EGM 5653 Advanced Mechanics of Materials CHAPTER 1 Introduction EGM 5653 Advanced Mechanics of Materials

Objectives Sections 1.1 Review of Elementary mechanics 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

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. The practical application of mathematics and science to create, design, test, improve, and develop knowledge, research, money, business, economics, and technology. This is why engineering is such a challenging and demanding field of study: It involves areas of expertise that continue to evolve independently yet are required to perform together as part of the engineering process. Thus, an engineer must be expert in many areas, must know how to communicate knowledge between those areas, and must apply that knowledge to create, design, study, research, and invent all kinds of things. It is not uncommon for engineers to begin their careers as mathematicians, applied scientists, or even economists

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. The practical application of mathematics and science to create, design, test, improve, and develop knowledge, research, money, business, economics, and technology. This is why engineering is such a challenging and demanding field of study: It involves areas of expertise that continue to evolve independently yet are required to perform together as part of the engineering process. Thus, an engineer must be expert in many areas, must know how to communicate knowledge between those areas, and must apply that knowledge to create, design, study, research, and invent all kinds of things. It is not uncommon for engineers to begin their careers as mathematicians, applied scientists, or even economists

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, . The practical application of mathematics and science to create, design, test, improve, and develop knowledge, research, money, business, economics, and technology. This is why engineering is such a challenging and demanding field of study: It involves areas of expertise that continue to evolve independently yet are required to perform together as part of the engineering process. Thus, an engineer must be expert in many areas, must know how to communicate knowledge between those areas, and must apply that knowledge to create, design, study, research, and invent all kinds of things. It is not uncommon for engineers to begin their careers as mathematicians, applied scientists, or even economists

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. 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. The practical application of mathematics and science to create, design, test, improve, and develop knowledge, research, money, business, economics, and technology. This is why engineering is such a challenging and demanding field of study: It involves areas of expertise that continue to evolve independently yet are required to perform together as part of the engineering process. Thus, an engineer must be expert in many areas, must know how to communicate knowledge between those areas, and must apply that knowledge to create, design, study, research, and invent all kinds of things. It is not uncommon for engineers to begin their careers as mathematicians, applied scientists, or even economists

1.1.3 Bending of Beams-2 Deflection Bending stress Shear stress For other cases, bending moment and depends on all the applied load (concentrated and distributed) and external moments. Deflection Bending stress The practical application of mathematics and science to create, design, test, improve, and develop knowledge, research, money, business, economics, and technology. This is why engineering is such a challenging and demanding field of study: It involves areas of expertise that continue to evolve independently yet are required to perform together as part of the engineering process. Thus, an engineer must be expert in many areas, must know how to communicate knowledge between those areas, and must apply that knowledge to create, design, study, research, and invent all kinds of things. It is not uncommon for engineers to begin their careers as mathematicians, applied scientists, or even economists Shear stress

We are interested in normal and shear stresses on a given section. 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-plastic, visco-elastic). We are interested in normal and shear stresses on a given section. The practical application of mathematics and science to create, design, test, improve, and develop knowledge, research, money, business, economics, and technology. This is why engineering is such a challenging and demanding field of study: It involves areas of expertise that continue to evolve independently yet are required to perform together as part of the engineering process. Thus, an engineer must be expert in many areas, must know how to communicate knowledge between those areas, and must apply that knowledge to create, design, study, research, and invent all kinds of things. It is not uncommon for engineers to begin their careers as mathematicians, applied scientists, or even economists

1.2 Method of Analysis-2 Several complex loads can be broken down into simple loads and the results superimposed. This is called the theory of superposition and is valid for all linear problems. Linear problems should have linearity in loading, boundary conditions AND constitutive equations.. The practical application of mathematics and science to create, design, test, improve, and develop knowledge, research, money, business, economics, and technology. This is why engineering is such a challenging and demanding field of study: It involves areas of expertise that continue to evolve independently yet are required to perform together as part of the engineering process. Thus, an engineer must be expert in many areas, must know how to communicate knowledge between those areas, and must apply that knowledge to create, design, study, research, and invent all kinds of things. It is not uncommon for engineers to begin their careers as mathematicians, applied scientists, or even economists

