aetsrhfjkkjlfkasdfgh

1. Mechanics of Materials
CONCEPT OF STRESS
Chapter I

2. Contents
 Review of Static
 Stress Analysis
 Design considerations
2

3. Recall that the necessary conditions for equilibrium of a body (rigid or deformable) are:
External Loads
The external loads acting on a deformable body are the known force and
moments that are applied to the body.
Support Reactions and Member Connections
The external loads that are applied to a member must generally be
transmitted to adjacent members that are connected to the given member,
or carried directly to some form of support.
Internal Resultants
In the study of mechanics of deformable bodies, we must
consider not only external forces and couples, that is, the
applied loads and reactions, but we also must consider
internal resultants, that is, forces and couples that are
internal to the original body.
Review of Statics
3

4. Review of Statics
4

5. The structure is designed to
support a 30 kN load
Perform a static analysis to
determine the reaction forces at
the supports and the internal
force in each structural member
The structure consists of a boom
AB and rod BC joined by pins
(zero moment connections) at
the junctions and supports
Review of Statics
5
Boom used to support a 30-kN load.

6. Structure is detached from supports and the
loads and reaction forces are indicated to
produce a free-body diagram
Ay and Cy cannot be determined from these equations
    
kN
30
0
kN
30
0
kN
40
0
kN
40
m
8
.
0
kN
30
m
6
.
0
0























y
y
y
y
y
x
x
x
x
x
x
x
C
C
A
C
A
F
A
C
C
A
F
A
A
M
Conditions for static equilibrium:
Structure Free-Body Diagram
6
Free-body diagram of boom showing
Applied load and reaction forces.

7. In addition to the complete structure, each
component must satisfy the conditions for
static equilibrium
 
0
m
8
.
0
0





y
y
B
A
A
M
Consider a free-body diagram of the boom AB:
kN
30


y
C
substitute into the structure equilibrium
equation
Results:





 kN
30
kN
40
kN
40 y
x C
C
A
Reaction forces are directed along the
boom and rod
Component Free-Body Diagram
7
Free-body diagram of member AB freed from structure.

8. The boom and rod are 2-force members, i.e., the
members are subjected to only two forces
which are applied at the ends of the members
For equilibrium, the forces must be parallel to an
axis between the force application points, equal
in magnitude, and in opposite directions
kN
50
kN
40
3
kN
30
5
4
0






BC
AB
BC
AB
B
F
F
F
F
F

Joints must satisfy the conditions for static
equilibrium which may be expressed in the
form of a force triangle:
Method of Joints
8
Free-body diagram of boom’s joint B and associated force triangle.
Free-body diagrams of two-force members AB and BC

9. The resultant of the internal forces for an axially
loaded member is normal to a section cut
perpendicular to the member axis.
A
P
ave 

The force intensity on that section is defined as
the normal stress.
The actual distribution of stresses is statically
indeterminate, i.e., can not be found from statics
alone.
The normal stress at a particular point may not be
equal to the average stress but the resultant of the
stress distribution must satisfy
Small area DA, at an arbitrary cross section
point carries/axial DF in this member.
Stress distributions at different sections along
axially loaded member.
Axial Loading: Normal Stress
9

10. The stress distributions in eccentrically loaded
members cannot be uniform or symmetric.
A uniform distribution of stress in a section
infers that the line of action for the resultant of
the internal forces passes through the centroid
of the section.
A uniform distribution of stress is only possible
if the line of action of the concentrated loads
P and P’ passes through the centroid of the
section considered. This is referred to as
centric loading.
If a two-force member is eccentrically loaded,
then the resultant of the stress distribution in a
section must yield an axial force and a
moment.
An example of simple eccentric loading.
Centric loading having resultant forces passing
through the centroid of the section.
Centric & Eccentric Loading
10

11. Conclusion: the strength of member BC is
adequate
MPa
165
all 

From the material properties for steel, the
allowable stress is
From a statics analysis
FAB = 40 kN (compression)
FBC = 50 kN (tension)
Can the structure safely support the 30 kN load
if rod BC has a diameter of 20 mm?
dBC = 20 mm
MPa
159
m
10
314
N
10
50
2
6
-
3





A
P
BC

At any section through member BC, the
internal force is 50 kN with a force intensity
or stress of
Stress Analysis
11
Boom used to support a 30-kN load.
Axial force represents the resultant of distributed elementary forces.

