1 Introduction

Thin-walled structural members of open- and/or closed-sections are widely used in aero- nautic,civil,and mechanical engineering. Examples include aircraft wing plates,helicopter rotor blades,wind turbine blades,tilt rotor blades,and steel girders of bridges. For prismatic bars with circular cross-sections,the torsional formulas are readily derived by the method of mechanics of materials. For members with noncircular cross- or closed-sections,the torsional problem is usually addressed in the subject of mechanics of elasticity and is often solved by the semi-inverse method of Saint-Venant or the Prandtl elastic-membrane analogy^{[1, 2, 3]}. However, neither the semi-inverse method nor the elastic-membrane analogy can be applied to structural members with multiple materials.

Due to their high specific stiffness in bending and torsion,thin-walled composite box-section beams have become the main structural members in aerospace applications. The use of ad- vanced composites offer additional advantages such as weight savings compared to traditional metal materials. During the past couple of decades,extensive research effort has been de- voted to model and analyze thin-walled composite beams,especially those with box sections. For example,Loughlan and Ata^[4] investigated the constrained torsional response of carbon- fibre composite beams of open and single-cell closed-sections by simplified engineering analysis methods,in which the non-isotropic nature of typical carbon-fibre composite material was taken into account by modifying conventional isotropic theories of elastic torsion. Detail comparisons among theoretical predictions,finite element analyses,and experimental data were provided in their work. Later,Loughlan and Ahmed^[5] further studied the structural performance of beams of multi-cell carbon fibre composite box,subjected to constrained torsional loadings. Kaiser et al.^[6] examined the behaviour of elastic beams of multi-cell closed-sections,made up of general composite laminates containing elastic couplings. By using the Vlasov-based theo- retical approach,the influence of in-plane warping caused due to the elastic couplings of the laminated branches of the cross-section as well as by the strains in the direction of the contour line of the cross-section profile was considered. Ferrero et al.^[7] studied the torsional response of thin-walled composite beams with mid-plane wall symmetry and with stiffness variation in the constituent walls around the sections. The analysis allows for shear stress variation through the thickness of the section walls. The stress predictions from their analysis for a beam with a single-cell box section appear to compare favourably with the finite element analysis. Bauchau et al.^[8] presented two comprehensive analysis methods for composite beams,and described the experimental results obtained from a beam with a thin-walled rectangular cross-section. The predictions from their theoretical work were found to be in good agreement with the twist and strain distributions observed in tests. It was found that the out-of-plane torsional warping of the cross-section was the key influential factor with regard to the accurate modelling of the torsional behaviour. Massa and Barbero^[9] proposed a simple method for the analysis of thin- walled composite beams of open or closed cross-sections,subjected to bending,torque,shear and axial forces. In the analysis,each laminated segment was modelled with the constitutive equations of the classical lamination theory accounting for a linear distribution of normal and shear strains through the thickness of the walls,which allowed for greater accuracy than the classical thin-walled theory. Jung et al.^[10] developed a set of mixed formulations for examining the key modelling issues associated with thin-walled composite beams with single and double- celled box-sections. The modelling issues included the mechanism of elastic couplings and the correct treatment of the hoop moment in the shell wall of the section.

In this paper,a new method is presented for solving the free torsion of thin-walled structural members of open- and/or closed-sections with multiple materials. By utilizing a simple statically indeterminate concept,torsional equations are derived based on the conditions of equilibrium and compatibility. The method presented here not only is very simple and easy to understand but also can be applied to thin-walled structural members of combined open- and closed-sections with multiple materials.

2 T_orsion of thin-walled open-sections

Consider a thin-walled open-section of three segments subjected to a torque T ,as shown in Fig. 1(a). In order to determine the shear stresses in the segments and the angle of twist of the section,let T₁,T₂,and T₃ be the torques carried by the segments 1,2,and 3,respectively,as shown in Fig. 1(b).

According to the equilibrium of twist moments between internal and external torques,the following equation holds:

For each segment,we have the following torsional equation:

where θ_k is the angle of twist,G_k is the shear modulus,Jk is the torsional constant,and the subscript k represents the k-th segment. Since all segments are connected together,they must have an identical angle of twist,that is,

where θ is the angle of twist of the section. Equation (3) is the compatibility equation in twist between segments. Substituting Eq. (2) into Eq. (3) yields Solving Eqs. (1) and (4) for θ and T_k (k = 1,2,3) yields Equations (5) and (6) indicate that an open-section of several segments behaves like a spring system,in which each segment can be modelled as a spring and all of the springs are considered to be connected in parallel. The shear stress in each segment can be easily calculated after the torque carried by each segment is determined.

3 T_orsion of thin-walled closed-sections

For the convenience of presentation,we consider a thin-walled closed-section having three compartments,as shown in Fig. 2(a). The section is subjected to a torque T . Similar to the open-section,the closed-section can also be split into three cells,each of which has a single compartment,as shown in Fig. 2(b). Let T₁,T₂,and T₃ be the torques carried by the cells 1,2, and 3,respectively. It is obvious that the equilibrium of twist moments between internal and external torques can also be expressed by Eq. (1).

