Structure and Architecture (AJ Macdonald) (1).pdf

Chapter 2 Structural requirements 2.1 Introduction horizontal force is applied to the wheelbarrow...

Chapter 2 Structural requirements 2.1 Introduction horizontal force is applied to the wheelbarrow by its operator it moves horizontally and is To perform its function of supporting a not therefore in a state of static equilibrium. building in response to whatever loads may be This occurs because the interface between the applied to it, a structure must possess four wheelbarrow and the ground is incapable of properties: it must be capable of achieving a generating horizontal reacting forces. The state of equilibrium, it must be stable, it must wheelbarrow is both a structure and a have adequate strength and it must have machine: it is a structure under the action of adequate rigidity. The meanings of these terms gravitational load and a machine under the are explained in this chapter. The influence of action of horizontal load. structural requirements on the forms which are Despite the famous statement by one adopted for structures is also discussed. The celebrated commentator, buildings are not treatment is presented in a non-mathematical machines1. Architectural structures must, way and the definitions which are given are not therefore, be capable of achieving equilibrium those of the theoretical physicist; they are under all directions of load. simply statements which are sufficiently precise to allow the significance of the concepts to structural design to be 2.3 Geometric stability appreciated. Geometric stability is the property which preserves the geometry of a structure and 2.2 Equilibrium allows its elements to act together to resist load. The distinction between stability and Structures must be capable of achieving a equilibrium is illustrated by the framework state of equilibrium under the action of shown in Fig. 2.1 which is capable of achieving applied load. This requires that the internal a state of equilibrium under the action of configuration of the structure together with gravitational load. The equilibrium is not the means by which it is connected to its stable, however, because the frame will foundations must be such that all applied collapse if disturbed laterally2. loads are balanced exactly by reactions generated at its foundations. The wheelbarrow provides a simple demonstration of the principles involved. When the 1 ‘A house is a machine for living.’ Le Corbusier. wheelbarrow is at rest it is in a state of static 2 Stability can also be distinguished from strength or equilibrium. The gravitational forces rigidity, because even if the elements of a structure have sufficient strength and rigidity to sustain the generated by its self weight and that of its loads which are imposed on them, it is still possible contents act vertically downwards and are for the system as a whole to fail due to its being exactly balanced by reacting forces acting at geometrically unstable as is demonstrated in the wheel and other supports. When a Fig. 2.1. 9 Structure and Architecture Fig. 2.1 A rectangular frame with four hinges is capable of achieving a state of equilibrium but is unstable because any slight lateral disturbance to the columns will induce it to collapse. The frame on the right here is stabilised by the diagonal element which makes no direct contribution to the resistance of the gravitational load. This simple arrangement demonstrates The parts of structures which tend to be that the critical factor, so far as the stability unstable are the ones in which compressive of any system is concerned, is the effect on it forces act and these parts must therefore be of a small disturbance. In the context of given special attention when the geometric structures this is shown very simply in Fig. 2.2 stability of an arrangement is being by the comparison of tensile and compressive considered. The columns in a simple elements. If the alignment of either of these is rectangular framework are examples of this disturbed, the tensile element is pulled back (Fig. 2.1). The three-dimensional bridge into line following the removal of the structure of Fig. 2.3 illustrates another disturbing agency but the compressive potentially unstable system. Compression element, once its initially perfect alignment occurs in the horizontal elements in the upper has been altered, progresses to an entirely parts of this frame when the weight of an new position. The fundamental issue of object crossing the bridge is carried. The stability is demonstrated here, which is that arrangement would fail by instability when this stable systems revert to their original state load was applied due to inadequate restraint following a slight disturbance whereas unstable of these compression parts. The compressive systems progress to an entirely new state. internal forces, which would inevitably occur Original alignment Fig. 2.2 The tensile Fig. 2.3 The horizontal element on the left here is elements in the tops of stable because the loads the bridge girders are pull it back into line subjected to following a disturbance. The compressive internal compressive element on force when the load is the right is fundamentally applied. The system is unstable. unstable and any eccentricity which is present initially causes an instability-type failure to develop. Compression 10 Tension  Structural requirements with some degree of eccentricity, would push Conversely, if an arrangement is not capable of the upper elements out of alignment and resisting load from three orthogonal directions cause the whole structure to collapse. then it will be unstable in service even though The geometric instability of the the load which it is designed to