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Optimization of Finite Dimensional Structures Makoto
Ohsaki Digital Instant Download
Author(s): Makoto Ohsaki
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Language: english

Optimization
of Finite
Dimensional
Structures
K11056_FM.indd 1 6/11/10 3:18:11 PM

K11056_FM.indd 2 6/11/10 3:18:11 PM

Optimization
of Finite
Dimensional
Structures
Makoto Ohsaki
Hiroshima University
Higashi-Hiroshima, Japan
K11056_FM.indd 3 6/11/10 3:18:11 PM

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Preface
The attempt to find mechanically efficient structural designs and shapes was
initiated mainly in the fields of mechanical engineering and aeronautical engi-
neering, which established the field known as structural optimization. Many
practically acceptable results have been developed for application to auto-
mobiles and aircraft. Some examples are structural components, including
the wings of aircraft and engine mounts of automobiles, which can be fully
optimized using efficient shape optimization techniques.
In contrast, regarding civil engineering and architectural engineering, struc-
tural optimization is difficult to apply because structures in these fields are
not mass products: structures are designed in accordance with their specific
design requirements. Furthermore, the structure’s shape and geometry are de-
termined by a designer or an architect in view of nonstructural performance,
including the aesthetic perspective. Therefore, the main role of structural en-
gineers is often limited to selection of materials, determination of member sizes
through structural analyses, planning details of the construction process, and
so on. However, for special structures, such as shells, membrane structures,
spatial long-span frames, and highrise buildings, the structural shape should
be determined in view of the responses against static and dynamic loads. In
truth, the beauty of the structural form is related closely to the mechanical
performance of the structure. Therefore, cooperation between designers and
structural engineers is very important in designing such structures.
Even for building frames, because of the recent trend of performance-based
design, optimization has been identified as a powerful tool for designing struc-
tures under constraints imposed on practical performance measures, including
elastic/plastic stresses and displacements under static/dynamic design loads.
Furthermore, recent rapid advancements in the areas of computer hardware
and software enabled us to carry out structural analysis many times to ob-
tain optimal or approximately optimal designs. Optimization of real-world
structures with realistic objective function and constraints is possible through
quantitative evaluation of nonstructural performance criteria, e.g., aesthetic
properties, and life-cycle costs, including costs of construction, fabrication,
and maintenance.
Many books describing structural optimization have been published since
the 1960s; e.g., Hemp (1973), Rozvany (1976), Haug and Cea (1981), Haftka,
Gürdal, and Kamat (1990), Papalambos and Wilde (2000), Bendsøe and Sig-
mund (2003), Arora (2004), etc. These books are mainly classifiable into the
following three categories:
v

vi Preface
1. Basic theories and methodologies for optimization with examples of
small structural optimization problems.
2. Continuum-based approaches for application to mechanical and aero-
nautical structures.
3. Theoretical and analytical results of structural optimization in earlier
times without the assistance of computer technology.
Using books of the first category, readers can learn only the concepts and
some difficult theories of structural optimization without application to large-
scale structures. On the other hand, for the books of the second category, a
good background in applied mathematics and continuum mechanics is needed
to fully understand the basic concepts and methods. Unfortunately, most
researchers, practicing engineers, and graduate students in the field of civil
engineering have no such background and are not strongly interested in the
basic theories or methods of structural optimization. Also, in mechanical en-
gineering, the finite element approach is used for practical applications, and
complex practical design problems are solved in a finite dimensional formula-
tion.
The derivatives of objective and constraint functions, called design sensitiv-
ity coefficients, should be computed if a gradient-based approach is used for
structural optimization. However, most methods of design sensitivity analy-
sis are developed mainly for a continuum utilizing variational principles, for
which sensitivity coefficients are to be computed for a functional, such as com-
pliance that can be formulated in an integral of a bilinear form of response.
For finite dimensional structures, including trusses and frames, variational
formulations are not needed, and sensitivity coefficients can be found simply
by differentiating the governing equations in a matrix-vector form.
Another important aspect of optimization in civil engineering is that the
design variables often have discrete values: the frame members are usually
selected from a pre-assigned list or catalog of available sections. Furthermore,
some traditional layouts are often used for plane and spatial trusses and for
latticed domes. Therefore, the optimization problem often turns out to be
a combinatorial problem, a fact that is not fully introduced into most books
addressing the study of structural optimization.
This book introduces methodologies and applications that are closely re-
lated to design problems of finite dimensional structures, to serve thereby as
a bridge between the communities of structural optimization in mechanical
engineering and the researchers and engineers in civil engineering. The book
provides readers with the basics of optimization of frame structures, such as
trusses, building frames, and long-span structures, with descriptions of various
applications to real-world problems.
Recently, many efficient techniques of optimization have been developed
for convex programming problems, e.g., semidefinite programming and inte-
rior point algorithms, which are extensions of the approaches used for linear

Preface vii
and quadratic programming problems. The book introduces application of
these methods to optimization of finite-dimensional structures. Approximate
methods resembling the conventional optimality criteria approaches have also
been developed with no reference to the pioneering papers in the 1960s and
1970s. Therefore, it is extremely important to describe their development his-
tory to young researchers so that similar methods are not re-developed with
no knowledge related to conventional approaches. For that reason, another
purpose of this book is to present the historical development of the method-
ologies and theorems on optimization of frame structures.
The book is organized as follows:
In Chapter 1, the basic concepts and methodologies of optimization of
trusses and frames are presented with illustrative examples. Traditional prob-
lems with constraints on limit loads, member stresses, compliance, and eigen-
values of vibration are described in detail. A brief introduction is also pre-
sented for multiobjective structural optimization, and the shape and topology
optimization of trusses.
In Chapter 2, the method of design sensitivity analysis, which is a necessary
tool for optimization using nonlinear programming, is presented for various
response quantities, including static response, eigenvalue of vibration, tran-
sient response for dynamic load, and so on. All formulations are written in
matrix-vector form without resort to variational formulation to support ready
comprehension by researchers and engineers.
In Chapter 3, details of truss topology optimization are described, including
historical developments and difficulties in problems with stress constraints and
multiple eigenvalue constraints. Recently developed formulations by semidef-
inite programming and mixed integer programming are introduced. Applica-
tions to plane and spatial trusses are demonstrated.
Chapter 4 presents methods for configuration optimization for simultane-
ously optimizing the geometry and topology of trusses. Difficulties in opti-
mization of regular trusses are extensively discussed, and an application to
generating a link mechanism is presented.
Chapter 5 summarizes various results of optimization of building frames.
Uniqueness of the optimal solution of a regular frame is first investigated,
and applications of parametric programming are presented. Multiobjective
optimization problems are also presented for application to seismic design,
and a simple heuristic method based on local search is presented.
In Chapter 6, as a unique aspect of this book, optimization results are pre-
sented for spatial trusses and latticed domes. Simple applications of nonlinear
programming and heuristic methods are first introduced, and the spatial varia-
tion of seismic excitation is addressed in the following sections. The trade-off
designs between geometrical properties and stiffness under static loads are
shown for arch-type frames and latticed domes described using parametric
curve and surface.
Mathematical preliminaries and basic methodologies are summarized in the
Appendix, so that readers can understand the details, if necessary, without the

viii Preface
exposition of tedious mathematics presented in the main chapters. Various
methodologies speciﬁcally utilized in some of the sections, e.g., the response
spectrum approach for seismic response analysis, are also explained in the
Appendix. Also, many small examples that can be solved by hand or using
a simple program are presented in the main chapters. Therefore, this book is
self-contained, and easily used as a textbook or sub-textbook in a graduate
course.
The author would like to deliver his sincere appreciation to Prof. Tsuneyoshi
Nakamura, Prof. Emeritus of Kyoto University, Japan, for supervising the
author’s study for master’s degree and Ph.D. dissertation on structural op-
timization. Supervision by Prof. Jasbir S. Arora of The University of Iowa
during the author’s sabbatical leave is also acknowledged.
The numerical examples in this book are a compilation of the author’s
work on structural optimization at Kyoto University, Japan, during the period
1985–2010. The author would like to extend his appreciation to researchers for
collaborations on the studies that appear as valuable contents in this book,
namely, Prof. Naoki Katoh of the Dept. of Architecture and Architectural
Engineering, Kyoto University; Prof. Shinji Nishiwaki of the Dept. of Me-
chanical Engineering and Science, Kyoto University; Prof. Hiroshi Tagawa of
the Dept. of Environmental Engineering and Architecture, Nagoya University;
Prof. Yoshihiro Kanno of the Dept. of Mathematical Informatics, University
of Tokyo; Prof. Peng Pan of the Dept. of Civil Engineering, Tsinghua Uni-
versity, P. R. China; Dr. Takao Hagishita of Mitsubishi Heavy Industries; Mr.
Yuji Kato of JSOL Corporation; Mr. Takuya Kinoshita, Mr. Shinnosuke Fu-
jita, and Mr. Ryo Watada, graduate students in the Dept. of Architecture
and Architectural Engineering, Kyoto University. The author would also like
to thank again Prof. Yoshihiro Kanno of University of Tokyo for checking the
details of the manuscript.
The assistance of Ms. Kari Budyk and Ms. Leong Li-Ming of CRC Press
and Taylor & Francis in bringing the manuscript to its ﬁnal form is heartily
acknowledged.
January 2010 Makoto Ohsaki

