Pre-requisite: Knowledge of Strength of Materials is Preferable

Course Description & Outline :

Introduction to design, Stress based design, Stress Tensor, Principal stresses, stress invariants, Static Failure theories.

Also covers Static Failure theories for ductile, brittle materials. Von-mises, tresca, mohr-coulomb and modified mohr theories. Some problems around static failure theories will be solved.

Concept of SN diagram, endurance limit, correction to endurance limit. Effect of mean tresses and finite and infinite life based designs and associated safety factors. Problems will be discussed.

Schedule for Lecture Delivery

Session 1 : 5-Aug-2015 (2-4 pm)

Session 2 : 12-Aug-2015 (2-4 pm)

Session 3 : 19-Aug-2015 (2-4 pm)

Teacher Forum

Introduction to Design and Static Failure Theories

Introduction:

The meaning of the word design is rather broad. One can design something for aesthetic, ergonomic or economic purposes for example. In this course, we will focus on design of a machine parts or component so that it does not actually fail during its service. We aim to at least have some sort of back of the envelope calculations to decide whether a component which is made of a certain material, is of a certain geometric shape can actually withstand the loads it is subjected to during service. A machine is a wise collection of interrealated parts (machine elements) in a certain manner, to modify force or motion. So these parts are naturally subjected to various loads. To design a machine element, various factors should be considered. Some of these are:

Material - It is important to select a suitable material for each part based on strength, suitability of fabrication and cost.

Load - The geometric parameters of the component should be such that stresses are not too high. Loads cause stresses in parts and over stressed parts may lead to failure of the component. We can categorize loads into Static Loads and Dynamic Loads. Static loads are those which are slowly applied and are essentially constant with time whereas Dynamic loads change with time (this change can be brought either by changing magnitude of force or direction of force or both).

Manufacturing and assembly - Various selected materials, size and shape of parts should be such that manufacturing and assembly of these can be done with ease and low cost.

Safety, maintenance and reliability - Maintenance and safety are interlinked. Any machine should be designed in such a way that regular maintenance can be done ensuring operator safety. Reliability is the probability that the machine will not fail in use.

Cost and aesthetics - Obviously the nal product to the customer should be economical and good looking.

Thus we see that design is a challenging task and can involve several tasks. Our main goal, in this course will be to focus on preventing failure during service of machine. So we should rst focus how to modify geometry of components and select suitable material. Before, we delve into design and theories based on which we can identify failure, we shall look at some of the fundamentals of basics of stress analysis.

Some fundamentals of Stress analysis

Stress

It is not the magnitude of force which is important while designing machine elements. One must look at the resistance offered by the material to the external force and deformation. The physical quantity that measures this resistance is called stress. Consider a body subjected to forces and moments shown in Fig. 1 below.

A stress component is positive if directions associated with both subscripts are either in positive or in negative direction of the coordinate axes chosen. Thus all stresses shown in Fig. 1 are positive. Also, note that due to complementary property of shear, shear stresses with reversed indices are equal e.g Txy = Tyx (there are six independent stress components). Now we express stress tensor using following matrix notation, where each element of the matrix is a component of the stress tensor.

Principal Stresses

This stress tensor, we defined above in Eqn. 1 is defined at an arbitrary point of the loaded body. Furthermore, it is defined with a definite co-ordinate axes. For a different set of coordinate axes the same stress tensor will have different components. For example, consider two sets of coordinate axes, with one rotated with respect to the other, as in Fig. 2.

The stress components in expressed in rotated coordinate axes be given by following equation

Note that both Eq. 2 and Eqn. 1 define the same physical quantity. Only because the co-ordinate systems are different, the two stresses ( Eq. 2 and Eqn. 1) have different components. This is similar, in spirit to a vector (say velocity) with different components in different co-ordinate systems. It is possible to show mathematically that for any given state of stress there exists at least three planes on which stress components are perpendicular (no shear component). It is always possible to find such three mutually perpendicular planes. Such planes are called principal planes. The stresses acting on these plane are called principal normal stresses or simply principal stresses. The normals to these plane can be chosen as axes for a coordinate system which is known as principal coordinate system and those normals are principal axes. Stress tensor when expressed in principal coordinate system say x"y"z" is of the following form

Let us assume that state of stress at a point is known for coordinate system xyz and is given by eq.(1). Then principal stresses can be determined by solving the following set of equations

Hydrostatic and Deviatoric components of Stress Tensor

Factor of Safety

Slides for Introduction to Design and Static Failure Theories

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Quiz - I

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Solutions

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Fatigue Failure Theories

Introduction

Fully reversed uniaxial stresses

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Fluctuating uniaxial stresses

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Assignment II

Problems[Problems have been taken from references at the end of the document]