MEGR 2144: Introduction to Solid Mechanics

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MEGR 2144 Introduction to Solid Mechanics

• Instructor Qiuming Wei
• Grader Matt Bolen
• Office DCH 362
• Phone [phone]
• Office Hours M, W, F 300-430pm
• Other times by appointment

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Course Objectives

• Apply the principles of equilibrium to the
  problems of solid mechanics to determine external
  and internal forces and moments, distinguish
  between statically determinate and indeterminate
  systems
• Explain the concepts of stress, strain, material
  behavior and distinguish between linear and
  nonlinear material behavior (elastic and
  inelastic)
• Formulate and solve mechanical and structural
  problems involving tension and torsion
• Formulate and solve mechanical and structural
  problems involving pure bending and transverse
  loading
• Formulate and solve mechanical problems involving
  pressure vessels
• Determine various modes of buckling and determine
  the critical loads of buckling for various
  boundary conditions
• Analyze the stress-state at a point and determine
  principal stresses and maximum shear-stress at
  any point in a simple structural problem
• Select, design and analyze a mechanical part
  based on stress-based and maximum
  deflection-based design criteria.

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Syllabus

• Textbook James M. Gere Mechanics of Materials,
  6th Edition, 2004, Thomson ISBN0-534-41793-0
• 22 Lectures
• 3 reviews classes (for the 3 tests)
• 1 comprehensive review for final exam
• 3 exams
• 1 comprehensive close book final
• BlackBoard Vista Domain available for this
  section.
• Course materials also available on Dr. Weis
  personal website http//www.coe.uncc.edu/qwei/

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Course Materials

• All lecture notes (ppt files) will be posted on
  my personal website, and Blackboard Vista
• http//www.mees.uncc.edu/Qwei.htm
• All homework solutions will be posted on my
  personal website, and Blackboard Vista.
• All exam problem solutions will be posted on my
  personal website, and Blackboard Vista .

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Rules of Classroom, Exams, Homework, and Grading
Policy

• Standard Grading Policy A, B, C, D, F.
• Homework/Quizzes 10 Quiz will be given at the
  end of each class except for the first.
• 3 Exams (total 60, 20 for each exam)
• Comprehensive Final 30.
• Cell phone use in classroom is prohibited.
• Homework
• is typically due at the beginning of class on the
  second class from the assignment date.
• may not be accepted late except for instances of
  sickness or death in the family (in the case of
  sickness or personal emergency, notify me via
  email and place your paper in my mailbox as soon
  as possible).
• must have your full name printed legibly on the
  top right corner.
• must be done on straight-cut paper (points may be
  deducted for uneven or frilly edges).
• must be stapled (for multiple pages) (or points
  may be deducted).
• must NOT be copied from another person or any
  other source (submitting copied work is a
  violation of the code of academic integrity,
  policy 105).
• must show coherent solution methods (writing the
  answer down will not earn credit).
• must have proper units on all answers, and all
  answers should be boxed.
• must have three or four significant figures for
  answers (and engineering notation is
  recommended).
• must be neat and legible.
• Course Attendance Policy
• Students are expected to punctually attend all
  scheduled lectures.
• Absences from class may be excused by the
  instructor for personal illness, religious
  holidays, or participation as an authorized
  University representative in an out-of-town
  event. Wherever possible, students are expected
  to seek permission of the instructor prior to
  absences.

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Chapter 01 Tension, Compression and Shear

• What is Mechanics of Materials? Or what is Solid
  Mechanics?
• Study of the behavior of solid bodies under
  different kinds of mechanical loading.
• Important concepts in Mechanics of Materials
• Stress force/area
• SI Unit is Pascal, or Pa, after the French
  scientist and philosopher Pascal (1623-1662)
• 1 Pa1Newton/1m2 1kPa103Pa 1MPa106Pa,
  1GPa109 Pa
• US Unit is PSI Pound per square inch.
• 1PSI6894.76 Pa.
• Strain (change in length/original length)
• Dimensionless (no unit)
• Displacements unit is meter (SI) or inch (US)

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1.1 Why Solid Mechanics?

• Upper Left A ship breaks from the middle. Why
  did this happen?
• Low Left A bridge is collapsed. Design problem?
• Right montage The twin towers collapsed about 2
  hours after the 911 terrorist attack. Why?

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1.2 Normal stress and strain examples

• In the tug-of-war game, the rope is pulled by two
  (groups) of people.
• An axial force is applied to the rope by the
  players.
• The rope is said to have a normal stress stress
  along the axis of the rope that tends to elongate
  the rope.

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1.2 Normal Stress and strain definition

• A prismatic member a straight structural member
  with a constant cross sections thru-out its
  length.
• If a prismatic member with cross-sectional area A
  is subjected to a normal force F, then we say
  that a normal stress, s is applied to this
  prismatic member, and we have
• sF/A (PaN/m2 , or psilb/in2 )
• We also say that the normal force is an axial
  force applied along the axial direction of the
  prismatic member.
• If the initial length of the member is l0, and
  under the normal force it is elongated to the
  final length l, we say that the normal strain of
  this prismatic member is
• e (l-l0)/l0 d/l0
• d l-l0

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1.3 Mechanical Properties of Materials

• With special mechanical testing equipment, we can
  measure how a solid responds to mechanical
  loading.
• The responses of a solid to mechanical loading
  are called the mechanical properties , or
  mechanical behavior of the solid (material).
• Generally speaking, mechanical testing is
  performed on standardized specimens ASTM
  (American Society for Testing and Materials).