Contents
Preface v
1 Various Formulations of Structural Optimization 1
1.1 Overview of structural optimization . . . . . . . . . . . . . . 1
1.2 History of structural optimization . . . . . . . . . . . . . . . 3
1.3 Structural optimization problem . . . . . . . . . . . . . . . . 5
1.3.1 Continuous problem . . . . . . . . . . . . . . . . . . . 5
1.3.2 Discrete problem . . . . . . . . . . . . . . . . . . . . . 10
1.4 Plastic design . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 Stress constraints . . . . . . . . . . . . . . . . . . . . . . . . 14
1.6 Fully-stressed design . . . . . . . . . . . . . . . . . . . . . . . 17
1.6.1 Stress-ratio approach . . . . . . . . . . . . . . . . . . . 17
1.6.2 Single loading condition . . . . . . . . . . . . . . . . . 20
1.6.3 Multiple loading conditions . . . . . . . . . . . . . . . 23
1.7 Optimality criteria approach . . . . . . . . . . . . . . . . . . 25
1.8 Compliance constraint . . . . . . . . . . . . . . . . . . . . . . 29
1.8.1 Problem formulation and sensitivity analysis . . . . . 29
1.8.2 Optimality conditions . . . . . . . . . . . . . . . . . . 31
1.8.3 Reformulation of the optimization problem . . . . . . 34
1.8.4 Convexity of compliance . . . . . . . . . . . . . . . . . 39
1.8.5 Other topics on compliance optimization . . . . . . . . 42
1.9 Frequency constraints . . . . . . . . . . . . . . . . . . . . . . 43
1.10 Conﬁguration optimization of trusses . . . . . . . . . . . . . 48
1.11 Multiobjective structural optimization . . . . . . . . . . . . . 50
1.11.1 Basic concepts . . . . . . . . . . . . . . . . . . . . . . 50
1.11.2 Problem formulation . . . . . . . . . . . . . . . . . . . 51
1.12 Heuristic approach . . . . . . . . . . . . . . . . . . . . . . . . 52
1.13 Simultaneous analysis and design . . . . . . . . . . . . . . . 55
2 Design Sensitivity Analysis 59
2.1 Overview of design sensitivity analysis . . . . . . . . . . . . . 59
2.2 Static responses . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.2.1 Direct diﬀerentiation method . . . . . . . . . . . . . . 62
2.2.2 Adjoint variable method . . . . . . . . . . . . . . . . . 66
2.3 Eigenvalues of free vibration . . . . . . . . . . . . . . . . . . 69
2.3.1 Simple eigenvalue . . . . . . . . . . . . . . . . . . . . . 69
2.3.2 Multiple eigenvalues . . . . . . . . . . . . . . . . . . . 73
ix

x Contents
2.4 Linear buckling load . . . . . . . . . . . . . . . . . . . . . . . 76
2.5 Transient responses . . . . . . . . . . . . . . . . . . . . . . . 78
2.5.1 Direct differentiation method . . . . . . . . . . . . . . 78
2.5.2 Adjoint variable method . . . . . . . . . . . . . . . . . 79
2.6 Nonlinear responses . . . . . . . . . . . . . . . . . . . . . . . 81
2.7 Shape sensitivity analysis of trusses . . . . . . . . . . . . . . 83
3 Topology Optimization of Trusses 85
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.2 Michell truss . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.3 Topology optimization problem . . . . . . . . . . . . . . . . . 88
3.4 Optimization methods . . . . . . . . . . . . . . . . . . . . . . 90
3.5 Stress constraints . . . . . . . . . . . . . . . . . . . . . . . . 93
3.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 93
3.5.2 Governing equations . . . . . . . . . . . . . . . . . . . 94
3.5.3 Discontinuity in stress constraint . . . . . . . . . . . . 95
3.5.4 Discontinuity due to member buckling . . . . . . . . . 98
3.5.5 Mathematical programming approach . . . . . . . . . 101
3.5.6 Problem with stress and local constraints . . . . . . . 106
3.6 Mixed integer programming for topology optimization with dis-
crete variables . . . . . . . . . . . . . . . . . . . . . . . . . . 113
3.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 113
3.6.2 Compliance minimization problem . . . . . . . . . . . 114
3.6.3 Stress constraints . . . . . . . . . . . . . . . . . . . . . 115
3.6.4 Numerical examples . . . . . . . . . . . . . . . . . . . 119
3.7 Genetic algorithm for truss topology optimization . . . . . . 122
3.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 122
3.7.2 Optimization considering nodal cost . . . . . . . . . . 123
3.7.3 Topological bit and fitness function . . . . . . . . . . . 123
3.8 Random search method using exact reanalysis . . . . . . . . 128
3.8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 128
3.8.2 Exact reanalysis . . . . . . . . . . . . . . . . . . . . . 128
3.8.3 Random search for topology optimization of trusses . 133
3.9 Multiple eigenvalue constraints . . . . . . . . . . . . . . . . . 136
3.9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 136
3.9.2 Multiple eigenvalues in optimal topology . . . . . . . . 138
3.9.3 Semidefinite programming for topology optimization . 140
3.9.4 Linear buckling constraint . . . . . . . . . . . . . . . . 142
3.10 Application of data mining . . . . . . . . . . . . . . . . . . . 149
3.10.1 Frequent item set of decent solutions . . . . . . . . . . 149
3.10.2 Topology mining of ground structures . . . . . . . . . 153

Contents xi
4 Conﬁguration Optimization of Trusses 159
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
4.2 General formulation and methodologies of conﬁguration opti-
mization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
4.3 Optimization of a regular grid truss . . . . . . . . . . . . . . 166
4.4 Generation of a link mechanism . . . . . . . . . . . . . . . . 174
4.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 174
4.4.2 Mechanical model of a link mechanism . . . . . . . . . 174
5 Optimization of Building Frames 185
5.1 Overview of optimization of building frames . . . . . . . . . 185
5.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 185
5.1.3 Continuum approach . . . . . . . . . . . . . . . . . . . 192
5.1.4 Semi-rigid connections and braces . . . . . . . . . . . 192
5.1.5 Formulation of cost function . . . . . . . . . . . . . . . 197
5.2 Local and global searches of approximate optimal designs . . 198
5.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 198
5.2.2 Optimization problem and optimality conditions . . . 200
5.2.3 Local search of approximate optimal solutions . . . . . 202
5.2.4 Global search of approximate optimal solutions . . . . 206
5.2.5 Numerical example of a regular plane frame . . . . . . 208
5.3 Parametric optimization of frames . . . . . . . . . . . . . . . 214
5.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 214
5.3.2 Two-level decomposition of frames . . . . . . . . . . . 216
5.3.3 General concept of decomposition to subsystems . . . 220
5.3.4 Parametric multidisciplinary optimization problem . . 222
5.3.5 Optimization of plane frames . . . . . . . . . . . . . . 224
5.3.6 Optimization of a three-dimensional frame . . . . . . . 228
5.4 Local search for multiobjective optimization of frames . . . . 234
5.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 234
5.4.2 Heuristic approaches to combinatorial multiobjective pro-
gramming . . . . . . . . . . . . . . . . . . . . . . . . . 235
5.4.3 Local search for multiobjective structural optimization 240
5.4.4 Properties of Pareto optimal solutions . . . . . . . . . 242
5.5 Multiobjective seismic design of building frames . . . . . . . 250
5.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 250
5.5.2 Formulation of the multiobjective programming prob-
lem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
5.5.3 Optimization method . . . . . . . . . . . . . . . . . . 253

xii Contents
6 Optimization of Spatial Trusses and Frames 259
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
6.2 Seismic optimization of spatial trusses . . . . . . . . . . . . . 261
6.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 261
6.2.2 Design sensitivity analysis . . . . . . . . . . . . . . . . 262
6.2.3 Optimization against seismic excitations . . . . . . . . 263
6.3 Heuristic approaches to optimization of a spatial frame . . . 266
6.4 Shape optimization considering the designer’s preference . . 271
6.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 271
6.4.2 Description of an arch-type frame using a Bézier curve 273
6.4.3 Shape optimization incorporating the designer’s prefer-
ence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
6.4.4 Sensitivity analysis with respect to control points . . . 277
6.5 Shape optimization of a single-layer latticed shell . . . . . . 280
6.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 280
6.5.2 Description of a latticed shell and formulation of the
optimization problem . . . . . . . . . . . . . . . . . . 281
6.6 Conﬁguration optimization of an arch-type truss with local ge-
ometrical constraints . . . . . . . . . . . . . . . . . . . . . . 288
6.6.1 Direct assignments of geometrical constraints . . . . . 288
6.6.2 Optimization using a genetic algorithm . . . . . . . . 291
6.7 Seismic design for spatially varying ground motions . . . . . 295
6.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 295
6.7.2 Response to spatially varying ground motions . . . . . 295
6.7.3 Problem formulation and design sensitivity analysis . 299
6.7.4 Postoptimal analysis . . . . . . . . . . . . . . . . . . . 301
6.8 Substructure approach to seismic optimization . . . . . . . . 305
6.8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 305
6.8.2 Frequency domain analysis for a secondary structure . 306
6.8.3 Optimization problem . . . . . . . . . . . . . . . . . . 309
Appendix 315
A.1 Mathematical preliminaries . . . . . . . . . . . . . . . . . . . 315
A.1.1 Positive deﬁnite matrix and convex functions . . . . . 315
A.1.2 Rayleigh’s principle . . . . . . . . . . . . . . . . . . . 316
A.1.3 Singular value decomposition . . . . . . . . . . . . . . 318
A.1.4 Directional derivative and subgradient . . . . . . . . . 319
A.2 Optimization methods . . . . . . . . . . . . . . . . . . . . . . 319

