Basic concepts. The Theory of Structures' is concerned with establishing an understanding of the behaviour of structures such as beams, columns, frames. II deals with statically indeterminate structures and contains ten chap- section B examinations in "Theory of Structures", has been added. Basic Theory of Structures provides a sound foundation of structural theory. This book presents the fundamental concepts of structural behavior. Organized into.

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Theory of the engineering structures is a fundamental science. Statements and meth- As this takes place, what is the role of classical theory of structures. Theory of Structures by BC Punmia. Identifier Identifier-arkark:// tsh The word structure has various meanings. ➢ By an engineering structure we mean roughly something constructed or built. ➢ The principal structures of concern.

Peter Marti. First published: Print ISBN: KG, Berlin. About this book This book provides the reader with a consistent approach to theory of structures on the basis of applied mechanics. It covers framed structures as well as plates and shells using elastic and plastic theory, and emphasizes the historical background and the relationship to practical engineering activities. This is the first comprehensive treatment of the school of structures that has evolved at the Swiss Federal Institute of Technology in Zurich over the last 50 years.

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The many worked examples and exercises make this a textbook ideal for in-depth studies. Each chapter concludes with a summary that highlights the most important aspects in concise form.

Specialist terms are defined in the appendix. There is an extensive index befitting such a work of reference. The structure of the content and highlighting in the text make the book easy to use. The notation, properties of materials and geometrical properties of sections plus brief outlines of matrix algebra, tensor calculus and calculus of variations can be found in the appendices.

This publication should be regarded as a key work of reference for students, teaching staff and practising engineers. Its purpose is to show readers how to model and handle structures appropriately, to support them in designing and checking the structures within their sphere of responsibility.

Reviews "Theory of Structures" is an excellent textbook. Clough, H. Martin, and L. Topp's of a paper on the "Stiffness and Deflection of Complex Structures". This paper introduced the name "finite-element method" and is widely recognised as the first comprehensive treatment of the method as it is known today.

Fazlur Khan designed structural systems that remain fundamental to many modern high rise constructions and which he employed in his structural designs for the John Hancock Center in and Sears Tower in Horizontal loads, for example wind, are supported by the structure as a whole. About half the exterior surface is available for windows. Framed tubes allow fewer interior columns, and so create more usable floor space. Where larger openings like garage doors are required, the tube frame must be interrupted, with transfer girders used to maintain structural integrity.

This laid the foundations for the tube structures used in most later skyscraper constructions, including the construction of the World Trade Center. Another innovation that Fazlur Khan developed was the concept of X-bracing, which reduced the lateral load on the building by transferring the load into the exterior columns. This allowed for a reduced need for interior columns thus creating more floor space, and can be seen in the John Hancock Center.

The first sky lobby was also designed by Khan for the John Hancock Center in The development of finite element programs has led to the ability to accurately predict the stresses in complex structures, and allowed great advances in structural engineering design and architecture. In the s and 70s computational analysis was used in a significant way for the first time on the design of the Sydney Opera House roof.

Finally, the properties of the material used to build the structure will enter the equations relating the strain in a member to the applied stress. These equations were, effectively, known by Navier , or more cer- tainly by Barre de Saint-Venant. Of course, although the equations are essentially simple, individual pieces of mathematics may become difficult. By the end of the nineteenth century, indeed, many problems had been for- mulated completely, but the equations were so complex that they could not usually be solved in closed form, and numerical computation was impossibly heavy.

This situation gave an exhilarating spur in the twentieth century to the development of highly ingenious approximate methods of solution, and also to a fundamental reappraisal of the whole basis of the theory of structures. That the equations may not be a good reflexion of reality, so that their solutions do not actually give the required information, is only slowly being realized.

This book is concerned with the basic equations and the way in which they should be used. The equations themselves have an intrinsic interest, as does their application to a whole range of structural problems.

