1. Solid Mechanics
Course No. ME213

2. Beam Theory Euler Bernoulli Beam
Euler–Bernoulli beam theory (also known as engineer's beam theory or classical beam theory) is a simplification of the linear theory of elasticity which provides a means of calculating the load carrying and deflection characteristics of beams.
ME213 Solid Mechanics

3. Beam Theory
Timoshenko Beam
The Timoshenko beam theory was developed by Stephen Timoshenko early in the 20th century.
The model takes into account shear deformation and rotational inertia effects, making it suitable for describing the behaviour of short beams, sandwich composite beams or beams subject to high-frequency excitation when the wavelength approaches the thickness of the beam. .
ME213 Solid Mechanics

4. Beam Theory
ME213 Solid Mechanics

5. Statically Determinate Beams
Three types of Statically Determinate Beams:
1. Simply supported beams:
3. Cantilever beams:
2. Overhanging beams:
ME213 Solid Mechanics

6. Statically Determinate Beams
ME213 Solid Mechanics

7. Statically Determinate Beams
ME213 Solid Mechanics

8. Statically Determinate Beams
Simply supported beam with central load
ME213 Solid Mechanics

9. Statically Determinate Beams
Simply supported beam with central load
ME213 Solid Mechanics

10. Statically Indeterminate Beams
Simply supported beam with asymmetric load
ME213 Solid Mechanics

11. Statically Determinate Beams
Simply supported beam with asymmetric load
ME213 Solid Mechanics

12. Statically Determinate Beams
Cantilever beam with end load
ME213 Solid Mechanics

13. Statically Determinate Beams
Cantilever beam with uniformly distributed load
ME213 Solid Mechanics

14. Statically Indeterminate Beams
The beam in which the number of reactions exceed the number of independent equations of equilibrium is known as statically indeterminate beam
In statics, a structure is statically indeterminate (or hyperstatic) when the static equilibrium equations are insufficient for determining the internal forces and reactions on that structure.
Based on Newton's laws of motion, the equilibrium equations available for a two dimensional body are
ME213 Solid Mechanics

15. Statically Indeterminate Beams
In the beam construction on the right, the four unknown reactions are VA, VB, VC and HA. The equilibrium equations are:
ME213 Solid Mechanics

16. Statically Indeterminate Beams
Since there are four unknown forces (or variables) (VA, VB, VC and HA) but only three equilibrium equations, this system of simultaneous equations does not have a unique solution.
The structure is therefore classified as statically indeterminate.
Additional techniques such as linear superposition are often used to solve statically indeterminate beam problems.
The superposition method involves adding the solutions of a number of statically determinate problems which are chosen such that the boundary conditions for the sum of the individual problems add up to those of the original problem.
Considerations in the material properties and compatibility in deformations are also taken to solve statically indeterminate systems or structures.
ME213 Solid Mechanics

17. Statically Indeterminate Beams
Determinacy: when all the forces in structure can be determined from equilibrium equation , the structure is referred to as statically determinate. Structure having more unknown forces than available equilibrium equations called statically indeterminate.
If n is number of structure parts & r is number of unknown forces:
r = 3n, statically determinate
r > 3n, statically indeterminate
ME213 Solid Mechanics

18. Statically Indeterminate Beams
Statically indeterminate to the 1st degree:
-- one redundant support
Statically indeterminate to the 2nd degree:
-- two redundant supports
ME213 Solid Mechanics

19. Statically Indeterminate Beams
Superposition Principle
We know that the differential equations for a deflected beam are linear differential equations, therefore the slope and deflection of a beam are linearly proportional to the applied loads.
This will always be true if the deflections are small and the material is linearly elastic.
Therefore, the slope and deflection of a beam due to several loads is equal to the sum of those due to the individual loads.
In other words, the individual results may be superimposed to determined a combined response, hence called the Principle of Superposition.
ME213 Solid Mechanics

20. Statically Indeterminate Beams
Superposition Method
Example 1
We can therefore determine:
1. The deflection at point C.
2. The slope at end A.
ME213 Solid Mechanics

21. Statically Indeterminate Beams
Superposition Method
Example 2
We can therefore determine :
1. The slope and deflection at point B.
ME213 Solid Mechanics

22. Statically Indeterminate Beams
The method of superposition is very useful for the reactions at the supports of statically indeterminate beams.
Example
ME213 Solid Mechanics

23. Statically Indeterminate Beams
Applying equations of equilibrium:
Conclusion:
-- This is a statically indeterminate problem.
-- Because the problem cannot be solved by means of equations of equilibrium
ME213 Solid Mechanics

24. Statically Indeterminate Beams
Treating the reaction at B as the redundant support, we have:
We can therefore determine :
1. The reaction force at location “B”.
2. The moment at location “A”.
ME213 Solid Mechanics

25. Statically Indeterminate Beams
ME213 Solid Mechanics

26. Statically Indeterminate Beams
The moment area method and the conjugate beam method can be easily applied for the analysis of statically indeterminate beams using the principle of superposition.
Depending upon the degree of indeterminacy of the beam, designate the excessive reactions as redundant and modify the support.
The redundant reactions are then treated as unknown forces.
The redundant reactions should be such that they produce the compatible deformation at the original support along with the applied loads.
ME213 Solid Mechanics

27. Statically Indeterminate Beams
Moment Area & Conjugate Beam Methods
Example 5.1 Determine the support reactions of the propped cantilever beam as shown in Figure 5.2(a).
Solution:
Let reaction at B is R acting in the upward direction as shown in Figure 5.2(b). The condition available is that the ∆B = 0.
ME213 Solid Mechanics

28. Statically Indeterminate Beams
Moment area method
The bending moment diagrams divided by EI of the beam are shown due to P and R in Figures 5.2(c) and (d), respectively.
ME213 Solid Mechanics

29. Statically Indeterminate Beams
Moment area method
Since in the actual beam the deflection of the point B is zero which implies that the deviation of point B from the tangent at A is zero. Thus,
Taking moment about A , the moment at A is given by
ME213 Solid Mechanics

30. Statically Indeterminate Beams
Moment area method
The vertical reaction at A is
The bending moment diagram of the beam is shown in Figure 5.2(e).
ME213 Solid Mechanics

31. Statically Indeterminate Beams
Conjugate beam method
The corresponding conjugate beam of the propped cantilever beam and loading acting on it are shown in Figure 5.2(f)
The unknown R can be obtained by taking moment about B i.e.
ME213 Solid Mechanics

32. Statically Indeterminate Beams
Example
Determine the reaction at the roller support B of the beam shown
ME213 Solid Mechanics

33. Statically Indeterminate Beams
Solution
ME213 Solid Mechanics

34. Statically Indeterminate Beams
Compatibility Equation
Conjugate beam
method
ME213 Solid Mechanics