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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Chapter 2, Problem 19E
A bar in the figure is under the uniformly distributed load q due to gravity. For a linear elastic material with Young’s modulus E and uniform cross-sectional area A, the governing differential equation can be written as
a. Starting from the above differential equation, derive an integral equation using the Galerkin method.
b. Write the expression of boundary conditions at
c. Derive the assembled finite element matrix equation, and solve it after applying boundary conditions.
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Solve the following differential equation for axial deformation of a bar of length 12mm using Galerkin Weighted Residual method
2022/09/
d²
dx²
= -0.75(4- x)²
One end of the bar is fixed whereas the displacement is 12 mm at the other end of the bar.
You may use Matlab for computations. Use the trial function û(x) = Co + ₂x + ₂x²
Question 2: Air at the temperature of T1 is being heated with the help of a cylindrical cooling
fin shown in the figure below. A hot fluid at temperature To passes through the pipe with
radius Rc.
a) Derive the mathematical model that gives the variation of the temperature inside the
fin at the dynamic conditions.
b) Determine the initial and boundary conditions to solve the equation derived in (a).
air
T1
Re
Assumptions:
1. Temperature is a function of onlyr direction
2. There is no heat loss from the surface of A
3. Convective heat transfer coefficient is constant
Differential equation
THE LAPLACE METHOD CANNOT BE USED.
When two springs in series, with constants k1 and k2 respectively,support a mass, the effective spring constant is calculated as:k = k1*k2 / k1+k2
A 420 g object stretches a spring 7 cm, and that same object stretches 2.8 cmanother spring. Both springs are attached to a common rigid support and then toa metal plate, as shown in the figure of springs in series. Then the object joins thecenter of the plate (figure of springs in series). Determine:
(a) the effective spring constant.(b) the position of the object at any time t, if the object is initially releasedfrom a point 60 cm below the position ofequilibrium and with an upward velocity of 1.2 m / s.Consider the acceleration of gravity as 9.8 m / s2.
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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