Homework No. 3
1. Analyze the same model as you developed in HW #1 with the following modifications
Change the material model to a perfectly plastic material with yield stress = 50 ksi. Conduct the
following analysis:
(i) Analyze the model for applied traction stress (in the normal direction to the free horizontal edge) of
50.1 ksi. Did the analysis proceed? What was the error message that you received, and what does it
mean to you.
(ii) Analyze the model for applied vertical displacement to the free horizontal edge of 1.2 in. Did the
analysis proceed? What results did you get? Plot the stress-strain relationship for any one finite
element. Is it similar to what you expected? What is the difference and why?
(iii) Change the boundary conditions to restrain the bottom horizontal edge in the vertical direction, and
the left vertical edge in the horizontal direction. Now apply vertical displacement at the top edge of
1.2 in., and simultaneously apply horizontal displacement at the right edge of 1.8 in. This is a case of
proportional loading in the same step.
a. Discuss the results that you get. What are the stresses in the 1 and 2 directions?
b. What are the strains and plastic strains in the 1 and 2 directions?
c. Where is the stress state on the von Mises ellipse?
d. Do the plastic strains make sense to you. Why?
(iv) Keep the boundary conditions of (iii). Apply horizontal displacement of 0.9 in. in the first step. Then
in the second step apply the following simultaneously: additional displacement of 0.9 in. (total
displacement of 1.8 in.) in the horizontal direction and vertical displacement of 1.2 in. in the vertical
direction.
a. Discuss the results that you get. What are the stresses and strains in the 1 and 2 directions at the
end of the step. Are they the same as (iii). Why not?
b. Did the stress states stay on the Von Mises ellipse. Prove it by plotting them on the Von Mises
Ellipse.
c. Did the incremental plastic strains follow the Prandtl – Reuss relations? Prove it by calculating.
What happened at the one point where it does not work?
d. Calculate the third plastic strain increment, and the effective plastic strain increment. Show that
the total effective plastic strain increment is the same as the value PEEQ output by ABAQUS.
e. Plot the effective stress-effective plastic strain curve for the material point. Is it the same as the
curve you input to ABAQUS.
(v) Keep the boundary conditions of (iii). Apply horizontal displacement of 0.9 in. in the first step. Then
in the second step apply the following simultaneously: additional displacement of 0.9 in. (total
displacement of 1.8 in.) in the horizontal direction and vertical displacement of negative (-1.2 in.)
compression in the vertical direction.
a. Discuss the results that you get. What are the stresses and strains in the 1 and 2 directions at the
end of the step. Are they the same as (iii). Why not?
b. Did the stress states stay on the Von Mises ellipse. Prove it by plotting them on the Von Mises
Ellipse.
c. Did the incremental plastic strains follow the Prandtl – Reuss relations? Prove it by calculating.
d. Calculate the third plastic strain increment, and the effective plastic strain increment. Show that
the total effective plastic strain increment is the same as the value PEEQ output by ABAQUS.
e. Plot the effective stress-effective plastic strain curve for the material point. Is it the same as the
curve you input to ABAQUS.
2. Develop and analyze a 2D finite element model of the structure below:
Using the results:
(a) Determine the reactions and check their values for equilibrium
(b) Plot the axial force diagram in the members and check the values using hand calculations
(c) Plot the shear force diagram and check the values
(d) Plot the bending moment diagram and check the values
3. Develop a Mathcad sheet that can take the stress state in x-y-z axes, and calculate the following:
(a) The stress vector on a plane normal to the vector
(b) The stress invariants
(c) The principal stresses
(d) Maximum shear stress
(e) Octahedral shear stress
Use the Mathcad sheet to assess the drive shaft coupling that the following stress components relative to
the x-y-z axes:
é 20 60 10 ù
ê
ú
ê 40 10 20 ú
êë 80 20 40 úû