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 aracer.mobi 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.

Personal information is secured with SSL technology. Free Shipping No minimum order. Description Basic Theory of Structures provides a sound foundation of structural theory. This book presents the fundamental concepts of structural behavior. Organized into 12 chapters, this book begins with an overview of the essential requirement of any structure to resist a variety of loadings without changing its shape. This text then examines the application of the laws of statics to structures as a means of determining the external reactions induced at supports due to loading. Other chapters consider the dependence of stress components on the choice of reference plane. This book discusses as well the method of determining the internal forces in the bars of a truss, which depends upon applying the conditions of equilibrium. The final chapter deals with the variety of factors affecting the strength of concrete. This book is intended to be suitable for civil engineering students.

For the purpose of the present discussion, equation 7. At collapse of the bent, fig. A single equation of statics may be written which relates the quantities marked in the figure: There is thus a profound difference between the plastic analysis of this simple space frame and corresponding analyses of plane frames.

Figure 7. The third equation of the theory of structures must be used: The extra necessary information is provided by the normality condition of plasticity theory. Collapse mechanism for the right-angle bent offig.

However, it may be noted that, just as for the plane frame, the collapse load, equation 7. Thus the single equilibrium equation 7. The same bent under different loading. The flow rule associated with equation 7. The difficulty with the problem of fig. As a first trial, the mechanism of fig. From equations 7. A statical examination of the frame shows at once that the yield condition, equation 7.

Collapse mechanism for fig. D, is quickly seen to be incorrect. There is no other arrangement of hinges, that includes a hinge at C, which gives a mechanism of the usual kind of one degree of freedom. The actual mode of collapse is one which involves hinges at the three locations A, C and D; see fig. At each of these hinge points two degrees of freedom are permitted; since the original frame had three redundancies, the resulting mechanism, viewed purely as a mechanism, appears to have three degrees of freedom.

However, the flow rule must be obeyed at each hinge, and exactly the right number of equations is obtained for the solution of the problem. A count of the variables involved shows how this comes about. If now one extra plastic hinge were inserted into this 'regular' collapse mechanism, then four extra unknowns M,T,09y would be introduced into the problem.

However, the yield condition and the flow rule may be written for the extra hinge, and by taking moments about two axes through the hinge, two extra equilibrium equations may also be written. Unlikely as it may seem, therefore, the 'overcomplete' mechanism of fig. It led to an upper bound on the value of the collapse load, that is, it gave an 'unsafe' estimate of the strength of the frame. Neverthe- less, assumed patterns of hinge rotations often lead to reasonably accurate estimates of strength.

If, for example, the value of Op in fig. If the hinges form according to the circular criterion, equation 7. Yield surface for a plastic hinge formed under bending and thrust. The yield surface could be imagined in three-dimensional space, and the conditions of normality and expressions for the flow rule would still hold — as they would in a hyperspace of four dimensions, which would be needed if the effect of a single axial load had also to be considered at a hinge.

The designer of plane frames is accustomed to making allowance, if necessary, for the effect of axial load on the formation of 'one-dimensional' plastic hinges. If such a hinge forms under the combined action of a bending moment M and a thrust P, then the yield condition takes the form -'- where Po is the value of the 'squash load' in the absence of bending. The exponent n in equation 7.

In either case, the general form of the yield surface is as sketched in fig. In the 'engineering' analysis of frames, the values of full plastic moment at the hinge points are simply reduced from Mo in accordance with equa- tion 7. However, the analysis appears to be no longer strictly 'one-dimensional'; the yield surface of equation 7.

It should be possible, therefore, to find collapse mechanisms of an unusual kind, in which hinges, formed under combined bending and axial load, give rise to both bending and stretching deformation. Such a hypothetical and, as will be seen, impossible mechanism for the simple portal frame of fig. Hypothetical but impossible collapse mechanism for the rectangular portal frame.

Thus, from the normality condition indicated in fig. Such ratios are not possible with the yield condition of equation 7. This is not a matter of a misleading scale in the plot of fig. Mechanisms such as that sketched in fig. While the conclusion is general, it is sharpened for the usual case in which the depths of structural members of plane frames are considered to be infinitesimal compared with their overall dimensions, so that centre-line geometry is used in the analysis.

In this case only infinitesimal axial movement is possible; reduced values of plastic moments can be calculated in the usual 'engineering' way from equation 7. The absolute minimum-weight design of frames The word 'frame' or 'beam' implies, as usual, that the primary structural action is that of bending. Secondary actions, such as those of shear and of axial load, may have to be taken into account in design — certainly their effects must be checked.

