Midterm TakeHome

10.04.2017

Midterm Exam MAK523, Foundations of Solid Mechanics Due Due Date Date:: (Apr (April il 17) 17)

1) What is the governing differential equations for 

w0  x  , φ  x  for the TBT if the beam

material obeys the following nonlinear material model:

σ x   Aε x  Bε xm where m is an integer .

2) The compatibility equations for the engineering strain εij is given by

ε ij ,kl  ε kl,ij  ε kj,il  ε il ,kj  0,

(i, j, k, l)= 1, 2, 3

(e.g., see J.Barker, Theory of Elasticity) Consider plane-stress large deformation strain field (namely, E11, E12 and E22) and derive the equivalent relations. Hint: See p.138 of Taber (Nonlinear Theory of Elasticity)

3) Determine the deflection at any point Q under the triangular loading acting on an infinite beam on an elastic foundation.

4) In order to find the equation of equilibrium for a Kirchhoff circular 

 plate loaded symmetrically by a force per unit area q(r), and fixed at the edges, we must extremize the following functional:

where D and ν are elastic constants, and w is the deflection (in the z direction) of the center plane of the plate. Show that the following is the proper governing differential equation for w:

5) Starting with a cubic distribution of the displacements through the plate thickness in terms of  unknown functions   f1 , f 2 , g1 , g 2, h1 , h 2 

determine the functions   f i , g i , hi  in terms of   w0 , φ x ,φ y  such that the following conditions are satisfied:

6) Consider the following circular plate having simply supported circumference. 2

R qo=constant

r Q 

r

h R

R z,wo(r) R

Show that the Kirchoff plate theory yields the following solution for  0  r   αR 4 qO R 4    r  

2  α 2    r    2               wO (r )  4 5 4 ( 2 ) log 2 1 4 ( 1 v ) 4 ( 1 v ) log α α α α α α         64 D   R  1  v    R    

What is σ rr  at r=0 ?

2

2

2