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Theory of Elasticity & Plasticity Examples (Part 2)
1. Given a function
−+
−+−=Φ
hy
hyqy
hy
hyqx
3
32
3
32 210
1344
.
a) Prove that the above function Φ is a valid stress function for an elastic thin rectangular plate shown in Fig. Q1, in absence of body forces.
b) Determine (i) the stress components and (ii) the resultant forces on four different edges of the plate, respectively.
2. If a function Φ can satisfy the condition 02 =Φ∇ , is the function ( )Φ+ 22 yx a
valid stress function for a 2-D elastic body with constant body forces? 3. For the simply supported narrow beam (i.e. the width is much small than the depth)
under loading shown in Fig. Q3, determine the stress function Φ for this beam and the stress field (i.e. stress components) throughout the beam?
4. A thick-wall cylindrical ring (i.e. in plane stress condition) has an inner radius a
and an outer radius b . It is subjected to the internal pressure p only. a) Determine the stress field in this cylindrical ring. b) If the initial uniaxial yield stress is ypσ and there is only the internal pressure p ,
at which location will the ring be subjected to yielding first? Determine the corresponding “yield” internal pressure. Assume the Tresca yield criterion.
Fig. Q1
a
h/2 x O
h/2
y
Fig. Q3
l/2
h/2
x O
h/2
y
l/2
ql/2 ql/2
q
h
1
5. Consider two short cylinders (i.e. under plane stress condition) made of the same
material shrink fit on each other. Before they were shrunk, the outer radius (b ) of the inner cylinder was bigger than the inner radius ( c ) of the outer cylinder by δ . The compound cylinder is then subjected to an internal pressure p . Denote that a and b are the inner and outer radii of the inner cylinder, respectively; c and d are the inner and outer radii of the outer cylinder. a) Determine the interface pressure between these two cylinders, and the
corresponding stress fields in the cylinders, before the internal pressure is applied.
b) Determine the final stress fields in these two cylinders after the internal pressure is applied.
6. Determine the typical displacement functions for the following four different types
of beams: a) simply supported beams, b) fixed-end beams, c) cantilever beams, d) beams with one fixed end and one simply-supported end. Assume the origin of coordinate system for each beam is located at the left-end support.
7. A rectangular plate is simply supported on four sides as show in Fig. Q7. The plate
is subjected to a set of body forces ( gFF yx ρ−== ,0 ). Determine the stress field (i.e. stress components) in this rectangular plate, using Rayleigh-Ritz method.
Fig. Q7 a
b
x O
Fx
y
Fy
Simply Supported (S.S.)
S.S. S.S.
S.S.