Load and Stress Analysis

1
Load and Stress Analysis
Section III
2
Talking Points

• Introduction about stresses
• Shearing force and bending moment diagrams
• Bending, Transverse, Torsional stresses
• Compound stresses and Mohrs circle
• Stress concentration
• Stresses in pressurized cylinders, rotating
  rings, curved beams, contact

3
Introduction about stresses
i. Static Equilibrium and Free-Body Diagram

• Assume downward force as negative and upward
  force as positive and counterclockwise moment as
  positive and clockwise as negative.
• Loads may act on multiple planes.

4
Introduction about stresses Cont.
ii. Direct Normal Stress Strain
Assuming elasticity

• The load is applied along the axis of the bar
  (perpendicular to the cross-sectional area) and
  it is uniformly distributed across the
  cross-sectional area of the bar.
• The normal stress can be tensile () or
  compressive (-) depending on the direction of the
  applied load P.
• The stress unit in N/m2 or Pa or multiple of this
  unit, i.e. MPa, GPa.

Hookes Law
5
Introduction about stresses Cont.

• Sometimes, a body is subjected to a number of
  forces acting on its outer edges as well as at
  some other sections, along the length of the
  body. In such case, the forces are split up, and
  their effects are considered on individual
  sections. The resulting deformation of the body
  is equal to the algebraic sum of the deformation
  of the individual sections. Such a principle of
  finding out the resultant deformation is called
  the principle of superposition.

Principle of Superposition
L1
L2
L3
d3
d1
d2
L3
L2
L1
P3
P1
P2
P4
d3
d1
d2
6
Introduction about stresses Cont.
Example on Principle of Superposition
A brass bar, having cross sectional area of
10 cm2 is subjected to axial forces as shown in
the figure. Find the total elongation of the bar
(dL). Take E 80 GPa.
dL -150 mm
7
Introduction about stresses Cont.
iii. Poissons Ratio
For 1D stress system ( )
1D stress system
or
From Hookes Law
For 2D stress system ( )
and

• For engineering materials, n 0.25 to 0.33.
• For a rounded bar, the lateral strain is equal to
  the reduction in the bar diameter divided by the
  original diameter.

8
Introduction about stresses Cont.
Example on Poissons Ratio
A 500 mm long, 16 mm diameter rod made of a
homogenous, isotropic material is observed to
increase in length by 300 mm, and to decrease in
diameter by 2.4 mm when subjected to an axial 12
kN load. Determine the modulus of elasticity and
Poissons ratio of the material.
E 99.5 GPa n 0.25
9
Introduction about stresses Cont.
iii. Direct Shear Stress Strain
Assuming elasticity

• The load, here, is applied in a direction
  parallel to the cross-sectional area of the bar.

Q
Q
G is known as modulus of rigidity
Relation between E, G, and n
Single Double Shear
The rivet is subjected to single shear
The rivet is subjected to double shear
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Shearing Force (S.F.) and Bending Moment (B.M.)
Diagrams
Sign Convention
Simply supported beam
Cantilever beam
Relationship between shear force and bending
moment
Or
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Shearing Force (S.F.) and Bending Moment (B.M.)
Diagrams - Examples
i. Concentrated Load Only
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Shearing Force (S.F.) and Bending Moment (B.M.)
Diagrams - Examples
ii. Distributed Load Only
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Shearing Force (S.F.) and Bending Moment (B.M.)
Diagrams - Examples
iii. Combination of Concentrated and Distributed
Load
14
Shearing Force (S.F.) and Bending Moment (B.M.)
Diagrams - Examples
iv. If Couple or Moment is Applied
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Bending, Transverse, Torsional stresses
i. Bending Stress
where, M is the applied bending moment (B.M.) at
a transverse section, I is the second moment of
area of the beam cross-section about the neutral
axis (N.A.), i.e. , s is
the stress at distance y from the N.A. of the
beam cross-section.
16
Bending, Transverse, Torsional stresses Cont.
ii. Transverse Stress
or
where Q is the applied vertical shear force at
that section A is the area of cross-section
above y, i.e. the area between y and the
outside of the section, which may be above or
below the neutral axis (N.A.) y is the distance
of the centroid of area A from the N.A. I is the
second moment of area of the complete
cross-section and b is the breadth of the
section at position y.
R
b
d
17
Bending, Transverse, Torsional stresses Cont.
iii. Torsional Stress
where T is the applied external torque r is the
radial direction from the shaft center J is the
polar second moment of area of shaft
cross-section r is the shaft radius and t is
the shear stress at radius r.
Note when torsion is present Ductile materials
tends to break in a plane perpendicular to its
longitudinal axis while brittle material breaks
along surfaces perpendicular to direction where
tension is maximum i.e. along surfaces forming
45o angle with longitudinal axis.
Solid section
Hollow shaft
Ductile material
Brittle material
18
Compound stresses and Mohrs circle

• Machine Design involves among other
  considerations, the proper sizing of a machine
  member to safely withstand the maximum stress
  which is induced within the member when it is
  subjected separately or to any combination of
  bending, torsion, axial, or transverse load.

