Fall 2014 AE 5301-005/ME5390-005

Advanced Mechanics of Materials

HW Assignment # 1

Solution

Your 10-digit UTA ID (also known as 1000 number) has the following form:

1000 – XY – LMNP

where X, Y, L, M, N and P are the digits between 0 and 9. Find X, Y, L, M, N and P from your UTA ID to solve some of the following problems:

1. Consider the following stress matrix:

[𝜎]= �𝑋 − 2 𝑌 0𝑌 𝑃 + 1 00 0 0

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Determine the components of the traction vector with respect to an area rotated θ about the z axis. Determine the components of stress transformed an angle θ about the same axis. Have you observed any relations between these two? 2. Determine the principal values and their orientations for the following stress matrix.

[𝜎]= �𝑋 − 2 𝑌 𝑀𝑌 𝑃 + 1 0𝑀 0 1

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3. Under what circumstances is the following symmetric stress field in static

equilibrium? 𝜎𝑥𝑥 = 3𝑥; 𝜎𝑦𝑦 = 4𝑦; 𝜏𝑥𝑦 = 𝐴𝑥 + 𝐵𝑦 (all in MPa)

Find stress vectors on the four sides of the rectangular body with side along x = 2 m and along y = 0.5 m. Using stress-strain relations for plane-stress, determine the deformed shape.

4. A cube of steel of slide 250 mm is loaded with a uniformly distributed pressure of

200 MPa on the four faces having normal in the x and y directions. Rigid constraints limit the total deformation of the cube in the z directions to 0.05 mm. Determine the normal stress, if any, which develops in the z direction.