Question

finite element problem

Tapered bar subjected to variable axial distributed load A Titanium tapered bar of 25 in. length has a variable cross-sectional area that decreases linearly from 20 in2 to 10 in2 . It is fixed at one end and subjected to an axial concentrated force F = 100 kip at the free end, as shown in the following figure. It’s also subjected to a linearly axial distributed load of variable intensity ?(?) = 0.1 (1 − ? ? ) kip/in. The problem is considered as one dimensional, and the aim of this project is to find, using Finite Element Method, the displacement ?(?) at any position on the x-axis.

The differential equation governing this elastic bars problem is given by: − ? ?? (??(?) ?? ??) − ?(?) = 0 ; 0 < ? < ? Where ? is the Titanium’s Young Modulus of 16. 106 ???; ?(?) is the variable cross-sectional area; and ?(?) is the intensity of the axial distributed load.

Part A: a) Give the expression of the differential equation governing this problem as a function of ?;

b) Give the approximate functions for a quadtratic element;

c) Give the elementary stifness matrix for a quadtratic element;

d) Give the elementary load vector for a quadtratic element;

Part B: We’ll calculate the displacement using a Finite Element Model of one quadratic element.

a) Give the elementary stiffness matrix of the element representing the whole bar

b) Give the elementary load vector of the element representing the whole bar;

c) Give the global matrix form of the Finite Element Model;

d) Give the boundary conditions on the nodal variables (primary as well as secondary variables)

e) Give the condensed equations of the Finite Element Model;

f) Calculate the displacements at ? = ? and ? = ? /2

g) Using the approximation functions, calculate the displacements at ? = ?/ 4 and ? = 3?/ 4

Part C: We’ll calculate the displacement using a Finite Element Model of two quadratic elements.

a) Give the elementary stiffness matrix of each element;

b) Give the elementary load vector of each element;

c) Give the global Matrix Form of the Finite Element Model;

d) Give the boundary conditions on the nodal variables (primary as well as secondary variables);

e) Give the condensed equations of the Finite Element Model;

f) Calculate the displacement at ? = ? /4 ; ? = ? /2 ; ? = 3? /4 and ? = ?..

Answer 1