Lecture Notes in: STRUCTURAL ENGINEERING

Lecture Notes in: STRUCTURAL ENGINEERING 30
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Draft DRAFT Lecture Notes in: STRUCTURAL ENGINEERING Analysis and Design Victor E. Saouma Dept. of Civil Environmental and Architectural Engineering University of Colorado, Boulder, CO 80309-04283 Draft PREFACE Whereas there are numerous excellent textbooks covering Structural Analysis, or Structural Design, I felt that there was a need for a single reference which  Provides a succinct, yet rigorous, coverage of Structural Engineering.  Combines, as much as possible, Analysis with Design.  Presents numerous, carefully selected, example problems. in a properly type set document. As such, and given the reluctance of undergraduate students to go through extensive verbage in order to capture a key concept, I have opted for an unusual format, one in which each key idea is clearly distinguishable. In addition, such a format will hopefully foster group learning among students who can easily reference misunderstood points. Finally, whereas all problems have been taken from a variety of references, I have been very careful in not only properly selecting them, but also in enhancing their solution through A appropriate gures and LT X typesetting macros. E Victor Saouma Structural EngineeringDraft Contents I ANALYSIS 29 1 A BRIEF HISTORY OF STRUCTURAL ARCHITECTURE 31 1.1 Before the Greeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.2 Greeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.3 Romans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.4 The Medieval Period (477-1492) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.5 The Renaissance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 1.5.1 Leonardo da Vinci 1452-1519 . . . . . . . . . . . . . . . . . . . . . . . . . 36 1.5.2 Brunelleschi 1377-1446 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 1.5.3 Alberti 1404-1472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 1.5.4 Palladio 1508-1580 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 1.5.5 Stevin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 1.5.6 Galileo 1564-1642 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 1.6 Pre Modern Period, Seventeenth Century . . . . . . . . . . . . . . . . . . . . . . 42 1.6.1 Hooke, 1635-1703 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 1.6.2 Newton, 1642-1727 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 1.6.3 Bernoulli Family 1654-1782 . . . . . . . . . . . . . . . . . . . . . . . . . . 45 1.6.4 Euler 1707-1783 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 1.7 The pre-Modern Period; Coulomb and Navier . . . . . . . . . . . . . . . . . . . . 47 1.8 The Modern Period (1857-Present) . . . . . . . . . . . . . . . . . . . . . . . . . . 48 1.8.1 Structures/Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 1.8.2 Ei el Tower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 1.8.3 Sullivan 1856-1924 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 1.8.4 Roebling, 1806-1869 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 1.8.5 Maillart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 1.8.6 Nervi, 1891-1979 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 1.8.7 Khan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 1.8.8 et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2 INTRODUCTION 55 2.1 Structural Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.2 Structures and their Surroundings . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.3 Architecture & Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.4 Architectural Design Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.5 Architectural Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.6 Structural Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.7 Structural Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57CONTENTS 7 Draft 6 INTERNAL FORCES IN STRUCTURES 113 6.1 Design Sign Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.2 Load, Shear, Moment Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.3 Moment Envelope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 6.4.1 Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 E 6-1 Simple Shear and Moment Diagram . . . . . . . . . . . . . . . . . . . . . 117 E 6-2 Sketches of Shear and Moment Diagrams . . . . . . . . . . . . . . . . . . 119 6.4.2 Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 E 6-3 Frame Shear and Moment Diagram . . . . . . . . . . . . . . . . . . . . . . 120 E 6-4 Frame Shear and Moment Diagram; Hydrostatic Load . . . . . . . . . . . 123 E 6-5 Shear Moment Diagrams for Frame . . . . . . . . . . . . . . . . . . . . . . 125 E 6-6 Shear Moment Diagrams for Inclined Frame . . . . . . . . . . . . . . . . . 127 6.4.3 3D Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 E 6-7 3D Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 6.5 Arches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 7 ARCHES and CURVED STRUCTURES 131 7.1 Arches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 7.1.1 Statically Determinate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 E 7-1 Three Hinged Arch, Point Loads. (Gerstle 1974) . . . . . . . . . . . . . . 134 E 7-2 Semi-Circular Arch, (Gerstle 1974) . . . . . . . . . . . . . . . . . . . . . . 135 7.1.2 Statically Indeterminate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 E 7-3 Statically Indeterminate Arch, (Kinney 1957) . . . . . . . . . . . . . . . . 137 7.2 Curved Space Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 E 7-4 Semi-Circular Box Girder, (Gerstle 1974) . . . . . . . . . . . . . . . . . . 