Note that for the thin-walled closed-section,it is more convenient to use shear flow than the torque. The shear flow is defined as the shear force per unit length of the cell wall. Note that the shear flow is parallel to the wall surface,and from the continuous condition,the shear flow must be constant within each cell. Also,the equilibrium between the shear flow and the torque in each cell requires the following equation^[1]:

where q_k is the shear flow,Ak is the cell cross-section area,and the subscript k represents the k-th cell. Substituting Eq. (7) into Eq. (1) yields Equation (8) is also an equilibrium equation between internal (shear flow) and external torques, which is not sufficient for determining the three shear flows involved. Similar to the open-section, the additional equations are the torsional equations of individual cells. Note that the work done by the torque T_k in the k-th cell can be expressed as follows: where θ_k is the angle of twist of the k-th cell. The strain energy stored in the k-th cell can be expressed as follows: where τ_k is the average shear stress, k is the average shear strain,t_k is the wall thickness,and G_k is the shear modulus. Note that the shear stress can be expressed in terms of the shear flow as follows: where t is the full wall thickness. S_k is the border that belongs only to the cell k,in which t = t_k. Skn is the border shared by the cell k and the cell n,in which t = t_k + t_n. (S_k + S_kn) is the perimeter of the cell k,and q_n is the shear flow in the cell n (see Fig. 3).

The first or second shear stress in Eq. (11) is defined based on the cell wall thickness of the k-th cell or the full wall thickness of the original closed-section. Substituting Eq. (11) into Eq. (10) yields

The conservation of energy requires W_k = U_k,which yields the following equation:

Similar to the open-section,since all cells are connected together,they must have an identical angle of twist,that is,Eq. (3) must also hold for the closed-section. Substituting Eq. (13) into Eq. (3) yields

Equation (14) is the compatibility equation in twist between cells. They together with the equilibrium equation (8) can be used for solving the shear flows in the three cells and the angle of twist of the section.

Let q_k^* = q_k/θ (k = 1,2,3),which represents the shear flow per unit angle of twist. In this way,Eq. (14) can be expressed as follows:

It is obvious that Eq. (15) can be used to solve for q_k^* (k = 1,2,3) directly. Then,the obtained q_k^* (k = 1,2,3) can be substituted into Eq. (8) for solving θ,that is, After q_k^* and θ are determined,the shear flow q_k and the corresponding shear stress in each cell can be calculated.

4 T_orsion of combined thin-walled open- and closed-sections

Consider the torsion of a combined open- and closed-sections subjected to a torque T . Assume that the torque carried by all open-segments be T_o and that by all closed-cells be T_c. Because of the compatibility in the twist between closed-cells,the shear flows per unit angle of twist in the closed-cells can be solved by Eq. (15) independently. Similarly,due to the compatibility in twist between the open- and closed-sections,we have the following equation:

where A_j and q_j^* are the cell cross-section area and the shear flow in the j-th closed-cell,G_k and Jk are the shear modulus and the torsional constant of the k-th open-segment. With the use of the equilibrium equation,T = T_o + T_c,Eq. (17) becomes Utilizing Eqs. (17) and (18),the torques carried by the open- and closed-sections can be written as After T_o and T_c are determined,the torques carried by individual segments in the open-section and by individual cells in the closed-section can be calculated by the torsional equations of the open- and closed-sections,respectively.

5 Example

As an example,we consider a thin-walled closed-section of three compartments with two different materials,subjected to a torque T ,as shown in Fig. 4. Applying Eq. (15) to each cell yields

where A₁ = A₃ = πr2/2 and A₂ = 4r₂ are the cross-section areas of cells 1,3 and cell 2, respectively (see Fig. 5),r is the radius,t is the wall thickness,and G₁ and G₂ are the shear moduli defined in Fig. 4,respectively. Solving Eq. (21) for q₁^*,q₂^*,and q₃^* yields

Substituting Eq. (22) into Eq. (8) and noting that q_k = q_k^*θ (k = 1,2,and 3) yield

The shear stresses in the cell wall are calculated as follows

Figures 6 and 7 numerically show the variations of the angle of twist and the shear stresses with the ratios of the two shear moduli. It can be seen from the figures that the angle of twist and the shear stresses in the two half circular segments decrease with the increase in the ratio of the two shear moduli,while the shear stresses in the four straight segments increase with the ratio of the two shear moduli.

6 Conclusions

This short paper has presented a new method to derive the torsional equations for thin-walled structural members of open- and/or closed-sections. The method has been developed based on a simple statically indeterminate concept,which,compared with the traditional membrane analogy,not only is very simple and easy to understand but also can be applied to the thin- walled structures of combined open- and closed-sections with multiple materials.