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MECHANICS - CASE STUDY SOLUTION


Distance Between Points Before
and After Loading

 

The distance between points A, B, C and O are measured using a laser device before and after the bracket is loaded. The results are listed in the table below.

    Distance (cm)
Pt 1 Pt 2 Initial Final
O A 1.0 1.00150
O B 1.41421 1.41775
O C 1.0 1.00200

These displacements can be used to find the normal strain in the direction of OA, OB, and OC. Then, the strains can be analyzed using the strain transformation equations. Finally, the maximum shear strain can be determined.

     
    Normal Strains


Normal Strain Between Points
 

Since the distance before and after loading are known, the normal strains can be determined as

     

The εOA and εOC are in the direction of the y-axis and x-axis, respectively. Also, the strain εOB is in the direction of the new rotated coordinate x´. This gives

      εOA = εy = 0.00150 = 1,500 μ

      εOB = ε = 0.00250 = 2,500 μ

      εOC = εx = 0.00200 = 2,000 μ

Since strain is generally a very small number, the symbol μ "micro-strain" is commonly used to represent 10-6. This symbol does not represent units, and strain is still unit-less.

     
    Rotating Strains

     
   

The stress element needs to be rotated 45o in the positive direction. Using the stress transformation equations, the stresses in the new x'-y' coordinate system will be,

    

     2,500 μ = 1,750 μ + 0 + γxy/2

     γxy = 750 μ = 0.000750

     
    Maximum Shear Strain

   

The maximum shear strain can be found by

     

     γmax = 450.7μ = 0.0004507

The maximum shear strain will on a plane that is rotated θγ-max. That angle is,

     

     θγ-max = -16.85o

     
   
 
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