96,99 €
Mehr erfahren.
- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Seismic Design and Analysis of Tanks A detailed view on the effects of seismic activity on tank structures As the use of above-ground and underground storage tanks (ASTs and USTs) continues to grow--with approximately 545,000 in the USA alone--the greatest threat to ASTs and USTs is earthquakes, causing the contamination of groundwater, a vital source of drinking water throughout the world. These tanks suffer a great deal of strain during an earthquake, as a complicated pattern of stress affects them, such that poorly designed tanks have leaked, buckled, or even collapsed during seismic events. Furthermore, in oil and gas industrial plants, the risk of damage is even more critical due to the effects of explosion, collapse, and air or soil contamination by chemical fluid spillages. Seismic Design and Analysis of Tanks provides the first in-depth discussion of the principles and applications of shell structure design and earthquake engineering analyses focused on tank structures, and it explains how these methodologies can help prevent the destruction of ASTs and USTs during earthquakes. Providing a thorough examination of the design, analysis, and performance of steel, reinforced concrete, and precast tanks, this book takes a look at tanks that are above-ground, underground, or elevated, anchored and unanchored, and rigid or flexible, and evaluates the efficacy of each method during times of seismic shaking--and it does so without getting bogged down in impenetrable mathematics and theory. Seismic Design and Analysis of Tanks readers will also find: * A global approach to the best analytical and practical solutions available in each region: * discussion of the latest US codes and standards from the American Society of Civil Engineers (ACSE 7), the American Concrete Institute (ACI 350,3, 371.R), the American Water Works Association (AWWA D100, D110, D115), and the American Petroleum Institute (API 650) * an overview of the European codes and standards, including Eurocode 8-4 and CEN-EN 14015 * Hundreds of step-by-step equations, accompanied by illustrations * Photographs illustrating real-world damage to tanks caused by seismic events Perfect for practising structural engineers, geotechnical engineers, civil engineers, and engineers of all kinds who are responsible for the design, analysis, and performance of tanks and their foundations--as well as students studying engineering--Seismic Design and Analysis of Tanks is a landmark text, the first work of its kind to deal with the seismic engineering performance of all types of storage tanks.
Sie lesen das E-Book in den Legimi-Apps auf:
Seitenzahl: 613
Veröffentlichungsjahr: 2023
Ähnliche
Table of Contents
Cover
Title Page
Copyright
Dedication
Preface
Acknowledgements
1 Appealing Shell Structures
1.1 Beams and Arches
1.2 Plates and Vaults
1.3 Rectangular and Cylindrical Tanks
1.4 Seismic Behaviour of Tanks
1.5 Field Observation of Damage to Tanks Induced by Seismic Events
1.6 Design Considerations
1.7 A Simplified Description of the Seismic Response of Tanks
1.8 Discussion of the Existing Codes
1.9 Structure of the Book
Notes
2 Above‐Ground Anchored Rigid Tanks
2.1 Introduction
2.2 Vertical Cylindrical Tanks Fully Anchored at the Base
2.3 Rectangular Tanks Fully Anchored at the Base
Notes
3 Above‐Ground Unanchored Rigid Tanks
3.1 Introduction
3.2 Vertical Cylindrical Tanks
3.3 Rectangular Tanks
Notes
4 Elevated Tanks
4.1 Introduction
4.2 Single Lumped‐Mass Model
4.3 Two Uncoupled Mass Model
4.4 Two Coupled Masses Model
Note
5 Flexible Tanks
5.1 Introduction
5.2 Impulsive Pressure Component
5.3 Effects of the Vertical Component of the Seismic Action
5.4 Periods of Vibration
5.5 Combination of Pressures
5.6 Tank Forces and Stresses
5.7 Effects of Rocking Motion
Notes
6 Other Peculiar Principles
6.1 Introduction
6.2 Effects of Soil–Structure Interaction
6.3 Flow‐Dampening Devices
6.4 Base‐Isolation Devices
6.5 Underground Rigid Tanks
6.6 Horizontal Tanks
6.7 Conical Tanks
Notes
7 General Design Principles
7.1 Introduction
7.2 Requirements for Steel Tanks
7.3 Requirements for Concrete Tanks
7.4 Detailing and Particular Rules
Notes
Appendix A: Dimensionless Design Charts
A.1 Introduction
Appendix B: Codes, Manuals, Recommendations, Guidelines, Reports
B.1 Introduction
References
Index
End User License Agreement
List of Tables
Chapter 1
Table 1.1 Importance Classes I to IV depending on the use and contents of th...
Table 1.2 Main categories of tanks considered in earthquake codes and standa...
Chapter 2
Table 2.1 Dimensionless coefficients , , and in order to evaluate ,
Table 2.2 Values of the kinematic viscosity in centistokes (1 cst 1 mm/s...
Table 2.3 Recommended value for constant (site‐independent) scale factors ...
Table 2.4 Response modification coefficients for impulsive and convective
Chapter 5
Table 5.1 Distinction between rigid (R) and flexible (F) tanks with respect ...
Table 5.2 Dimensionless coefficient in order to evaluate the first impulsi...
Table 5.3 Dimensionless coefficients , , , , , , and for deflectio...
Table 5.4 Dimensionless coefficient in order to evaluate the fundamental i...
Table 5.5 Effective non‐dimensional mass coefficient used to evaluate the ...
Table 5.6 Coefficients and , impulsive and convective masses and heights,...
Chapter 6
Table 6.1 Effective damping factor for horizontal impulsive mode as a func...
Table 6.2 Brief summary of scientific literature regarding seismic isolated ...
Table 6.3 Impulsive () and convective () masses with respect to the liquid...
Chapter 7
Table 7.1 Minimum annular plate thickness in mm according to Table 5‐1a in ...
Table 7.2 Minimum bottom plate thickness in mm according to Table 11.1 in Po...
Table 7.3 Minimum thickness in mm (wall and bottom) for ground‐supported or ...
Table 7.4 Minimum thickness of the wall in mm for ground‐supported or other ...
Table 7.5 Design displacement (Point E.7.3, API 650) in the case of piping f...
Table 7.6 Boundary conditions for shells, according to Point 5.2.2, UNI ENV ...
Table 7.7 The effective width of the cylinder wall , according to Schmidt a...
