[SOLVED] AA530 - SOLID MECHANICS  - HW 2

Description

Equilibrium & Cauchy Stress) The figure below shows an infinitesimal triangular component taken from a 2D solid in equilibrium. The slanted surface has an angle  with respect to the vertical line.

• Derive Cauchy’s formula by considering equilibrium of forces (i.e., express T₁ and T₂ in

terms of given stresses and ).

• Calculate normal and shear tractions (i.e., stresses) applied to the slanted surface.

• In which , do we obtain the maximum normal stress? Given ₁ = 30 MPa, ₂ = 10 MPa, and ₁₂ = ₂₁ = –10 MPa, what is this  value and the corresponding maximum stress (0  

< 180)?

• In which , do we obtain the maximum shear stress? Given ₁ = 30 MPa, ₂ = 10 MPa, and

₁₂ = ₂₁ = –10 MPa, what is this  value and the corresponding maximum stress (0   <

180)?

• What is the relationship between the two ’s obtained in 1.3. and 1.4?

• Given ₁ = 30 MPa, ₂ = 10 MPa, and ₁₂ = ₂₁ = –10 MPa, plot the trajectory of normal (xaxis) and shear (y-axis) stresses in an x-y Cartesian coordinate under the variations of  from

0 to 180 degrees (Use Matlab).

• Show that the normal and shear stresses derived in 1.2. are following a circular trajectory under the variation of  (i.e., mathematically derive Mohr’s circle relationship). What are the principal stresses and maximum shear stress?

1

2. [50 points] (Cauchy stress) The stress tensor at a point is given by:

2.1. Find the stress component perpendicular and parallel to the plane with the unit normal vector:

                                                                                   nˆ =(1,  1,   1)/   3

2.2. Determine the principal stresses and the corresponding directions (you can use Matlab).

2.3. Find the maximum shear stress (hint: use relationship between principal normal stresses and

maximum shear stresses, e.g., the information in Problem 1.7).

2.4. Find hydrostatic and von-Mises stresse