We were discussing, Stress analysis of bars of composite sections, Stress and strain in strength of materials, Strain energy stored due to impact loading in our previous posts.

Today we will understand here the theories of
failure, in strength of material, with the help of this post.

As we know very well that when a body or component
or material will be subjected with an external load, there will be developed
stresses and strains in the body or component.

As per hook’s
law, stress will be directionally proportional to the strain within the
elastic limit or we can say in simple words that if an external force is
applied over the object, there will be some deformation or changes in the shape
and size of the object. Body will secure its original shape and size after
removal of external force.

Within the elastic limit, there will be no permanent
deformation in the body i.e. deformation will be disappeared after removal of
load.

If external load is applied beyond the elastic
limit, there will be a permanent deformation in the body i.e. deformation will
not be disappeared after removal of load. Component or material or body will be
said to be failed, if there will be developed permanent deformation in the body
due to external applied load.

Theories of failure help us in order to calculate
the safe size and dimensions of a machine component when it will be subjected
with combined stresses developed due to various loads acting on it during its
functionality.

There are following theories as listed here for
explaining the causes of failure of a component or body subjected with external
loads.

3. The maximum shear stress theory

4. The maximum strain energy theory

5. The maximum shear strain energy theory

We will now understand here the maximum shear stress
theory with the help of this article. The maximum shear stress theory is also
termed as Guest and Tresca’s theory and this theory is only used for ductile
materials.

According to the theory of maximum shear stress,
“The failure of a material or component will occur when the maximum value of shear
stress developed in the body exceeds the limiting value of shear stress i.e.
value of shear stress corresponding to the yield point of the material”.

Let us explain the maximum shear stress theory by
considering here one component which is subjected with an external load and we
have drawn here the stress-strain curve as displayed in following figure.

Point A – It is proportionality limit; up to this
point hooks law will be followed.

Point B – Elastic limit, up to this point the
deformation will be elastic.

Point C – Lower yield stress.

Point D – Ultimate stress, it is the maximum value
of stress in stress – strain diagram.

Point E- It is the fracture point, up to this
point the material will have only elastic & plastic deformation ,but at
this point fracture or rupture take place.

If maximum value of shear stress developed in
the body exceeds the value of shear stress corresponding to the point D,
failure will take place.

Therefore in order to avoid the condition of failure
of the component, maximum value of shear stress developed in the body must be
below than the value of shear stress corresponding to the point D.

###
*
Condition
of failure*

*Condition of failure*

Maximum value of shear stress developed in the body
> Yield strength in shear under tensile test i.e. value of shear stress
corresponding to the yield point of the material

Let us consider that Ïƒ

_{1}, Ïƒ_{2}and Ïƒ_{3}are the principle stresses at a point in material and Ïƒ_{t}is the principle stress in simple tension at elastic limit.
Now as we know that maximum shear stress at a point
in the material will be equal to the half of difference between maximum and
minimum principle stress and therefore we will have following equation.

Ï„

_{Max}= (1/2) x (Ïƒ_{1}- Ïƒ_{3})
Let us determine the value of shear stress
corresponding to the yield point of the material. In case of simple tension, Stress
will be available in one direction only and therefore at elastic limit, principle
stresses will be Ïƒ

_{t}, 0 and 0.
Value of shear stress corresponding to the yield
point of the material = (1/2) x Ïƒ

_{t}

*Let us write here the condition of failure*
(1/2) x (Ïƒ

_{1}- Ïƒ_{3}) > (1/2) x Ïƒ_{t}
(Ïƒ

_{1 }- Ïƒ_{3}) > Ïƒ_{t}###
*
Condition
for safe design*

*Condition for safe design*

Maximum value of shear stress developed in the body ≤ Permissible shear stress
(Ï„

_{Per})
Permissible shear stress is basically defined as the
ratio of yield strength in shear under tensile test i.e. value of shear stress
corresponding to the yield point of the material to the factor of safety.

Permissible shear stress = Yield strength in shear
under tensile test / F.O.S

Permissible shear stress = Value of shear stress
corresponding to the yield point of the material /
F.O.S

Permissible shear stress =
(Ïƒ

_{t}/2)/ F.O.S
Permissible shear stress =
Ïƒ

_{t}/ (2 x F.O.S)

*For tri-axial state of stress*
For tri-axial state of stress, we will
have following equation

Do you have suggestions? Please write in comment
box.

We will now discuss the maximum strain energy theory,
in the category of strength of material, in our next post.

###
**Reference:**

Strength of material, By R. K. Bansal

Image Courtesy: Google

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