Structural Analysis” Structural Engineering Handbook Ed
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Richard Liew, J.Y.; Shanmugam, N.W. and Yu, C.H. “Structural Analysis” Structural Engineering Handbook Ed. Chen Wai-Fah Boca Raton: CRC Press LLC, 1999 StructuralAnalysis 2.1 FundamentalPrinciples 2.2 FlexuralMembers 2.3 Trusses 2.4 Frames 2.5 Plates 2.6 Shell 2.7 InfluenceLines 2.8 EnergyMethodsinStructuralAnalysis 2.9 MatrixMethods 2.10TheFiniteElementMethod 2.11InelasticAnalysis J.Y.RichardLiew, 2.12FrameStability N.E.Shanmugam,and C.H.Yu 2.13StructuralDynamic DepartmentofCivilEngineering 2.14DefiningTerms TheNationalUniversityof References Singapore,Singapore FurtherReading 2.1 FundamentalPrinciples Structuralanalysisisthedeterminationofforcesanddeformationsofthestructureduetoapplied loads.Structuraldesigninvolvesthearrangementandproportioningofstructuresandtheircompo- nentsinsuchawaythattheassembledstructureiscapableofsupportingthedesignedloadswithin theallowablelimitstates.Ananalyticalmodelisanidealizationoftheactualstructure.Thestructural modelshouldrelatetheactualbehaviortomaterialproperties,structuraldetails,andloadingand boundaryconditionsasaccuratelyasispracticable. Allstructuresthatoccurinpracticearethree-dimensional.Forbuildingstructuresthathave regularlayoutandarerectangularinshape,itispossibletoidealizethemintotwo-dimensional framesarrangedinorthogonaldirections.Jointsinastructurearethosepointswheretwoormore membersareconnected.Atrussisastructuralsystemconsistingofmembersthataredesignedto resistonlyaxialforces.Axiallyloadedmembersareassumedtobepin-connectedattheirends.A structuralsysteminwhichjointsarecapableoftransferringendmomentsiscalledaframe.Members inthissystemareassumedtobecapableofresistingbendingmomentaxialforceandshearforce.A structureissaidtobetwodimensionalorplanarifallthememberslieinthesameplane.Beams arethosemembersthataresubjectedtobendingorflexure.Theyareusuallythoughtofasbeing inhorizontalpositionsandloadedwithverticalloads.Tiesaremembersthataresubjectedtoaxial tensiononly,whilestruts(columnsorposts)arememberssubjectedtoaxialcompressiononly. c 1999byCRCPressLLC 2.1.1 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 and to translate freely in the direction parallel to the foundation surface (Figure 2.1b) No translational movement in any other direction is allowed. A fixed support (Figure 2.1c) does not allow rotation or translation in any direction. A rotational spring represents a support that provides some rotational restraint but does not provide any translational restraint (Figure 2.1d). A translational spring can provide partial restraints along the direction of deformation (Figure 2.1e). 2.1.2 Loads and Reactions Loads may be broadly classified as permanent loads that are of constant magnitude and remain in one position and variable loads that may change in position and magnitude. Permanent loads are also referred to as dead loads which may include the self weight of the structure and other loads such as walls, floors, roof, plumbing, and fixtures that are permanently attached to the structure. Variable loads are commonly referred to as live or imposed loads which may include those caused by construction operations, wind, rain, earthquakes, snow, blasts, and temperature changes in addition to those that are movable, such as furniture and warehouse materials. Ponding load is due to water or snow on a flat roof which accumulates faster than it runs off. Wind loads act as pressures on windward surfaces and pressures or suctions on leeward surfaces. Impact loads are caused by suddenly applied loads or by the vibration of moving or movable loads. They are usually taken as a fraction of the live loads. Earthquake loads are those forces caused by the acceleration of the ground surface during an earthquake. A structure that is initially at rest and remains at rest when acted upon by applied loads is said to be in a state of equilibrium. The resultant of the external loads on the body and the supporting forces or reactions is zero. If a structure or part thereof is to be in equilibrium under the action of a system c 1999 by CRC Press LLC of loads, it must satisfy the six static equilibrium equations, such as P P P Fx = 0, Fy = 0, Fz = 0 P P P Mx = 0, My = 0, Mz = 0 (2.1) The summation in these equations is for all the components of the forces (F ) and of the moments (M) about each of the three axes x,y, and z. If a structure is subjected to forces that lie in one plane, say x-y, the above equations are reduced to: X X X Fx = 0, Fy = 0, Mz = 0 (2.2) Consider, for example, a beam shown in Figure 2.2a under the action of the loads shown. The FIGURE 2.2: Beam in