CHAPTER IX.
=103. Lateral Wind Pressure on Trusses.=—The duties of a bridge structure are not confined entirely to the supporting of vertical loads. There are some horizontal or lateral loads of considerable magnitude which must be resisted; these are the wind loads resulting from wind pressure against both structure and moving train. In order to determine the magnitudes of these loads it is assumed in the first place that the direction of the wind is practically or exactly at right angles to the planes of the trusses and the sides of the cars. This assumption is essentially correct. There is probably nothing else so variable as both the direction and pressure of the wind. These variations are not so apparent in the exposure of our bodies to the wind, for the reason that we cannot readily appreciate even considerable changes either in direction or pressure. As a matter of fact suitable measuring apparatus shows that there is nothing steady or continued in connection with the wind unless it be its incessant variability. Its direction may be either horizontal or inclined, or even vertical, while within a few seconds its pressure may vary between wide limits. Under such circumstances the wind is as likely to blow directly against both bridge and train as in any other direction, and inasmuch as such a condition would subject the structure to its most severe duty against lateral forces, it is only safe and proper that the assumption should be made. The open work of bridge-trusses enables the wind to exert practically its full pressure against both trusses of a single-track bridge, or against even three trusses if they are used for a double-track structure. Hence it is customary to take the exposed surface of bridge-trusses as the total projected area on a plane throughout the bridge axis of both trusses if there are two, or of three trusses if there are three. Inasmuch as the floor of a bridge from its lowest point to the top of the rails or other highest point of the floor is practically closed against the passage of the wind, all that surface between the lowest point and the top of the rail or highest floor-member is considered area on which wind pressure may act.
Many experimental observations show that on large surfaces, greater perhaps than 400 or 500 square feet in area, the pressure of the wind seldom exceeds 20 or 25 pounds per square foot, while it may reach 80 or 90 pounds, or possibly more on small surfaces of from 2 to 40 or 50 square feet in area. This distinction between small and large exposed areas in the treatment of wind pressures is fundamental and should never be neglected.
This whole subject of wind pressures has not yet been brought into a completely definite or well-defined condition through lack of sufficient experimental observations, but in order to be at least reasonably safe civil engineers frequently, and perhaps usually, assume a wind pressure acting simultaneously on both bridge and train at 30 pounds per square foot of exposed surface and 50 pounds per square foot of the total exposed surface of a bridge structure which carries no moving load. This distinction arises chiefly from the fact that a wind pressure of 30 pounds per square foot on the side of many railroad trains, particularly light ones, will overturn them, and it would be useless to use a larger pressure for a loaded structure. There have been wind pressures in this country so great as to blow unloaded bridges off their piers; indeed in one case a locomotive was overturned which must have resisted a wind pressure on its exposed surface of not less than 90 pounds and possibly more than 100 pounds per square foot.
The consideration of wind pressure is of the greatest importance in connection with the high trusses of long spans, as well as in long suspension and cantilever bridges, and in the design of high viaducts, all of which structures receive lateral wind pressures of great magnitude.
Some engineers, instead of deducing the lateral wind loads from the area of the projected truss surfaces, specify a certain amount for each linear foot of span, as in ”The General Specifications for Steel Railroad Bridges and Viaducts” by Mr. Theodore Cooper it is prescribed that a lateral force of 150 pounds for each foot of span shall be taken along the upper chords of through-bridges and the lower chords of deck-bridges for all spans up to 300 feet in length; and that for the same spans a lateral force of 450 pounds for each foot of span shall be taken for the lower chords of through-spans and the upper chords of deck-spans, 300 pounds of this to be treated as a moving load and as acting on a train of cars at a line 8⁵/₁₀ feet above the base of rail.
When the span exceeds 300 feet in length each of the above amounts of load per linear foot is to be increased by 10 pounds for each additional 30 feet of span.
Special wind-loadings and conditions under which they are to be used are also prescribed for viaducts.