Contents xiii
A.2.1 Classification of optimization problems . . . . . . . . . 319
A.2.2 Nonlinear programming . . . . . . . . . . . . . . . . . 321
A.2.3 Dual problem . . . . . . . . . . . . . . . . . . . . . . . 334
A.2.4 Semidefinite programming . . . . . . . . . . . . . . . . 336
A.2.5 Combinatorial problem . . . . . . . . . . . . . . . . . 338
A.3 Heuristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
A.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 340
A.3.2 Single-point-search heuristics . . . . . . . . . . . . . . 341
A.4 Multiobjective programming . . . . . . . . . . . . . . . . . . 345
A.4.1 Definition of multiobjective programming . . . . . . . 345
A.4.2 Constraint approach . . . . . . . . . . . . . . . . . . . 347
A.4.3 Linear weighted sum approach . . . . . . . . . . . . . 348
A.4.4 Goal programming . . . . . . . . . . . . . . . . . . . . 349
A.5 Parametric structural optimization problem . . . . . . . . . . 350
A.6 Parametric curves and surfaces . . . . . . . . . . . . . . . . . 353
A.6.1 Bézier curve . . . . . . . . . . . . . . . . . . . . . . . . 353
A.6.2 Bézier surface . . . . . . . . . . . . . . . . . . . . . . . 356
A.6.3 Adjoint curve . . . . . . . . . . . . . . . . . . . . . . . 357
A.7 Response spectrum approach . . . . . . . . . . . . . . . . . . 359
A.7.1 SRSS method . . . . . . . . . . . . . . . . . . . . . . . 359
A.7.2 CQC method . . . . . . . . . . . . . . . . . . . . . . . 361
A.7.3 Design response spectrum . . . . . . . . . . . . . . . . 362
A.7.4 Sensitivity analysis of mean maximum response . . . . 363
A.8 List of available standard sections of beams and columns . . 364
References 367
Index 407
Author Index 417

Chapter 1
Various Formulations of Structural
Optimization
Various formulations of optimization of finite dimensional structures are pre-
sented in this chapter. The concepts of structural optimization are first pre-
sented in Sec. 1.1 followed by historical review in Sec. 1.2. The basic formula-
tions are presented in Sec. 1.3 with an illustrative example. The simple opti-
mization approach to plastic design that is formulated as a linear programming
problem is presented in Sec. 1.4. Optimization results under stress constraints
are shown in Sec. 1.5. The approximate method called fully-stressed design
(FSD) is presented in Sec. 1.6 with investigation of the relation between op-
timum design and FSD. The optimality criteria approach to a problem with
displacement constraints is presented in Sec. 1.7. Problems concerning the
compliance and frequency of free vibration as measures of static and dynamic
stiffness are extensively studied in Secs. 1.8 and 1.9, respectively. An example
of shape and topology optimization of a truss is presented in Sec. 1.10 as an
introduction to Chaps. 3 and 4. The basic formulation of multiobjective struc-
tural optimization programming and various methodologies of heuristics are
shown in Secs. 1.11 and 1.12, respectively, as an introduction to several sec-
tions in the following chapters. Finally, developments in simultaneous analysis
and design are summarized in Sec. 1.13.
1.1 Overview of structural optimization
In the process of designing structures in various fields of engineering, the
designers and engineers make their best decisions at every step in view of
structural and non-structural aspects such as stiffness, strength, serviceability,
constructability, and aesthetic property. In other words, they make their
optimal decisions to realize their best designs; hence, the process of structural
design may be regarded as an optimum design even though optimality is not
explicitly pursued.
Structural optimization is regarded as an application of optimization meth-
ods to structural design. The typical structural optimization problem is for-
mally formulated to minimize an objective function representing the structural
1

2 Optimization of Finite Dimensional Structures
Optimization algorithm
function values
gradients of functions
update variables
Sensitivity analysis
compute gradients
of responses
Structural analysis
evaluate responses
FIGURE 1.1: Relations among structural analysis, optimization algo-
rithm, and design sensitivity analysis for optimization using a nonlinear pro-
gramming approach.
cost under constraints on mechanical properties of the structure. The total
structural weight or volume is usually used for representing the structural
cost. Even for the case in which the structural weight is not strongly related
to the cost, it is very important that a feasible solution satisfying all the
design requirements can be automatically found through the process of op-
timization. The mechanical properties include nodal displacements, member
stresses, eigenvalues of vibration, and linear buckling loads. The structural
optimization problem can be alternatively formulated to maximize a mechan-
ical property under constraint on the structural cost.
Although there are many possible formulations for structural optimization,
e.g., minimum weight design and maximum stiffness design, the term struc-
tural optimization or optimum design is usually used for representing all types
of optimization problems corresponding to structural design.
In this book, we consider finite dimensional structures, such as frames and
trusses, which are mainly used in civil and architectural engineering. In the
typical process of structural optimization of finite dimensional structures, the
cross-sectional properties, nodal locations, and member locations are chosen
as design variables. There are many methods for structural optimization that
are classified into
• Nonlinear programming based on the gradients (sensitivity coefficients
or derivatives) of the objective and constraint functions, which is the
most popular and straightforward approach.
• Heuristic approaches, including genetic algorithm and simulated anneal-
ing, that do not need gradient information.
In a nonlinear programming approach, the design variables are updated in
the direction defined by the sensitivity coefficients of the objective function
and constraints. The relations among structural analysis, optimization al-
gorithm, and design sensitivity analysis for optimization using a nonlinear
programming approach are illustrated in Fig. 1.1, where the arrows represent
the direction of data flow; i.e., sensitivity analysis is carried out at each step of

Various Formulations of Structural Optimization 3
optimization to provide gradients of responses for the optimization algorithm,
and structural analysis is needed for sensitivity analysis and function evalua-
tion at an optimization step (see Chap. 2 and Appendix A.2.2 for details of
sensitivity analysis and nonlinear programming, respectively).
There are several approaches to the classification of structural optimization
problems. In the field of continuum structural optimization, shape optimiza-
tion means the optimization of boundary shape, whereas the addition and/or
removal of holes are allowed in topology optimization (Bendsøe and Sigmund
2003). In this book, we present various methodologies and results for opti-
mization of finite dimensional structures, including rigidly jointed frames and
pin-jointed trusses. Since optimization of trusses and frames was developed
gradually in 1960s and 1970s by academic groups in different geographical
locations, several different terminologies, e.g., configuration, geometry, and
layout, were used for representing the similar processes of shape and topology
optimization; see, e.g., Dobbs and Felton (1969), Svanberg (1981), Lin, Che,
and Yu (1982), Imai and Schmit (1982), Zhou and Rozvany (1991), Twu and
Choi (1992), Bendsøe, Ben-Tal, and Zowe (1994), Dems and Gatkowski (1995),
Ohsaki (1997b), Bojczuk and Mróz (1999), Stadler (1999), Evgrafov (2006),
and Achtziger (2007). On the other hand, optimization of cross-sectional ar-
eas of trusses was traditionally called optimum design, design optimization, or
structural optimization (Hu and Shield 1961; Prager 1974a; Rozvany 1976).
However, the term sizing optimization was often used recently to distinguish it
from shape optimization (Grierson and Pak 1993; Lin, Che, and Yu 1982; Zou
and Chan 2005; Schutte and Groenwold 2003), and structural optimization
covers all areas related to optimization of structures.
In this chapter, we present a historical review and various formulations of
optimization of finite dimensional structures.
1.2 History of structural optimization
The origin of structural optimization is sometimes credited to Galileo Galilei
(1638), who investigated the optimal shape of a beam subjected to a static
load. However, his approach was rather intuitive, and he did not establish
any theoretical foundation of structural optimization.
The intrinsic properties of minimizing or maximizing functions or function-
als in physical phenomena in nature were noticed from ancient times as various
minimum/maximum principles. The theoretical basis of minimum principles
as a foundation of modern optimization was investigated in the 18th century
and established as the calculus of variation. The principle of minimum po-
tential energy that leads to the shape of a hanging cable called catenary is
extensively used nowadays for the design of flexible structures, e.g., cable nets

