Design for Thermal Stresses

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Library of Congress Cataloging-in-Publication Data:

Barron, Randall F.

Design for thermal stresses / Randall F. Barron, Brian R. Barron.

p. cm.

Includes index.

ISBN 978-0-470-62769-3 (hardback); ISBN 978-1-118-09316-0 (ebk); ISBN 978-1-118-09317-7 (ebk); ISBN 978-1-118-09318-4 (ebk); ISBN 978-1-118-09429-7 (ebk); ISBN 978-1-118-09430-3 (ebk); ISBN 978-1-118-09453-2 (ebk)

1. Thermal stresses. I. Barron, Brian R. II. Title.

TA654.8.B37 2011

620.1′1296—dc23

2011024789

Preface

Situations involving thermal stresses arise in many engineering areas, from aerospace structures to zirconium-clad nuclear fuel rods. It is important for the engineer to recognize the importance of alleviation of thermal stresses and to have the tools to carry out this task. For example, in the design of cryogenic fluid transfer systems, the vacuum-jacketed transfer lines must accommodate differential contractions as large as an inch or a couple of centimeters when the inner pipe is cooled from ambient temperature to the cryogenic temperature range. In process industry systems, such as shell-and-tube heat exchangers, differential thermal expansion between the tube and the heat exchanger shell would result in mechanical failure if design approaches were not use to alleviate this situation. Even the seemingly mundane or everyday problem of buckling of concrete slabs in highways due to summer heating is a thermal stress problem.

Thermal stresses arise in each of these systems when the system components undergo a change in temperature while the component is mechanically constrained and not free to expand or contract with the temperature change. Often thermal stresses cannot be changed by making the part bigger. Thermal stresses arise as a result of constraints and thermal stresses may be controlled safely by reducing the extent of the restraint. For example, flexible expansion bellows are used in cryogenic fluid transfer lines to reduce the constraint and forces between the cold inner line and the warm outer vacuum jacket to acceptable levels. The problem of thermal buckling of concrete slabs can be alleviated by providing a sufficiently large gap to allow some unconstrained expansion of the highway slab.

Some of the treatments of thermal stresses concentrate on the analysis of the thermomechanical system, which is an important consideration. However, the design process involves an interaction of both the analysis and synthesis processes, and not simply an application of formulas. This design process is emphasized in this text. Example problems are included to illustrate the application of the principles for practicing engineers and student study, and homework problems are included to allow practice in applying the principles.

This text evolved from the authors' academic and industrial experiences. One of us (RFB) has taught senior undergraduate and graduate level mechanical engineering courses in the areas of thermal stresses, directed MS and PhD thesis and dissertation research projects, including studies of thermal stresses in the thermal shroud of a space environmental simulation chamber, and conducted continuing education courses involving thermal stress applications for practicing engineers for more than 3 decades. He has first-hand industrial experience in the area of thermal stress design, including application of thermal design principles in design and manufacture of cryogenic liquid storage and transfer systems (dewars and vacuum-jacketed transfer lines), heat exchangers, and space environmental simulation chambers. BRB has conducted research involving the development of a hybrid finite element and finite difference numerical technique for solving thermal problems involving ultrashort laser pulses in layered media, in which thermal stresses can present severe design challenges.

The first part of the text (Chapters 1–4) covers thermal stress design in bars, beams, and trusses, which involves a strength-of-materials approach. Both analytical and numerical design methods are presented. The second part of the book (Chapters 5–9) covers more advanced thermal design for plates, shells, and thick-walled vessels, which involves a theory of elasticity approach. The final chapter (Chapter 10) covers the problem of thermal buckling in columns, beams, plates, and shells. Material on thermal viscoelastic problems (creep, etc.) is not included because of space limitations. The material included in the appendixes includes a discussion of the SI units used for quantities in thermal stress problems, tables of material properties relating to thermal stresses, brief coverage of mathematical functions (Bessel and Kelvin functions), and the characteristics of matrices and determinants required for design and analysis of plates and shells.

The book is written for use in junior- or senior-level undergraduate engineering elective courses in thermal design (mechanical, chemical, or civil engineering) and for graduate-level courses in thermal stresses. The proposed text is intended for use as a textbook for these classes, and a sufficient number of classroom-tested homework problems are included for a one-semester course in thermal stress design.

In addition, the book is intended for use by practicing engineers in the process industries, cryogenic and space-related fields, heat exchanger industries, and other areas where consideration of thermal stresses is an important part of the design problem. Detailed example problems are included with emphasis on practical engineering systems. The proposed text would serve as a reference and a source of background material to help engineers accomplish their design tasks.

Our most heartfelt thanks and appreciation is extended to each of our spouses, Shirley and Kitty, who generously gave their support and encouragement during the months of book preparation.

Nomenclature

Greek letters

Chapter 1

Introduction

1.1 Definition of Thermal Stress

Thermal stresses are stresses that result when a temperature change of the material occurs in the presence of constraints. Thermal stresses are actually mechanical stresses resulting from forces caused by a part attempting to expand or contract when it is constrained.