140 7.2.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 7.2.1.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 7.2.1.2 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 E 7-5 Internal Forces in an Helicoidal Cantilevered Girder, (Gerstle 1974) . . . 144 8 DEFLECTION of STRUCTRES; Geometric Methods 149 8.1 Flexural Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 8.1.1 Curvature Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 8.1.2 Di erential Equation of the Elastic Curve . . . . . . . . . . . . . . . . . . 151 8.1.3 Moment Temperature Curvature Relation . . . . . . . . . . . . . . . . . . 152 8.2 Flexural Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 8.2.1 Direct Integration Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 E 8-1 Double Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 8.2.2 Curvature Area Method (Moment Area) . . . . . . . . . . . . . . . . . . . 154 8.2.2.1 First Moment Area Theorem . . . . . . . . . . . . . . . . . . . . 154 8.2.2.2 Second Moment Area Theorem . . . . . . . . . . . . . . . . . . . 154 E 8-2 Moment Area, Cantilevered Beam . . . . . . . . . . . . . . . . . . . . . . 157 E 8-3 Moment Area, Simply Supported Beam . . . . . . . . . . . . . . . . . . . 157 8.2.2.3 Maximum De ection . . . . . . . . . . . . . . . . . . . . . . . . 159 E 8-4 Maximum De ection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 E 8-5 Frame De ection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 E 8-6 Frame Subjected to Temperature Loading . . . . . . . . . . . . . . . . . . 162 Victor Saouma Structural EngineeringCONTENTS 9 Draft 11 APPROXIMATE FRAME ANALYSIS 227 11.1 Vertical Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 11.2 Horizontal Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 11.2.1 Portal Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 E 11-1 Approximate Analysis of a Frame subjected to Vertical and Horizontal Loads236 12 KINEMATIC INDETERMINANCY; STIFFNESS METHOD 253 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 12.1.1 Sti ness vs Flexibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 12.1.2 Sign Convention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 12.2 Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 12.2.1 Methods of Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 12.3 Kinematic Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 12.3.1 Force-Displacement Relations . . . . . . . . . . . . . . . . . . . . . . . . . 257 12.3.2 Fixed End Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 12.3.2.1 Uniformly Distributed Loads . . . . . . . . . . . . . . . . . . . . 260 12.3.2.2 Concentrated Loads . . . . . . . . . . . . . . . . . . . . . . . . . 260 12.4 Slope De ection; Direct Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 12.4.1 Slope De ection Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 261 12.4.2 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 12.4.3 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 12.4.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 E 12-1 Propped Cantilever Beam, (Arbabi 1991) . . . . . . . . . . . . . . . . . . 263 E 12-2 Two-Span Beam, Slope De ection, (Arbabi 1991) . . . . . . . . . . . . . . 264 E 12-3 Two-Span Beam, Slope De ection, Initial De ection, (Arbabi 1991) . . . 265 E 12-4 dagger Frames, Slope De ection, (Arbabi 1991) . . . . . . . . . . . . . . . 267 12.5 Moment Distribution; Indirect Solution . . . . . . . . . . . . . . . . . . . . . . . 269 12.5.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 12.5.1.1 Sign Convention . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 12.5.1.2 Fixed-End Moments . . . . . . . . . . . . . . . . . . . . . . . . . 269 12.5.1.3 Sti ness Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 12.5.1.4 Distribution Factor (DF) . . . . . . . . . . . . . . . . . . . . . . 270 12.5.1.5 Carry-Over Factor . . . . . . . . . . . . . . . . . . . . . . . . . . 271 12.5.2 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 12.5.3 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 12.5.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 E 12-5 Continuous Beam, (Kinney 1957) . . . . . . . . . . . . . . . . . . . . . . . 272 E 12-6 Continuous Beam, Simpli ed Method, (Kinney 1957) . . . . . . . . . . . . 275 E 12-7 Continuous Beam, Initial Settlement, (Kinney 1957) . . . . . . . . . . . . 277 E 12-8 Frame, (Kinney 1957) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 E 12-9 Frame with Side Load, (Kinney 1957) . . . . . . . . . . . . . . . . . . . . 283 E 12-10Moment Distribution on a Spread-Sheet . . . . . . . . . . . . . . . . . . . 285 13 DIRECT STIFFNESS METHOD 287 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 13.1.1 Structural Idealization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 13.1.2 Structural Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 13.1.3 Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 Victor Saouma Structural EngineeringCONTENTS 11 Draft II DESGIN 347 14 DESIGN PHILOSOPHIES of ACI and AISC CODES 349 14.1 Safety Provisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 14.2 Working Stress Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 14.3 Ultimate Strength Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 14.3.1 The Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 14.3.2 Reliability Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 14.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 14.