and membrane structures (Krishna 1979). The surface of the minimum area
for the specified boundary shape in three-dimensional space is called minimal
surface, which is equivalent to the surface with vanishing mean curvature, and
can be achieved by a membrane with a uniform tension field without external
load or pressure. Therefore, the minimal surface is effectively used as the ideal
self-equilibrium shape for designing a membrane structure that does not have
bending stiffness (Otto 1967, 1969).
Papers by Michell (1904), Maxwell (1890), and Cilly (1900) are often cited
as the first paper that mentioned the basic idea of topology optimization;
see Sec. 3.1 for the history of topology optimization. However, the so-called
Michell truss or Michell structure has an infinite number of members; hence,
it did not lead to any practical development until the 1950s, when the prop-
erties of the optimal plastic design of frames were investigated (Foulkes 1954;
Drucker and Shield 1961; Heyman 1959). We do not discuss the history of
optimization of continuum structures such as plates and shells, because the
scope of this book is limited to finite dimensional structures. A comprehensive
literature review of early developments of structural optimization is found in
Bradt (1986), which was originally published by the Polish Academy of Sci-
ence, and includes about 300 entries up to the 1950s starting with the book by
Galileo Galilei (1638), and more than 1800 entries for the period 1960–1980.
In the 1950s, optimality conditions were studied for the plastic design of
frames (Foulkes 1954; Drucker and Shield 1961). In the 1960s, conditions or
criteria of optimality were derived utilizing minimum principles for several
performance measures of structures (Sewell 1987). Hu and Shield (1961) in-
vestigated the uniqueness of optimal plastic design. Taylor (1967) derived the
optimality condition for a vibrating rod with specified natural frequency using
Hamilton’s principle or the principle of least action. Prager and Taylor (1968)
developed optimality conditions for sandwich beams considering constraints
on compliance, natural frequency, buckling load, and plastic limit load, using
minimal total potential energy, Rayleigh’s principle, and lower- and upper-
bound theorems of limit analysis, respectively. Prager (1972, 1974a) summa-
rized the optimality conditions corresponding to various types of constraints,
including the case of multiple constraints.
Plastic design of frames was extensively studied in the 1960s and 1970s, be-
cause analytical and/or computationally inexpensive methods can be used for
this problem. Prager (1971) developed conditions for an optimal frame, sub-
jected to alternative loads, exhibiting the so-called Foulkes mechanism. Adeli
and Chyou (1987) presented a kinematic approach using automatic generation
of independent mechanisms (see Hemp (1973) for various early developments
in optimal plastic design).
In the 1970s, when the computer power was still not strong enough to use
mathematical programming approaches to optimization of real-world struc-
tures, optimality criteria (OC) approaches were widely used for finite dimen-
sional structures. The modern discrete OC approaches to trusses and frames
were initiated by Venkayya, Khot, and Berke (1973). Dobbs and Nelson (1975)

Various Formulations of Structural Optimization 5
developed the OC approach to truss design. Reviews of OC approaches are
found in Berke and Venkayya (1974) and Venkayya (1978).
Owing to the rapid development of computer hardware and software tech-
nologies, many numerical approaches were developed in the 1980s and 1990s
to obtain optimization results for practical problems. Developments in this
period can be found in many books, e.g., Arora (2007), Adeli (1994), Burns
(2002), and Haftka, Gürdal, and Kamat (1990).
It should be noted that the preferred terminologies for structural optimiza-
tion vary with age. As noted earlier, structural optimization of trusses covered
only optimization of cross-sectional properties in the 1950s and 1960s. How-
ever, sizing optimization was recently used to distinguish it from shape and
topology optimization. Optimality conditions were first called Kuhn-Tucker
conditions; however, the name was corrected to Karush-Kuhn-Tucker condi-
tions in the 1980s. Multiple load sets for formulation of constraints on static
responses were called alternative loads until the 1970s; however, they are now
usually called multiple loading conditions or multiple load sets. Furthermore,
framed structure was used for representing finite dimensional structures, in-
cluding pin-jointed trusses and rigidly jointed frames; however, they are clas-
sified into trusses and frames, respectively, in recent literature. In this book,
we use up-to-date terminology, for consistency, even for describing the results
of papers in the early stages of development.
1.3 Structural optimization problem
1.3.1 Continuous problem
If the design variables can vary continuously, i.e., can have real values, and
the objective and constraint functions are continuous and differentiable with
respect to the variables, the structural optimization problem can be formu-
lated as a nonlinear programming (NLP) problem. Let A = (A1, . . . , Am)⊤
denote the vector of m design variables. For a sizing design optimization
problem, A represents the cross-sectional areas of truss members, heights of
the sections of frame members, etc. For a geometry optimization problem, A
may represent the nodal coordinates of trusses and frames. All vectors are
assumed to be column vectors throughout this book.
The number of design variables is often reduced using the approach called
design variable linking, utilizing, e.g., the symmetry properties of the struc-
ture. The requirements to be considered in practical applications can also be
used for reducing the number of variables; e.g., the beams in the same story
of a building frame should have the same section. However, in the following,
we assume that each variable can vary independently, and, for trusses and
frames, Ai belongs to member i, for simplicity.

Consider an elastic finite dimensional structure subjected to static loads.
The vector of state variables representing the nodal displacements is denoted
by U = (U1, . . . , Un)⊤
, where n is the number of degrees of freedom. In most of
the design problems in various fields of engineering, the design requirements
for responses such as stresses and displacements are given with inequality
constraints specified by design codes:
Hj(U(A), A) ≤ 0, (j = 1, . . . , nI
) (1.1)
where nI
is the number of inequality constraints. Generally, there exist equal-
ity constraints on the response quantities; e.g., an eigenvalue of vibration
should be exactly equal to the specified value. However, we consider inequal-
ity constraints only, for simple presentation of formulations.
The constraint function Hj(U(A), A) depends on the design variables im-
plicitly through the displacement (state variable) vector U(A) and also di-
rectly on the design variables. For example, the axial force Ni of the ith
member of a truss is given using a constant n-vector bi, defining the stress-
displacement relation as
Ni = Aib⊤
i U(A) (1.2)
which depends explicitly on Ai and implicitly on A through U(A).
The upper and lower bounds, which are denoted by AU
i and AL
i , respectively,
are usually given for the design variable Ai due to the restriction in fabrication
and construction. The objective function, e.g., the total structural volume, is
denoted by F(A). Then the structural optimization problem is formulated as
Minimize F(A) (1.3a)
subject to Hj(U(A), A) ≤ 0, (j = 1, . . . , nI
) (1.3b)
AL
i ≤ Ai ≤ AU
i , (i = 1, . . . , m) (1.3c)
Problem (1.3) is classified as an NLP problem, because U(A) is a nonlinear
function of A; see Appendix A.2.2 for details of NLP. The constraints (1.3c)
are called side constraints, bound constraints, or box constraints, which are
treated separately from the general inequality constraints (1.3b) in most of
the optimization algorithms.
As is seen from the definition of constraints in (1.3b), the differential coef-
ficients of U(A) with respect to A, called design sensitivity coefficients, are
needed when solving Problem (1.3) using a gradient-based NLP algorithm.
For convenience in deriving the conditions to be satisfied at the optimal solu-
tion, the constraint function with respect to A only is defined as
e
Hj(A) = Hj(U(A), A) (1.4)
If the side constraints are treated separately from the general inequality
constraints, the conditions for optimality are derived using the Lagrangian

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RODNEY ESBEL ALLISON, OF PERRY, OHIO. 117 Bolles)
Mason, of Perry, Lake Co., Ohio, and grandson of Elijah Mason. He
was born in Perry, Nov. 26, 1822, where he lived twenty-nine years,
and since then in Painesville, Ohio. He is a farmer, and a Republican
in politics. He resides some three miles from the village, in
Painesville. CHILDREN r.OKN IX PAINESVILLE, OHIO. SSL Katie Mary
Mason," b. Nov. 9, 1854; m., Jan. 17, 1884, Henry Neil; res.
Painesville, Oliio. 382, Jessie Allison Mason," b. Dec. 27, 1869; res.
Painesville, Ohio. 383. Rodney Esbel Allison ^ [230] (Samuel,*
James,^ Capt. Samuel,^ Samuel^). He was born in Weathersfield,
Vt., July 16, 1829; married, Dec. 1, 1853, Malvina Tyler, daughter of
Ralph and Maria (Gordon) Tyler. Her father was born in Marcellus, N.
Y., Sept. 23, 1810, and died Nov. 17, 1871. She was born in Mayfield,
Ohio, June 16, 1833, and was residing in Perry, Lake Co., when
married, Mr. Allison lived in Weathersfield eight years, in Conneaut,
Ohio, ten years, and in Perry, Ohio, forty-three years. P. ().,
Painesville, Ohio. In earl}^ life he was a teacher; is now a farmer
and a justice of the peace ; does public business to some extent,
and settles many estates ; residence, Perry, Lake Co., Ohio.
CHILDREX EORX IN PEKRY, LAKE CO., OHIO. 384. Genevieve Maria
Allison," b. Dec. 28, 1864; m., July 18, 1889, Harry Graves, b. March
24, 1866; merchant; res. Geneva, Ashtabula Co., Ohio. 385. John
Tyler Allison," b. May 8, 1870; d. May 17, 1872. 386. Gertrude Mary
Allison," b. Jan. 23, 1872. 387. Orman Button Allison ^ [231]
(Samuel,* James,^ Capt. Samuel,^ Samuel^). He was bori], Feb. 3,
1831, in Weathersfield, Windsor Co., Vt. ; married, April 15, 1857,
Mary Elnora Hause, daughter of Harris E. and Lucinda (Mavnard)
Hause. Her father was born in New York, Jan. 15, 1816 ; died, Feb,
12, 1879, at Six Mile, Jenning Co., Ind. Mrs, Allison was born at the
latter place, April 8, 1840. Mr. Allison lived in Perry, O., for seven
years ; twelve in Spencer, Jennings Co., Ind, ; two in Noble, Richland
Co,, 111. ; eight in Frankfort, Kan.; four in Montrose, Henry Co.,
Missouri; one year in Live Oak, Sutter Co., Cal. Farmer. Residence,
Eight Mile, Morrow Co., Oregon, which has been his home for eight
years.

