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About Mechanical Library Our goal is to provide all the required Mechanical Engineering materials for free to anyone in the world, because no one owns education, it is equal right to everyone and no one should suffer the lack of books. Post a comment. Powered by Blogger. This book is written as a text for advanced undergraduates and graduate students in aerospace, civil, and mechanical engineering and applied mechanics.

It is also intended as a reference for practitioners. The book contains topics sufficient for two academic semesters or three quarters. Thus, there is enough variety that instructors of a one-semester course or one- or two-quarter courses can choose topics of interest to students.

Engineering structures and machines, such as airplanes, automobiles, bridges, spacecraft, buildings, electric generators, gas turbines, and so forth, are usually formed by connecting various parts or members.

In most structures or machines, the primary function of a member is to support or transfer external forces loads that act on it, without failing. Failure of a member may occur when it is loaded beyond its capacity to resist fracture, general yielding, excessive deflection, or instability. These types of failure depend on the nature of the load and the type of member. In elementary mechanics of materials, members subjected to axial loads, bending moments, and torsional forces are studied.

Simple formulas for the stress and deflection of such members are developed Gere, Some of these formulas are based on simplifying assumptions and as such must be subjected to certain restrictions when extended to new problems. In this book, many of these formulas are used and extended to applications of more complex problems.

But first we review, without derivation, some of the basic formulas from mechanics of materials and highlight the limitations to their application. We include a review of bars under axial load, circular rods subjected to torsion, and beams loaded in shear and bending.

In the equations that follow, dimensions are expressed in terms of force [F], length [L], and radians [rad]. In this book, we derive relations between load and stress or between load and deflection for a system or a component a member of a system. Our starting point is a description of the loads on the system, the geometry of the system including boundary conditions , and the properties of the material in the system.

Generally the load-stress relations describe either the distributions of normal and shear stresses on a cross section of the member or the stress components that act at a point in the member.

For a given member subjected to prescribed loads, the load-stress relations are based on the following requirements:. The equations of equilibrium or equations of motion for bodies not in equilibrium. The compatibility conditions continuity conditions that require deformed volume. Elements in the member to fit together without overlap or tearing Two different methods are used to satisfy requirements 1 and 2: the method of mechanics of materials and the method of general continuum mechanics.

Often, load-stress and loaddeflection relations are not derived in this book by general continuum mechanics methods. Instead, the method of mechanics of materials is used to obtain either exact solutions or reliable approximate solutions. In the method of mechanics of materials, the load-stress relations are derived first.

They are then used to obtain load-deflection relations for the member. A simple member such as a circular shaft of uniform cross section may be subjected to complex loads that produce a multiaxial state of stress.

However, such complex loads can be reduced to several simple types of load, such as axial, bending, and torsion. Each type of load, when acting alone, produces mainly one stress component, which is distributed over the cross section of the member. If the deformations of the member that result from one type of load do not influence the magnitudes of the other types of loads and if the material remains linearly elastic for the combined loads, the stress components.

In a complex member, each load may have a significant influence on each component of the state of stress. Then, the method of mechanics of materials becomes cumbersome, and the use of the method of continuum mechanics may be more appropriate.

To derive load-stress and load-deflection relations for specified structural members, the stress components must be related to the strain components. Consequently, in Chapter 3 we discuss linear stress-strain-temperature relations. These relations may be employed in the study of linearly elastic material behavior. In addition, they are employed in plasticity theories to describe the linearly elastic part of the total response of materials.

Because experimental studies are required to determine material properties e. To obtain needed isotropic elastic material properties, we employ a tension specimen Figure 1.

If lateral as well as longitudinal strains are measured for linearly elastic behavior of the tension specimen, the resulting stress-strain data represent the material response for obtaining the needed elastic constants for the material. The fundamental. The stress-strain-temperature relations presented in Chapter 3 are limited mainly to small strains and small rotations.

The reader interested in large strains and large rotations may refer to Boresi and Chong When a structural member is subjected to loads, its response depends not only on the type of material from which it is made but also on the environmental conditions and the manner of loading. Depending on how the member is loaded, it may fail by excessive dejection, which results in the member being unable to perform its design function; it may fail by plastic deformation general yielding , which may cause a permanent, undesirable change in shape; it may fail because of afracture break , which depending on the material and the nature of loading may be of a ductile type preceded by appreciable plastic deformation or of a brittle type with little or no prior plastic deformation.

Fatigue failure, which is the progressive growth of one or more cracks in a member subjected to repeated loads, often culminates in a brittle fracture type of failure. Another manner in which a structural member may fail is by elastic or plastic instability. In this failure mode, the structural member may undergo large displacements from its design configuration when the applied load reaches a critical value, the buckling load or instability load.

This type of failure may result in excessive displacement or loss of ability because of yielding or fracture to carry the design load. In addition to the failure modes already mentioned, a structural member may fail because of environmental corrosion chemical action. Most load-resisting members are designed on the basis of stress in the main body of the member, that is, in portions of the body not affected by the localized stresses at or near a surface of contact between bodies.