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 Strain Engineering Stress True Strain True Stress The practical application of mathematics and science to create, design, test, improve, and develop knowledge, research, money, business, economics, and technology. This is why engineering is such a challenging and demanding field of study: It involves areas of expertise that continue to evolve independently yet are required to perform together as part of the engineering process. Thus, an engineer must be expert in many areas, must know how to communicate knowledge between those areas, and must apply that knowledge to create, design, study, research, and invent all kinds of things. It is not uncommon for engineers to begin their careers as mathematicians, applied scientists, or even economists Relationship between engineering and true quantities

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 The practical application of mathematics and science to create, design, test, improve, and develop knowledge, research, money, business, economics, and technology. This is why engineering is such a challenging and demanding field of study: It involves areas of expertise that continue to evolve independently yet are required to perform together as part of the engineering process. Thus, an engineer must be expert in many areas, must know how to communicate knowledge between those areas, and must apply that knowledge to create, design, study, research, and invent all kinds of things. It is not uncommon for engineers to begin their careers as mathematicians, applied scientists, or even economists

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. The practical application of mathematics and science to create, design, test, improve, and develop knowledge, research, money, business, economics, and technology. This is why engineering is such a challenging and demanding field of study: It involves areas of expertise that continue to evolve independently yet are required to perform together as part of the engineering process. Thus, an engineer must be expert in many areas, must know how to communicate knowledge between those areas, and must apply that knowledge to create, design, study, research, and invent all kinds of things. It is not uncommon for engineers to begin their careers as mathematicians, applied scientists, or even economists

Failure by excessive deflection Elastic deflection 1.4.1 Modes of Failure Failure by excessive deflection Elastic deflection Deflection caused by creep Failure by yielding Failure by fracture Sudden failure of brittle material Failure of cracked (flawed) members Progressive fracture (fatigue) Stress- corrosion cracking Failure by instability The practical application of mathematics and science to create, design, test, improve, and develop knowledge, research, money, business, economics, and technology. This is why engineering is such a challenging and demanding field of study: It involves areas of expertise that continue to evolve independently yet are required to perform together as part of the engineering process. Thus, an engineer must be expert in many areas, must know how to communicate knowledge between those areas, and must apply that knowledge to create, design, study, research, and invent all kinds of things. It is not uncommon for engineers to begin their careers as mathematicians, applied scientists, or even economists

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. The practical application of mathematics and science to create, design, test, improve, and develop knowledge, research, money, business, economics, and technology. This is why engineering is such a challenging and demanding field of study: It involves areas of expertise that continue to evolve independently yet are required to perform together as part of the engineering process. Thus, an engineer must be expert in many areas, must know how to communicate knowledge between those areas, and must apply that knowledge to create, design, study, research, and invent all kinds of things. It is not uncommon for engineers to begin their careers as mathematicians, applied scientists, or even economists

Solution 1.23 The practical application of mathematics and science to create, design, test, improve, and develop knowledge, research, money, business, economics, and technology. This is why engineering is such a challenging and demanding field of study: It involves areas of expertise that continue to evolve independently yet are required to perform together as part of the engineering process. Thus, an engineer must be expert in many areas, must know how to communicate knowledge between those areas, and must apply that knowledge to create, design, study, research, and invent all kinds of things. It is not uncommon for engineers to begin their careers as mathematicians, applied scientists, or even economists

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 The practical application of mathematics and science to create, design, test, improve, and develop knowledge, research, money, business, economics, and technology. This is why engineering is such a challenging and demanding field of study: It involves areas of expertise that continue to evolve independently yet are required to perform together as part of the engineering process. Thus, an engineer must be expert in many areas, must know how to communicate knowledge between those areas, and must apply that knowledge to create, design, study, research, and invent all kinds of things. It is not uncommon for engineers to begin their careers as mathematicians, applied scientists, or even economists

Solution 1.28, page 23 The practical application of mathematics and science to create, design, test, improve, and develop knowledge, research, money, business, economics, and technology. This is why engineering is such a challenging and demanding field of study: It involves areas of expertise that continue to evolve independently yet are required to perform together as part of the engineering process. Thus, an engineer must be expert in many areas, must know how to communicate knowledge between those areas, and must apply that knowledge to create, design, study, research, and invent all kinds of things. It is not uncommon for engineers to begin their careers as mathematicians, applied scientists, or even economists