Table 7.8 Fluid tightness classification for concrete retaining and containm...
Table 7.9 Minimum thickness of solid non‐pre‐stressed one‐way slabs, accordi...
Table 7.10 Values for coefficients ,, , and used in the evaluation of...
List of Illustrations
Chapter 1
Figure 1.1 (a) Three‐hinged arch with a uniform load on top; (b) two‐hinged ...
Figure 1.2 Barrel vault response: (a) continuous support on two edges; (b) e...
Figure 1.3 Flat roof plate on side beams.
Figure 1.4 Qualitative distribution of internal stresses in a transversal ar...
Figure 1.5 Qualitative disposition of post‐tensioned cables (a) along the sh...
Figure 1.6 Qualitative variation of membrane action (hoop force) and bendi...
Figure 1.7 (a) Empty (cylindrical or rectangular) tank with the correspondin...
Figure 1.8 Qualitative distribution of , and .
Figure 1.9 Qualitative trend of the convective wave and variation of the dyn...
Figure 1.10 Equivalent mechanical model for a tank full of water under seism...
Figure 1.11 New Zealand Recommendation – NZSEE‐09: (a) percentages of total ...
Figure 1.12 Acceleration spectrum having the typical shape adopted for stiff...
Figure 1.13 Burned tanks following the Izmit earthquake in the Kocaeli Provi...
Figure 1.14 Global overview of observed damage in tanks in past earthquakes....
Figure 1.15 Observed damage to tanks in past earthquakes: (a), (c), (e) Emil...
Figure 1.16 Observed damage to tanks in past earthquakes: (a) Kern County, C...
Figure 1.17 Observed damage to tanks in past earthquakes: (a)–(d) and (f) Em...
Figure 1.18 (a), (b) and (d) Kern County, California earthquake (21 July 195...
Figure 1.19 Hydrodynamic pressure distribution on the wall and base of a upr...
Figure 1.20 Equivalent mechanical model for response of contents of a rigid ...
Chapter 2
Figure 2.1 Tank‐liquid geometry investigated: coordinate system , ground ac...
Figure 2.2 Coefficient for impulsive pressure variation, normalized to , ...
Figure 2.3 Impulsive hydrodynamic pressures exerted on the wall and the base...
Figure 2.4 Nondimensional impulsive mass (a) and heights , and as a f...
Figure 2.5 (a) Vertical distribution of the dimensionless impulsive pressure...
Figure 2.6 Slosh wave shapes for first four antisymmetrical modes of a cylin...
Figure 2.7 Vertical variation of convective pressure corresponding to (a) th...
Figure 2.8 Convective hydrodynamic pressures exerted on the wall and the bas...
Figure 2.9 Convective components of liquid mass (, ) (a) and heights of co...
Figure 2.10 Equivalent mechanical model for vertical response of a rigid tan...
Figure 2.11 Dimensionless coefficients (or impedance functions so called by ...
Figure 2.12 Tank‐liquid geometry investigated: coordinate system , ground a...
Figure 2.13 Heightwise distributions of the impulsive (a) and first/second m...
Figure 2.14 Calexico earthquake (California, 14 June 2010): a 7.2 magnitude ...
Figure 2.15 Horizontally excited cylindrical tank with conical roof under an...
Figure 2.16 (a) Horizontally excited cylindrical tank with flat roof under a...
Figure 2.17 Types of ground‐supported joints between the wall and its founda...
Figure 2.18 Membrane and bending stress resultants for the design of cylindr...
Figure 2.19 Simplified pressure distribution in circumferential direction on...
Figure 2.20 (a) Coefficient for impulsive pressure variation in rigid (and...
Figure 2.21 (a) Coefficient for impulsive pressure variation, normalized t...
Figure 2.22 Dimensionless convective pressure coefficients on rectangular ...
Figure 2.23 Static and hydrodynamic pressure distribution acting on tank wal...
Figure 2.24 Membrane and bending stress resultants for design of rectangular...
Figure 2.25 Equivalent linear distribution of impulsive (a) and convective (...
Chapter 3
Figure 3.1 Damaged tanks observed during the Emilia earthquake in Italy, 20 ...
Figure 3.2 Response to overturning moment in plan and elevation (source: ada...
Figure 3.3 Unanchored tank‐liquid geometry investigated: coordinate system
Figure 3.4 Fluid pressure distribution analytically derived by Taniguchi and...
Figure 3.5 Contact area at base of an unanchored tank.
Figure 3.6 Plastic rotation of base plate of uplifting tank.
Figure 3.7 Fluid pressure distribution analytically derived by Taniguchi and...
Figure 3.8 Uplifting mechanism, in plan and in elevation, and restoring forc...
Chapter 4
Figure 4.1 A wide range of configuration of elevated tanks damaged during ea...
Figure 4.2 Elevated tanks may be categorized into several different types: (...
Figure 4.3 Large‐capacity water elevated steel tank near Urbana, Illinois....
Figure 4.4 Staging configurations to be used in case of frame‐supported rein...
Figure 4.5 Single‐pedestal tank built by Adriano Rivoli S.p.A. from Monopoli...
Figure 4.6 Single lumped‐mass model according to the Indian seismic code Poi...
Figure 4.7 Equivalent two masses uncoupled system, according to Point C3.11,...
Figure 4.8 Two coupled masses dynamic model for elevated fluid container (Po...
Figure 4.9 Equivalent two degrees of freedom system corresponding to the two...
Figure 4.10 Displacement and angular amplitude of free oscillations at t...
Chapter 5
Figure 5.1 (a) Vertical nodal pattern corresponding to a beam‐type mode of v...
Figure 5.2 Circumferential nodal pattern corresponding to the first three mo...
Figure 5.3 Coefficient (Equation (5.8)) [206] for the first mode flexible ...
Figure 5.4 Equivalent mechanical model for a flexible tank full of water und...
Figure 5.5 Equivalent mechanical model of rectangular flexible tanks as sugg...
Figure 5.6 Non‐dimensional impulsive and convective masses , (with ) and...
Figure 5.7 Pressure distribution over the rectangular wall surface of the ...
Figure 5.8 Dimensionless coefficient for a steel roofless cylindrical tank...
Figure 5.9 Coefficient used for the evaluation of the impulsive horizontal...
