Graphical Methods for Continuous Beams and Frames
Historical, Pedagogical, and Design Approaches
This dissertation revisits historical graphical methods for the elastic analysis of continuous beams and frames, a neglected branch of graphic statics. While graphic statics is celebrated for its clarity in axial-force systems, its application to flexural,
statically indeterminate structures has been largely overlooked. Between the late 19th and mid-20th centuries, several methods were developed, notably the fixed-points method and trial-closing-string methods, but they have mostly faded from practice and scholarship, and their potential for teaching and design remains unexplored.
This research reviews their historical development and rationale, focusing on the fixed-points and characteristic-points methods within the trial-closing-string family. Building on this, two graphical methods are proposed: Method I, an enhanced
characteristic-points method, and Method II, a synthesis of the fixed-points method and Method I, both providing a more intuitive, bidirectional representation of stiffness– moment relationships. They are accessible to architects and versatile for complex frame structures.
Within interactive parametric environments, the pedagogical and design potential is explored. Method I offers a dynamic graphical interpretation of classical displacementbased analysis, while Method II develops intuition for stiffness–moment relationships. Both methods support parametric feasibility assessment in structure design, offering greater transparency, real-time bidirectional control, and clearer geometric constraints than conventional finite element method–based tools.
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- Publisher
- Presses universitaires de Louvain
- Author
- Shuyuan Han,
- Set
- | n° 31
- Language
- English
- BISAC Subject Heading
- ARC024000 ARCHITECTURE / Buildings
- BIC subject category (UK)
- AM Architecture > TN Civil engineering, surveying & building
- Onix Audience Codes
- 06 Professional and scholarly
- CLIL (Version 2013-2019)
- 3075 Bâtiment > 3076 Architecture/Urbanisme
- Title First Published
- 20 October 2025
- Subject Scheme Identifier Code
- : Architecture et génie civil
Paperback
- Publication Date
- 20 October 2025
- ISBN-13
- 9782390616269
- Extent
- Main content page count : 370
- Code
- 109224
- Dimensions
- 16 x 24 cm
- Weight
- 588 grams
- Packaging Type
- No outer packaging
- List Price
- 41.00 €
- ONIX XML
- Version 2.1, Version 3
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Contents
CHAPTER 1 INTRODUCTION 1
1.1 Background 3
1.2 Problem statement and existing research 14
1.3 Objectives of the research 18
1.4 Organisation of the content 20
CHAPTER 2 THE PRECURSOR AND EMERGENCE OF CONTINUOUS BEAMS AND
FRAMES 23
2.1 The functions of continuous beams and frames 25
2.2 Knee-braced timber and cast-iron frames: the predecessors of rigid frames 28
2.3 The emergence of rigid frame and engineered continuous beam 42
2.4 Vierendeel girder: transition between rigid frame and continuous frames 84
2.5 The continuous frame full-fledged in both construction, function, and engineering 94
2.6 Summary 104
CHAPTER 3 A REVIEW OF THE FIXED-POINTS METHOD 107
3.1 The fixed-points method 109
3.2 The evolution of the fixed-points method 110
3.3 The theoretical foundation of fixed-points method 126
3.4 The principles and determination of the fixed points 132
3.5 The fixed-points method for continuous beams 140
3.6 The fixed-points method for continuous frames 146
3.7 Robert Maillart's application in the analysis of bridge frames 151
3.8 Discussion 157
3.9 Summary 159
CHAPTER 4 A REVIEW OF THE TRIAL-CLOSING-STRING METHODS 161
4.1 The trial-closing-string method 163
4.2 The history of the trial-closing-string method 164
4.3 The theoretical foundation of the trial-closing-string methods 170
4.4 Greene's semi-graphical method 175
4.5 Filder’s characteristic points method 178
4.6 Wolfe’s method 185
4.7 Comparison and discussion 192
4.8 Summary 196
CHAPTER 5 METHOD I – DEVELOPING CHARACTERISTIC-POINTS METHOD 199
5.1 The advantages and limitations of Filder’s characteristic points method 201
5.2 Enhancing the visual distinction of the characteristic points method diagram 204
5.3 The location of the characteristic points 207
5.4 The characteristic moment: formula, meaning, and sign 215
5.5 Applications for the analysis of structures 218
5.6 Reverse approach: deriving member stiffness from predefined inner force 233
5.7 Summary 236
CHAPTER 6 METHOD II – SYNTHESIZING FIXED-POINTS METHOD WITH METHOD I 241
6.1 Complementary potential of Method I and fixed-points method 243
6.2 Load-induced bending moment and its distribution 246
6.3 Bending moment induced by transverse displacement at joint 256
6.4 Reverse approach: deriving member stiffness from predefined bending moment 260
6.5 Comparison 262
6.6 Summary 267
CHAPTER 7 APPLICATION: EXPLORING THE PEDAGOGICAL RELEVANCE OF METHOD I AND II 269
7.1 Pedagogical potential of method I and II 271
7.2 Method I: interactive diagram for intuitively illustrating deflection methods 274
7.3 Method II: interactive diagram for intuitive stiffness-moment exploration 297
7.4 Summary and discussion 302
CHAPTER 8 APPLICATION: EXPLORING THE DESIGN RELEVANCE OF METHOD I AND II 305
8.1 Introduction 307
8.2 Implementing in a parameterized environment 311
8.3 Re-sizing of the L-frame of the Garstatt Bridge 315
8.4 Re-sizing the frame of the Weissensteinstrasse Overpass 322
8.5 Summary and discussion 327
CHAPTER 9 GENERAL CONCLUSION 329
9.1 Summary of results 330
9.2 Contributions 332
9.3 Limitations and future work 337
REFERENCES 341