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Civil - Strength of Materials - State of Stress in Three Dimensions

__STATE OF STRESS IN THREE DIMENSIONS__

1.
What are the types of failures?

Brittle failure:

i.Failure of a
material represents direct separation of particles from each other, accompanied
by considerable deformation.

ii. Ductile failure:

Slipping of particles accompanied, by
considerable plastic deformations.

2.List out different theories of failure

i.
Maximum Principal Stress
Theory. ( Rakine's theory)

ii.
Maximum
Principal Strain Theory. ( St. Venant's theory)

iii. Maximum Shear Stress Theory. ( Tresca's theory or Guest's theory )

iv.
Maximum Shear Strain Theory. (Von -Mises- Hencky theory or Distortion
energy theory)

v.
Maximum Strain Energy Theory. (Beltrami Theory or Haigh's theory)

3. Define: Maximum Principal Stress Theory. (Rakine's theory)

According to this theory, the failure of the
material is assumed to take place when
the

value of the
maximum Principal Stress (? _{1}) reaches a value to that of the elastic limit
stress( f _{y}) of the material. ? _{1} = f _{y.}

4. Define: Maximum Principal Strain
Theory. ( St. Venant's theory)

According to this
theory, the failure of the material is assumed to take place when the value of
the maximum Principal Stain (e _{1}) reaches a value to that of the
elastic limit strain( f _{y} / E) of the material.

5. Define : Maximum Shear Stress
Theory. ( Tresca's theory)

According to this
theory, the failure of the material is assumed to take place when the maximum
shear stress equal determined from the simple tensile test.

6. Define : Maximum
Shear Strain Theory (Von -Mises- Hencky theory or Distortion energy theory)

According to this
theory, the failure of the material is assumed to take place when the maximum
shear strain exceeds the shear strain determined from the simple tensile test.

In 3D, shear strain energy due to
distortion U = (1/ 12G)[ ( ? _{1} - ? _{2})^{2} + ( ? _{2} - ? _{3}) ^{2} + ( ? _{3} - ? _{1}) ^{2} ]

Shear strain energy due to simple tension, U = f
_{y} ^{2} / 6G

7. Define: Maximum Strain Energy Theory
(Beltrami Theory)

According to this
theory, the failure of the material is assumed to take place when the maximum
strain energy exceeds the strain energy determined from the simple tensile
test.

8.
What are the theories used for
ductile failures?

1.
Maximum Principal Strain Theory. ( St. Venant's theory)

2.
Maximum Shear Stress
Theory. ( Tresca's theory)

9.
Write the limitations of Maximum
Principal Stress Theory. (Rakine's theory)

1. This theory
disregards the effect of other principal stresses and effect of shearing
stresses on other planes through the element.

2. Material in
tension test piece slips along 45^{0} to the axis of the test piece,
where normal stress is neither maximum nor minimum, but the shear stress is
maximum.

3.Failure is not a brittle, but it is a cleavage
failure.

10. Write the limitations of Maximum Shear
Stress Theory. ( Tresca's theory).

This theory does
not give the accurate results for the state of stress of pure shear in which
the maximum amount of shear is developed (in torsion test).

11.Write the
limitations of Maximum Shear Strain Theory.(Von -Mises- Hencky theory or Distortion energy theory).

It cannot be
applied for the materials under hydrostatic pressure.

12. Write the limitations of Maximum Strain
Energy Theory. ( Beltrami Theory).

This theory does
not apply to brittle materials for which elastic limit in tension and in
compression are quite different.

13.
Write the failure theories and its
relationship between tension and shear.

1.
Maximum Principal Stress Theory. ( Rakine's theory) ? _{y} = f _{y}

2.Maximum Principal Strain Theory. (
St. Venant's theory) ? _{y} = 0.8 f _{y}

3. Maximum Shear Stress Theory. (
Tresca's theory) ? _{y} =0.5 f _{y}

4.Maximum Shear
Strain Theory ( Von- Mises - Hencky theory or Distortion energy theory) ? _{y}= 0.577 f _{y}

5.
Maximum Strain Energy Theory. (
Beltrami Theory) ? _{y}=
0.817f _{y} .

14.
Write the volumetric strain per unit
volume.

f _{y} ^{2} / 2E

20. Define : Octahedral Stresses

A plane, which is
equally inclined to the three axes of reference, is called octahedral plane. The
normal and shearing stress acting on this plane are called octahedral stresses.

20.
Define: Plasticity ellipse.

The graphical
surface of a Maximum Shear Strain Theory (Von -Mises- Hencky theory or Distortion energy theory) is a straight
circular cylinder. The equation in 2D is

? _{1}^{2} - ? _{1}? _{2} + ? _{2}^{2} = f _{y} ^{2} which is called the
Plasticity ellipse

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