118 WALTER SCOTT ALLISON, OF VERNON, IND.
CHILDREN. 388. Carrie Bell Allison,* b. Perry, Lake Co., Ohio, June 4,
1858 ; m., Oct. 6, 1874, Franklin P. Vauglian, farmer. Members of
Christian church. Res., Frankfort, Kansas, and res., 1890, Eight Mile,
Morrow Co., Oregon. Children: I. Mertie M. Vaughan,' b. Sept. 15,
1875 ; d. May 4, 1880. II. NeUie G. Vaughan,' b. Sept. 9, 1877. III.
John Vaughan,' b. June 18, 1883. IV. Charles Vaughan,' b. Feb. 24,
1887. 389. William Orman Allison," b. Madison, Lake Co., Ohio, Jan.
7, 1860 ; m., Dec. 1, 1880, Aurilla Snow. Farmer ; res. Montrose, Mo.
They res., 1890, Eight Mile, Morrow Co., Oregon. Children: I. Walter
Allison,' b. Nov. 10, 1881. II. Pearl Allison,' b. Nov. 12, 1883. 390.
Emma Ann Allison," b. Six Mile, Ind., Sept, 27, 1862 ; m., Jan. 10,
1882, Alfred Doolittle. He is a carpenter. Children: I. Elmer Allison
Doolittle,' b. April 22, 1883. II. Ermie Doolittle,' b. Oct. 9, 1885. III.
Lester Doolittle,' b. Jan. 10, 1887. Mr. Doolittle res. North Bend, King
Co., Washington. 391. Gertrude Allison," b. Six Mile, Ind., March 28,
1864 ; d. there May 4, 1864. 392. Oscar Hause Allison," b. Noble,
Richland Co., 111., Oct. 13, 1867 ; farmer ; res. Eight Mile, Morrow
Co., Oregon. 393. Cora Lucinda Allison," b. Frankfort, Kansas, June
25, 1873 ; res. Eight Mile, Morrow Co., Oregon. 394. Walter Scott
Allison & [232] (Samuel,* James,^ Capt. Samuel,^ Samuel^). He
was born in Weatliersfield, Vt., July 9, 1832 ; married, Oct. 9, 1857,
Rebecca McConnell, born at Hardenburg, Ind. He went to Ohio when
eiglit years of age, where he lived fifteen years, and in Vernon,
Jennings Co., Ind., nine years. Machinist. He was a soldier in the
Union ami}'' in Sixth Regiment Indiana volunteers, and died at
Nashville, Tenn., July 27, 186-4. CHILDREN BORN IN NORTH
VERNON, .JENNINGS CO., IND. 395. Frank Ellsworth Allison," b. June
9, 1861 ; farmer ; res. Pittsburgh, Kan. ; m. Clara Ann Hoffman, b.
Jersey Co., 111., Nov. 8, 186L They were married at Gerard, Kansas,
Aug. 16, 1882. Children: I. Bessie Blanche Allison,' b. Pittsburgh,
Kan., Jan. 16, 1884. II. Ellsworth George Allison,' b. Leon, Butler Co.,
Kan., Aug. 21, 1885. III. Walter M. Allison,' b. Leon, Kan., Oct. 10,
1886. 396. Flora Dell Allison," b. June 2, 1863 ; m., Oct. 10, 1882, at
Hardenburg, Ind., Morton Oathout, b, Hardenburg, Ind., Oct. 2,

1861. Res. Ewing, Jackson Co., Ind. Children: I. Walter Oathout,' b.
Queensville, Jennings Co., Ind., June 12, 1883. II. Ralph Logan
Oathout,' b. Queensville, Ind., March 21, 1888. III. Hazel May
Oathout,' b. Ewing, Jackson Co., Ind., Oct. 5, 1890. 397. Roland Hill
xllison 5 [234] (Sam uel,* James,^ Capt. Samuel,^ Samuel^). He
was born in Weatliersfield, Vt., July 5, 1836 ; married, Oct. 5, 1862,
Theodocia W., daughter

ROLAND HILL ALLISON, OF CLINTON, MO. 119 of Rev.
Martin E. and Clarissa (Toiisley) Cook, and granddaughter of Josiah
Cook of Windham, Vt. Her family lived in Massachusetts, Bellville, N.
Y., Dayton, O., and her father died in Streetsboro', O., Oct. 4, 1841.
She was born at Dayton, O., Dec. 27, 1837. Mr. Allison left Ohio in
1854, and lived in Jennings Co., Ind., with his brother, Clinton J.,
until 1856; removed to St. Louis, Mo., living there until '59; then was
in trade in Ottawa, 111., until Aug., 1861, when he enlisted in
Company B., Fifty-third regiment, Illinois volunteers ; was promoted
to first lieutenant, then to captain, and two years later was
commissioned major of the same regiment, and resigned in 1865.
He participated in the siege of Corinth, Miss., of Vicksburg and of
Atlanta, and was in Sherman's March to the Sea, ending at
Savannah, Ga., in Dec, 1864. He is a Republican in politics, is
engaged in the sale of machinery and agricultural implements, and
he and his family are Baptists in their religious faith ; res. Clinton,
Henry Co., Mo. No children. 398. Alfred Bixby Quinton^ [362] (Royal
Bellows Quintou,^ Samuel Quinton,^ Margaret Allison,^ Capt.
Samuel,^ Samuel 1). He v/as born in Denmark, Iowa, Jan. 26, 1865;
married, Jan. 25, 1882, Georgie Helen, daughter of George A. and
Helen M. (Crane) Hoffman, of Topeka. Her father was born in Lyons,
N. Y., in 1830, a son of Charles Ogden Hoffman, who died in New
York city, in 1885, and grandson of Ogden Hoffman. She was born in
Rochester, N. Y., Sept. 9, 1867. Mr. Quinton graduated at Michigan
University, at Ann Arbor, in 1876. He then located in Topeka, Kan.,
and has been in the active practice of his profession as an attorney.
He has been county judge for four years; resides at Topeka, Kan.
CHILDREN BORN IN TOPEKA, KAN. 399. Helen Hoffman Qninton,^
b. April 5, 1882. 400. Georgie Fay Quinton,' b. Oct. 24, 1885. 401.
Eugenie Quinton,' b. Jan. 1.5, 1888. 402. Alfred Bixby Quinton, Jr.,'
b. Aug. 17, 1890. ALLISONS, OR ELLISONS, OF NEW HAMPSHIRE.
408. Mrs. Mary Allison (or probably Ellison), of Nottingham, N. H.,
died Jan. 17, 1869, in the 109th year of her age. She was born in
Lee, N. H., May 20, 1750. She, at her death, had eight daughters

living. The youngest was sixty years of age, three of them were over
eighty years of age.

120 ALLISONS, OR ELLISONS, OF NEW HAIVEPSHIKE. and
the eldest was in her eighty-sixth year. (N. E. Hist. Reg., 1859, vol.
13.) 404. Richard Allison (or Ellison), of New Hampshire, was
arrested on suspicion of conspiring against the state during the War
of the Revolution. On June 9, 1777, a committee of the General
Assembly was chosen to investigate, and they reported in favor of
sending him to jail for safe keeping. (N. H. Town Papers, vol. 8, p.
580.) 405. Joseph Allison enlisted April 26, 1781, for three years, or
for the war, in the army of the Revolution. 405a. Ebenezer Allison,
(or Ellison), of Deerfield, N. H., refused to sign the Association List,
in 1776.

