Chapter 3
Load and Stress
Analysis
Shigley’s Mechanical Engineering Design
Objectives of Chapter
One of the main objectives of this book is to describe how specific
machine components function and how to design or specify them so
that they function safely without failing structurally.
Machine components transmit forces and motion from one point to
another. Force distributed over a surface leads to the concept of
stress, stress components, and stress transformations
(Mohr’s circle) for all possible surfaces at a point.
Equilibrium and Free-Body Diagrams
For static equilibrium, the forces and moments acting on the system
balance such that
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Shear Force and Bending Moments in Beams

Cut beam at any location x1

Internal shear force V and bending moment M must ensure equilibrium

Shear force and bending moment are related by the equation:
Fig. 3−2
Sign Conventions for Bending and Shear
Fig. 3−3
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Distributed Load on Beam

q(x): load intensity: force per unit length and is positive in the positive y
direction.
Fig. 3−4
Relationships between Load intensity, Shear, and Bending
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Shear-Moment Diagrams
In your textbook, Check the V-x and M-x diagrams
given in Fig. 3−5
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Moment Diagrams – Two Planes
and Combining Moments from Two Planes
Fig. 3−24
Add moments from two planes as
perpendicular vectors
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Singularity Functions
Check theTable 3-1
(Singularity Functions)
in your textbook
Table 3−1
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The examples that follow show how these functions are used:
Example 3-2
Check the solution in your textbook
Fig. 3-5
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Motivation of Mohr’s Circle
Figure : Stress in a loaded deformable material body assumed as a continuum.
^ Parry, Richard Hawley Grey (2004). Mohr circles, stress paths and geotechnics (2 ed.). Taylor & Francis. pp. 1–30. ISBN 0-415-27297-1.
In engineering, e.g., structural or mechanical, the stress distribution within an object,
for instance stresses in a rock mass around a tunnel, airplane wings, or building
columns, is determined through a stress analysis. Calculating the stress distribution
implies the determination of stresses at every point (material particle) in the object.
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3–5 Cartesian Stress Components
Shear stress is often
resolved into
perpendicular
components.
First subscript indicates
direction of surface
normal
Second subscript
indicates direction of
shear stress
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Cartesian Stress Components

In general, a complete state of stress is defined by nine stress
components;
In most cases, “cross shears” are equal
Plane stress occurs when stresses on one surface are zero
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3–6 Mohr’s Circle for Plane Stress-Plane-Stress
Transformation Equations

Cutting plane stress element at an arbitrary angle and balancing stresses gives
plane-stress transformation equations
Fig. 3−9
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Principal Stresses for Plane Stress


The two values of 2p are the principal directions.
Substituting Eq. (3-10) into Eq. (3-8) gives expression for the
non-zero principal stresses.
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Extreme-value Shear Stresses for Plane Stress

The two extreme-value shear stresses are
Eq. (3-14) will not give the maximum shear
stress in cases where there are two nonzero principal stresses that are both
positive or both negative
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Maximum Shear Stress


All stress elements are actually 3-D.
Plane stress elements simply have one surface with zero
stresses.
 Always three extreme shear values:


Fig. 3−12
Maximum Shear Stress is the largest
Principal stresses are usually ordered
such that 1 > 2 > 3,
in which case max = 1/3
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Mohr’s Circle Diagram


A graphical method for visualizing the stress state at a point
Center of circle:
C = (, ) = [(x + y)/2, 0 ]
and radius of
2
 x   y 
2

R 

xy

2


Check the Mohr’ Circle Diagram (Fig. 3−10)
in your textbook
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Mohr’s Circle Diagram
Hibbeler Strength of Materials Textbook
Example 3-4
Check the solution in your textbook
Fig. 3−11
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Uniformly Distributed Stresses

For tension and compression,

For direct shear (no bending present),
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Normal Stresses for Beams in Bending
The beam is subjected to positive
bending.
x axis is neutral axis (have zero stress)

Bending stress varies linearly with distance from neutral axis, y
Maximum bending stress is
where y is greatest.
Zero stress
c is the magnitude of the greatest y
Fig. 3−14
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The calculation of the neutral axis location;
Hibbeler Strength of Materials Textbook
The calculation of I
(Moment of Inertia for
Bending);
Hibbeler Strength of Materials Textbook
Example 3-5
Check the solution in your textbook
Fig. 3−15
Dimensions in mm
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Two-Plane Bending
Quite often, in mechanical design, bending occurs in
both xy and xz planes.
Considering cross sections with one or two planes of
symmetry only, the bending stresses are given by;
For solid circular cross section, the maximum bending stress is
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Example 3-6
Check the solution in your textbook
Fig. 3−16
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Transverse Shear Stress

Transverse shear stress is
always accompanied with
bending stress.
Fig. 3−18
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Transverse Shear Stress in a Rectangular Beam
It is particularly interesting and significant here to observe
that;
• the shear stress is maximum at the bending neutral
axis, where the normal stress due to bending is zero,
and
• that the shear stress is zero at the outer surfaces,
where the bending stress is a maximum.
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Check the solution in your textbook
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Torsion
Angle of twist, in radians, for a solid round bar

Torsional Shear Stress
For round bar in torsion, torsional shear stress is
proportional to the radius 

Maximum torsional shear stress is at the outer surface
Fig. 3−21
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Example 3-9
Check the solution in your textbook
Fig. 3−24
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Stress Concentration


Localized increase of stress near discontinuities
Kt is Theoretical (Geometric) Stress Concentration Factor
Stress
concentration
factor
used for normal
stresses
Stress concentration
factor
used for shear stresses
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Theoretical Stress Concentration Factor
Check Figure A-15-9
(THEORETICAL CONCENTRATION FACTOR
GRAPH)
in your textbook!!!
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Example 3-13
Check the solution in your textbook
Fig. 3−30
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