Plate with Applied Loads
and Displacement

A 0.5 cm thick rectangular plate is pulled in tension by two loads in the x and y directions as shown. The total deflection in the x and y direction is 0.021 cm and 0.009 cm, respectively. What is the Poisson's ratio of the material? The z direction has no load and the deflection is not known.


Stresses and Strains on Plate


This problem involves loading from two directions, and thus requires at least the 2-D Hooke's Law. The 3-D Hooke's Law could be used, but since σz is zero, those equations will reduce to the 2-D equations. The equations are,

     εx = (σx - ν σy)/E
     εy = (σy - ν σx)/E

The strains and stresses in the x and y direction need to be calculated.

     σx = Px/Ax = 5/[(0.05)(0.005)] = 20 MPa
     σy = Py/Ay = 9/[(0.10)(0.005)] = 18 MPa

     εx = 0.021/10 = 0.0021 cm/cm
     εy = 0.009/5 = 0.0018 cm/cm

Substituting the stresses and strains into the 2-D Hooke's Law equations, gives

     0.0021 E = 20 - ν 18
     0.0018 E = 18 - ν 20

Solving for ν gives,

     ν = 0.1876

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