equilibrium. reaction at support B must act perpendicular to the surface on which the rollers are constrained to roll upon. The support reactions and the applied loads, which are resolved in vertical and horizontal directions, are shown in Figure 2.2b. √ From geometry, it can be calculated that By = 3Bx. Equation 2.2 can be used to determine the magnitude of the support reactions. Taking moment about B gives 10Ay − 346.4x5 = 0 from which Ay = 173.2 kN. P Equating the sum of vertical forces, Fy to zero gives 173.2 + By − 346.4 = 0 and, hence, we get By = 173.2 kN. Therefore, √ Bx = By/ 3 = 100 kN. c 1999 by CRC Press LLC P Equilibrium in the horizontal direction, Fx = 0 gives, Ax − 200 − 100 = 0 and, hence, Ax = 300 kN. There are three unknown reaction components at a fixed end, two at a hinge, and one at a roller. If, for a particular structure, the total number of unknown reaction components equals the number of equations available, the unknowns may be calculated from the equilibrium equations, and the structure is then said to be statically determinate externally. Should the number of unknowns be greater than the number of equations available, the structure is statically indeterminate externally; if less, it is unstable externally. The ability of a structure to support adequately the loads applied to it is dependent not only on the number of reaction components but also on the arrangement of those components. It is possible for a structure to have as many or more reaction components than there are equations available and yet be unstable. This condition is referred to as geometric instability. 2.1.3 Principle of Superposition The principle states that if the structural behavior is linearly elastic, the forces acting on a structure may be separated or divided into any convenient fashion and the structure analyzed for the separate cases. Then the final results can be obtained by adding up the individual results. This is applicable to the computation of structural responses such as moment, shear, deflection,etc. However, there are two situations where the principle of superposition cannot be applied. The first case is associated with instances where the geometry of the structure is appreciably altered under load. The second case is in situations where the structure is composed of a material in which the stress is not linearly related to the strain. 2.1.4 Idealized Models Any complex structure can be considered to be built up of simpler components called members or elements. Engineering judgement must be used to define an idealized structure such that it represents the actual structural behavior as accurately as is practically possible. Structures can be broadly classified into three categories: 1. Skeletal structures consist of line elements such as a bar, beam, or column for which the length is much larger than the breadth and depth. A variety of skeletal structures can be obtained by connecting line elements together using hinged, rigid, or semi-rigid joints. Depending on whether the axes of these members lie in one plane or in different planes, these structures are termed as plane structures or spatial structures. The line elements in these structures under load may be subjected to one type of force such as axial force or a combination of forces such as shear, moment, torsion, and axial force. In the first case the structures are referred to as the truss-type and in the latter as frame-type. 2. Plated structures consist of elements that have length and breadth of the same order but are much larger than the thickness. These elements may be plane or curved in plane, in which case they are called plates or shells, respectively. These elements are generally used in combination with beams and bars. Reinforced concrete slabs supported on beams, box-girders, plate-girders, cylindrical shells, or water tanks are typical examples of plate and shell structures. 3. Three-dimensional solid structures have all three dimensions, namely, length, breadth, and depth, of the same order. Thick-walled hollow spheres, massive raft foundation, and dams are typical examples of solid structures. c 1999 by CRC Press LLC Recent advancement in finite element methods of structural analysis and the advent of more powerful computers have enabled the economic analysis of skeletal, plated, and solid structures. 2.2 Flexural Members One of the most common structural elements is a beam; it bends when subjected to loads acting transversely to its centroidal axis or sometimes by loads acting both transversely and parallel to this axis. The discussions given in the following subsections are limited to straight beams in which the centroidal