CTC / MTC 222 Strength of Materials

1
CTC / MTC 222Strength of Materials

• Final Review

2
Final Exam

• Wednesday, December 12, 1015 -1215
• 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

3
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.

4
Chapter One Basic Concepts

• SI metric unit system and U.S. Customary unit
  system
• Unit conversions
• Basic definitions
• Mass and weight
• Stress, direct normal stress, direct shear stress
  and bearing stress
• Single shear and double shear
• Strain, normal strain and shearing strain
• Poissons ratio, modulus of elasticity in tension
  and modulus of elasticity in shear

5
Direct Stresses

• Direct Normal Stress , ?
• s Applied Force/Cross-sectional Area F/A
• Direct Shear Stress, ?
• 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 cross-section of the member
• Double shear applied shear force is resisted by
  two cross-sections of the member

6
Direct Stresses

• Bearing Stress, sb
• sb 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

7
Chapter Two Design Properties

• Basic Definitions
• Yield point, ultimate strength, proportional
  limit, and elastic limit
• Modulus of elasticity and how it relates strain
  to stress
• Hookes Law
• Ductility - ductile material, brittle material

8
Chapter Three Direct Stress

• Basic Definitions
• 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
• 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

9
Chapter Four Axial Deformation and Thermal
Stress

• Axial strain e,
• e d / L , where d total deformation, and L
  original length
• Axial deformation, d
• d F L / A E
• If unrestrained, thermal expansion will occur due
  to temperature change
• d a x L x ?T
• If restrained, deformation due to temperature
  change will be prevented, and stress will be
  developed
• s E a (?T)

10
Chapter Five Torsional Shear Stress and
Deformation

• For a circular member, tmax Tc / J
• T applied torque, c radius of cross section,
  J polar moment of inertia
• Polar moment of Inertia, J
• Solid circular section, J p D4 / 32
• Hollow circular section, J p (Do4 - Di4 ) /
  32
• Expression can be simplified by defining the
  polar section modulus, Zp J / c, where c r
  D/2
• Solid circular section, Zp p D3 / 16
• Hollow circularsection, Zp p (Do4 - Di4 ) /
  (16Do)
• Then, tmax T / Zp

11
Chapter Six Shear Forces and Bending Moments in
Beams

• Sign Convention
• Positive Moment M
• Bends segment concave upward ? compression on
  top

12
Relationships Between Load, Shear and Moment

• Shear Diagram
• 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.

13
Relationships Between Load, Shear and Moment

• Moment Diagram
• 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.

14
Chapter Seven Centroids and Moments of Inertia
of Areas

• Centroid of complex shapes can be calculated
  using
• AT Y ? (Ai yi ) where
• AT total area of composite shape
• Y 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 Y ? (Ai yi ) / AT
• Perform calculation in tabular form
• See Examples 7-1 7-2

15
Moment of Inertia ofComposite Shapes

• Perform calculation in tabular form
• 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,
  Y from some reference axis
• Compute the moment of inertia of each part with
  respect to its own centroidal axis, Ii
• Compute the distance, di Y - 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
• IT ? (Ii Ai di2)
• See Examples 7-5 through 7-7

16
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)
• smax 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
• stop M ct / I
• sbot M cb / I

17
Section Modulus, S

• Maximum Stress due to bending
• smax M c / I
• Both I and c are geometric properties of the
  section
• Define section modulus, S I / c
• Then smax M c / I M / S
• Units for S in3 , mm3
• Use consistent units
• Example if stress, s, 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
• Stop I / ctop
• Sbot I / cbot

18
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
• ? 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 cross-section between the axis where
  the shear stress is calculated and the top (or
  bottom) of the beam
• Q is also called the first moment of the area
• Mathematically, Q AP y , 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
• y distance to the centroid of AP from the
  overall centroidal axis
• Units of Q are length cubed in3, mm3, m3,

19
Shear Stress in Common Shapes

• The General Shear Formula can be used to develop
  formulas for the maximum shear stress in common
  shapes.
• 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
• t thickness of web, h depth of beam

20
Chapter Fifteen Pressure Vessels

• If Rm / t 10, pressure vessel is considered
  thin-walled
• Stress in wall of thin-walled sphere
• s p Dm / 4 t
• Longitudinal stress in wall of thin-walled
  cylinder
• s p Dm / 4 t
• Longitudinal stress is same as stress in a sphere
• Hoop stress in wall of cylinder
• s 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