Graphical Analysis: A Text Book on Graphic Statics

389 Pages·1921·22.118 MB·English

by  William Sidney Wolfe

GRAPHICAL ANALYSIS
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GRAPHICAL ANALYSIS
A TEXT BOOK ON
GRAPHIC STATICS
BY
WILLIAM S. WOLFE, M.S.
HeadofStructuralDepartment, Smith, Hinchman & Grills. Formerly
Instructor in Architectural Engineering, University ofIllinois.
AssociateMemberAmericanSocietyofCirilEngineers.
AssociateMemberSocietyofNavalArchitects
andMarine Engineers
FirstEdition
Fourth Impression
McGRAW-HILL BOOK COMPANY,
Inc.
NEW YORK: 370 SEVENTH AVENUE
LONDON: 6&8BOUVERIE ST.,E.C.4
1921
re
no
•2.
W64-
Copyright1921, bythe
McGkaw-Hill Book Company, Inc.
PRINTEDINTHEUNITEDSTATESOFAMERICA
PREFACE
This book has been developed from notes and blue prints
prepared by the writer and used in his classes at the University
of Illinois. Certain additional material has been added, a part
of which has been briefly presented in a number of articles con-
tributed by the writer to the technical press. The notes referred
to above were modeled after a set prepared by Dr. N. C. Ricker,
severalyearsbefore, towhomthewriterisgreatlyindebted.
Many additional problems might have been added and the
discussion greatly extended. This is especially true of Chapters
X
IV and VII to inclusive. However, it has been thought best
to keep the book to smaller proportions, and the writer believes
that a thorough mastery of the constructions and solutions
here presented will give the student an excellent grasp of the
subject.
The object has been to deal with the analysis of stresses rather
than with design or the computation of loads. Nevertheless it
has seemed desirable to give some attention to the determination
of loads, and Chapter IX has been devoted almost exclusively to
design. Material and ideas have, of course, been drawn from
many sources.
The writer is especially indebted to Prof. C. R.Clark forassist-
ance in connection with the early part of the work and to Prof.
L. H. Provineforencouragement and constructivecriticism.
w. s. w.
Detroit, Mich.
October, 1921.