However, the first task of a designer is to establish a distribution of bending moments for the structure. Thereafter, calculations may proceed in one of two basic ways. If a plastic design is being made, then each member will be designed so that its full plastic moment just exceeds, at each critical section, the value of the bending moment established by the designer for that section. If an elastic approach is used, then the stresses at each section must be kept within certain specified limiting values.

The initial bending-moment distribution established by the designer is, of course, not unique; it is one of infinitely many that can be found for a hyperstatic structure. However, as has been seen in chapter 1, both the elastic and the plastic approach outlined above are, by the plastic theorems, safe. The plastic design is, in general, direct, whereas the elastic design is iterative, since the bending-moment distribution is affected by the elastic properties of the members, and these are not known a priori.

Prismatic members are commonly used in steelwork design, since it is inconvenient, except for heavy fabricated construction, to vary the cross- section of a member within its length. The use of reinforced concrete gives more possibility of variation of cross-section, but in both steel and reinforced- concrete design some material will be 'wasted', since the full potential strength of a member whether assessed elastically or plastically will be realized at only a few cross-sections.

The minimum would in fact only be relative, since such a design would have been matched to a particular, and perhaps arbitrary, set of equilibrium moments. However, among this class of designs there will be one, and possibly more than one, equilibrium distribution which gives an absolute minimum weight. This 'absolute minimum' design may not be easy to fabricate, and is likely to be more expensive than a design which wastes material; however, the study is of interest for its own sake, and because it leads to a target against which a practical design can be judged.

The 'strength' will not be defined for the time being — that is, it could be measured in terms of full plastic moment or in terms of a limiting elastic stress. Had the beam been simply supported, then the bending moments Mw would be statically determinate fig. As it is, the bending moments Mw are the free bending moments for the beam, and the actual bending moments of fig.

Thus, from fig. Fixed-ended beam under general loading. Since the strength of the beam follows exactly the distribution of bending moments M, this implies that the section of the beam will vanish at two locations; in practice, some material must be provided to transmit shear forces.

From equations 8. The beam loaded as in fig. The results implied by equation 8. The minimum-weight design of any single-span beam is that sketched in fig. However, the basic result is independent of the material properties, and arises from the linear assumption of equation 8.

The result was obtained by mechanical differentiation of equation 8. As usual, the bending moments M may be related to the external loading by consideration of a compatible pattern of deformation.

Minimum-weight configuration for thefixed-endedbeam. The following theorem holds: However, the quantity Jo may of course be evaluated for any other equilib- rium distribution of bending moments M, from equation 8. Combining equations 8. Equation 8. The requirement of no discontinuity of slope at inflexion points is necessary to prevent extra terms arising in equation 8.

The minimum-weight theorem, as stated above, requires the bending mo- ments of the minimum-weight design to be conformable with a deflected form of the frame consisting of arcs of constant curvature. Deflexions of the minimum-weight propped cantilever. Minimum-weight design of a two-span beam. Since the signs of the bending moments must be the same as the signs of the curvatures everywhere in the beam, the reactant line is located immediately, fig.

The same uniqueness of solution is apparent for the propped cantilever, fig. This independence holds only for the simplest beams and frames. The two-span beam on simple supports, for example fig. It is possible to establish certain general geometrical relationships to help with the solution of such problems, and numerical techniques of solution have been established.

The general theorem, however, applies only to plastic structures, whereas the use of virtual work in establishing equation 8. Indeed, there is no necessity for the frame actually to deflect into a shape consisting of arcs of constant curvature. This displacement pattern was used in the equation of virtual work to establish regions for which the bending moments are 'hogging' and 'sagging', and to locate the inflexion points. The frame is then designed so that its strength at each section follows exactly the minimum-weight bending- moment distribution and is zero at the inflexion points.

For an elastic design, the strength constraint would be the attainment of a maximum limiting stress. If the beam were of rectangular section, for example, and of constant depth, then the width at each cross-section should be made proportional to the bending moment there — the limiting stress would be reached at the outer fibres of the beam at all cross-sections. Further, the total weight of the beam would satisfy the linear assumption of equation 8. These same arguments apply also with the introduction of a shape factor for the section to fully plastic design.

Inverse design of grillages A structure will in general be hyperstatic, so that its elastic analysis requires the simultaneous solution of all three of the master equations. A continuous beam, for example, may rest on several supports, and initially unknown redundant forces will act on the system.

Such a beam may carry different loads in adjacent spans, and, as a consequence, the cross-section of the beam may well vary from span to span while perhaps being prismatic within each span. There are no formal difficulties in the elastic calculations associated with this continuous beam; the loads, the section properties and the boundary conditions could be introduced into a computer program, for example, and the required results will be produced.

