Mechanics of Materials_Chapter III_Pure Bending.pdf

Mechanics of materials_Chapter II

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

2. Chapter 3
Pure Bending
2

3. Lasha Talakhadze
Pure bending
3

4. free-body diagram of the bar
Pure bending
4

5. Pure bending
Clamp used to glue lumber
pieces together
5

6. Pure bending
Pure bending refers to flexure of a
beam under a constant bending
moment
A positive sign is assigned to M when
the concavity of the beam faces
upward and a negative sign otherwise.


Definition:
6

7. Symmetric members in pure bending
Stresses resulting from pure bending moment M.
a couple M actually consists of two equal and opposite forces
x components: ??????
��??????=0
Moments about y axis: �??????
��??????=0
Moments about z axis: (−�??????
��??????)=�
tensile stress (?????? > 0) leads to a negative moment (clockwise)
Internal Moment and Stress Relations
7

8. Symmetric members in pure bending
Deformations
The member bends uniformly
Line AB along the upper face of the member
intersecting the plane of the couples will have
a constant curvature
Any cross section perpendicular to the axis
of the member remains plane the plane of
the section passes through C
8

9. Symmetric members in pure bending
For pure bending ??????
�� =0 and ??????
�� =0
only nonzero stress component exerted on any of
the small cubic elements considered here is the
normal component ??????
�
Deformations
For M > 0, strain ??????
� and the stress ??????
� are negative in
the upper portion of the member (compression) and
positive in the lower portion (tension)
9

10. Deformations
Symmetric members in pure bending
There must exist a surface where ??????
� and ??????
� = 0
 Called neutral surface
Intersects :
and a transverse section along a straight line
called the neutral axis of the section
the plane of symmetry along an arc of circle DE
10

11. Symmetric members in pure bending
Deformations
11

12. Origin of coordinates is now selected on the neutral
surface—rather than on the lower face of the member
Symmetric members in pure bending
Deformations
 the distance from any point to the neutral
surface is measured by its coordinate y
We have:
�=????????????
�
′
=(??????−�)??????
length L of the undeformed member
length L’ of arc JK :
Since the original length of arc JK was
equal to L, the deformation of JK is
??????=�−�
′
=−�??????
12

13. Symmetric members in pure bending
Deformations
The longitudinal strain ??????
� in the elements of JK
??????
�=
??????
�
=
−�??????
????????????

??????
�=
−�
??????

minus sign is due to the fact that it is assumed
the bending moment is positive.
13

14. Symmetric members in pure bending
Deformations
Denoting the largest distance from the neutral surface as c
maximum absolute value of the strain as ??????
??????
??????
??????=
−�
??????

??????
�=
−�
c
??????
??????
14

15. Stresses and deformations
in the elastic range
Normal stresses in the member remain below the yield strength ??????
??????
Assuming the material to be homogeneous with modulus of elasticity E
??????
�=�??????
�
??????
�=
−�
c
??????
?????? Recall:
Multiplying both sides with E: ????????????
�=
−�
c
????????????
??????
??????
�=
−�
c
??????
??????
??????
?????? : maximum absolute value of the stress
(in the elastic range, the normal stress varies
linearly with the distance from the neutral
surface)
15

16. Stresses and deformations
in the elastic range
Determine: location of the neutral surface and the maximum value ??????
?????? of the
stress
??????
��??????=0
??????
��??????=
−�
c
??????
??????�??????=
−??????
??????
c
��??????=0
��??????=0 (first moment of the cross section)
16

17. First Moment of an Area
first moment of the area A with respect to the
x axis is the integral
�
�= ��??????
??????

first moment of the area A with respect to the
y axis is the integral
�
�= ��??????
??????

17

18. First Moment of an Area
��??????=??????�
??????

The centroid of the area A is the point C
of coordinates � and � satisfies the
relationship
��??????=??????�
??????

