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“Frontmatter” Structural Engineering Handbook Ed. Chen Wai-Fah Boca Raton: CRC Press LLC, 1999 Structural Engineering Contents 1 Basic Theory of Plates and Elastic Stability Eiki Yamaguchi 2 Structural Analysis J.Y. Richard Liew, N.E. Shanmugam, and C.H. Yu 3 Structural Steel Design 1 E. M. Lui 4 Structural Concrete Design 2 Amy Grider and Julio A. Ramirez and Young Mook Yun 5 Earthquake Engineering Charles Scawthorn 6 Composite Construction Edoardo Cosenza and Riccardo Zandonini 7 Cold-Formed Steel Structures Wei-Wen Yu 8 Aluminum Structures Maurice L. Sharp 9 Timber Structures Kenneth J. Fridley 10 Bridge Structures Shouji Toma, Lian Duan, and Wai-Fah Chen 11 Shell Structures Clarence D. Miller 12 Multistory Frame Structures J. Y. Richard Liew and T. B alendra and W. F. Chen 13 Space Frame Structures Tien T. Lan 14 Cooling Tower Structures Phillip L. Gould and Wilfried B. Krätzig 15TransmissionStructuresShu-jinFang,SubirRoy,andJacobKramer c 1999byCRCPressLLC 16 Performance-Based Seismic Design Criteria For Bridges LianDuanandMarkReno 15B Tunnel Structures Birger Schmidt, Christian Ingerslev, Brian Brenner, and J N. Wang 17 Effective Length Factors of Compression Members Lian Duan and W.F. Chen 18 Stub Girder Floor Systems Reidar Bjorhovde 19 Plate and Box Girders MohamedElgaaly 20 Steel Bridge Construction Jackson Durkee 21 Basic Principles of Shock Loading O.W. Blodgett and D.K. Miller 22 Welded Connections O.W. Blodgett and D. K. Miller 23 Composite Connections Roberto Leon 24 Fatigue and Fracture Robert J. Dexter and John W. Fisher 25 Underground Pipe J. M. Doyle and S.J. Fang 26 Structural Reliability 3 D. V. Rosowsky 27 Passive Energy D issipation and Active Control T.T. Soong and G.F. Dargush 28 An Innnovative Design For Steel Frame Using Advanced Analysis 4 Seung-Eock Kim and W. F. Chen 29 Welded Tubular Connections—CHS Trusses Peter W. Marshall c 1999 by CRC Press LLC 30 Earthquake Damage to Structures Mark Yashinsky Yamaguchi, E. “Basic Theory of Plates and Elastic Stability” Structural Engineering Handbook Ed. Chen Wai-Fah Boca Raton: CRC Press LLC, 1999 BasicTheoryofPlatesandElastic Stability EikiYamaguchi DepartmentofCivilEngineering, KyushuInstituteofTechnology, Kitakyusha,Japan 1.1 Introduction 1.2 Plates BasicAssumptions • GoverningEquations • BoundaryCon- ditions • CircularPlate • ExamplesofBendingProblems 1.3 Stability BasicConcepts • StructuralInstability • Columns • Thin- WalledMembers • Plates 1.4 DefiningTerms References FurtherReading 1.1 Introduction Thischapterisconcernedwithbasicassumptionsandequationsofplatesandbasicconceptsofelastic stability.Herein,weshallillustratetheconceptsandtheapplicationsoftheseequationsbymeansof relativelysimpleexamples;morecomplexapplicationswillbetakenupinthefollowingchapters. 1.2 Plates 1.2.1 BasicAssumptions WeconsideracontinuumshowninFigure1.1.Afeatureofthebodyisthatonedimensionismuch smallerthantheothertwodimensions: t<<L x ,L y (1.1) wheret,L x ,andL y arerepresentativedimensionsinthreedirections(Figure1.1).Ifthecontinuum hasthisgeometricalcharacteristicofEquation1.1andisflatbeforeloading,itiscalledaplate.Note thatashellpossessesasimilargeometricalcharacteristicbutiscurvedevenbeforeloading. ThecharacteristicofEquation1.1lendsitselftothefollowingassumptionsregardingsomestress andstraincomponents: σ z = 0 (1.2) ε z = ε xz =ε yz =0 (1.3) WecanderivethefollowingdisplacementfieldfromEquation1.3: c 1999byCRCPressLLC FIGURE 1.1: Plate. u(x,y,z) = u 0 (x, y) − z ∂w 0 ∂x ν(x,y, z) = ν 0 (x, y) − z ∂w 0 ∂y (1.4) w(x,y,z) = w 0 (x, y) whereu, ν, andw aredisplacementcomponentsinthedirections of x-, y-, and z-axes,respectively. AscanberealizedinEquation1.4,u 0 andν 0 aredisplacementcomponentsassociatedwith the plane ofz = 0. Physically,Equation1.4impliesthatthelinearfilamentsoftheplateinitiallyperpendicular to the middle surface remain straight and perpendicular to the deformed middle surface. This is knownastheKirchhoffhypothesis. AlthoughwehavederivedEquation1.4fromEquation1.3inthe above,onecanarriveatEquation1.4startingwiththeKirchhoffhypothesis: theKirchhoffhypothesis is equivalent to the assumptions of Equation 1.3. 1.2.2 Governing Equations Strain-Displacement Relationships Using the strain-displacement relationships in the continuum mechanics, we can obtain the following strain field associated with Equation 1.4: ε x = ∂u 0 ∂x − z ∂ 2 w 0 ∂x 2 ε y = ∂ν 0 ∂y − z ∂ 2 w 0 ∂y 2 (1.5) ε xy = 1 2 ∂u 0 ∂y + ∂ν 0 ∂x − z ∂ 2 w 0 ∂x∂y This constitutes the strain-displacement relationships for the plate theory. Equilibrium Equations In the plate theory, equilibrium conditions are considered in terms of resultant forces and moments. This is derived by integrating the equilibrium equations over the thickness of a plate. Because of Equation 1.2, we obtain the equilibrium equations as follows: c 1999 by CRC Press LLC ∂N x ∂x + ∂N xy ∂y + q x = 0 (1.6a) ∂N xy ∂x + ∂N y ∂y + q y = 0 (1.6b) ∂V x ∂x + ∂V y ∂y + q z = 0 (1.6c) where N x ,N y , and N xy are in-plane stress resultants; V x and V y are shearing forces; and q x ,q y , and q z are distributed loads per unit area. The terms associated with τ xz and τ yz vanish, since in the plate problems the top and the bottom surfaces of a plate are subjected to only vertical loads. We must also consider the moment equilibrium of an infinitely small region of the plate, which leads to ∂M x ∂x + ∂M xy ∂y − V x = 0 ∂M xy ∂x + ∂M y ∂y − V y = 0 (1.7) where M x and M y are bending moments and M xy is a twisting moment. The resultant forces andthe moments are defined mathematically as N x = z σ x dz (1.8a) N y = z σ y dz (1.8b) N xy = N yx = z τ xy dz (1.8c) V x = z τ xz dz (1.8d) V y = z τ yz dz (1.8e) M x = z σ x zdz (1.8f) M y = z σ y zdz (1.8g) M xy = M yx = z τ xy zdz (1.8h) The resultant forces andthe moments are illustrated in Figure 1.2. Constitutive Equations Since the thickness of a plate is small in comparison with the other dimensions, it is usually acceptedthattheconstitutiverelationsforastateofplanestressareapplicable. Hence,thestress-strain relationships foran isotropic plate are given by σ x σ y τ xy = E 1 − ν 2 1 ν 0 ν 10 00(1 − ν)/2 ε x ε y γ xy (1.9) c 1999 by CRC Press LLC FIGURE 1.2: Resultant forces and moments. whereE andν areYoung’smodulusandPoisson’sratio,respectively. UsingEquations1.5,1.8,and1.9, theconstitutiverelationshipsforanisotropicplateintermsofstressresultantsanddisplacementsare described by N x = Et 1 − ν 2 ∂u 0 ∂x + ν ∂ν 0 ∂y (1.10a) N y = Et 1 − ν 2 ∂ν 0 ∂y + ν ∂u 0 ∂x (1.10b) N xy = N yx Et 2(1 + ν) ∂ν 0 ∂x + ∂u 0 ∂y (1.10c) M x =−D ∂ 2 w 0 ∂x 2 + ν ∂ 2 w 0 ∂y 2 (1.10d) M y =−D ∂ 2 w 0 ∂y 2 + ν ∂ 2 w 0 ∂x 2 (1.10e) M xy = M yx =−(1 − ν)D ∂ 2 w 0 ∂x∂y (1.10f) where t is the thickness of a plate and D is the flexural rigidit y defined by D = Et 3 12(1 − ν 2 ) (1.11) InthederivationofEquation1.10,wehaveassumedthattheplatethicknesst is constantandthatthe initial middle surface lies in the plane of Z = 0. Through Equation 1.7, we can relate the shearing forces to the displacement. Equations 1.6, 1.7, and 1.10 constitute the framework of a plate problem: 11 equations for 11 unknowns, i.e., N x ,N y ,N xy ,M x ,M y ,M xy ,V x ,V y ,u 0 ,ν 0 , and w 0 . In the subsequent sections, we shall drop thesubscript 0 thathas been associated with the displacements for the sake of brevity. In-Plane and Out-Of-Plane Problems As may be realized in the equations derived in the previous section, the problem can be de- composed into two sets of problems which are uncoupled with each other. 