The design of such a beam, as opposed to its analysis, is not so straight- forward; the section properties cannot be introduced numerically into the calculations, since it is precisely the determination of the section or sections of the beam which is the object of the design process. Design proceeds, in fact, by trial and error, whether this process is done manually or by com- puter. A guess is made of the cross-sectional properties of the members; an analysis is made to determine stresses and deflexions throughout the beam; and finally a check is made as to whether the criteria of strength and stiffness are satisfied.

In this trial-and-error process, one design variable can be allowed, namely a scale factor. If, for example, all the section properties were increased in proportion, then all the numerical results would be decreased in proportion. However, this single 'degree of freedom' will allow only one of the structural criteria to be satisfied; if the stress condition governs, then the structure may be too stiff; if deflexions are given their limiting values, then the structure may be understressed.

Moreover, in either case limiting conditions will be reached at isolated portions of the structure, and it may be a complex problem to alter the sections of the design so that larger portions of the structure contribute to the satisfactory and economic performance of the whole. Simply supported beam. More importantly, tapering members may be used without difficulty, since the mathematical analysis of members of continuously varying section is not needed.

To continue the discussion with reference to beams, the steps in the inverse method are: The function w satisfies the boundary conditions of slope and deflexion. The bending moments M must satisfy the equilibrium equations; thus, since M and K are known, the flexural rigidity El may be determined. The depth of the beam may be fixed at each section so that the stress does not exceed the permitted value. Two simple examples will illustrate the above steps. From equation 9. If a beam of constant depth h were used, then the upper and lower fibres of the beam would have the same stress at all cross-sections.

Equations 9. Propped cantilever. If, for example, the beam is made from concrete having a rectangular cross- section of variable width b and constant depth h, then equation 9. The calculations will be inexact since the simple theory of bending has been used to derive the above equations, and no allowance has been made for the effects of reinforcement, shift of neutral axis, and so on.

Equation 9. The beam tapers parabolically from mm width at the centre to zero at each end; no allowance has been made for the effect of shearing force. This design is likely to be uneconomical since, as noted, a beam of constant depth will be fully stressed at only one cross-section. If the constriction of constant depth is maintained, then the beams of constant curvature discussed in chapter 8 may be used — for the particular case of the propped cantilever, the deformation pattern of fig.

The bending- moment diagram must be arranged to give the indicated inflexion point. However, this particular problem will not be discussed further, although it is straightforward to extend the ideas to multi-span beams, and to deal with complex patterns of loading — indeed, bending-moment distributions can be determined graphically rather than analytically. The examples of simple beams have been given as an introduction to the problem of the inverse design of grillages — it will be seen that analagous steps may be applied to such two-dimensional systems of beams.

When the flexural rigidity of a flat plate has been determined, to correspond with a given loading, the plate is replaced by beams forming a grillage of a specific shape. The material of the plate is 'concentrated' along the lines of the beams, each beam replacing some width of the notional plate.

With a large number of beams in the grillage, little error will be introduced by this replacement of the theoretically continuous plate by discrete beams; even if the number of beams is small, the calculations will be reasonably accurate.

The approximation is analagous to the replacement of a distributed load by a number of concentrated loads. A deflected form of the grillage w x,y is assumed which satisfies the boundary conditions. The principal curvatures are calculated from equations 9. Finally, the equilibrium equation must be satisfied. Some simple examples will make the process clear. A typical panel is shown in fig. Array of columns supporting a floor. I cxcy The last of equations 9.

From equations 9. The assumed deflexion function, equation 9. Beams must be provided at the edges of this rectangle to transmit the load into the columns. Possible design of a clamped circular grillage. It is convenient to work in polar coordinates; the principal curvatures are 9. Design of beams to cover a circular area. At the periphery, the total width of the beams is therefore 0.

These beams taper uniformly to zero at the centre of the circular area; the arrangement is shown in fig. The equations giving the lines of zero twist must be integrated numerically, and the resulting shape of the grillage is sketched in fig.

Designs of grillages to cover a square area with ends of beams a simply supported and b clamped. Figure 9. The calculation of deflexions plays no part in simple plastic design, and much of the work involved in the inverse method described above can be avoided. For grillages, for example, the beams may be run along orthogonal curves of zero twist, as for the inverse elastic design. However, once equation 9. The problem is to determine the orthogonal curves of zero twist, and this can certainly be done by assuming a deflexion function satisfying the boundary conditions.