The later chapters of this book give a tiny sample, from the almost infinite number of topics in the theory of structures, for which the results are important, or startling, or simply amusing. The theory of structures A structure from the Latin struere is anything built: Before the Renaissance all these structures were built without calculation, but not without 'theory', or what today would be called a 'code of practice'. Mignot's statement in , at the expertise held in Milan, that ars sine scientia nihil est practice is nothing without theory , testifies to the existence of a medieval rule-book for the construction of cathedrals; the few pages of a builder's manual bound in with the book of Ezekiel in about BC show that there were yet earlier rules.

These rules, for construction in the two available materials, stone and wood, were essentially rules of proportion and, as such, are effectively correct. Stresses in ancient structures are low, and this has helped to ensure their survival.

The stone in a medieval cathedral, or in the arch ring of a masonry bridge, is working at a level of one or two orders of magnitude below its crushing strength. Similarly, deflexions of such structures due to loading are negligibly small although the movements imposed by warping of the material or by slow movements of foundations may often be seen.

What is necessary is that ancient structures should be of the right form; a flying buttress must be of the right shape, an arch ring must have a certain depth, and a river pier must have a minimum width. Correct form is a matter of correct geometry, and the ancient and medieval rules of proportion were established empirically to give satisfactory designs. The introduction two centuries ago of iron, and then steel, as a structural material generated new structural forms, such as the large-span lattice girder, or truss, and the high-rise framed building; and the advent of reinforced concrete made possible such attenuated construction as the thin shell roof.

Further, higher stresses will lead to larger elastic deformations of the components of the structure; these may be small in themselves, but overall deformations may now be of importance. This higher 'performance', coupled with the increase in complexity of structural form, dictated the need for a structural theory no longer based on empirical rules of proportion — a theory which could assess whether or not a particular construction would fulfil its design criteria.

There are three main criteria which must be satisfied if a structure is to be successful and a large number of other minor criteria that the modern designer will also take into account ; they are those of strength, stiffness and stability. The homely example of a four-legged table may make clear the three aspects of performance that are being examined. The legs of the table must not break when a normal weight is placed on top, and the table top itself must not deflect unduly.

Both these criteria will usually be satisfied easily by the demands imposed by a dinner party. Finally, the stability criterion may be manifest locally, or overall. If the legs of the table are slender, they may buckle when the overall load on the table is increased. Alternatively, if the legs are not at the four corners, but situated so that the top overhangs them, then placing a heavy weight near an edge may result in the whole table overturning. I The strength of beams to Galileo was concerned with the fracture strength of beams.

His structure was the simple cantilever, with a value of bending moment at the root of the cantilever calculable by statics; in modern terms, he wished to evaluate the maximum value of the bending moment that could be imposed on a beam made of material having a known tensile stress at fracture.

This is a problem which falls into what is now known as 'the strength of materials'; that is, it is concerned with the calculation of internal stresses in a structural member and the relation of those stresses to an observed limiting strength. Galileo deduced correctly that the strength of a beam of rectangular cross- section is proportional to the breadth and the square of the depth of the beam; he assumed that, at fracture, the whole of the cross-section was in a state of uniform tension, so that the 'neutral' axis lay in the surface of the beam.

There were numerical mistakes in this analysis, repeated by James Bernoulli , and it fell to Parent to derive the first proper account of the position of the neutral axis in elastic bending. Parent's work was not well known, and Coulomb is often credited with the solution of the elastic problem; he does indeed use explicitly a linear stress-strain relationship to derive a linearly varying stress distribution over a cross-section in flexure.

However, Coulomb, like Galileo, was interested in the problem of fracture, and a full grasp of the proper basis of elastic analysis is only glimpsed in his work. Navier gave in with some correction and much expansion by Saint- Venant in a full account of the elastic bending of beams, that is, of the determination of the elastic stresses in a cross-section resulting from the flexure of that section by a specified bending moment.