resist will be arrangements in Figures 2.1 and 2.3 would applied from only one direction. have been obvious if their response to It frequently occurs in architectural design horizontal load had been considered (Fig. 2.4). that a geometry which is potentially unstable This demonstrates one of the fundamental must be adopted in order that other requirements for the geometric stability of any architectural requirements can be satisfied. For arrangement of elements, which is that it must example, one of the most convenient structural be capable of resisting loads from orthogonal geometries for buildings, that of the directions (two orthogonal directions for plane rectangular frame, is unstable in its simplest arrangements and three for three-dimensional hinge-jointed form, as has already been shown. arrangements). This is another way of saying Stability can be achieved with this geometry by that an arrangement must be capable of the use of rigid joints, by the insertion of a achieving a state of equilibrium in response to diagonal element or by the use of a rigid forces from three orthogonal directions. The diaphragm which fills up the interior of the stability or otherwise of a proposed frame (Fig. 2.5). Each of these has arrangement can therefore be judged by disadvantages. Rigid joints are the most considering the effect on it of sets of mutually convenient from a space-planning point of perpendicular trial forces: if the arrangement is view but are problematic structurally because capable of resisting all of these then it is they can render the structure statically stable, regardless of the loading pattern which indeterminate (see Appendix 3). Diagonal will actually be applied to it in service. elements and diaphragms block the framework and can complicate space planning. In multi- panel arrangements, however, it is possible to (a) (b) produce stability without blocking every panel. The row of frames in Fig. 2.6, for example, is stabilised by the insertion of a single diagonal. (a) (b) (c) Fig. 2.5 A rectangular frame can be stabilised by the insertion of (a) a diagonal element or (b) a rigid diaphragm, or (c) by the provision of rigid joints. A single rigid joint is in fact sufficient to provide stability. Fig. 2.4 Conditions for stability of frameworks. (a) The two-dimensional system is stable if it is capable of achieving equilibrium in response to forces from two mutually perpendicular directions. (b) The three- dimensional system is stable if it is capable of resisting forces from three directions. Note that in the case illustrated the resistance of transverse horizontal load is Fig. 2.6 A row of rectangular frames is stable if one panel achieved by the insertion of rigid joints in the end bays. only is braced by any of the three methods shown in Fig. 2.5. 11 Structure and Architecture Fig. 2.7 These elements. Arrangements which do not require frames contain bracing elements, either because they are the minimum number of fundamentally stable or because stability is braced panels provided by rigid joints, are said to be self- required for bracing. stability. Most structures contain bracing elements whose presence frequently affects both the initial planning and the final appearance of the building which it supports. The issue of stability, and in particular the design of bracing systems, is therefore something which affects the architecture of buildings. Where a structure is subjected to loads from different directions, the elements which are used solely for bracing when the principal load is applied frequently play a direct role in resisting secondary load. The diagonal elements in the frame of Fig. 2.7, for example, would be directly involved in the resistance of any horizontal load which was applied, such as might occur due to the action of wind. Because real structures are usually subjected to loads from different directions, it is very rare for elements to be used solely for bracing. The nature of the internal force in bracing components depends on the direction in which the instability which they prevent occurs. In Fig. 2.8, for example, the diagonal element will be placed in tension if the frame sways to the Where frames are parallel to each other the right and in compression if it sways to the left. three-dimensional arrangement is stable if a Because the direction of sway due to instability few panels in each of the two principal cannot be predicted when the structure is directions are stabilised in the vertical plane being designed, the single bracing element and the remaining frames are connected to would have to be made strong enough to carry these by diagonal elements or diaphragms in either tension or compression. The resistance the horizontal plane (Fig. 2.7). A three- of compression requires a much larger size of dimensional frame can therefore be stabilised cross-section than that of tension, however, by the use of diagonal elements or diaphragms especially if the element is long3, and this is a in a limited number of panels in the vertical critical factor in determining its size. It is and horizontal planes. In multi-storey normally more economical to insert both arrangements these systems must be provided diagonal elements into a rectangular frame at every storey level. None of the components which are added to stabilise the geometry of the rectangular 3 This is because compression elements can suffer from frame in Fig. 2.7 makes a direct contribution to the buckling phenomenon. The basic principles of this the resistance of gravitational load (i.e. the are explained in elementary texts on structures such as Engel, H., Structural Principles, Prentice-Hall, Englewood carrying of weight, either of the structure itself Cliffs, NJ, 1984. See also Macdonald, Angus J., Structural or of the elements and objects which it Design for Architecture, Architectural Press, Oxford, 1997, 12 supports). Such elements are called bracing Appendix 2.  