If this function involves inflexion points then, as usual, the bending moments must change sign at these points. Alternative grillage to cover a circular area. Thus the deflexion function is of the form of equation 9. The principal curvatures are constant, that is, no inflexion points occur within the circle; equation 9.

One solution of the equilibrium equation 9. It may be noted that, if an arbitrary set of orthogonal curves is sketched, then the value of cot 2tp in equation 9. It must be checked that this function w satisfies the boundary conditions of the problem but, with care, almost any 'reasonable' form of orthogonal grillage may be sketched by the designer such as those of figs.

This is a remarkable feature of the plastic approach; such orthogonal systems have been used in reinforced-concrete construction, notably by Nervi The relation between incremental and static plastic collapse A structure will in practice be acted upon by a number of independent loads superimposed floor loads, snow, wind, crane loads, and so on.

Moreover, each load will vary between limits which will be specified — the wind may blow from east to west, or not at all, or from west to east. The engineer making an elastic check of a given design will arrange for calculations to be made separately for each load; for any particular cross-section, the value of each load is then chosen to give the greatest and least action at that section. The designer is then able to determine the range of stress at the cross-section due to all the loads, and to make an assessment, according to given elastic rules of design, of the safety of the structure.

By contrast, the engineer making an estimate of the static plastic carrying capacity of a framed structure must arrange all the loading in the way expected to be most critical before the single plastic-collapse calculation is made.

In practice there is no great difficulty in arriving at the worst combination of loads although even for the simple portal frame the most critical position of a crane crab is not immediately obvious. However, if loads on a frame do act randomly and independently within specified limits, then there is the possibility of incremental collapse of the frame. Moreover, it is shown below that incremental collapse is always a more critical phenomenon than the corresponding static collapse of the frame under the worst combination of steady loads having their maximum values.

Incremental collapse occurs in the following way. Under a certain com- bination of the independently varying loads it may be that a plastic hinge develops at one or more cross-sections of the frame. Any plastic de- formation that does occur will be constrained by those portions of the frame that remain elastic. Under a different combination of loads it is possible that another hinge or hinges could form at different cross-sections.

A yet further combination of loads could lead to yet further hinges, and so on. Now, although collapse will not occur under any of these individual com- binations of loads, it may be that had all the hinges formed simultaneously, then they would have given rise to a mechanism of collapse. If this is so, then incremental collapse is possible. A first combination of loads will lead to small irrecoverable hinge rotations; when the combination is removed, the frame is left slightly deformed.

A second combination will lead to permanent deformations elsewhere in the frame, and so on. If the loading combinations follow in a more or less cyclic order, then, after a very few such cycles, the frame will exhibit pronounced overall deformation and will look, in fact, as if it were failing by a plastic collapse mechanism of the usual static type. This sort of incremental collapse will occur if the magnitudes of the applied loads exceed certain values to be calculated.

For smaller values of the loads, then it is possible that some plastic deformation could occur in thefirstfew cycles of loading, but that after a time all further variation of load would be resisted purely elastically by the frame.

If this happens, then the frame is said to have shaken down under its specified set of variable loads. The limit factor that is, the shakedown limit , which divides incremental collapse from shakedown behaviour, can be calculated. In order to assign a numerical value to the limit, it is convenient to retain the idea of a load factor applied to the specified values of the loads.

This load factor is applied not to the current value of a load, but to the range within which that load acts. As the value of X is imagined to be slowly increased, then a state will be reached at which a single critical section or, by accident of geometry, two or more critical sections will just begin to yield under the most unfavourable combination of the independent loads. The corresponding value of X is the conventional safety factor of stress calculated by the elastic designer.

However, just as for the static case, the attainment of the yield stress at one or more cross-sections does not imply collapse; the value of X may be increased and yet the frame might still be able to shake down. If the value of X is above that of the conventional safety factor then yield is, by definition, occurring somewhere in the frame. Such yield implies in turn that, if all the loads were removed, the frame would no longer be stress free; the frame has been distorted, and residual self-stressing moments will have been induced.

The restriction on the value of X may be calculated as follows. The values of Mi will be functions of the loads; as individual loads vary between their prescribed limits, inequalities With the value of load factor applied to the limits on the loads, inequalities Then the necessary conditions for shakedown to occur are that The counterpart to the usual lower-bound theorem for static loading is the statement that if any set of residual moments m; can be found to satisfy inequalities Indeed, inequalities such as Such an analysis would automatically indicate which of the inequalities were 'critical' at the value As, and would thus indicate the incremental collapse mechanism.

