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MLE5102 Problem Set 1

Semester II AY 2023/24

Submission Deadline: Week 6 Before Tutorial (Friday, 23rd Feb 2024, 10.59 am)

Problem set 1 submission:

• This Problem Set comprises of 10% of your final grade.

• You should submit your answers to Canvas > Assignment > Problem set 1 .

• You can either type your answers in a word document or write your answers on a physical piece of paper and scan it into a digital PDF copy.

• The due date is 23rd Feb 2024, 10.59 am (before tutorial). Any submission later than this will not be considered for grading.

• There are 4 pages and 4 Ruestions. "Answer all questions and show your workings clearly.

Problem 1 (2 marks)

Figure 1.1 shows a biaxial stress state that involves a tensile stress and a compressive stress of equal magnitude.

(a) Determine the normal stresses on the ↵ and the β surfaces.

(b) Determine the shear stress on the ↵ surface and in the β direction.

Problem 2 (3 marks)

Figure 3.1 depicts a trapezoidal plate of which Young’s modulus is E and Poisson’s ratio is v. The plate is subject to uniform distributions of surface tractions on four of the exterior surfaces of the plate, where the surface tractions in the x-y-z coordinates are given by

The surface tractions result in uniform stresses in the plate.

(a) Determine the stress component σxy.

(b) Write down the stress components that are zero in the x-y-z coordinates.

(c) Determine the stress tensor σ in the x-y-z coordinates.

Problem 3 (3 marks)

The figure below plots a cube of size L of which the displacement u is given by

(a) Determine the strain tensor ✏ in the cube.

(b) Calculate the change of length of the segment AB due to the displacement.

(c) Evaluate the change of angle θ between CD and DE.

Problem 4 (2 marks)

Figure 4.1 shows a cylinder in a smooth rigid container. The elastic properties of the cylinder material are i sotropic, characterized by µ and v. The cylinder is subject to pressure P along the z axis within the linear elastic limit. Because of the rigid container, t he side surface of the cylinder is constrained in the transverse direction, but the surface can slide freely along the wall of the container without friction. Determine the following quantities.

(a) The strains ✏xx, ✏yy, and ✏zz.

(b) The stresses σxx and σyy.