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 E 14-1 LRFD vs ASD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 15 LOADS 359 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 15.2 Vertical Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 15.2.1 Dead Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 15.2.2 Live Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 E 15-1 Live Load Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 15.2.3 Snow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 15.3 Lateral Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 15.3.1 Wind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 E 15-2 Wind Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 15.3.2 Earthquakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 E 15-3 Earthquake Load on a Frame . . . . . . . . . . . . . . . . . . . . . . . . . 374 E 15-4 Earthquake Load on a Tall Building, (Schueller 1996) . . . . . . . . . . . 375 15.4 Other Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 15.4.1 Hydrostatic and Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 E 15-5 Hydrostatic Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 15.4.2 Thermal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 E 15-6 Thermal Expansion/Stress (Schueller 1996) . . . . . . . . . . . . . . . . . 378 15.4.3 Bridge Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 15.4.4 Impact Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 15.5 Other Important Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 15.5.1 Load Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 15.5.2 Load Placement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 15.5.3 Structural Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 15.5.4 Tributary Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 16 STRUCTURAL MATERIALS 387 16.1 Steel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 16.1.1 Structural Steel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 16.1.2 Reinforcing Steel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 16.2 Aluminum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 16.3 Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 16.4 Masonry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 16.5 Timber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 16.6 Steel Section Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 16.6.1 ASCII File with Steel Section Properties . . . . . . . . . . . . . . . . . . . 404 Victor Saouma Structural EngineeringCONTENTS 13 Draft E 19-6 Design of a Column, (?) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 E 19-7 Column Design Using AISC Charts, (?) . . . . . . . . . . . . . . . . . . . 465 20 BRACED ROLLED STEEL BEAMS 467 20.1 Review from Strength of Materials . . . . . . . . . . . . . . . . . . . . . . . . . . 467 20.1.1 Flexure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467 20.1.2 Shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470 20.2 Nominal Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 20.3 Flexural Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 20.3.1 Failure Modes and Classi cation of Steel Beams . . . . . . . . . . . . . . 471 20.3.1.1 Compact Sections . . . . . . . . . . . . . . . . . . . . . . . . . . 472 20.3.1.2 Partially Compact Section . . . . . . . . . . . . . . . . . . . . . 473 20.3.1.3 Slender Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 20.3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475 E 20-1 Shape Factors, Rectangular Section . . . . . . . . . . . . . . . . . . . . . 475 E 20-2 Shape Factors, T Section . . . . . . . . . . . . . . . . . . . . . . . . . . . 475 E 20-3 Beam Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477 20.4 Shear Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 20.5 De ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480 20.6 Complete Design Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480 21 UNBRACED ROLLED STEEL BEAMS 483 21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 21.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 21.3 AISC Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484 21.3.1 Dividing values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484 21.3.2 Governing Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 21.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486 21.4.1 Veri cation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486 E 21-1 Adequacy of an unbraced beam, (?) . . . . . . . . . . . . . . . . . . . . . 486 E 21-2 Adequacy of an unbraced beam, II (?) . . . . . . . . . . . . . . . . . . . . 488 21.4.2 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 E 21-3 Design of Laterally Unsupported Beam, (?) . . . . . . . . . . . . . . . . . 490 21.5 Summary of AISC Governing Equations . . . . . . . . . . . . . . . . . . . . . . . 495 22 Beam Columns, (Unedited) 497 22.1 Potential Modes of Failures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 22.2 AISC Speci cations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 22.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 22.3.1 Veri cation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 E 22-1 Veri cation, (?) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 E 22-2 8.2, (?) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500 22.3.2 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503 E 22-3 Design of Steel Beam-Column, (?) . . . . . . . . . . . . . . . . . . . . . . 505 Victor Saouma Structural EngineeringCONTENTS 15 Draft 26.1.4 Tendon Con guration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546 26.1.5 Equivalent Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546 26.1.6 Load Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548 26.2 Flexural Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548 E 26-1 Prestressed Concrete I Beam . . . . . . . . . . . . . . . . . . . . . . . . . 