Torsion of Non Circular Bars Saint Venants Theory

Dr. Mathew John
Assistant Professor
Department of Mechanical Engineering
Government Engineering College Barton Hill
S4 ME202: ADVANCED MECHANICS OF SOLIDS
Online Class June 2020 –L3
1
Module 5 – Part B
(KTU Syllabus)

Module 5 – Part B
Text Books:
Dr. Mathew John S4 ME - AMOS GEC Barton Hill
2

TORSION OF
GENERAL PRISMATIC BARS – SOLID SECTIONS
3
Ref:S.Jose,Advanced Mechanics of Materials

St. Venant’s Theory
4Ref:S.Jose,Advanced Mechanics of Materials

5Ref:S.Jose,Advanced Mechanics of Materials

Saint Venant’s Method
6Ref:S.Anil Lal ,Advanced Mechanics of Solids

Ref:Advanced Mechanics of Solids by L.S Srinath
The components of displacements are;
7
Sub sin α = y/r and cos α = x/r
Angle ß =θz

The components of displacements are;
Where,
is calledWarping function orTorsion function. It is a scalar function.
8

The components of strain are;
Substitute eqns (a), (b) and © in the above relations,We get
The non-zero strain
components
Ref:Advanced Mechanics of Solids by L.S Srinath 9

Now apply Hooke’s Law
Since,
Now, the non-zero stress components corresponding to the non-zero
strain components are,
Sub. Strain components; ie. eqns. (1) and (2) above;
We get the non-zero stress
components as,

Governing Equation (GE)
from the Equations of Equillibrium
The stress components should satisfy the equations of equilibrium
Sub. the non-zero stress components, eqns.
(3) and (4) in the above third eqn.(5);We get,
(Eq. 5)
11

Governing Equation (GE) forTorsion of arbitrary cross-section
Warping function must satisfy the above GE; the Laplace Equation.
i.e.TheWarping function should be Harmonic
It is a boundary value problem. Hence formulation of a Boundary
Condition (BC) is required for its solution.
(Eq. 6)
12

Boundary Condition ofWarping Function
Equate the resisting traction and external traction
On sub. the non-zero stress components above;
BC in terms of stress
components
(Eq. 7)
13

Substitute direction cosines nx and ny, we get the boundary
condition as,
(Eq. 8)

Torque (T) in terms ofWarping function
16
Shear stresses on the
shaded area are shown
Ref:S.Anil Lal ,Advanced Mechanics of Solids

17

18Ref:S.Anil Lal ,Advanced Mechanics of Solids

19
Torsion Equation

Equation forTorque (any cross section)
20
Assignment/Home work : Obtain the solutions
for (a) Circular (b) Elliptical cross-sections
Hint:Take a suitable warping function and find out J for each case- circular or elliptical

21

Torsion of Non Circular Bars Saint Venants Theory

More Related Content

What's hot

Similar to Torsion of Non Circular Bars Saint Venants Theory

Recently uploaded

Torsion of Non Circular Bars Saint Venants Theory