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Vibrations of
Shells and Plates


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Page B

MECHANICAL ENGINEERING
A Series of Textbooks and Reference Books
Founding Editor

L. L. Faulkner
Columbus Division, Battelle Memorial Institute
and Department of Mechanical Engineering
The Ohio State University
Columbus, Ohio

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23.

Spring Designer’s Handbook, Harold Carlson
Computer-Aided Graphics and Design, Daniel L. Ryan
Lubrication Fundamentals, J. George Wills
Solar Engineering for Domestic Buildings, William A. Himmelman
Applied Engineering Mechanics: Statics and Dynamics, G. Boothroyd and
C. Poli

Centrifugal Pump Clinic, Igor J. Karassik
Computer-Aided Kinetics for Machine Design, Daniel L. Ryan
Plastics Products Design Handbook, Part A: Materials and Components;
Part B: Processes and Design for Processes, edited by Edward Miller
Turbomachinery: Basic Theory and Applications, Earl Logan, Jr.
Vibrations of Shells and Plates, Werner Soedel
Flat and Corrugated Diaphragm Design Handbook, Mario Di Giovanni
Practical Stress Analysis in Engineering Design, Alexander Blake
An Introduction to the Design and Behavior of Bolted Joints, John H.
Bickford
Optimal Engineering Design: Principles and Applications, James N.
Siddall
Spring Manufacturing Handbook, Harold Carlson
Industrial Noise Control: Fundamentals and Applications, edited by Lewis
H. Bell
Gears and Their Vibration: A Basic Approach to Understanding Gear
Noise, J. Derek Smith
Chains for Power Transmission and Material Handling: Design and Applications Handbook, American Chain Association
Corrosion and Corrosion Protection Handbook, edited by Philip A.
Schweitzer
Gear Drive Systems: Design and Application, Peter Lynwander
Controlling In-Plant Airborne Contaminants: Systems Design and Calculations, John D. Constance
CAD/CAM Systems Planning and Implementation, Charles S. Knox
Probabilistic Engineering Design: Principles and Applications, James N.
Siddall


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Page C

24. Traction Drives: Selection and Application, Frederick W. Heilich III and
Eugene E. Shube
25. Finite Element Methods: An Introduction, Ronald L. Huston and Chris E.
Passerello
26. Mechanical Fastening of Plastics: An Engineering Handbook, Brayton
Lincoln, Kenneth J. Gomes, and James F. Braden
27. Lubrication in Practice: Second Edition, edited by W. S. Robertson
28. Principles of Automated Drafting, Daniel L. Ryan
29. Practical Seal Design, edited by Leonard J. Martini
30. Engineering Documentation for CAD/CAM Applications, Charles S. Knox
31. Design Dimensioning with Computer Graphics Applications, Jerome C.
Lange
32. Mechanism Analysis: Simplified Graphical and Analytical Techniques,
Lyndon O. Barton
33. CAD/CAM Systems: Justification, Implementation, Productivity
Measurement, Edward J. Preston, George W. Crawford, and Mark E.
Coticchia
34. Steam Plant Calculations Manual, V. Ganapathy
35. Design Assurance for Engineers and Managers, John A. Burgess
36. Heat Transfer Fluids and Systems for Process and Energy Applications,
Jasbir Singh
37. Potential Flows: Computer Graphic Solutions, Robert H. Kirchhoff
38. Computer-Aided Graphics and Design: Second Edition, Daniel L. Ryan
39. Electronically Controlled Proportional Valves: Selection and Application,
Michael J. Tonyan, edited by Tobi Goldoftas

40. Pressure Gauge Handbook, AMETEK, U.S. Gauge Division, edited by
Philip W. Harland
41. Fabric Filtration for Combustion Sources: Fundamentals and Basic Technology, R. P. Donovan
42. Design of Mechanical Joints, Alexander Blake
43. CAD/CAM Dictionary, Edward J. Preston, George W. Crawford, and Mark
E. Coticchia
44. Machinery Adhesives for Locking, Retaining, and Sealing, Girard S.
Haviland
45. Couplings and Joints: Design, Selection, and Application, Jon R.
Mancuso
46. Shaft Alignment Handbook, John Piotrowski
47. BASIC Programs for Steam Plant Engineers: Boilers, Combustion, Fluid
Flow, and Heat Transfer, V. Ganapathy
48. Solving Mechanical Design Problems with Computer Graphics, Jerome
C. Lange
49. Plastics Gearing: Selection and Application, Clifford E. Adams
50. Clutches and Brakes: Design and Selection, William C. Orthwein
51. Transducers in Mechanical and Electronic Design, Harry L. Trietley
52. Metallurgical Applications of Shock-Wave and High-Strain-Rate
Phenomena, edited by Lawrence E. Murr, Karl P. Staudhammer, and
Marc A. Meyers
53. Magnesium Products Design, Robert S. Busk


