2' Stress Analysis

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2. Stress Analysis

• ENGR 310 Mechanics of Materials
• Fall, 2007
• Tomasz Arciszewski

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Experiments

• Necessary to determine mechanical properties of
  materials
• Necessary to verify and modify analytical/mathemat
  ical models
• Computer simulation will never completely replace
  experiments

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Tension and Compression Test

• Our focus on tension test
• Objective to determine the relationship between
  the average normal stress and average normal
  strain in engineering materials
• Usually conducted using standard specimens
• Two punch marks establish gauge-length (base)
• Initial measurement
• X-sectional area A0
• Gauge- length L0
• Minimization of bending - ball-and socket-joints
  used

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Loading

• Static - applied very slowly
• Measured and recorded at frequent intervals

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Elongation

• Measured between punch marks
• d L - L0
• Measured using
• a caliper, or an extensometer

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Strain Direct Measurement
Directly measured using electrical resistance
strain gauges
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Strain Direct Measurement
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Stress-Strain Diagram

• Product of a tension or compression test
• Graphical representation of a the relationship
  between average normal stress and average normal
  strain is called the stress-strain diagram

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Conventional Stress-Strain Diagram

• Nominal (engineering) stress applied load P
  divided by the original x-sectional area A0
• ? P/ A0
• Nominal (engineering) strain
• ? ? /L0
• Conventional stress-strain diagram is for nominal
  stress and nominal strain

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Conventional Stress-Strain Diagram

• Produced from a tension test
• Static loading
• Mild, low-carbon structural steel (0.2 carbon
  content)

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Major Behavior Types

• Elastic behavior - no permanent/plastic
  deformations occur after unloading
• Plastic behavior - permanent/plastic deformations
  occur after unloading
• Yielding - relatively constant stresses
• Strain hardening - further stress increase
• Necking - nominal stress decrease

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Elastic Behavior

• occurs when after unloading the specimen comes
  back to its initial shape with no plastic
  deformations
• Proportional limit ? pl is the maximum stress
  when the stress-strain relationship is still
  linear (proportional)
• Elastic limit ?e is the maximum stress when
  specimen still retains its initial shape after
  unloading

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Plastic Behavior

• occurs when the specimen exhibits
  permanent/plastic deformations after unloading

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Yielding and Yield Point

• A type of plastic behavior when the specimen
  undergoes deformations under a relatively
  constant stress, called yield stress, or stress
  changes within the range between upper yield
  point and lower yield point

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Strain Hardening

• A type of plastic behavior which occurs after
  yielding and the further strain increases are
  accompanied by stress increases
• Ultimate stress - the maximum nominal stress
  recorded during the test

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Necking

• A type of plastic behavior which occurs after
  strain hardening, the nominal stress decreases
  while strain is still growing
• It is localized in the central part of specimen
• Fracture stress - the nominal stress associated
  with the fracture of the specimen

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True Stress-Strain Diagram

• Conventional diagram is related to the initial
  x-sectional area and it has decreasing stress
  paradox
• True diagram is related to the actual x-sectional
  area and it reflects actual normal stresses
  (raising)

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Mild Steel Stress-Strain Diagram

• Mild steel, 0.1 - 0.2 carbon content
• Both upper and lower yield points occur
• Plastic shelf show in light color for an
  exaggerated strain scale
• Foundation for plastic analysis and design of
  steel structures

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Carbon Content versus Yield Point
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Yield Point Offset Method
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Ductile Materials

• Undergo large deformations before rupture
• Structural, mild steel
• Ductility measures
• Percent elongation (Lf - L0)/L0 100
• (35-40 for mild steel)
• Percent reduction of area (A0 - Af)/A0 100
• (about 60 for mild steel)

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Brittle Materials

• Little or no yielding before fracture
• Small deformations before fracture
• Cast iron, plastics, concrete

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Behavior various Factors

• Material behavior changes
• Various factors
• Carbon content for steel
• Temperature for steel, plastics
• Aging
• Loading speed
• Radiation (plastics)
• Loading/unloading history

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Ductile Material Fracture Bowl
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Temperature Impact
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Loading Speed Impact

• 500 C degrees
• Left - high speed
• Right - static loading

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Creep Fracture Various Conditions
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Hookes Law

• Robert Hooke, 1676
• It describes a linear relationship between stress
  and strain in within a part of elastic region
• ? ? E
• ? - normal stress
• ? - normal strain
• E - the constant of proportionality

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Constant of Proportionality

• Modulus of elasticity
• Youngs modulus
• Unit - force per area
• Slope of the stress-strain line
• 29 103 ksi for steel

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Hookes Law for Shear

• ? G
• - shear stress
• - shear strain
• G - shear modulus, modulus of rigidity, modulus
  in shear

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Poissons Ratio

• S.D. Poisson, French, early 1800s
• Axial loading
• Isotropic material
• Elongation/contraction in one direction is
  accompanied by a contraction/elongation in a
  transverse direction
• Approximately 0.29 for steel

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Poissons Ratio

• ? absol. value of (?transverse/ ?longitudinal)

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Relationship among E, ? and G

• G E / (2(1 ?))
• G - modulus in shear
• E - Youngs modulus
• ? - Poissons coefficient

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Axial Loading Elongation/Shortening
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Axial Loading Elongation/Shortening

• ? ? E
• ? ?/E P/(AE)
• ?l ?l (Pl)/(AE)

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Principle of Superposition

• Fundamental concept in structural analysis and
  design
• Resultant stress or displacement at the point can
  be
• determined considering separately various
  component
• loads and adding stresses or displacements caused
  by
• them

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Principle of Superposition Requirements

• The loading must be linearly related to the
  stress
• or displacement that is to be determined
• The loading must not significantly change
• the original geometry or configuration of the
  member
• or structure

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Thermal Elongation

• Temperature changes in materials cause
  deformations and stresses (in statically
  indeterminate structures)
• A linear relationship is assumed
• ?T ? ?T L
• ? - linear coefficient of thermal expansion (1/C
  degree, 1/K degree)
• ?T - temperature change
• L - initial length

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