Normal Strain and Stress Normal Strain and Stress, Stress strain diagram, Hooke’s...

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Transcript of Normal Strain and Stress Normal Strain and Stress, Stress strain diagram, Hooke’s...

  • Slide 1
  • Normal Strain and Stress Normal Strain and Stress, Stress strain diagram, Hookes Law 1
  • Slide 2
  • Strain When a body is subjected to load, it will deform and can be detected through the changes in length and the changes of angles between them. The deformation is measured through experiment and it is called as strain. The important of strain: it will be related to stress in the later chapter 2
  • Slide 3
  • Normal Strain Normal strain is detected by the changes in length. 3 (epsilon) l: length after deformed l: original length. Note : dimensionless very small (normally is m (=10 -6 m)) 480(10) -6 m/m = 480 m/m = 480 micros = 0.0480 %
  • Slide 4
  • Example 1 4 When load P is applied, the RIGID lever arm rotates by 0.05 o. Calculate the normal strain of wire BD Foundation: L/L Knowledge required: geometrical equation Rigid: no deformation on the lever
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  • Geometry: The mathematics 5 Sine and Cosine Rule
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  • Example 1 6 L BD after deformed is DB Cosine rule can be applied here Strain: When force P is applied to the rigid lever arm
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  • Example 1 7 L BD after deformed is DB Cosine rule can be applied here Strain: When force P is applied to the rigid lever arm
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  • Example 2 8 The force applied to the handle of the rigid lever the arm to rotate clockwise through an angle of 3 o about pin A. Determine the average normal strain developed in the wire. Originally, the wire is unstretched. Discuss the approach?
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  • Solution 9 L BD = 0.6155 m = 0.0258 m/m
  • Slide 10
  • Simple Tensile Test 10 Strength of a material can only be determined by experiment The test used by engineers is the tension or compression test This test is used primarily to determine the relationship between the average normal stress and average normal strain in common engineering materials, such as metals, ceramics, polymers and composites
  • Slide 11
  • Nominal or engineering stress is obtained by dividing the applied load P by the specimens original cross-sectional area. Nominal or engineering strain is obtained by dividing the change in the specimens gauge length by the specimens original gauge length. Conventional StressStrain Diagram
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  • Elastic Behaviour A straight line Stress is proportional to strain, i.e., linearly elastic Upper stress limit, or proportional limit; pl If load is removed upon reaching elastic limit, specimen will return to its original shape Yielding Material deforms permanently; yielding; plastic deformation Yield stress, Y Once yield point reached, specimen continues to elongate (strain) without any increase in load Note figure not drawn to scale, otherwise induced strains is 10-40 times larger than in elastic limit Material is referred to as being perfectly plastic
  • Slide 14
  • Conventional StressStrain Diagram 14 Strain Hardening. Ultimate stress, u While specimen is elongating, its x-sectional area will decrease Decrease in area is fairly uniform over entire gauge length Necking At ultimate stress, cross-sectional area begins to decrease in a localized region of the specimen. Specimen breaks at the fracture stress.
  • Slide 15
  • StressStrain Behavior of Ductile and Brittle Materials Ductile Materials Material that can subjected to large strains before it ruptures is called a ductile material. Brittle Materials Materials that exhibit little or no yielding before failure are referred to as brittle materials.
  • Slide 16
  • StressStrain Behavior of Ductile and Brittle Materials Yield Strength 0.02% strain for ductile material Strain hardening When ductile material is loaded into the plastic region and then unloaded, elastic strain is recovered. The plastic strain remains and material is subjected to a permanent set.
  • Slide 17
  • Hookes Law Hookes Law defines the linear relationship between stress and strain within the elastic region. E can be used only if a material has linear elastic behaviour. = stress E = modulus of elasticity or Youngs modulus = strain E can be derived from stress and strain graph. What is it?
  • Slide 18
  • Strain Energy When material is deformed by external loading, it will store energy internally throughout its volume. Energy is related to the strains called strain energy. Modulus of Resilience When stress reaches the proportional limit, the strain-energy density is the modulus of resilience, u r :
  • Slide 19
  • Example The stressstrain diagram for an aluminum alloy that is used for making aircraft parts is shown. When material is stressed to 600 MPa, find the permanent strain that remains in the specimen when load is released. Also, compute the modulus of resilience both before and after the load application. Approach to the problem: Parallel to elastic line Both slope is equal Distance CD can be calculated based on the slope Permanent strain: 0.023 distance CD
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  • Solution When the specimen is subjected to the load, the strain is approximately 0.023 mm/mm. The slope of line OA is the modulus of elasticity, From triangle CBD,
  • Slide 21
  • This strain represents the amount of recovered elastic strain. The permanent strain is Computing the modulus of resilience, Note that the SI system of units is measured in joules, where 1 J = 1 Nm Solution:
  • Slide 22
  • Modulus of Toughness Modulus of toughness, u t, represents the entire area under the stressstrain diagram. It indicates the strain-energy density of the material just before it fractures.
  • Slide 23
  • Example 23 The bar DA is rigid and is originally held in the horizontal position when the weight W is supported from C. If the weight causes B to be displaced downward 0.625mm, determine the strain in wires DE and BC. Also if the wires are made of A-36 steel and have a cross- sectional area of 1.25 mm 2, determine the weight W. Discuss the approach????
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  • 24 1) Calculate the displacement of D. 2) Based on displacement on D, calculate the strain and normal stress * strain in mm/mm, stress and E in MPa, F in N and length in mm
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  • 25 3) Based on normal stress at wire DE, calculate the T of wire D 4) Calculate W, based on FBD of bar DA 5) Calculate normal stress of wire CB and strain of wire CB Strain can not be calculated as normal stress goes beyond yield stress (Sy = 250 MPa), elastic property is no more applied. Therefore it requires the stress and strain curve to predict the strain
  • Slide 26
  • Poissons Ratio (nu), states that in the elastic range, the ratio of these strains is a constant since the deformations are proportional. Negative sign since longitudinal elongation (positive strain) causes lateral contraction (negative strain), and vice versa. Poissons ratio is dimensionless. Typical values are 1/3 or 1/4.
  • Slide 27
  • Example A bar made of A-36 steel has the dimensions shown. If an axial force of P is applied to the bar, determine the change in its length and the change in the dimensions of its cross section after applying the load. The material behaves elastically.
  • Slide 28
  • Discuss the approach Approach: Property A-36: E, = P/A z = / E L z = L * z x = y = - z L x = L * x L y = L * y
  • Slide 29
  • 1) The normal stress in the bar : 2) From the table for A-36 steel, E st = 200 GPa 3) The axial elongation of the bar is therefore 4) The contraction strains in both the x and y directions are 5) The changes in the dimensions of the cross section are Solution