.. . 1
TABLE OF CONTENTS
PAGE
Preface v
Notation
xiii
CHAPTER
I
GENERAL METHODS
ART.
1 Introduction 1
2. Definitions 2
3. Graphical Construction and Measurement ofan Angle 3
4. Representation of Forces 4
5. Composition of Forces 6
6. Resolution of Forces 8
7. Concurrent-CoplanarForces,ConditionsforEquilibrium 10
8. Unknowns 11
9. Non-Concurrent Coplanar Forces 13
10. Funicularor Equilibrium Polygon 14
11 Resultant a Couple 18
12. Non-ConcurrentCoplanarForces,ConditionsforEquilibrium 19
13. Parallel Forces 19
14. Resultantof a Numberof Parallel Forces 21
15. Reactionsfora Beam 23
16. Reactionsfor a Beam witha Distributed Load 25
.
17. Reactions fora Beam with Inclined Loads 26
18. Reactionsfora Truss, Vertical Loads 27
19. Reactions fora Trusswith Vertical and Inclined Loads 27
20. HorizontalComponentDividedEquallybetweentheTwoReactions 3
21 ReactionsofaRafterSupportedonPurlins 32
.
22. Stresses in the Members ofa Frame 33
23. Stresses in a Bicycle Frame 35
24. Passing a Funicular Polygon through Three Given Points 37
CHAPTER
II
CENTROIDS
25. Centroid ofa Broken Line 40
26. Centroid ofan Arc 42
27. Centroidofa Curve 44
28. CentroidsofAreas 45
vii
TABLE OF CONTENTS
viii
ART. PAGE
29. Centroidofa Triangle 45
30. Centroidofa Quadrilateral 46
31. Centroidofa Trapezoid 47
32. Given the Centroid, Area and Distance between the Two Parallel
SidesofaTrapezoidtoFindtheLengthsoftheParallelSides 48
33. Centroidofa Sector 49
34. Centroidofa CircularSegment 50
35. Centroidsof IrregularAreas 51
36. CentroidsofVolumes 53
37. Centroidofa TriangularPyramid 53
38. Complex Volumeswhichcanbe Divided into a Numberof Regular
Volumes 54
39. IrregularVolumes, Division into Slices 57
40. CentroidLocatedbytheUseofSectionsandAreaCurves 59
41. StressVolumes 61
CHAPTER
III
MOMENTS
42. First Momentof a Forceabouta Point 64
43. FirstMomentofaNumberof Forcesabouta Point 65
44. First Momentof an Area abouta Given Axis 67
45. Moment Diagram fora Beam 68
46. Second Moments 69
17 Second Momentofa Numberof Parallel Forces 70
48. Momentof Inertiaofan Area 72
19 Mohr's and Culman's Methods 73
50. Radiusof Gyration 74
.";1, RadiusofGyrationofRectangles, ParallelogramsandTriangles 75
52. Momentof Inertia, Exact Method 76
53. Higher Moments 77
54. Central Circle or Ellipseof [nertia 79
55. Complicated Problems 80
CHAPTER IV
BEAMS
56 ConstructionoftheElastic( lurve 81
57 Simple Beams s">
antilever Beams 's'->
.".'i BeamswithanOverhangingEnd '.»1
60 BeamswithaVariable1 94
61 Beams uiili <tae End l ix< d 96
62 B< am I ixed al Both Ends 99
1 ontinuout Beam ofTwoSpan* K)2
64. ContinuousBeam ofThreeSpanc 107
TABLE OF CONTENTS
ix
CHAPTER V
TRUSSES
ART. pAGE
65. Weight ofTrusses 113
66. Other Weights. 114
67. Snow Load 115
68. Wind Loads 115
69. Notation 116
70. Stresses Obtained Analytically 116
71. Stresses by Analytical Moments 117
72. Stresses by Graphical Moments 117
73. Stresses Obtained Graphically by Joints 118
74. Stresses Obtained by Stress Diagram 120
75. Stress Diagram, Upper and Lower Chord Loads 122
76. Wind Loads; Reactions and Stress Diagrams 124
77. Stress Diagram Combined Loads 126
78. Maximum and Minimum Stresses, Reversals 128
79. Cantilever Trusses 134
80. Mill Bent 137
81. Three-Hinge Arch 141
82. A Large Three-Hinge Arch 145
83. A Large Mill Bent 147
84. Cantilever Truss with Four Supports 151
85. Combination Truss, Three-Hinge Arch and Mill Bent 154
86. Bridge Trussesof the KType 156
87. Trussed Dome 158
88. Ring Dome, Dead Loads 159
89. Ring Dome, Wind Loads 163
CHAPTER VI
MOVING LOADS
90. Single Concentrated Moving Load 168
91. SingleConcentratedMovingLoadandaUniformDeadLoad 171
92. Uniform Moving Load Longer than the Span 175
93. Moving Uniform Load Shorter,than the Span 177
94. Uniform Dead Load, and Uniform Moving LoadShorter than the
Span 179
95. Two Concentrated Moving Loads 179
96. Three Concentrated Moving Loads 184
97. FourConcentratedMovingLoadsandaLTniformDeadLoad 187
98. A Large Numberof Concentrated Moving Loads 190
99. Maximum Shears and Moments in a Turntable 193
100. Moving Loads on Trusses 196
1
x TABLE OF CONTENTS
CHAPTER VII
MASONRY
ART. PAGE
101. Stresses in Rectangular Piers 201
102. Stress Volumes 204
103. Problems, Rectangular Piers 205
104. Problems, Irregular Piers 206
105. Kerns 209
106. LocationofPointsontheEdgeoftheKernAnalytically 214
107. Pressureon Wall Footings 216
108. Retaining Walls 218
109. Line of Pressure in a Pier 222
110. Masonry Chimneys 224
111. Line of Pressure in an Arch 228
112. Three-Hinged Arches 231
113. Three-Hinged Arch Symmetrically Loaded 237
114. Two-Hinged Arches 238
115. Two-Hinged Arch, Methodof Least Work 238
116. Hingeless Arches, Method of Least Work 242
117. Hingeless Arches, General Discussion 242
I 18 Solution ofanArchUsingTheoryofLeastCrown Pressure 245
119. Investigation of a Gothic Vault 245
120. A Study of Domes 250
CHAPTER VIII
RE1\I'< >R('ED CONCRETE
121. Simple Rectangular Beams 254
122 For Rectangular Beams 259
I
r-Beamfi 2(10
12A I of T-Beams 264
125 Double Reinforced ( Joncrete Beams 264
126 Bending Stresses in 'lomplex Sections -t>7
i_'7 Combined Stresses J7(>
128 Eccentrically LoadedColumns J7
129 Reinforced ( kmcrete ( Ihimneys 275
130, Deflection uf Reinforced Concrete Beams 278
(II U'TKlt IX
DESIGN
131 D ign of Beams 281
n of Plate < iirdeni 297
'..ri mi . I), ignand Details '{(>(>
1.;I Wooden Tru D< ignand Details 305
i.. Diagramfor DesigningWooden Compression Members 313
D
136 Diagramfor igning Casl IronO-GeeWashers :;i I