Mechanics of Materials_Chapter IV_Analysis and Design of Beams for Bending.pdf

Mechanics of Materials

1. Mechanics of Materials
Lecturer: Van Tang NGUYEN
Email: [email protected]
Aeronautics
Department
1

2. Chapter 4
Analysis and
Design of Beams
for Bending
2

3. free-body diagram of the bar
Bending
3
C D
C D
Transverse loading causes only
bending and shear in the beam.

4. The transverse loading of a beam may consist of
Bending
Concentrated loads
Distributed loads
Or the combination of both

5. Beams are classified according to the way they are supported:
Beam Classification
distance L is called the span.

6. Bending
When a beam is subjected to transverse loads, the
internal forces in any section of the beam consist
of a shear force V and a bending couple M
The bending couple M creates normal
stresses in the cross section, while the shear
force V creates shearing stresses

7. Distribution of the normal stresses in a given
section :
Bending
??????
�=
�??????
??????
=
�
??????
the normal maximum stress:
??????
�=−
��
??????
stress at any given point:
I: is the moment of inertia
S: Elastic section modulus
For rectangular shape �=
??????
�
=
1
6
�ℎ
2
=
1
6
??????ℎ

8. ??????
�=
�??????
??????
=
�
??????

Bending
??????
� ~ �
 determination of the location and magnitude of the largest M
 A bending-moment diagram where the bending moment M is
determined at various points of the beam and plotted against the distance x
measured from one end.
C D
x
 A shear diagram is drawn by plotting the shear V against x.
C D

9. Shear and Bending moment diagram
the shear and bending-moment diagrams
are obtained by:
 Determining the values of V and M at
selected points of the beam
by passing a section through
the point to be determined
 Considering the equilibrium of the
portion of beam located on either side
of the section

10. Sign Conventions
Shear and Bending moment diagram
A positive shear force acts clockwise against the
material
and a negative shear force acts counterclockwise
against the material
A positive bending moment compresses the
upper part of the beam
and a negative bending moment compresses
the lower part.

11. Sign Conventions
Shear and Bending moment diagram
(+)
(+)
(+)

12. Application
Shear and Bending moment diagram
Draw the shear and bending-moment diagrams for a simply supported
beam AB of span L subjected to a single concentrated load P at its
midpoint C

13. Shear and Bending moment diagram
1. Determine the reactions at the supports from the free-body diagram of
the entire beam
The magnitude of each
reaction is equal to P/2
2. Cut the beam at a point D
between A and C and draw the
free-body diagrams of AD and DB
Assuming that the shear and
bending moment are positive
Application

14. Shear and Bending moment diagram
Application
3. Consider the free body AD
??????=+??????/2
M=+??????�/2
Plot V and M between A and C
L
C A
C A

15. Cutting the beam at a point E
between C and B and considering
the free body EB
Shear and Bending moment diagram
Application
??????′=−??????/2
M′=??????(�−�)/2
C
C B
B
the shear is negative, and the bending
moment positive

16. Shear and Bending moment diagram
Application

17. Shear and Bending moment diagram
Application
Draw the shear and bending-moment diagrams for a cantilever beam AB
of span L supporting a uniformly distributed load w

18. Cut the beam at a point C, located
between A and B, and draw the free-
body diagram of AC
Application
Shear and Bending moment diagram
Using the distance x from A to C
and replacing the distributed load
over AC by its resultant wx applied at
the midpoint of AC

19. Application
Shear and Bending moment diagram
↑ ??????
�=0;
−��−??????=0
??????=−��
↺ �
�=0; w�
�
2
+�=0;
�=−
1
2
��
2
;

20. Application
Shear and Bending moment diagram
??????=−��
�=−
1
2
��
2
;

21. 1. Draw the shear and bending-moment diagrams, (b) determine the
equations of the shear and bending-moment curves.
EXERCISES

22. 2. Draw the shear and bending-moment diagrams, (b) determine the
equations of the shear and bending-moment curves.
EXERCISES

23. 3. Draw the shear and bending-moment diagrams, (b) determine the
equations of the shear and bending-moment curves.
EXERCISES

24. Relationships Between Load,
Shear, and Bending moment
The construction of the shear diagram and, especially, of the bending-moment
diagram will be greatly facilitated if certain relations existing between load,
shear, and bending moment can be established
Consider a supported beam AB is carrying a
distributed load � per unit length
Shear and bending moment at C is denoted
by V and M, respectively, assumed to be
positive.
 shear and bending moment at C’ is
denoted by V+∆V and M+ ∆ M.

