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Problem Set 1

MECH4900 - T3 2022

Problem 1:

A 200-mm-long rod with a diameter of 2.5 mm is loaded with a 2000 N weight. If the diameter decreases to 2.2 mm, compute the following:

a) The final length of the rod.

b) The true stress and true strain at this load.

c) The engineering stress and strain at this load.

Hint: You can solve this using the given information, but you will need to make an assumption about what kind of deformation is happening.

Problem 2:

a) Compare the engineering and true strain for a rod deformed elastically from 10 mm to 10.1 mm in length.

b) Compare the engineering and true strain for a rod deformed plastically from 10 mm to 16.5 mm in length.

c) Please comment on when it is appropriate to use the engineering strain and why.

Problem 3:

A platform is suspended by two parallel rods as shown in the sketch. The platform is constrained so that each rod experiences the same strain. Rod A is 4340 steel [Q + T (650°)] (E = 210 GPa, σys = 855 MPa). Rod B is 7075-T6 aluminum alloy (E = 70 GPa, σys = 505 MPa). Both rods are 1.28 cm in diameter.

a) What load can be applied to the platform before yielding will occur?

b) Which rod will be the first to yield?

Problem 4:

A 2024-T3 aluminum alloy possesses yield and tensile strengths of 345 and 485 MPa, respectively, with a modulus of elasticity of 70 GPa. An unknown load causes a 400 mm long and 15 mm diameter rod to extend by 4 mm. If possible, calculate the load. If not, give an estimate or bound and state why an exact value can’t be given.

Problem 5:

Consider a 2-D material element with stresses applied such that

Gij = [5(15) 5(5)] MPa

Plotted as a 2D stress element, this looks like:

G22 = 5 MPa

G12

= G21

G11 = 15 MPa

a) What are the principal stresses?

b) What is the maximum shear stress?

c) At what angles are the principal stresses and maximum shear stress oriented with respect to the original coordinate axes?

Problem 6:

For equations 2.8, 2.9, and 2.11 in your textbook, the elastic stress concentration kt increases toward infinity as the notch root radius or minor ellipse diameter approaches zero. In practice, such high kt values are never experienced. Why?

Problem 7:

a) A 2 m wide panel has a circular through hole with a radius of 1 cm and an elliptical hole with major and minor radii of 1 cm and 0.5 cm respectively. Assuming they are far enough from each other to not interact, which produces the highest stress concentration factor for uniaxial tension?