This part, and it is only a part, of the general structural problem had therefore been solved — the criterion of strength could be examined. Daniel Bernoulli demonstrated in that the resulting elastic curve gave minimum strain energy in bending, and he proposed to Euler that the calculus of variations should be applied to the inverse problem of finding the shape of the curve of given length, satisfying given end-conditions of position and direction, so that the strain energy was minimized. Euler made this analysis of the 'elastica' in , and his work is summa- rized in chapter 6.

Euler found nine separate classes of solution, and the first class, which deals with very small excursions from the linear form, is of great practical importance.

Euler showed that the excursions were sinusoidal, and that they could be maintained only in the presence of a calculable value of the axial load; in practical terms, the elastica buckles in the presence of the 'Euler buckling load'.

Lagrange gave the first satisfactory account of higher buckling modes.


Thus, provided that the values of the internal structural forces were known say, the values of the bending moments from point to point of the structure , then each of the three main structural criteria strength, stiffness, stability could be examined with some confidence by the time of the first quarter of the nineteenth century. If in fact the internal forces can be found at once from the equations of statical equilibrium, then the structure is statically determinate.

Navier treated also statically indeterminate beams. He found that equa- tions of statics alone may not be sufficient to determine the internal forces. In order to calculate these internal forces for a hyperstatic structure, the two remaining groups of equations must be used; statements of compati- bility of deformation must be made, and some material laws stress-strain relationships, effects of temperature must be postulated.

Thus, for a two- dimensional loaded beam resting on two simple supports, the reactions on those supports can be found by statics. If the same beam rests on three supports, however, only two equations are available to determine the values of three reactions. In order to proceed with the analysis, the very small elastic displacements of the beam must be examined, and — most importantly — boundary conditions must be specified that the supports are rigid, for example.

All this is necessary, not to compute displacements which will be determined by the analysis, but which may or may not be of interest , but in order to solve the main structural problem, that of finding the internal forces in the hyperstatic structure resulting from given external loading.

The stresses arising from these forces can then be calculated, and the strength criterion examined. Indeed, it was Navier who formulated clearly the elastic method of design, which is based on the actual working values of the internal forces; the resulting internal stresses should not exceed a proportion of the limiting stress for the material.

Now for the three-dimensional table only three appropriate equations of overall equilibirum can be written; the forces in the legs of a tripod can be found from statics, but not those in the legs of the four-legged table.

To calculate these, the flexure of the table-top and the elastic compressions of the legs must be taken into account, and the problem at once becomes extremely complex. A finite-element package may be able to deal with the very large number of equations, but neither the computer nor the engineer making hand calculations will have exact knowledge of the boundary conditions; both will probably assume, unthinkingly, that the table legs are of the same length and that the floor is completely rigid.

A real table has legs of unequal length and stands on an uneven floor. Its 'actual' observable state is an accidental product of its history — it will be standing on three of its four legs, and the fourth will be unloaded. At any moment, however, a small shift of loading may rock the table or a passing waiter may nudge it to a new position so that the unloaded leg now carries weight, while one of the other three legs is relieved.

All that can truly be said of the table is that the load in a leg lies somewhere between zero and a value calculable by simple statics. What is certain is that the sum of the forces in the four legs must exactly equal the weight of the table and its imposed load, but to ask for the value of the 'actual' load in an individual leg is meaningless; there are an infinite number of equilibrium states for a hyperstatic structure subjected to a given loading.

Of the three groups of master equations, those of equilibrium are the most exact. They would appear to be perfectly exact, but they are in fact influenced marginally by the assumptions made by a structural engineer. For example, the leg of a table will be represented by a line for the purpose of calculating the load in that leg, and the precise way in which this is done affects the calculations.

The second group of equations, those defining the material properties, are less exact. The elastic moduli for steel do not show much variation, but the yield stress varies with very small changes in the chemical composition.

Concrete shows a much wider variation of both elastic and strength properties, and in addition is markedly time-dependent; similarly, timber is not only anisotropic but it has variable properties from section to section. However, it is the third group of equations, those of compatibility and particularly the boundary conditions which may be in many cases as for the four-legged table essentially unknowable.