Figure 5.10 Notations for the solution in term of deflections, bending momen...
Figure 5.11 Elastic–plastic buckling or “elephant foot buckling”: (a) Liverm...
Figure 5.12 Elastic buckling or “diamond buckling” during the Livermore eart...
Figure 5.13 Buckling along and at the top of the tank's wall: (a)/(b) San Fe...
Figure 5.14 Qualitative representation of three different modes of superposi...
Figure 5.15 Damaged tanks observed during the Emilia earthquake, Italy (20 a...
Figure 5.16 (a) earthquake event of Magnitude 5.9, 24 January 1980 in Liverm...
Figure 5.17 Equivalent mechanical model, due to Veletsos and Yang [626, 646]...
Figure 5.18 Equivalent mechanical model, due to Haroun, Housner, and Ellaith...
Figure 5.19 Equivalent heights and in Haroun, Housner, and Ellaithy proc...
Figure 5.20 Equivalent mechanical model, due to Malhotra [370], for flexible...
Figure 5.21 Evaluation of the weighted average thickness in the case of a ta...
Figure 5.22 Foundations for stell cylindrical tanks: (a) circular concrete r...
Figure 5.23 Variation of the rotational parameter as a function of the sle...
Chapter 6
Figure 6.1 Soil–structure interaction effects: (a) schematic showing effects...
Figure 6.2 Effective damping factor of the tank–foundation system for hori...
Figure 6.3 Effective damping factor of the tank–foundation system for vert...
Figure 6.4 Dimensionless damping factors , and of rigid circular footin...
Figure 6.5 Sloshing suppression devices: (a) ring horizontal rigid baffles f...
Figure 6.6 Rigid (a)–(c) and flexible baffles (d); (e) hinged‐spring baffles...
Figure 6.7 A combination of ring and sectored baffles.
Figure 6.8 Tank and baffle system geometry: (a) cylindrical tank with a sing...
Figure 6.9 Flexibility functions , , , , and as a function of the fl...
Figure 6.10 Damping ratio between flexible and rigid ring baffle as a func...
Figure 6.11 Equivalent mechanical model for response of contents of a tank i...
Figure 6.12 Overhead view of the isolation system and cross‐section of the L...
Figure 6.13 Overview of the isolation bearings in Melchorita (Peru).
Figure 6.14 (a) Isolated model of liquid storage tank and (b) the correspond...
Figure 6.15 Typical dynamic pressure distributions proposed in seismic codes...
Figure 6.16 Nomenclature for horizontal axis circular cylindrical tank in lo...
Figure 6.17 Natural frequency in rad/sec for the longitudinal (a) and tran...
Figure 6.18 Radial impulsive pressures on half‐full horizontal cylinder ta...
Figure 6.19 Equivalent mechanical model representing the impulsive and first...
Figure 6.20 Equivalent mechanical model, due to El Damatty and Sweedan [576,...
Figure 6.21 Equivalent mechanical quantities [576, 577]: three concentrated ...
Figure 6.22 Equivalent mechanical quantities [576, 577]: three heights as ...
Figure 6.23 The first impulsive flexible component of the deformational wa...
Figure 6.24 The fundamental sloshing frequency [567, 577] (rigid wall conf...
Chapter 7
Figure 7.1 Typical floor arrangements with (a) bottom and (b) annular plates...
Figure 7.2 Buckling mechanism in correspondence to an opening during the Emi...
Figure 7.3 Reinforcing details: addition of a thickened shell insert plate, ...
Figure 7.4 Spherical shallow shell roof subjected to external uniform pressu...
Figure 7.5 Buckling of (a) a spherical arc and different kinds of loss of st...
Figure 7.6 Buckled locally surface (
snap‐through
) in domes on a square...
Figure 7.7 Factors used for the evaluation of the elastic critical bucklin...
Figure 7.8 Typical tank foundations: (a) pad, (b) ring beam, and (c) raft fo...
Figure 7.9 Overturning check in case of unanchored tanks: wind actions (Poin...
Figure 7.10 Typical arrangement for anchor systems: (a) holding‐down strap; ...
Figure 7.11 Basic examples of collapse/buckling modes for a ring‐stiffened c...
Figure 7.12 Two buckling modes of a clamped
T
‐section ring: (a) distorsional...
Figure 7.13 (a) Full circular ring under uniform external pressure in its in...
Figure 7.14 Series of the possible shape of buckled circular rings under uni...
Figure 7.15 A complete ring buckled (out of plane) under the action of a rad...
Figure 7.16 Steel‐stiffened cylindrical tank under uniform external pressure...
Figure 7.17 Possible geometries of ring cross‐section with (a)
L
, (b)
T
, (c)...
Figure 7.18 Limiting stiffness of ring‐stiffened cylinder with one to five...
Figure 7.19 Geometric limits for distortional buckling strength of
T
‐section...
Figure 7.20 Global and local geometry of longitudinally stiffened cylindrica...
Figure 7.21 Three possible buckling modes: (a) local buckling or torsional b...
Figure 7.22 Possible geometry of the longitudinal stiffeners in case of (a) ...
Figure 7.23 Cylindrical stresses and action loads for the design against buc...
Figure 7.24 Capacity curve of a tank shell as a function of the following fa...
Figure 7.25 Membrane stress state inside the von Mises envelope when the fai...
Figure 7.26 Examples of “diamond buckling”.
Figure 7.27 Examples of “elephant foot buckling”: (a) Alaska earthquake (27 ...
Figure 7.28 Maximum bar diameters and spacings for crack control in members ...
Figure 7.29 Minimum thickness in slabs (without drop panels on top of the co...
Figure 7.30 Effective span length in the case of different support conditi...
Figure 7.31 Anchorage of bottom reinforcement at end supports used in the ev...
Figure 7.32 values for slabs (and beams) according to expressions given in...
Figure 7.33 Bond conditions: (a) and (b) good bond conditions for all bars; ...
Figure 7.34 Anchorage in case of curved bar.
Figure 7.35 Minimum value of the cover for slabs (and beams).
Figure 7.36 (a) Anchorage at intermediate supports ( is the diameter of the...
Figure 7.37 Slab corner reinforcement, according to DIN 1045‐1.
Figure 7.38 Edge reinforcement for a slab, according to Point 9.3.1.4, UNI E...