However, just as for static collapse, an 'upper-bound' approach through the examination of mechanisms will lead to simpler calculations. Suppose that the correct mechanism for incremental collapse is known; that is, there is a pattern of hinges which, if they formed simultaneously, would give a mechanism of the usual type.

The virtual mechanism will now be examined which has the same hinge locations as the incremental collapse mechanism, but for which all deformation occurs only at the hinges, the members of the frame itself remaining straight. A simple mechanism of this sort is sketched in fig. In either case the product MP6 on the right-hand side is positive.

Equations Thus the basic equation for incremental collapse is derived: The relationship between the shakedown load factor Xs and the static load factor 2C for maximum values of the same loads may be determined by further examination of equation Incremental and static plastic collapse 99 The two equations are, of course, almost identical; the only difference is that all loads in equation If the two equations are divided by their load factors, and subtracted, then nin j Thus any load of fixed magnitude e.

Dead loads will of course play their part in the evaluation of the static collapse load factor Ac, but do not enter the shakedown analysis if it is done according to equation Second, the whole of the right-hand side of equation The difference between As and Ac arises only from loads which would produce a positive elastic bending moment at a section where there is a negative hinge rotation, or which would produce a negative elastic bending moment at a section where there is a positive hinge rotation.

Equation Elastic bending moments for afixed-baseportal frame, and possible collapse mechanisms. Incremental and static plastic collapse As an example, the collapse of an idealized fixed-base portal frame will be examined. Figure The values of V and H are supposed to vary randomly and independently within the ranges Also shown in fig. The following equations may be derived: It will be noted that the 'yield surface' for incremental collapse lies entirely within the corresponding yield surface for static collapse, as it must from inequality Interaction diagram for fixed-base portal frame, showing loading limits for static and for incremental collapse.

In fact, it is seen from the example of fig. The bending of a beam of trapezoidal cross-section Section 7 of Tredgold's Practical essay on the strength of cast iron deals with the strength and deflexion of cast iron when it resists pressure or weight — that is, it deals with the elastic bending of beams. In the edition Tredgold considered only beams of cross-section having two axes of symmetry, but in the second edition of he added a paragraph 85a - together with a footnote - which investigates a problem of bending with only one axis of symmetry: Hitherto we have only considered those forms where the neutral axis divides the section into identical figures; but there are some interesting casesa where this does not happen, such, for example, as the triangular section.

The triangular section considered by Tredgold is shown in fig. Tredgold's immediate problem is to locate the neutral axis MN, and he does not have Navier's formal statement of that the neutral axis must pass through the centre of gravity of the cross-section.

Navier was in fact the first to make this statement, but the result is implicit in earlier work, for example that of Parent and of Coulomb Both Parent and Coulomb were aware that the net longitudinal force on a cross-section in pure bending must be zero. This requirement of statical equilibrium, coupled with the idea of a linear distribution of strain, and hence of stress for a linear-elastic material, leads at once to the correct position of the neutral axis.

Tredgold was not aware of this requirement. A trapezoidal cross-section derived from an isosceles triangle. Thus the second moment of area about MN of the portion of the cross-section lying above MN, divided by the ordinate nd, must equal the equivalent expression obtained for the portion lying below MN.

Tredgold derives an equation expressing this equality: This equation is actually a quadratic in n, and Tredgold obtains the general solution. If the cross-section has two axes of symmetry then Tredgold's procedure leads to the correct answer, but the results are in error for the trapezoidal section of fig. Tredgold has a footnote to the effect that this result was first given in the Philosophical Magazine of Indeed, T.

Finally, for the triangle, T. Saint-Venant ascribes the principe singulier of equal 'strengths' above and below the neutral axis to Tredgold; as has been noted, Tredgold's principle happens to give the correct result for doubly symmetric sections. However, Tredgold is interested in the curious behaviour of his incorrect expression for the section modulus of the trapezium.

As the value of m is increased from zero, so that the cross-section is reduced, so the value of the section modulus increases. He notes that William Emerson first announced this seeming paradox in The edition of Emerson's The principles of mechanics discusses the problem of the bending of a beam of arbitrary cross-section, and in particular he deals with the section of fig.

However as remarked by Tredgold Emerson places his neutral axis at the base of the cross-section, and adopts in ! From this faulty analysis he deduces that if one-ninth of the triangular prism is cut away i.

Thus the removal of about one-eighth of the depth of the triangle results in an increase of elastic strength of about 9 per cent. This might be a fair measure of strength if the material were brittle, and liable to fracture in the presence of such a limiting stress — the material would not then, however, be suitable for general structural use.

