Design Analysis and Review of Stresses at a Point
Need for Design Analysis:
 To verify the design for safety of the structure and the users.
 To understand the results obtained in FEA, it is necessary to
have the theoretical knowledge of the stresses developed in
various loading conditions.
A detailed and comprehensive discussion of the followings can be
found in:
1. “Mechanical Engineering Design”, 4th edition, by Shigley
and Mischke. McGraw Hill, 1989.
2. “Advanced Strength and Applied Stress Analysis”, Second
Edition, by Budynas. McGraw Hill, 1999.
3. “Formulas for Stress and Strain” Roark and Young. McGraw Hill.
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Some important terms that frequently occur in
Design Analysis
Uniaxial stresses
Biaxial Stresses
3-D Stresses
Torsional Stress
Bending Stress
Normal Stress
Combined Stresses
Mohr’s Circle
Failure Theories
Von-Mises Stress
Yield Strength
Factor of safety
Margin of safety
Buckling
Fatigue Stresses
Modal Analysis
Impact Loading
Resonance
Stresses in Cylinders
Rotating Rings
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Stresses at a Point
3-D Stresses
y
σy
τyz
τyx
τxy
τzy
τzx
σx
τxz
x
σz
z
2-D Stress Distribution
σy
τyx
τxy
τxy
σx
σx
τyx
σy
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Principal Stresses
 In the above figure, the stresses at a point are given in the
Cartesian coordinate (x, y, z) directions. Stresses in any
direction φ (measured from the x-axis) are given by the
following equation,
σ = (1/2) (σx + σy) + (1/2) (σx – σy) cos 2φ + τxy sin 2φ
Τ = (1/2) (σx – σy) sin 2φ + τxy cos 2φ
 The normal stress σ and shearing stress Τ vary in magnitude
with angle φ. When the shearing stress Τ = 0, the normal
stresses become maximum and minimum in magnitude.
These stresses are called principal stresses.
 Mohr’s Circle is used to convert stresses at a point into
principal stresses and the relationship is given as,
σ1 = (1/2) (σx + σy) + [(1/2) (σx – σy)2 + τ2xy]1/2
σ2 = (1/2) (σx + σy) - [(1/2) (σx – σy)2 + τ2xy]1/2
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Uniaxial Stress
Uniaxial stress occurs when a bar or a plate
structure is either stretched or compressed.
The resulting stress in the bar is given as,
L
σ = F/A
Where: F is the applied force, and A is the
cross-section area of the bar.
F
Stress-strain relations:
σ=Eε
F
Where: E is the Young’s modulus and ε is the strain.
L
Strain-deflection relationship:
ε = Change in length/original length = ∆L/L
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Biaxial Stresses
When a plate-like structure is loaded in its plane, there are biaxial
stresses and strains,
Y
σx
σx
X
 The plate shown is loaded in the x-direction, but the strains
will occur in both, x and y-directions.
 The plate stretches in the x-direction, but due to the poison’s
effect, it will shrink in the y-direction, thus creating bidirectional strains.
σY
σx
 If loads are applied in both x and y-directions, the plate will
have a bi-directional stress distribution.
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Stress-Strain Relations
Assuming that the x and y axes are principal directions, stresses
and strains are related by the following equations:
σ1 = E (ε1 + νε2)/(1 – ν2)
ε1 = σ1/E - νσ2/E
σ2 = E (ε2 + νε1)/(1 – ν2)
ε2 = σ2/E – νσ1/E
σ3 = 0
ε3 = -νσ1/E – νσ2/E
Beam in Bending
Normal Stress: For a beam in pure bending, the normal stress is
given as
σ = Mc/I
Where, I is the moment of inertia about the axis of beam rotation.
Transverse Shear: The transverse shear is given as
Τ = VQ/Ib
Where,
V is the shearing force
c
Q = ∫ y dA
y
I = Moment of inertia about the bending axis
b = width of the beam
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 If a beam is slender, the transverse shear is negligible and
ignored. However, if a beam is not very slender, the
transverse shear becomes significant and can’t be ignored.
