47 Theory and Analysis of Structures 47.1 Fundamental Principles Boundary Conditions • Loads and Reactions • Principle of Superposition 47.2 Beams Relation between Load, Shear Force, and Bending Moment • Shear Force and Bending Moment Diagrams • Fixed-End Beams • Continuous Beams • Beam Deflection • Curved Beams 47.3 Trusses Method of Joints • Method of Sections • Compound Trusses 47.4 Frames Slope Deflection Method • Frame Analysis Using Slope Deflection Method • Moment Distribution Method • Method of Consistent Deformations 47.5 Plates Bending of Thin Plates • Boundary Conditions • Bending of Rectangular Plates • Bending of Circular Plates • Strain Energy of Simple Plates • Plates of Various Shapes and Boundary Conditions • Orthotropic Plates 47.6 Shells Stress Resultants in Shell Element • Shells of Revolution • Spherical Dome • Conical Shells • Shells of Revolution Subjected to Unsymmetrical Loading • Cylindrical Shells • Symmetrically Loaded Circular Cylindrical Shells 47.7 Influence Lines . Influence Lines for Shear in Simple Beams • Influence Lines for Bending Moment in Simple Beams • Influence Lines for Trusses • Qualitative Influence Lines • Influence Lines for Continuous Beams 47.8 Energy Methods Strain Energy Due to Uniaxial Stress • Strain Energy in Bending • Strain Energy in Shear • The Energy Relations in J.Y. Richard Liew Structural Analysis • Unit Load Method National University of Singapore 47.9 Matrix Methods Flexibility Method • Stiffness Method • Element Stiffness N.E. Shanmugam Matrix • Structure Stiffness Matrix • Loading between Nodes • National University of Singapore Semirigid End Connection © 2003 by CRC Press LLC 47-2 The Civil Engineering Handbook, Second Edition 47.10 Finite Element Method Basic Principle • Elastic Formulation • Plane Stress • Plane Strain • Choice of Element Shapes and Sizes • Choice of Displacement Function • Nodal Degrees of Freedom • Isoparametric Elements • Isoparametric Families of Elements • Element Shape Functions • Formulation of Stiffness Matrix • Plates Subjected to In-Plane Forces • Beam Element • Plate Element 47.11 Inelastic Analysis An Overall View • Ductility • Redistribution of Forces • Concept of Plastic Hinge • Plastic Moment Capacity • Theory of Plastic Analysis • Equilibrium Method • Mechanism Method • Analysis Aids for Gable Frames • Grillages • Vierendeel Girders • Hinge-by-Hinge Analysis 47.12 Stability of Structures Stability Analysis Methods • Column Stability • Stability of Beam-Columns • Slope Deflection Equations • Second-Order Elastic Analysis • Modifications to Account for Plastic Hinge Effects • Modification for End Connections • Second-Order Refined Plastic Hinge Analysis • Second-Order Spread of Plasticity Analysis • Three-Dimensional Frame Element • Buckling of Thin Plates • Buckling of Shells 47.13 Dynamic Analysis Equation of Motion • Free Vibration • Forced Vibration • Response to Suddenly Applied Load • Response to Time- Varying Loads • Multiple Degree Systems • Distributed Mass Systems • Portal Frames • Damping • Numerical Analysis 47.1 Fundamental Principles The main purpose of structural analysis is to determine forces and deformations of the structure due to applied loads. Structural design involves form finding, determination of loadings, and proportioning of structural members and components in such a way that the assembled structure is capable of supporting the loads within the design 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, loading, and boundary conditions as accurately as is practicable. Structures often appear in three-dimensional forms. For structures that have a regular layout and are rectangular in shape, subject to symmetric loads, it is possible to idealize them into two-dimensional frames arranged in orthogonal directions. A structure is said to be two-dimensional or planar if all the members lie in the same plane. Joints in a structure are those points where two or more members are connected. Beams are members subjected to loading acting transversely to their longitudinal axis and creating flexural bending only. Ties are members that are subjected to axial tension only, while struts (columns or posts) are members subjected to axial compression only. A truss is a structural system consisting of members that are designed to resist only axial forces. A structural system in which joints are capable of transferring end moments is called a frame. Members in this system are assumed to be capable of resisting bending moments, axial force, and shear force. Boundary Conditions A hinge or pinned joint does not allow translational movements (Fig. 47.1a). It is assumed to be frictionless and to allow rotation of a member with respect to the others. A roller permits the attached structural part to rotate freely with respect to the rigid surface and to translate freely in the direction parallel to the surface (Fig. 47.1b). Translational movement in any other direction is not allowed. A fixed support (Fig. 47.1c) does not allow rotation or translation in any direction. A rotational spring provides some © 2003 by CRC Press LLC Theory and Analysis of Structures 47-3 (a) Hinge support (b) Roller support (d) Rotational spring (c) Fixed support (e) Translational spring FIGURE 47.1 Various boundary conditions. rotational restraint but does not provide any translational restraint (Fig. 47.1d). A translational spring can provide partial restraints along the direction of deformation (Fig. 47.1e). Loads and Reactions Loads that are of constant magnitude and remain in the original position are called permanent loads. They 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. Loads that may change in position and magnitude are called variable loads. They 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 loads are due to water or snow on a flat roof that 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 is to be in equilibrium under the action of a system of loads, it must satisfy the six static equilibrium equations: (47.1) Â=Fxyz0, Â=F 0, Â=F 0 Â=Mxy0, Â=M 0, Â=M z0 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: Â=Fxy0, Â=F 0, ÂM z =0 (47.2) Consider a beam under the action of the applied loads, as shown in Fig. 47.2a. The reaction at support B must act perpendicular to the surface on which the rollers are constrained to roll upon. The support © 2003 by CRC Press LLC 47-4 The Civil Engineering Handbook, Second Edition 400 kN ° A 60 B 30° C 5 m 5 m (a) Applied load 400 Sin 60° = 346.4 kN A x 400 Cos 60° = 200 kN Bx A y By (b) Support reactions FIGURE 47.2 Beam in equilibrium. reactions and the applied loads, which are resolved in vertical and horizontal directions, are shown in Fig. 47.2b. From geometry, it can be calculated that By = 3Bx 1. Equation (47.2) can be used to determine the magnitude of the support reactions. Taking moment about B gives 10Ay – 346.4 ¥ 5 = 0 from which Ay = 173.2 kN Equating the sum of vertical forces, SFy, to zero gives 173.2 + B y – 346.4 = 0 and hence we get By = 173.2 kN Therefore BBxy==3 100 kN . Equilibrium in the horizontal direction, SFx = 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 equal 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. © 2003 by CRC Press LLC Theory and Analysis of Structures 47-5 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. 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.