12. Design of new structures requires selection of
appropriate materials and component dimensions
to meet performance requirements
For reasons based on cost, weight, availability, etc.,
the choice is made to construct the rod from
aluminum all= 100 MPa). What is an
appropriate choice for the rod diameter?
  mm
2
.
25
m
10
52
.
2
m
10
500
4
4
4
m
10
500
Pa
10
100
N
10
50
2
2
6
2
2
6
6
3






















A
d
d
A
P
A
A
P
all
all
An aluminum rod 26 mm or more in diameter is
adequate
Design
12
Boom used to support a 30-kN load.

13. Forces P and P’ are applied transversely to the
member AB.
A
P

ave

The corresponding average shear stress is,
The resultant of the internal shear force distribution
is defined as the shear of the section and is equal
to the load P.
Corresponding internal forces act in the plane of
section C and are called shearing forces.
Shear stress distribution varies from zero at the
member surfaces to maximum values that may be
much larger than the average value.
The shear stress distribution cannot be assumed to be
uniform.
Opposing transverse loads creating shear on member AB
This shows the resulting internal shear force
on a section between transverse forces
Shearing Stress
13

14. Single Shear
A
F
A
P


ave

Double Shear
A
F
A
P
2
ave 


Bolt subject to single shear. Bolt subject to double shear.
(a) Diagram of bolt in single shear
(b) Section E-E’ of the bolt
(a) Diagram of bolt in double shear
(b) Section K-K’and L-L’ of the bolt.
Shearing Stress Examples
14

15. Bolts, rivets, and pins create
stresses on the points of contact
or bearing surfaces of the
members they connect.
d
t
P
A
P


b

Corresponding average force
intensity is called the bearing
stress,
The resultant of the force
distribution on the surface is
equal and opposite to the force
exerted on the pin.
Equal and opposite forces between plate
and bolt, exerted over bearing surfaces.
Dimensions for calculating bearing stress area.
Bearing Stress in Connections
15

16. Would like to determine the
stresses in the members and
connections of the structure
shown.
Must consider maximum
normal stresses in AB and
BC, and the shearing stress
and bearing stress at each
pinned connection
From a statics analysis:
FAB = 40 kN (compression)
FBC = 50 kN (tension)
Components of boom used to support 30 kN load.
Stress Analysis & Design Example
16

17. 40
20
25
  
2
6
m
10
300
mm
25
mm
40
mm
20
)
(







 Diameter
Width
thickness
Area
We need to calculate the
minimum Area to obtain the
maximum stress
Find the smallest cross-sectional area
17

18. The rod is in tension with an axial force of 50 kN.
The boom is in compression with an axial force of
40 kN and average normal stress of –26.7 MPa.
The sections of minimum area at A and B are not under
stress, since the boom is in compression, and therefore
pushes on the pins.
  
MPa
167
m
10
300
10
50
m
10
300
mm
25
mm
40
mm
20
2
6
3
,
2
6











N
A
P
A
end
BC

At the flattened rod ends, the smallest cross-sectional
area occurs at the pin centerline,
At the rod center, the average normal stress in the circular
cross-section (A = 314x10-6m2) is BC = +159 MPa.
Rod & Boom Normal Stresses
18

19. The cross-sectional area for pins at A, B,
and C,
2
6
2
2
m
10
491
2
mm
25 









 
r
A
MPa
102
m
10
491
N
10
50
2
6
3
, 



 
A
P
ave
C

The force on the pin at C is equal to the
force exerted by the rod BC,
The pin at A is in double shear with a total
force equal to the force exerted by the
boom AB,
MPa
7
.
40
m
10
491
kN
20
2
6
, 


 
A
P
ave
A

Diagrams of the single shear pin at C.
Free-body diagrams of the double shear pin at A.
Pin Shearing Stresses
19

20. Divide the pin at B into sections to determine
the section with the largest shear force,
(largest)
kN
25
kN
15


G
E
P
P
MPa
9
.
50
m
10
491
kN
25
2
6
, 


 
A
PG
ave
B

Evaluate the corresponding average shearing
stress,
Free-body diagrams for various sections at pin B.
Pin Shearing Stresses
20