550 26.3 Case Study: Walnut Lane Bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . 552 26.3.1 Cross-Section Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552 26.3.2 Prestressing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554 26.3.3 Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555 26.3.4 Flexural Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555 27 COLUMNS 557 27.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557 27.1.1 Types of Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557 27.1.2 Possible Arrangement of Bars . . . . . . . . . . . . . . . . . . . . . . . . . 558 27.2 Short Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558 27.2.1 Concentric Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558 27.2.2 Eccentric Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559 27.2.2.1 Balanced Condition . . . . . . . . . . . . . . . . . . . . . . . . . 559 27.2.2.2 Tension Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562 27.2.2.3 Compression Failure . . . . . . . . . . . . . . . . . . . . . . . . . 563 27.2.3 ACI Provisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563 27.2.4 Interaction Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564 27.2.5 Design Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564 E 27-1 R/C Column, c known . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564 E 27-2 R/C Column, e known . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566 E 27-3 R/C Column, Using Design Charts . . . . . . . . . . . . . . . . . . . . . . 571 28 ELEMENTS of STRUCTURAL RELIABILITY 573 28.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573 28.2 Elements of Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573 28.3 Distributions of Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 575 28.3.1 Uniform Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575 28.3.2 Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575 28.3.3 Lognormal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577 28.3.4 Beta Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577 28.3.5 BiNormal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577 28.4 Reliability Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577 28.4.1 Performance Function Identi cation . . . . . . . . . . . . . . . . . . . . . 577 28.4.2 De nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577 28.4.3 Mean and Standard Deviation of a Performance Function . . . . . . . . . 578 28.4.3.1 Direct Integration . . . . . . . . . . . . . . . . . . . . . . . . . . 579 28.4.3.2 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . 579 28.4.3.3 Point Estimate Method . . . . . . . . . . . . . . . . . . . . . . . 582 28.4.3.4 Taylor's Series-Finite Di erence Estimation . . . . . . . . . . . . 583 28.4.4 Overall System Reliability . . . . . . . . . . . . . . . . . . . . . . . . . . . 584 28.4.5 Target Reliability Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . 584 28.5 Reliability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584 Victor Saouma Structural EngineeringCONTENTS 17 Draft 33.3.2.1 Portal Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639 E 33-1 Approximate Analysis of a Frame subjected to Vertical and Horizontal Loads641 33.4 Lateral De ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651 33.4.1 Short Wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653 33.4.2 Tall Wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654 33.4.3 Walls and Lintel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654 33.4.4 Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655 33.4.5 Trussed Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656 33.4.6 Example of Transverse De ection . . . . . . . . . . . . . . . . . . . . . . . 657 33.4.7 E ect of Bracing Trusses . . . . . . . . . . . . . . . . . . . . . . . . . . . 661 Victor Saouma Structural EngineeringDraft List of Figures 1.1 Hamurrabi's Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.2 Archimed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.3 Pantheon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.4 From Vitruvius Ten Books on Architecture, (Vitruvius 1960) . . . . . . . . . . . 34 1.5 Hagia Sophia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 1.6 Florence's Cathedral Dome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 1.7 Palladio's Villa Rotunda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 1.8 Stevin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 1.9 Galileo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 1.10 Discourses Concerning Two New Sciences, Cover Page . . . . . . . . . . . . . . . 41 1.11 \Galileo's Beam" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 1.12 Experimental Set Up Used by Hooke . . . . . . . . . . . . . . . . . . . . . . . . . 43 1.13 Isaac Newton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 1.14 Philosophiae Naturalis Principia Mathematica, Cover Page . . . . . . . . . . . . 44 1.15 Leonhard Euler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 1.16 Coulomb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 1.17 Nervi's Palazetto Dello Sport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.1 Types of Forces in Structural Elements (1D) . . . . . . . . . . . . . . . . . . . . . 