Indeed, it became clear in the s, with the publications of the Reports of the Steel Structures Research Committee, that the stresses in real structures bore very little relation to those calculated confidently by the elastic designer. It was this observation which led to the development of plastic methods of design. Thus glass, which would shatter, is not used to form a load- bearing structure; wrought iron or steel, however, can suffer a certain amount of permanent deformation, as can timber, reinforced concrete and, as it turns out, masonry.

Such materials allow internal forces in a structure to redistribute themselves; as loads are slowly increased, their final collapse values are predictable, and reproducible, with spectacular accuracy. The small imperfections of fabrication and erection of a hyperstatic structure, which alter so markedly the elastic distribution of internal forces, have no effect on ultimate carrying capacity. Figure 1. Application of the three master equa- tions leads to the conventional elastic solution sketched in fig.

This calculation relies, of course, on the assumption that the three supports are completely rigid, or, at least, if they do settle, then they all settle by the same amount. Any very small differential settlement, which must inevitably occur in practice, will have a large effect on the computed values of the bending moments.

In particular, the largest bending moment at the central prop may be either reduced or increased, but it is on this single value that the elastic design of this uniform beam would be based. The plastic designer imagines a hypothetical increase in loading; in fig. Whatever the 'actual' bending-moment distribution one value somewhere in the beam will finally exhaust the carrying capacity of the cross-section — a plastic hinge will form in a steel beam.

This mechanism is not at all dependent on the order of formation of the hinges, and so is not influenced by those practical imperfections which so alter the elastic solution.

The bending-moment diagram for the continuous beam which is collapsing by the mechanism of fig. More precisely, the bending moments in fig. It must be emphasized that this increase is hypothetical.

Theory Of Structures

An alternative way of regarding the design process is to note that figs. The elastic designer believes fig. Thus, for this simple example, the 'elastic' designer will note that the largest bending moment in fig.

The 'plastic' designer in effect goes through the same process, but uses now fig. The factor of 1. Ancient structures of stone or wood have very much larger strength factors, but their behaviour is governed by other criteria as has been noted above. It is not necessarily correct, however, to identify a factor such as 1. A clear definition is needed of what constitutes, and what does not constitute, a structural failure, when related to the design of that structure.

Destruction of a building by bombing in time of war represents a class of loading which lies outside the normal brief of the designer although at another time a specific design of an air-raid shelter may be needed , and such destruction is not a true structural failure.

Similarly, if the ultimate strength of a particular bridge is assessed at say tonne superimposed load, and as a consequence a tonne load limit is placed on the bridge, then the passage of a tonne load which actually causes collapse corresponds to wilful destruction rather than structural failure. It is always possible, in fact, to imagine loading conditions which will cause collapse, but it is usually wrong to group those conditions with the 'normal' types of loading, and to pursue quasi-probabilistic calculations along the tail of some normal or skew distribution.

The empirical assignment of values to load factors in this way is sensible; in the absence of precise information it is right to make use of experience. But it is wrong to forget that the numerical work has been arranged empirically, and to come to believe that the values of load factor found to give good practical designs actually correspond to a real state of loading. The collapse mechanism of fig. If the design is based on fig. Neither bending-moment distribution, however, can be said to represent the 'actual' state of the beam, which is essentially unknowable.

If the beam happens to be stiff, so that any deflexion criterion is easily satisfied, then either estimate will do. However, if the flexibility of the beam begins to govern the design, then the designer would like some guidance as to how to proceed. Such guidance cannot be given — or, rather, the designer should be aware that any estimate of deflexion may not be observable closely in the real structure. Ways of making estimates of deflexions for large and complex structures are discussed in chapter 3.

The problem is equally acute if stability calculations are pursued. For example, if the forces acting on the ends of a member of a structure are known, then that member may be checked for, say, lateral-torsional buckling. A range of such equilibrium distributions should therefore be considered if the stability criterion is likely to be of significance to the design.

