3D problems with axial symmetry

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Chapter VII

• 3-D problems with axial symmetry

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3-D problems with axial symmetryPressurized
thick-walled cylinder
Balance equation in terms of displacements
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3-D problems with axial symmetryPressurized
thick-walled cylinder

• Gradient and divergence in cylindrical
  coordinates
• Displacements

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3-D problems with axial symmetryPressurized
thick-walled cylinder

• Strains

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3-D problems with axial symmetryPressurized
thick-walled cylinder

• Stresses (Hookes law in plane strain state)

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3-D problems with axial symmetryPressurized
thick-walled cylinder

• Boundary conditions

r a (inner wall) r b (outer wall)
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3-D problems with axial symmetryPressurized
thick-walled cylinder

• Constants

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3-D problems with axial symmetryPressurized
thick-walled cylinder

• Complete results in plane strain state

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3-D problems with axial symmetryPressurized
thick-walled cylinder

• Plane strain state ? Plane stress state

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3-D problems with axial symmetryPressurized
thick-walled cylinder
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3-D problems with axial symmetryParticular case
thin-walled cylinder
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3-D problems with axial symmetrySpherical
coordinates
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3-D problems with axial symmetryPressurized
thick-walled sphere
spherical coordinates r j q
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3-D problems with axial symmetryParticular case
thin-walled sphere
Thin-wall
a, b, c are constant
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3-D problems with axial symmetrySpherical
coordinates with symmetry of revolution
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3-D problems with axial symmetrySolution to
Kelvins and Boussinesqs problems
Balance equation in terms of displacements Homo
thety ? form of the solution is Equilibrium ?
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3-D problems with axial symmetrySolution to
Kelvins and Boussinesqs problems
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3-D problems with axial symmetrySolution to
Kelvins problem
KELVIN
a ? ?? balance of vertical forces
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3-D problems with axial symmetrySolution to
Kelvins problem
Vertical balance
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3-D problems with axial symmetrySolution to
Boussinesqs problem
BOUSSINESQ
Vertical balance
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3-D problems with axial symmetryBoussinesqs
problem applied to a distributed load
Principle Substitute P with q dA then
integrate (generally numerical integration)
? one gets the general solution to the problem
of stress determination under a foundation

• Assumptions
• the ground is a linear elastic material and
  follows Hookes law
• the vertical load distribution q is known

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3-D problems with axial symmetryBoussinesqs
problem applied to a distributed load
Practically, it can be hard to determine the load
distribution only M and P are known

• Flexible foundation slab linear distribution of
  pressure
• Rigid foundation slab

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3-D problems with axial symmetryParticular case
load distributed over a circle
In plane view OM ? r
Vertical displacement in M (surface)
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3-D problems with axial symmetryHertzs problem
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3-D problems with axial symmetryHertzs problem
Centers of both spheres are getting closer by a
Integral equation for p p(r)
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3-D problems with axial symmetryHertzs problem
Solution diagram of p hemisphere the radius of
which is a
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3-D problems with axial symmetryHertzs problem
hence
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3-D problems with axial symmetryHertzs problem
One can express p0 as a function of the
compression force P and the solution is

• diagram of p is known
• everything is determined (BOUSSINESQ)

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3-D problems with axial symmetryHertzs problem
1st case 2 identical balls (same E,n,R) n
0.3 2nd case Ball on a planar surface
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3-D problems with axial symmetryHertzs problem
2nd case evolution of the stresses along axis z
? the largest tmax is located at z a / 2