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

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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 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

17. 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