Stress at any section is internal resistance offered by metal against the deformation caused by applied load. It is Internal resistance pre-unit area. When a metal is subjected to a load, it is deformed, no matter how strong the metal. If the load is small, the distortion will probably disappear when the load is removed. If the distortion disappears and the metal returns to its original dimensions upon removal of the load, the strain is called elastic strain. If the distortion disappears and the metal remains distorted, the strain type is called plastic strain

1. Theory of Elasticity and Plasticity
By Ashish Vivek Sukh
M.Tech Structural Engineering
SGT UNIVERSITY

2. Stress
• Stress at any section is internal resistance offered by metal against the deformation caused by applied load.
• It is Internal resistance pre-unit area.
• When a metal is subjected to a load, it is deformed, no matter how strong the metal.
• If the load is small, the distortion will probably disappear when the load is removed.
• If the distortion disappears and the metal returns to its original dimensions upon removal of the load, the strain is
called elastic strain.
• If the distortion disappears and the metal remains distorted, the strain type is called plastic strain.

3. Types of Stress
• There are six types of stress: compression, tension, shear, bending,
torsion, and fatigue.
• Each of these stresses affects an object in different ways and is
caused by the internal forces acting on the object.
• Stresses occur in any material that is subject to a load.

4. stress
Normal Stress σ
(Perpendicular)
Tangential Stress τ (Parallel) or
Shear stress
Direct Normal Stress σ
(Axial Force)
Bending Normal
Stress σ
(B.M.)
Tensil
(+)
Tensil
(+)
Compressive
(-)
Compressive
(-)
Direct Shear Stress τ
(Due to S.F.)
Torsional Shear Stress
τ (B.M.)
Tensil
(+)
Compressive
(-)
Tensil
(+)
Compressive
(-)

5. Direct Normal Stress σ (Axial Force)
• A member is loaded by an axial force.
• The value of the normal force for any prismatic section is simply the
force divided by the cross sectional area.
• A normal stress will occur when a member is placed in tension or
compression.

6. Normal Stress σ (Axial Force)

7. Tensile Stress
• Tensile stress is that type of stress in which the two sections of
material on either side of a stress plane tend to pull apart or elongate
as illustrated in Figure
• Tensile stress is a quantity associated with stretching or tensile forces

8. Tensile Stress

9. +
Beam σ
A
C/A

10. Normal tensile stress elements in x direction
σ
σ

11. Compressive Stress
• Compressive stress is the reverse of tensile stress.
• Adjacent parts of the material tend to press against each other
through a typical stress plane as illustrated in Figure

12. Compressive
Stress

13. -
Beam σ
A
C/A

14. σσ
• Normal Compressive stress elements in x direction

15. Bending Normal Stress (σ) or Flexural
• For simply supported beam.
• Upper surface of the bending beam is in compression.
• Bottom surface is in tension.
• The neutral axis (NA) is a region of zero stress.

16. • For cantlever beam.
• Upper surface of the bending beam is in tension.
• Bottom surface is in compression.
• The neutral axis (NA) is a region of zero stress.

19. Tangential Stress (τ) or Shear stress
• Direction of the deforming force or external force is parallel to the
cross-sectional area,
• Stress experienced by the object is called shearing stress or tangential
stress.
• This results in the change in the shape of the body.

20. Direct Shear Stress τ
• Direct Shear Stress is the component of stress which lies in the same
plane of the geometry of the material.
• Shear stress arises from the force vector component that is parallel to
the cross section of the material.

21. Shear Stress in beam

23. Shear Stress Element

24. Pure Shear Condition

26. Torsional Shear Stress
• Torsional stress is the shear stress produced in the shaft due to the
twisting.
• This twisting in the shaft is caused by the couple acting on it.
• A couple is Two equal and opposite parallel forces acting upon a body
with a different line of acting points said as a couple.
• When a machine member is under the twisting force then it is said to
be the shaft is subjected to torsion.

27. • Consider a shaft is fixed at one end and another end is subjected to
the torque as shown in the figure.
• each and every cross section of the shaft is subjected to the Torsional
shear stress.
• Due to the Circular section of the shaft,
• It has been considered that the shear stress at the centre axis will be
zero and it is maximum at the outer surface of the shaft.

28. From the Torsion equation for a circular member is
Where
• τ = Torsional stress induced at the outer surface of the shaft (Maximum
Shear stress)
• r = Radius of the shaft
• T = Twisting Moment or Torque
• J = Polar moment of inertia
• C = Modulus of rigidity for the shaft material.
• l = Length of the shaft
• θ = Angle of twist in radians on a length “l”

31. Torsional Shear Stress element

32. Fatigue
• Fatigue is damage caused by repetitive stress and strain.
• Fatigue strengh is the point after applying fatigue stress a material
fails or breaks.
• Usually fatigues strength is measured in "cycles to failure".
• A familiar example of fatigue failure is bending a copper wire back
and forth until it breaks.

33. THE END