19
Compound stresses and Mohrs circle Cont.
Maximum Minimum Normal Stresses
For 2D Case
Stress State
3D General Stress State

• Where
• sx is a stress at a critical point in tension or
  compression normal to the cross section under
  consideration, and may be either bending or axial
  load, or a combination of the two.
• sy is a stress at the same critical point and in
  direction normal to the sx stress.
• txy is the shear stress at the same critical
  point acting in the plane normal to the Y axis
  (which is the XZ plane) and in a plane normal to
  the X axis (which is the YZ plane). This shear
  stress may be due to a torsional moment,
  transverse load, or a combination of the two.
• sn(max) and sn(min) are called principal stresses
  and occurs on planes that are at 90 to each
  other, called principle planes also planes of
  zero shear.

Note sx, sy, sz all ve, txy, tyx, tzy, tyz,
txz, tzx all ve. Due to static balance, txy
tyx, tzy tyz, and txz tzx.
2D Stress State
Clockwise (CW)
Counterclockwise (CCW)
20
Compound stresses and Mohrs circle Cont.
Maximum Shear Stresses (tmax)

• tmax at the critical point being investigated is
  equal to half of the greatest difference of any
  of the three principal stresses. For the case of
  two-dimensional loading on a particle causing a
  two-dimensional stresses The planes of maximum
  shear are inclined at 45 with the principal
  planes.

The planes of maximum shear are inclined at
45 with the principal planes.

• The angle between the principal plane and the X-Y
  plane is defined by

21
Compound stresses and Mohrs circle Cont.
Mohrs Circle

• It is a graphical method to find the maximum and
  minimum normal stresses and maximum shear stress
  of any member.

From the diagram sx OA, txy AB, sy OC, and
tyx CD. The line BED is the diameter of Mohr's
circle with center at E on the s axis. Point B
represents the stress coordinates sx, txy on the
X faces and point D the stress coordinates sy,
tyx on the Y faces. Thus EB corresponds to the
X-axis and ED to the Y-axis. The maximum
principal normal stress smax occurs at F, and the
minimum principal normal stress smin at G. The
two extreme-value shear stresses one clockwise
and one counterclockwise, occurs at H and
I, respectively. We can construct this diagram
with compass and scale and find the required
information with the aid of scales. A
semi-graphical approach is easier and quicker
and offer fewer opportunities for error.
2-D
22
Compound stresses and Mohrs circle Cont.

• tmax is equal to half of the greatest difference
  of any of the three principal stresses. In the
  case of the below figure

where,
Principal Element
Min
Max
True views on the various faces of the principal
element
3-D
23
Compound stresses and Mohrs circle Examples.
Example
A machine member 50 mm diameter by 250 mm
long is supported at one end as a cantilever. In
this example note that sy 0 at the critical
point.
Case 2 Bending only
Case 1 Axial load only
In this case all points in the member are
subjected to the same stress.
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Compound stresses and Mohrs circle Examples.
Case 3 Torsion only
Case 4 Bending Axial Load
In this case the critical point occur along the
outer surface of the member.
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Compound stresses and Mohrs circle Examples.
Case 5 Bending Torsion
Case 6 Torsion Axial Load
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Compound stresses and Mohrs circle Examples.
Case 7 Bending, Axial Load, and Torsion
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Compound stresses and Mohrs circle Examples.
Example on Mohrs circle
The stress element shown in figure has sx
80 MPa and txy, 50 MPa (CW). Find the principal
stresses and directions.
Locate sx 80 MPa along the s axis. Then from
sx, locate txy 50 MPa in the (CW) direction of
the t axis to establish point A. Corresponding to
sy 0, locate tyx 50 MPa in the (CCW)
direction along the t axis to obtain point D. The
line AD forms the diameter of the required circle
which can now be drawn. The intersection of the
circle with the s axis defines smax and smin as
shown.
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Stress Concentration

• Occurs when there is sudden changes in
  cross-sections of members under consideration.
  Such as holes, grooves, notches of various kinds.
• The regions of these sudden changes are called
  areas of stress concentration.
• Stress-concentration factor (Kt or Kts)
• The analysis of geometric shapes to determine
  stress-concentration factors is a difficult
  problem, and not many solutions can be found.

Theoretically
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• Stresses in pressurized cylinders, rotating
  rings, curved beams, contact