A ductile material will exhibit some plastic behaviour, and if the 'true' strength of the cross-section is assessed by a calculation of the full plastic modulus, then common sense is not offended. The plastic modulus has value and this has its largest value, 0. The simple plastic bending of beams As will have been noted from the last chapter, the theory of bending of beams seems always to have given some difficulty.

The first key requirement of statics, that there should be no net thrust across a cross-section in pure bending, was recognized in the eighteenth century; but it was only in that Navier stated explicitly that as a consequence the neutral axis must pass through the centre of gravity of the cross-section.

However, even Navier was not aware of the consequences of a second statical requirement; moments of the forces acting on a cross-section lead to the notion of principal axes of bending.

Thus Navier gave wrong expressions for the bending of a rectangular cross-section about an axis not parallel to one of its sides, and it fell to Saint-Venant in his edition of Navier to discuss fully the question of principal second moments of area. Saint-Venant extended his analysis to cover non-linear behaviour of the material, but confined his work in this connexion to symmetrical cross- sections. Ewing again discussed only the rectangular section bent about a principal axis, and indeed most of the modern standard texts on plastic theory do not treat the unsymmetrical problem.

Brown seems to be the first to have recorded the general features of plastic unsym- metrical bending. He gives no specific solutions, but notes that the principal axes of elastic and plastic bending need not coincide this is a property also of sections having only one axis of symmetry , and that the principal axes of plastic bending are not necessarily orthogonal.

A rectangular cross-section bent about an inclined axis. In elastic bending the neutral axis and the axis of the applied bending moment do not necessarily coincide; similarly, the zero-stress axis for fully plastic bending makes an angle a in fig. The parameter z is marked in fig. Yield surface for the rectangular cross-section. The value of a marked infig. Further, it will be seen fromfig. General cross-section. The condition of no net axial thrust requires the zero-stress axis to divide the cross-section into two equal areas.

As the axis of the applied bending moment changes the zero-stress axis will shift; it will remain an equal-area axis but, for the general asymmetrical cross-section, it will not pass through a fixed point. Just as for the rectangular section, the components Mx and My of the bending moment M may be used as coordinate axes to plot a yield surface.

This yield surface must be skew-symmetric, since the numerical values of Mx and My will be the same if the angle 6 in fig. A typical yield surface of this kind is sketched in fig. The vector M to a general point P makes an angle 9 with the reference direction, while the normal to the curve makes an angle a corresponding with the angle of the zero-stress axis. The axes are indicated in fig. As sketched, the plastic principal axes are clearly not orthogonal — nor need there be any coincidence with an elastic principal axis.

The elastic orthogonality condition, not explored by Navier, arises from the assumed linear stress distribution across the section, whatever its shape. If this neutral axis is taken also to be the axis of x, then the bending moment about the axis Gy at right angles must be zero.

Skew-symmetrical yield surface for the general cross-section. Plastic principal axis Elastic principal axis Fig. Elastic and plastic principal axes of bending will differ for the general cross-section. By contrast, the plastic principal axis sketched in fig. Elastic principal axes for the angle section. The line CT must be perpendicular to a plastic principal axis, and there are, in general, two solutions of equation There will be a "strong" and a "weak" principal axis of plastic bending, but no condition of orthogonality between these axes is imposed by equation Elastic principal axes are sketched in fig.

If, for example, the angle section were used as a cantilever with one leg vertical, as indicated in fig. 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. Similar behaviour occurs when the section becomes fully plastic. The plastic principal axes do not coincide with the corresponding elastic axes, but they are certainly not horizontal and vertical.

Thus the cantilever beam of fig. The skew-symmetric yield surface for the unequal angle will be calculated for an idealized section composed of two thin rectangles. Angle in fully plastic state bent about the 'strong' axis. Geometrically, the centre of tension T and the centre of compression C cf. In fact, the bending moments Mx and My acting about the x and y axes may be computed for the fully plastic state of fig. However, the inclination 6 of the axis of the applied bending moment is fixed from equations Angle in fully plastic state bent about the 'weak' axis.

The value of 6 given by equation Table It will be seen that there are only small differences between the calculated directions of the strong plastic principal axes and the corresponding elastic values from the tables. The inclination of the axis is given by A second expression for tana2 may be found from the condition that the axis of the applied bending moment is parallel to the plastic principal axis.

Yield surface for the unequal angle. For the position of the axis shown in fig. The yield surface may be plotted for the cross-section, and the above analysis gives the information necessary to discuss plastic bending about any axis, not necessarily a principal plastic axis. The angle tested as a cantilever: At the point B the value of z fig. For the approximation of very thin rectangles this means that the value of Mx is indeterminate, and BAf and B1 A are straight lines parallel to the axis of Mx.