58 2.2 Basic Aspects of Cable Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.3 Basic Aspects of Arches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.4 Types of Trusses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.5 Variations in Post and Beams Con gurations . . . . . . . . . . . . . . . . . . . . 62 2.6 Di erent Beam Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 2.7 Basic Forms of Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 2.8 Examples of Air Supported Structures . . . . . . . . . . . . . . . . . . . . . . . . 65 2.9 Basic Forms of Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 2.10 Sequence of Structural Engineering Courses . . . . . . . . . . . . . . . . . . . . . 67 3.1 Types of Supports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.2 Inclined Roller Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.3 Examples of Static Determinate and Indeterminate Structures . . . . . . . . . . . 72 3.4 Geometric Instability Caused by Concurrent Reactions . . . . . . . . . . . . . . . 74 4.1 Types of Trusses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.2 Bridge Truss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.3 A Statically Indeterminate Truss . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.4 X and Y Components of Truss Forces . . . . . . . . . . . . . . . . . . . . . . . . 85LIST OF FIGURES 21 Draft 9.2 Strain Energy De nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 9.3 De ection of Cantilever Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 9.4 Real and Virtual Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 9.5 Torsion Rotation Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 9.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 9.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 9.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 9.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 9.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 9.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 9.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 9.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 9.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 9.15 (correct 42.7 to 47.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 10.1 Statically Indeterminate 3 Cable Structure . . . . . . . . . . . . . . . . . . . . . . 200 10.2 Propped Cantilever Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 10.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 10.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 10.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 10.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 10.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 10.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 10.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 10.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 10.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 10.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 10.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 10.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 10.15De nition of Flexibility Terms for a Rigid Frame . . . . . . . . . . . . . . . . . . 220 10.16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 10.17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 11.1 Uniformly Loaded Beam and Frame with Free or Fixed Beam Restraint . . . . . 228 11.2 Uniformly Loaded Frame, Approximate Location of In ection Points . . . . . . . 229 11.3 Approximate Analysis of Frames Subjected to Vertical Loads; Girder Moments . 230 11.4 Approximate Analysis of Frames Subjected to Vertical Loads; Column Axial Forces231 11.5 Approximate Analysis of Frames Subjected to Vertical Loads; Column Moments 232 11.6 Horizontal Force Acting on a Frame, Approximate Location of In ection Points . 233 11.7 Approximate Analysis of Frames Subjected to Lateral Loads; Column Shear . . . 235 11.8 Approximate Analysis of Frames Subjected to Lateral Loads; Girder Moment 235 11.9 Approximate Analysis of Frames Subjected to Lateral Loads; Column Axial Force236 11.10Example; Approximate Analysis of a Building . . . . . . . . . . . . . . . . . . . . 237 11.11Free Body Diagram for the Approximate Analysis of a Frame Subjected to Vertical Loads237 11.12Approximate Analysis of a Building; Moments Due to Vertical Loads . . . . . . . 239 11.13Approximate Analysis of a Building; Shears Due to Vertical Loads . . . . . . . . 241 11.14Approximate Analysis for Vertical Loads; Spread-Sheet Format . . . . . . . . . . 242 11.15Approximate Analysis for Vertical Loads; Equations in Spread-Sheet . . . . . . . 243 Victor Saouma Structural EngineeringLIST OF FIGURES 23 Draft 15.13Load Life of a Structure, (Lin and Stotesbury 1981) . . . . . . . . . . . . . . . . 381 15.14Concept of Tributary Areas for Structural Member Loading . . . . . . . . . . . . 382 15.15One or Two Way actions in Slabs . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 15.16Load Transfer in R/C Buildings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 15.17Two Way Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 15.18Example of Load Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 16.1 Stress Strain Curves of Concrete and Steel . . . . . . . . . . . . . . . . . . . . . . 388 16.2 Standard Rolled Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 16.3 Residual Stresses in Rolled Sections . . . . . . . . . . . . . . . . . . . . . . . . . 390 16.4 Residual Stresses in Welded Sections . . . . . . . . . . . . . . . . . . . . . . . . . 390 16.5 In uence of Residual Stress on Average Stress-Strain Curve of a Rolled Section . 