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

2. Symmetric Bending of Beams
A beam is any long structural member on which loads act perpendicular to the longitudinal axis.
Learning objectives
Understand the theory, its limitation and its applications for strength based design and analysis of symmetric bending of beams.
Develop the discipline to visualize the normal and shear stresses in symmetric bending of beams.

3. Pure Bending
Independent of
material model

4. Deformation of symmetric member
Under action of M and M’, the member will bend but will remain symmetric with respect to the plane containing the couples.
except

5. compressed (compression) elongated (tension)
There exists a surface // to the upper and lower faces of the member
(see as a line on the cross-section) where no elongation and the bending
normal stress is zero. This surface is called the neutral axis.

6. The length of arc DE for both deformed and undeformed
L = rq
Consider an arc some distance y above the neutral surface
The length of arc JK can be expressed as
L’ = (r- y)q

7. The deformation d of JK
d = L’-L = (r- y)q - rq = - yq
The normal strain is max when y is the largest

8. Derive flexure formula
Because
From earlier,

9. Derive flexure formula (continued)
This equation can be satisfied only if
The first area moment of the cross section about its NA = 0

10. Derive flexure formula (continued)
Taking the moment about the z axis = 0
is the 2nd area moment of the cross section w.r.t. z axis

11. Discuss flexure stress
Compression
Top Surface (+y)
Tension
Bottom Surface (-y)

12. Not only bending about z axis produces a normal
stress in x direction but also bending about y axis.
If the bending moment is about the y axis, a similar
relationship exists.

13. Example

14. Compute the area moment of inertia

15. A normal stress in the x direction due to Mz

16. A normal stress in the x direction due to My

17. Transverse Deformations due to bending

18. Member of several materials
Assume a bar is made of two different materials bonded together. The bar will deform as previously shown.
The normal strain in x direction will vary linearly with distance from the N.A.

20. Method of transformed section
The resistance to bending would be
the same if each section were made
of the same material ,where the 2nd
material was multiplied by n

21. Example – transformed section to all aluminum

23. Try it for yourself at home Transform section to all steel
Season’s Greeting!
Try it for yourself at home
Transform section to all steel

26. Beams bending analysis
Beams carry loads perpendicular to their longitudinal axis.
Internal shear forces and
bending moments develop
along the span of a beam.
In designing a beam, it is critical to determine the internal
shear force (V) and bending moment distribution (M). This
is accomplished by constructing shear and bending moment
diagrams.

27. Steps in constructing a V and M diagram
In general, the load distribution across the width of the
beam is assumed to be applied uniformly. Therefore,
a beam can be analyzed in 2 dimensions rather than 3.
1. Determine the reactions at each support.

28. Review of support conditions

29. Shear forces and bending moments in beams
Neither half is in equilibrium

30. The force imbalance that exists must be counteracted
so that static equilibrium is maintained. This is done
Through internal forces and moments.

31. Calculating internal V and M as a function of x by isolating
a segment of beam a distance x from the left end whose
width is dx.

32. =
eliminate last term = 0

33. Diagram by inspection V = constant M = f(x) V = f(x) M = f(x2)
V = no effect
M = spike

35. Faculty of Engineering Mahidol University, Thailand
Dr. Wonsiri Punurai (Bo)