Virtual work The theory of structures deals with the mechanics of slightly deformable bodies. The 'slight' deformations are such that, viewed overall, the geometry of the structure does not appear to alter, so that, for example, equilibrium equations written for the original structure remain valid when the structure is deformed.

A familiar example for pin-jointed trusses arises in the resolution of forces at nodes; the inclinations of the bars are assumed to remain fixed with respect to a set of reference axes. For beams and frames, deflexions within the length of a member are 'small' compared with the length of that member, and so on.

Thus when overall equilibrium equations are written for slightly de- formable structures they are identical with those obtained by rigid-body statics. The use of the equation of virtual work to obtain such relations between the external forces acting on rigid bodies can be traced back to Aristotle BC , to Archimedes BC with his formulation of the laws of the lever, and, in more recent times, to Jordan of Nemore in the thirteenth century AD. The insight obtained by the use of virtual work in the study of rigid bodies is deep, but the application is straightforward.

In effect, the simplifying concept of an energy balance is used to obtain rela- tions between external forces by the study of a small rigid-body displacement which need not be a possible displacement for the structure in question — hence 'virtual'. The equation of virtual work is altered profoundly when the body being studies suffers slight deformations. Not only will external forces acting on the body be involved; account must somehow be taken of the internal forces in equilibrium with those external forces.

In the general application of the equation of virtual work, full use is made of the implication of the words 'in equilibrium with'; the internal forces may be any one of the infinitely many sets which satisfy the equilibrium equations.

There has been some misunderstanding on this point. As a single example, Sokolnikoff states the theorem of virtual work in terms of an equation between the work done by the surface forces and the change in internal strain energy.

As such, he remarks that 'the Theorem of Virtual Work and that of Minimum Potential Energy are seen to be merely different ways of stating the same principle'. There seems, therefore, to be no particular virtue in retaining a separate concept of 'virtual work', and indeed both the theorem and the remark are omitted from Sokolnikoff's second edition of In general it may be noted that any presentation of virtual work which involves material properties elastic constants and the like is at best a highly specialized formulation.

The essential feature of the broad statement of the principle of virtual work is that it is not concerned with material properties, but with the other two master equations of the theory of structures. The principle relates two completely independent statements about a structure, namely an expression of equilibrium of forces and an expression of compatibility of deformation.

There is no requirement that the deformations are those which arise from the forces if they do so arise, and are elastic, then the principle reduces to that of Minimum Potential Energy, as noted by Sokolnikoff. In fig. Pieu 2. Thus the internal forces P will be identified with bending moments M, and internal deformations with changes of curvature K and also discrete 'kinking' of a member by hinge rotations 6.

Thus the bending moments in the member are not determined uniquely by the equilibrium equation 2. Since the whole of equation 2. Thus a free bending- moment diagram for the fixed-ended beam of fig.

The complete bending-moment diagram results from the superposition of figs. The functions ar are linear functions of the coordinate x which is used to define the location of the sections of the frame, so that equation 2.

Originally straight member deformed with elastic curvatures K and hinge discontinuities 6. In words, equations 2. Bending moments M are in equilibrium with external loads w. Bending moments Mw are in equilibrium with external loads w.

The last of these three statements implies that the bending moments arMr are self-stressing or residual moments; these cannot arise in statically determinate structures. These deformations cause lateral deflexions y of the member. Then, with the usual assumption of small deformations, and the deflexions y are said to be compatible with the hinge discontinuities 6 and the curvatures K. On the left-hand side the summation includes all point loads W, and the integral extends over all the distributed loads; on the right-hand side the summation includes all hinge discontinuities 6, and the integral extends over the rest of the frame.

Equation 2. It may be repeated that there is no necessary interdependence of the equilibrium statement W,w;M and the compatibility statement y;6,K.

