MAK 410 — Summer 2020
Midterm Examination 2
24.07.2020
Examination duration: 120 minutes
This is a closed-book examination. You are not allowed to use any books or lecture notes.
Question 1 (10 points)
Circle the correct answer: TRUE or FALSE. One point is awarded for each correct answer, but one
point is taken off for 2 wrong answers.
TRUE / FALSE
1-)
Element by element lumping of distributed loads is more accurate
than node by node lumping.
TRUE / FALSE
2-)
A stiffness matrix is singular before boundary conditions are imposed.
TRUE / FALSE
3-)
In structural stress analyses, quadrilateral elements provide more
accurate results compared to triangular elements in general.
TRUE / FALSE
4-)
The displacements obtained by the finite element method exactly
satisfy the natural boundary conditions.
TRUE / FALSE
5-)
The total value of the shape functions of an element depend on the
element size and element type.
TRUE / FALSE
6-)
The stiffness matrix of an element obtained by a variational
formulation leads to more accurate results compared to the stiffness
matrix of the same element obtained by using the mechanics of
materials approach.
TRUE / FALSE
7-)
When discretizing two- and three-dimensional problems, it is better to
use finite elements of high aspect ratios.
TRUE / FALSE
8-)
The strain displacement matrix has the same value at every point of a
linear triangular element.
TRUE / FALSE
9-)
If the analytical deflection profile v(x) of a beam is a 4’th order
polynomial, a single Bernoulli-Euler beam finite element is sufficient
to obtain the exact solution.
TRUE / FALSE
10-)
The constitutive equations relate strains to displacements.
Question 2 (30 points)
The three-node bar element shown at right has
quadratic shape functions, N(x) = 0 + 1 x + 2 x2.
For L = 1 m, Young’s modulus E = 200109 Pa,
nodal displacements u1 = 0, u2 = 510−6 m, and u3 =
1510−6 m, find:
a)
b)
c)
d)
The displacement u at x = 0.25 m.
The strain as a function of x.
The strain at x = 0.25 m.
The stress at x = 0.25 m.
1/3
Question 3 (30 points)
For the constant strain triangle plane-stress finite element shown below, the Young’s modulus is E
= 70 GPa and the Poisson’s ratio is  = 0.3. The x-y coordinates of the nodes are given in
parenthesis next to node numbers, and the nodal displacements are ux1 = 0.005, uy1 = 0.002, ux2 = 0,
uy2 = 0, ux3 = 0.005, uy3 = 0. All the dimensions are in millimeters (mm). Determine: (a) the strain
values at the centroid of the triangle, (b) the stress values at the centroid of the triangle.
Question 4 (30 points)
Use the minimum number of finite elements to analyze the problem given in the figure below.
Determine: (a) the displacements and rotations of all the nodes of the beam element(s), (b) reactions
acting on the fixed support at the left end of the beam. Number the elements and the nodes that you
use in your solution.
Some formulas that might be useful:
1-) For a spring finite element:
2/3
2-) For an Euler-Bernoulli beam finite element:
̅𝐞 =
𝐊
𝐟 𝐞𝐚 =
(q is positve upward)
3-) For a constant strain triangle finite element:
4-) For an isotropic material with elastic modulus E and Poisson’s ratio ν, the constitutive matrix E
for the plane-stress problem reduces to:
𝐄=
5-) For a plane-stress finite element:
𝑒𝑥𝑥
𝜎𝑥𝑥
𝐞 = { 𝑒𝑦𝑦 } , 𝛔 = {𝜎𝑦𝑦 } ,
𝜎𝑥𝑦
2𝑒𝑥𝑦
6-) The 3 × 2 symmetric-gradient operator D is
𝐃=
3/3