391 16.6 Concrete Stress-Strain curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 16.7 Concrete microcracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 16.8 W and C sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 16.9 prefabricated Steel Joists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 17.1 Stress Concentration Around Circular Hole . . . . . . . . . . . . . . . . . . . . . 410 17.2 Hole Sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 17.3 E ect of Staggered Holes on Net Area . . . . . . . . . . . . . . . . . . . . . . . . 411 17.4 Gage Distances for an Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 17.5 Net and Gross Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 17.6 Tearing Failure Limit State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 18.1 Stability of a Rigid Bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 18.2 Stability of a Rigid Bar with Initial Imperfection . . . . . . . . . . . . . . . . . . 439 18.3 Stability of a Two Rigid Bars System . . . . . . . . . . . . . . . . . . . . . . . . 439 18.4 Two DOF Dynamic System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 18.5 Euler Column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 18.6 Simply Supported Beam Column; Di erential Segment; E ect of Axial Force P . 444 18.7 Solution of the Tanscendental Equation for the Buckling Load of a Fixed-Hinged Column448 18.8 Column E ective Lengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 18.9 Frame E ective Lengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 18.10Column E ective Length in a Frame . . . . . . . . . . . . . . . . . . . . . . . . . 450 18.11Standard Alignment Chart (AISC) . . . . . . . . . . . . . . . . . . . . . . . . . . 451 18.12Inelastic Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 18.13Euler Buckling, and SSRC Column Curve . . . . . . . . . . . . . . . . . . . . . . 453 19.1 SSRC Column Curve and AISC Critical Stresses . . . . . . . . . . . . . . . . . . 456 19.2 F versus KL=r According to LRFD, for Various F . . . . . . . . . . . . . . . . 459 cr y 20.1 Bending of a Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468 20.2 Stress distribution at di erent stages of loading . . . . . . . . . . . . . . . . . . . 469 20.3 Stress-strain diagram for most structural steels . . . . . . . . . . . . . . . . . . . 469 20.4 Flexural and Shear Stress Distribution in a Rectangular Beam . . . . . . . . . . 470 20.5 Local ( ange) Buckling; Flexural and Torsional Buckling Modes in a Rolled Section, (Lulea University)472 20.6 W Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 20.7 Nominal Moments for Compact and Partially Compact Sections . . . . . . . . . . 474 20.8 AISC Requirements for Shear Design . . . . . . . . . . . . . . . . . . . . . . . . . 479 Victor Saouma Structural EngineeringLIST OF FIGURES 25 Draft 29.3 Deformation, Shear, Moment, and Axial Diagrams for Various Types of Portal Frames Subjected to Vertical and Horizontal Loads589 29.4 Axial and Flexural Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590 29.5 Design of a Statically Indeterminate Arch . . . . . . . . . . . . . . . . . . . . . . 591 29.6 Normal and Shear Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594 30.1 Ei el Tower (Billington and Mark 1983) . . . . . . . . . . . . . . . . . . . . . . . 603 30.2 Ei el Tower Idealization, (Billington and Mark 1983) . . . . . . . . . . . . . . . . 605 30.3 Ei el Tower, Dead Load Idealization; (Billington and Mark 1983) . . . . . . . . . 605 30.4 Ei el Tower, Wind Load Idealization; (Billington and Mark 1983) . . . . . . . . . 606 30.5 Ei el Tower, Wind Loads, (Billington and Mark 1983) . . . . . . . . . . . . . . . 607 30.6 Ei el Tower, Reactions; (Billington and Mark 1983) . . . . . . . . . . . . . . . . 607 30.7 Ei el Tower, Internal Gravity Forces; (Billington and Mark 1983) . . . . . . . . . 609 30.8 Ei el Tower, Horizontal Reactions; (Billington and Mark 1983) . . . . . . . . . . 609 30.9 Ei el Tower, Internal Wind Forces; (Billington and Mark 1983) . . . . . . . . . . 610 31.1 Cable Structure Subjected to p(x) . . . . . . . . . . . . . . . . . . . . . . . . . . 612 31.2 Longitudinal and Plan Elevation of the George Washington Bridge . . . . . . . . 614 31.3 Truck Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615 31.4 Dead and Live Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616 31.5 Location of Cable Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617 31.6 Vertical Reactions in Columns Due to Central Span Load . . . . . . . . . . . . . 617 31.7 Cable Reactions in Side Span . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618 31.8 Cable Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619 31.9 Deck Idealization, Shear and Moment Diagrams . . . . . . . . . . . . . . . . . . . 620 32.1 Magazzini Generali; Overall Dimensions, (Billington and Mark 1983) . . . . . . . 622 32.2 Magazzini Generali; Support System, (Billington and Mark 1983) . . . . . . . . . 622 32.3 Magazzini Generali; Loads (Billington and Mark 1983) . . . . . . . . . . . . . . . 623 32.4 Magazzini Generali; Beam Reactions, (Billington and Mark 1983) . . . . . . . . . 623 32.5 Magazzini Generali; Shear and Moment Diagrams (Billington and Mark 1983) . . 624 32.6 Magazzini Generali; Internal Moment, (Billington and Mark 1983) . . . . . . . . 624 32.7 Magazzini Generali; Similarities Between The Frame Shape and its Moment Diagram, (Billington and Mark 1983)625 32.8 Magazzini Generali; Equilibrium of Forces at the Beam Support, (Billington and Mark 1983)625 32.9 Magazzini Generali; E ect of Lateral Supports, (Billington and Mark 1983) . . . 