Theory of structures I module 1

theory of structures I

1. THEORY OF STRUCTURES I
MODULE 1

2. CONTENTS
Analysis of bars of varying sections
Thermal stresses
Stress analysis
Principal Planes & Principal Stresses
Mohr’s Circle

3. ANALYSIS OF BARS OF VARYING
SECTIONS
A bar of different lengths and different diameters is shown in figure. Let
this bar is subjected to an axial load P.
Though each section is subjected to the same axial load P, yet the stresses,
strains and change in lengths will be different. The total change in length will
be obtained by adding changes in length of the individual section.

4. Let, P = Axial load acting on the bar
L
1
= length of section 1
A
1
= Cross sectional area of section 1
L
2
& A
2
= length and cross sectional area of section 2
L
3
& A
3
= length and cross sectional area of section 3
E = Young’s modulus of the bar

5. Then, Stress for the section 1,
σ
1
= ( load / Area of the section 1 ) =
Similarly, Stress for the section 2 and section 3 are given as,
σ
2
= σ
3
=
Strain of section 1,
Strain of section 2 and 3 are,

6. But strain in section 1 = Change in length of section 1 or
Length of section 1
Where dL
1
= Change in length of section 1 = e
1
L
1

=
Similarly, changes in length of section 2 and of section 3 are obtained as
dL
2
= Change in length of section 2 = e
2
L
2
=
dL
3
= Change in length of section 3 = e
3
L
3
=

7. Total change in the length of the bar =
dL = dL
1
+ dL
2
+ dL
3
=
Above equation is used when E of different sections is same. If, E of
different sections is different, then total change in length of the bar is
given by,

8. THERMAL STRESSES

9. THERMAL STRESSES
Thermal stress are the stress induced in a body due to change
in temperature. Let,
L = Original length of the body
T = Rise in temperature
E = Young’s modulus
α = Coefficient of linear expansion
dL = Extension of rod
If the rod is free to expand, then extension of the rod is given
by,
dL = α . T . L

11. Compressive strain = Decrease in length
Original length
= α . T . L
L + α . T . L
≈ α . T . L
L
= α . T
But, Stress = E
Strain
Stress = Strain x E = α . T . E
Load on the rod = Stress x Area = α . T . E . A

12. ASSIGNMENT – SUBMISSION
( 13/07/2015)
Types of external loads
Different types of beams
Different types of supports
relationship between elastic constants
Relationship between modulus of elasticity and modulus of
rigidity
Relationship between Young’s modulus and bulk modulus

13. STRESS ANALYSIS

14. PRINCIPAL PLANES & PRINCIPAL
STRESSES
The planes which have no shear stresses, are known as
PRINCIPAL PLANES
These planes carry only normal stresses
The normal stresses acting on a principal plane, are
known as PRINCIPAL STRESSES

15. METHODS FOR DETERMINING
STRESSES ON OBLIQUE SECTION
Analytical method
Graphical method

16. ANALYTICAL METHOD FOR
DETERMINING STRESSES ON OBLIQUE
SECTION
A member subjected to a direct stress in one plane
The member is subjected to like direct stresses in two
mutually perpendicular directions

17. A MEMBER SUBJECTED TO A DIRECT
STRESS IN ONE PLANE
Normal stress 
n
=  cos
2
θ
Tangential stress 
t
=

 sin2θ
2

18. THE MEMBER IS SUBJECTED TO LIKE
DIRECT STRESSES IN TWO MUTUALLY
PERPENDICULAR DIRECTIONS
Normal stress 
n
= 
1
+ 
2
+ 
1
- 
2
cos2θ
2 2
Tangential stress 
t
= 
1
- 
2
sin2θ
2

19. MOHR’S CIRCLE
Mohr’s circle is a graphical method of finding
normal, tangential and resultant stresses on
an oblique plane

20. MOHR’S CIRCLE WHEN A BODY IS
SUBJECTED TO TWO MUTUALLY
PERPENDICULAR PRINCIPAL TENSILE
STRESSES OF UNEQUAL INTENSITIES

21. MOHR’S CIRCLE WHEN A BODY IS
SUBJECTED TO TWO MUTUALLY
PERPENDICULAR STRESSES WHICH ARE
UNEQUAL AND UNLIKE