1. In-plane problems The problem may be also called a stretching problem of a plate andis governed by c 1999 by CRC Press LLC ∂N x ∂x + ∂N xy ∂y +q x =0 ∂N xy ∂x + ∂N y ∂y +q y =0 (1.6a,b) N x = Et 1−ν 2 ∂u ∂x +ν ∂ν ∂y N y = Et 1−ν 2 ∂ν ∂y +ν ∂u ∂x N xy = N yx = Et 2(1+ν) ∂ν ∂x + ∂u ∂y (1.10a∼c) Herewehavefiveequationsforfiveunknowns.Thisproblemcanbeviewedandtreated inthesamewayasforaplane-stressprobleminthetheoryoftwo-dimensionalelasticity. 2.Out-of-planeproblems Thisproblemisregardedasabendingproblemandisgovernedby ∂V x ∂x + ∂V y ∂y +q z =0 (1.6c) ∂M x ∂x + ∂M xy ∂y −V x =0 ∂M xy ∂x + ∂M y ∂y −V y =0 (1.7) M x =−D ∂ 2 w ∂x 2 + ∂ 2 w ∂y 2 M y =−D ∂ 2 w ∂y 2 + ∂ 2 w ∂x 2 M xy = M yx =−(1−ν)D ∂ 2 w ∂x∂y (1.10d∼f) Herearesixequationsforsixunknowns. EliminatingV x andV y fromEquations1.6cand1.7,weobtain ∂ 2 M x ∂x 2 +2 ∂ 2 M xy ∂x∂y + ∂ 2 M y ∂y 2 +q z =0 (1.12) SubstitutingEquations1.10d∼fintotheabove,weobtainthegoverningequationintermsofdis- placementas D ∂ 4 w ∂x 4 +2 ∂ 4 w ∂x 2 ∂y 2 + ∂ 4 w ∂y 4 =q z (1.13) c 1999byCRCPressLLC or ∇ 4 w = q z D (1.14) where the operator is defined as ∇ 4 =∇ 2 ∇ 2 ∇ 2 = ∂ 2 ∂x 2 + ∂ 2 ∂y 2 (1.15) 1.2.3 Boundary Conditions Since the in-plane problem of a plate can be treated as a plane-stress problem in the theory of two-dimensional elasticity, the present section is focused solely on a bending problem. Introducing the n-s-z coordinate system alongside boundaries as shown in Figure 1.3, we define the moments and the shearing force as M n = z σ n zdz M ns = M sn = z τ ns zdz (1.16) V n = z τ nz dz In the plate theor y, instead of considering these three quantities, we combine the twisting moment and theshearing force by replacing the action of the twisting momentM ns with that of the shearing force, as can be seen in Figure 1.4. We then define the joint ver tical as S n = V n + ∂M ns ∂s (1.17) The boundary conditions are therefore given in general by w = w or S n = S n (1.18) − ∂w ∂n = λ n or M n = M n (1.19) where the quantitieswith a barare prescribed valuesand are illustrated in Figure 1.5. These two sets of boundary conditions ensure the unique solution of a bending problem of a plate. FIGURE 1.3: n-s-z coordinate system. The boundary conditions for some practical cases are as follows: c 1999 by CRC Press LLC [...]... Structural Analysis” Structural Engineering Handbook Ed Chen Wai-Fah Boca Raton: CRC Press LLC, 1999 Structural Analysis J.Y Richard Liew, N.E Shanmugam, and C.H Yu Department of Civil Engineering The National University of Singapore, Singapore 2.1 2.1 Fundamental Principles 2.2 Flexural Members 2.3 Trusses 2.4 Frames 2.5 Plates 2.6 Shell 2.7 Influence Lines 2.8 Energy Methods in Structural Analysis... E.M 1987 Structural Stability Theory and Implementation, Elsevier, New York [2] Chen, W.F and Lui, E.M 1991 Stability Design of Steel Frames, CRC Press, Boca Raton, FL [3] Galambos, T.V 1988 Guide to Stability Design Criteria for Metal Structures, 4th ed., Structural Stability Research Council, John Wiley & Sons, New York c 1999 by CRC Press LLC Richard Liew, J.Y.; Shanmugam, N.W and Yu, C.H Structural. .. Analysis 2.9 Matrix Methods 2.10 The