CHAPTER YI. ALLISONS OF PENNSYLVANIA. The name
Allison occurs quite frequently among the Scotch-Irish who settled in
the south-western part of Chester county, Pennsylvania, from 1718
to 1740, at about the same dates as the emigrations from the same
localities in the north of Ireland occurred to New Hampshire,
Massachusetts, and to Maine. (See Futhey & Cope's Hist, of Chester
Co., Penn.) The surnames, with the same Christian names of the
early Scotch-blooded settlers in New Hampshire, were often
duplicated at the same dates in the Scotch settlements in
Pennsylvania, and among them are Allison, Park, Morrison, Cochran,
Boyd, Dickey, McAllister, Stewart, Wilson, Mitchell, Steele, Campbell,
and others. Nor is this strange when we remember " that as early as
1718 no less than five vessels of immigrants from the north of
Ireland arrived on the coast of New England, but, forbidden to land
at Boston by the intolerant Puritans, the immigrants moved up the
Kennebec and there settled. The winter of 1718-'19 being one of
unusual severity, the great majority of these settlers left the
Kennebec and came overland into Pennsylvania, settling in
Northampton count}'." — Letter of Wm. H. Egle, M. b., of
Harrisburg, Penn., dated April 13, 1878. He is the author of the "•
Illustrated History of the Commonwealth of Pennsylvania," published
in 1876. ALLISONS OF ALLEN TOWNSHIP, PENN. 406. James Allison,
Sr., in 1780, lived in the Scotch-Irish settlement of Allen toivnship^
Northampton county, Penn., and was there taxed. He lived on the
property owned a few years ago by Daniel Saegar. This settlement
included Weaversville and the adjacent localities. In relation to this
settlement, Rev. J. C. Clyde, D. D., in his ''History of the Allen
Township Presbyterian Church, Northampton County, Penn.," says,
that "as early as 1717 [it was 1718] no less

122 FRANCIS ALISON, OF PHILADELPHIA, PENN. than five
vessels of immi'grauts from the north of Ireland arrived on the coast
of New England, but forbidden to land at Boston by the intolerant
Puritans, the emigrants moved up the Kennebec and there settled.
The winter of 1717-18, being one of unusual severity, the great
majority of these settlers left the Kennebec, and came overland into
Pennsylvania, settling in Northampton county." (See p. 44, note to
Samuel Allison, No. 1, of Londonderry, N. H.) It was at this very time
that one portion of those emigrants went from the Kennebec, and
founded the Scotch settlement of Londonderry, N. H. In the Scotch
settlement of Allen township were the following Allisons, all
presumably the children of James Allison, Sr. Mr. Allison was a
farmer. CHILDKEN. 407. James Allison. Jr. He was a farmer; res. in
Allen townsliip, and was taxed in 1780. 408. John Allison. He was a
farmer; a resident of Allen townsliip, and was taxed in 1780. 409.
Sarah Allison, m. Joseph Horner. 410. Mary Allison, m. Joseph Hays.
411. Jeannie Allison, m. William Scott. 412. Margaret Allison. 413.
Ann Allison, m. James Wilson. REV. FRANCIS ALISON AND HIS
DESCENDANTS. 414. Rev. Francis Alison, D. D., was perhaps the
most influential person of this family name in Chester county at that
early period. He was born in 1705, in the parish of Leck,i county of
Donegal, Ireland ; educated at the University at Glasgow, Scotland;
emigrated to America in 1735; licensed as a Presbyterian minister in
1736 or 1737 ; installed over the church in New London, Chester
county. May 25, 1736, and remained fifteen years ; went to
Philadelphia in 1752, took charge of the academy there, and became
viceprovost of the college of Pennsylvania, afterwards University of
Pennsylvania, on its establishment in 1755. He was professor of
moral philosophy and assistant pastor of the First Presbyterian
church in Philadelphia, Penn. In 1756 the degree of A. M. was given
him by Yale college, and in 1758 the degree of D. D. was conferred
upon him by the University of Glasgow, Scotland. It is asserted that
he was the ' Leek is a parish on the direct road between Letter
Kenney and Raphel, and some three miles from Letter Kenney. There
is a church there, and Rev. A. W. Smyth was the incumbent in 1892.

FRANCIS ALISON, OF CHATHAM, PENN. 123 first clergyman
in tliis country to receive the degree of D. D. He married Hannah,
daughter of James Armitage, of Newcastle, Delaware, and died Nov.
28, 1779, in his 74th year. The father of Mrs. Alison was a son of
Benjamin and jIary Armitage, who came from Holmfirth Parish,
Yorkshire, Eng., and resided near Bristol, Penn. CHILDREN. 415.
Francis Alison, 2 d. in infancy. 416. Ezekiel Alison,^ d. in infancy.
417. Benjamin Alison,^ d. unman-ied about 1782. 418. Frances
Alison, Jr. 2 (421), b. in 1751; res. Chatham, Chester Co., Penn., May
11, 1813. He m. Mary Mackey. 419. Mary Alison,- d, unmarried. (See
foot note.') 421. Francis Alison, Jr.,^ [418] (Francis i). He was born
in Chester county, Penn., in 1751 ; married Mary Mackey, who was
born in Chester county, Penn., in 1757. She died in Chatham, Penn.,
in 1827.^ Mr. Alison was graduated in arts from the University of
Pennsylvania (then the college of Philadelphia) in 1770 ; studied
medicine, and was a surgeon during the Revolution. He was a
physician of eminence, and died in Chatham, Chester county, Penn.,
where he resided May 11, 1813. CHILDKEN BOKX IN CUESTER
COUNTY, PENN., PROBABLY IN CHATHAM. 422. Francis Alison,' d.
1794, aged 14 years. 423. Rachel Alison,' d. April 13, 1843, aged 62
years ; single. 424. Sarah Alison' ('^•^2), m. Alexander Adams; res.
Chester Co., Penn. ; d. June, 1843, aged 60 years. 425. Horatio
Tates Alison,3 d. 1808, aged 25 years; single. 426. Agnes Alison,' d.
1800, aged 13 years. 427. Oliver Alison,3 d. Oct. 14", 1855, aged 66
years; single. 428. Robert Alison ^ (435), b. 1789; m., May 27,
1839, Elizabeth Aitken. He d. May 4, 1854. 429. Maria Alison,' m.
William Hesson; res. Chester Co., Penn., and d. in 1811, aged 21
years. They had a son, Horatio Hesson,^ who married Margaret
Downing. They had children who went west and married there,
namely, — 1 Other Allisons. 420. Anne Allison, of Donegal, Penn. ;
m. Thomas Anderson, Nov. 30, 1774. 420 a. Miss Allison, of Bemis's
Valley, Penn. ; m. Oct. 30, 1879, Frank Stewart, a. in Bellefonte,
Penn. 420 b. Robert Alison was made a lieutenant in a Pennsylvania
regiment, Feb. 8, 1747'48. He was a nephew of Rev. Francis Alison,
on the authority of Dr. Robert S. Alison, of Ardmore, Penn. 420 c.

James Allison was a resident of Pennsylvania, June 6, 17.58. 2 She
was the daughter of John and Rachel (Elder) Mackey, who lived near
New London, of Chester county, Penn., and granddaughter of Robert
Mackey, of the same place. Pier grandfather was lieutenant of the
Provincial forces of 1747-"48, and her fatlier, John Mackey, was a
member of the constitutional convention in 1776.

124 ROBERT ALISON, OF CHATHAM, PENN. I. William
Hesson.s II. Jeanette Hesson,^ III. AVright Hesson.5 IV. Madge
Hesson." 430. Louisa Alison,^ d. , aged 70 years; single. 43L Julia
Alison, « d. June 27, 1854, aged 49 years; single. 432. Sarah Alison^
[424] (Francis,^ Francis, i). She was born in Chester county, Penn.,
and married Alexander Adams. CHILDEEN. 433. Thomas Adams,*
(439) b. Feb. 24, 1810, in Londonderry, Chester Co., Penn.; m.,
1835, Ruth A. England. 4.34. Mary Adams,^ b. Feb. 24, 1810; m.
Samuel Ramsey. Children: I. Margaret Ramsey.^ II. Adams
Ramsey.^ III. Francis Ramsey.^ IV. Horatio Ramsey. = V. Lucetta
Ramsey." 435. Robert Alison ^ [428] (Francis,^ Francis i). Dr. Alison
was born in Chester county in 1789. He graduated in medicine at the
Universit}^ of Pennsylvania in 1819, and practised his profession
until his death. May 4, 1854. He resided in Chatham, and Jennerville,
Chester county, Penn. He married, May 27, 1839, Elizabeth Aitken,
daughter of John and Jane Aitken, of Chester county. Jane Aitken
was the daughter of Capt. James and Sarah (Gettys) McDowell of
Chester county, Penn. (See Futhey & Cope's history of Chester
county, Penn.). She was born in 1807; died Aug. 21, 1851, in
Wilmington, Delaware. CHILDREN BOKN IN JENNEKVILLE, CIIESTEE
CO., PENN. 436. Louisa Jane Alison,^ b. 1841; d. Aug. 21, 1850.
437. Francis John Alison,^ (446) b. May 16, 1843; lawyer; res.
Philadelphia, Penn. 438. Robert Henry Alison,* (450) b. June 8,
1845; physician; res. Ardmore, Penn. 439. Thomas Adams ^ [433]
(Sarah Alison,^ Francis,^ Francis 1). He was born in Londonderry,
Chester county, Penn., Feb. 24, 1810. He married, 1835, Ruth A.
England. CHILDKEN. 440. Sarah Adams.e 441. Mary Adams,"' m.
Joseph Pratt. Children: I. Nathaniel Pratt." II. Adams Pratt." 442.
Robert Adams,'' m. Elizabeth Strawbridge. Children: I. Anna Adams."
II. Sarah Adams."