18

19. When an area possesses an axis of symmetry, the first moment of the
area with respect to that axis is zero
First Moment of an Area
if an area A possesses an axis of symmetry, its centroid C is located on
that axis.
�
�= ��??????
??????
=0
19

20. Stresses and deformations
in the elastic range
��??????=0 (first moment of the cross section)
the neutral axis passes through the centroid of the section
(−�??????
��??????)=�
Recall:
z axis coincides with the neutral axis of the cross section
Moments about z axis:
(−�)(
−�
c
??????
??????)�??????=�
??????
??????
c
�
2
�??????=�
20

21. Stresses and deformations
in the elastic range
??????
??????=
????????????
??????

??????
�=−
??????�
??????

(*), (**) : elastic flexure formulas
??????
�=
−�
c
??????
??????
(*)
(**) (flexural stress)
I: is the moment of inertia
21

22. Elastic section modulus
Stresses and deformations
in the elastic range
??????=
??????
�

??????
??????=
??????
??????

beams should be designed with as large a value of S as is practical
22

23. For example
Stresses and deformations
in the elastic range
a wooden beam with a rectangular cross section of width b and depth h has
??????=
??????
�
=
1
12
�ℎ
3
ℎ/2
=
1
6
�ℎ
2
=
1
6
??????ℎ
A is the cross-sectional area of the beam
23

24. Stresses and deformations
in the elastic range
Two types of steel beam cross sections:
(a) American Standard beam (S) (b) wide flange beam (W).
24

25. Stresses and deformations
in the elastic range
The deformation of the member caused by the bending
moment M is measured by the curvature of the neutral
surface
The curvature is defined as the reciprocal of the
radius of curvature ??????
1
??????
=
??????
??????
�

In the elastic range: ??????
�=
??????
�
�

1
??????
=
??????
�
��
=
1
��
????????????
??????

1
??????
=
????????????
??????

25

26. Deformations in a transverse
cross section
transverse cross section of a member in pure bending remains plane
there is the possibility of deformations within the plane of the section
26

27. Deformations in a transverse
cross section
For pure bending, ??????
�� =0 and ??????
�� =0, ??????
�≠0
According to Poisson’s ratio:
??????
�=−????????????
� ??????
�=−????????????
�
??????
�=??????
�
??????
??????
�=??????
�
??????

 expand or contract in both the y and z directions
27

28. various horizontal lines in the section is bent into
arcs of circle
Deformations in a transverse
cross section
??????
�=??????
�
??????
??????
�=
−�
??????

Compare
neutral axis of the transverse section is bent into a
circle of radius
??????′=
??????
??????

??????
�=??????
�
??????

28

29. The reciprocal of the radius of curvature ??????′
represents the curvature of the transverse cross
section and is called the anticlastic curvature.
Deformations in a transverse
cross section
�??????�??????�??????���??????� ���������=
1
??????′
=
??????
??????

29

30. 1. Using an allowable stress of 155 MPa, determine the largest bending
moment M that can be applied to the wide-flange beam shown. Neglect the effect
of fillets.
EXERCISES
30

31. Stress concentrations
??????
??????=
????????????
??????

is applicable for a member with a plane of
symmetry and a uniform cross section
Higher stresses also occur if the cross section of the
member undergoes a sudden change
??????
??????=??????
????????????
??????

31

32. Stress concentrations
32

33. Eccentric axial loading in a
plane of symmetry
Eccentric loading: the load does not
go through the centroid of the
member
 distribution of stress on cross
section of the member is not
uniformed
Walkway light Clamp used to glue lumber
pieces together
33

34. Eccentric axial loading in a
plane of symmetry
members that possess a plane of symmetry
loads are applied in the plane of symmetry
of the member
Consider:
Apply the conditions of equilibrium
�=� M=��
34

35. Eccentric axial loading in a
plane of symmetry
??????
�=(??????
�)
���??????????????????�+(??????
�)
����??????�??????=
�
??????
−
��
??????

The stress distribution due to the original eccentric is the superposition of
uniform stress distribution of axial loading and linear stress distribution of bending
35

36. General case of eccentric axial
loading analysis
??????
�=
�
??????
−
�
��
??????
+
�
��
??????

The eccentric force P is statically equivalent
to the system consisting of:
a centric force P
the two couples M
y = Pa, M
z = Pb
Consider a more general case when the axial
load is not applied in a plane of symmetry.
36

37. 37