University of Florida/Eas4200c.f08.radsam.d

HW 4

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For simplification purposes, we look at a a 1-dimensional model first. Equating all forces on the x-direction, we obtain:

 

 

Applying Taylor series to the term inside the brackets, it becomes:  

Recall:  

Neglecting higher order terms:

 

Now, we may look at a non-uniform 3-D field without applied loads, and focusing on the x-direction.

"Picture is worth a thousand words" - Vu-Quoc

The figure shows an infinitesimal element in which the stress is not uniform. However, the element must remain in equilibrium, therefore the six equations of equilibrium must be satisfied. For example, forces along the x-direction are:

  Facets with normal X

  Facets with normal Y

  Facets with normal Z

 

HW 6

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Consider plate of dimensions:

a = dimension along the x-axis
b = dimension along the y-axis

First, we look at a 1-D case:

 


Therefore:

 

 

 

 



From book...pg 71

The vector   vanishes because no loads are applied on the lateral surface:
 

this way, the stress vector may be evaluated on the lateral surface, knowing that  , thus:

 

Therefore, we have:  

 

If we look at Figure 2, we may easily derive:

 

 

So we may express   as:

 

So, the free boundary condition   is given by:   on the lateral surface.

Note: For solid sections with a single contour boundary, this constant may be approximated to zero.

We are interested in the shear stresses   and the resultant torque. Considering a small area  , the torque is:

 

 

  1. "Stiffness". Retrieved 2008-09-10.
  2. "Yield (Engineering)". Retrieved 2008-09-10.
  3. "Yield (Engineering)". Retrieved 2008-09-10.
  4. Sun, C.T. (2006). Mechanics of Aircraft Structures. pp. 2-4.