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Page D

54. How to Integrate CAD/CAM Systems: Management and Technology,
William D. Engelke
55. Cam Design and Manufacture: Second Edition; with cam design software
for the IBM PC and compatibles, disk included, Preben W. Jensen
56. Solid-State AC Motor Controls: Selection and Application, Sylvester
Campbell
57. Fundamentals of Robotics, David D. Ardayfio
58. Belt Selection and Application for Engineers, edited by Wallace D.
Erickson
59. Developing Three-Dimensional CAD Software with the IBM PC, C. Stan
Wei
60. Organizing Data for CIM Applications, Charles S. Knox, with contributions
by Thomas C. Boos, Ross S. Culverhouse, and Paul F. Muchnicki
61. Computer-Aided Simulation in Railway Dynamics, by Rao V. Dukkipati
and Joseph R. Amyot
62. Fiber-Reinforced Composites: Materials, Manufacturing, and Design, P. K.
Mallick
63. Photoelectric Sensors and Controls: Selection and Application, Scott M.
Juds
64. Finite Element Analysis with Personal Computers, Edward R. Champion,
Jr., and J. Michael Ensminger
65. Ultrasonics: Fundamentals, Technology, Applications: Second Edition,
Revised and Expanded, Dale Ensminger
66. Applied Finite Element Modeling: Practical Problem Solving for
Engineers, Jeffrey M. Steele
67. Measurement and Instrumentation in Engineering: Principles and Basic
Laboratory Experiments, Francis S. Tse and Ivan E. Morse
68. Centrifugal Pump Clinic: Second Edition, Revised and Expanded, Igor J.

Karassik
69. Practical Stress Analysis in Engineering Design: Second Edition, Revised
and Expanded, Alexander Blake
70. An Introduction to the Design and Behavior of Bolted Joints: Second
Edition, Revised and Expanded, John H. Bickford
71. High Vacuum Technology: A Practical Guide, Marsbed H. Hablanian
72. Pressure Sensors: Selection and Application, Duane Tandeske
73. Zinc Handbook: Properties, Processing, and Use in Design, Frank Porter
74. Thermal Fatigue of Metals, Andrzej Weronski and Tadeusz Hejwowski
75. Classical and Modern Mechanisms for Engineers and Inventors, Preben
W. Jensen
76. Handbook of Electronic Package Design, edited by Michael Pecht
77. Shock-Wave and High-Strain-Rate Phenomena in Materials, edited by
Marc A. Meyers, Lawrence E. Murr, and Karl P. Staudhammer
78. Industrial Refrigeration: Principles, Design and Applications, P. C. Koelet
79. Applied Combustion, Eugene L. Keating
80. Engine Oils and Automotive Lubrication, edited by Wilfried J. Bartz
81. Mechanism Analysis: Simplified and Graphical Techniques, Second
Edition, Revised and Expanded, Lyndon O. Barton
82. Fundamental Fluid Mechanics for the Practicing Engineer, James W.
Murdock


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83. Fiber-Reinforced Composites: Materials, Manufacturing, and Design,
Second Edition, Revised and Expanded, P. K. Mallick
84. Numerical Methods for Engineering Applications, Edward R. Champion, Jr.
85. Turbomachinery: Basic Theory and Applications, Second Edition,
Revised and Expanded, Earl Logan, Jr.
86. Vibrations of Shells and Plates: Second Edition, Revised and Expanded,
Werner Soedel
87. Steam Plant Calculations Manual: Second Edition, Revised and
Expanded, V. Ganapathy
88. Industrial Noise Control: Fundamentals and Applications, Second Edition,
Revised and Expanded, Lewis H. Bell and Douglas H. Bell
89. Finite Elements: Their Design and Performance, Richard H. MacNeal
90. Mechanical Properties of Polymers and Composites: Second Edition,
Revised and Expanded, Lawrence E. Nielsen and Robert F. Landel
91. Mechanical Wear Prediction and Prevention, Raymond G. Bayer
92. Mechanical Power Transmission Components, edited by David W. South
and Jon R. Mancuso
93. Handbook of Turbomachinery, edited by Earl Logan, Jr.
94. Engineering Documentation Control Practices and Procedures, Ray E.
Monahan
95. Refractory Linings Thermomechanical Design and Applications, Charles
A. Schacht
96. Geometric Dimensioning and Tolerancing: Applications and Techniques
for Use in Design, Manufacturing, and Inspection, James D. Meadows
97. An Introduction to the Design and Behavior of Bolted Joints: Third
Edition, Revised and Expanded, John H. Bickford
98. Shaft Alignment Handbook: Second Edition, Revised and Expanded,
John Piotrowski
99. Computer-Aided Design of Polymer-Matrix Composite Structures, edited

by Suong Van Hoa
100. Friction Science and Technology, Peter J. Blau
101. Introduction to Plastics and Composites: Mechanical Properties and
Engineering Applications, Edward Miller
102. Practical Fracture Mechanics in Design, Alexander Blake
103. Pump Characteristics and Applications, Michael W. Volk
104. Optical Principles and Technology for Engineers, James E. Stewart
105. Optimizing the Shape of Mechanical Elements and Structures, A. A.
Seireg and Jorge Rodriguez
106. Kinematics and Dynamics of Machinery, Vladimír Stejskal and Michael
Valásek
107. Shaft Seals for Dynamic Applications, Les Horve
108. Reliability-Based Mechanical Design, edited by Thomas A. Cruse
109. Mechanical Fastening, Joining, and Assembly, James A. Speck
110. Turbomachinery Fluid Dynamics and Heat Transfer, edited by Chunill Hah
111. High-Vacuum Technology: A Practical Guide, Second Edition, Revised
and Expanded, Marsbed H. Hablanian
112. Geometric Dimensioning and Tolerancing: Workbook and Answerbook,
James D. Meadows


