(z,s)andεz1(z,s)are strain caused by primary and secondary warping,respectively,which are calculated by:

The circumferential shear strain, γ_sz, can be expressed as:

The transverse shear strain, γ_nz, is shown as:

2.3 Equivalent constitutive equation

The geometric relationship between (x, y, z) and (z, s, n) of arbitrary point on the midline is:

The normal stress σ_z, shear stresses τ_sz and τ_nz are integrated along the cross section, and the expressions of the internal force and moment can be obtained as:

where N_z is the axial force; V_x and V_y are the shear forces in the x and y directions, respectively; M_x, M_y, and M_z are the bending moments around the x, y, and z axis, respectively; M_w is the bimoment generated by the constrained torsional normal stress. The equivalent constitutive equation of composite laminated box beam can be obtained by simplifying Eq. (9):

a_ij are called the equivalent stiffness coefficients and the method of calculating a_ij is shown in Section 2.

2.4 Equivalent stiffness calculation

Circumferentially uniform stiffness (CUS) is a typical composite tube configuration with angular symmetry of symmetric wall. The composite laminated box beam of CUS configurations can be formed by lamination or winding method. The CUS configuration can reduce the number of times of cutting lamina, the initial defects are small, and the global mechanical properties are superior. Therefore, this paper focused on the stiffness performance of composite laminated box beam with CUS configuration. According to the symmetry of θ(y) = θ(−y) of the CUS configuration, the internal force-displacement relationship, Eq. (10), can be simplified as:

It can be found that for the composite laminated box beam of CUS configurations, the stiffness coefficient a₁₇ characterizing the tensile-torsional coupling effect and the stiffness coefficients a₂₅, a₃₄ characterizing the bending-shear coupling effect are not zero. Assume that the box beam works under pure bending, the equivalent bending stiffnesses of the x-axis and y-axis are obtained by inverting Eq. (12):

Assume that it works under pure torsion, the equivalent torsional stiffness is obtained by inverting Eq. (11):

The calculation method of the equivalent stiffness coefficients a_ij in Eqs. (13) and (14) are expressed in Section 2.

4.1 Test verification

In order to verify the correctness of the refined analytical model of the shear-deformable beam, the three-point bending and the torsion test were carried out on the specimens in Group B and T, respectively. The specimens are made of T300/QY8911, the mechanical properties parameters are shown in Table 1. The lay-ups and geometric parameters of specimens are shown in and Table 2, respectively.

In the engineering, the layered equivalent superposition method based on classical laminate theory is often used to calculate the equivalent stiffnesses of composite thin-walled beam structures:

where t_k is the thickness of a single layer; E_k is the longitudinal elastic modulus of a single layer, which is calculated by:

where b_k and h_k are the width and height of the k-th single layer section, θ is the ply angle.

The equivalent stiffnesses are calculated by the layered equivalent superposition method and the refined analytical model using shear-deformable beam theory respectively, and compared with the experimental results.

4.1.1 Three-point bending test

A three-point bending loading test was carried out on the specimens of Group B. The test was conducted on the INSTRON 5982 universal testing machine, as shown in Figure 3(a). A lateral concentrated load P was applied on the upper middle surfaces of the span, and the loading speed was 1 mm/min. The strain ϵ at mid-span of the specimens was monitored [20, 21]. It is known from the traditional material mechanics that the relationship between P and ϵ is:

Figure 3

Three-point bending test process of specimen BCS4

(26)P=8KεL′h

where L^′ is the distance between the two supported points, h is the height of the section, K is the bending stiffness of the specimens. In order to avoid inaccurate results caused by the excessive local deformation on the upper bottom surface, the test were only carried out in the elastic range. The initial and final state of the three-point bending test are shown in Figures 3(b) and (c)(take specimen BCS4 as an example).

Table 3 compares the theoretical and experimental results of the equivalent bending stiffness of four specimens in Group B. Figure 4(a)-(d) are theoretical and experimental P − ϵ curves of the four specimens in Group B.

Table 3

Comparison of theoretical and experimental results of equivalent bending stiffness

No.	Experiment/ Theory	Bending stiffness/ (N·mm2)	De/%
BCS1	Experiment	54580205	–
	〈EI〉x	45536750	−16.6
	[EI]x	58547250	7.3
BCS2	Experiment	162759101	–
	〈EI〉x	169383531	4.1
	[EI]x	168580503	3.6
BCS4	Experiment	74004918	–
	〈EI〉x	45536750	-38.5
	[EI]x	78930367	6.7
BCS5	Experiment	139189698	–
	〈EI〉x	125002347	−10.2
	[EI]x	143815866	3.3

Figure 4

Theoretical and experimental P − ϵ curves of the four specimens in Group B.