25. Relationships Between Load,
Shear, and Bending moment
Detach the portion of beam CC’ and draw
its free-body diagram
The forces exerted on the free body include a
load of magnitude w∆x and internal forces and
couples at C and C’

26. Relationships between Load and Shear.
Relationships Between Load,
Shear, and Bending moment
The sum of the vertical components of the
forces acting on the free body CC’ is zero
↑ ??????
�=0;
??????−??????+∆??????−�∆�=0
∆??????=−��
Dividing both members of the equation by
∆x and then letting ∆x approach zero,
�??????
��
=−�
(is not valid at a point where a
concentrated load is applied)

27. Integrating between points C and D,
Relationships Between Load,
Shear, and Bending moment
??????
�−??????
�=− ���
�??????
�??????

is not valid when concentrated loads are
applied between C and D, since they do not
take into account the sudden change in shear
caused by a concentrated load
??????
�−??????
�= -( area under load curve between C and D)

28. Relationships between Shear and Bending Moment.
Relationships Between Load,
Shear, and Bending moment
↺ �
�=0;
�+∆�−�−??????∆�+�∆�
∆�
2
=0
∆�=??????∆�−�
1
2
(∆�)
2

��
��
=??????
slope dM/dx of the bending-
moment curve is equal to the value
of the shear.
(no concentrated load is applied)

29. �
�−�
�= ??????��
�
??????
�
??????

Relationships between Shear and Bending Moment.
Relationships Between Load,
Shear, and Bending moment
�
�−�
�= area under shear curve between C and D

30. Application
Relationships Between Load,
Shear, and Bending moment
Draw the shear and bending-moment diagrams for the simply
supported beam shown in figure and determine the maximum value of
the bending moment.

31. Relationships Between Load,
Shear, and Bending moment
Application
magnitude of the reactions at the supports:
�
�=�
�=
1
2
��
the shear V at any distance x from A is ??????
�−??????
�=− ���
�
??????
�
??????
Using
??????−??????
�=− ���=−��
�
0

32. ??????=??????
�−��=
1
2
��−��=�(
1
2
�−�)
Relationships Between Load,
Shear, and Bending moment
Application

33. Relationships Between Load,
Shear, and Bending moment
Application
The value M of the bending moment at any distance x from A is:
From �
�−�
�= ??????��
�
??????
�
??????

�−�
�= ??????��
�
0

�= �
1
2
�−���=
1
2
�(��−�
2
)
�
0

�
????????????�=��
2
/8

34. 4. Draw the shear and bending-moment diagrams for the beam and
loading shown, and determine the maximum absolute value (a) of the shear,
(b) of the bending moment.
EXERCISES

35. Design of prismatic beams for bending
Prismatic means the cross-section is the same at any location across the
long axis of the beam.

36. Design of prismatic beams for bending
The design of a beam depends on the maximum absolute value |M|
max of the
bending moment.
The largest normal stress ??????
� in the beam is found at the surface of the beam in
the critical section where |M|
max occurs:
??????
�=
|M|
max??????
??????
=
|M|
max
??????

A safe design requires that
??????
�≤??????
??????��
??????
??????��: allowable stress

37. solving for S yields the minimum allowable value of the section modulus for the
beam being
Design of prismatic beams for bending
�
�??????�=
|M|
max
??????
??????��

Once the type and material are chosen, beams with the smallest weight per
unit length—and, thus, the smallest cross-sectional area should be selected,
since this beam will be the least expensive.

38. The design procedure generally includes the following steps
Design of prismatic beams for bending
Step 1 determine the value of ??????
??????��
Step 2 Draw the shear and bending-moment diagrams  |M|
max
Step 3  �
�??????�
the design procedure described here takes into account only the
normal stresses occurring on the surface of the beam
Short beams, especially those made of timber, may fail in shear under a
transverse loading
The determination of shearing stresses in beams will be discussed in next Chapter

39. Application
Design of prismatic beams for bending
Select a wide-flange beam to support the 60-kN load as shown in
the Figure . The allowable normal stress for the steel used is 165 MPa.

40. Design of prismatic beams for bending
Application
The allowable normal stress is given: ??????
??????�� = 165 MPa.
The shear is constant and equal to 60 kN.
The bending moment is maximum at B.
|M|
max = (60kN)(2.4)=144 kN.m
The minimum allowable section modulus is
�
�??????�=
|M|
max
??????
??????��
=
��� ��.�
��� �????????????
=���.����
�
�
�

41. Design of prismatic beams for bending
The most economical is the W460 x 52 shape since it weighs only
52 kg/m,