On the Torsional Rigidity of Orthotropic Beams with Rectangular Cross Section

. The paper deals with the torsional rigidity of homogenous and orthotropic beam with rectangular cross section. The torsional rigidity of the considered beam is defined in the framework of the Saint-Venant theory of uniform torsion. Exact and approximate solutions are given to the determination of the torsional rigidity. The shape of cross section is determined which gives maximum value of the torsional rigidity for a given cross-sectional area. The dependence of torsional rigidity as a function of the ratio shear moduli of beam is also studied


Introduction
shows the beam with rectangular cross section which is subjected to torsional load. The material of the beam is elastic, homogenous and Cartesian orthotropic with shear moduli = and = . Although the exact solution is known for the twisted orthotropic beam with a rectangular cross section [1,2,3,4], the basic formulae are given for the simplicity, which are directly connected to the torsional stiffness. The Prandtl stress function = ( , ) for Cartesian orthotropic beam with solid cross section satisfies the following Dirichlet type boundary-value problem ( Figure 1) where denotes the cross section and is the boundary curve of . In present problem The solution of the boundary-value problem for = ( , ) is as follows [1,2] ( , ) = ∑ 32 The expression of the torsional rigidity can be represented as according to [1,2,3,4].

Lower bound for the torsional rigidity
Nowinski [5] gave a lower bound formula for the torsional rigidity of homogeneous and Cartesian anisotropic beam. In the present problem the form of this lower bound expression is as follows where ̃=̃( , ) is a statically admissible stress function. ̃=̃( , ) satisfies the boundary condition and it is twice continuously differentiable function of the variables and .
Application of formula (7) to the function 2. Upper bound for the torsional rigidity Nowinski [5] derived an upper bound formula for the torsional rigidity of homogenous and Cartesian anisotropic beam. In the present problem, for Cartesian orthotropic beam this formula gives where = ( , ) is a kinematically admissible torsion function whose second order partial derivatives with respect to and are continuous functions.
Substitution for = ( , ) the function into (11) gives For the cross section shown in Figure 1 In formula (14) for rectangular cross section (see Figure 1) are used.
According to the inequality relation which is valid for harmonic mean and arithmetic mean of two positive real numbers = and = we can write that is

Equality in inequality relation (16) is valid only if
is a possible upper bound for , it is weaker than as .

The bounding formulae as a function of the ratio of shear moduli
In this section the lower and the upper bounds of torsional rigidity with the exact value of torsional rigidity as a function of ratio of shear moduli = / is analysed. The following numerical data are used = 0.045 m, = 0.025 m, = 10 × 10 10 Pa. 4. Determination of the cross section whose torsional rigidity has maximum value for a given cross sectional area The geometric dimensions of the torsional rigidity with the maximum value for given cross-sectional area is calculated in two steps. Firstly, the approximate value of the geometric dimensions is obtained by the application of lower bound formula (10). Secondly, the obtained value will be made accurate by the application of formula (6). The cross-sectional area in terms of and is and 1 = 4 1 = √ 2 √ 4 variables 1 and 1 can be considered as a first approximation of the geometrical dimension of the optimal cross section. It is very easy to prove that The exact values of cross sectional dimensions of optimal cross section is computed for the following numerical data = 7 × 10 10 Pa, = 9.5 × 10 10 Pa, = 0.0045 m 2 .
For this data the following results are achieved The plot of function = ( ) for 0.03 ≤ ≤ 0.0315 is shown in Figure 3. Expression of ( ) is as follows The plot of function 0 ( ) is shown in Figure 4.

Conclusions
Some properties of torsional rigidity of homogenous and Cartesian orthotropic beam are studied. The cross section of the beam is a rectangle. The dimensions of the cross section is determined which gives the maximum value of torsional rigidity for given a cross-sectional area.