If an unloaded structure is given an arbitrary set y, 9, K of compatible deformations, then equation 2. Reactant lines are known to be straight, so that if residual moments THA, ms, rnj and niE are defined at the corners of the frame, the whole bending-moment diagram may be drawn. The question arises as to whether the values of the four residual moments can be assigned arbitrarily, or whether there might exist relationships between their values.

Figure 2. IUj Thus only three of the four residual moments can have arbitrary values; the fourth is then calculable from equation 2. It is of course known from general structural theory that the fixed-base arch form, such as a portal frame, has three redundancies, and these may be regarded as any three of the four moments m marked in fig. Simple plastic collapse will occur when enough plastic hinges have formed to turn the structure or part of the structure into a mechanism of one or by accident more than one degree of freedom.

Under the action of the working value of the loads, an equilibrium distribution of bending moment in the frame may be written cf. At collapse, in a mechanism with hinge rotations 0, the bending moments at the hinge positions will have values Mp , or, more precisely, Mp ,, since the values may be different at different positions. A mechanism of collapse of a simple portal frame is shown schematically in fig.

For the purpose of analysis, this actual mechanism is replaced by the 'virtual' mechanism of fig. Moreover, the sign of a hinge rotation will accord with the sign of the bending moment at that hinge, so that equation 2. Thus the fixed-ended beam offig.

Tree' bending moments of fig. The 'virtual' collapse mechanism is shown in fig. The reactant line has not of course been positioned arbitrarily in fig. Betti, Maxwell, Muller-Breslau, Melchers The equation of virtual work will be used to derive some of the familiar elas- tic theorems.

As usual, there is no necessary connection between the equilibrium statement Wk,M and the compatibility statement yk,K. Thus in fig. Then equation 3. Kdx, 3. If therefore the roles of figs. That Betti's theorem applies to statically indeterminate frames is clear from its derivation.

Maxwell's reciprocal theorem: Similarly, an elastic deflexion atj results at section i from the application of a unit load at section j fig. Inserting these two statements into equation 3. Influence lines According to Charlton, the reciprocal properties of linear statically deter- minate structures were known to Mohr.

Specifically, Mohr considered the simply supported beam of fig. Mohr's reciprocal theorem. Muller-Breslau's discussion of a statically indeterminate beam. This is, of course, a special statically determinate application of equation 3. Moreover, he concluded that the shape of the deflected beam sketched in fig.

Muller-Breslau extended these ideas to the statically indeterminate struc- ture. It is required to find, say, the reaction C due to the external load P. If equation 3. Extension of Muller-Breslau's principle. Once again fig. This is Muller-Breslau's principle, and it may be used to determine internal forces in a frame.

Once again, the deflected form of the beam in fig. It may be noted that imaginary internal discontinuities of this sort will produce linear influence lines for statically determinate structures. Such structures cannot sustain self-stress, and small imperfections, of manufacture or assembly for example, do not introduce internal forces.

The influence line for the reaction at support B is found by moving the support through a unit distance; see fig. The influence line for the shear force at X is found by introducing a unit lateral discontinuity and no discontinuity in slope ; see fig.

Since equation 3. All that is required is that the scale model should have flexural rigidities that are the same constant proportion from section to section as those of the original. Beggs proposed that, instead of imaginary deformations, real deformations should be imposed on a carefully made and properly scaled celluloid model.

Such a model can be cut from a plastic sheet of uniform thickness, the depths of the members being varied to ensure correct values of the flexural rigidities.

Theory of Structures

The required coefficients e. Observations of this kind can be very accurate, and it is possible to obtain acceptable estimates if the frame is cut from cardboard and tested pinned to a drawing board.

Melchers The equation of virtual work may be used to calculate the elastic deflexions of structures by means of the device of a unit dummy load. The elastic bending moments in the frame are denoted Ji. Then, with fig. This is the conventional way of using virtual work for deflexion calculations; the data required are the actual elastic solution under the real loading, and any equilibrium solution for the unit dummy load.