It can be seen from Table 3 and Figure 4 that the maximum error of results calculated by the layered equivalent superposition method is −38.5%, while the maximum error calculated by the shear deformation beam theory is only 7.3%. This is because the layered equivalent superposition method is based on the one-dimensional constitutive equation of composite materials, the transverse shear deformation and the bending-shear coupling effects are not considered, while the shear- deformable beam theory is based on three-dimensional constitutive equation and the actual deformation state of the component is considered.

4.1.2 Torsion test

The torsion test was carried out on the specimens in Group T. The torsion testing machine can record the relative torsion angle φ between the two chucks. The relationship between φ and torque M_z is:

(27)φ=180∘πMzLKG

where L is the total length of the specimens, K_G is the torsional stiffness of the specimens. The torsion test was carried out in the linear elastic range of the specimens. The specimens did not undergo nonlinear strength failure of the material. The initial and final state of the torsion test are shown in Figures 5(a) and (b)(take specimen TCS4 as an example). Table 4 compares the theoretical and experimental results of the equivalent bending stiffness of four specimens in Group T. Figure 6(a)-(d) are the M_z-φ curves of four specimens in Group T.

Figure 5

Torsion test process of specimen BCS4.

Figure 6

Theoretical and experimental M_z-φ curves of the four specimens in Group T.

Table 4

Comparison of theoretical and experimental results of equivalent torsional stiffness.

No.	Experiment/Theory	Twisting stiffness/ (N·mm2)	De /%
TCS1	Experiment	44087964	–
	<GJ>	106484765	141.5
	[GJ]	45815345	3.9
TCS3	Experiment	92078987	–
	<GJ>	108910048	18.2
	[GJ]	96489393	4.7
TCS4	Experiment	159784264	–
	<GJ>	165162237	3.3
	[GJ]	165162237	3.3
TCS5	Experiment	126678364	–
	<GJ>	144476626	14.1
	[GJ]	129924277	2.6

It can be seen from Table 4 and Figure 6 that the maximum error of the results calculated by the layered equivalent superposition method is up to 141.53%. The reason for error is that the layered equivalent superposition method ignores the cross-sectional warping deformation and the torsional-tensile coupling. The maximum error calculated by the analytical model of shear-deformable beam is 4.78%, and the calculation accuracy can meet the requirements of engineering design. For the TCS4 specimen, the results calculated by the two methods are the same, the reason is that TCS4’s lay-up is balanced oblique symmetric, the effect of lay-ups on the equivalent stiffnesses of composite laminated box beam will be discussed in Section 4.1.

4.2 Finite element verification

In order to analyze the calculation accuracy of the theoretical model, a composite laminated box beam with the cross section width b=164 mm, height h=230 mm, span l=2600 mmand wall thickness T=10 mm was selected as a numerical example. The number of layers is 50, single layer thickness is 0.2 mm. The material of the example is the same as the specimens in Section 3.1. The finite element method for calculating the equivalent bending stiffness of composite structures is given by:

where N_z is the applied axial tensile load, I_x is section moment of inertia respect to the x-axis, A is the cross-sectional area, ϵ_z is the axial strain. Establish a finite element model of the example in ABAQUS. C3D8R three-dimensional solid element is selected [17, 18]. The C3D8R element has 8 nodes, each node has 6 degrees of freedom. Set 5 and 300 elements in the thickness and length direction, respectively. Material properties are assigned to four

walls using the “Composite layup” function in ABAQUS. Apply a completely fixed constraint on the right end face, establish a reference point RP1 at the left end, and establish a coupling constraint between the RP1 and the left end face.Apply an axial tensile load N_z=7480Non the RP1, and ensure that the composite laminated box beam only under-goes axial deformation within the linear elastic range. After the calculation is completed, a strain cloud image can be obtained, and the axial strain ϵ_z is extracted.

In the similar way,replace tensile load N_z=7480 N with the torque M_z=10 N·m on RP1, and the historical output of the torsion angle, φ, is established. The equivalent torsional stiffness of the composite laminated box beam can be calculated as:

The finite element model of composite laminated box beam is shown in Figure 7. Assume that the numerical example adopts a uniform angle lay-ups, and the equivalent bending stiffness and equivalent torsional stiffness of the example are calculated by the finite element method, the layered equivalent superposition method and the shear deformation beam theory respectively. The results are shown in Figures 8 and 9.

