Statically Indeterminate Truss
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Fig. 1.13 Support reactions in a cantilever beam subjected to an inclined load at its free end
When loads are applied to a structure, reactions are generated in the supports and in many structural analysis problems the first step is to calculate their values. It is important, therefore, to identify correctly the type of reaction associated with a particular support. Thus, supports that prevent translation in a particular direction produce a force reaction in that direction while supports that prevent rotation induce moment reactions. For example, in the cantilever beam of Fig. 1.13, the applied load W has horizontal and vertical components which induce horizontal (/?A H) and vertical (RA v) reactions of force at the builtin end A, while the rotational effect of W is balanced by the moment reaction MA. We shall consider the calculation of support reactions in detail in Section 2.5.
1.4 Statically determinate and indeterminate structures
In many structural systems the principles of statical equilibrium (Section 2.4) may be used to determine support reactions and internal force distributions; such systems
Fig. 1.14 (a) Statically determinate truss; (b) statically indeterminate truss are called statically determinate. Systems for which the principles of statical equilibrium are insufficient to determine support reactions and/or internal force distributions, i.e. there are a greater number of unknowns than the number of equations of statical equilibrium, are known as statically indeterminate or hyperstatic systems. However, it is possible that even though the support reactions are statically determinate, the internal forces are not, and vice versa. Thus, for example, the truss in Fig. 1.14(a) is, as we shall see in Chapter 4, statically determinate both for support reactions and forces in the members whereas the truss shown in Fig. 1.14(b) is statically determinate only as far as the calculation of support reactions is concerned.
Another type of indeterminacy, kinematic indeterminacy, is associated with the ability to deform, or the degrees of freedom of, a structure and is discussed in detail in Section 16.3. A degree of freedom is a possible displacement of a joint (or node as it is often called) in a structure. Thus a joint in a plane truss has three possible modes of displacement or degrees of freedom, two of translation in two mutually perpendicular directions and one of rotation, all in the plane of the truss. On the other hand a joint in a threedimensional space truss or frame possesses six degrees of freedom, three of translation in three mutually perpendicular directions and three of rotation about three mutually perpendicular axes.
1.5 Analysis and design
Some students in the early stages of their studies have only a vague idea of the difference between an analytical problem and a design problem. It will be instructive, therefore, to examine the various steps in the design procedure and to consider the role of analysis in that procedure.
Initially the structural designer is faced with a requirement for a structure to fulfil a particular role. This may be a bridge of a specific span, a multistorey building of a given floor area, a retaining wall having a required height, and so on. At this stage the designer will decide on a possible form for the structure. In the case of a bridge, for example, the designer must decide whether to use beams, trusses, arches or cables to support the bridge deck. To some extent, as we have seen, the choice is governed by the span required, although other factors may influence the decision. Thus, in Scotland, the Filth of Tay is crossed by a multispan bridge supported on columns, whereas the road bridge crossing the Firth of Forth is a suspension bridge.
In the latter case a large height clearance is required to accommodate shipping. In addition it is possible that the designer may consider different schemes for the same requirement. Further decisions are required as to the materials to be used: steel, reinforced concrete, timber, etc.
Having decided on a form for the structure, the loads on the structure are calculated. These arise in different ways. Dead loads are loads that are permanently present, such as the structure's selfweight, fixtures, cladding, etc. Live or imposed loads are movable or actually moving loads, such as temporary partitions, people, vehicles on a bridge, snow, etc. Wind loads are live loads but require special consideration since they are affected by the location, size and shape of the structure. Other live loads may include soil or hydrostatic pressure and dynamic effects produced, for example, by vibrating machinery, wind gusts, wave action or, in some parts of the world, earthquake action.
In some instances values of the above loads are given in Codes of Practice. Thus, for floors in office buildings designed for general use, CP3: Chapter V: Part I specifies a distributed load of 25 kN/m2 together with a concentrated load of 27 kN applied over any square of side 300 mm, while CP3: Chapter V: Part 2 gives details of how wind loads should be calculated.
When the loads have been determined, the structure is analysed, i.e. the external and internal forces and moments are calculated, from which are obtained the internal stress distributions and also the strains and displacements. The structure is then checked for safety, i.e. that it possesses sufficient strength to resist loads without danger of collapse, and for serviceability, which determines its ability to carry loads without excessive deformation or local distress; Codes of Practice are used in this procedure. It is possible that this check may show that the structure is underdesigned (unsafe and/or unserviceable) or overdesigned (uneconomic) so that adjustments must be made to the arrangement and/or the sizes of the members; the analysis and design check are then repeated.
Analysis, as can be seen from the above discussion, forms only part of the complete design process and is concerned with a given structure subjected to given loads. Thus, generally, there is a unique solution to an analytical problem whereas there may be one, two or more perfectly acceptable solutions to a design problem.
1.6 Structural idealization
Generally, structures are complex and must be idealized or simplified into a form that can be analysed. This idealization depends upon factors such as the degree of accuracy required from the analysis because, usually, the more sophisticated the method of analysis employed the more time consuming, and therefore more costly, it is. Thus a preliminary evaluation of two or more possible design solutions would not require the same degree of accuracy as the check on the finalized design. Other factors affecting the idealization include the type of load being applied, since it is possible that a structure will require different idealizations under different loads.
We have seen in Section 1.3 how actual supports are idealized. An example of structural idealization is shown in Fig. 1.15 where the simple roof truss of Fig. 1.15(a) is supported on columns and forms one of a series comprising a roof structure. The roof cladding is attached to the truss through purlins which connect
each truss, and the truss members are connected to each other by gusset plates which may be riveted or welded to the members forming rigid joints. This structure possesses a high degree of statical indeterminacy and its analysis would probably require a computerbased approach. However, the assumption of a simple support system, the replacement of the rigid joints by pinned or hinged joints and the assumption that the forces in the members are purely axial, result, as we shall see in Chapter 4, in a statically determinate structure (Fig. 1.15(b)). Such an idealization might appear extreme but, so long as the loads are applied at the joints and the truss is supported at joints, the forces in the members are predominantly axial and bending moments and shear forces are negligibly small.
At the other extreme a continuum structure, such as a folded plate roof, would be idealized into a large number of finite elements connected at nodes and analysed using a computer; the finite element method is, in fact, an exclusively computerbased technique. A large range of elements is available in finite element packages including simple beam elements, plate elements, which can model both inplane and outofplane effects, and threedimensional 'brick' elements for the idealization of solid threedimensional structures. A wide range of literature devoted to finite element analysis is available but will not be considered here as the method is outside the scope of this book.
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