Melchers proposed in an inversion of this process. Second, the elastic moments Jt must be expressible as any equilibrium distribution Mw on which is superimposed an appropriate self-stressing distribution m. Thus equation 3. Equation 3. Thus the central deflexion of a fixed-ended beam carrying a uniformly distributed load, fig. A uniformly distributed load produces b elastic bending moments or d collapse bending moments.

A central point load produces c 'free' bending moments or e elastic bending moments. Such a sequence cannot be followed for a hyperstatic structure.

The calculations require the simultaneous solution of all three groups of the master structural equations; the work is much heavier, and considerable attention has been given in recent years to the derivation of economical computational schemes. It is of interest that approximate calculations, which abandon the attempt to solve all the equations simultaneously, can lead to meaningful estimates of deflexions.

Thus the calculations are much simplified if equations of compatibility are put on one side, and attention concentrated on the equilibrium equations; similarly, deformation patterns can be assumed, without attempting to satisfy the equilibrium equations. Thus, in fig. The actual bending moments in the body taken to be a framed structure for this discussion are Jt, and the corresponding elastic curvatures are K.

Figure 3. Thus 2 —dx, 3. As a very simple example, the fixed-ended beam of fig. Table 3. For example, the sketched deflexions of fig. Introducing equation 3. The equilibrium set W,Jt of fig. Jettied construction A standard undergraduate problem in the theory of statically determinate structures concerns the 'optimum' way to lift a uniform beam. The corresponding bending-moment diagram is sketched in fig.

Using b Fig. A beam on two supports. A two-storey timber-framed house. A two-storey house is shown schematically infig. Thus, infig. The bending- moment diagram for the beam has the same general features of that shown infig.

First-floor beam of fig.

The optimum amount of jetty may be determined by equating expressions 4. From equation 4. Knee braces were often used in timber construc- tion to help support long-span beams, and, as for jettied construction, their use can lead to a great reduction in stresses. Figure 4. In either case the essential load bearing member may be represented as the beam on four supports sketched in fig.

There are several assumptions implicit in the sketch of fig.

Durham Cathedral: Great Dormitory. Durham, however, the beam is acted upon by more or less point loads transmitted by the purlins.

The walls provide support forces R, and these are shown as equal in fig. In the same way the braces are represented in fig. There are other equally important and essentially unjustifiable assumptions which are noted below.

However, a first analysis will be made on the basis of the sketch of fig. By analogy with the 'undergraduate problem' of the statically determinate beam, the effects of varying the value of a will be studied, and an attempt made to optimize the placing of the knee braces. St Albans Abbey: Beam on four supports. Bending-moment diagram for fig. As a is increased further the value of Ms increases above the relatively least value of equation 4.

A conventional elastic analysis will be made, and a further range of more or less doubtful assumptions will be built into the calculations.

For example, unless the working is to become excessively tedious, the beam must be taken to have uniform section throughout its length, and the basic elasticity of the wood will also be taken as uniform.

Both of these assumptions are clearly only very rough representations of eleventh- and fifteenth-century timbers. Further, it is as usual the boundary conditions which influence critically the results of any attempted elastic analysis; some decision must be made, in order to proceed with the calculations at all, about the possible displacements of the four support points R and S in fig. It has been noted already that symmetry has been assumed; any 'usual' elastic analysis a computer package, say will assume further, and without remark, that all four supports are rigid.

In fact, any small differential settlement of the supports will have a marked effect on the values of the bending moments in the beam. However, no assumption other than rigidity of the supports is really possible, since the support conditions are, in essence, unknowable.

They depend on such factors as compressibility of the raking braces, shrinkage of connexions between the timbers, possible decay of the wall plates, and so on. It is of interest that there is no value of a in the range 0. This 'undergraduate' optimum design does not depend so critically on the doubtful assumptions that have to be made for an elastic analysis, since the system is statically determinate.

