Chapter 3
Load and
Stress Analysis
Shear Force and Bending
Moments in Beams
Internal shear force V & bending moment M
must ensure equilibrium
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2015
Fig. 3−2 Mohammad Suliman Abuhaiba, Ph.D., P.E.
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Sign Conventions for Bending
and Shear
Mohammad Suliman Abuhaiba, Ph.D., P.E.Fig. 3−3
Distributed Load on Beam
Distributed load q(x) = load intensity
Units of force per unit length
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2015
Fig. 3−4 Mohammad Suliman Abuhaiba, Ph.D., P.E.
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Relationships between Load, Shear,
and Bending
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Plane stress occurs = stresses on one surface
are zero
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Fig. 3−8
Cartesian Stress Components
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Plane-Stress Transformation Equations
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Fig. 3−9Mohammad Suliman Abuhaiba, Ph.D., P.E.
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Principal Stresses for Plane Stress
principal directions
principal stresses
Zero shear stresses at principal surfaces
Third principal stress = zero for plane stress
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2015
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Extreme-value Shear Stresses for
Plane Stress
Max shear stresses: on surfaces that are
±45º from principal directions
Two extreme-value shear stresses:
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2015
Mohammad Suliman Abuhaiba, Ph.D., P.E.
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Mohr’s Circle Diagram
Relation between x-y stresses and principal
stresses
Relationship is a circle with center at
C = (s, t) = [(sx + sy)/2, 0]
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2015
2
2
2
x y
xyRs s
t
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Mohr’s
Circle
Diagram
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2015
Fig. 3−10
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For a stress element undergoing sx, sy, and
sz, simultaneously,
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Elastic Strain
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Hooke’s law for shear:
Shear strain g = change in a right angle of astress element when subjected to pure
shear stress.
G = shear modulus of elasticity
For a linear, isotropic, homogeneous
material,
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2015
Elastic Strain
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For tension and compression,
For direct shear (no bending present),
Uniformly Distributed Stresses
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2015
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Normal Stresses for Beams in Bending
Straight beam in positive bending
x axis = neutral axis
xz plane = neutral plane
Neutral axis is coincident with centroidal
axis of the cross section
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2015
Fig. 3−13
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Bending stress varies linearly with distance
from neutral axis, y
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2015
Fig. 3−14
Normal Stresses for Beams in
Bending
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Transverse Shear Stress (TSS)
TSS is always accompanied
with bending stress
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2015
Fig. 3−18
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Transverse Shear Stress in a
Rectangular Beam
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Torsion
Angle of twist for a solid round bar
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Fig. 3−21
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Stress Concentration
Localized increase
of stress near
discontinuities
Kt = Theoretical
(Geometric) Stress
Concentration
Factor
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Theoretical Stress
Concentration Factor
A-15 and A-16
Peterson’s Stress-Concentration
Factors
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2015
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Stress Concentration for Static
and Ductile Conditions
With static loads and ductile materials
Highest stressed fibers yield (cold work)
Load is shared with next fibers
Cold working is localized
Overall part does not see damage unless
ultimate strength is exceeded
Stress concentration effect is commonly
ignored for static loads on ductile
materials
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2015
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Stresses in Pressurized Cylinders
Tangential and radial stresses
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Fig. 3−31
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Special case of zero outside pressure, po = 0
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Stresses in Pressurized Cylinders
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If ends are closed, then longitudinal stresses
also exist
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Stresses in Pressurized Cylinders
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Thin-Walled Vessels
Cylindrical pressure vessel with wall
thickness 1/10 or less of the radius
Radial stress is quite small compared to
tangential stress
Average tangential stress
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2015
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Thin-Walled Vessels
Maximum tangential stress
Longitudinal stress (if ends are closed)
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Curved Beams in Bending
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Fig. 3−34
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Location of neutral axis
Stress distribution
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Curved Beams in Bending
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Stress at inner and outer surfaces
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Curved Beams in Bending
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Example 3-15
Plot the distribution of stresses across
section A–A of the crane hook shown in Fig.
3–35a. The cross section is rectangular, with
b = 0.75 in and h = 4 in, and the load is F =
5000 lbf.
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Example 3-15
Fig. 3−35
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Formulas for Sections of
Curved Beams (Table 3-4)
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