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Structural Analysis

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    The word structure describes much of what is seen in nature. Living plants, from the frailest of ferns to the most rugged of trees, possess a structural form consistent with their needs. Insects and animals play a more active role in building the structures that they need, e. g. the delicate web of the spider, etc. Humans, too, are builders of structures; but more than that, they are conceivers and designers.

    When the structures of humans began to reflect their ability to conceive and design them as well as to construct them, structural engineering was born, and it has grown in sophistication as it has endeavoured to meet the demands of humanity.

    Determinate Beams

    A beam is defined as a structural member predominantly subjected to bending moment. In this chapter, only beams of symmetrical section, the centroidal axis of such beams being a straight line will be discussed. Furthermore, the beam is acted on by only transverse loading and moment loading and that all the loads and reactions lie in the plane of symmetry.

    For these, it follows that such a beam will be subjected to bending and shear in the plane of loading without axial stretching and twisting. Statically determinate beams may be classified as:

    Simple beam: A beam that is supported at its two ends with a hinged and a roller or their equivalents , is termed a simply supported beam or simple beam.

    Cantilever beam: a cantilever beam, which is fixed or built-in at one end and free at the other end. , in which the three elements of reaction of fixed support are provided by a hinge and roller closely placed at one end segment.

    Simple beam with overhang : A beam may be simply supported at two points and have its end portion( or portions) extend beyond the support; it is then called a simple beam with overhang.

    Compound beam: The beams indicated above may be connected by internal hinges or rollers to form a compound beam.

    Care must be used in providing the connections so that instability is not produced.

    Sign Conventions

    The sign conventions for beam shear and moments are as follows: Shear is considered positive at a section when it tends to rotate the portion of the beam in the clockwise direction about an axis through a point inside the free body and normal to the plane of loading; otherwise, it is negative  Bending moment is considered positive at a section when it tends to bend the member concave downward; other wise, it is negative.

    Numerical Examples

    The following examples will serve to illustrate the construction of shear and bending moment diagrams of transversely loaded beams.  Analyse the loaded compound beam as shown below, which is composed of a simple beam and cantilever connected by an internal hinge.

    Determinate Trusses

    A truss, such as the one shown in Fig(1. 3), may be defined as a plane structure composed of a number of members jointed together at their ends by smooth pins so as to form a rigid framework the external forces and reactions of which are assumed to lie in the same plane and to act only at the pins.

    Furthermore, the centroidal axis of each member coincides with the line connecting the joint centers at the ends of member and that the weight of each member is negligible in comparison to the other external forces acting on the truss. From these conditions it follows that each member is a truss is a two-force member and is subjected only to direct axial forces (tension or compression). A truss is completely analyzed when the internal axial forces in all members have been determined. It is customary to use a plus sign to designate tension and a minus sign to designed compression.

    Types of Trusses

    Common trusses may be classified according to their formations as simple, compound, and complex.

    Simple truss: A rigid plane truss can always be formed by beginning with three bars pinned together at their ends in the form of a triangle and then extending from this two new bars for each new joint. The new joint and the two joints to which it is connected should never lie along the same straight line, to avoid geometric instability.  The trusses are all simple trusses. The shaded triangle abc in each truss diagram is the base figure from which the form was extended by using two additional bars to connect each of the new joint in alphabetical order.

    Compound truss: If two or more simple trusses are connected together to form one rigid framework, the composite truss is called a compound truss. One simple truss can be rigidly connected to another simple truss at certain joints by three links neither parallel nor concurrent or by the equivalent of this type of connection.

    Determinate Rigid Frames

    A rigid frame may be defined as a structure composed of a number of members connected together by joints some or all of which are rigid, that is, capable of resisting both force and moment as distinguished from a pin-connected joint which offers no moment resistance. In the analysis of rigid frames, the centroidal axis of each member is assumed to coincide with the line connecting the joint centers of the ends f the member. The socalled joint centre is therefore the concurrent point of all centroidal axes of members meeting at the joint. With the joint rigid the end of all connected members must not only translate but also rotate identical amounts at the joint.

    Analysis of Statically Determinate Rigid Frames

    Rigid frames are usually built to be highly statically indeterminate. The analysis of determinate rigid frames in this chapter is primary of academic interest rather than of practical use and serves as a prelude to the analysis of indeterminate frames.

    To analyze a statically determinate rigid frame, we start by finding the reaction components from statistical equations for the entire structure. This done, we are able to determinate the shear, moment, and axial force at any cross section of the frame by taking the free body cut through that section and by using the equilibrium equations. Based on the centroidal axis of each member, we can plot the shear, bending moment, and the direct force diagrams for the rigid frame. However, it is the bending moment diagram with which we are mainly concerned in the analysis of a rigid frame.

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