21. To determine the bearing stress at A in the boom AB, we
have t = 30 mm and d = 25 mm,
  
MPa
3
.
53
mm
25
mm
30
kN
40



td
P
b

To determine the bearing stress at A in the bracket, we
have t = 2(25 mm) = 50 mm and d = 25 mm,
  
MPa
0
.
32
mm
25
mm
50
kN
40



td
P
b

Pin Bearing Stresses
21

22. Axial or transverse forces may produce
both normal and shear stresses with
respect to a plane other than one cut
perpendicular to the member axis.
Axial forces on a two force member
result in only normal stresses on a
plane cut perpendicular to the
member axis.
Transverse forces on bolts and pins
result in only shear stresses on the
plane perpendicular to bolt or pin
axis.
Axial forces on a two-force member. (a) Section plane
perpendicular to member away from load application. (b)
Equivalent force diagram models of resultant force acting at
centroid and uniform normal stress.
(a) Diagram of a bolt from a single shear joint with a section
plane normal to the bolt. (b) Equivalent force diagram
model of the resultant force acting at the section centroid and
the uniform average shear stress.
Stress in Two Force Members
22

23. Stress on an Oblique Plane
23
Pass a section through the member forming an
angle q with the normal plane.
q
q
q
q

q
q
q

q
q
cos
sin
cos
sin
cos
cos
cos
0
0
2
0
0
A
P
A
P
A
V
A
P
A
P
A
F






The average normal and shear stresses on
the oblique plane are
q
q sin
cos P
V
P
F 

Resolve P into components normal and
tangential to the oblique section,
From equilibrium conditions, the distributed
forces (stresses) on the plane must be
equivalent to the force P.

24. The maximum normal stress occurs when the
reference plane is perpendicular to the member
axis,
0
0
m 

 

A
P
The maximum shear stress occurs for a plane at +
45o with respect to the axis,

 



0
0 2
45
cos
45
sin
A
P
A
P
m
q
q

q
 cos
sin
cos
0
2
0 A
P
A
P


Normal and shearing stresses on an oblique
plane
Selected stress results for axial loading.
Maximum Stresses
24

25. Stress components are defined for the planes
cut parallel to the x, y and z axes. For
equilibrium, equal and opposite stresses are
exerted on the hidden planes.
Only six components of stress are required to
define the complete state of stress
The combination of forces generated by the
stresses must satisfy the conditions for
equilibrium:
0
0












z
y
x
z
y
x
M
M
M
F
F
F
   
yx
xy
yx
xy
z a
A
a
A
M





D

D


 0
zy
yz
zy
yz 


 
 and
similarly,
Consider the moments about the z axis:
State of Stress
25

26. Structural members or machines
must be designed such that the
working stresses are less than
the Ultimate/Yield strength of
the material.
Factor of safety considerations:
• uncertainty in material properties
• uncertainty of loadings
• uncertainty of analyses
• number of loading cycles
• types of failure
• maintenance requirements and
deterioration effects
• importance of member to integrity of
whole structure
• risk to life and property
• influence on machine function
Factor of Safety
26

27. Ultimate stress:
Is the maximum amount of the stress a material can withstand before
it breaks or begins to carry less load.
This stress is determined in the lab for a given material by conducting
normal tensile/compression load or shear load using a machine
know as the Universal Testing Machine (UTM)
When a normal load is applied, the resulting stress is known as the
ultimate tensile/compression stress. Similarly, when a shear load is
applied the resulting stress in known as the ultimate shear stress
Allowable stress:
Is the maximum stress that a structural member or a machine
component will be allowed to carry under normal conditions.
Allowable stress should be always less than the ultimate stress
Ultimate Stress and Allowable Stress
27

28. UTM machine
Shear Test for a bolt on a UTM
Tensile Test on a UTM
Determining the ultimate stress of a material
28

29. Problem 1.1
29

30. Problem 1.1
30

31. Problem 1.1
31

32. Problem 1.1
32

33. Problem 1.1
33

34. Problem 1.2
34

35. Problem 1.2
35

36. Problem 1.2
36

37. Problem 1.2
37

38. 38

39. 39

40. 40

41. 41

42. 42