Some typical bending-moment diagrams are sketched in fig. As a matter of interest, although not of present concern, it may be noted that in the range 0. Moreover, since a plastic solution to the present problem is independent of the essentially unknowable conditions of support of the beam, it has at least the relevance and applicability of the elastic solution given above.

Bending-moment diagrams for various values of a. Mmax Fig. End span of the beam of fig. Equations 4. Plastic collapse of the beam of fig.

The optimum value of a for the plastic analysis is 0. However, such braces would be long and might be uneconomical. At Durham the braces are positioned to provide support at a value of a roughly equal to 0. Thus the braces at Durham should ensure that the wall plates are largely relieved of direct bearing, the roof loads being transmitted down the braces to the feet of the wall posts.

Indeed it is evident for some of the roof beams at Durham that their ends are barely in contact with the walls. Beam subject to transverse load.

Thus both of the functions of equations 5. Equation 5. Had the beam been clamped at its ends, for example, then the value of R would not be known from statics, and an initially unknown bending moment acting at the left-hand end of the beam would also enter the calculations. It is shown later, in equations 5.

An essential requirement for the solution of equations 5. With Macaulay's notation, equations 5. Thus equation 5. Had separate bending equations been written for each of the n segments of the beam, then 2n constants of integration would have been involved.

Macaulay's method was taught at Cambridge and popularized by Case , and was of obvious help in the manual solution of problems in the bending of beams. Wittrick took up the matter in , and extended the analysis to deal with the problem of bending in the presence of axial load. He also studied by the same technique the question of beams on elastic foundations, and certain problems in shells.

The matter is not so straightforward when bending with axial load is considered. Figure 5. The beam of fig. The question to be answered is how equations 5. For the case of a uniform beam for which El is constant, both of equations 5. Thus the complementary function will be the same throughout the beam, but the constants b, c, d and e will be different for each segment.

Thus, from equation 5. Thus the complete general solution of a beam-column loaded transversely by a number of loads Fi, Fj, For the problem of fig. For the solution of the beam problem in the absence of axial load, equation 5. A more general approach is to start from the equation where p represents the intensity of any distributed transverse loading on the beam.

The most general complementary function for equation 5. The elastica The history of mathematics in the seventeenth century has been well studied, and indeed perhaps too much attention has been paid to the quarrel between Newton and Leibnitz, and their supporters, about the invention of the calculus. What is incontrovertible is the explosive influence that invention had on the development of applied mathematics and mechanics.

By the middle of the next century, however, progress had been so rapid that only professional mathematicians can now contribute to the history of the subject. Newton's fluxions will be understood by the historian, but Euler's calculus of variations moved the subject to a different level of learning. There is a corresponding gap in the history of the development of mathematics in the eighteenth century, with some notable exceptions, such as those provided by Truesdell.

Euler's equation is, in fact, simple and easily memorable. If equation 6. Daniel Bernoulli, one of perhaps a dozen mathematicians in the world who knew and appreciated what Euler had done, was quite capable of applying equation 6.

The difficulty was that Daniel Bernoulli, having obtained his fourth-order differential equation, could not solve it.This calculation relies, of course, on the assumption that the three supports are completely rigid, or, at least, if they do settle, then they all settle by the same amount. Using b Fig. Navier, C. Thus the deflexion function is of the form of equation 9. If this happens, then the frame is said to have shaken down under its specified set of variable loads.

If the resulting deflexions are superimposed, it is seen that the tip of the cantilever will move both horizontally and vertically under the action of a purely vertical load. The approximation is analagous to the replacement of a distributed load by a number of concentrated loads. Simple plastic collapse will occur when enough plastic hinges have formed to turn the structure or part of the structure into a mechanism of one or by accident more than one degree of freedom.

Similarly, deflexions of such structures due to loading are negligibly small although the movements imposed by warping of the material or by slow movements of foundations may often be seen.

LATISHA from Port Saint Lucie
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