Figure 7

Finite element model of composite laminated box beam.

Figure 8

The equivalent bending stiffness of composites laminated box beam calculated by three methods.

Figure 9

The equivalent torsional stiffness of composites laminated box beam calculated by three methods.

It can be seen from Figure 8 that compared with the layered equivalent superposition method, the equivalent bending stiffness calculated by the refined analytical model using shear-deformable beam theory is closer to the finite element simulation, which further verifies the correctness of the refined analytical model. For composite laminated box beam with uniform angle lay-ups, the layered equivalent superposition underestimates the equivalent bending stiffness in the range of 0^∘<θ≤ 50^∘, and the equivalent bending stiffness value decreases rapidly in this range; at θ=0^∘ and above 50^∘, the results calculated by the three methods are not much different, indicating that the non-classical effects like shear deformation, section warping etc. have little effect on overall deformation. It can be seen from Figure 9 that the equivalent torsional stiffness takes the maximum value at θ=45^∘. The layered equivalent superposition method seriously overestimates equivalent torsional stiffness, which is far from the results of refined analytical model and finite element simulation.

5.1 Selection of lay-ups

In order to study the effect of the lay-ups on the equivalent stiffnesses, the numerical example in Section 2.2 is also taken as the research object. Under the premise that the number of layers also is 50, the three most commonly used lay-ups in engineering were selected: (i) the uniform angle, [θ]₅₀; (ii) the balanced oblique symmetric, [(θ/−θ)₁₂/θ]_s; and (iii) the balanced anti-symmetric, [θ/−θ]₂₅. The equivalent bending stiffness and equivalent torsional stiffness of numerical example with these three different lay-ups were calculated by the layered equivalent superposition method and the shear-deformable beam theory, the results are shown in Figure 10 and Figure 11.

Figure 10

Equivalent bending stiffness of composite laminated box beam in three kinds of lay-ups change with ply angle.

Figure 11

Equivalent torsional stiffness of composite laminated box beam in three kinds of lay-ups change with ply angle.

It can be seen from Figure 10 that for numerical example with three different lay-ups in this section, the results of the equivalent bending stiffnesses calculated by the layered equivalent superposition method are the same. The reason is that the equivalent longitudinal elastic modulus E_k of each single layer is an even function with respect to the ply angle θ. The actual situation is that the composite laminated box beam with balanced symmetrical or balanced anti-symmetric wall has higher equivalent bending stiffness in the range of 0^∘<θ<45^∘, which is due to the layered equivalent superposition method ignores the interlayer interaction and the tensile-bending coupling effect of composite structures.

It can be seen from Figure 11 that the equivalent torsional stiffness takes the maximum value at θ=45^∘. If the number of layers is the same, equivalent torsional stiffness of composite laminated box beam with the balanced oblique symmetric and the balanced anti-symmetric layups is higher. The results of the layered equivalent super-position method and the shear deformation beam theory are the same. This is because that the wall plate is balanced oblique symmetric lay-ups, it can be known from the classical laminate theory that the tensile-shear coupling coefficient A₁₆ and A₂₆, the tensile-bending coupling stiffness coefficients B₁₁ and B₁₂, the tensile-torsional coupling stiffness coefficient B₁₆ and the shear coupling stiffness coefficient A₄₅ of the balanced oblique symmetric layups are all equal to zero. According to Eq. (21), a₂₅ = a₃₄ = a₁₇ = 0, the Eqs. (13) and (14) are reduced to:

That is to say, the composite laminated box beam with the balanced oblique symmetric laminated wall has no bending-shear coupling and tensile-torsional coupling effect, so the equivalent bending stiffness and the equivalent torsional stiffness are larger even if the number of layers is constant.

Compared with the balanced oblique symmetric layups, the existence of tensile-torsional coupling stiffness coefficient B₁₆ of the balanced anti-symmetric lay-ups leads to a slightly smaller equivalent bending stiffness of the composite laminated box beam, but B₁₆ will decrease as the number of layers is increased. When the number of layers is 50, B₁₆ ≈ 0, the equivalent bending stiffness and the equivalent torsional stiffness of composite laminated box beam with the balanced oblique symmetric and the balanced anti-symmetric are approximately equal.