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Types and Equipment for Mechanical Testing
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What do we measure in a mechanical testing?

• During a mechanical testing, we measure the force
  applied to the specimen, and use a strain gage or
  an extensometer to measure the elongation of the
  specimen.
• From the force and the cross-section area of the
  specimen, we calculate the stress of the
  specimen.
• From the elongation of the specimen and the
  original length of the specimen, we calculate
  the strain of the specimen.
• We then plot the stress vs. strain we obtain the
  stress-strain curve of the specimen.
• The stress-strain curve reflects the mechanical
  properties of the solid.
• If the specimen recovers its original dimension
  when the mechanical load (force) is removed, we
  say the deformation is elastic.
• If the specimen can not recover its original
  dimension when the force is removed, we say the
  specimen has gone through plastic deformation,
  and the strain that remains after the load is
  removed is called plastic strain.
• Important parameters in a tensile stress-strain
  curve of metal
• Proportional limit the stress beyond which the
  stress-strain relation is no longer linear.
• Elastic limit the stress beyond which plastic
  deformation begins.
• Yield point the stress beyond which material
  starts to yield (to have permanent deformation ).
• Quite often the elastic limit and the yield point
  are hard to differentiate.
• Fracture point the point at which the specimen
  breaks.
• Tensile strength the maximum stress on the
  stress-strain curve.
• Percent elongation El (L1-L0)/L0 x (100),
  where L0 is the original gage length, and L1 is
  the distance between the gage marks at fracture.

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Definition of s0.2

• In practice, it is hard to determine the exact
  yield strength (yield point).
• We use the offset method to define a point on the
  stress-strain curve, and use the stress at that
  point to represent the yield strength for
  engineering design.
• The most common way is to draw a straight line on
  the stress-strain curve parallel to the elastic
  part (initial linear part), but offset by a
  strain of 0.002 (or 0.2).
• The stress at the intersection of the offset line
  and the stress-strain curve define the yield
  strength s0.2.

• Offset method to define the yield strength.
• Please note that the offset strain 0.002 is
  plastic (permanent strain)
• The 0.2 offset yield strength of a material is
  extremely important for engineering design.

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Two types of elasticity

• If the specimen recovers its original dimension
  upon unloading, we say the deformation is
  elastic.
• If during elastic deformation, the stress is
  proportional to the strain, we say the material
  is linearly elastic, and Hookes law applies
• s E e
• E is a materials constant, called the elastic
  modulus, or Youngs modulus.
• The unit of E is Pa. For most solids, E is from a
  few GPa to 1000 GPa (diamond)
• If during elastic deformation, the stress is not
  proportional to the strain, we say the material
  has non-linear elasticity.

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Typical values of Youngs modulus

• Aluminum 70 GPa
• Steels 210 GPa
• Copper 130 GPa
• Plastics 100 MPa10 GPa
• Tungsten 400 GPa.
• Ceramics 300-600 GPa
• Diamond 1000 GPa.

• Note
• The Youngs modulus, or any elastic constant of a
  solid, does not depend on the heat-treatment
  history, microstructure, etc. of the solid.
• It only depends on the inter-atomic bonds of the
  solid.
• The stronger the bond, the greater the Youngs
  modulus.

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Poissons Ratio of a Solid

• When a round bar (or any prismatic member) is
  pulled by a force F, it gets longer (normal
  strain (or axial strain) ezgt 0), and thinner
  (lateral strain (or transverse strain) ex lt0)
• The ratio of the lateral strain to the normal
  strain is a material constant, and
• n -(ez /ex) Poissons ratio (After Poisson,
  the French scientist, 1781-1840)
• Most metals n is 0.3
• Diamond n 0.17
• Some ceramics n 0.2
• Can n be negative?
• When the round bar is compressed by a force F, it
  gets shorter (normal strain ezlt 0) and thicker
  (lateral strain ex gt0), and the definition of
  Poissons ratio stays the same, and is positive.

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General mechanical behavior of solid materials

• Ductile materials
• If the specimen undergoes significant amount of
  plastic deformation before it breaks (fractures),
  we call the material ductile.
• The blue stress-strain curve represents that of a
  ductile material.
• Most metals (Cu, Ag, Ni, Al, etc.), some steels,
  some alloys, are typical ductile metals.
• Brittle materials
• If the specimen undergoes little or no plastic
  deformation before it breaks (fractures), we call
  the material brittle .
• The red stress-strain curve represents that of a
  brittle material.
• Most ceramics, some hardened steels, metal-metal
  compounds (intermetallic compounds) are brittle.

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Summary of Lecture 01

• Prismatic member structural member with constant
  cross-sectional area
• Normal stress s (Pa, KPa, MPa, GPa, or PSI) F
  (N or lb)/A (m2 or in2)
• Normal strain e (m/m or in/in) d (m or in)/L0
  (m or in)
• Mechanical behavior of solids
• Stress-strain curves
• Elastic and inelastic deformation
• Plastic (permanent) deformation
• Hookes law for linear elasticity
• Youngs modulus (E, GPa) Poissons ratio (n)
• Yield strength elastic limit proportional
  limit
• Ductile and brittle materials.

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Homework Problems

• 1.2-4
• 1.3-6
• 1.4-4
• 1.5-4.
• Homework due on Thursday, August 23, 2007.