ME350 Static Machine Components

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ME 350 Static Machine Components
Lecture Note 5 Beam Deflection Dr. Y.B.
Guo Mechanical Engineering The University of
Alabama
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Topics

• Introduction
• Spring Rate
• Bending Deflection
• Superposition Method
• Castiglianos Method

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Introduction

• Deflection Induced Design Problems

Curved roller
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Spring Rate

• Elasticity
• Component regains its orginal geometry after
  releasing load
• Rigid
• No geometry change under loading
• Spring Rate
• The derivative of load w.r.t. displacement
  (generic sense)

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Spring Rate

• Beam Spring Rate
• Spring Constant

dF
dy
Linear Spring Constant
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Spring Rate

• Spring Rate _at_ Tension, Compression, and Torsion
• Torsional Spring Constant

?
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Spring Rate

• Spring Rate _at_ Bending

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Bending Deflection

• Derivation

Recall
?
F
y f(x)
(derived in note 4)
0
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Bending Deflection

• Derivation

Then
Deflection of Beam
Slope of Beam
Moment in Beam
Shear in Beam
Intermediate/Distributed Load
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Bending Deflection

• Derivation
• Given q, V, or M, Determine y by Integration
• Boundary Conditions (BC)

Fixed End -Clamped
Simply-Supported (Pinned)
Free
A
A
x
y(0) 0
y(A) 0
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Bending Deflection

• Example

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Superposition Method

• The total deflection _at_ a given point is equal to
  the sum of the deflections by each load acting
  separately (Only works for linear problems)

Superposition of Applied Moments
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Superposition Method

• Example

1. Distributed Load, Table A-9-3 (p. 970) At
free end y ymax
(down)
2. Intermediate Load, Table A-9 (p. 993) At
free end y ymax
(down)
Total Deflection 0.324 0.584 0.908 in
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Castiglianos Method

• Strain Energy
• Force Displacement
• Regular Force Linear Displacement
• Torque Angular Displacement
• Moment (Linear Angular) Displacement
• The product of average force and displacement

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Castiglianos Method

• Strain Energy _at_ Different Loading Modes

Cross-section shape Factor c Rectangular
1.2 Circular 1.11 Thin-walled round tube
2.00
Others Table 4.1
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Castiglianos Method

• Castiglianos Theorem
• Small displacement
• Displacement corresponding to any force

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Castiglianos Method

• Example
• Total Strain Energy
• Displacement in the middle

Moment Shear
Deform Modes bending transverse shear
Rectangular bar, c1.2
Displacement
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Castiglianos Method

• More Examples
• In-class exercises

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Castiglianos Method

• Dummy Load
• Find displacement _at_ any position without loading
  there
• Place an imaginary load Q in the desired
  direction
• Set Q0 after taking the derivative

Example Find the vertical deflection at the free
end due to P if transverse shear is neglected.
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Castiglianos Method
Step 1 Add a fictitious force, Q, in the
vertical direction at the free end.
Step 2 Compute strain energies a. Axial load in
AB ? FQ b. Bending in AB ? MPy c. Axial load in
BC ? FP d. Bending in BC ? MPhQx
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Castiglianos Method
Step 4 Compute deflection