1. Chapter 9
Deflection of Beams

2. 9.1 Introduction -- Concerning about the “deflection” of a beam
-- Special interest: the maximum deflection
-- Design: to meet design criteria

3. 9.1 Introduction or If M is not a constant, i.e. M=M(x) (4.21)
M = bending moment
E = modulus I = moment of inertia
If M is not a constant, i.e. M=M(x)
(9.1) or
-- will be explained in Sec. 9.3

4. y = y(x)

5. 9.2 Deformation of a Beam under Transverse Loading
(9.1)
Since M(x) = -Px
(9.2)

6. Example: a beam subjected to transverse loads
Moment Diagram and the Deformed Configuration:
Mmax occurs at C

7. In addition to M(x) and 1/, we need further information on:
1. Slope at various locations
2. Max deflection of a beam
3. Elastic curve: y = y(x)

8. 9.3 Equation of the Elastic Curve
(9.4)
(9.5)

9. 9.3 Equation of the Elastic Curve
The curvature of a plane curve at Point Q(x,y) is
(9.2)
For small slope dy/dx  0, hence
Therefore,
(9.3)
(9.4)
Fially,

10. -- the governing diff. equation of a beam
-- the governing diff. equation of the “elastic curve”
EI = flexural rigidity

11. (9.4)
Integrating Eq. (9.4) once
(9.5)
Since,
It follows,
(9.5’)

12. (9.5)
Integrating Eq. (9.5) once again,
(9.6) or

13. C1 & C2 are determined from the B.C.s
Boundary conditions:
-- the conditions imposed on the beam by its supports
Examples:

14. Three types of Statically Determinate Beams:
1. Simply supported beams:
2. Overhanging beams:
3. Cantilever beams:

15. 9. 4 Direct Determination of the Elastic Curve
9.4 Direct Determination of the Elastic Curve from the Load Distribution
(9.4)
(9.31)
(9.32)

17. (9.33)

18. 9.5 Statically Determinate Beams
Applying equations of equilibrium:
(9.37)
Conclusion:
-- This is a statically indeterminate problem.
-- Because the problem cannot be solved by means of equations of equilibrium

19. 9.5 Statically Determinate Beams
By adding (1) deflection y = y(x) and  = (x), the problem can be solved.
i.e. five unknowns with six equations

20. Statically indeterminate to the 1st degree:
-- one redundant support
Statically indeterminate to the 2nd degree:
-- two redundant supports

21. 9. 6 Using Singular functions to Determine the
9.6 Using Singular functions to Determine the Slope and Deflection of a Beam
(9.44)
(9.45)

23. (9.46)
(9.47)

24. 9.7 Method of Superposition

25. 9.8 Application of Superposition to Statically Indeterminate Beams

26. 9.9 Moment-Area Theorems
(9.54)
(9.55)

27. (9.57)
(9.56)
(9.59)
(9.60)

28. 9.10 Application To Cantilever Beams And Beams With Symmetric Loads

29. 9.11 Bending-Moment Diagrams By Parts

30. 9.12 Application Of Moment-area Theorems To Beams With Unsymmetric Loadings
(9.61)
(9.62)
(9.63)

31. 9.13 Maximum Deflection

32. 9.14 Use Of Moment-Area Theorems With Statically Indeterminate Beams