Figure 7.39 Single or multiple‐leg stirrup‐type (closed bar) slab shear rein...
Figure 7.40 Recommended details: intersections between members subjected to ...
Figure 7.41 Tensile stress distribution across joint sections with a positiv...
Figure 7.42 Recommended details: intersections between members subjected to ...
1
Figure A.1 Dimensionless hoop force and bending moment for hydrostatic (...
Figure A.2 Dimensionless hoop force and bending moment for hydrostatic (...
Figure A.3 Dimensionless hoop force and bending moment for impulsive (ri...
Figure A.4 Dimensionless hoop force and bending moment for impulsive (ri...
Figure A.5 Dimensionless hoop force and bending moment for convective (
Figure A.6 Dimensionless hoop force and bending moment for convective (
Guide
Cover Page
Title Page
Copyright
Dedication
Preface
Acknowledgements
Table of Contents
Begin Reading
Appendix A Dimensionless Design Charts
Appendix B Codes, Manuals, Recommendations, Guidelines, Reports
References
Index
Wiley End User License Agreement
Pages
iii
iv
v
xi
xii
xiii
xv
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
277
278
279
280
281
282
283
285
286
287
288
289
290
291
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
331
332
333
334
335
336
337
Seismic Design and Analysis of Tanks
Gian Michele CalviRoberto Nascimbene
IUSS - University School for Advanced StudiesPaviaItaly
Copyright © 2023 by John Wiley & Sons, Inc. All rights reserved.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey.
Published simultaneously in Canada.
No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per‐copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750‐8400, fax (978) 750‐4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748‐6011, fax (201) 748‐6008, or online at http://www.wiley.com/go/permission.
Trademarks
Wiley and the Wiley logo are trademarks or registered trademarks of John Wiley & Sons, Inc. and/or its affiliates in the United States and other countries and may not be used without written permission. All other trademarks are the property of their respective owners. John Wiley & Sons, Inc. is not associated with any product or vendor mentioned in this book.
Limit of Liability/Disclaimer of Warranty
While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.
For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762‐2974, outside the United States at (317) 572‐3993 or fax (317) 572‐4002.
Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com.
Library of Congress Cataloging‐in‐Publication Data applied for:
Hardback ISBN: 9781119849810
Cover Design: Wiley
Cover Images: Courtesy of Gian Michele Calvi and Roberto Nascimbene
To our dads
Preface
Not many books dealing with designing shell structures are available in the international literature. This was the main motivation inducing us to write a book on this subject, published in 2011, in Italian. That book found its roots in classical texts and in well‐established university courses. First of all, the fundamental text, Stresses in Shells, published by W. Flügge in 1960, possibly a compendium of the monumental Statik und Dynamik der Schalen, written when he was still living in Germany. Together with Vlasov, Reissner, Dischinger, and a few others, he had a fundamental role in developing the membrane and flexural solutions for most kinds of shells between the 1930s and the 1950s.
In the Preface to his first book in English, mentioned above and written at Stanford, where he moved before the Second World War, he wrote: “At first sight it may look to many people like a mathematics book, but it is hoped that the serious reader will soon see that it has been written by an engineer and for engineers … The author wishes to assure his readers that nowhere in this book has an advanced mathematical tool been used just for the sake of displaying it. No matter which mathematical tool has been used, it had to be used to solve the problem at hand.”
This book contains all the fundamental equations to solve any static problem of simple and complex shell structures, being clearly and overtly not to be used as class notes, but rather to find specific solutions or as a basis for further research. The kind of book that cannot be ignored by designers of complex shells that hide the complicated mathematical nature of their structural responses behind their apparent intuitive simplicity and their aesthetic appeal.
Quite to the contrary, another fortunate book, Thin Shell Concrete Structures, by D. Billington, had been expressly written as a textbook for a graduate course, allowing an easier and faster reading. This obviously came at a price, sometimes leaving the reader with an impression of vague or missing information, or with the feeling of some missing link between theory and practice.
Obviously, it was easy for good teachers to bridge this gaps. A. Scordelis, at the University of California, Berkeley, integrated this text with his notes and papers, but he had participated in the design and analysis of spectacular shell structures, such as the elliptical paraboloid of the Oklahoma State Fair Arena (120 m), the circular paraboloid of the Arizona State Fairgrounds Coliseum at Phoenix (114 m), the reverse dome of the Garden State Art Center in New Jersey (78 m), the roof of the San Juan Coliseum in Puerto Rico (94 m), the roof of St. Mary Cathedral in San Francisco, conceived by Pier Luigi Nervi, and made of eight hyperbolic paraboloids with a height of 42 m. It is not often that a student has a teacher with such experience.
Our book in Italian was something in between, with an extensive presentation of the mathematical apparatus and a number of design examples presented in some detail. However, part of its success (it still sells well) was due to the total absence of any competing reference in Italian.
When we started considering the preparation of an English version, it immediately became clear that there was much less point in revisiting what was available in other books, while the section on seismic design and assessment could have been profitably expanded, since very little information is available on the subject.
The relevance of the subject had recently been confirmed by the occurrence of two shocks in northern Italy, on 20 and 29 May 2012 (with a magnitude MW = 6.11 and 5.96). The affected region, in the Po Valley, is one of the most industrialized zones of Northern Italy. The majority of structures severely damaged were industrial facilities: one‐storey pre‐cast reinforced concrete structures and nearby storage steel tanks, causing the economic loss of approximately 5 billion Euros, mostly due to the interruption of industrial production. The large number of industrial facilities in the stricken area, in combination with their intrinsic deficiencies, induced damage and losses disproportionately high, compared to the relatively moderate seismic intensity of the events.
In the aftermath of the earthquakes, a large reconnaissance effort was undertaken and a clearinghouse (http://www.eqclearinghouse.org/2012-05-20-italy/), hosted by the Eucentre Foundation and the Earthquake Engineering Research Institute (EERI), was prepared. The most common types of failures observed in tanks were fracture of anchors and elephant's foot buckling near the base of the tanks. In general, elephant's foot buckling was experienced in squat tanks, while some of the slender tanks surveyed developed diamond‐shaped buckling. Total and partial collapse of legged tanks was another common occurrence, induced by shear failure and/or buckling of their legs due to axial forces, resulting from the overturning moment. In some cases, flat‐bottomed, steel cylindrical tanks, typically larger than legged tanks, failed in tension at the bottom of the tank wall, where they met the anchor rods or massive concrete pads.