Structural requirements (cross-bracing) than a single element and to (a) design both of them as tension-only elements. When the panel sways due to instability the element which is placed in compression simply buckles slightly and the whole of the restraint is provided by the tension diagonal. (b) Fig. 2.8 Cross-bracing is used so that sway caused by instability is always resisted by a diagonal element acting (c) in tension. The compressive diagonal buckles slightly and carries no load. It is common practice to provide more bracing elements than the minimum number Fig. 2.9 In practical bracing schemes more elements required so as to improve the resistance of than are strictly necessary to ensure stability are provided three-dimensional frameworks to horizontal to improve the performance of frameworks in resisting horizontal load. Frame (a) is stable but will suffer load. The framework in Fig. 2.7, for example, distortion in response to horizontal load on the side walls. although theoretically stable, would suffer Its performance is enhanced if a diagonal element is considerable distortion in response to a provided in both end walls (b). The lowest framework (c) horizontal load applied parallel to the long side contains the minimum number of elements required to of the frame at the opposite end from the resist effectively horizontal load from the two principal horizontal directions. Note that the vertical-plane bracing vertical-plane bracing. A load applied parallel to elements are distributed around the structure in a the long side at this end of the frame would also symmetrical configuration. cause a certain amount of distress as some movement of joints would inevitably occur in the transmission of it to the vertical-plane bracing at the other end. In practice the performance of the frame is more satisfactory if vertical-plane bracing is provided at both ends (Fig. 2.9). This gives more restraint than is necessary for stability and makes the structure statically indeterminate (see Appendix 3), but results in the horizontal loads being resisted close to the points where they are applied to the structure. Fig. 2.10 In practice, bracing elements are frequently Another practical consideration in relation confined to a part of each panel only. to the bracing of three-dimensional rectangular frames is the length of the diagonal elements which are provided. These sag in response to Figures 2.11 and 2.12 show typical bracing their own weight and it is therefore systems for multi-storey frameworks. Another advantageous to make them as short as common arrangement, in which floor slabs act possible. For this reason bracing elements are as diaphragm-type bracing in the horizontal frequently restricted to a part of the panel in plane in conjunction with vertical-plane which they are located. The frame shown in bracing of the diagonal type, is shown in Fig. Fig. 2.10 contains this refinement. 2.13. When the rigid-joint method is used it is 13 Structure and Architecture Fig. 2.11 A typical bracing scheme for a multi-storey framework. Vertical-plane bracing is provided in a limited number of bays and positioned symmetrically on plan. All other bays are linked to this by diagonal bracing in the horizontal plane at every storey level. normal practice to stabilise all panels individually by making all joints rigid. This eliminates the need for horizontal-plane bracing altogether, although the floors normally act to distribute through the structure any unevenness in the application of horizontal load. The rigid-joint method is the normal method which is adopted for reinforced concrete frames, in which continuity through junctions between elements can easily be achieved; diaphragm bracing is also used, however, in both vertical and horizontal planes in certain types of reinforced concrete frame. Loadbearing wall structures are those in which the external walls and internal partitions serve as vertical structural elements. They are normally constructed of masonry, reinforced Fig. 2.12 These drawings of floor grid patterns for steel frameworks show typical locations for vertical-plane bracing. Fig. 2.13 Concrete floor slabs are normally used as horizontal-plane bracing of the diaphragm type which acts 14 in conjunction with diagonal bracing in the vertical planes.  Structural requirements concrete or timber, but combinations of these materials are also used. In all cases the joints between walls and floors are normally incapable of resisting bending action (in other words they behave as hinges) and the resulting lack of continuity means that rigid-frame action cannot develop. Diaphragm bracing, provided by the walls themselves, is used to stabilise these structures. A wall panel has high rotational stability in its own plane but is unstable in the out-of- plane direction (Fig. 2.14); vertical panels must, therefore, be grouped in pairs at right Fig. 2.15 Loadbearing masonry buildings are normally multi-cellular structures which contain walls running in two orthogonal directions. The arrangement is inherently stable. in two orthogonal directions is normally straightforward (Fig. 2.15). It is unusual therefore for bracing requirements to have a significant effect on the internal planning of this type of building. The need to ensure that a structural framework is adequately braced is a factor that Fig. 2.14 Walls are can affect the internal planning of buildings. unstable in the The basic requirement is that some form of out-of-plane direction bracing must be