FRANCIS JOHN ALISON, OF PHILADELPHIA, PENN. 125 III.
Louisa Adams.' IV. Robert Adams.» V. Edwin Adams." 443. Louisa
Adams, ^ m. Andrew J. Young; res. 1330 Spring Garden St.,
Philadelphia, Penn. Children: I. Edwin Stanton Young." II. James
Thomas Young." 444. Oliver Adams. ^ 445. Emmeline Adams. s 446.
Francis John Alison * [437] (Robert,^ Francis,^ Francis^). He was
born in Jennerville, Chester county, Penn., May 16, 1843 ; married,
Sept. 6, 1877, Sophia Dallas Dixon, who was born in Philadelphia,
Penn., Dec. 28, 1853. She was the daughter of Fitz Eugene and
Catherine Chew (Dallas) Dixon. Her father was born in Amsterdam,
Sept. 4, 1820 ; resided in Farley, Bucks county, Penn., and died in
Philadelphia, Penn., Jan. 22, 1880. He was the son of Thomas Dixon,
Jr., and his wife, Mary B. Dixon, who was born Jan. 26, 1781, in
Westminster, London, Eng. ; resided in Boston, Mass , where he died
Sept. 15, 1849. He was the son of Thomas Dickson (or Dixon), born
Nov. 6, 1739, in Perthshire, Scotland ; resided in Amsterdam ;
married Elizabeth Mann, and died in Amsterdam, Oct. 25, 1824. Mr.
Alison graduated from the academic department of Harvard
University, Cambridge, Mass., in 1865 ; was admitted to the bar of
the city of Philadelphia, Penn., June 7, 1875, and practises his
profession as a lawyer at 216 South 4th St., of that city ; resides at
327 South 18th St., Philadelphia, Penn. CHILDREN BOKIf IN
PHILADELPHIA, PENN. 447. Catherine Dallas Alison,^ b. June 11,
1878. 448. Mary Elizabeth Alison,^ b. June 17, 1880. 449. Frances
Armitage Alison,*^ b. March 27, 1889. 450. Robert Henry Alison ^
[438] (Robert,-^ Francis,^ Francis^). He was born in Jennerville,
Chester county, Penn., June 8, 1845. He graduated in arts at Yale
College, New Haven, Conn., in 1867, and in medicine, from the
University of Pennsylvania, in 1869. He is a physician. From May,
1871, to Oct., 1872, he was a resident physician of the Pennsylvania
Hospital in Philadelphia. From Feb., 1883, to Nov., 1884, when he
resigned, he was port physician of the port of Philadelphia. He
removed to Ardmore, Montgomery county, Penn., Nov. 4, 1884;
unmarried; resides at Ardmore, Penn.

126 JAMES ALISON, OF PITTSBURGH, PENN. OTHER
ALLISONS OF PENNSYLVANIA. Rev. James Allison, of Pittsburgh,
Penn., in a personal letter, Dec. 17, 1890, says: "Part of the Allison
family, to which I belong, went to Mecklenburg count}^. North
Carolina, nearly one hundred and fifty years ago. Many of the
descendants are still there (see sketch of Allisons of North Carolina,
No. 579). Another part went to Virginia, and thence passed on into
Indiana (see Allisons of Indiana, No. 463). One family of the part
that went to North Carolina returned to Cecil county, Maryland, and
afterwards removed to Washington county, Pennsylvania. One of its
number (Hon. James Allison, No. 485), afterward going to Beaver
county, Pennsylvania, served in the Eighteenth congress and was
reelected to the Nineteenth, but declined on account of ill-health.
The late Hon. John Allison, register of the United States Treasury,
was his son (see sketches of Hon. James Allison, and of his son,
Hon. John Allison, No. 486). The father of Hon. William B. Allison,
United States senator from Iowa, removed from the Cumberland
valley, Pennsylvania, to Bellafonta, Pennsylvania, and then to- the
Western Reserve, Ohio, where William B. Allison was born (see
sketch of Hon. William B. Allison, No. 489). One of my grandfather's
brothers went from the Cumberland valley to Erie, Pennsylvania,
where his descendants still live (James Allison and his descendants
of Lake Pleasant, Erie county, Pennsylvania, may be of this family.
See notice of them. No. 490). Another went to Butler ; and my
grandfather himself removed to the south side of the Monongahela
river, near this city, in 1810, and afterward to a place ten miles north
of this city, where he resided until his death." 460. James Allison*
(James,'5 George,^ Allison i).i Rev. James Allison was born in
Pittsburgh, Penn., September 27, 1823 ; married, August 6, 1851,
Mary Jane, daughter of Robert Anderson, who was born in Lancaster
county, Penn., and who lived in Washington, Washington county, and
in Sewickley, Alleghany county, Penn. Mrs. Allison was born and died
in the latter place. He married, second, November 6, 1855, Caroline,
daughter of Hon. Jolui M. Snowden. She was born in Pittsburgh,
Penn. Mr. Allison graduated at Jefferson College, Penn., in the class

of 1845, taking the first honor. He studied theology in the *He is the
son of .James and Elizabeth (Brickett) Allison, grandson of George
and Susan (McKoberts) Allison, son of Allison, an emigrant from the
north of Ireland.

ALLISONS OF INDIANA. 127 Western Theological Seminary,
Alleghany, Penn. In 1848 he took charge of the Presbyterian church
at Sewickley, Penn., fourteen miles from Pittsburgh, where he
continued to be pastor until 1864. During his pastorate the church
had grown to be the strongest in the county outside of Pittsburgh. In
1864 he resigned, and became editor and proprietor, in connection
with the late Robert Patterson, of The Presbyterian Banner^ at
Pittsburgh, of which from 1856 to 1861 he had been one of the
editors and proprietors. This paper was started in Chillicothe, Ohio,
July 5, 1814, — one of the very first religious newspapers of its kind,
— and is very widely circulated. Mr. Allison is its editor in 1891. He
was one of the original signers of the memorial on the subject of the
reunion between the old and new school Presbyterian churches in
1864, and was the author in 1868 of the platform by which the
union was effected in 1869. Much of the time during the War of the
Rebellion he was in the field with the Pennsylvania troops, though
not a soldier. From 1865 to 1890 he served on the General
Assembly's Board for Freedmen, acting as treasurer, without salary,
from 1870 to 1889. CIIILDKEX BORN IN SEWICKLEY, PENN. 461.
Lizzie Allison, = b. in 1852; m., in 1875, S. W. Reinliart; res.
Brookline, Mass. 462. John M. S. Allison,^ b. in 1857; m. Miss M. B.
Lauglilin; was an editor; res. Pittsburgh, Penn.; d. Dec. 27, 1877.
ALLISONS OF INDIANA — A BRANCH OF THE PENNSYLVANIA
FAMILY. The account of the Allisons as furnished by this family is
that there were six brothers : 463. George Allison^ settled in Iredell
Co., N. C. 464. William Allison i settled in Charlotte, N. C. 465. John
Allison* settled in North Carolina. 466. Thomas Allison ' settled in
North Carolina; was a teacher. 467. Robert Allison * settled in North
Carolina; see sketch of North Carolina Allisons. 468. James Allison, '
(46y) m. Miss Young; res. Donegal, Penn. 469. James Allison 1(468).
He settled in Donegal township, Penn., near where Harrisburg now
stands. He was an elder in the Presbyterian church. He married Miss
Young. He had three sons, and perhaps other children.

128 WILLIAM ALLISON, OF DONEGAL, PENN. CHILDREN.
470. "William Allison! (473). Settled near Staunton, Va. 471. John
Allison.^ He was a colonel in the Kevolution, it is said. 472. James
Allison.^ William Allison 2 [470] (James i)- He left his fa,tlier's home
ill township of Donegal, Penn., and settled near Staunton, Va. In the
Revolutionary army he was a lieutenant, and was with General
Washington in his retreat through New Jersey. After his settlement in
Virginia he was for many years an elder in the Presbyterian church,
for like most of the Allisons he was a pronounced adherent of that
church. CHILDREN. 474. James Allison' (470). Deceased. 475. John
Allison. ' Deceased. 476. James Allison ^ [474] (William,2 James i).
He went West, and married, near Cincinnati, O., Sarah Cox, a lady of
German descent, who died before Mr. Allison. He had several
children. Among them were : CHILDREN. 477. William Allison.^ Ees.
Toledo, Ohio. 478. Mary Ann Allison, < m. James Shevoel; res.
Lawrenceburg, Ind. 479. James Young Allison,* (480) b. in Jefferson
Co., Ind.; res. Madison, Ind. 480. James Young Allison * [479]
(James,^ William,2 James ^). Hon. James Y. Allison was born in
Jefferson county, Ind., Aug. 20, 1823 ; married Antoinette Mclntire.
He was educated at Hanover college, Jefferson county, Ind.; studied
law with Joseph G. Marshall, of Madison, Ind.; was admitted to the
bar in Sept., 1847; served three terms as prosecuting attorney, one
term as a state senator, and was elected judge of the fifth judicial
circuit in Oct., 1873, for six years, and was reelected for six years
more in 1878. He resided in Madison, Ind., in that year. CHILDREN.
48L Edward Allison.B 482. James Graham Allison.^ 483. Antoinette
M. Alliscm.'' 484. Charles B. Allison.^ 485. Hon. James Allison. He
was born in Cecil county, Maryland, Oct. 14, 1772; lawyer. Acquired
a high legal position in western Pennsylvania, was elected to 18th
con

HON. WM. B. ALLISON, UNITED STATES SENATOR FROM
IOWA.

