An Stress-Strain Analysis for Elasticity Equations Assignment

Assignment Task

Question

1. For the plane-stress condition given below:

σx = 400 MPa σy = 500 MPa τxy = - 30 MPa

• Draw the element clearly showing its coordinate system and indicate all acting stresses 
• Construct a Mohr’s circle of stress and find the principal stresses and the orientation of the principal axes relative to the x,y axes 
• determine the stresses on an element, rotated in the x-y plane 60° counter- clockwise from its original position as shown above
• Show these stresses on a sketch of an element oriented in comparison to the original element drawn in part (a) of this question
• Using the matrix transformation law to compare the principal stresses with the Mohr’s circle ones 
• Obtain the corresponding strain matrix considering that it is a plane stress condition and that the linear elastic material has Young Modulus E=195 GPa and Poisson’s ratio n=0.32 
• Explain whether the strain matrix is 3x3 or 2x2 and justify your
• Based on the above answer, sketch the strain element

2. The state of stress at a point of an elastic solid is given in the x-y- z coordinates by:

• What type of state is this – plane stress or plane strain? Justify your answe
• Using the matrix transformation law, determine the state of stress at the same point for an element rotated about the x-axis 60° clockwise from its original position;
• Obtain the respective strain matrix if the linear elastic material has Young Modulus E=195 GPa and Poisson’s ratio n=0.32;
• Calculate the strain invariants and write the characteristic equation for the strain at the new rotated position
• Is the transformed strain equation different from the on in the original position? Justify your answer
• Calculate the principal stresses and the absolute maximum shear stress at the point
• What are the stress invariants and the characteristic equation for the transformed state of stress
• What are the principal strains and how many are they?

3. The 3-dimensional state of strain at a material point in x, y, z coordinates is given by:

• Explain what type of strain state is represented by this matrix 
• Draw the element clearly indicating the coordinate axes
• Calculate the volumetric strain and the deviatoric strain invariants 
• Write the characteristic equation of strain for this element 
• Draw the Mohr’s circle for the strain state clearly indicating the coordinate axes and their respective Report the principal strains and maximum shear strain
• Calculate the mean stress and the deviatoric stress tensor if the linear elastic material has Young Modulus E=195 GPa and Poisson’s ratio n=0.32
• Write the characteristic equation of stress.

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