provided in three orthogonal and must be grouped into orthogonal planes. If diagonal or diaphragm bracing is arrangements for used in the vertical planes this must be stability. accommodated within the plan. Because vertical-plane bracing is most effective when it is arranged symmetrically, either in internal cores or around the perimeter of the building, angles to each other so that they provide this can affect the space planning especially in mutual support. For this to be effective the tall buildings where the effects of wind loading structural connection which is provided in the are significant. vertical joint between panels must be capable of resisting shear4. Because loadbearing wall structures are normally used for multi-cellular 2.4 Strength and rigidity buildings, the provision of an adequate number of vertical-plane bracing diaphragms 2.4.1 Introduction The application of load to a structure generates internal forces in the elements and 4 See Engel, H., Structural Principles, Prentice-Hall, external reacting forces at the foundations (Fig. Englewood Cliffs, NJ, 1984 for an explanation of shear. 2.16) and the elements and foundations must 15 Structure and Architecture Fig. 2.16 The structural elements of a building conduct the loads to the foundations. They are subjected to internal forces that generate stresses the magnitudes of which depend on the intensities of the internal forces and the sizes of the elements. The structure will collapse if the stress levels exceed the strength of the material. have sufficient strength and rigidity to resist these. They must not rupture when the peak load is applied; neither must the deflection which results from the peak load be excessive. The requirement for adequate strength is satisfied by ensuring that the levels of stress which occur in the various elements of a structure, when the peak loads are applied, are within acceptable limits. This is chiefly a matter of providing elements with cross- sections of adequate size, given the strength of the constituent material. The determination of the sizes required is carried out by structural calculations. The provision of adequate rigidity is similarly dealt with. Structural calculations allow the strength and rigidity of structures to be controlled precisely. They are preceded by an assessment indeterminate structures (see Appendix 3), the of the load which a structure will be required two sets of calculations are carried out to carry. The calculations can be considered to together, but it is possible to think of them as be divisible into two parts and to consist firstly separate operations and they are described of the structural analysis, which is the separately here. evaluation of the internal forces which occur in the elements of the structure, and secondly, 2.4.2 The assessment of load the element-sizing calculations which are The assessment of the loads which will act on carried out to ensure that they will have a structure involves the prediction of all the sufficient strength and rigidity to resist the different circumstances which will cause load internal forces which the loads will cause. In to be applied to a building in its lifetime (Fig. 16 many cases, and always for statically 2.17) and the estimation of the greatest  Structural requirements Fig. 2.17 The prediction of the maximum load which will occur is one of the most problematic aspects of structural calculations. Loading standards are provided to assist with this but assessment of load is nevertheless one of the most imprecise parts of the structural calculation process. magnitudes of these loads. The maximum load structure when the most unfavourable load could occur when the building was full of conditions occur. To understand the various people, when particularly heavy items of processes of structural analysis it is necessary equipment were installed, when it was exposed to have a knowledge of the constituents of to the force of exceptionally high winds or as a structural force systems and an appreciation of result of many other eventualities. The concepts, such as equilibrium, which are used designer must anticipate all of these to derive relationships between them. These possibilities and also investigate all likely topics are discussed in Appendix 1. combinations of them. In the analysis of a structure the external The evaluation of load is a complex process, reactions which act at the foundations and the but guidance is normally available to the internal forces in the elements are calculated designer of a structure from loading from the loads. This is a process in which the standards5. These are documents in which data structure is reduced to its most basic abstract and wisdom gained from experience are form and considered separately from the rest presented systematically in a form which of the building which it will support. allows the information to be applied in design. An indication of the sequence of operations which are carried out in the analysis of a 2.4.3 The analysis calculations simple structure is given in Fig. 2.18. After a The purpose of structural analysis is to preliminary analysis has been carried out to determine the magnitudes of all of the forces, evaluate the external reactions, the structure is internal and external, which occur on and in a subdivided into its main elements by making ‘imaginary cuts’ (see Appendix 1.7) through the junctions between them. This creates a set of 5 In the UK the relevant standard is BS 6399, Design ‘free-body-diagrams’ (Appendix 1.6) in which Loading for Buildings, British Standards Institution, 1984. the forces that act between the elements are 17 Structure and Architecture Uniformly distributed geometry of the structure. The