1. BDA30303
Solid Mechanics II

2. Lecture’s Information
Saifulnizan Jamian E

3. Contents Plane strain analysis Beam deflection Buckling of column
Strain energy
Thick-walled cylinder
Theories of elastic failure

4. Chapter 1
Plane Strain Analysis

5. Transformation of Plane Strain
Plane strain - deformations of the material take place in parallel planes and are the same in each of those planes.
Plane strain occurs in a plate subjected along its edges to a uniformly distributed load and restrained from expanding or contracting laterally by smooth, rigid and fixed supports
Example: Consider a long bar subjected to uniformly distributed transverse loads. State of plane stress exists in any transverse section not located too close to the ends of the bar.

6. Transformation of Plane Strain
State of strain at the point Q results in different strain components with respect to the xy and x’y’ reference frames.
Applying the trigonometric relations used for the transformation of stress,

7. Principal Strains
The previous equations (Eq. 1 and Eq. 3) are combined to yield parametric equations for a circle.
To obtain this property,  is eliminate from both equations. This is done by transposing first term in right side of Eq. 1 to left side and squaring both members of the equation, then squaring both member of Eq. 3, and finally adding member to member the two equations.

8. Principal Strains
Principal strains occur on the principal planes of strain with zero shearing strains.

9. Maximum Shearing Strain
Maximum shearing strain occurs for

10. Example 7.04
In a material in a state of plane strain, it is known that the horizontal side of a 10 x 10-mm square elongates by 4 m, while its vertical side remains unchanged, and that the angle at the lower left corner increases by 0.4 x 10-3 rad. Determine
The principal axes and principal strains, and
The maximum shearing strain and the corresponding normal strain.

11. Answer

12. Exercise
The following state of strain has been measured on the surface of a thin plate. Knowing that the surface of the plate is unstressed, determine
(a) the direction and magnitude of the principal strains,
(b) the maximum in-plane shearing strain,
(c) the maximum shearing strain. (Use v = 1/3 )
 x = -260 y = -60  xy = +480

13. Mohr’s Circle for Plane Strain
The equations for the transformation of plane strain are of the same form as the equations for the transformation of plane stress - Mohr’s circle techniques apply.
Abscissa for the center C and radius R ,
Principal axes of strain and principal strains,
Maximum in-plane shearing strain,

14. Three-Dimensional Analysis of Strain
Previously demonstrated that three principal axes exist such that the perpendicular element faces are free of shearing stresses.
By Hooke’s Law, it follows that the shearing strains are zero as well and that the principal planes of stress are also the principal planes of strain.
Rotation about the principal axes may be represented by Mohr’s circles.

15. Three-Dimensional Analysis of Strain
For the case of plane strain where the x and y axes are in the plane of strain,
the z axis is also a principal axis
the corresponding principal normal strain is represented by the point Z = 0 or the origin.
If the points A and B lie on opposite sides of the origin, the maximum shearing strain is the maximum in-plane shearing strain, D and E.
If the points A and B lie on the same side of the origin, the maximum shearing strain is out of the plane of strain and is represented by the points D’ and E’.

16. Three-Dimensional Analysis of Strain
Consider the case of plane stress,
Corresponding normal strains,
Strain perpendicular to the plane of stress is not zero.
If B is located between A and C on the Mohr-circle diagram, the maximum shearing strain is equal to the diameter CA.

17. Example 7.05
As a result of measurements made on the surface of a machine component with strain gages oriented in various ways, it has been established that the principal strains on the free surface are a = +400  and b = -50 . kNowing that Poisson’s ratio for the given material is v = 0.3, determine (a) the maximum in-plane shearing strain, (b) the true value of the maximum shearing strain near the surface of the component.

18. Answer

19. Latihan
For the given state of plane strain, use Mohr’s circle to determine
(a) the orientation and magnitude of the principal strains,
(b) the maximum in-plane shearing strain,
(c) the maximum shearing strain.
ε x = +60μ , ε y = +240μ , γ xy = −50μ

20. Measurements of Strain: Strain Rosette
Strain gages indicate normal strain through changes in resistance.
With a 45o rosette, ex and ey are measured directly. gxy is obtained indirectly with,
Normal and shearing strains may be obtained from normal strains in any three directions,

21. Sample Problem 7.7
Using a 60 rosette, the following strains have been determined at point Q on the surface of a steel machine base:
1 = 40 2 = 980 3 = 330
Using the coordinate axes shown, determine at point Q,
The strain components x , y and xy,
The principal strains, and
The maximum shearing strain.
Use v = 0.29

23. Latihan

24. Stress-Strain Relationship
For linear elastic materials, the stress-strain relations come from the generalized Hooke’s law. For isotropic materials, the two material properties are Young’s modulus and Poisson’s ratio. Considering an elemental cubic inside the body, Hooke’s law gives