Stress state

Homework 1 (Stress state) All calculations were performed in Mathematica 5.2, the computer code necessary to solve complicated calculations is provided.
Task 1.1 For the state of plane stress in the(x,y) coordinates MPa determine a) the principal stresses. The eigenvalue of the stress matrix is defined by the following equation:

Solving the preceding equation we obtain the followingprincipal stresses

b) the orientation of principal planes and The orientation of the principal planes are the eigenvectors of the stress matrix, solving the stress matrix for the eigenvectors values, wehave:

c) the maximum shearing stress. The Maximum shear stress is defined by the following equation:

Task 1.2 For the state of plane stress in the (x,y) coordinates MPa determine

a) the normaland shear stress on the plane with the unit normal vector that is rotated by angle counterclockwise to the x axis, Since the magnitude of the By definition: angle is not given, a general solutionshould be given

Analogously for the other normal stresses and shear stress we obtain:

b) sketch the graph of the normal and shear stress as functions of angle in the interval and [0,180°] and

c)estimate from the graphs the principal stresses and the orientation of principal planes As can be seen in the graph, the maximum values found in the functions of normal stresses and the value of theyare, provides us with the magnitudes of the principal stresses and their angle. where

According to the graph, the value of where the shearing stress is cero is approximated 53° . The Maximum valueof the functions can be seen and have the value of 70 MPa for normal stress and 30 MPa for shear stress Task 1.3 For the three-dimensional stress state in the (x,y,z) coordinates, the stresscomponents (in MPa) are:

a) Show that the normal component of the stress vector on a plane with normal in the direction

has magnitude 1.

First of all, the normal vector must be normalized, the...