哈工大计算力学考试题及答案.docx

（1）For a plane stress state, if the y-coordinate is regarded as a symmetric axis, try to make a sketch and write down the displacement boundary conditions at the symmetric axis in finite element modeling. (6 points)
Solution:
B
A
x
y

As shown in the figure, for a symmetric problem, we may define at point A; and at point B,
（2）Try to use the Castigliano’s first theorem to obtain the matrix equilibrium equations for the system of springs shown in the following Figure. (10 points)
Solution：
For the spring element, the strain energy is given by
In which, k – stiffness of the spring, - deflection of the spring.
The total strain energy of the system of four springs is expressed by means of the nodal displacements and spring constants as
By application of the Castigliano’s first theorem for each element
The system stiffness matrix can be written as
（3）The interpolation functions for a beam element of length L are
write down a polynomial representation of the displacement field in terms of the above interpolation functions and show that = constant for the beam element subject to pure bending. (12 points)
Solution：
The displacement field for a beam element is
It can also be expressed in terms of interpolation functions and nodal variables as
Substitute the interpolation functions into the above equation and after a few manipulations, we have
for the beam element subject to pure bending, we have
So,
（4）For a 2D problem, if the mid-points of each side of a triangular element are also defined as nodal points, try to write down an appropriate polynomial representation of the displacement field variable, and discuss its convergence conditions. (14 points）
Solution：
The polynomial representation of the displacement field variable can be written as
The convergence conditions include:
(1) the compatibility conditions.
Since the above equations are continuous within the element, so the displacement field is continuous in the element.
On the common boundary, the side line is a quadratic function that has three independent constants. And since there are three nodes, the boundary curve can be uniquely determined by the quadratic function, so on the common boundary, there is no void, no material overlap either.
(2) the completeness condition.
The rigid body motion can be determined by the constants and
The rigid body rotation can be realized by , and
The constant strain condition can be satisfied by
, , and
In summary, convergence conditions are satisfied for the element.
（5）Considering a beam element, Denoting the element length by L and the moment of inertia of the cross-sectional area by , write down an appropriate function to express the displacement field, and finally, derive the finite element equation and nodal forces of the element by using the Galerkin’s method. (18 points)
Solution：
The governing equation for the problem of beam flexure is
The displacement solution can be written as
Therefore, the element residual equations are
Integrating the derivative term by parts and assuming a constant , we obtain
and since
Integrating again by parts and rearranging gives
The shear forces and bending moments at element nodes now explicitly appear in the element equations.
The above equation can be written in the matrix form where the terms of the stiffness matrix are defined by
The terms of the element force vector are defined by
or,
where the integral term represents the equivalent nodal forces and moments produced by the distributed load.
（6）Consider the three-node line element with interpolation functions
Use the element as the parent element in the isoparametric mapping
with but otherwise arbitrary nodal coordinates.
a. How does the x coordinate vary between nodes of the isoparametric element?
b. Has the basic element geometry changed from that of the parent element?
c. Determine the Jacobian matrix for the transformation，and calculate the Jacobian matrix for the basic element with nodal coordinates .
d. Find the inverse of the Jacobian matrix, and calculate its value for the above basic element too.
e. Calculate the value of determinate at a point with .
(20 points)
Solution：
a.
It can be seen that the x coordinate vary as a quadratic function between nodes of the isoparametric element
b. the basic element geometry may change from that of the parent element. The basic element is still a straight line element, however, its length may change. The length of the parent element is 1, and the length of the basic element is .
c. the Jocabian matrix can be written as
That is
for the basic element with nodal coordinates
we have
d. the inverse of the Jocabian matrix is
for the basic element with nodal coordinates
we have
e. The value of determinate at a point with is calculated by
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