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113. Handbook of Materials Selection for Engineering Applications, edited by

G. T. Murray
114. Handbook of Thermoplastic Piping System Design, Thomas Sixsmith and
Reinhard Hanselka
115. Practical Guide to Finite Elements: A Solid Mechanics Approach, Steven
M. Lepi
116. Applied Computational Fluid Dynamics, edited by Vijay K. Garg
117. Fluid Sealing Technology, Heinz K. Muller and Bernard S. Nau
118. Friction and Lubrication in Mechanical Design, A. A. Seireg
119. Influence Functions and Matrices, Yuri A. Melnikov
120. Mechanical Analysis of Electronic Packaging Systems, Stephen A.
McKeown
121. Couplings and Joints: Design, Selection, and Application, Second Edition,
Revised and Expanded, Jon R. Mancuso
122. Thermodynamics: Processes and Applications, Earl Logan, Jr.
123. Gear Noise and Vibration, J. Derek Smith
124. Practical Fluid Mechanics for Engineering Applications, John J. Bloomer
125. Handbook of Hydraulic Fluid Technology, edited by George E. Totten
126. Heat Exchanger Design Handbook, T. Kuppan
127. Designing for Product Sound Quality, Richard H. Lyon
128. Probability Applications in Mechanical Design, Franklin E. Fisher and Joy
R. Fisher
129. Nickel Alloys, edited by Ulrich Heubner
130. Rotating Machinery Vibration: Problem Analysis and Troubleshooting,
Maurice L. Adams, Jr.
131. Formulas for Dynamic Analysis, Ronald L. Huston and C. Q. Liu
132. Handbook of Machinery Dynamics, Lynn L. Faulkner and Earl Logan, Jr.
133. Rapid Prototyping Technology: Selection and Application, Kenneth G.
Cooper
134. Reciprocating Machinery Dynamics: Design and Analysis, Abdulla S.
Rangwala

135. Maintenance Excellence: Optimizing Equipment Life-Cycle Decisions,
edited by John D. Campbell and Andrew K. S. Jardine
136. Practical Guide to Industrial Boiler Systems, Ralph L. Vandagriff
137. Lubrication Fundamentals: Second Edition, Revised and Expanded, D. M.
Pirro and A. A. Wessol
138. Mechanical Life Cycle Handbook: Good Environmental Design and
Manufacturing, edited by Mahendra S. Hundal
139. Micromachining of Engineering Materials, edited by Joseph McGeough
140. Control Strategies for Dynamic Systems: Design and Implementation,
John H. Lumkes, Jr.
141. Practical Guide to Pressure Vessel Manufacturing, Sunil Pullarcot
142. Nondestructive Evaluation: Theory, Techniques, and Applications, edited
by Peter J. Shull
143. Diesel Engine Engineering: Thermodynamics, Dynamics, Design, and
Control, Andrei Makartchouk
144. Handbook of Machine Tool Analysis, Ioan D. Marinescu, Constantin
Ispas, and Dan Boboc


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145. Implementing Concurrent Engineering in Small Companies, Susan
Carlson Skalak
146. Practical Guide to the Packaging of Electronics: Thermal and Mechanical

Design and Analysis, Ali Jamnia
147. Bearing Design in Machinery: Engineering Tribology and Lubrication,
Avraham Harnoy
148. Mechanical Reliability Improvement: Probability and Statistics for
Experimental Testing, R. E. Little
149. Industrial Boilers and Heat Recovery Steam Generators: Design,
Applications, and Calculations, V. Ganapathy
150. The CAD Guidebook: A Basic Manual for Understanding and Improving
Computer-Aided Design, Stephen J. Schoonmaker
151. Industrial Noise Control and Acoustics, Randall F. Barron
152. Mechanical Properties of Engineered Materials, Wolé Soboyejo
153. Reliability Verification, Testing, and Analysis in Engineering Design, Gary
S. Wasserman
154. Fundamental Mechanics of Fluids: Third Edition, I. G. Currie
155. Intermediate Heat Transfer, Kau-Fui Vincent Wong
156. HVAC Water Chillers and Cooling Towers: Fundamentals, Application,
and Operation, Herbert W. Stanford III
157. Gear Noise and Vibration: Second Edition, Revised and Expanded, J.
Derek Smith
158. Handbook of Turbomachinery: Second Edition, Revised and Expanded,
edited by Earl Logan, Jr., and Ramendra Roy
159. Piping and Pipeline Engineering: Design, Construction, Maintenance,
Integrity, and Repair, George A. Antaki
160. Turbomachinery: Design and Theory, Rama S. R. Gorla and Aijaz Ahmed
Khan
161. Target Costing: Market-Driven Product Design, M. Bradford Clifton, Henry
M. B. Bird, Robert E. Albano, and Wesley P. Townsend
162. Fluidized Bed Combustion, Simeon N. Oka
163. Theory of Dimensioning: An Introduction to Parameterizing Geometric
Models, Vijay Srinivasan