In summary, in the case of the same number of layers, the composite laminated box beam with the balanced oblique symmetric or the balanced anti-symmetric lay-ups can obtain a greater equivalent stiffness. According to the design principle of the laminate, the balanced oblique symmetric laminate can also avoid the local out-of-plane warping deformation caused by tensile-bending and the bending-torsion coupling of the asymmetric laminate.Moreover, so as to reduce the intra-layer cracking and edge separation between the two directional layers [22], the number of layers of same ply angle in laminates is often less than 4 in practical engineering applications. In order to study the influences of the ply angle and the lay-ups on equivalent bending stiffness and equivalent torsional stiffness of composite laminated box beam, the example in 3.2 was also taken as the object. The five kinds of layups shown in Table 5 were selected on the basis of satisfying the design principle of laminates mentioned above. Calculate the equivalent bending stiffness and equivalent torsional stiffness using Eqs. (13) and (14), the results are shown in Figure 12.

Figure 12

Equivalent bending and torsional stiffnesses of composite laminated box beam in five lay-ups change with ply angle.

5.2 Prediction of equivalent stiffness under five lay-ups

It can be seen from Figure 12 that the equivalent stiffness of the composite laminated box beam has extremely high designability. Dividing the values of [EI]_x and [GJ] in each figure in Figure 12 by the section bending moment of inertia I_x or the torsional moment of inertia J_d respectively, the equivalent elastic modulus E_eq and the equivalent shear modulus G_eq of the composite laminated box beam under five kinds of lay-ups can also be predicted. The change of E_eq and G_eq is the same as that of the curves in Figure 12, but the numerical values and units are different, which is not repeated here.

Compared with the balanced anti-symmetric lay-up, the equivalent bending stiffness of the CRL1 and CRL2 layups are larger, and the equivalent torsional stiffness is smaller; the CRL1 lay-ups can obtain the maximum equivalent bending stiffness at any angle. However, the CRL1 layup is continuously laid 8 layers of 0^∘ at the mid-plane of the wall laminate, which does not meet the design principle of the laminate, and should be avoided in engineering. The equivalent bending stiffness of the CRL2 lay-up is greater than the balanced anti-symmetric lay-up in the range of 0^∘<θ≤12^∘, which can be used for the composite structures that require high equivalent bending stiffness.

The 90^∘ layers ratio of the CRL3 lay-up is high, and the equivalent bending stiffness and equivalent torsional stiffness are lower than the balanced oblique symmetric lay-up. The CRL3 lay-ups can fully exert the hoop tightening effect of the 90^∘ layer. The CRL4 and CRL5 lay-ups can obtain greater equivalent torsional stiffness than the balanced oblique symmetric lay-ups, and the equivalent torsional stiffness is not sensitive to the change of θ. Within the range of 0^∘<θ<45^∘, the value of the equivalent bending stiffness is larger, so if both the equivalent bending stiffness and equivalent torsional stiffness of the composite laminated box beam have maximum design requirements, θ should be less than 45^∘.

6.1 Global buckling equilibrium differential equation

For composite laminated thin-walled box beam, the effect of transverse shear deformation should be fully considered when establishing the global buckling calculation model. For simple supported beams, let w_M and w_S be the deflections caused by the bending moment and the shear force respectively, the total deflection is:

Find the second derivative of Eq. (32), the formula for calculating the curvature can be defined by:

The curvature produced by the bending moment is:

where F is the axial compression load, [EI]_y is the equivalent bending stiffness with respect to the y-axis, calculated by Eq. (13). The shear angle can be calculated according to Shearing Hooke’s law:

where A is the cross-sectional area, k_s is the shear section coefficient, for a box cross-sections, k_s = 2; G_eq is the equivalent shear modulus, which can be obtained by dividing the equivalent torsional stiffness [GJ] by the torsional moment of inertia J_d, [GJ] is calculated by Eq. (14). The additional curvature produced by the shear deformation is:

Substitute Eqs. (36) and (34) into Eq. (33), the differential equation of the deflection curve considering both the bending moment and the shear force can be defined by:

The general solution of Eq. (37) is:

where C₁, C₂ are constant. The boundary condition of the simply supported beam is:

Substitute Eq. (39) into Eq. (38), the formula for calculating the global buckling is given by:

By introducing the length factor μ_y, the formula for calculating the global buckling critical load of composite laminated box beam with y-axis instability under different boundary conditions is obtained as:

where α is the shear effect coefficient, defined by:

It can be found that α ≤ 1, and if α = 1, F_cr = P_y, the result of the Eq. (41) is the same as the Euler critical load formula in traditional material mechanics.