Without constraints, there would be no thermal stresses. For example, consider the bar shown in Figure 1.1. If the bar were subjected to a temperature change ΔT of 20°C and the ends were free to move, the stress in the bar would be zero. On the other hand, if the same bar were subjected to the same temperature change and the ends were rigidly fixed (no displacement at the ends of the bar), stresses would be developed in the bar as a result of the forces (tensile or compressive) on the ends of the bar. These stresses are called thermal stresses.

Figure 1.1. Illustration of external constraints. (A) No constraint—the bar is free to expand or contract. Thermal stresses are not present. (B) External constraint—the bar has both ends rigidly fixed and no motion is possible. Thermal stresses are induced when the bar experiences a change in temperature.

There are two types of constraints as far as thermal stresses are concerned: (a) external constraints and (b) internal constraints. External constraints are restraints on the entire system that prevent expansion or contraction of the system when temperature changes occur. For example, if a length of pipe were fixed at two places by pipe support brackets, this constraint would be an external one.

Internal constraints are restraints present within the material because the material expands or contracts by different amounts in various locations, yet the material must remain continuous. Suppose the pipe in the previous example were simply supported on hangers, and the inner portion of the pipe were suddenly heated 10°C warmer than the outer surface by the introduction of a hot liquid into the pipe, as shown in Figure 1.2. If the outer surface remains at the initial temperature, the outer layers would not expand, because the outer temperature did not change, whereas the inner layers would tend to expand due to a temperature change. Thermal stresses will arise in this case because the inner layer of material and outer layer of material are not free to move independently. This type of constraint is an internal one.

Figure 1.2. Internal constraints. The inner surface is heated by the fluid and tends to expand, but the outer (cool) surface constrains the free motion. Thermal stresses are induced by this constraint.

1.2 Thermal–Mechanical Design

The design process involves more than solving the problem in a mathematical manner [Shigley and Mischke, 1989]. Ideally, there would be no design limitations other than safety. However, usually multiple factors must be considered when designing a product. A general design flowchart is shown in Figure 1.3.

Figure 1.3. General design flowchart.

Initially, there is usually a perceived need for a product, process, or system. The specifications for the item required to meet this need must be defined. Often this specification process is called preliminary design. The input and output quantities, operating environment, and reliability and economic considerations must be determined. For example, anticipated forces that would be applied to the system must be specified.

After the design problem has been defined, the next step involves an interaction between synthesis, analysis, and optimization. Generally, there are many possible design solutions for a given set of specifications. (Not everyone drives the same model of car, for example, although all car models provide a solution to the problem of transportation from one place to another.) Various components for a system may be proposed or synthesized. An abstract or mathematical model is developed for the analysis of the system. The results of the analysis may be used to synthesize an improved approach to the design solution. Based on specific criteria defining what is meant by the best system, the optimum or best system is selected to meet the design criteria.

In many cases, the optimal design emerging from the synthesis/analysis design phase is evaluated or tested. A prototype may be constructed and subjected to conditions given in the initial specifications for the system. After the evaluation phase has been completed successfully, the design then moves into the manufacturing and marketing arena.

When including consideration of thermal stresses in the design process, there are many cases in which the stresses are weakly dependent or even independent of the dimensions of the part. In these cases, the designer has at least three alternatives to consider: (a) materials selection, (b) limitation of temperature changes, and (c) relaxation of constraints.

For identical loading and environmental conditions, different materials will experience different thermal stresses. For example, a bar of 304 stainless steel, rigidly fixed at both ends, will experience a thermal stress that is about eight times that for Invar under the same conditions. Many factors in addition to thermal stresses dictate the final choice of materials in most design situations. Cost, ease of fabrication, and corrosion resistance are some of these factors. The designer may not have complete freedom to select a material based on thermal stress considerations alone.

A reduction of the temperature change will generally reduce thermal stresses. For a bar with rigidly fixed ends, if the temperature change is 50°C instead of 100°C, the thermal stress will be reduced to one-half of the thermal stress value for the larger temperature difference. In some steady-state thermal conditions, the temperature change of the part may be reduced by using thermal insulation. The design temperature change is often determined by factors that cannot be changed by the designer, however.

In many cases, the most effect approach to limit thermal stresses in the design stage is to reduce or relax the constraints on the system. The system may be made less constrained by introducing more flexible elements. This approach will be illustrated in the following chapters.

1.3 Factor of Safety in Design

In general, a part is designed such that it does not fail, except under desired conditions. For example, fuses must fail when a specified electric current is applied so that the electrical system may be protected. On the other hand, the wall thickness for a transfer line carrying liquid oxygen is selected such that the pipe does not rupture during operation of the system.

One issue in the design process is the level at which the part would tend to fail. This issue is addressed in the factor of safety fs. It is defined as the ratio of the failure parameter of the part to the design value of the same parameter. The first decision that the designer must make is to define what constitutes failure for the component or system under consideration. There are several failure criteria, including

a. breaking (rupture) of the part

b. excessive permanent deformation (yielding) of the part

c. breaking after fluctuating loads have been applied for a period of time (fatigue)

d. buckling (elastic instability)

e. excessive displacement or vibration

f. intolerable wear of the part

g. excessive noise generation by the part

The selection of the proper failure criteria is often the key to evaluating and planning for safety considerations.

