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Studies in Mechanics of Solids
Homework 5
1.- Show that the partial differential equation
φ,xxxx + 2 φ,xxyy + φ,yyyy = 0
where the stress function is defined such that
σxx = φ,yy ; σyy = φ,xx and σxy = -φ,xy
satisfies the equilibrium and compatibility equations in rectangular Cartesian coordinates under
plane stress conditions.
2.- Show how the superposition of the stresses resulting from a fourth order polynomial
expression for φ and a pure shear lead to the solution of the problem of a cantilever beam loaded
at one end.
3.- Consider a cantilever beam with the following characteristics: L=1 m; b=1 m; h = 0.1 m; E =
2e11 Pa; ν = 0.3; P = 104 N.
a) Use the solution above to explore the displacement field ux, uy and explain your findings.
b) Use the small strain-displacement relationships to obtain expressions for the strain tensor
components exx, eyy, exy . Explore the strain field and explain your findings.
c) Use Hooke’s law to obtain expressions for the stress tensor components σxx, σyy, σxy. Explore
the stress field and explain your findings.
d) Obtain an expression for the stress function, explore it and explain your findings.
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