1. Draw a stress-strain diagram to represent the behavior outlined below.  Before starting, read further down in order to choose an appropriate scale.

a) a linearly elastic part from the origin O to point A.

b) a perfectly plastic part AB whose plastic flow has a strain range equal to 10 times the elastic one.

c) a parabolic strain hardening part BC and a necking part CD characterized by the following:

    c1) an ultimate stress equal to twice the elastic limit.

    c2) a fracture stress equal to one and a half times the elastic limit.

    c3) a strain hardening range equal to twice the plastic flow range.

    c4) a necking range equal to one and a half times the plastic flow range.

2. Let the elastic limits for stress and strain in the graph of question 1 be s = 200 MPa and e = 0.001, respectively.
a) Determine the modulus of elasticity.

b) Is "Modulus of Elasticity" synonymous with "Young's Modulus"?  

c) Determine the modulus of resilience = Area under the elastic part of the curve.. 

d) Determine the modulus of toughness = Area under the curve up to the fracture point.  Note that the area under a parabolic segment symmetric about the axis of the parabola is equal to two thirds of the area of the rectangle constructed on the base and height of the segment. 

e) What is approximately the energy expended to perform a tensile test up to fracture of a specimen having a gauge length of 30 cm and a diameter of 5 cm?  Recall that a Joule = 1 N.m.

f) If the specimen is unloaded from the point at the onset of strain hardening what will be the final gauge length?

Statically Determinate Problems

3.   In the figure, bar bd is rigid, and cable ac has the properties shown.

a) Name the equilibrium equation that yields the force in cable ac.

b) Name the type of motion that bar bd can have.

c) How does knowledge of the force in cable ac allow to determine the displacement at a?

d) How does knowledge of the displacement of point a allow to determine the displacement of any other point of the bar?

e) Describe the sequence of operations by which to determine the displacement at d caused by a given load P. 

4.  In the figure, bar ab is rigid, and cables ac and bd have the properties shown. How should the cable flexibilities be related if
a) bar ab is to remain level?

b) bar ab is to tilt clockwise?

c) bar ab is to remain remain level, and P is applied at the quarter point from a?

Statically Indeterminate Problems

5.  The steel rod and aluminum sleeve assembly may be modeled as two springs in parallel whose axial stiffness k is the sum of the constituent stiffnesses k_st and k_al.  Assuming that the cross sectional areas and the moduli of elasticity are related as follows, 

A_st = 1.5A_al

E_st = 3E_al

a) What is the ratio of  each spring stiffness to the stiffness of the assembly ?

b) If an axial load P = 10 k. is applied at the ends, what will be the axial forces in the steel tube and the aluminum sleeve?

6.  For the three-bar truss,

a) Write the equilibrium equations governing the forces in the three bars.

b) Write the deformation-displacement relations expressing the bar elongations in terms of the displacements (u, v) of joint A.

c) Write the force deformation relations, assuming the bars have the same EA.

d) The equations written above should form a system of as many equations as unknowns.

 How does the force method proceed?

How does the displacement method proceed?

Thermal Stresses

7.   The two bar system is maintained between two rigid walls.  If the temperature of the left bar is increased,

a) Will the left bar be in tension or compression?   Will its length increase or decrease?

b) Answer the same questions for the right bar.

c) If the temperature of both bars is increased by the same amount, and assuming A₁ > A₂, will the joint of  the two bars move to the left or to the right?

8.   The sleeve-rod assembly is maintained between two rigid plates. If the temperature of the system is increased by a uniform amount, and given that the coefficient of thermal expansion of aluminum is larger than that of steel, will the aluminum sleeve be in tension or in compression? What will be the state of the steel rod?