6.2 Analysis of influencing factors of global buckling critical load

6.2.1 Effect of shear deformation

The Eq. (42) can be reduced to:

(43)α=11+ksπ2EeqGeqλy2

where λ_y is the slenderness ratio of y axis; E_eq is the equivalent elastic modulus, which can be obtained by dividing the equivalent bending stiffness [EI]_y by the moment of inertia I_y. It can be found that if the span of the composite laminated box beam is fixed, the shear effect coefficient is related to the ratio of E_eq and G_eq. The numerical examples in Section 3.2 with balanced oblique symmetric [(θ/−θ)₁₂/θ]_s lay-up is selected, the shearing effect coefficient of the composite laminated box beam under the simple supported and cantilever conditions are calculated by the Eq. (43), and compared with the isotropic material box beam, as shown in Figure 13. It can be seen that the effect of transverse shear deformation for isotropic material box beam is less than 5%; for composite laminated box beam, transverse shear deformation will reduce the global buckling critical load by 27% at θ=0^∘. The different of calculating the global buckling critical load with consideration of shear deformation or not is shown in Figure 14. Compared with the cantilever beam, the simply supported beam is more affected by the transverse shear deformation; at θ = 10^∘, the global buckling critical load of the simply supported beam takes the maximum value; in the range of 0^∘<θ<20^∘, neglecting the transverse shear deformation to calculate the global buckling critical load of the simply supported beam will produce a large error, the maximum error can reach more than 20%.

Figure 13

Shear effect coefficients of simply supported beam and cantilever beam change with ply angle.

Figure 14

Global buckling load of composite box beam calculated by the two methods change with ply angle.

6.2.2 Effect of the lay-ups

It can be seen from Eqs. (40) that F_cr is related to [EI]_y, [GJ] of composite laminated box beam if the box beam geometry is fixed. As can be seen from Section 4, it is possible to obtain a larger [EI]_y, [GJ] by increasing the ratio of 0^∘ and ±45^∘ laminates ,and laying 0^∘ and ±45^∘ monolayers on the outside of the wall as much as possible. According to the above conclusions, combined with the design principle of laminate ply, select three lay-ups: (i) [±θ]₂₅; (ii) [90/(04/±θ)₄]_s; (iii) [90/(±45)₁₁/±θ]_s. The F_cr of the numerical example in the three lay-ups are calculated by Eqs. (40) and Euler formula respectively, and the results are shown in Figure 15. It can be seen that increasing the 0^∘ layers ratio can significantly improve F_cr. Regardless of the layups, the reduction of F_cr is most obvious in the range of 0^∘<θ<45^∘; increasing the 45^∘ layers ratio reduces the effect of lateral shear deformation, but also reduces F_cr.

Figure 15

Global buckling load of composite laminated box beam in 3 different lay-ups.

6.2.3 Effect of slenderness ratio

Select the three lay-ups of section 5.2.2, calculate the F_cr by Eq. (40). The change of F_cr with the slenderness ratio λ_y and the ply angle θ is shown in Figure 16. It can be seen that the F_cr of the lay-ups [90/(04/±θ)₄]_s is the largest. 0^∘<θ<40^∘, the three layers are arranged in order of F_cr from large to small: [90/(04/±θ)₄]_s, [±θ]₂₅, [90/(±45)₁₁/±θ]_s; 40^∘≤ θ<90^∘, the order is: [90/(04/±θ)₄]_s, [90/(±45)₁₁/±θ]_s, [±θ]₂₅. λ_y > 80, the difference of F_cr under the three lay-ups is less than 5%, indicating that the enhancement of F_cr by increasing the number of 0^∘ layers is weakened with the increase of slenderness ratio. Therefore, for large-to-fine-ratio composite laminated box beam, the λ_y should be reduced by increasing the cross-sectional size and increasing the thickness of the wall, thereby increasing F_cr.

Figure 16

Global buckling critical load of composite laminated box beam in three different lay-ups change with the ply angle and the slenderness ratio.

The shear effect coefficient α is calculated by the Eq. (43), and the change of 1/α with λ_y and θ is as shown in Figure 17. It can be seen from that the three types of layups are ordered according to the degree of effect of shear deformation as: [90/(04/±θ)₄]_s, [±θ]₂₅, [90/(±45)₁₁/±θ]_s; α increases to 1 with the increase of λ_y and θ. λ_y > 80, 1/α < 1.05, the effect of transverse shear deformation is small, which can be neglected in engineering.

Figure 17

1/α of composite laminated box beam in three different lay-ups change with the ply angle and the slenderness ratio