Assienment: Measurement with strain gauge Background The strain state at a point

Question

Assienment: Measurement with strain gauge Background The strain state at a point on a surface is given by three strain components. Usually these are denoted a, ey, and " The numerical values depend on the orientation of the coordinate system. Change the orientation and you get other values even though the values represent the same state of strain. This is analysed and visualized with Mohr's circle. Using Mohr's strain circle it is possible to calculate the strain state in an arbitrarily oriented coordinate system at a point where& Gy, and oy are known in one coordinate system. Techniques to measure the state of strain are based on these facts. It is possible measure the normal strain with a strain gauge, cf.http://en.wikipedia.org/wiki/Strain gauge If we use a strain gauge with a specific orientation we measure the value of the normal strain in that direction. It is obviously not possible to determine three strain components&y;, and hy from one strain component, cf. the course in Linear Algebra. We only get one equation from Mohr's strain circle to determine the three strain components. The trick is to use three strain gauges at the point. For each measured value we get one equation from Mohr's strain circle. With three measured values, we get three linear equations for the three unknowns & &y;, and Fy Assignments 1. Write a MATLAB function that can be used to calculate the state of strain from measurements of three normal strain components measured in three directions at a point. Input to the program is three normal strain components(a, , &) and the corresponding directions (a, , ) in which the strains are measured. The angles easured in degrees counter clockwise relative the x axis. Output of the program is the state of strain &, y, and y Write a MATLAB function that, for a linearly elastic material and a state of plane 2. atrix representation of the stress tensor if the strains in the xy plane and material parameters are known. Input to the program is the three strain components (aBy, M), Young's modulus E and Poisson's ratio v. Output from the program is the matrix representation of the stress tensor S a MATLAB function that for a given state of stress calculates the effective stress due to von Mises. Input to the program is the matrix representation of the stress tensor S. Output from the program is the von Mises effective stress or S. Output from the program is the von Mises effective stress o.

Explanation / Answer

1) Matlab Code

  

function [ strain_xy ] = strain( e1,e2,e3,a1,a2,a3 )
% strain_matrix stores input strain values
% transform_matrix stores vector transformation values
% strain_xy stores output strain values

strain_matrix = [e1 e2 e3]';
transform_matrix = [0.5*(1+cos(2*a1)) 0.5*(1-cos(2*a1)) sin(2*a1);0.5*(1+cos(2*(a1+a2))) 0.5*(1-cos(2*(a1+a2))) sin(2*(a1+a2));0.5*(1+cos(2*(a1+a2+a3))) 0.5*(1-cos(2*(a1+a2+a3))) sin(2*(a1+a2+a3))];
strain_xy = inv(transform_matrix)*strain_matrix;
end

2) Matlab code

function [ ] = plane( exx,eyy,txy )
E = 208*10^9; %youngs modulus
v = 0.29;      %poissons ratio
sigmax = E/(1-v^2) * (exx + v*eyy);   %plane stress in x direction
sigmay = E/(1-v^2) * (v*exx + eyy);   %plane stress in y direction
sigmaxy = E/(2*(1+v))*txy;   %plane stress in xy direction
disp(sigmax);
disp(sigmay);
disp(sigmaxy);
end

3) Matlab Code

function [ vonmises_stress ] = vonmises( str_ten )
% str_ten is the input stress tensor
%sigma1 and sigma2 are the principal stresses
%vonmises_stress is the equivalent von mises stress
sigma1 = 0.5*(str_ten(1)+str_ten(2)) + sqrt((0.5*(str_ten(1)-str_ten(2)))^2+str_ten(3)^2);
sigma2 = 0.5*(str_ten(1)+str_ten(2)) - sqrt((0.5*(str_ten(1)-str_ten(2)))^2+str_ten(3)^2);
vonmises_stress = sqrt((sigma1^2 + sigma2^2 + (sigma1-sigma2)^2)/2);
end

4) Attach/send the link to the railwy_data.txt file with the question as well.

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