reason for this is explained in Appendix 3. In these circumstances the analysis and element-sizing calculations are carried out together in a trial and error process which is only feasible in the context of computer-aided design. The different types of internal force which can occur in a structural element are shown in Fig. 2.19. As these have a very significant influence on the sizes and shapes which are specified for elements they will be described briefly here. In Fig. 2.19 an element is cut through at a particular cross-section. In Fig. 2.19(a) the forces which are external to one of the (a) Fig. 2.18 In structural analysis the complete structure is broken down into two-dimensional components and the (b) internal forces in these are subsequently calculated. The diagram shows the pattern forces which result from gravitational load on the roof of a small building. Similar breakdowns are carried out for the other forms of load and a complete picture is built up of the internal forces which will occur in each element during the life of the structure. (c) exposed. Following the evaluation of these inter-element forces the individual elements are analysed separately for their internal forces Fig. 2.19 The investigation of internal forces in a simple beam using the device of the ‘imaginary cut’. The cut by further applications of the ‘imaginary cut’ produces a free-body-diagram from which the nature of the technique. In this way all of the internal forces internal forces at a single cross-section can be deduced. in the structure are determined. The internal forces at other cross-sections can be In large, complex, statically indeterminate determined from similar diagrams produced by cuts made structures the magnitudes of the internal in appropriate places. (a) Not in equilibrium. (b) Positional equilibrium but not in rotational equilibrium. (c) forces are affected by the sizes and shapes of Positional and rotational equilibrium. The shear force on the element cross-sections and the properties the cross-section 1.5 m from the left-hand support is of the constituent materials, as well as by the 15 kN; the bending moment on this cross-section is 18 magnitudes of the loads and the overall 22.5 kNm.  Structural requirements resulting sub-elements are marked. If these were indeed the only forces which acted on the sub-element it would not be in a state of equilibrium. For equilibrium the forces must balance and this is clearly not the case here; an additional vertical force is required for equilibrium. As no other external forces are present on this part of the element the extra (a) (b) force must act on the cross-section where the cut occurred. Although this force is external to the sub-element it is an internal force so far as the complete element is concerned and is called the ‘shear force’. Its magnitude at the cross-section where the cut was made is simply the difference between the external (c) (d) forces which occur to one side of the cross- section, i.e. to the left of the cut. Once the shear force is added to the Fig. 2.20 The ‘imaginary cut’ is a device for exposing diagram the question of the equilibrium of internal forces and rendering them susceptible to the sub-element can once more be equilibrium analysis. In the simple beam shown here shear examined. In fact it is still not in a state of force and bending moment are the only internal forces equilibrium because the set of forces now required to produce equilibrium in the element isolated by the cut. These are therefore the only internal forces which acting will produce a turning effect on the act on the cross-section at which the cut was made. In the sub-element which will cause it to rotate in a case of the portal frame, axial thrust is also required at the clockwise sense. For equilibrium an anti- cross-section exposed by the cut. clockwise moment is required and as before this must act on the cross-section at the cut because no other external forces are present. plane, do not balance. Shear force and bending The moment which acts at the cut and which moment occur in structural elements which are is required to establish rotational bent by the action of the applied load. Beams equilibrium is called the bending moment at and slabs are examples of such elements. the cross-section of the cut. Its magnitude is One other type of internal force can act on obtained from the moment equation of the cross-section of an element, namely axial equilibrium for the free-body-diagram. Once thrust (Fig. 2.20). This is defined as the amount this is added to the diagram the system is in by which the external forces acting on the a state of static equilibrium, because all the element to one side of a particular location do conditions for equilibrium are now satisfied not balance when they are resolved parallel to (see Appendix 1). the direction of the element. Axial thrust can Shear force and bending moment are forces be either tensile or compressive. which occur inside structural elements and In the general case each cross-section of a they can be defined as follows. The shear force structural element is acted upon by all three at any location is the amount by which the internal forces, namely shear force, bending external forces acting on the element, to one moment and axial thrust. In the element-sizing side of that location, do not balance when they part of the calculations, cross-section sizes are are resolved perpendicular to the axis of the determined that ensure the levels of stress element. The bending moment at a location in which these produce are not excessive. The an element is the amount by which the efficiency with which these internal forces can moments of the external forces acting to one be resisted depends on the shape of the cross- side of the location, about any point in their section (see Section 4.2). 