164. Handbook of Mechanical Alloy Design, edited by George E. Totten, Lin
Xie, and Kiyoshi Funatani
165. Structural Analysis of Polymeric Composite Materials, Mark E. Tuttle
166. Modeling and Simulation for Material Selection and Mechanical Design,
edited by George E. Totten, Lin Xie, and Kiyoshi Funatani
167. Handbook of Pneumatic Conveying Engineering, David Mills, Mark G.
Jones, and Vijay K. Agarwal
168. Clutches and Brakes: Design and Selection, Second Edition, William C.
Orthwein
169. Fundamentals of Fluid Film Lubrication: Second Edition, Bernard J.
Hamrock, Steven R. Schmid, and Bo O. Jacobson
170. Handbook of Lead-Free Solder Technology for Microelectronic
Assemblies, edited by Karl J. Puttlitz and Kathleen A. Stalter
171. Vehicle Stability, Dean Karnopp
172. Mechanical Wear Fundamentals and Testing: Second Edition, Revised
and Expanded, Raymond G. Bayer


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174.
175.
176.
177.
178.
179.
180.

181.
182.
183.
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Liquid Pipeline Hydraulics, E. Shashi Menon
Solid Fuels Combustion and Gasification, Marcio L. de Souza-Santos
Mechanical Tolerance Stackup and Analysis, Bryan R. Fischer
Engineering Design for Wear, Raymond G. Bayer
Vibrations of Shells and Plates: Third Edition, Revised and Expanded,
Werner Soedel
Refractories Handbook, edited by Charles A. Schacht
Practical Engineering Failure Analysis, Hani M. Tawancy, Anwar UlHamid, and Nureddin M. Abbas
Mechanical Alloying and Milling, C. Suryanarayana
Mechanical Vibration: Analysis, Uncertainties, and Control, Second
Edition, Revised and Expanded, Haym Benaroya
Design of Automatic Machinery, Stephen J. Derby
Practical Fracture Mechanics in Design: Second Edition, Revised and
Expanded, Arun Shukla
Practical Guide to Designed Experiments, Paul D. Funkenbusch

Additional Volumes in Preparation
Mechanical Engineering Software
Spring Design with an IBM PC, Al Dietrich
Mechanical Design Failure Analysis: With Failure Analysis System Software for
the IBM PC, David G. Ullman



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Page i

Vibrations of
Shells and Plates
Third Edition,
Revised and Expanded

Werner Soedel
Dept. of Mechanical Engineering
Purdue University
West Lafayette, Indiana

Marcel Dekker, Inc.

New York


Although great care has been taken to provide accurate and current information,
neither the author(s) nor the publisher, nor anyone else associated with this
publication, shall be liable for any loss, damage, or liability directly or indirectly
caused or alleged to be caused by this book. The material contained herein is not
intended to provide specific advice or recommendations for any specific situation.

Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe.
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A catalog record for this book is available from the Library of Congress.
ISBN: 0-8247-5629-0
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Neither this book nor any part may be reproduced or transmitted in any form or by any
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Current printing (last digit):
10 9 8 7 6 5 4 3 2 1
PRINTED IN THE UNITED STATES OF AMERICA


To my grandchildren: Amber, Nicholas, Ashleigh, April, Jackson,

Broderic, Thomas, Carter, Kathleen and the yet unborn



Preface to the Third Edition

The third edition of Vibrations of Shells and Plates contains a significant
amount of new material, in part fundamental type, and in part it consists of
important application examples. Several of the added topics were suggested
by readers of the earlier editions.
In Chapter 2, on deep shell equations, Section 2.12 describes how to
obtain radii of curvature for any shell geometry analytically if they cannot
easily be determined by inspection. To Section 3.5, Other Geometries, an
example of a parabolic cylindrical shell has been added. To Chapter 4, on
nonshell structures, Section 4.5 was added to show that Love’s equations
can also be reduced to the special case of a circular cylindrical tube that
oscillates in torsional motion. This equation is further reduced to the
classical torsion shaft. The reduction is not obvious because transverse
shear deformation is assumed to be small in the standard Love’s theory. It
is therefore illustrative from an educational viewpoint that this reduction is
possible without resorting to the material of Chapter 12, where transverse
shear deformation is considered.
A significant amount of new material has been added to Chapter 5, on
natural frequencies and modes. Section 5.14 describes the in-plane vibration
of rectangular plates and 5.15 discusses a case of in-plane vibration of
circular plates, because of the importance of this type of vibration to
piezoelectric crystals and spur gears, for example. The new Section 5.16
describes the closed-form solution of the natural frequencies and modes
of a circular cylindrical shell segment, which supplements Section 5.5,
which examines the closed cylindrical shell. Finally, a relatively substantial

Section 5.17 has been added on natural frequency and mode solutions
v


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Preface to the Third Edition

by power series because of the importance of this approach to solving
differential vibration equations where the solutions cannot be expressed in
terms of trigonometric, hyperbolic, Bessel, Legendre, or other functions.
This approach is usually not discussed in typical standard textbooks on
vibration, despite its potential usefulness and historical importance.
Three more cases of technical significance were added to Chapter 6,
on simplified shell equations. Section 6.16 was added to present the case of
a closed-form solution of a special type of toroidal shell, which is limited
in its application but useful from a theoretical viewpoint. While the barrelshaped shell is discussed in Section 6.13 using a Donnell-Mushtari-Vlasov
simplification, a more exact solution is now also given in the new Section
6.17, where the importance of avoiding shells of zero Gaussian curvature,
if higher natural frequencies are desired, is now clearly illustrated. Finally,
an example of a doubly curved plate solution based on the DonnellMushtari-Vlasov theory is now given in Section 6.18. Again, all new cases
are significant from an applications viewpoint and should be helpful to
researchers and practicing engineers.
While the first and second editions identify strain energy expressions
in a general way, and can be worked out for any case, the new edition
includes explicit strain energy equations for a variety of standard cases,
for the purpose of quick reference. These expressions, now given in the
new Section 7.7, are typically used in energy methods of vibration analysis,
notably in the Rayleigh-Ritz method.
In forced vibrations, an initial value example has been added as