It appears that we are still struggling to reach an acceptable quality in design, assessment, and strengthening of tanks and silos, and “competing against time”, as G.W. Housner entitled the report on the Loma Prieta earthquake (17 Oct. 1989) to the Governor of California. The damage to infrastructures, freeways, industrial plants had been severe and the scope of the report had been extended from what happened to the measures to be taken to prevent such destruction in future earthquakes. After some thirty years, it is evident that the report title still applies: we can still state “earthquakes will occur, whether they are catastrophes or not depends on our actions”, but our actions in the past three decades have not been as effective as they should have been.
This book is based on the evidence emerging from a number of structures surveyed following earthquake events, on some significant consulting activity developed in the field of industrial plants, on research developed and published by the authors and other colleagues.
The design and assessment of the expected performance of tanks and silos are presented, considering the following cases:
above‐ground cylindrical and rectangular anchored rigid tanks;
above‐ground cylindrical and rectangular unanchored tanks;
underground rigid tanks;
elevated tanks on shaft and frame‐type towers;
flexible tanks.
This possibly artificial categorization has been found to be convenient with reference to the main response parameters to be considered.
The effects of liquid viscosity, non‐homogeneous liquids, soil–structure interaction, the introduction of damping devices and isolation systems are presented and discussed.
This book is intended primarily for teaching courses on seismic design and analysis of tanks to graduate students and for professional training courses. However, it is expected that this text can be effective and practical as a design and analysis reference for researchers and practising engineers.
Gian Michele CalviRoberto NascimbeneIUSS ‐ University School for Advanced StudiesPaviaItaly
Acknowledgements
This book emerges from a long story that dates back to one of the first Italian university courses on design of shells, taught at the University of Pavia in the early 2000s. At that time, one of the authors of this book was the professor in charge, the other one his assistant. A friendly relationship between the two of them survived the challenges of life and made this book possible, favouring a continuous, enjoyable atmosphere over the long years of its gestation and completion.
Many students have been fundamental in providing criticism and suggestions to improve the class notes that were used as the origin of this endeavour and for several years. The list of their names cannot be made explicit here, but each one of them is felt to be part of this effort.
We mention here two names only, not of students or colleagues, but rather of the two people who have been essential to complete a decent product: Giulia Fagà and Gabriele Ferro, for their continuous support in drawing, refining, and commenting on the figures, which make this book more understandable and reading it more pleasurable.
1Appealing Shell Structures
After reading this chapter you should be able to:
List the main subsectors and components of a tank's design and analysis
Explain the function of each element
Identify the behaviour related to seismic and static performance
1.1 Beams and Arches
The structural design process has traditionally been, and still is, essentially carried out for elements subjected to bending actions (beams and slabs), generally controlled by a flexural behaviour, upon which the design is based. Once flexural resistance has been ensured, these members are verified to prevent excessive deformation or shear failure. In the case of members loaded by a combination of bending moments and axial compressive loads (columns or walls), the preliminary design is often based on the axial component only and the combination with flexural action is then verified (in the case of tanks, bending is sometimes considered just to check the possibility of buckling).
These simplified approaches assume the presence of structural components able to resist either tensile or compressive stresses, such as concrete and steel bars in reinforced concrete elements. The design is thus based on an estimate of the loads to be carried by each member, and subsequently on the design of sections where the maximum resulting bending moment is expected. As an example, consider a beam where the moment acting on a section is estimated as , where is the applied load per unit length, is the length of the element and is a coefficient that depends on the end constraints. This acting moment has to be balanced by a couple, estimated as the result of internal tensile and compressive actions, equal to each other and multiplied by the distance between their approximate points of application to compute a resisting moment.
It is thus quite understandable that in ancient times the material mainly used for roofing systems was timber, combined in boards, joists, beams, and girders with progressively increasing capacity.
The only viable, though more complex, alternative was to resort to an arch, which was able to cover long spans using materials able to resist only compressive actions, such as brick or stone masonry. Though widely applied in ancient times, it was Robert Hooke1 in 1675 who first clearly expressed the basic concept that allows the design of a fully compressed arch. His statement was simple, though very comprehensive: a perfectly compressed arch shall have a shape in reverse and identical to that which a suspended cable would assume under the same load combination. For example, under a uniformly distributed load, the cable would assume a parabolic shape, with an upward concavity and that should be the geometry of a compressed arch, with the concavity oriented downward.
As stated, this conceptual solution does not offer a relevant clue as to how to design an arch when several different load configurations are considered, nor takes into account the effects of abutment constraints, etc. The problem is thus far more complex and, as common in the past (and present) building engineering practice, crucial simplifications were adopted for sizing and preliminary design, e.g. assuming that the horizontal reaction at the abutment () can be approximated by the equation applicable to a three‐hinged arch case:
where and are the load per unit length and the span length, and represents the height of the arch.
It is interesting to compare an arch and a beam used to span a similar length supporting similar weights. Consider thus a three‐hinged arch, assume the horizontal forces are eliminated at the abutments by means of some horizontal tie, and compare it to a beam of equal span, simply supported at both ends, assuming that it is made by an elastic material with a similar behaviour in tension and in compression.
Immediately one notes that the same external moment induced at midspan by the applied loads (, assuming a uniformly distributed load per unit length), must be equilibrated by internal action couples characterized by quite different arms. For the case of an arch, the internal couple results from forces located in the centre of the mass of the arch (compression) and of the tie (tension), while for the case of a elastic beam, they are applied at points located at a distance of two‐thirds of the beam height. When a fully plastic response is assumed, and thus a constant value is assumed for both tensile and compressive stresses, the distance between the resultant forces is one half of the section depth (, Figure 1.1(a)). In this case, the beam's internal action would be:
Consequently, assuming an identical strength in compression and in tension, , and a given width of the beam section, , the required section depth () could be derived as::
For the sake of simplicity, assume now that the arch and the tie are also made with materials with the same compression capacity (the arch) and tensile capacity (the tie). Assuming that both compression and tensile forces will act at the centre of the corresponding element, each force can be derived from Equation (1.1), and consequently the required depth of arch () and tie () would be:
Figure 1.1 (a) Three‐hinged arch with a uniform load on top; (b) two‐hinged parabolic arch and simply supported beam.
