16.26 Problem Set #4	3/14/97
	Due 3/21/97

1. Develop the constitutive laws (load-strain in-plane and moment-curvature out-of-plane, including DT(z)) for the following beams-
   
   
   1. a "plane stress" beam- start with the three-D relations, and assume that all 2 and 3 components of stress are zero
   2. a "plate" beam- start with the plate relations and assume that Ny Nxy and My and Mxy are zero
   3. a "large deflection plate" beam- as b, but assume Ny Nxy ky and kxy are zero
      
      

   Develop them generically for anisotropic material, then reduce in terms of E and n for isotropic materials
   
   

2. Use one of your models above to analyze the following. Justify your choice in light of the answer.
   
   A steel beam 1 m long, .1 m wide, and .01 m thick, is cantilevered from one end. It is initially at 400°C. Cold water hits the top surface, T = 10°C, hc = 5000 w/m2K; assume the other surface is insulated. Describe what happens. Plot the axial stress as a function of z at a number of representative times. Plot the tip deflection as a function of time.
   
   
3. A thin, spinning disk (frisbee) of uniform thickness may be easily analyzed by assuming that it is stationary but seeing a body force (the "centrifugal" force)
   
   
   
   The solution is well known (Timoshenko & Goodier, Theory of Elasticity, Art. 32 pp. 80-83) and looks like:
   
   
   
   
   where b is the outer radius of the disk. Now considier the same disk, not spinning, but with a temperature distribution
   
   
   
   Find the stress state in the disk. The equilibrium Eq. in the r directionis below; the stress-strain Eqs. are the same in the r-q plane as they are in the 1-2. A final hint: the shear stresses and strains are zero by symmetry.