Section 8.17 that shows the response of a plate to an initial displacement
that is equal to static sag due to the weight of the plate. The concept of
modal mass, stiffness, damping, and forcing is now introduced in Section
8.18. Explicitly introduced in Section 8.19 is the response of shells to
periodic forcing. The general solution for shells is illustrated by the special
case of a plate in Section 8.20. Finally, in Section 8.21, the phenomenon of
beating is discussed by way of an example.
In Section 9.9, plate examples illustrating the application of the
dynamic Green’s function have been added. Also solved by way of the
dynamic Green’s function is the case of a ring that is impacted by a point
mass.
The response of a ring on an elastic foundation to a harmonic point
moment excitation is solved in Section 10.6. Following this, for the first time
a moment loading dynamic Green’s function is formulated for shells and
plates in general in Section 10.7 and illustrated by an example.
Added to the subchapter on complex receptance is a description of
how to express such complex receptances in terms of magnitudes and phase
angles. The new Section 13.12 shows how one can subtract systems from


Preface to the Third Edition

vii

each other, which is more subtle than reversing additions by changing plus
signs to minus signs in the receptance expressions. The receptance treatment
of three or more systems connected by one displacement each was added in
Section 13.13 and illustrated on hand of connected plates in Section 13.14.
Also in this chapter is the solution of a continuous plate on two interior
knife edges by way of a receptance formulation of three plates connected

by moments.
While Chapter 14 in the first edition pointed out that the complex
modulus model of hysteretic damping is valid for harmonic forcing,
Section 14.4 now describes how it is also used for steady-state periodic
response calculations.
Added as Section 15.9 is the analysis of shells composed of
homogeneous and isotropic lamina (so-called sandwich shells), because of
their technical importance, and examples are presented in Section 15.10.
Also, because of their general technical importance as a class of cases,
the equations or motion of shells of revolution that spin about their axes are
now derived explicitly in Section 16.7 and a reduced example, the spinning
disk, is discussed in Section 16.8.
A significant amount of important new material has been added to
the chapter on elastic foundations. The force transmission into the base of
the elastic foundation is analyzed in Section 18.6, and a special illustration
taken from simplified tire analysis—namely, the vertical force transmission
through a rigid wheel that supports by way of an elastic foundation an
elastic ring—was added in Section 18.7. This case has implications beyond
the tire application, however. The general response of shells on elastic
foundations on base excitations is now presented in Section 18.8 and plate
examples are given in Section 18.9. As stimulated by tire applications,
Section 18.10 shows how natural frequencies and modes of a ring on
an elastic foundation in point ground contact may be obtained from the
natural frequencies and rings not in ground contact. This leads indirectly
to the results of Section 18.11 in which the ground contact motion creates
a harmonic point excitation. Important resonance effects are discussed.
In closing, the goals of the first and second editions are preserved
by the additions made in this third edition, namely: (1) to present the
foundation of the theory of vibration of shells and other structures, (2)
to present analytical solutions that illustrate the behavior of vibrating

shells and other structures and to give important general information to
designers of such structures, (3) to present basic information needed for
the development of finite element and finite difference programs (see also
Chapter 21), and (4) to allow such programs to be checked out against some
of the exact results collected in this book.
The remarks in the Preface to the Second Edition on how to use this
book in teaching are still valid. The book contains too much material to be


viii

Preface to the Third Edition

covered in a standard three-credit course. Chapters 2 to 8 should be treated
in depth, comprising a major part of the 45 lectures per semester that are
typically available. Then, approximately six chapters can typically be added,
with the topics to be treated a function of the interests of the graduate
students and/or lecturer. The new material that has been added to the third
edition contributes to the range of choices, and certain new examples will
enrich the fundamental lectures. The book may, of course, also be used in
a mode of self-study.
I am indebted to Prof. J. Kim of the University of Cincinnati
and his students, who scrutinized the second edition carefully and gave
me a list of (fortunately minor) transcription, typing, and typesetting
corrections (which were corrected in the second printing without thanking
them in print). I am also grateful for contributions by D. T. Soedel,
F. P. Soedel, and S. M. Soedel and by various students of my graduate
course ME 664, “Vibrations of Continuous Systems,” who either checked
many of the new additions by way of independent assignments, made
suggestions, or found a small remnant of typesetting errors. In my 2003

class, they were: S. Basak, N. Bilal, R. Deng, M. R. Duncan, J. C. Huang,
R. J. Hundhausen, J. W. Kim, U. J. Kim, A. A. Kulkarni, A. Kumar,
T. Puri, L. B. Sharos, T. S. Slack, M. C. Strus, D. N. Vanderlugt, A. Vyas,
F. X. Wang, C. L. Yang, and K. H. Yum. Also contributing, from the
2001 class, were: H. V. Chowdhari, R. S. Grinnip III, Y. J. Kim, Y. Pu,
B. H. Song, S. J. Thorpe, and M. R. Tiller. Earlier classes contributed
also, directly or indirectly, and I regret that the names of these individuals
have not been recorded by me. Finally, general assistance in preparing
lecture notes and generating from them this third edition was provided by
D. K. Cackley, M. F. Schaaf-Soedel, and A. S. Greiber-Soedel.
Werner Soedel