626 33.1 Flexible, Rigid, and Semi-Flexible Joints . . . . . . . . . . . . . . . . . . . . . . . 627 33.2 Deformation of Flexible and Rigid Frames Subjected to Vertical and Horizontal Loads, (Lin and Stotesbury 1981)628 33.3 Deformation, Shear, Moment, and Axial Diagrams for Various Types of Portal Frames Subjected to Vertical and Horizontal Loads629 33.4 Axial and Flexural Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 630 33.5 Design of a Shear Wall Subsystem, (Lin and Stotesbury 1981) . . . . . . . . . . . 632 33.6 Trussed Shear Wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634 33.7 Design Example of a Tubular Structure, (Lin and Stotesbury 1981) . . . . . . . . 635 33.8 A Basic Portal Frame, (Lin and Stotesbury 1981) . . . . . . . . . . . . . . . . . . 636 33.9 Approximate Analysis of Frames Subjected to Vertical Loads; Girder Moments . 637 33.10Approximate Analysis of Frames Subjected to Vertical Loads; Column Axial Forces638 33.11Approximate Analysis of Frames Subjected to Vertical Loads; Column Moments 638 33.12Approximate Analysis of Frames Subjected to Lateral Loads; Column Shear . . . 640 33.13Approximate Analysis of Frames Subjected to Lateral Loads; Girder Moment 640 33.14Approximate Analysis of Frames Subjected to Lateral Loads; Column Axial Force641 Victor Saouma Structural EngineeringDraft List of Tables 3.1 Equations of Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.1 Static Determinacy and Stability of Trusses . . . . . . . . . . . . . . . . . . . . . 83 8.1 Conjugate Beam Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . 165 9.1 Possible Combinations of Real and Hypothetical Formulations . . . . . . . . . . . 175 9.2 k Factors for Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 9.3 Summary of Expressions for the Internal Strain Energy and External Work . . . 198 Z L 10.1 Table of g (x)g (x)dx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 1 2 0 10.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 10.3 Displacement Computations for a Rectangular Frame . . . . . . . . . . . . . . . 219 11.1 Columns Combined Approximate Vertical and Horizontal Loads . . . . . . . . . 250 11.2 Girders Combined Approximate Vertical and Horizontal Loads . . . . . . . . . . 251 12.1 Sti ness vs Flexibility Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 12.2 Degrees of Freedom of Di erent Structure Types Systems . . . . . . . . . . . . . 255 13.1 Example of Nodal De nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 13.2 Example of Element De nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 13.3 Example of Group Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 13.4 Degrees of Freedom of Di erent Structure Types Systems . . . . . . . . . . . . . 293 14.1 Allowable Stresses for Steel and Concrete . . . . . . . . . . . . . . . . . . . . . . 351 14.2 Selected values for Steel and Concrete Structures . . . . . . . . . . . . . . . . . 355 14.3 Strength Reduction Factors,  . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 15.1 Unit Weight of Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 15.2 Weights of Building Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 15.3 Average Gross Dead Load in Buildings . . . . . . . . . . . . . . . . . . . . . . . . 361 15.4 Minimum Uniformly Distributed Live Loads, (UBC 1995) . . . . . . . . . . . . . 362 15.5 Wind Velocity Variation above Ground . . . . . . . . . . . . . . . . . . . . . . . . 366 15.6 C Coecients for Wind Load, (UBC 1995) . . . . . . . . . . . . . . . . . . . . . 367 e 15.7 Wind Pressure Coecients C , (UBC 1995) . . . . . . . . . . . . . . . . . . . . . 367 q 15.8 Importance Factors for Wind and Earthquake Load, (UBC 1995) . . . . . . . . . 368 15.9 Approximate Design Wind Pressure p for Ordinary Wind Force Resisting Building Structures368 15.10Z Factors for Di erent Seismic Zones, ubc . . . . . . . . . . . . . . . . . . . . . . 372Draft Part I ANALYSISDraft Chapter 1 A BRIEF HISTORY OF STRUCTURAL ARCHITECTURE If I have been able to see a little farther than some others, it was because I stood on the shoulders of giants. Sir Isaac Newton 1 More than any other engineering discipline, Architecture/Mechanics/Structures is the proud outcome of a of a long and distinguished history. Our profession, second oldest, would be better appreciated if we were to develop a sense of our evolution. 1.1 Before the Greeks 2 Throughout antiquity, structural engineering existing as an art rather than a science. No record exists of any rational consideration, either as to the strength of structural members or as to the behavior of structural materials. The builders were guided by rules of thumbs and experience, which were passed from generation to generation, guarded by secrets of the guild, and seldom supplemented by new knowledge. Despite this, structures erected before Galileo are by modern standards quite phenomenal (pyramids, Via Appia, aqueducs, Colisseums, Gothic cathedrals to name a few). 3 The rst structural engineer in history seems to have been Imhotep, one of only two com- moners to be dei ed. He was the builder of the step pyramid of Sakkara about 3,000 B.C., and yielded great in uence over ancient Egypt. 4 Hamurrabi's code in Babylonia (1750 BC) included among its 282 laws penalties for those \architects" whose houses collapsed, Fig. 1.1. 1.2 Greeks 5 The greek philosopher Pythagoras (born around 582 B.C.) founded his famous school, which was primarily a secret religious society, at Crotona in southern Italy. At his school he allowed1.3 Romans 33 Draft Figure 1.2: Archimed conqueror of Syracuse. 