Assuming that all considered elements have the same width (), then the depth of the arch and the tie can be computed as a function of the depth of the beam, combining Equations (1.3) and (1.4):
It can immediately be verified that for reasonable values of the rise of the arch compared to its span (, e.g. ) and of the height of the beam, compared to its span (, e.g. ), the depth required for the arch and the tie is at least 10 times less than the one required for the beam.
Applying the same uniform load on a two‐hinged parabolic arch (as shown in Figure 1.1(b)) and on a simply supported beam with the same span (both with a rectangular section ), the deflection of the arch at the keystone (Point A) and of the beam at midspan (Point B) can be calculated as follows:
The apparent overall stiffness differs by two or three orders of magnitude.
This rather trivial example is just a first case study in which the superiority of curved geometry structures is shown in terms of the required material to obtain similar strength or deformation capacities under gravity loads, when compared to similar structures based on straight geometry. The more complex case of cylindrical vs. rectangular tanks will offer more, possibly not as trivial, evidence.
1.2 Plates and Vaults
As already mentioned, a common technology to cover a rectangular area was based on the properties of timber, a material readily available, easy to work, and structurally attractive. This technology is made by a combination of linear elements, overlaying girders, beams and joists, until reasonable span dimensions are achieved to apply boards of reasonable thickness.
Covering the same area using a single plate would require some homogeneous material capable of carrying shear and bending moments in two directions. Clearly this is feasible, though impossible in practical terms, using a steel plate, but became a viable alternative only with the advent of reinforced concrete. Its potential for an isotropic (or rather orthotropic) behaviour, the possibility of shaping its geometry and tapering its thickness, the separation of the internal elements countering compression and tensile stresses, appear to be an ideal combination to build an efficient horizontal slab.
Consider first a simple comparison between a simply supported beam and a similar one‐way slab of indefinite width. The bending moment will be expressed by the same equation, while the slab stiffness will increase because of the hindered transversal dilatation. This effect will be accounted for by a correction factor to be applied to the beam stiffness equal to , where is the Poisson coefficient, in the range of 0.15 for concrete. The correction will thus be in the range of 2% not as relevant.
A much more relevant effect will become evident if comparing a one‐way and a two‐way response, particularly when the two sides of the slab will not differ much.
Take the example of a simply supported square plate, with a uniform load , and assume an isotropic elastic response (i.e. in the case of concrete, neglecting any cracking phenomenon). In the case of a two‐way response, the maximum bending moment and the deflection at the centre of the plate will be calculated as:
where is the side of the plate, its thickness and its flexural stiffness.
Considering the same geometry and the same load, but hinged supports on two opposite sides only, the bending moment and flexural deflection will be those of a simply supported beam (possibly with the minor stiffness correction mentioned above, not applied in the equations):
The values of bending moments and deflection calculated for the beam are thus approximately three times those obtained for the bidirectional plate.
It is easy to observe that a barrel vault sustained by continuous supports on two sides can be regarded as a tri‐dimensional transformation of an arch with the corresponding transversal section. Its basic structural behaviour under gravity loads can thus be derived from that of an arch (Figure 1.2(a)). Things are quite different when a barrel vault is not sustained along the support lines perpendicular to the arch section, but rather along the other two sides (Figure 1.2(b)) or even by punctual vertical support locatedat the corners (Figure 1.2(c)). The structural system becomes much more complex, with the barrel vaults forced to act somehow as a longitudinal beam, coupled with transversal arches. This response can be compared with that of a structure made of longitudinal beams and a transverse slab (Figure 1.3).
Figure 1.2 Barrel vault response: (a) continuous support on two edges; (b) edge beam without continuous lateral support, with columns and (c) edge beam with columns.
Figure 1.3 Flat roof plate on side beams.
The structural response of the barrel vault can be decomposed into two interacting mechanisms: an arch action in the transversal direction and a beam action in the longitudinal direction.
While in the case of slab and beams the components' responses are essentially decoupled, the behaviour of the barrel vault is more complex, because the two systems interact between them, with a resulting combined response that depends mainly on the ratio between the longitudinal length of the vault () and its radius of curvature ().
The traditional beam theory can be applied as a first approximation to calculate internal forces and deformation of the vault structure in the longitudinal direction. The applied bending moment can thus be readily computed and section equilibrium will allow the determination of stresses along the arch‐shaped section of the beam. In general, if free rotation is assumed at the ends, the stress distribution will look like that shown in Figure 1.4(a). Unfortunately, one fundamental assumption of beam theory is that the transversal plane section will remain plane in the deformed configuration and this is not always an acceptable approximation for barrel vaults. Actually, refined numeral analysis has shown that this assumption is valid only for relative values of the fundamental parameters mentioned above, i.e. essentially for . Obviously, this can be obtained also by taking measures to restrain section deformations, such as, for example, inserting transversal walls or ties [499]. It is clear that in these cases all the advantages of a deep beam will apply, with a large lever arm between compression and tension resultants and small displacements.
Figure 1.4 Qualitative distribution of internal stresses in a transversal arch section of a barrel vault, for (a) large or (b) small ratios.
In the case of unrestrained relatively short vaults, the section deformation may differ significantly from a straight line and the related internal stress distribution will be affected (Figure 1.4(b)).
This problem is no different from that of any deep beam with a complex transversal section, which counts on shape more than on mass material to resist the applied forces. As pointed out, a main design problem in these cases is to take measures to control the section deformations. A second fundamental problem (in the case of reinforced concrete) is related to the high potential for significant cracking of the parts subjected to tension, which may not be adequately controlled by means of a proper reinforcement distribution. A possible viable solution may be found in a rational application of post‐tensioning, which will not only be effective in controlling cracking, but also in reducing the deformation. Possible qualitative arrangements of cables are illustrated in Figure 1.5, as a function of the presence of beam‐like elements at the edges of the vault. It is evident in Figure 1.5(a) that a rational longitudinal disposition of the cables in a vault will imply potentially significant effects on the transversal response since the sectional arch sections will be subjected to varying transverse forces and bending moments, not necessarily negligible nor necessarily favourable.