In other words, most failures by excessive elastic deflection, yielding, and fracture of members are associated with stresses and strains in portions of the body far removed from the points of application of the loads. In certain cases, however, the contact stresses created when surfaces of two bodies are pressed together by external loads are the significant stresses; that is, the stresses on or somewhat beneath the surface of contact are the major cause of failure of one or both of the bodies.

For example, contact stresses may be significant at the area 1. We note that, in each of these examples, the members do not necessarily remain in fixed contact. In fact, the contact stresses are often cyclic in nature and are repeated a very large number of times, often resulting in a fatigue failure that starts as a localized fracture crack associated with localized stresses.

The fact that contact stresses frequently lead to fatigue failure largely explains why these stresses may limit the load carrying capacity of the members in contact and hence may be the significant stresses in the bodies. This fracture progresses outwardly under the influence of the repeated wheel loads until the entire rail cracks or fractures. This fracture is called a transverse fissure failure. In contrast, bearings and gear teeth sometimes fail as a result of formation of pits pitting at the surface of contact.

The bottom of such a pit is often located approximately at the point of maximum shear stress. Steel tappets have been observed to fail by initiation of microscopic cracks at the surface that then spread and cause flaking. Chilled cast-iron tappets have failed by cracks that start beneath the surface, where the shear stress is highest, and spread to the surface, causing pitting failure.

The principal stresses at or on the contact area between two curved surfaces that are pressed together are greater than at a point beneath the contact area, whereas the maximum shear stress is usually greater at a point a small distance beneath the contact surface. Several investigators have attempted to solve this problem. Hertz was the first to obtain a satisfactory solution, although his solution gives only principal stresses in the contact area. As noted in, failures occur in many mechanical systems.

Failures Caused by fatigue culminate in cracks or fracture after a sufficient number of fluctuations of load. Fracture of a structural member as the result of repeated cycles of load or fluctuatingloads is commonly referred to as a fatigue failure orfatigue fracture.

The corresponding number of load cycles or the time during which the member is subjected to these loads before fracture occurs is referred to as thefatigue life of the member.

The fatigue life of a member is affected by many factors ASM, For example, it is affected by 1. Gauthier and Petrequin, ; Buxbaum et. In practice, accurate estimates of fatigue life are difficult to obtain, because for many materials, small changes in these conditions may strongly affect fatigue life. The designer may wish to rely on testing of full-scale members under in-service conditions.

However, testing of full-scale members is time consuming and costly. Therefore, data from laboratory tests of small material specimensare used to establish fatigue failure criteria, even though these data may not be sufficient to dethnine the fatigue life of the real member.

Nevertheless, laboratory tests are useful in determining the effect of load variables on fatigue life and in comparing the relative fatigue resistance of various materials and establishing the importance of fabrication methods, surface finish, environmental effects, etc.

As noted, the formulas for determining stresses in simple structural members and machine elements are based on the assumption that the distribution of stress on any section of a member can be expressed by a mathematical law or equation of relatively simple form.

For example, in a tension member subjected to an axial load the stress is assumed to be distributed uniformly over each cross section; in an elastic beam the stress on each cross section is assumed to increase directly with the distance from the neutral axis; etc. The assumption that the distribution of stress on a section of a simple member may be expressed by relatively simple laws may be in error in many cases.

The conditions that may cause the stress at a point in a member, such as a bar or beam, to be radically different from the value calculated from simple formulas include effects such as 1. These discontinuities cause sudden increases in the stress stress peaks at points near the stress raisers. The term stress gradient is used to indicate the rate of increase of stress as a stress raiser is approached. The stress gradient may have an influence on the damaging.

Often, large stresses resulting from discontinuities are developed in only a small portion of a member. Hence, these stresses are called localized stresses or simply stress concentrations.

In many cases, particularly in which the stress is highly localized, a mathematical analysis is difficult or impracticable. Then, experimental, numerical, or mechanical methods of stress analysis are used.

The solution for the values of stress concentrations by the theory of elasticity applied to members with known discontinuities or stress raisers generally involves differential equations that are difficult to solve. However, the elasticity method has been used with success to evaluate stress concentrations in members containing changes of section, such as that caused by a circular hole in a wide plate see Section In addition, the use of numerical methods, such as finite elements, has lead to approximate solutions to a wide range of stress concentration problems.

Experimental methods of determining stress concentrations may also prove of value in cases for which the elasticity method becomes excessively difficult to apply. Some experimental methods are primarily mechanical methods of solving for the significant stress; see for example, the first three of the list of methods given in the next paragraph.

These three methods tend to give values comparable with the elasticity method. Likewise, when a very short gage length is used over which the strain is measured with high precision, the elastic strain strain-gage method gives values of stress concentration closely approximating the elasticity value.

In the other methods mentioned, the properties of the materials used in the models usually influence the stress concentration obtained, causing values somewhat less than the elasticity values.

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