Download 4 Solved Problems on Finite Element Analysis - Final Exam | AE 420 and more Exams Aerospace Engineering in PDF only on Docsity!
AAE320 – Finite Element Methods for Aerospace Structures Final exam – Spring 1999 Duration : 3 hours Closed notes/closed books
Problem 1: Theory (12 points)
- Give the three (sufficient) conditions for convergent and show that the 8-node C^0 quad element used in the 2D Poisson problem satisfies these conditions?
- Write the semi-discrete equation for linear transient thermal problems. What are the two methods that can be used to solve it? Explain the basic idea of the two methods.
- Give the name of (and describe) three residual methods.
- What is the difference between Euler and Mindlin beam theories? Which beam element locks and why? What can be done to solve the locking problem and why does it work?
Problem 2: Beam on an elastic foundation (8 points)
Consider the beam bending problem shown in figure 1.
x
0 L
q(x)
k
P P
Figure 1
The expression of the potential energy for such a problem can be shown to be
E I
d^2 w dx 2
2 − P
dw dx
2
dx 0
L ∫ −^ q ( x ) w d x 0
L ∫
where w(x) is the beam deflection E is Young’s modulus I is the beam’s moment of inertia P is compressive axial load applied at the end of the beam
k is the stiffness of the distributed spring symbolizing the elastic foundation of the beam L is the length of the beam q(x) is the distributed load acting on the beam
a) What differential equation does the beam deflection w(x) satisfy? (2 points)
b) Derive the local stiffness matrix [k] and local load vector {r} for a 2-node beam element of length l. Sketch the shape functions to be used with this particular element. Describe how you would derive the expressions of these shape functions (without actually deriving them!). Write the expression of each component kij and ri of [k] and {r} (without actually performing the integrations!). (6 points)
Problem 3. Heat conduction in 2D (15 points)
Consider the classical heat conduction problem (^2) T
x^2
2 T
y^2
= 0 on ,
with, three types of boundary conditions :
- imposed temperature : (^) T = T*^ along (^) T ,
- imposed normal heat flux : (^) qn = −
T
n
= qn*^ along (^) q,
- imposed convection condition : (^) qn = −
T
n
= h T( − Tf) along c.
In the previous relations, is the coefficient of thermal conductivity, Tf is the (known) temperature of the surrounding fluid, h is an experimentally determined coefficient, and
T^ and (^) qn^ denote known functions.
The corresponding variational formulation can be shown to be:
T
x
2
T
y
^2
∫ d^ −^ qn
q
∫ T d^ q^ −^ h Tf^ T^ −^
T^2
d^ c c
Derive the finite element formulation for a general M-node global element (i.e., for which the shape functions are written in terms of x and y). Make sure to indicate the continuity requirement. Write the local stiffness matrix and the local load vectors in matrix and component forms. (8 points)
Specialize your previous results for the case of the isoparametric six-node triangular element. Write the expressions of all the quantities entering the local stiffness matrix (not the local load vector), including all the shape functions. (7 points)
