ME350 Static Machine Components

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ME 350 Static Machine Components
Lecture Note 4 Stress _at_ Axial, Bending,
Torsion Load Dr. Y.B. Guo Mechanical
Engineering The University of Alabama
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Topics

• Stress _at_ Axial Loading
• Stress _at_ Bending Beam
• Normal Stress
• Shear Stress
• Shear Stress _at_ Torsion

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Stress _at_ Axial Loading

• Pure Tension/Compression
• Pure Shear

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Stress _at_ Bending Beam Normal Stress

• Straight Beam 2-axes symmetry
• Straight Beam
• Loading in symmetrical plane
• Elastic deformation
• Homogeneous material properties
• Point-of-interest (POI) far away from stress
  concentration/loading location

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Stress _at_ Bending Beam Normal Stress

• Straight Beam 1-axes symmetry
• Straight Beam
• Loading in symmetrical plane
• Elastic deformation
• Homogeneous material properties
• POI far away from stress concentration

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Stress _at_ Bending Beam Normal Stress

• Find Neutral Bending Axis

0.9
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Stress _at_ Bending Beam Normal Stress

• Normal Stress for Straight Beam

M applied moment I Area moment of inertia y
POI from neutral axis
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Stress _at_ Bending Beam Normal Stress

• Normal Stress for Straight Beam

2nd moment of Inertia I can be found in Table A-18
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Stress _at_ Bending Beam Normal Stress

• Parallel Axis Theorem
• I I1 c2A
• Example
• I2 I1(h/2)2?(bh)
• bh3/12bh2/2
• bh3/3

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Stress _at_ Bending Beam Normal Stress

• Example

12,000 lb Pipe
4 ft
6 ft
3. Max Stress
2. Find Maximum Moment
1. Find Reaction Forces
smax Mmaxy/I 28800 lb/in2 yh/2 Ibh3/
12 y/I1 (Given b1/6, h6)
12,000 lb
12,000 lb
Mx
V
RA
RA
RB
0?x?4 ?M 0 -Mx RAx 4?x?10 ?M 0
-Mx RAx - 12,000(x-4)
?Fy 0 RA-12,000RB ?MA 0 12,0004-RB10
RA 7,200 lb and RB 4,800 lb
Mmax 28,800 lb-ft
x0
x10
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Stress _at_ Bending Beam Normal Stress

• Normal Stress for Curved Beam
• Neutral surface shift e to inner surface

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Stress _at_ Bending Beam Normal Stress

• Normal Stress for Curved Beam
• Force and Moment Equilibrium
• c0 distance from outer fiber to neutral axis
• ci distance from inner fiber to neutral axis
• e distance from centroidal axis to neutral axis
• r0 radius of outer fiber
• ri radius of inner fiber
• rn radius of neutral axis
• rc radius of centroidal axis

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Stress _at_ Bending Beam Normal Stress

• Normal Stress for Curved Beam
• Example

16.9 kpsi
s
Tension
Tension
s
?
Compression
-5.6 kpsi
4?
M
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Stress _at_ Bending Beam Shear Stress

• Shear Stress for Straight Beam

Near Surface

• ______________, Net Force

No shear on top free surf
Far Surface

• ____________, Net Force

• ________________, Net Force

Bottom Surface
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Stress _at_ Bending Beam Shear Stress

• Shear Stress for Straight Beam
• Force equilibrium

First Moment of Area
Area
Distance from y axis to the area centroid
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Stress _at_ Bending Beam Shear Stress

• Example

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Shear Stress _at_ Torsion

• Torsion Assumption
• Torsion twisting about the axis
• Elastic deformation homogeneous matl
• Analysis performed away from force
• Bar with circular cross-section
• Parallel planar cross-sections remain
• parallel and planar
• Radial Lines remain straight
• Angles of rotation are Small.

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Shear Stress _at_ Torsion

• Shear Strain
• Max shear strain

Unwrap
________ the curved triangle ACC?
Since g is small
l
A
C
g
rq
C?
?max
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Shear Stress _at_ Torsion

• Shear Stress

From Hookes Law tr Ggr and tmax Ggmax
Moment Arm
Total Torque
Force
J polar 2nd moment of area
Circular Cross-Section
Hollow Cross-Section
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Shear Stress _at_ Torsion

• Shear Stress

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Shear Stress _at_ Torsion

• Twist Angle

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Shear Stress _at_ Torsion

• Example

A 250 mm long shaft, which is 10 mm in diameter,
carries a torque of 4.10 Nm. If the shaft is made
of steel, G 80 GPa - What is the maximum
shear stress due to torsion? - What is the
angle of twist?
Need
Twist angle