These wind loads are resisted in the bridges on which they act by a truss formed between each two upper chords for the upper portion of the bridge, and between each two lower chords for the lower portion of the structure.
[Illustration: FIG. 26.]
=104. Upper and Lower Lateral Bracing.=—Fig. 26 shows what are called the upper and lower lateral bracing for such trusses as are shown in the preceding figures. The wind is supposed to act in the direction shown by the arrow. _DERA_ and _KLBC_ are the two portals at the ends of the structure, braced so as to resist the lateral wind pressures. It will be observed that the systems of bracing between the chords make an ordinary truss, but in a horizontal plane, except in the case of inclined chords like that of Fig. 24. In the latter case the lateral trusses are obviously not in horizontal planes, but they may be considered in computations precisely as if they were. These lateral trusses are then treated with their horizontal panel wind loads just as the vertical trusses are treated for their corresponding vertical loads, and the resulting stresses are employed in designing web and chord members precisely as in vertical trusses. The wind stresses in the chords, in some cases, are to be added to those due to vertical loading, and in some cases subtracted. In other words, the resultant stresses are recognized and the chord members are so designed as properly to resist them. At the present time it is the tendency in the best structural work to make all the web members of these lateral trusses of such section that they can resist both tension and compression, as this contributes to the general stiffness of the structure. On account of the great variability of the wind pressures and the liability of the blows of greatest intensity to vary suddenly, some engineers regard all the wind load on structure or train as a moving load and make their computations accordingly. It is an excellent practice and is probably at least as close an approximation to actual wind effects as the assumption of a uniform wind pressure on a structure.
Both the lateral and transverse wind bracing of railroad bridges have other essential duties to perform than the resistance of lateral wind pressures. Rapidly moving railroad trains produce a swaying effect on a bridge, in consequence of unavoidable unevenness of tracks, lack of balance of locomotive driving-wheels, and other similar influences. These must be resisted wholly by the lateral and transverse bracing, and these results constitute an important part of the duties of that bracing. These peculiar demands, in connection with the lateral stability of bridges, make it the more desirable that the lateral and transverse bracing should be as stiff as practicable.
=105. Bridge Plans and Shopwork.=—After the computations for a bridge design are completed in a civil engineer’s office they are placed in the drawing-room, where the most detailed and exact plans of every piece which enters the bridge are made. The numerical computations connected with this part of bridge construction are of a laborious nature and must be made with absolute accuracy, otherwise it would be quite impossible to put the bridge together in the field. The various quantities of bars, plates, angles, and other shapes required are then ordered from the rolling-mill by means of these plans or drawings. On receipt of the material at the shop the shopwork of manufacture is begun, and it involves a great variety of operations. The bridge-shop is filled with tools and engines of the heaviest description. Punches, lathes, planers, riveters, forges, boring and other machines of the largest dimensions are all brought to bear in the manufacture of the completed bridge.
=106. Erection of Bridges.=—When the shop operations are completed the bridge members are shipped to the site where the bridge is to be erected or put in place for final use. A timber staging, frequently of the heaviest timbers for large spans, called false works, is first erected in a temporary but very substantial manner. The top of this false work, or timber staging, is of such height that it will receive the steelwork of the bridge at exactly the right elevation. The bridge members are then brought onto the staging and each put in place and joined with pins and rivets. If the shopwork has not been done with mathematical accuracy, the bridge will not go together. On the accuracy of the shopwork, therefore, depends the possibility of properly fitting and joining the structure in its final position. The operations of the shop are so nicely disposed and so accurately performed that it is not an exaggeration to state that the serious misfit of a bridge member in American engineering practice at the present time is practically impossible. This leads to rapid erection so that the steelwork of a pin-connected railroad bridge 500 feet long can be put in place on the timber staging, or false works, and made safe in less than four days, although such a feat would have been considered impossible twenty years ago.
[Illustration: FIG. 27.]