Figure 1.5 Qualitative disposition of post‐tensioned cables (a) along the shell and (b) along the edge beams.
1.3 Rectangular and Cylindrical Tanks
The vertical walls of a rectangular tank can be regarded as a series of vertical plates, usually fixed on three sides and free, simply supported or fixed on the fourth one (the side shared with the roof). Each plate is generally subjected to its own weight (in plane) and to the forces generated by the contained material (mainly out of plane), which generate a variable pressure along the height. In the absence of internal friction, the maximum horizontal pressure at the base of each plate is , where is the weight per volume unit of the fluid and is the height of the tank. This pressure varies linearly with the height, becoming zero at the top. As such, the pressure along the height can be evaluated as [184]:
For squat and large tanks, the acting forces are mainly equilibrated by a vertical cantilever action while the contribution of the plate effect in the horizontal direction can be ignored. If this is the case, and it is further assumed that no constraint is provided at the top side, the maximum bending moment per unit length at the base is:
With a progressive reduction of the horizontal measure of each plate with respect to their height, the contribution of the transverse reaction becomes progressively more important. For example, when the height is equal to the horizontal span (i.e. the tank assumes a cubic shape), the same maximum moment around the base line of of each plate is reduced to less than one‐fifth, becoming:
The estimation of this maximum bending moment is one of the crucial issues when the flexural response controls the design of the structure and is often used as a basic preliminary parameter to define the required wall thickness at the base of the tank.
As an example, with the aim of getting some feeling about figures, consider the case of a cubic tank with side length of 20 m, with a consequent total capacity of 8000 m of liquid, assumed to be water. The resulting bending moment at the base of each wall is:
Assuming that the design moment at the ultimate limit state is factorized to 1.3 times ( kNm/m) and that the neutral axis depth is approximately equal to , the required concrete thickness and the corresponding amount of vertical reinforcement can be estimated as:
where is the concrete design strength, is the yield strength of steel, is the unitary width of the wall, and is the depth of the section excluding the concrete cover, assumed to be 35 mm. The resulting vertical reinforcement percentage is 1.14%. The acting bending moment reduces to about 1000 kNm/m at midheight, hence the section can be tapered, the reinforcement can be reduced, or both. A possible solution is to reduce the wall thickness to 550 mm and the vertical reinforcement ratio to about 1%. Considering a linear tapering, the wall depth at the top will be 200 mm. The bending moment acting at midheight around a vertical axis is about 1400 kNm/m and will require a horizontal reinforcement ratio approximately equal to 1.4%. Cracking width and distance at the base can be checked by applying one of the several formulations proposed in different codes of practice. Considering as an example the equations recommended by the Italian code, NTC 2018, the serviceability limit state bending moment ( kN/m) will be used, estimating the maximum distance between cracks as and their width as approximately 0.2 mm.
Simple calculations can estimate the total amount of concrete required to cast the four walls of the tank, of about 880 m. A proper evaluation of the required reinforcement will imply some assumptions about appropriate minimum percentages for the compression sides of the sections, for corner detailing, for low stress regions. A reasonable total amount will be around 2200 kN, which corresponds to an average steel weight of 2.5 kN/m or to an average reinforcement percentage in both directions of about 1.6%.
A tank with a similar volume capacity can be designed as a cylindrical structure, with a diameter of 20 m and a height of 25 m. In order to complete some simplified design as done above in the case of a square tank, and thus be able to compare the required material quantities, it is necessary to anticipate the basic features of the response of cylindrical tanks. This can be regarded as one of the simplest cases of a shell structure, which can be used as a case study for the development of a more general theory, to be applied to more complex geometrical shapes.
For this purpose, a number of key assumptions have to be accepted (similar to those applied to the simplified theory of plates):
the thickness is small compared to the smallest radius of curvature;
the resulting displacements will be small with respect to the shell thickness;
the cross‐sections normal to the midsurface of the shell will remain straight and normal to the midsurface after deformation;
the component of internal stress normal to the midsurface is negligible if compared to the tangential components;
the stress distribution along the shell thickness is linear, and its value is zero at mid‐surface.
Under these assumptions, the response of a cylindrical tank can be simplified in the interaction of two different reaction systems: a flexural response of vertical cantilevers, and a retaining action of a system of horizontal rings. The solution of this redundant problem must obviously respect both equilibrium and compatibility and was ingeniously developed between the end of the nineteenth and the beginning of the twentieth century.
To solve the problem it was first assumed one should ignore any flexural reaction at the base of the tank, i.e. to allow the base of the cylinder to rotate and to move freely along the horizontal plane. In this case, the horizontal acting forces will have to be equilibrated by the ring action only, while the vertical action will be simply equilibrated by a vertical reaction component. This situation is defined as a membrane solution, since bending is excluded, and shear in the plane tangent to the shell must also be zero because of the axi‐symmetric condition of both loads and structure. Considering a tank containing a liquid with unit weight with no internal friction or cohesion, with height with no internal friction or cohesion, with height , radius , and a total self weight (or total applied vertical load) , the solution described can be expressed by the following equations, as a function of internal vertical () and horizontal () actions per unit length:
where is the vertical coordinate, being equal to zero at the base and to at the top.
The horizontal displacement () and rotation around the tangent horizontal line () required by compatibility in this specific case were also calculated, as:
where is the thickness of the shell and is the Young modulus of its constituent material. Note that the rotation is constant along the height and the displacement, consistently, varies linearly, from a maximum value at the base to zero at the top. Assuming a concrete shell, with a Young modulus MPa, a thickness of 200 mm and uncracked sections, the displacement at the base would be 4.2 mm and the constant rotation rad, which will lead to zero displacement at the top. In the case of some flexural and shear constraint at the base that would not allow either relative displacement or rotation or both, some flexural action will occur, and the displacement and rotation resulting from this action will have to properly combine with the membrane results to assure compatibility.
