Printer-Friendly - Properties of Materials - Stress and Strain

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Oct 30, 2023

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Properties of Materials Stress and Strain Structures I R. Taher Properties of Materials - Stress and Strain 1 Warning The information contained in this lecture is protected by intellectual property laws. The content may be used only by the students enrolled in this course. Do not reproduce, transmit, publish, rewrite, sell or post the content on internet sites without a written permission from the instructor teaching this course. Properties of Materials - Stress and Strain 2 Textbook Reading • Simplified Structural Analysis and Design for Architects, Revised 2 nd Edition, by Rima Taher, published by Cognella, 2021. • Chapter 4: Properties of Materials - Stress and Strain • Important: See Problem 4.9.1 in Chapter 4 and Table 4.1. Properties of Materials - Stress and Strain 3 Properties of Materials Stress and Strain • Stress • Strain • Properties of Materials • Stress-Strain Diagram • Modulus of Elasticity • Poisson’s Ratio • Modulus of Rigidity • Thermal Expansion • Thermal Stress Properties of Materials - Stress and Strain 4 Stress • Is the body’s internal resistance to deformation. • Unit Stress: is the resistance developed by a unit area of cross-section of a member. • Units: psi, ksi. • Types of Stresses: tension, compression, torsion, shear… • Indirect Stresses: bending, torsion… Properties of Materials - Stress and Strain 5 • Stress formula: f = F/A • f: stress • F: force • A: area of stress • Example A steel rod has a diameter of 2 in. and is subjected to a tension force of 8,000 lb. What is the average stress in the rod? Properties of Materials - Stress and Strain 6

Solution Calculate the rod’s cross-sectional area: Divide the tension force by the cross-sectional area to determine the average stress: Properties of Materials - Stress and Strain 7 Strain • Strain is the deformation per unit length. • Assume a rod’s original length is L and assume that a tension force (F) is applied to the rod. If the elongation due to the tension force is ∆L. Then the strain ( ε ) is given by: ε = ∆L / L Properties of Materials - Stress and Strain 8 Properties of Materials • It is important to know the mechanical and physical properties of structural materials in order to design structural members. • In designing members, there are 2 important factors: 1- Safety 2- Economy • Lab experiments are generally conducted to determine material properties. • ASTM, the American Society for Testing and Materials is known for material testing. Properties of Materials - Stress and Strain 9 Stress-Strain Diagram • This diagram is a plot of stress values versus strain for a material based on experimental testing. • Different materials have different stress-strain diagrams. • The diagram in this figure is for steel. Properties of Materials - Stress and Strain 10 • The first linear phase on the diagram is the “elastic phase”. Here the stress is proportional to the strain up to a certain value represented by point B on the diagram which marks the “elastic limit” of the material. • In this phase, the tested rod will return to its original dimensions when the load is removed. Properties of Materials - Stress and Strain 11 • Beyond the elastic limit, the “yield point” is reached, point C, on the diagram. • At this point, the rod continues to elongate without much increase in the load. Properties of Materials - Stress and Strain 12

• Point D is the “ultimate strength” of the material, and it is the maximum stress that the rod can sustain. • Beyond this point, the diagram exhibits a downward curve until failure occurs at point E. Properties of Materials - Stress and Strain 13 • The portion shown beyond the elastic limit is called the “plastic phase”. • In this phase, the tested rod will not return to its original dimensions when the load is removed, and some residual deformation will remain. Properties of Materials - Stress and Strain 14 Modulus of Elasticity (E) • It is defined as the slope of the line in the linear phase of the stress- strain diagram and is given by: • E = f / ε • The modulus of elasticity is often a constant. • The unit is: psi or ksi. • For steel, this value is close to 30 x 10 6 psi. Properties of Materials - Stress and Strain 15 Poisson’s Ratio • When a rod is subjected to a force in the longitudinal direction, a deformation occurs in the transverse direction, in addition to the longitudinal direction. • If the force is tensile, the rod will elongate and the transverse dimensions will decrease. • The effects are reversed for compression. Properties of Materials - Stress and Strain 16 • Poisson’s ratio ( μ ) is the ratio of the transverse strain to the longitudinal strain. • Values of this ratio are generally published. Properties of Materials - Stress and Strain 17 Modulus of Rigidity (G) • This modulus connects the modulus of elasticity and Poisson’s ratio and is given by: Properties of Materials - Stress and Strain 18

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Example • A steel rod has a diameter of 3 in. and is subjected to a tension force of 50,000 lb. Determine both the longitudinal and transverse strains. For steel, E = 30 x 10 6 psi and μ = 0.288. Properties of Materials - Stress and Strain 19 Solution • Calculate the cross-sectional area: • Determine the longitudinal strain (ε) using the equation: Properties of Materials - Stress and Strain 20 • Determine the transverse strain using Poisson’s ratio (μ): • μ = Transverse Strain/ Longitudinal Strain Properties of Materials - Stress and Strain 21 Thermal Expansion • Temperature changes generally lead to changes in the dimensions of materials. • The “coefficient of thermal linear expansion (α)” for a material is the change in the length of the material for each unit of original length and per degree of temperature change. • The unit of this coefficient is inches per inch and per degree Fahrenheit (in/in/⁰F). • The total deformation or change in length due to a temperature change is given by the equation: Properties of Materials - Stress and Strain 22 Properties of Materials - Stress and Strain 23 δ: total deformation (in) α: coefficient of thermal linear expansion (In/in/⁰ F) L: original length (in) ∆t: change in temperature (⁰ F) Thermal Stress • If temperature changes and the material is prevented from deforming freely, the restraining forces produce stresses in the material. • An equation for calculating the thermal stress can be developed as follows: Properties of Materials - Stress and Strain 24

• f: thermal stress (psi) • E: modulus of elasticity (psi) • ε: strain (in/in) • L: original length (in) • ∆L: change in length (in) • ∆t: change in temperature (⁰F) • α: coefficient of thermal linear expansion (In/in/⁰ F) Properties of Materials - Stress and Strain 25