User:Eas4200c.f08.gator.edwards/ Wk5 Class Notes
4 zero strain components
4 zero stress components
To verify they are inverses, multiply together to get identity matrix
Now we move on to part C of the roadmap: Equilibrium Equation for stresses
Similarly for x.
Moment of Inertia:
|Notes on Determinants -team gator|
A determinant is the scale factor for volume when matrix A is regarded as a linear transformation. To find the determinant of the 2 X 2 matrix A:
Note: the denominator of the above equation is the determinant of the I matrix.
is the determinant of
Neutral AxisBending Moment Stress = 0
Our goal is to prove:
Now let's use indicial notation for the above equation, where x=1, y=2, and z=3:
In indicial notation equation 3 becomes:
Derivation of 
First take the sum of all of the forces in the x direction of our 1-D model.
The term multiplied by A is expanded using a Taylor series expansion. The higher order terms are ignored so it is replaced with the equivalent below.
This is then substituted back into the equation.
The dx can be factored out and the final equation is shown below.
Derivation of Equation 4
Now, let's investigate the non-uniform stress field in 3-D, but without applied load and focusing on the x-direction only (i.e. w/o the other stress components to avoid cluttering the figure)
Note: is the normal stress normal to the i axis facet in the j axis direction.
Prandtl Stress function Φ
The Prandtl stress function is a special case of the Morera stress functions, in which it is assumed that A=B=0 and C is a function of x and y only.
Φ acts as a potential function. (, ) are components of Φ with respect to (y,z)
A brief mention should be made of the Maxwell stress functions. Maxwell stress functions are defined by assuming that the Beltrami stress tensor tensor is restricted to be of the form :
Substituting the expressions for the stress into the Beltrami-Michell equations yields the expression of the elastostatic problem in terms of the stress functions:
These must also yield a stress tensor which obeys the specified boundary conditions.
The stress tensor which automatically obeys the equilibrium equation may now be written as:
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