EGM 5653 CHAPTER 1 Introduction EGM 5653 Advanced

EGM 5653 CHAPTER 1 Introduction EGM 5653 Advanced Mechanics of Materials Namas Chandra Advanced Mechanics of Materials Chapter 1 -1

EGM 5653 Objectives Review some of the mechanics principles How do you formulate problems using mechanics of materials approach How does this approach compare with continuum mechanics approach How do you evaluate mechanical properties What are the various modes of failure/limits on design Sections 1. 1 Review of Elementary mechanics 1. 2 Method of Analysis 1. 3 Stress-strain relations 1. 4 Failure theories and design criteria Namas Chandra Advanced Mechanics of Materials Chapter 1 -2

EGM 5653 1. 1 Introduction Using equations of equilibrium, the balance of forces and moments we can obtain the state of stress due to applied load and moment. Similarly the change in length as a function of total length can also be determined. We will use specific examples of uniaxial bar, torsion of bar and bending of beams as examples. Most of the problems can be viewed as a combination of these loading conditions, but simply much more complex. Namas Chandra Advanced Mechanics of Materials Chapter 1 -3

EGM 5653 1. 1. 1 Axially Loaded Members This is 1 -D problem with the applied load at the end uniquely determining the stress, strain and elongation. The stress assumes that the load is uniformly loaded on the constant area. The elongation, relates the applied load to deformation. The strain can be defined as the elongation over the total length, thus It also assumes that the stress-strain relationship is linear, thus retaining a linear relation. Namas Chandra Advanced Mechanics of Materials Chapter 1 -4

EGM 5653 1. 1. 2 Torsionally loaded members Here a bar/beam/any section is loaded under torsion. Torsion is a moment with the axis of the moment coinciding with the longitudinal axis. We need to find out the shear stress , in the member, angle of twist, and the shear strain, . Namas Chandra Advanced Mechanics of Materials Chapter 1 -5

EGM 5653 1. 1. 3 Bending of Beams Beam is a structural member whose length is large compared to other dimensions. Also the beam primarily carries moment. If torque is applied, then member is called shaft typically of circular cross-section. If the same member is subjected to axial load then it is a bar. Here, the stress varies from zero at the neutral axis to the maximum and minimum at the two surfaces For pure bending for a symmetric beam, the neutral axis lies on the centroid with maximum tensile and maximum compressive at the surface to oppose the external moment. Namas Chandra Advanced Mechanics of Materials Chapter 1 -6

EGM 5653 1. 1. 3 Bending of Beams-2 For other cases, bending moment and depends on all the applied load (concentrated and distributed) and external moments. Bending stress Deflection Shear stress Namas Chandra Advanced Mechanics of Materials Chapter 1 -7

EGM 5653 1. 2 Method of Analysis We need to establish a relation between (1) load and stress, and (2) load and deflection. Define the geometry of the problem, include all the boundary conditions, specify the material properties. We are interested in normal and shear stresses on a given section. Basic equations that need to be satisfied include Force and moment equilibrium (for static)for dynamics use. Displacement continuity (compatibility conditions) Constitutive equation (e. g. linear elastic isotropic, linear elastic anisotropic, elastic-plastic, visco-elastic). We are interested in normal and shear stresses on a given section. Namas Chandra Advanced Mechanics of Materials Chapter 1 -8

EGM 5653 1. 2 Method of Analysis-2 Several complex loads can be broken down into simple loads and the results superimposed. This is called theory of superposition and is valid for all linear problems. Linear problems should have linearity in loading, boundary conditions AND constitutive equations. . Namas Chandra Advanced Mechanics of Materials Chapter 1 -9

EGM 5653 1. 3. 1 Elastic and Inelastic Response Tensile tests are carried out where in circular or dog-bone type specimens are subjected to uniaxial load. The change in length is measured by clip gages attached to the specimen or by using strain gages. Engineering Stress True Stress Engineering Strain True Strain Relationship between engineering and true quantities Namas Chandra Advanced Mechanics of Materials Chapter 1 -10

EGM 5653 1. 3. 2 Mechanical Testing of Materials The above curve gives a number of important material properties. Some of the properties include Yield Strength Ultimate tensile strength Modulus of Elasticity Percent of elongation % reduction in area Other properties that are of interest include: Modulus of resilience Modulus of toughness Modulus of rupture Poisson’s ratio Necking point in ductile material Namas Chandra Advanced Mechanics of Materials Chapter 1 -11

EGM 5653 1. 3. 2 Engineering vs. true stress-strain response Note that the true stress-true strain does not have ultimate tensile value and there is no peak point. Namas Chandra Advanced Mechanics of Materials Chapter 1 -12

EGM 5653 1. 4. 1 Modes of Failure by excessive deflection o. Elastic deflection o. Deflection caused by creep Failure by yielding Failure by fracture o. Sudden failure of brittle material o. Failure of cracked (flawed) members o. Progressive fracture (fatigue) o. Stress- corrosion cracking Failure by instability Namas Chandra Advanced Mechanics of Materials Chapter 1 -13

EGM 5653 Problem 1. 23, page 23 Use mechanics of materials approach to derive the load-stress and load-displacement relations for a solid circular rod of constant radius r and length L subjected to a torsional load T as shown. Namas Chandra Advanced Mechanics of Materials Chapter 1 -14

EGM 5653 Solution 1. 23 Namas Chandra Advanced Mechanics of Materials Chapter 1 -15

EGM 5653 Problem 1. 28, page 23 A steel shaft of circular cross section is subjected to twisting moment T. The controlling factor in the design of the shaft is the angle of twist per unit length, . The maximum allowable twist is 0. 005 rad/m, and the maximum shear stress, = 30 Mpa. Determine the diameter at which the maximum allowable twist, and the not the shear stress is the controlling factor. For steel G= 77 GPa Namas Chandra Advanced Mechanics of Materials Chapter 1 -16

EGM 5653 Solution 1. 28, page 23 Namas Chandra Advanced Mechanics of Materials Chapter 1 -17