=107. Statically Determinate Trusses.=—The bridge structures which have been treated require but the simplest analysis, based only on statical equations of equilibrium of forces acting in one plane, i.e., the plane of the truss. It is known from the science of mechanics that the number of those equations is at most but three for any system of forces or loads, viz., two equations of forces and one of moments. This may be simply illustrated by the system of forces _F₁_, _F₂_, etc., in Fig. 27. Let each force be resolved into its vertical and horizontal components _V_ and _H_. Also let _l₁_, _l₂_, etc. (not shown in the figure), be the normals or lever-arms dropped from any point _A_ on the lines of action of the forces _F₁_, _F₂_, etc., so that the moments of the forces about that point will be _F₁l₁_, _F₂l₂_, etc. The conditions of purely statical equilibrium are expressed by the three general equations
_H₁ + H₂_ + etc. = _F₁_ cos _a₁_ + _F₂_ cos _a₂_ + etc. = 0; (35) _V₁ + V₂_ + etc. = _F₁_ sin _a₁_ + _F₂_ sin _a₂_ + etc. = 0; (36) _Fl = F₁l₁ + F₂l₂_ + etc. = 0. (37)
If all the forces except three are known, obviously those three can be found by the three preceding equations; but if more than three are unknown, those three equations are not sufficient to find them. Other equations must be available or the unknown forces cannot be found. In modern methods of stress determinations those other needed equations express known elastic relations or values, such as deflections or the work performed in stressing the different members of structures under loads. A few fundamental equations of these methods will be given.
In Figs. 19, 20, and 21 let the truss be cut or divided by the imaginary sections _QS_. Each section cuts but three members, and as the loads and reactions are known, the stresses in the cut members will yield but three unknown forces, which may be found by the three equations of equilibrium (35), (36), (37). If more than three members are cut, however, as in the section _TV_ of Figs. 22 and 23, making more than three unknown equations to be found, other equations than the three of statical equilibrium must be available. Hence the general principle that _if it is possible to cut not more than three members by a section through the truss, it is statically determinate_, but _if it is not possible to cut less than four or more, the stresses are statically indeterminate_.
At each joint in the truss the stresses in the members meeting there constitute, with the external forces or loads acting at the same point, a system in equilibrium represented by the two equations (35) and (36). If there are _m_ such joints in the entire structure, there will be 2_m_ such equations by which the same number of unknown quantities may be found. Since equilibrium exists at every joint in the truss, the entire truss will be in equilibrium, and that is equivalent to the equilibrium of all the external forces acting on it. This latter condition is expressed by the three equations (35), (36), and (37), and they are essentially included in the number 2_m_. Hence there will remain but 2_m_ - 3 equations available for the determination of unknown stresses or external forces.
If, therefore, all the external forces (loads and reactions) are known, the 2_m_ - 3 equations of static equilibrium can be applied to the determination of stresses in the bars of the truss or other structure. It follows, therefore, that the greatest number of bars that a statically determinate truss can have is
_n_ = 2_m_ - 3. (38)
In Fig. 19 there are twelve joints and twenty-one members, omitting counter web members and the verticals _ab_ and _fl_, which are, statically speaking, either superfluous or not really bars of the truss. Hence
_m_ = 12 and 2_m_ - 3 = 21. (39)
Again, in Fig. 21 there are fifteen joints. Hence
_m_ = 15, 2_m_ - 3 = 27,
and there are twenty-seven bars or members of the truss. The number of joints and bars in actual, statically determinate trusses, therefore, confirm the results.
=108. Continuous Beams and Trusses—Theorem of Three Moments.=—These considerations find direct application to what are known as ”continuous beams,” i.e., beams (or trusses) which reach continuously over two or more spans, as shown in Fig. 28.
[Illustration: FIG. 28.]