19 Structure and Architecture The shapes of bending moment, shear force and axial thrust diagrams are of great significance for the eventual shapes of structural elements because they indicate the locations of the parts where greatest strength will be required. Bending moment is normally large in the vicinity of mid-span and near rigid joints. Shear force is highest near support joints. Axial thrust is usually constant along the length of structural elements. 2.4.4 Element-sizing calculations The size of cross-section which is provided for a structural element must be such as to give it adequate strength and adequate rigidity. In other words, the size of the cross-section must allow the internal forces determined in the analysis to be carried without overloading the structural material and without the occurrence of excessive deflection. The calculations which are carried out to achieve this involve the use of the concepts of stress and strain (see Appendix 2). In the sizing calculations each element is considered individually and the area of cross- section determined which will maintain the stress at an acceptable level in response to the peak internal forces. The detailed aspects of the calculations depend on the type of internal Fig. 2.21 The magnitudes of internal forces normally vary force and, therefore, the stress involved and on along the length of a structural element. Repeated use of the properties of the structural material. the ‘imaginary cut’ technique yields the pattern of internal As with most types of design the evolution forces in this simple beam. of the final form and dimensions of a structure is, to some extent, a cyclic process. If the element-sizing procedures yield cross-sections The magnitudes of the internal forces in which are considered to be excessively large or structural elements are rarely constant along unsuitable in some other way, modification of their lengths, but the internal forces at any the overall form of the structure will be cross-section can always be found by making undertaken so as to redistribute the internal an ‘imaginary cut’ at that point and solving the forces. Then, the whole cycle of analysis and free-body-diagram which this creates. element-sizing calculations must be repeated. Repeated applications of the ‘imaginary cut’ If a structure has a geometry which is stable technique at different cross-sections (Fig. and the cross-sections of the elements are 2.21), allows the full pattern of internal forces sufficiently large to ensure that it has adequate to be evaluated. In present-day practice these strength it will not collapse under the action of calculations are processed by computer and the loads which are applied to it. It will the results presented graphically in the form of therefore be safe, but this does not necessarily bending moment, shear force and axial thrust mean that its performance will be satisfactory 20 diagrams for each structural element. (Fig. 2.22). It may suffer a large amount of  Structural requirements separate issue and is considered separately in the design of structures. 2.5 Conclusion In this chapter the factors which affect the basic requirements of structures have been reviewed. The achievement of stable equilibrium has been shown to be dependent largely on the geometric configuration of the structure and is therefore a consideration which affects the Fig. 2.22 A structure with adequate strength will not determination of its form. A stable form can collapse, but excessive flexibility can render it unfit for its almost always be made adequately strong and purpose. rigid, but the form chosen does affect the efficiency with which this can be accomplished. So far as the provision of adequate strength is deflection under the action of the load and any concerned the task of the structural designer is deformation which is large enough to cause straightforward, at least in principle. He or she damage to brittle building components, such must determine by analysis of the structure the as glass windows, or to cause alarm to the types and magnitudes of the internal forces building’s occupants or even simply to cause which will occur in all of the elements when the unsightly distortion of the building’s form is a maximum load is applied. Cross-section shapes type of structural failure. and sizes must then be selected such that the The deflection which occurs in response to a stress levels are maintained within acceptable given application of load to a structure limits. Once the cross-sections have been depends on the sizes of the cross-sections of determined in this way the structure will be the elements6 and can be calculated once adequately strong. The amount of deflection element dimensions have been determined. If which will occur under the maximum load can the sizes which have been specified to provide then be calculated. If this is excessive the adequate strength will result in excessive element sizes are increased to bring the deflection they are increased by a suitable deflection within acceptable limits. The amount. Where this occurs it is the rigidity detailed procedures which are adopted for requirement which is critical and which element sizing depend on the types of internal determines the sizes of the structural force which occur in each part of the structure elements. Rigidity is therefore a phenomenon and on the properties of the structural which is not directly related to strength; it is a materials. 6 The deflection of a structure is also dependent on the properties of the structural material and on the overall configuration of the structure. 21