Combined effect of bending and torsion

IIf a body is subjected to Shear Force, Bending moment and Twisting moments, then the combined stress in the shaft will be calculated by using of principle of superposition.

1.Shear stress due to direct shear:

The average Shear Stress due to direct shear force q = F/A

Where F= Shear Force at the section

A= Area of the cross section, 

But the shear stress distribution in the section will get by using the equaction

qd = FQ/Ib

Where Q= Ay

Where, A = Area between the extreme face of beam and the plane at which the shear stress is q

y = Distance of the centroid of area from N.A

F = Shear force at the cross-section.

I = Moment of Inertia of the beam cross section about N.A.

b = Width of the fiber at the plane at which shear stress is q

2.Shear stress due to Torsion:

For solid or hollow shafts of uniform circular cross-section and constant wall thickness, the torsion relations are:

where:
R is the outer radius of the shaft.
τ is the maximum shear stress at the outer surface.
φ is the angle of twist in radians.
T is the torque (N·m ).
ℓ is the length of the object the torque is being applied to or over.
G  is the shear modulus or modulus of rigidity
J is the polar moment of inertia for a round shaft or concentric tube only.
The shear stress at a point within a shaft is:

qS = TR/J

Note: The combined shear stress in body due to direct shear and torsion is                   q = qd + qs

3.Stresses due to Bending Moment:

Stresses caused by the bending moment are known as flexural or bending stresses. These stresses are calculated by using bending equation,

M/I = f/y = E/R

Where
M = Bending Moment
I = Moment of inertia about Neutral axis (N.A.)
f = Bending stress
y = Distance of the fiber from N.A.
R = Radius of Curvature
E = Young's Modulus

2.Combined effect of bending and torsion

In some applications the shaft are simultaneously subjected to bending moment M and Torque T. 

The Bending moment comes on the shaft due to gravity or Inertia loads. So the stresses are set up due to bending moment and Torque.

Equivalent twisting moment :

A twisting moment which, if acting alone, would produce in a circular shaft a shear stress of the same magnitude as the shear stress produced by a given twisting moment and a given bending moment acting simultaneously.

Equivalent bending moment :

A bending moment which, acting alone, would produce in a circular shaft a normal stress of the same magnitude as the maximum normal stress produced by a given bending moment and a given twisting moment acting simultaneously.