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Assignment 2 – ME and IE Groups

Structural Analysis of a Steel Cantilever Beamusing Higher-Order Differential Equations

Background

This project investigates the mathematical modelling of abeam which is either imbedded at both ends or free at one end. The model involves a higher-order ODE together with boundary conditions which depend on the manner in which the beam is supported. Analytical solutions are obtained for a number of testcases.

The particular case of a steel cantilever beam subjected to an end load is then investigated by using both analytical and numerical techniques. The deflection of the beam obtained by an analytical approach is validated by using finite difference methods and suggestions are given on possible finite element approaches.

Consider a horizontal beamAB, as shown in the figure, with the assumption that the beam is uniform in cross section and of homogeneous material. If there is no load, the axis of symmetry is the straight line which is indicated by the solid line.

However, when there are external loadings, the beam is distorted, and the result is a curve called the deflection curve or elastic curve. Beams can be supported in many ways. A cantilever beam has one end rigidly fixed while the other end is free to move as shown in Figure B3.2(a). A beam which is supported at both ends A and B is called a simply supported beam.

Problem Statement

Using calculus and the Bernoulli-Euler law (the out-of-plane displacement y of abeam is governed by the Euler-Bernoulli Beam Equation), it is possible to develop a governing differential equation to represent the deflection of the beam. Also, it is possible to formulate boundary conditions associated with this differential equation which depend on the manner in which the beam is supported.

Consider first the general problem of deflection of beams.

(a) Develop a governing Find the differential equation to represent the deflection y(x) of abeam of length l subjected to a vertical load w(x) such that x denotes the distance from one end.

(b) Discuss the possible boundary conditions for problems of this type. (c) Determine the deflection y(x) of the beam in the following cases:

(i) A constant load w0 is distributed uniformly along its length 0 ≤ x ≤ land the beam is imbedded atx = 0 andfree atx = l.

(ii) A constant load w0 is distributed uniformly along its length 0 ≤ x ≤ land the beam is imbedded at both ends x = 0 and x = l.

Formulate a mathematical model stating clearly any assumptions made. Hence find the maximum deflection, ymax , of the cantilever beam together with the bending moment, M, and the maximum stress, σmax . Comment on the accuracy of this value of ymax and suggest how it can be improved. Validate the model by using a finite difference approximation and suggest possible finite element approaches.

The data for steel cantilever beam subjected to an end load is given in part (b) of the assignment.

Bonus Points: Use Ansys APDL to analyse stress, deflection, and shear force for stress (SFD) of Cantilever Beam.