The flexural solution of the problem is fully derived and explained in [66], here we will simply mention and use it, as expressed by the following fourth‐order differential equation and its solution:
where is the horizontal pressure, is the flexural stiffness of a plate with thickness equal to that of the shell and is the ratio between the horizontal stiffness of the rings () and the flexural stiffness of the plate, expressed by the following equations:
In a mathematical context, the function is defined as “particular integral” and can here be represented by the membrane solution found earlier, which is “particular” in the sense that corresponds to the specific situation described above with reference to the base constraints. Again, in mathematical terms, the first term of the solution is the “general integral” of the associated homogeneous equation (i.e. without any applied pressure). This part of the solution depends on the boundary conditions (in this case, from the base restraints, which will determine the values of the constant and ).
It is intuitive that the parameter , i.e. the ratio between the cantilever and the ring stiffness, determines which one of the two systems will dominate the structural response. Equation (1.17b) is thus indicating that large radius, thicker tanks are more affected by the flexural response, while small, thin shells are dominated by the ring's reaction. It is interesting to observe that this is not the only relevant role played by the parameter .
Actually, the wavelength of the damped sinusoid expressed by the general integral is and the damping factor expressed by the negative exponent of is a linear function of . Both effects have a clear physical meaning: if the stiffness of the rings system prevails on the flexural response, is higher and consequently the period of the sinusoid is shorter and its effect is damped more rapidly. The damping factor can be expressed as a function of the wavelength, replacing with , and obtaining . This expression indicates the reduction factor to be applied to the sinusoid is around 5, a distance equal to one quarter of the wavelength (i.e. at ), around 23 at a distance of half wavelength (), and around 535 at .
In the case study we were considering, it is immediately obvious that to calculate that m and m (a Poisson ratio has been adopted), the wavelength of the sinusoid is thus around 6 metres, and the flexural effect can certainly be ignored at a distance of about 3 metres from the base. While the values of bending moment () and shear force () along the height can be calculated by applying the usual relations between displacement, curvature, bending moment and shear, i.e.:
It appears that only the maximum values of bending moment (Figure 1.6) and shear force are relevant to design the required amount of vertical reinforcement at the base, a reinforcement that will be needed for a few metres only.
As already pointed out, the combination of the membrane and the flexural solutions aims to obtain total displacement and rotation at the base compatible with the restraint of each specific case. For a fixed‐base tank, the displacements and rotations generated by the flexural effect ( and ) must be equal and opposite to those generated by the membrane solution, i.e.:
Figure 1.6 Qualitative variation of membrane action (hoop force) and bending moment or along the height of the container.
In the case study, the base bending moment and shear, calculated by applying again a 1.3 protection factor, are kNm/m and kN/m. These values are compatible with the assumed section depth (0.20 m), requiring a total amount of reinforcement of 3254 mm, or a geometrical steel percentage %. At heights above 5 m from the base and on the compression side, the minimum allowed reinforcement percentage can be adopted (possibly 0.3%). The bending moment computed at the base of the cylindrical tank is about seventeen times lower than that obtained for the cubic tank, but unfortunately this is not the governing action to dimension the shell. Actually, when considering the circumferential membrane forces, it is immediately realized that a relevant crack pattern will be induced. For example, at a height of about 3 metres the membrane action is kN/m (Figure 1.6), which would require about 7670 mm of steel, corresponding to a very high geometrical percentage of 3.8%. The resulting theoretical crack opening width would be 0.4 mm with an average crack maximum distance of mm. This crack pattern would not be acceptable in most cases.
The most obvious correction of the preliminary design to redirect the result to performances similar to those obtained in the case of a rectangular tank with 900 mm thick walls will simply be to increase the shell thickness. In fact, increasing the shell depth from 200 to 400 mm will induce a modification of the coefficient m, thus increasing the flexural stiffness with respect to the membrane one. The previous calculations will be repeated, obtaining kNm/m, kN/m and kN/m. While bending moment and shear action will be easily absorbed by the increased resisting section, the theoretical mean crack width will now be 0.17 mm. The total required amounts of concrete and steel will still be around 30% lower than those estimated in the case of a rectangular tank, thus resulting in a competitive design.
A second, more sophisticated, design solution could be based on the insertion of post‐tensioning cables to reduce the tensile membrane forces and displacement along the circumferential direction. A possible way of estimating an effective post‐tensioning force could be to equilibrate the tensile circumferential resultant due to the internal fluid pressure kN/m acting on each ring of unitary height. In this case, a preliminary estimate of the required post‐tensioning steel area () can be obtained as follows:
where is the steel stress after prestress losses occurred, estimated around 65% of the steel cable yield strength. A reasonable corresponding concrete section is then obtained assuming that at yield conditions of the steel cables the concrete average stress is still acceptable, i.e., lower than the compression design strength:
For a strip of unitary height of the vertical wall, this corresponds to a thickness of about 25 cm. As a consequence, in this case the total amount of concrete required is 393 m, i.e. less than one‐third of the amount estimated for the rectangular tank, with some 55 tons of steel reinforcement. These calculations are obviously preliminary estimates, since a more refined solution would require reasoning in terms of induced deformations rather than brutally equating forces.
An additional further alternative would be to modify the connection between the foundation and the cylinder, allowing relative displacement and rotation. In such conditions, the membrane boundary constraints would be respected and the flexural response would be eliminated. The consideration related to the circumferential stresses will still apply, with the consequent need of post‐tensioning or adequate thickness. To allow rotation and displacement, a rubber pad should be designed and inserted, properly connecting it to both sides.
1.4 Seismic Behaviour of Tanks
A cylindrical tank is evidently a stiff structure, the simplest assumption to define its response to a ground motion is to consider it as a rigid body. As such, the tank will be possibly modelled as a mass, equal to a fraction of the total mass of structure and contents, rigidly connected to the ground. The mass relative displacement will be assumed as null, the maximum base shear () equal to the mass multiplied by the peak ground acceleration () and the maximum overturning moment () equal to the base shear multiplied by some equivalent height.
In many cases, and particularly for empty tanks, this model will provide acceptable results, provided that reasonable values have been assumed for the fraction of the total mass and for the equivalent height at which the mass is lumped.
Considering first the case of an empty tank with a light roof, a possible choice is to consider the mass of the wall () only, to ignore the mass of the base plate () and to assume an equivalent height (), equal to one‐half of the height of the tank:
Considering once more the examples of the rectangular and cylindrical tanks previously analyzed (Section 1.4) and a peak ground acceleration , with reference to Figure 1.7(a), the base shear () and the total bending moment () at the base of the wall will be:
