Historical Analysis of Arches and Modern Shells

181 Pages·2015·17.58 MB·English

Historical Analysis of Arches and Modern Shells
by
R. Georg
B.S., University of Wisconsin - Platteville, 2012
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillment
of the requirements for the degree of
Master of Science
Department of Civil, Environmental and Architectural Engineering
2015
This thesis entitled:
Historical Analysis of Arches and Modern Shells
written by R. Georg
has been approved for the Department of Civil, Environmental and Architectural Engineering
Prof. Victor Saouma
Prof. George Hearn
Prof. Abbie Liel
Date
The final copy of this thesis has been examined by the signatories, and we find that both the
content and the form meet acceptable presentation standards of scholarly work in the above
mentioned discipline.
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Georg, R. (M.S. Civil Engineering)
Historical Analysis of Arches and Modern Shells
Thesis directed by Prof. Victor Saouma
Prior 20th century, the study of masonry arches and domes consumed years of work with
some of the greatest minds in history. However, with the advancements of materials, such as steel
and concrete, and the use of computer analysis programs, the art of masonry arch and shell design
has become stagnant. With various design methods seen throughout literature, an appreciation for
the development of these methods, through basic hand calculations, must be understood to first,
ensure correct design principles are applied and second, aid in the further development of these
methods.
Thisthesisstartswithanextensiveliteraturereviewofhistoricalanalysisanddesignmethod-
ology, starting in the 16th century and continuing on through the mid-20th century and today’s
current practices. The review focuses first on masonry arch design, including principles of geo-
metric design, wedge theory, line of thrust and the ultimate load theorem. The second part views
the design and analysis of domes and vaults, concluding with a case study of St. Peter’s Dome in
Rome.
The thesis continues by reviewing the derivations of a beam and plate subjected to flexure,
prior to the thin shell derivation. In all three cases, equilibrium, compatibility and stress-strain
relationships are considered to develop the differential equation relating transverse displacement to
the load. This methodology is chosen in order to introduce the shell gradually by building upon
the initial derivations of the beam and plate.
Tying the historical design methods and derivation of the shell equation, the design and
analysis of a circular cylindrical shell will be conducted. The derived shell equation will first be
simplified to membrane theory, followed by the derivations of the governing equations for shells
through the theory of shallow shells. The analysis of the cylindrical shell will hold similarity to the
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analysis of statically indeterminate beams.
Finally, the methodology of shallow shells will be incorporated into the development of a
reinforced concrete design and analysis program. The development of this program will simplify
future analyses of circular cylindrical shells and improve design efficiency. The resulting design
methodology will be recorded to aid in the future design of shells and the inspection of current
structures. The thesis concludes by offering future studies to further develop the field of masonry
arch and dome design.
Dedication
This thesis is dedicated Professor Saouma due to his passion for the topic and continuous
guidance. MaytheworkIhavecompletedhelpfuelhisfutureworkanddevelopmentofhisstructural
analysis book.
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Acknowledgements
I would first like to thank Professor Saouma for selecting myself to perform this thesis and
work under him. He has continually guided my efforts and offered aid and insight when necessary.
Also, I would like to thank Prof. Hearn and Prof. Liel for agreeing to participate as members of
the thesis committee and so graciously evaluating the work that I have performed.
Second, I would like to thank my thesis work group members Trupti Sonavane and Kyle
Prusinski. We spent many meetings together collaborating and discussing our work. Both individ-
uals have offered much aid and support in the development and finalization of my thesis.
Finally I would like to the University of Colorado and the administration, specifically Pamela
Williams, and all her continuous help and guidance through the process of completing my master’s
thesis.
Contents
Chapter
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Evolution of Modern Era Shell Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Historical Review 5
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 The Origin of Masonry Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.1 Structural Theory of Masonry . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.2 Modern Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Masonry Arch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3.1 Geometric Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3.2 Wedge Theory/ Collapse Analysis . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.3 Line of Thrust Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3.4 Stress Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3.5 Elastic Theory - Winkler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3.6 Ultimate Load Theory - Heyman . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.4 Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
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2.4.2 The Hen’s Egg - Viviani . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.4.3 Structural Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.4.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.4.5 Vault Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.5 St. Peter’s Basilica, Rome - Case Study . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.5.2 Three Mathematicians Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.5.3 Poleni’s Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.5.4 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3 Beams, Plates and Shells 63
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.2 Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.2.1 Curvature Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.2.2 Differential Equation of the Elastic Curve . . . . . . . . . . . . . . . . . . . . 66
3.3 Plates (Adapted from Pilkey & Wunderlich) . . . . . . . . . . . . . . . . . . . . . . . 67
3.3.1 Fundamental Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.3.2 Plate Theories: Kirchhoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.4 Thin Shell Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.4.1 Definitions and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.4.2 Derivation of Governing Differential Equation . . . . . . . . . . . . . . . . . . 83
3.4.3 Simplifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.4.4 Shallow Shells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.4.5 Membrane Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
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3.4.6 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
3.4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4 Analysis of Circular Cylindrical Shells 111
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.2 Circular Cylindrical Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.3 Simply Supported Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.3.1 Membrane Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.3.2 Theory of Shallow Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.3.3 Shallow Shells vs. Membrane Theory . . . . . . . . . . . . . . . . . . . . . . . 121
4.3.4 Bending Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.3.5 Edge Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
4.3.6 Prestressing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5 Circular Cylindrical Shell Design Tool 138
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.1.1 Preliminary Design Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.2 Matlab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.2.1 Cylindrical Shell Roof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.2.2 Edge Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6 Conclusions and Recommendations for Future Studies 149
6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.2 Recommendations for Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
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Bibliography 153
Appendix
A Design Code 155
A.1 Shell Design - Part One . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
A.1.1 Shell Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
A.1.2 Shell Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
A.2 Edge Beam - Part Two. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164