The beam shown is continuous over three spans, but a beam or truss may be continuous over any number of spans. In general the ends of the beam or girder may be fixed or held at the ends _A_ and _D_, so that bending moments _M_ and _M₃_ at the same points may have value. The bending moments at the other points of support are represented by _M₁_, _M₂_, etc. The points of support may or may not be at the same elevation, but they are usually assumed to be so in engineering practice. Finally, it is ordinarily assumed that the continuous structure is straight before being loaded, and that in that condition it simply touches the points of support. Whether the preceding assumptions are made or not, a perfectly general equation can be written expressing the relation between the bending moments over each set of three consecutive points of support, as _M_, _M₁_, and _M₂_, or _M₁_, M₂, and M₃. Such an equation expresses what is called the ”Theorem of Three Moments.” It is not necessary to give the most general form of this theorem, as that which is ordinarily used embodies the simplifying assumptions already described. This simplified form of the ”Theorem of Three Moments” applied to the case of Fig. 28 will yield the following two equations:
1 ₁ _Ml₁ + 2M₁_(_l₁ + l₂_) + _M₂l₂_ + ---- ∑ _W_(_l₁² - z²_)_z_ _l₁_
1 ₂ + ----- ∑ _W_(_l₂² - z²__)z_ = 0. (40) _l₂_
1 ₂ _M₁l₂ + 2M₂_(_l₂ + l₃_) + _M₃l₃_ + ------ ∑ _W_(_l₂² - z²_) _l₂_ 1 ₃ + ---- ∑ _W_(_l₃² - z²_)_z_ = 0. (41) _l₃_
The figure over the sign of summation shows the span to which the summation belongs. If there is but one weight or load _W_ in each span, the sign of summation is to be omitted. In an ordinary bridge structure or beam the ends are simply supported and _M = M₃ = 0_. In any case if the number of supports be _n_, there will be _n_ - 2 equations like the preceding.
If the end moments _M_ and _M₃_ are not zero, they will be determinable by the local conditions in each instance. In any event, therefore, they will be known, and there will be but _n_ - 2 unknown moments to be found by the same number of equations. When the moments are known the reactions follow from simple formulæ.
=109. Application to Draw- or Swing-bridges.=—In general the reactions or supporting forces of the beams and trusses of ordinary civil-engineering practice are vertical, and all their points of application are known. Hence there are but two equations of equilibrium, equations (36) and (37), for external forces. These two equations for the external forces and the _n_ - 2 equations derived from the theorem of three moments are therefore always sufficient to determine the _n_ reactions. After the reactions are known all the stresses in the bars or members of the trusses can at once be found. The preceding equations and methods as described are constantly employed in the design and construction of swing- or drawbridges.
=110. Special Method for Deflection of Trusses.=—The method of finding the elastic deflections produced by the bending of solid beams has already been shown, but it is frequently necessary to determine the elastic deflections of bridge-trusses or other jointed or so-called articulate frames or structures. It is not practicable to use the same formulæ for the latter class of structures as for the former. The elastic deflection of a bridge- or roof-truss depends upon the stretching or compressions of its various members in consequence of the tensile or compressive forces to which they are subjected. Any method by which the deflection is found, therefore, must involve these elastic changes of length. There are a number of methods which give the desired expressions, but probably the simplest as well as the most elegant procedure is that which reaches the desired expression through the consideration of the work performed in the truss members in producing their elastic lengthenings and shortenings.
The general features of this method can readily be shown by reference to Fig. 29. It may be supposed that it is desired to find the deflection of any point, as _J_, of the lower chord produced both by the dead and live load which it carries. It is known from what has preceded that every member of the upper chord will be shortened and that every member of the lower chord will be lengthened; and also that generally the vertical web members will be shortened and the inclined web members lengthened. If there can be obtained an expression giving that part of the deflection of _J_ which is due to the change of length of any one member of the truss independently of the others, then that expression may be applied to every other member in the entire truss, and by taking the sum of all those effects the desired deflection will at once result. While this expression will be found for some one particular truss member, it will be of such a general form that it may be used for any truss member whatever; it will be written for the upper-chord member _BC_ in Fig. 29.