As noted above, equations The general shape of fig. The strong and weak axes are marked; the latter is, of course, normal to the yield surface.

The normality rule applies in general cf. If this angle were mounted as a cantilever carrying a tip load, and orientated so that bending occurs about an axis close to the weak axis, then both vertical and lateral movement of the tip would occur, in general.

If, however, the angle were mounted so that the value of a in the inset figure in fig. If, on the other hand, the test were arranged so that the value of a in fig.

Results from a test on a cantilevered angle. The results themselves are shown in a different way in fig. The predicted slope of the curve in the plastic region is obtained from the direction of the normal at the appropriate point of the yield surface of fig.

Leaning walls; domes and fan vaults; the error function Masonry may be regarded as an assemblage of dry stones or bricks or other similar material , some squared and fitted and some not, placed together to form a stable structure. Any mortar that may have been used will have been weak, and may have decayed with time; it cannot be assumed to add strength to the construction.

Stability of the whole is assured, in fact, by the compaction under gravity of the various elements; a general state of compressive stress can exist, but only feeble tensions can be resisted. This is the unilateral model of masonry; the material can resist compres- sion, but has zero tensile strength.

Further, compressive stresses in masonry structures are usually very low indeed, so that the material may be assumed, effectively, to be infinitely strong in compression. To add a final and impre- cise assumption, it is clear that individual components of a masonry structure must in fact possess tensile strength, even if the overall assemblage has none.

As an example, it is easy to envisage a dry stone wall in which the stones can indeed be lifted away, but which, in the absence of such interference, will retain its structural shape. The stones must, however, have a certain shape and size; an attempt to build a vertically sided wall from small particles sand will be unsuccessful.

A monolithic rectangular block, of height H and width b, may be tilted on its base until its centre of gravity is vertically above one corner; slight further movement will cause the block to topple. A tilted block of masonry. A tilted block for which the support force falls outside the middle third of the original width. A fracture has developed whose shape is to be determined.

The calculations are not so simple for unilateral masonry. At this condition, according to simple elastic theory, the left-hand bottom corner will be just free of stress, and the block will be supported by linearly increasing compressive forces along the bottom surface.

An 'actual' fracture cf.

A thin slice of masonry. A slight further inclination will transfer some of the 'bricks' just above the fissure to the passive pile below. The shape of the free surface in fig.

At the general section, distant X from the origin, the total weight W of the masonry is supported by a force again acting at the limit of the middle third of a base of reduced dimension Y. Using equation From equation The final results may be collected together: The dimensionless height h of a wall that is just becoming unstable is given, from the second of equations Thus the height H of the actual wall is related to the width b by that is 0.

These solutions may be applied to the discussion of the problem of leaning towers, of which perhaps the most famous is that of Pisa. There are, however, many other examples of such towers, both in Italy in Venice and the islands of the lagoon and in other parts of the world. Stress resultants acting on an element of a domical shell. As will be seen, this approach must finally be abandoned for the real dome made of unilateral material.

An element of a shell of revolution is shown in fig. Stress resultants for a hemispherical dome under its own weight, according to simple shell theory. I These expressions are sketched in fig.

Such tensile stresses are inadmissible for the unilateral masonry structure. If a hemispherical dome is to behave as dictated by the membrane analysis, then the tensile behaviour implied by the sketch of fig. For example, encircling chains could be used in an attempt to prevent the 'bursting' of the dome. In practice, the slight outward spread of the base of the dome, induced by the outward thrust of the masonry, will lead at once to meridional cracking, rising from the base and dying out twoards the crown.

Hoop stresses Ne of fig. A quasi two-dimensional arch formed by slicing a hemispherical dome along neighbouring meridians. Poleni noted that the cracks had already divided the dome into portions approximating half spherical lunes orange slices ; for the purpose of his analysis, he sliced the dome hypothetically into 50 such lunes.

He then considered the stability of, effectively, a two-dimensional arch composed of two lunes leaning against each other at the crown, as shown in fig. If such an arch were in itself stable, then he argued, correctly, that the whole dome, cracked or not, would also be stable. In order to confirm the stability of the arch, Poleni made use of Hooke's theorem of If this line lies within the material of the arch, then stability will be assured. From a drawing of the cross-section of the dome of St Peter's, Poleni calculated the weight of a lune, dividing it into 16 segments for this purpose and making due allowance for the weight of the lantern.

