Here an efficient displacement potential formulation based finite difference technique is used to solve the elastic field of a simply supported beam of orthotropic composite materials. A simply supported beam made of orthotropic composite material under uniformly distributed loading is considered and its elastic behaviors under such loading conditions are analyzed considering plane stress condition. The solutions of the problem satisfy the force equilibrium conditions as well as boundary conditions. For understanding the elastic behavior of a simply supported beam, the displacement and stress components of some important sections of the beam are shown graphically. Effects of different orthotropic composite materials on the solutions are also analyzed. Besides, at a particular section of the beam, the comparative analysis of the elastic field is carried out by using the FDM and FEM methods.
Strengthtoweight ratio of a fiber reinforced composite material is usually higher than that of the corresponding isotropic material. The use of composite materials gradually increases with time, especially to satisfy the demand of dynamic vehicles such as aircrafts, rockets and so forth, to reduce energy consumptions and to absorb vibrational energy during moving. The use of composite simply supported beams in the construction of engineering structures is quite extensive. It is known that the mechanical properties like strength and toughness of a fiber reinforced composite differ significantly from those of the isotropic materials, which eventually play an important role in defining the state of stress and displacements of the corresponding structure under loading. It is noted that stress analysis of composite structures is usually carried out by the numerical method like the finite element method (FEM) but computations are extremely higher rather than the finite difference based displacement potential numerical approach [
In the solution of structures, the physical conditions of bounding surfaces are mathematically modeled in terms of a mixed mode of boundary conditions, where one of the components of stress as well as displacement on the boundary is assumed to be known. However, the earlier mathematical models of elasticity were inadequate in handling the practical stress problems, as most of them are of mixedboundaryvalue type. Since the numerical solution of mixedboundaryvalue problems considering low computations, especially with nonisotropic materials, is beyond the scope of the existing mathematical models of elasticity, the use of a new mathematical formulation is investigated to analyze the elastic field of a simply supported beam under uniformly distributed loading.
Although the stress analysis has now become a classical subject in the field of solid mechanics, somehow these stress analysis problems are still suffering from many shortcomings and thus are being constantly looked into [
Here, a rectangular simply supported beam of orthotropic composite material under uniformly distributed loading is shown in Figure
(a) Boundary conditions of different boundary segments of the simply supported beam as observed in Figure
Boundary segment  Given boundary conditions  Correspondence between mesh points and given boundary conditions  

Mesh point on the physical boundary conditions 
 






At point 









Corner points  Given boundary conditions  Used boundary conditions  Correspondence between mesh points and given boundary conditions  

Mesh point on the physical boundary conditions  Mesh point on the imaginary boundary conditions  




















Properties of composites used to obtain numerical results.
Material  Property  Boron/epoxy 

Fiber 

414 

0.20  


Resin 

3.45 

0.35  


Composite 

282.9 

24.2  

10.4  

0.27  

0.023 
A simply supported beam of orthotropic composite material under uniformly distributed loading.
Application of the normal and tangential displacement, stress components, and the governing equation as finite difference forms.
With reference to a rectangular coordinate system
The values of
When the values of the constant
The present computational scheme involves evaluation of a single function,
In finite difference method, the region occupied by the structure under consideration is divided into fine meshes to get nodal points as shown in Figure
The boundary conditions are discretized in such a way that their applications does not involve additional points outside the imaginary boundary. The whole body of the beam is divided into four segments. To satisfy this requirement, the same boundary conditions must be required to discretize in different forms such as
Consider
The geometry of the problem is shown in Figure
Distribution of the normalized displacement component,
Distribution of the normalized displacement component,
Deformed shape of the cantilever beam under uniformly distributed loading.
Figure
Distribution of the normalized normal stress component,
Distribution of the normalized normal stress component,
Distribution of the normalized shear stress component,
Figure
Distribution of the normalized displacement component,
Distribution of normalized displacement component,
Distribution of the normalized axial stress component,
Distribution of the normalized lateral stress component,
Threedimensional beam is considered twodimensional problem considering plane stress condition. The number of elements used to mesh the geometry of the simply supported beam as observed in Figure
Distribution of the normalized axial displacement component,
Distribution of the normalized lateral displacement component,
Distribution of the normalized axial stress component,
Distribution of the normalized lateral stress component,
Distribution of the normalized shear stress component,
Applying the finite difference technique on the displacement potential function formulation, a simply supported orthotropic composite beam is solved to obtain the elastic field. Simply supported composite beams are widely used in many structures. The solutions of a simply supported orthotropic composite beam using FDM technique identify the critical sections of the beam under uniformly distributed loading. Effect of different composites on the solutions is also analyzed by the present finite difference technique. The displacement and stress components at a particular section of a simply supported beam are analyzed by using FDM and FEM methods in a comparative fashion. From the comparison between the present and finite element solutions, it is observed that there are some differences in some solutions of the present problem obtained by the FDM and FEM although the trend of the curves of the solutions obtained by both of the methods is similar. The present studies will be helpful for the designing of simply supported beams made of orthotropic composite materials.
Displacement potential function
Longitudinal elastic modulus
Transverse elastic modulus
Shear modulous
Major Poisson’s ratio
Minor Poisson’s ratio
Displacement component in
Displacement component in
Normal stress in
Normal stress in
Shear stress
Normal displacement component
Tangential displacement component
Normal stress component
Tangential displacement component
Applying uniformly distributed stress
Width of the beam
Length of the beam
Finite difference method
Finite element method.
The author declares that there is no conflict of interests regarding the publication of this paper.