If the failure criterion is the breaking or rupture of the part when stress is applied and the temperature is not high enough for creep effects to be significant, the failure parameter would be the ultimate strength Su for the material. On the other hand, if the failure criterion is yielding, then the yield strength Sy would be the failure parameter selected. In either case, the design parameter would be the maximum applied stress σ for the part. The factor of safety may be written as follows, for these cases:

1.1 1.1

The factor of safety may be defined in a similar manner for the other failure criteria.

The factor of safety may be prescribed, as is the case for such codes as the ASME Code for Unfired Pressure Vessels, Section VIII, Division 1, in which the factor of safety for design of cylindrical pressure vessels is set at 3.5. When the factor of safety is not prescribed, the designer must select it during the early stages of the design process. It is generally not economical to use a factor of safety that assures that absolutely no failure will occur under the worst possible combination of conditions. As a result, the selection of the factor is often based on the experience of the designer in related design situations.

In general, the value of the factor of safety reflects uncertainties in many factors involved in the design. Some of these uncertainties are as follows:

a. Scatter (uncertainty) in the material property data

b. Uncertainty in the maximum applied loading

c. Validity of simplifications (assumptions) in the model used to estimate the stresses or displacements for the system

d. The type of environment (corrosive, etc.) to which the part will be exposed

e. The extent to which initial stresses or deformations may be introduced during fabrication and assembly of the system

One of the more important factors in selection of the factor of safety is the extent to which human life and limb would be endangered if a failure of the system did occur or the possibility that failure would result in costly or unfavorable litigation.

The probabilistic or reliability-based design method [Shigley and Mischke, 1989] attempts to reduce the uncertainty in the design process; however, the disadvantage of this method lies in the fact that there is uncertainty in the uncertainty (probabilistic) data and the data is not extensive.

The uncertainty in the value of the strength parameter (ultimate or yield strength) may be alleviated somewhat by understanding the causes of the scatter in the data for the strength parameter. The values of the ultimate and yields strengths reported in the literature are generally average or mean values. In this case, 50 percent of the data lies above the mean value and 50 percent of the data lies below the reported value. A 1-in-2 chance would be excellent odds for a horse race, but this is not what one would likely employ in the design of a mechanical part. The value for the strength for which the probability of encountering a strength less than this value may be found from the normal probability distribution tables, if the standard deviation images/c01_I0002.gif is known from the strength data. The ultimate strength for this case is given by the following expression:

1.2 1.2

The quantity 1.2 is the average ultimate strength, and the factor kS is defined by

1.3 1.3

Values for the probability factor Fp are given in Table 1.1. Similar expressions may be used for the yield strength and fatigue strength.

Table 1.1. Probability Factor Fp for Various Probabilities of Survival

aThe survival rate is the probability that the actual strength value is not less than the S value given by eq. (1.2).

bThe failure rate is (1 − survival rate) or the probability that the actual strength is less than the S value given by eq. (1.2).

cFp is used in eq. (1.3).

Information on the standard deviation for the strength data is not readily available for all materials. If no specific standard deviation data are available, the following approximation may be used for the ratio images/c01_I0005.gif : 0.05 for ultimate strength; 0.075 for yield strength; and 0.10 for fatigue strength or endurance limit. The designer has the task of deciding what risk is acceptable for the minimum strength used in the design.

The reliability of the maximum anticipated loading (either mechanical or thermal) used in the design affects the value of the factor of safety selected. If there are safeguards (pressure relief valves, for example) on the system to prevent the loading from exceeding a selected level, then the factor of safety may be smaller than for the case in which the loading is more uncertain.

The validity of the mathematical model (set of assumptions or simplifications) used in the design has a definite influence on the factor of safety selected. It may be noted that a very complicated numerical analysis (or, as is commonly stated, a sophisticated analysis) is not precisely accurate, despite the opinions of some overly enthusiastic novice computer analysist. The estimated uncertainty in the analysis may be used as a guide in selecting the factor of safety.

Example 1-1

304 stainless steel is to be used in a design. A factor of safety of 2.5 is selected, based on yielding as the failure criterion. It is desired that the uncertainty (failure rate) for the yield strength be 0.1%, and the standard deviation for the yield strength data is 7.5 percent of the mean yield strength. Determine the stress to be used in the design.

The average yield strength for 304 stainless steel is found in Appendix B:

For a 0.1 percent failure rate or 99.9 percent survival rate, the probability factor from Table 1.1 is Fp = 3.09. The factor kS may be found from eq. (1.3):

The yield strength value that will be exceeded 99.9 percent of the time for 304 stainless steel is found from eq. (1.2):

The design stress is found from the definition of the factor of safety:

1.4 Thermal Expansion Coefficient

One of the important material properties related to thermal stresses is the thermal expansion coefficient. There are generally two thermal expansion coefficients that we will consider: (a) the linear thermal expansion coefficient, α, and (b) the volumetric thermal expansion coefficient, βt.

The linear thermal expansion coefficient is defined as the fractional