The line of thrust for fig. Collapse mechanism for the arch of fig. This shape indeed lay between the surfaces of the masonry. Poleni had established a solution for the dome for which the hoop stress is zero. He had found a way in which a spherical dome could satisfy the conditions of equilibrium, but in which the forces are no longer compelled to lie in a spherical surface as they are by the simple membrane analysis leading to equations Rather the forces follow the line of the 'hanging chain', and a certain thickness of masonry is necessary to contain this line.

It is of interest to examine the minimum thickness of spherical lunes which will just be stable. Portion PP of the dome near the crown remains coherent, while the hinges at P, Q and R ensure separation of the lunes so that no hoop stress can be generated. If the arch of fig. Least thicknesses for incomplete spherical domes.

For a full hemisphere the minimum thickness is just over 4 per cent of the radius. The results quoted above may be obtained from the statics of fig. Solutions are sought for which the hoop stress Ne is zero, but the dome is no longer assumed to be spherical; rather, the radius of curvature r, defining the generating curve, is allowed to be a variable.

Using the second of equations The rectangular coordinate system of fig. The 'perfect' shape of dome in which the form follows the line of thrust; the curve in the figure for the portion between 0,0 and 1,1 can be fitted within the circles of fig. It will be seen that xo is a scale factor, and may be thought of as some particular dimension of the dome say half the diameter at its base.

If such a profile can be found, then the complete conoid may be cut in two by a vertical plane passing through the axis, and the resulting half-shell used as a model for the real fan vault.

The shell of fig. Notation for the shape of the fan vault. Three 'perfect' shapes for the fan vault in which the form follows the line of thrust. Conditions at the edge of the fan are of interest. The horizontal thrust exerted by each vault has value WXQ, independent of the value of the constant C. The upper edges of the vault require horizontal and vertical forces as shown in order that overall equilibrium may be maintained.

Forces generated by the shapes of fig. A study was made by Anderson et al. Effectively, a thin membrane was proposed as a sort of parachute, and the governing design criterion was that, to avoid wrinkling of the membrane, the stresses everywhere should be tensile. The problem is an almost exact 'Hookean' analogue of that of the masonry vault and, although the loading conditions are different, the parachute proposed is remarkably like a complete fan vault entering the atmosphere point first.

At the widest part of the fan, the parachute edge must be reinforced by a stiffening ring; no solution exists for a free edge. This, again, is analagous to the compressive edge load required for the masonry fan vault. Ars sine scientia nihil est, Gothic theory of architecture at the Cathedral of Milan. The Art Bulletin, 31, Bernoulli, D De vibrationibus et sono laminarum elasticarum. Commentarii Academiae Scientiarum Petropolitanae, 13 —3 , Bernoulli, James Specimen alterum calculi differentialis.

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Explicationes, annotationes et additiones, Acta eruditorum Lipsiae. The exposed beam soffits were protected against fire by plaster. This mill at Belper was the world's first attempt to construct fireproof buildings, and is the first example of fire engineering. This was later improved upon with the construction of Belper North Mill , a collaboration between Strutt and Bage, which by using a full cast iron frame represented the world's first "fire proofed" building.

The Forth Bridge was one of the first major uses of steel, and a landmark in bridge design. Also in , the wrought-iron Eiffel Tower was built by Gustave Eiffel and Maurice Koechlin, demonstrating the potential of construction using iron, despite the fact that steel construction was already being used elsewhere.

During the late 19th century, Russian structural engineer Vladimir Shukhov developed analysis methods for tensile structures , thin-shell structures , lattice shell structures and new structural geometries such as hyperboloid structures. Pipeline transport was pioneered by Vladimir Shukhov and the Branobel company in the late 19th century.

Maillart noticed that many concrete bridge structures were significantly cracked, and as a result left the cracked areas out of his next bridge design - correctly believing that if the concrete was cracked, it was not contributing to the strength. This resulted in the revolutionary Salginatobel Bridge design. He went on to demonstrate that treating concrete in compression as a linear-elastic material was a conservative approximation of its behaviour.

Freyssinet constructed an experimental prestressed arch in and later used the technology in a limited form in the Plougastel Bridge in France in He went on to build six prestressed concrete bridges across the Marne River , firmly establishing the technology.

The possibility of creating structures with complex geometries, beyond analysis by hand calculation methods, first arose in when Alexander Hrennikoff submitted his D. Sc thesis at MIT on the topic of discretization of plane elasticity problems using a lattice framework. This was the forerunner to the development of finite element analysis.

In , Richard Courant developed a mathematical basis for finite element analysis. This led in to the publication by J. Turner, R. 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