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2.11 Inelastic Analysis 2-106

An Overall View • Ductility • Redistribution of Forces • Concept of Plastic Hinge • Plastic Moment Capacity • Theory of Plastic Analysis • Equilibrium Method • Mechanism Method • Gable Frames • Analysis Aids for Gable Frames • Grillages • Vierendeel Girders • Hinge-by-Hinge Analysis

### 2.12 Structural Stability 2-130

Stability Analysis Methods • Column Stability • Stability of Beam-Columns • Beam-Column Subjected to Transverse Loading • 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

J. Y. Richard Liew Thin Plates • Buckling of Shells

Department of Civil ^gm^n^ 2.13 Structural Dynamic 2-157

National University of Singapore, Equation of Motion • Free Vibration • Forced Vibration •

Singnpoie Response to Suddenly Applied Load • Response to Time

N E Shanmugam Varying Loads • Multiple Degree Systems • Distributed Mass

Systems • Portal Frames • Damping • Numerical Analysis

Department of Civil Engineering,

National University of Singapore, References 2-169

Singapore Further Reading 2-172

### 2.1 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. The 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.

Structures often appear in three-dimensional form. It is possible to idealize structures that have a regular layout, are rectangular in shape, and are subjected to symmetric loads 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 transverse 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.

### 2.1.1 Boundary Conditions

A hinge or pinned joint does not allow translational movements (Figure 2.1a). 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 (Figure 2.1b). Translational movement in any other direction is not allowed. A fixed support (Figure 2.1c) does not allow rotation or translation in any direction. A rotational spring 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).

FIGURE 2.1 Various boundary conditions: (a) hinge support, (b) roller support, (c) fixed support, (d) rotational support, and (e) translational spring.

### 2.1.2 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 objects 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:

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

Consider a beam under the action of the applied loads as shown in Figure 2.2a. The reaction at support B must act perpendicular to the surface on which the rollers are constrained to roll. The support reactions and the applied loads, which are resolved in vertical and horizontal directions, are shown in Figure 2.2b.

With geometry, it can be calculated that By = Equation 2.2 can be used to determine the magnitude of the support reactions. Taking the moment about B gives

FIGURE 2.2 Beam in equilibrium: (a) applied load and (b) support reactions.

from which we get

Equating the sum of vertical forces, Y1 , to 0 gives

and hence we get Therefore,

Bx = Byp = 100 kN Equilibrium in the horizontal direction, Y1 Fx = 0, gives

and hence,

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.

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