Given the following three -dimensional stress state in matrix form determine the magnitudes of the principal stresses. If it is too lengthy , show the steps of how it can be calculated. (14 points) (2.9f) [sigma ]=[} 20&5&0 5&-10&10 0&10&-5 1&(20-lambda ) ] MPa Answer: vert } 20-lambda &5&0 5&-10-lambda &10 0&10&-5-lambda vert (20-lambda )[(-10-lambda )(-5-lambda )-100]-5[5(-5-lambda )]=0

To find the magnitudes of the principal stresses, we need to calculate the eigenvalues of the given stress matrix. The eigenvalues of a matrix A are the values of λ that satisfy the equation:det(A - λI) = 0where I is the identity matrix.In this case, the stress matrix is given by: We can rewrite this matrix in the form A - λI: Now we can calculate the determinant of this matrix:det( )Expanding the determinant along the first row, we get: Simplifying this equation, we get: Expanding and simplifying further, we get: This is a cubic equation, which can be solved numerically or using specialized software. Solving this equation, we find that the eigenvalues (principal stresses) are:λ1 ≈ 5.52 MPaλ2 ≈ -10.52 MPaλ3 ≈ 20.00 MPaTherefore, the magnitudes of the principal stresses are approximately 5.52 MPa, 10.52 MPa, and 20.00 MPa.