Transcript CTC / MTC 222 Strength of Materials Final Review

CTC / MTC 222
Strength of Materials
Final Review
Final Exam
• Wednesday, December 12, 10:15 -12:15
• 30% of grade
• Graded on the basis of 30 points in increments of ½ point
• Open book
• May use notes from first two tests plus two additional sheets
of notes
• Equations, definitions, procedures, no worked examples
• Also may use any photocopied material handed out in class
Work problems on separate sheets of engineering
paper
• Hand in test paper, answer sheets and notes stapled to back
of answer sheets
Course Objectives
• To provide students with the necessary
tools and knowledge to analyze forces,
stresses, strains, and deformations in
mechanical and structural components.
• To help students understand how the
properties of materials relate the
applied loads to the corresponding
strains and deformations.
Chapter One – Basic Concepts
• SI metric unit system and U.S. Customary
unit system
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Unit conversions
• Basic definitions
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Mass and weight
Stress, direct normal stress, direct shear stress
and bearing stress
• Single shear and double shear
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Strain, normal strain and shearing strain
Poisson’s ratio, modulus of elasticity in tension
and modulus of elasticity in shear
Direct Stresses
• Direct Normal Stress , 
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σ = Applied Force/Cross-sectional Area = F/A
• Direct Shear Stress, 
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Shear force is resisted uniformly by the area of the part in
shear
 = Applied Force/Shear Area = F/As
Single shear – applied shear force is resisted by a single crosssection of the member
Double shear – applied shear force is resisted by two crosssections of the member
Direct Stresses
• Bearing Stress, σb
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σb = Applied Load/Bearing Area = F/Ab
Area Ab is the area over which the load is transferred
For flat surfaces in contact, Ab is the area of the smaller of the
two surfaces
For a pin in a close fitting hole, Ab is the projected area,
Ab = Diameter of pin x material thickness
Chapter Two – Design Properties
• Basic Definitions
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Yield point, ultimate strength, proportional
limit, and elastic limit
Modulus of elasticity and how it relates
strain to stress
Hooke’s Law
Ductility - ductile material, brittle material
Chapter Three – Direct Stress
• Basic Definitions
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Design stress and design factor
Understand the relationship between design stress,
allowable stress and working stress
Understand the relationship between design factor, factor of
safety and margin of safety
• Design / analyze members subject to direct stress
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Normal stress – tension or compression
Shear stress – shear stress on a surface, single shear and
double shear on fasteners
Bearing stress – bearing stress between two surfaces,
bearing stress on a fastener
Chapter Four – Axial Deformation
and Thermal Stress
• Axial strain ε,
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ε = δ / L , where δ = total deformation, and L = original
length
• Axial deformation, δ
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δ =F L
/
AE
• If unrestrained, thermal expansion will occur due to
temperature change
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δ = α x L x ∆T
• If restrained, deformation due to temperature change
will be prevented, and stress will be developed
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σ = E α (∆T)
Chapter Five – Torsional Shear
Stress and Deformation
• For a circular member, τmax = Tc / J
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T = applied torque, c = radius of cross section, J = polar moment
of inertia
• Polar moment of Inertia, J
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Solid circular section, J = π D4 / 32
Hollow circular section, J = π (Do4 - Di4 ) / 32
• Expression can be simplified by defining the polar
section modulus, Zp= J / c, where c = r = D/2
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Solid circular section, Zp = π D3 / 16
Hollow circularsection, Zp = π (Do4 - Di4 ) / (16Do)
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Then, τmax = T / Zp
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Chapter Six – Shear Forces and
Bending Moments in Beams
• Sign Convention
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Positive Moment M
Bends segment concave upward 
compression on top
Relationships Between Load,
Shear and Moment
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Shear Diagram
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Application of a downward concentrated load
causes a downward jump in the shear diagram.
An upward load causes an upward jump.
The slope of the shear diagram at a point
(dV/dx) is equal to the (negative) intensity of
the distributed load w(x) at the point.
The change in shear between any two points on
a beam equals the (negative) area under the
distributed loading diagram between the points.
Relationships Between Load,
Shear and Moment
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Moment Diagram
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Application of a clockwise concentrated moment
causes an upward jump in the moment diagram.