[Illustration: FIG. 29.]
The general problem is to determine the deflection of the point _J_ when the bridge carries both dead and moving load over the entire span, as shown in Fig. 29. The general plan of procedure is first to find the stresses due to this combined load in every member of the truss, so that the corresponding lengthening or shortening is at once shown. The effect of this lengthening and shortening for any single member _BC_ in producing deflection at _J_ is then determined; the sum of all such effects for every member of the truss is next taken, and that sum is the deflection sought. In this case the vertical deflection will be found, because that is the deflection generally desired in connection with bridge structures, but precisely the same method and essentially the same formulæ are used to find the deflection in any direction whatever. The following notation will be employed:
Let _w_ = deflection in inches at any panel-point or joint of the truss;
” _P_ = any arbitrary load or weight supposed to be hung at the point where the deflection is desired and acting as if gradually applied. This may be taken as unity;
” _Z_ = stress produced in any member of truss by _P_;
” _S_ = stress produced in any member of truss by the combined dead and moving loads;
Let _l_ = length in inches of any member of the truss in which _Z_ or _S_ is found;
” _A_ = area of cross-section of same member in square inches;
” _E_ = coefficient of elasticity.
_S_ or _Z_ may be either tension or compression, and the formulæ will be so expressed that tension will be made positive and compression negative.
The change of length of the chord member _BC_ produced by a stress gradually increasing from zero to _S_ is
_S_ ----_l_. _AE_
If it be supposed that _BC_ is a spring of such stiffness that it will be compressed by the gradual application of _Z_ exactly as much as the shortening of the actual member by the stress _S_, the deflection of the point 4 with the weight _P_ hung from it, and due to that compression alone, will be precisely the same as that due to the actual shortening of _BC_ by the combined dead and moving loads.
It is known by one of the elementary principles of mechanics that, since _P_ acts along the direction of the vertical deflection _w_, the work performed by the weight _P_ over that deflection is equal to the work performed by _Z_ over the change of length _l_. Hence
_l l Sl_ --- _Pw_ = --- _Z_ ----, or 2 2 _AE_
_Z Sl_ _w_ = --- ----. (42) _P AE_
The quantity _Z÷P_ is the stress produced in the member by a unit load applied at the joint or point where the deflection is desired. Again, _S÷A_ is the stress per unit of area, i.e., intensity of stress, in the member considered by the actual dead and moving loads. For brevity let these be written
_Z S_ --- = _z_ and --- = _s_; _P A_
then
_zsl_ _w_ = ------. (43) _E_
If the influence of every member of the truss is similarly expressed, the value of the total deflection produced by the dead and moving loads will be
_zsl_ _w_ = ∑ ------. (44) _E_
The sign of summation ∑ indicates that the summation is to extend over all the web and chord members of the truss.
=111. Application of Method for Deflection to Triangular Frame.=—Before applying those equations to the case of Fig. 29 it is best to consider a simpler case, i.e., that of the triangular frame shown in Fig. 18. The reactions are
_l₂ l₁_ _R_ = --- _W_ and _Rʹ_ = --- _W_. (45) _l l_
The stresses in the various members are:
_l₁_ In _CB_, _S_ = ---- _W_ sec α. _l_
_l₂_ ” _CA_, _S_ = ---- _W_ sec β. _l_
_l₂ l₁_ ” _AB_, _S_ = --- _W_ tan β = --- _W_ tan α. _l l_
Also: _CB = h_ sec α; area of section = _A₁_. _CA = h_ sec β; ” ” ” = _A₂_. _AB = l_; ” ” ” = _A₃_.
