Analytical solution of a composite beam with finger joints and incomplete interaction between the layers

The joint bending of two Bernoulli–Euler’s beams is considered. Each beam has one end fixed and the other free. The beams have the different lengths and thicknesses. The long beam is loaded by the concentrated force. This beam is composite as it includes the internal joint. There is the frictionless unilateral contact between the beams. The elastic lines of the beams are to be found. This problem is reduced to finding of the density of forces of interaction between the beams and the constant that describes the unknown term in the displacement of the unrestrained part of the composite beam. The mathematical formulation of this contact problem is propounded. The density is assumed to be the sum of piecewise continuous function and delta-functions describing the concentrated forces. The uniqueness of the solution of the problem is proved and the analytical solution is constructed. Two possible contact patterns are found out. The former is contact at one point at the end of the short beam. The latter is contact at the same point and at one more point located at the unrestrained part of the composite beam. The coordinate of this point is the root of the cubic equation. The obtained analytical solution is used for the optimization of the structure. The optimization problem is to find the beams thicknesses that minimize the maximum stress for the given loading, beams lengths and the overall deflection. This problem is solved numerically for some values of the given parameters. The hypothesis of the equal-stressed optimum structure is set up on the basis of the numerical results. This hypothesis enables to construct the analytical solution of the optimization problem.

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An Analytical Method for Elastic-Plastic Stress in a Thermoplastic Composite Cantilever Beam Subjected to Bias-Axis Tension Load

Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science ◽

10.1243/095440605x31878 ◽

2005 ◽

Vol 219 (10) ◽

pp. 1017-1026 ◽

Cited By ~ 1

Author(s):

X Wang ◽

K Dong ◽

J Xiao

Keyword(s):

Finite Element ◽

Analytical Solution ◽

Cantilever Beam ◽

Composite Beam ◽

Steel Fibre ◽

Thermoplastic Composite ◽

Hardening Material ◽

Elastic Plastic ◽

Tension Load ◽

Plastic Stress

The paper presents an elastic-plastic stress analysis for a woven steel fibre-reinforced thermoplastic composite cantilever beam subjected to bias-axis tension load. A polynomial stress function is utilized to satisfy both the governing differential equation in the plane stress case and the corresponding boundary conditions for elastic-plastic deformation. An analytical solution for the elastic-plastic problem of the composite beam is obtained by means of Tsai-Hill strength theory of anisotropic material. The composite beam is composed of a steel fibre reinforced low-density polyethylene thermoplastic matrix with exponent hardening material property. In contrast, an elastic-plastic finite element for the same problem is also carried out by applying ANSYS finite element analysis system. Comparing the analytical solution with the finite element solution, it can be found that two kinds of results obtained by making use of two different solving methods are suitably approached. Finally, some examples for thermoplastic composite cantilever beams with two kinds of volume percentage of fibres are carried out and the corresponding results are discussed.

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