Finite Element Method 2.11 Inelastic Analysis 2.12 Frame Stability 2.13 Structural Dynamic 2.14 Defining Terms References Further Reading Fundamental Principles Structural analysis is the determination of forces and deformations of the structure due to applied loads Structural design involves the arrangement and proportioning of structures and their components in such... as a good approximation Since the aspect ratio of a component of a steel structural member such as a web plate is large in general, we can often assume k is simply equal to 4.0 1.4 Defining Terms The following is a list of terms as defined in the Guide to Stability Design Criteria for Metal Structures, 4th ed., Galambos, T.V., Structural Stability Research Council, John Wiley & Sons, New York, 1988 Bifurcation:... deformation is negligible and structural behavior is linear before the buckling load is reached The way we have obtained Equation 1.44 in the above is a typical application of the linear buckling analysis In mathematical terms, Equation 1.43 is called a characteristic equation and Equation 1.44 an eigenvalue The linear buckling analysis is in fact regarded as an eigenvalue problem 1.3.2 Structural Instability... as the initial crookedness of a member and the eccentricity of loading, we can rarely observe the bifurcation Instead, an actual structural behavior would be more like the one indicated in Figure 1.11a However, the bifurcation load is still an important measure regarding structural stability and most instabilities of a column and a plate are indeed of this class In many cases we can evaluate the bifurcation... phenomenon is the softening: as Figure 1.11c illustrates, there exists a peak load-carrying capacity, beyond which the structural strength deteriorates We often observe this phenomenon when yielding takes place To compute the associated equilibrium path, we need to resort to nonlinear structural analysis Since nonlinear analysis is complicated and costly, the information on stability limit and ultimate... assembled structure is capable of supporting the designed loads within the allowable limit states An analytical model is an idealization of the actual structure The structural model should relate the actual behavior to material properties, structural details, and loading and boundary conditions as accurately as is practicable All structures that occur in practice are three-dimensional For building structures... arranged in orthogonal directions Joints in a structure are those points where two or more members are connected A truss is a structural system consisting of members that are designed to resist only axial forces Axially loaded members are assumed to be pin-connected at their ends A structural system in which joints are capable of transferring end moments is called a frame Members in this system are assumed... Boundary Conditions A hinge represents a pin connection to a structural assembly and it does not allow translational movements (Figure 2.1a) It is assumed to be frictionless and to allow rotation of a member with FIGURE 2.1: Various boundary conditions respect to the others A roller represents a kind of support that permits the attached structural part to rotate freely with respect to the foundation . “Frontmatter” Structural Engineering Handbook Ed. Chen Wai-Fah Boca Raton: CRC Press LLC, 1999 Structural Engineering Contents 1 Basic Theory. Stability Eiki Yamaguchi 2 Structural Analysis J.Y. Richard Liew, N.E. Shanmugam, and C.H. Yu 3 Structural Steel Design 1 E. M. Lui 4 Structural Concrete Design 2 Amy