In this instance it is simplest to take _P = W_. Equation (44) then gives
( _l₁² h_ sec³ α _l₂² h_ sec³ _β_ _w_ = (----- ---------- + ----- ------------- ( _l² A₁ l² A₂_
_l₂² l_ tan² _β_ ) _W_ + ---- ------------) ------. (46) _l² A₃_ ) _E_
Let it be supposed that
_l_ = 25 feet = 300 inches; _h_ = 8 feet 4 inches = 100 inches; _l₂_ = 16 feet 8 inches = 200 inches and _l₁_ = 100 inches; tan β = 1; sec β = 1.414; sec α = 2.24; _W_ = 10,000 pounds.
If the bars are all supposed to be of yellow-pine timber, there may be taken
_E_ = 1,000,000 pounds; _A₁_ = 10″ × 12″ = 120 square inches; _A₂_ = 10″ × 10″ = 100 square inches; _A₃_ = 10″ × 12″ = 120 square inches.
The insertion of these quantities in equation (46) gives the deflection
_w_ = .01042 + .01253 + .01111 = 0.034. (47)
Equation (47) is so written as to show the portion of the deflection due to each member of the frame.
In applying either equation (43) or equation (44) care must be taken to give each stress and its corresponding strain (lengthening or shortening) the proper sign. As the formulæ have been written and used, a tensile stress and its resulting stretch must each be written positive, while a compressive stress must be written negative. This holds true for both the stresses _Z_ and _S_ (or _z_ and _s_). The magnitude of the assumed load _P_ is a matter of indifference, since the stress _Z_ will always be proportional to it and the ratio _P ÷ Z_ will therefore be constant. _P_ is frequently taken as unity; or, as in the case just given, it may have any value that the conditions of the problem make most convenient.
=112. Application of Method for Deflection to Truss.=—In making application of the deflection formulæ to any steel railroad truss similar to that shown in Fig. 29, it will first be necessary to determine the stresses in all its members due to the dead and moving loads, since the deflection under the moving load is sought. These loads will be considered uniform, and that is sufficiently accurate for any railroad bridge. The moving train-load will be taken as covering the entire span, assumed, for a single-track railroad, 240 feet in length between centres of end pins. There are eight panels of 30 feet each, and the depth of truss at centre is 40 feet. Other truss dimensions are as shown in Fig. 29. The dead loads, or own weight, are taken at 400 pounds per linear foot of span for the rails and other pieces that constitute the track; at 400 pounds per linear foot for the steel floor-beams and stringers, and 1600 pounds per linear foot for the weight of trusses and bracing. The moving train-load will be taken at 4000 pounds per linear foot. This will make the panel-loads for each truss as follows:
Lower-chord dead load, 30 × 800 = 24,000 pounds per panel. Lower-chord moving load, 30 × 2000 = 60,000 ” ” ” ------ Total load on lower chord = 84,000 ” ” ” Upper-chord dead load, 30 × 400 = 12,000 ” ” ”
The structure is a “through” bridge, hence all moving loads rest on the lower chord.
[Illustration: FIG. 30.]
The stresses in the truss members due to the combined uniform dead and moving load are best found by the graphical method. One diagram only is needed to determine all the stresses, and it is shown in Fig. 30. This diagram is drawn accurately to scale, and the stresses measured from it are shown in the table on page 136.
The stresses in all the truss members due to the unit load hung at _J_ are readily found by the single diagram shown in Fig. 31, also carefully drawn to scale. These stresses measured from the diagram are given in the table as indicated by the column _z_; they are also represented in equation (44) by the letter _z_. The quantity _s_ in equation (44) is the intensity of the stress (pounds per square inch of cross-section of member) produced by the combined dead and moving loads in each member. As shown, these stresses are least in the web members near the centre of the span, and greatest in the chord members. The lengths in inches of the truss members are shown in the proper column of the table. It will be observed that all counter web members are omitted, as they are not needed for the uniform load. The coefficient of elasticity (_E_) is taken at 28,000,000 pounds. The quantities represented by the second member of equation (44) are computed from these data, and they appear in the last column of the table, the sum of which gives the desired deflection in inches. The elements of the table show how much of the deflection is due to the chords and to the web members, and they show that disregarding the latter would lead to a considerable error.