A counter-clockwise moment causes a
downward jump.
The slope of the moment diagram at a point
(dM/dx) is equal to the intensity of the shear at
the point.
The change in moment between any two points
on a beam equals the area under the shear
diagram between the points.
Chapter Seven – Centroids and
Moments of Inertia of Areas
• Centroid of complex shapes can be calculated
using:
• AT ̅Y̅ = ∑ (Ai yi ) where:
• AT = total area of composite shape
• ̅Y̅ = distance to centroid of composite shape from some
reference axis
• Ai = area of one component part of shape
• yi = distance to centroid of the component part from the
reference axis
• Solve for ̅Y̅ = ∑ (Ai yi ) / AT
• Perform calculation in tabular form
• See Examples 7-1 & 7-2
Moment of Inertia of
Composite Shapes
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Perform calculation in tabular form
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Divide the shape into component parts which are simple shapes
Locate the centroid of each component part, yi from some
reference axis
Calculate the centroid of the composite section, ̅Y̅ from some
reference axis
Compute the moment of inertia of each part with respect to its
own centroidal axis, Ii
Compute the distance, di = ̅Y̅ - yi of the centroid of each part
from the overall centroid
Compute the transfer term Ai di2 for each part
The overall moment of inertia IT , is then:
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IT = ∑ (Ii + Ai di2)
See Examples 7-5 through 7-7
Chapter Eight – Stress Due to
Bending
• Positive moment – compression on top, bent concave
upward
• Negative moment – compression on bottom, bent
concave downward
• Maximum Stress due to bending (Flexure Formula)
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σmax = M c / I
Where M = bending moment, I = moment of inertia, and c
= distance from centroidal axis of beam to outermost fiber
• For a non-symmetric section distance to the top fiber,
ct , is different than distance to bottom fiber cb
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σtop = M ct / I
σbot = M cb / I
Section Modulus, S
• Maximum Stress due to bending
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σmax = M c / I
Both I and c are geometric properties of the section
• Define section modulus, S = I / c
• Then σmax = M c / I = M / S
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Units for S – in3 , mm3
Use consistent units
• Example: if stress, σ, is to be in ksi (kips / in2 ), moment, M, must be
in units of kip – inches
• For a non-symmetric section S is different for the top and the
bottom of the section
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Stop = I / ctop
Sbot = I / cbot
Chapter Nine – Shear Stress in
Beams
• The shear stress,  , at any point within a beams cross-section
can be calculated from the General Shear Formula:
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 = VQ / I t, where
• V = Vertical shear force
• I = Moment of inertia of the entire cross-section about the centroidal
axis
• t = thickness of the cross-section at the axis where shear stress is to
be calculated
• Q = Statical moment about the neutral axis of the area of the crosssection between the axis where the shear stress is calculated and the
top (or bottom) of the beam
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Q is also called the first moment of the area
Mathematically, Q = AP ̅y̅ , where:
• AP = area of theat part of the cross-section between the axis where the
shear stress is calculated and the top (or bottom) of the beam
• ̅y̅ = distance to the centroid of AP from the overall centroidal axis
• Units of Q are length cubed; in3, mm3, m3,
Shear Stress in Common Shapes
• The General Shear Formula can be used to develop
formulas for the maximum shear stress in common
shapes.
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Rectangular Cross-section
• max = 3V / 2A
Solid Circular Cross-section
• max = 4V / 3A
Approximate Value for Thin-Walled Tubular Section
• max ≈ 2V / A
Approximate Value for Thin-Webbed Shape
• max ≈ V / t h
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t = thickness of web, h = depth of beam
Chapter Fifteen – Pressure
Vessels
• If Rm / t ≥ 10, pressure vessel is considered thinwalled
• Stress in wall of thin-walled sphere
• σ = p Dm / 4 t
• Longitudinal stress in wall of thin-walled cylinder
• σ = p Dm / 4 t
• Longitudinal stress is same as stress in a sphere
• Hoop stress in wall of cylinder
• σ = p Dm / 2 t
• Hoop stress is twice the magnitude of longitudinal stress
• Hoop stress in the cylinder is also twice the stress in a
sphere of the same diameter carrying the same pressure