Preface to the Second Edition

The second edition of Vibration of Shells and Plates contains some revisions
and a significant amount of new material. The new material reflects the
latest developments in this field and meets the need of graduate students
and practicing engineers to become acquainted with additional topics such
as traveling modes in rotating shells, thermal effects, and fluid loading.
Love’s theory remains the fundamental theory for deep shell
equations (Chapter 2) since it can be shown that all the other linear
thin shell theories (Flugges,
ă
Novozhilovs, etc.) are based on relatively
minor—in a practical sense, most likely unimportant—extensions. This
edition includes a new section on other deep shell theories and another
section shows that the derived equations are also valid for shells of
nonuniform thickness, except where bending and membrane stiffnesses
become functions of the surface coordinates. Because Hamilton’s principle

is used for derivations throughout the book, a discussion of it and a simple
example are now included in Chapter 2.
While there are obviously a very large number of potential shell
geometries, two more have been added to the chapter on equations of
motion for commonly occurring geometries (Chapter 3). Torodial shells
occur in engineering as aircraft and automobile tires, space station designs,
and segments of such shells from impellers of pumps and fluid couplings.
The equations for a cylindrical shell of noncircular cross section have been
added in order to have one example of a shell that is not a shell of
revolution, and also because it occurs quite commonly in pressure vessels.
In Chapter 5, where natural frequencies and modes are discussed,
a formal separation of space and time variables has been added based
ix


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Preface to the Second Edition

on the observations that the common way of arguing that the motion
in time is obviously harmonic at a natural frequency is not necessarily
accepted by new students of shell behavior. For optimal learning, frequent
detailed explanations are provided. Also included now is a section on how
simultaneous partial differential equations of the type treated here can be
uncoupled.
Because inextensional approximation is particularly useful for rings, a
section on this topic has been added to Chapter 6 on simplified equations.
Chapter 7 covers approximate solution techniques. The discussion of
the Galerkin technique, which in the first edition had been condensed to a
point where clarity suffered, is now significantly expanded in Chapter 7.

Again based on teaching experiences, it has become desirable to
discuss more extensively the use of the Dirac delta function when describing
point forces in space and impulses in time. Also added to Chapter 8
is a discussion of the necessary two orthogonal sets of natural modes
for shells of revolution, described by two different phase angles. Finally,
two relatively detailed examples for a circular cylindrical shell have been
included, one dealing with a harmonic response, the other with an initial
value problem.
Chapter 9 has been expanded to include the harmonic Green’s
function as an introduction to transfer function techniques such as the
receptance method.
A significant amount of new material can be found in Chapter
13, on combinations of structures, because of a strong interest in modal
synthesis by industry. Sections added show the forced response of combined
structures—how to treat systems joined by springs (important from a
vibration isolation point of view) and how to approach displacement
excitations—and discuss receptances that are complex numbers. The section
on dynamic absorbers is now expanded to include the forced behavior.
As additional examples of composite structures, two examples on the
vibration of net or textile sheets have been added to Chapter 15.
Because Coriolis effects in spinning shells of revolution create the
phenomenon of traveling modes, Chapter 16, on rotating structures, has
been added to develop the theory and give several illustrative examples
of significance. The subject is introduced by way of spinning strings and
beams, and the rotating ring is discussed extensively because of its many
practical applications. Also given is an example of a rotating circular
cylindrical shell.
Heating can influence or excite vibrations; thus the new Chapter 17
extends the basic theory to include thermal effects.
At times, one encounters shells or plates that are supported by

an elastic medium. Often, the elastic medium can be modeled as an


Preface to the Second Edition

xi

elastic foundation consisting of linear springs, as presented in Chapter 18.
This chapter introduces similitude arguments and is therefore also an
introduction to Chapter 19.
In Chapter 19, because of the importance of scaling to practical
engineers who often study small models of structures, and because of its
importance to rules of design, specialized similitudes for various structural
elements are presented. Exact and approximate scaling relationships are
derived. Also, the proper way of nondimensionalizing results is discussed.
Shell and plate structures often contain or are in contact with liquids
or gases. The equations of motion of liquids or gases are derived by
reduction from the equations of motion of three-dimensional elastic solids,
and the necessary boundary conditions are discussed. One section gives an
introduction to noise radiation from a shell by way of an example. The
study of engineering acoustics is closely related to the vibration of shell
structures, and Chapter 20 is a natural lead-in to this tropic. Also discussed
by way of an example is the topic of the interaction of structures with
incompressible liquids having free surfaces.
A new Chapter 21, on discretizing approaches, discusses finite
difference and finite element techniques for obtaining natural modes
and frequencies and also the forced response from the resulting matrix
equations. Also included is an example of a finite element for shells of
revolution.
At this point it is appropriate to suggest how the second edition is