+-------+----------+---------+--------+-----+---------+ | | _S_ | _s_ | _z_ | _l_ | _w_ | +-------+----------+---------+--------+-----+---------+ | _L₁_ | +373,300 | +12,000 | +.555 | 360 | +.08563 | | _L₂_ | +373,300 | +12,000 | +.555 | 360 | +.08563 | | _L₃_ | +480,000 | +12,000 | +.833 | 360 | +.1284 | | _L₄_ | +540,000 | +12,000 | +1.125 | 360 | +.1736 | | _P₁_ | -502,300 | -9,000 | -.748 | 472 | +.1132 | | _U₁_ | -501,000 | -9,500 | -.870 | 376 | +.1108 | | _U₂_ | -544,800 | -10,000 | -1.135 | 363 | +.1472 | | _U₃_ | -576,000 | -10,000 | -1.50 | 360 | +.1928 | | _T₁_ | +84,000 | +9,000 | 0 | 324 | -- | | _T₂_ | +143,500 | +10,000 | +.3738| 472 | +.0629 | | _P₂_ | -12,000 | -1,000 | -.250 | 432 | +.00386 | | _T₃_ | +93,720 | +7,400 | +.456 | 562 | +.0677 | | _P₃_ | +12,000 | +1,000 | -.35 | 480 | -.0060 | | _T₄_ | +60,000 | +4,800 | +.625 | 600 | +.0643 | | _P₄_ | -12,000 | -1,000 | 0 | 480 | -- | +-------+----------+---------+--------+-----+---------+
Deflection for ½ truss members = 1.2300 inches. Deflection at _J_ = 2 × 1.2300 = 2.4600 inches.
[Illustration: FIG. 31.]
As the deflection is usually desired in inches, the lengths of members must be taken in the same unit.
=113. Method of Least Work.=—The so-called theorem or principle of “Least Work” is closely related to the subject of elastic deflections just considered in its availability for furnishing equations of condition in addition to those of a purely statical character in cases where indetermination would result without them. This principle of least work is expressed in the simple statement that when any structure supports external loading the work performed in producing elastic deformation of all the members will be the least possible. Although this principle may not be susceptible of a complete and general demonstration, it may be shown to hold true in many cases if not all. The hypothesis is most reasonable and furnishes elegant solutions in many useful problems.
The application of this principle requires the determination of expressions for the work performed in the elastic lengthening and shortening of pieces subjected either to tension or compression, and for the work performed in the elastic bending of beams carrying loads at right angles to their axes. Both of these expressions can be very simply found.
Let it be supposed that a piece of material whose length is _L_ and the area of whose cross-section is _A_ is either stretched or compressed by the weight or load _S_ applied so as to increase gradually from zero to its full value. The elastic change of length will be _SL/AE_, _E_ being the coefficient of elasticity. The average force acting will be ½_S_, hence the work performed in producing the strain will be
1 _S²L_ --- -----. (48) 2 _AE_
It will generally be best, although not necessary, to take _L_ in inches. The expression (48) applies either to tension or compression precisely as it stands.