Shear and Bending Moment Diagrams
 For a beam, the maximum stress occurs at the point where
the bending moment is maximum. When the location of the
maximum moment is not obvious, we need to draw a shear
and bending moment diagram to find the magnitude and
location.
Sign Convention for Shear and Bending Moment
Diagrams
Positive Shear and moments
Torsion
The shearing stress in pure torsion is given as
τ = Tr/J
where r is the radial distance from the center
of the shaft, and J is the polar moment of inertia.
r
T
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Stress Concentration
 In the stress equations for axial, bending, and torsion, it is
assumed that the cross section is uniform and no
discontinuities exist. If there are holes, notches, fillets,
change in cross section, etc., these equations cannot be used.
Stresses in the irregular regions are always higher than the
uniform sections – as much as by a factor of 3 or more.
The stress concentration factor is given as,
Kt = σmax/σaverage
Where,
σmax = Maximum stress at some point
σaverage = Stress at the above point, calculated as if there is no
stress concentration, and the area is the net area.
Uniform cross section
Stress concentration
 The stress concentration factor Kt can be found in
engineering handbooks and texts (see the books by Shigley,
Martin, Roark and Young, etc.)
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Deflections
 Deflection occurs when a static load is applied on a structure
at rest (in equilibrium).
 All deflections are assumed as deformation in the structure.
 Rigid body motion doesn’t result in stresses, and is not
important in FEA, unless it is due to either buckling or
warping.
 Buckling and warping require separate analysis, which is
carried out after conduction the static analysis.
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Elastic Structures
All elastic structures follow the Hook’s law,
F=kx
Most engineering structures are considered as elastic and follow
the Hook’s law.
Tension, Compression, and Torsion
Deflection for tension and compression loading is given as,
δ = FL/AE
The stiffness can be found by,
k = F/x = (F)/ δ = (F)/(FL/AE) = AE/L
Torsion
Angular deflection in torsion is,
θ = TL/JG
The stiffness is,
k = T/ θ = GJ/L
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Bending
 Since there are numerous loading and support conditions that
can be applied to a beam, the deflection equation can be
found in an engineering textbook or a handbook. Shigley,
Norton, and Roark and Young are good source of
information.
Buckling
 Buckling occurs in long columns that are loaded by
compressive forces.
 A beam structure, that has several load members, should
always be checked for buckling.
 In buckling, a structure can fail even before reaching the
yield stress point, and therefore, the standard failure criteria
are not valid.
 To check for buckling, the critical buckling load should be
calculated, which will determine the maximum allowable
load magnitude.
Euler’s formula gives the critical buckling load equation,
PCR = Cπ2EI/L2
C depends on the end-constraint of the part under buckling. It has
the following values,
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Column End
Condition
Recommended
Value
Fixed-Free
1/4
Pin-Pin
1
Fixed-Pin
1.2
Fixed-Fixed
1.2
 Some textbooks use the term ‘Load Factor’, which is similar
to the ‘Factor of Safety’ in stress analysis, and is defined as,
LF = Pcr/Papplied
 For safety, the Load Factor (LF) must be greater than 1.
Failure of Engineering Structures
 Generally, there are two types of engineering materials that
are used in FEA: ductile and brittle.
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Ductile Materials
 According to the ductile material failure theory, failure
occurs when the Von-Mises stress (VMS) in the structure
exceeds the yield strength of the material.
 The VMS is calculated from the principal stresses by the
formula,
σVM = [(σ1)2 + (σ2)2 - σ1 σ2]1/2
Brittle Materials
 Brittle materials, such as, cast iron and concrete are governed
by Modified Mohr or Coulomb-Mohr theories.
 In FEA, computer is unable to distinguish between a ductile
and a brittle material. Since the expected results are seldom
exact answers, FEA software uses the Von-Mises stresses for
checking failure in structures, regardless of the applicable
theory for the material used.
Structure Loads
 Generally, all loads applied in FEA are static loads.
 The structure is assumed to be in an equilibrium condition
when the loads are applied.
 For other types of loads, such as, impact, vibrations, and
fatigue, there are more advanced FEA software, which can
handle these loads.
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