best used for teaching. The prerequisites remain: an introductory vibration
course and some knowledge of boundary value problem mathematics. Also,
it is still true that Chapters 2 through 8 should be treated in depth. My
usual way of operating, considering a full semester of 45 lectures, is to
accomplish this in approximately half of the available time. Then I select
approximately six chapters of additional material from the remaining 13
chapters, with the topics changing from year to year (depending to some
extent on the interest areas of the students). I treat these in relative depth
and then allow myself three lectures at the end of the semester to survey the
rest of the chapters.
Paradoxically, the material presented in this edition has also been used
by me several times in two- and three-day courses for practicing engineers
in industry, without requiring an appreciable amount of mathematics. In
this case I use the book to outline the mathematical developments but dwell
extensively on the physical principles and on the practical implications of
the results. I have found that this is very useful to engineers who work
mainly with ready-made finite element codes, work purely experimentally,
or are designers of shell structures, and even to engineering managers who
need an overview of the subject. Those who have the proper background,


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Preface to the Second Edition

and are so inclined, seem able to use the book later in a program of
self-study.
Several persons need to be mentioned for their direct or indirect help
on this second edition. They are, in no particular order, S. M. Soedel, F. P.
Soedel, D. T. Soedel, J. Alfred, R. Zadoks, Y. Chang, L. E. Kung, J. Blinka,

D. Allaei, J. S. Kim, J. Kim, H. W. Kim, S. Saigal, S. C. Huang, J. L. Lin,
M. P. Hsu, R. M. V. Pidaparti, D. S. Stutts, D. Huang, D. C. Conrad, H.
J. Kim, S. H. Kim, Z. Liu and G. P. Adams. I apologize if I have forgotten
someone. I would also like to express my appreciation to former students
in my graduate course who were able to detect nagging small errors (all
of them, I hope) that occurred in the writing and proofreading stages of
the original notes used in my lectures and on which the new material in
this second edition is based. I also thank those whose persistent questions
helped me determine how the material should be organized and presented.
Just as the first edition did, the second edition attempts to provide
information that is useful to the practicing engineer without losing sight of
the fact that the primary purpose is graduate education. Its usefulness as a
reference book has also been enhanced.
Werner Soedel


Preface to the First Edition

This book attempts to give engineering graduate students and practicing
engineers an introduction to the vibration behavior of shells and plates. It
is also hoped that it will prove to be a useful reference to the vibration
specialist. It fills a need in the present literature on this subject, since it is
the current practice to either discuss shell vibrations in a few chapters at
the end of texts on shell statics that may be well written but are too limited
in the selection of material, or to ignore shells entirely in favor of plates
and membranes, as in some of the better known vibration books. There
are a few excellent monographs on very specialized topics, for instance,
on natural frequencies and modes of cylindrical and conical shells. But a
unified presentation of shell and plate vibration, both free and forced, and
with complicating effects as they are encountered in engineering practice, is

still missing. This collection attempts to fill the gap.
The state of the art modern engineering demands that engineers have
a good knowledge of the vibration behavior of structures beyond the usual
beam and rod vibration examples. Vibrating shell and plate structures are
not only encountered by the civil, aeronautical, and astronautical engineer,
but also by the mechanical, nuclear, chemical, and industrial engineer. Parts
or devices such as engine liners, compressor shells, tanks, heat exchangers,
life support ducts, boilers, automotive tires, vehicle bodies, valve read
plates, and saw disks, are all composed of structural elements that cannot
be approximated as vibrating beams. Shells especially exhibit certain effects
that are not present in beams or even plates and cannot be interpreted by
engineers who are only familiar with beam-type vibration theory. Therefore,
this book stresses the understanding of basic phenomena in shell and plate
xiii


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Preface to the First Edition

vibrations and it is hoped that the material covered will be useful in
explaining experimental measurements or the results of the ever-increasing
number of finite element programs. While it is the goal of every engineering
manager that these programs will eventually be used as black boxes, with
input provided and output obtained by relatively untrained technicians,
reality shows that the interpretation of results of these programs requires a
good background in finite element theory and, in the case of shell and plate
vibrations, in vibration theory of greater depth and breadth than usually
provided in standard texts.
It is hoped that the book will be of interest to both the stress analyst

whose task it is to prevent failure and to the acoustician whose task it
is to control noise. The treatment is fairly complete as far as the needs
of the stress analysts go. For acousticians, this collection stresses those
applications in which boundary conditions cannot be ignored.
The note collection begins with a historical discussion of vibration
analysis and culminates in the development of Love’s equations of shells.
These equations are derived in Chapter 2 in curvilinear coordinates.
Curvilinear coordinates are used throughout as much as possible, because
of the loss of generality that occurs when specific geometries are singled
out. For instance, the effect of the second curvature cannot be recovered
from a specialized treatment of cylindrical shells. Chapter 3 shows the
derivation by reduction of the equations of some standard shell geometries
that have a tendency to occur in standard engineering practice, like the
circular cylindrical shell, the spherical shell, the conical shell, and so on.
In Chapter 4 the equations of motion of plates, arches, rings, beams, and
rods are obtained. Beams and rings are sometimes used as supplementary
examples in order to tie in the knowledge of beams that the reader may
have with the approaches and results of shell and plate analysis.
Chapter 5 discusses natural frequencies and modes. It starts with the
transversely vibrating beam, followed by the ring and plate. Finally, the
exact solution of the simply supported circular cylindrical shell is derived.
The examples are chosen in such a way that the essential behavior of these
structures is unfolded with the help of each previous example; the intent
is not to exhaust the number of possible analytical solutions. For instance,
in order to explain why there are three natural frequencies for any mode
number combination of the cylindrical shell, the previously given case of
the vibrating ring is used to illustrate modes in which either transverse or
circumferential motions dominate.
In the same chapter, the important property of orthogonality of
natural modes is derived and discussed. It is pointed out that when two or

more different modes occur at the same natural frequency, a superposition
mode may be created that may not be orthogonal, yet is measured by


