Consider the two-dimensional case described in Exercise 3.22 with no body forces. Show that equilibrium equations are

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Consider the two-dimensional case described in Exercise 3.22 with no body forces. Show that equilibrium equations are identically satisfied if the stresses are expressed in the form:

Ox 22 12: 22  07 Tr 22

where ∅(x, y) is an arbitrary stress function. This stress representation will be used in Chapter 7 to establish a very useful solution scheme for two-dimensional problems.

Data from exercise 3.22

Consider the equilibrium of a two-dimensional differential element in Cartesian coordinates, as shown in the following figure. Explicitly sum the forces and moments and develop the two-dimensional equilibrium equations:

6+ dox + Otxx + Fx = 0 ax  Txy Jtxy + ddy + Fy = 0  y Txy = Tyx doy dy +  %+. Tyx Tyx+ Fy +-- y Fx atyx dy

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