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Lecture Strength of Materials I: Chapter 3 - PhD. Tran Minh Tu

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Chapter 3 - Axially loaded members. The following will be discussed in this chapter: Normal stress and normal strain, tension and compression test, poisson’s ratio, shearing strain, allowable stress – factor of safety, statically indeterminate problem.

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Nội dung Text: Lecture Strength of Materials I: Chapter 3 - PhD. Tran Minh Tu

STRENGTH OF MATERIALS 1/10/2013 TRAN MINH TU - University of Civil Engineering, 1 Giai Phong Str. 55, Hai Ba Trung Dist. Hanoi, Vietnam
Axially loaded CHAPTER members 3 1/10/2013 2
Contents 3.1. Introduction 3.2.Normal Stress and Normal Strain 3.3. Tension and Compression Test 3.4. Poisson’s ratio 3.5. Shearing Strain 3.6. Allowable Stress – Factor of Safety 3.7. Statically Indeterminate Problem 1/10/2013 3
3.1. Introduction • Suitability of a structure or machine may depend on the deformations in the structure as well as the stresses induced under loading. Statics analyses alone are not sufficient. • Considering structures as deformable ones allows us to determinate the member forces and reactions which are statically indeterminate. • Determination of the stress distribution within a member also requires the consideration of deformations in the member. • Chapter 3 is concerned with the stress and deformation of a structural member under axial loading. Later chapters will deal with torsional and pure bending loads. 1/10/2013 4
3.1. Introduction • Prismatic bar: Straight structural member with the same cross- section throughout its length • Axial force: Load directed along the axis of the member • Axial force can be tensile or compressive Axially loaded members are structural components subjected only to axial force (tension or compression) 1/10/2013 5
3.1. Introduction 1/10/2013 6
3.1. Introduction  Axial force diagram Using the method of section , the internal axial force is obtained from the equilibrium as a function of coordinate z Z  0  N z  ...  Kinematic assumptions Before deformation After deformation 1/10/2013 7
3.2. Normal stress and normal strain  Kinematic assumptions 1. The axis of the member remains straight 2. Cross sections which are plane and are perpendicular to the axis before deformation, remain plane and remain perpendicular to the axis after deformation. And the cross sections do not rotate about the axis  Normal stress Nz z   z  const A  – normal stress at any point on the cross-sectional area Nz – internal resultant normal force A – cross-sectional area 1/10/2013 8
3.2. Normal stress and normal strain  Elongation of the bar: Consider the bar, which has a cross-sectional area that gradually varies along its length L. The bar is subjected to concentrated loads at its ends and variable external load distributed along its length. L N ( z )dz L     0 EA( z ) 1/10/2013 9
3.2. Normal stress and normal strain  Elongation of the bar – constant load and cross-sectional area: Nz L L    EA  Normal Strain – elongation per unit length Nz   EA – stiffness of axially loaded bar L EA 1/10/2013 10
3.2. Normal stress and normal strain n N zi Li N zi  const L    EA i i 1  EA i 1/10/2013 11
3.4. Poisson’s Ratio Poisson’s Ratio • For a slender bar subjected to axial loading: z z  x y  0 E • The elongation in the x-direction is accompanied by a contraction in the other directions. Assuming that the material is isotropic (no directional dependence), x   y  0 • Poisson’s ratio is defined as lateral strain x y    axial strain z z 1/10/2013 12
3.3. Tension and Compression Test • Strength of a material can only be determined by experiment • One 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 the average normal strain in common engineering materials, such as metals, ceramics, polymers and composites Performing the tension or compression test • Specimen of material is made into “standard” shape and size • Before testing, 2 small punch marks are identified along the specimen’s length • Measurements are taken for both the specimen’s initial x-sectional area A0 and the gauge-length distance L0; between the two marks 1/10/2013 13
3.3. Tension and Compresion Test Performing the tension or compression test • Seat the specimen into a testing machine as shown below - The machine will stretch the specimen at a slow constant rate until the breaking point - At frequent intervals during test, the data is recorded of the applied load P. • The Elongation δ = L − L0 is measured by using either a caliper or an extensometer • δ is used to calculate the normal strain in the specimen • Sometimes, the strain can also be read directly by using an electrical-resistance strain gauge 1/10/2013 1
3.3. Tension and Compresion Test • A stress-strain diagram is obtained by plotting the various values of the stress and corresponding strain in the specimen Conventional stress-strain diagram • Using recorded data, we can determine the nominal or engineering stress by P σ= A0 • Likewise, the nominal or engineering strain is found directly from strain gauge reading, or by δ = L0 By plotting σ (ordinate) against  (abscissa), we get a conventional stress-strain diagram 1/10/2013 15
3.3. Tension and Compression Test Conventional stress-strain diagram • This Figure shows the characteristic stress-strain diagram for steel, a commonly used material for structural members and mechanical elements 1/10/2013 16
3.3. Tension and Compresion Test Conventional stress-strain diagram Elastic behavior. • the straight line • The stress is proportional to the strain, i.e., linearly elastic • Upper stress limit, or proportional limit; σpl • If the load is removed upon reaching the elastic limit , the specimen will return to its original shape Yielding. • The material deforms permanently; yielding; plastic deformation • Yield stress, σY • Once the yield point is reached, the specimen continues to elongate (strain) without any increase in load 17
3.3. Tension and Compresion Test Conventional stress-strain diagram Strain hardening. • Ultimate stress, σu • While the specimen is elongating, its z-sectional area will decrease • Decrease in area is fairly uniform Figure 3-4 over entire gauge length Necking. • At ultimate stress, x-sectional area begins to decrease in a localized region • As a result, a constriction or “neck” tends to form in this region as the specimen elongates further • The Specimen finally breaks at fracture stress, σf 1/10/2013 18
3.3. Tension and Compresion Test Stress – Strain Diagram: Ductile materials • Defined as any material that can be subjected to large strains before it rupture, e.g., mild steel • Such materials are used because of its capacity of absorbing shock or energy so that it, will exhibit a large deformation before failing • Ductility of material is to report its percent elongation or percent reduction in area at time of fracture 1/10/2013 19
3.3. Tension and Compression Test Ductile materials • Percent elongation is the specimen’s fracture strain expressed as a percent Lf − L0 Percent elongation = (100%) L0 • Percent reduction in area is defined within necking region as A0 − Af Percent reduction in area = (100%) A0 • Most metals do not exhibit constant yielding behavior beyond the elastic range, e.g. aluminum • It does not have a well-defined yield point, thus it is standard practice to define its yield strength using a graphical procedure called the offset method 1

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