Preface to the First Edition

xv

the experimenter as the governing mode shape. Ways of dealing with this
phenomenon are also pointed out.
For some important applications, it is possible to simplify the
equations of motion. Rayleigh’s simplification, in which either the bending
stiffness or the membrane stiffness is ignored, is presented. However, the
main thrust of Chapter 6 is the derivation and use of the Donnell-MushtariVlasov equations.
While the emphasis of Chapter 5 was on so-called exact solutions
(series solutions are considered exact solutions), Chapter 7 presents
some of the more common approximate techniques to obtain solutions
for geometrical shapes and boundary condition combinations that do
not lend themselves to exact analytical treatment. First, the variational
techniques known as the Rayleigh-Ritz technique and Galerkin’s method
and variational method are presented. Next, the purely mathematical
technique of finite differences is outlined, with examples. The finite element
method follows. Southwell’s and Dunkerley’s principles conclude the
chapter.
The forced behavior of shells and plates is presented in Chapters 8,
9, and 10. In Chapter 8, the model analysis approach is used to arrive
at the general solution for distributed dynamic loads in transverse and
two orthogonal in-plane directions. The Dirac delta function is then used
to obtain the solutions for point and line loads. Chapter 9 discusses
the dynamic Green’s functions approach and applies it to traveling load

problems. An interesting resonance condition that occurs when a load
travels along the great circles of closed shells of revolution is shown.
Chapter 10 extends the types of possible loading to the technically
significant set of dynamic moment loading, and illustrates it by investigating
the action of a rotation point moment as it may occur when rotating
unbalanced machinery is acting on a shell structure.
The influence of large initial stress fields on the response of shells and
plates is discussed in Chapter 11. First, Love’s equations are extended to
take this effort into account. It is then demonstrated that the equations
of motion of pure membranes and strings are a subset of these extended
equations. The effect of initial stress fields on the natural frequencies of
structures is then illustrated by examples.
In the original derivation of Love’s equations, transverse shear strains,
and therefore shear deflections, were neglected. This becomes less and less
permissible as the average distance between node lines associated with the
highest frequency of interest approaches the thickness of the structure. In
Chapter 12, the shear deformations are included in the shell equations. It is
shown that these equations reduce in the case of a rectangular plate and the
case of a uniform beam to equations that are well known in the vibration


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Preface to the First Edition

literature. Sample cases are solved to illustrate the effect shear deformation
has on natural frequencies.
Rarely are practical engineering structures simple geometric shapes. In
most cases the shapes are so complicated that finite element or difference
methods have to be used for accurate numerical results. However, there is

a category of cases in which the engineering structures can be interpreted
as being assembled of two or more classic shapes or parts. In Chapter 13,
the method of receptance is presented and used to obtain, for instance, very
general design rules for stiffening panels by ring- or beam-type stiffeners.
It is also shown that the receptance method gives elegant and easily
interpretable results for cases in which springs or masses are added to the
basic structure.
The formulation and use of equivalent viscous damping was
advocated in the forced vibration chapters. For steady-state harmonic
response problems a complex modulus is often used. In Chapter 14, this
type of structural damping, also called hystereses damping, is presented and
tied in with the viscous damping formulation.
Because of the increasing importance of composite material structures,
the equations of motion of laminated shells are presented and discussed in
Chapter 15, along with some simple examples.
This book evolved over a period of almost ten years from lecture
notes on the vibration of shells and plates. To present the subject in a
unified fashion made it necessary to do some original work in areas where
the available literature did not provide complete information. Some of
it was done with the help of graduate students attending my lectures,
for instance, R. G. Jacquot, U. R. Kristiansen, J. D. Wilken, M. Dhar,
U. Bolleter, and D. P. Powder. Especially talented in detecting errors were
M. G. Prasad, F. D. Wilken, M. Dhar, S. Azimi, and D. P. Egolf. Realizing
that I have probably forgotten some significant contributions, I would
like to single out in addition O. B. Dale, J. A. Adams, D. D. Reynolds,
M. Moaveni, R. Shashaani, R. Singh, J. R. Friley, J. DeEskinazi, F. Laville,
E. T. Buehlmann, N. Kaemmer, C. Hunckler, and J. Thompson, and extend
my appreciation to all my former students.
I would also like to thank my colleagues on the Purdue University
faculty for their direct or indirect advice.

If this book is used for an advanced course in structural vibrations of
about forty-five lectures, it is recommended that Chapters 2 through 8 be
treated in depth. If there is time remaining, highlights of the other chapters
can be presented. Recommend prerequisites are a first course in mechanical
vibrations and knowledge of boundary value problem mathematics.
Werner Soedel