1. Strength of Materials I EGCE201 กำลังวัสดุ 1
Instructor: ดร.วรรณสิริ พันธ์อุไร (อ.ปู)
ห้องทำงาน: 6391 ภาควิชาวิศวกรรมโยธา
โทรศัพท์: 66(0) ต่อ 6391

2. Beam Deflections As a beam is loaded,
different regions are subjected to V and M
the beam will deflect x x
Recall the curvature equation
The slope (q)
& deflection (y) at any spanwise location
can be derived

3. A cantilever with a point load
The moment is M(x) = -Px
The curvature equation for this case is
The curvature is at A and finite at B
Obviously, the curvature can be related to displacement

4. The slope & deflection equation
The following relations are established
For small angles,

5. Double Integration method
Multiply both sides of this equation by EI and integrate
Slope eqn
Where C1 is a constant of integration, Integrating again
Deflection eqn
C1 and C2 must be determined from boundary conditions

6. Boundary conditions Cantilever beam Overhanging beam
Simply supported beam

7. Statically indeterminate beam
The reactions at the supports cannot be completely determined from the 3 eqns.
The degree to which a beam is statically indeterminate is identified by a no. of reactions that can’t be determined.
To determine all reactions, the beam deformations must be considered.

8. Statically indeterminate to 1st degree
From the FBD shown, one can write
Obviously not enough to solve for Ay, MA, and B

9. (1) Begin by drawing a FBD diagram for an arbitrary segment of beam
For beam segment AC, summing moment about C we write
The curvature equation is determined from
Integrating twice in x, we have

10. (2) Using the B.C.s x=0, q=0, and y=0 the constants C1 = C2 = 0
Using the 3rd B.C. (x=L, y=0) the above equation becomes or
(2)

11. The 2 equations generated from the original FBD of the beam and the equation generated by considering a BC at x=L from the curvature equation:
2 equations and 2 unknowns, they are now solved to determine the reactions at all supports
Ans

12. Statically indeterminate to 2nd degree
From the FBD shown, one can write
Obviously not enough to solve for Ay, MA, By and MB

13. For beam segment AC, summing moment about C we write
The curvature equation is determined from
Integrating twice in x, we have

14. Using the B.C.s x=0, q=0, and y=0 the constants C1 = C2 = 0
Using the 3rd B.C. (x=L, q=0, y=0) the above equation becomes or

15. 2 equations and 2 unknowns, they are now solved to determine the reactions at all supports
Ans

16. Superposition
The slope & deflection at any point of the beam may be determined by superposing the slopes and deflections caused by the distributed and concentrated loads.

17. Superposition (cont.)

18. Superposition (cont.)

19. Superposition (cont.)
The method of superposition can be used for many loading conditions, provided the equations for the slope and deflection associated with a specific type of load are known. For example,

20. Tables
Click on each beam in which u are interested to see the curvature equation, max deflection, and slope for each loading condition (You should start working on it before seeing!)

22. Singularity functions
Instead of using the curvature equation, one can also use what so called the singularity functions.
For the beam shown above, the bending moment in
terms of singularity functions by

23. Integrate twice to get

25. Next Week 1. Quiz 2. Continuation of Beam Deflections
- theorems of area moment method
- moment diagrams by parts