Introduction To Finite Element Analysis And Design
2nd Edition
ISBN: 9781119078722
Author: Kim, Nam H., Sankar, Bhavani V., KUMAR, Ashok V., Author.
Publisher: John Wiley & Sons,
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Textbook Question
Chapter 2, Problem 26E
Consider the uniform bar in the figure. Axial load q is linearly distributed along the length of the bar according to
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Figure below shows the bar with three equal elements. Use the finite element method and calculate:
2.1 the global stiffness matrix
2.2 the displacement on node 2, 3, 4
2.3 the Strain in each element
2.4 the stresses in each element using Hook's law and compare with theoretical stresses (o=F/A)
(1)
(2)
(3)
-50 N
10 mm
10 mm
10 mm
A1=50 mm?
A2=20 mm?
A:=10 mm?
E=200 GPa
A very thick structure is subjected to certain traction boundary conditions
on its surface. The cross-section and the applied load do not vary with the z-coordinate. The
following stress function is proposed for this problem:
-y
p(x,y) = Sin (x) (A x²e + B e")
(i) use the biharmonic equation to find restrictions, if any, on values of A and B
(ii) calculate all stress components
(iii) calculate all strain components in terms of A, B, and C as well as the Young modulus and
Poisson's ratio E and y, respectively.
(iv) check that the equilibrium equations are satisfied
(v) determine the traction boundary conditions at x =± a and
y=+b
A bar of a uniform cross-section is rigidly fixed at one end and loaded by an off-centre tensile point load F = 2.5 kN at the other end, see Fig. Q1a. The Figure Q1b presents the view of the beam from the free end where the point of application of the load F is indicated as A. The allowable stress [σ]=269 MPa
The geometrical parameters are given as follow
h=17 mm , b=48 mm
the magnitude of the maximum normal stress in the beam and where within the beam it is achieved.
a) Find the bending moment relative to z-axis Mz on the cross section
b) Find the bending moment relative to y-axis My on the cross section
c)The second moment of area relatve to z-axis
d)The second moment of area relative to y-axis
e) The magnitude of the maximum normal stress in the beam
f)The magnitude of the minimum normal stress in the beam
Chapter 2 Solutions
Introduction To Finite Element Analysis And Design
Ch. 2 - Answer the following descriptive questions.
a....Ch. 2 - Use the Galerkin method to solve the following...Ch. 2 - Solve the differential equation in problem 2 using...Ch. 2 - Prob. 4ECh. 2 - Using the Galerkin method, solve the following...Ch. 2 - A one-dimensional heat conduction problem can be...Ch. 2 - Solve the one-dimensional heat conduction problem...Ch. 2 - Prob. 8ECh. 2 - Solve the differential equation in problem 8 for...Ch. 2 - Prob. 10E
Ch. 2 - Prob. 11ECh. 2 - Prob. 12ECh. 2 - Using the Galerkin method, calculate the...Ch. 2 - The boundary-value problem for a clamped-clamped...Ch. 2 - The boundary-value problem for a cantilevered beam...Ch. 2 - Prob. 16ECh. 2 - Consider a finite element with three nodes, as...Ch. 2 - A vertical rod of elastic material is fixed at...Ch. 2 - A bar in the figure is under the uniformly...Ch. 2 - Prob. 20ECh. 2 - A tapered bar with circular cross section is fixed...Ch. 2 - The stepped bar shown in the figure is subjected...Ch. 2 - A bar shown in the figure is modeled using three...Ch. 2 - Consider the tapered bar in problem 17. Use the...Ch. 2 - Consider the tapered bar in problem 21. Use the...Ch. 2 - Consider the uniform bar in the figure. Axial load...Ch. 2 - Determine shape functions of a bar element shown...Ch. 2 - Consider a finite element with three nodes, as...
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