1.3 Romans 10 Science made much less progress under the Romans than under the Greeks. The Romans apparently were more practical, and were not as interested in abstract thinking though they were excellent ghters and builders. 11 As the roman empire expanded, the Romans built great roads (some of them still in use) such as the Via Appia, Cassia, Aurelia; Also they built great bridges (such as the third of a mile bridge over the Rhine built by Caesars), and stadium (Colliseum). 12 One of the most notable Roman construction was the Pantheon, Fig. 1.3. It is the best- Figure 1.3: Pantheon preserved major edi ce of ancient Rome and one of the most signi cant buildings in architectural history. In shape it is an immense cylinder concealing eight piers, topped with a dome and fronted by a rectangular colonnaded porch. The great vaulted dome is 43 m (142 ft) in diameter, and the entire structure is lighted through one aperture, called an oculus, in the center of the dome. The Pantheon was erected by the Roman emperor Hadrian between AD 118 and 128. Victor Saouma Structural EngineeringDraft Chapter 2 INTRODUCTION 2.1 Structural Engineering 1 Structural engineers are responsible for the detailed analysis and design of: Architectural structures: Buildings, houses, factories. They must work in close cooperation with an architect who will ultimately be responsible for the design. Civil Infrastructures: Bridges, dams, pipelines, o shore structures. They work with trans- portation, hydraulic, nuclear and other engineers. For those structures they play the leading role. Aerospace, Mechanical, Naval structures: aeroplanes, spacecrafts, cars, ships, submarines to ensure the structural safety of those important structures. 2.2 Structures and their Surroundings 2 Structural design is a ected by various environmental constraints: 1. Major movements: For example, elevator shafts are usually shear walls good at resisting lateral load (wind, earthquake). 2. Sound and structure interact:  A dome roof will concentrate the sound  A dish roof will di use the sound 3. Natural light:  A at roof in a building may not provide adequate light.  A Folded plate will provide adequate lighting (analysis more complex).  A bearing and shear wall building may not have enough openings for daylight.  A Frame design will allow more light in (analysis more complex). 4. Conduits for cables (electric, telephone, computer), HVAC ducts, may dictate type of oor system. 5. Net clearance between columns (unobstructed surface) will dictate type of framing.2.6 Structural Analysis 57 Draft 2.6 Structural Analysis 12 Given an existing structure subjected to a certain load determine internal forces (axial, shear, exural, torsional; or stresses), de ections, and verify that no unstable failure can occur. 13 Thus the basic structural requirements are: Strength: stresses should not exceed critical values:  f Sti ness: de ections should be controlled:   max Stability: buckling or cracking should also be prevented 2.7 Structural Design 14 Given a set of forces, dimension the structural element. Steel/wood Structures Select appropriate section. Reinforced Concrete: Determine dimensions of the element and internal reinforcement (num- ber and sizes of reinforcing bars). 15 For new structures, iterative process between analysis and design. A preliminary design is made using rules of thumbs (best known to Engineers with design experience) and analyzed. Following design, we check for Serviceability: de ections, crack widths under the applied load. Compare with acceptable values speci ed in the design code. Failure: and compare the failure load with the applied load times the appropriate factors of safety. If the design is found not to be acceptable, then it must be modi ed and reanalyzed. 16 For existing structures rehabilitation, or veri cation of an old infrastructure, analysis is the most important component. 17 In summary, analysis is always required. 2.8 Load Transfer Elements 18 From Strength of Materials, Fig. 2.1 Axial: cables, truss elements, arches, membrane, shells Flexural: Beams, frames, grids, plates Torsional: Grids, 3D frames Shear: Frames, grids, shear walls. Victor Saouma Structural EngineeringDraft Chapter 3 EQUILIBRIUM & REACTIONS To every action there is an equal and opposite reaction. Newton's third law of motion 3.1 Introduction 1 In the analysis of structures (hand calculations), it is often easier (but not always necessary) to start by determining the reactions. 2 Once the reactions are determined, internal forces are determined next; nally, deformations 1 (de ections and rotations) are determined last . 3 Reactions are necessary to determine foundation load. 4 Depending on the type of structures, there can be di erent types of support conditions, Fig. 3.1. Roller: provides a restraint in only one direction in a 2D structure, in 3D structures a roller may provide restraint in one or two directions. A roller will allow rotation. Hinge: allows rotation but no displacements. Fixed Support: will prevent rotation and displacements in all directions. 3.2 Equilibrium 5 Reactions are determined from the appropriate equations of static equilibrium. 2 6 Summation of forces and moments, in a static system must be equal to zero . 1 This is the sequence of operations in the exibility method which lends itself to hand calculation. In the sti ness method, we determine displacements rsts, then internal forces and reactions. This method is most suitable to computer implementation. 2 In a dynamic system F = ma where m is the mass and a is the acceleration.

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