WILLIAM B. ALLISON, OF DUBUQUE, IOWA. 129 gress from
Pennsylvania, reelected to 19th. After practising his profession for
fifty years he died in June, 1854. 486. Hon. John Allison, son of the
foregoing James Allison, was born in Pennsylvania Aug. 5, 1812.
Studied law but never practiced. Was elected to the assembly of
Pennsylvania in 1846-'47 and '49, and was a member of the 33d and
34th congresses, house of representatives, from Pennsylvania. He
was appointed registrar of the treasury of the United States in 1869;
and died while in office, March 23, 1873. 487. John Allison, his son,
is living on a ranch in Montana. (See letter of Rev. James Allison,
preceding No. 460.) 488. Hon. Robert Allison was born in
Pennsylvania, and was a representative to congress from that state
from 1831'33. (From Charles Lanman's "Biographical Annals of the
United States Government." The sketch of W. B. Allison is from
Harper's Weekly, March 17, 1888). 489. Hon. William B. Allison,^
United States senator from Iowa. He was born in Perry, Wayne
county, Ohio, March 2, 1829, and is the son of John Allison,^ who
was born in Bellefonte (or its neighborhood), Penn., in 1798, and
who removed to Ohio about 1824, and resided on a farm in Perry.
John Allison^ was the son of Archibald Allison^ who migrated from
the county of Monaghan, Ireland, in 1783, and settled in Centre
county, Penn. Senator Allison spent his early years upon a farm and
was educated at Allegheny college, Penn., and at Western Reserve
college, Ohio. He studied law, and practiced in Ohio till 1857, when
he located in Dubuque, la., which has been his home since April,
1857. He began his public career when the war broke out as a
member of the staff of the governor, and his first task was to aid in
the organization of the volunteer regiments that were destined to
serve in the War of the Rebellion. He was sent to congress while the
war was going on, and has been representative and senator from
that time to the present, except between 1871 and 1873, when he
declined an election, so that he has participated in all the legislation
that has been enacted during and since the great conflict. He has
done his full share in it all, and his impress is on the statutes which
have framed and modified our fiscal and banking systems, our

methods of taxation, as it was on the laws which gave to Mr. Lincoln
the power to put down the rebellion, and which readjusted their
relations to the Union of the once insurrectionary states. His
biography is 9

130 WILLIAM B. ALLISON, OF DUBUQUE, IOWA. part of the
history of the times in which he has lived. Through them Mr. Allison
has accurately represented the sentiments and opinions of his
section and of his party. He has performed the duties imposed upon
him with calmness and caution. He was one of the congressmen
depended upon by the president and secretar}'- of the treasury to
devise ways and means needed for the support of the government.
After the war he continued to be a radical Republican, always acting
with his party, opposed to Johnson, and a believer in the
reconstruction measures which were intended to revolutionize the
political complexion of the conquered South, and to make the
freedman a citizen and a voter. Senator Allison is one of the safe
men of the Republican party. He is without passion, prejudice, or
very strong friendships. He has not made the mistake, so common of
recent j^ears, of allying himself to a faction. He is not weak, nor a
trimmer, nor a man of undecided views. It is not for any one of
these qualities that he fails to make enemies ; it is because he is
never carried away by the passions of the moment, but is so moved
and dominated by his judgment that the public men who know him
and have been associated with him realize that his action is always
the result of his matured opinion. There are very few men who have
been so long in public life as he who are so scrupulously devoted to
their work. Men like him are oftener found in the British parliament,
where tenure of place is more secure. Practically, Mr. Allison's tenure
has been as strong as theirs, and his familiarity with the business of
legislation is as accurate and thorough as that of the under-
secretaries of the British cabinet. This is especially true of his
acquaintance with fiscal matters. On his first entrance into
congressional life he came to the front in the consideration of all
questions affecting the treasury, the banks, and taxation. He was a
member of the ways and means committee of the house of
representatives very early, if we take into consideration the very
large majority which his party had in congress at the time, and the
number of able men in both houses. The reputation that he then
made for himself for accurate information and sound judgment has

not been lost. He has not been tempted to endeavor to shine in the
discussion of other questions. He has been content to be easily the
first authority on all bills relating to expenditures. Some of his short
speeches have indicated that he might have been a leading debater
on questions of constitutional law and on taxation and bank policy.
So far as the

WILLIAM B. ALLISON, OF DUBUQUE, IOWA. 131 last two
subjects are concerned, he has been prominent, and there are very
few public men of his party whose opinions on all fiscal matters are
more respected than Mr. Allison's ; but of recent years he has been
chairman of the appropriations committee, and none but the most
reckless undertake to question his statements of fact concerning the
expenditures of the government. As chairman of the appropriations
committee he has been of very important service to the cause of
sound administration. He is a wise economist. This means judicious
liberality as opposed to an extravagant saving. The modern
deficiency bill, and the urgency bill, which has only recently become
one of the appropriation bills to be reckoned with at every session of
congress, would not exist, or would involve inconsiderable amounts
of money, if Mr. Allison's views about the regular and stated bills
always prevailed. The chairman of the senate appropriations
committee knows what each branch of the public service needs for
its proper maintenance, and is willing to take the responsibility of
advocating its appropriation. The spirit in which he . performs this
vital public function is directly opposed to that which moves very
many members of congress, who do not appreciate their
responsibilities to refuse appropriations, and thus lower the
aggregate, when the refusal will not attract public attention and
arouse popular protest. Not many years ago the member of the
house committee who had charge of the diplomatic appropriation bill
refused to allow the secretary of state any money for postage or
cable charges, and thus threatened to cut off the state department
from all correspondence with our representatives in foreign
countries. This incident illustrates the tendency and attitude of
certain persons who seek to figure before the country as savers of
the people's money, and who have wider reputations as economists
than Mr. Allison ; but Mr. Allison is neither sordid nor extravagant. He
does not advocate loose and unguarded expenditure, and he is
always desirous that every department and division of the
government shall have all that it needs. It is not exaggerating to say
that when he is ready to sign a report of his committee on an

appropriation bill he knows as much of the requirements of the
objects for which the proposed expenditures are to be made as the
executive officer who is at the head of the department. And in all the
years during which he has acted in his present capacity there has
not been a whisper injuriously affecting his reputation. Mr. Allison's
influence on general legislation has been felt

132 WILLIAM B. ALLISON, OF DUBUQUE, IOWA. because
he insists on havint;^ a reason for his votes. He is largely influenced
by the feeling and opinions of his section of the country. This has
made him an advocate of lower rates of tariff duties, and a
consistent friend of the land-grant railroads. In 1870, after he had
declined a reelection to the house of representatives, and just before
he was chosen to be Senator Harlan's successor, he took a very
prominent part in the debate on Mr. Schenck's tariff bill. In the
course of a speech on that measure, he said, — " The tariff of 1846,
although confessedly and professedly a tariff for revenue, was, so
far as regards all the great interests of the country, as perfect a tariff
as any that we have ever had." Perhaps the following extract from
the same speech will best illustrate his tariff views of that time : "
Our policy should be so to cheapen manufactured products that we
can revive our export trade, now swept away because we cannot
compete with other nations in the markets of the world. If we could
restore what we have lost, and in addition greatly enlarge our
exportations of manufactures, we would then have an enlarged
home market for our agricultural products in a concentrated form, in
exchange for other commodities which we do not and cannot
produce." He is really the author of the existing silver law, although
he did not bring forward and advocate the measure as an original
proposition. As tlie Bland bill passed the house of representatives it
was a free coinage measure, and the senate finance committee was
equally divided for and against it, Mr. Allison neither approving nor
opposing it. Some silver legislation was inevitable, and Mr. Allison
suggested the measure which was adopted. He is a bimetallist, but
not of the Bland kind, and the law as it stands to-day (1888) ought
to bear Mr. Allison's rather than Mr. Bland's name. The measure was
probably the most conservative that could have been adopted at the
time it became a law. Mr. Allison's friendship for the land-grant
roads, which came into existence during the beginning of his service
in the house, was shown by his opposition to the Thurman act.
There was no question as to the sincerity of his position, however.
He voted and spoke against the bill because he believed that it

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