To obtain the expression for the work performed by the stresses in a beam bent by loads acting at right angles to its axis, a differential length (_dL_) of the beam is considered at any normal section in which the bending moment is _M_, the total length being _L_. Let _I_ be the moment of inertia of the normal section, _A_, about an axis passing through the centre of gravity of the latter, and let _k_ be the intensity of stress (usually the stress per square inch) at any point distant _d_ from the axis about which _I_ is taken. The elastic change produced in the indefinitely short length _dL_ when the intensity _k_ exists is (_k/E_)_dL_. If _dA_ is an indefinitely small portion of the normal section, the average force or stress, either of tension or compression, acting through the small elastic change of length just given, can be written by the aid of equation (5) as
_Md_ ½_k.dA_ = ----- ._dA._ (49) 2_I_
Hence the work performed in any normal section of the member, for which _M_ remains unchanged, will be, since ∫_k.dA.d_ = _M_,
⌠ _M M²_ ⌡ ------ _kd.dA.dL_ = ----- _dL._ (50) 2_IE_ 2_IE_
The work performed throughout the entire piece will then be
⌠ _M²_ ⌡ ------ _dL._ (51) 2_IE_
Each of the expressions (48) and (51) belongs to a single piece or member of the structure. The total work performed in all the pieces subjected either to direct stress or to bending, and which, according to the principle of least work, must be a minimum, is found by taking the summation of the two preceding expressions:
1 ⎲ _S²L_ 1 ⎲ ⌠ _M²_ _e_ = ----⎳ ------ + ----- ⎳ ⌡ ----- _dL_ = minimum. (52) 2_E_ _A_ 2_E_ _I_
In making an application of equation (52) it is to be remembered that _S_ is the direct stress of tension or compression in any member, and that _M_ is the general value of the bending moment in any bent member expressed in terms of the length _L_.
=114. Application of Method of Least Work to General Problem.=—The problem which generally presents itself in the use of equation (52) is the finding of an equation which expresses the condition that the work expended in producing elastic deformation shall be a minimum, some particular stress in the structure or some external load or force being the variable. If _t_ represent that variable, then the desired equation of condition will be found simply by placing the first differential coefficient of _e_ in equation (52) equal to zero:
_de_ 1 (⎲ _S dS_ ⎲ ⌠ _M dM_ ) ----- = ----(⎳ --- --- _dL_ + ⎳ ⌡ ---- --- _dL_) = 0. (53) _dt_ _E_( _A dt_ _I dt_ )
The solution of equation (53) will give a value of _t_ which will make the work performed as expressed in equation (52) a minimum. This method is not a difficult one to employ in such cases as those of drawbridges and stiffened suspension bridges. In the latter case particularly it is of great practical value.
=115. Application of Method of Least Work to Trussed Beam.=—The method of least work may be illustrated by the application of the preceding equations to the simple truss shown in Fig. 32. The pieces _BC_ and _GD_ are supposed to be of yellow-pine timber, the former 10 inches by 14 inches (vertical) in section and the latter 8 inches by 10 inches, while each of the pieces _BD_ and _DC_ are two 1⅝-inch round steel bars. The coefficient of elasticity _E_ will be taken at 1,000,000 pounds for the timber and 28,000,000 for the steel. The length of _BC_ is 360 inches; _GD_ 96 inches; _BD_ = 96 × 2.13 = 204.5 inches.
tan α = 1.875 and sec α = 2.13.
The weight _W_ resting at _G_ is 20,000 pounds. A part of this weight is carried by _BC_ as a simple timber beam, while the remainder of the load will be carried on the triangular frame _BCD_ acting as a truss, the elastic deflection of the latter throwing a part of the load on _BC_ acting as a beam. According to the principle of least work the division of the load will be such as to make the work performed in straining the different members of the system a minimum.
That part of _W_ which rests on _BC_ as a simple beam may be represented by _W₁_, while _W₂_ represents the remaining portion carried by the triangular frame. As _G_ is at the centre of the span, the beam reaction at either _B_ or _C_ is ½_W₁_. Hence the general value of the bending moment in either half of the beam at any distance _x_ from either _B_ or _C_ is
_M_ = ½_W₁x._ Hence _M²dL_ = ¼_W₁²x²dx_.
As there is but one member acting as a beam, whose moment of inertia _I_ is constant, the second term of the second member of equation (52) becomes, by the aid of the preceding equation,
1 1 ¹/₂ 1 _W₁²l³_ ----- ⌠ _M²dL_ = ---- ⌠ _¼W₁²x²dx_ = ---- --------. (54) 2_EI_ ⌡ _EI_ ⌡₀ _EI_ 96
[Illustration: FIG. 32.]
The numerical elements of the expression for the work done in the members of the triangular frame are:
Member. Stress. Length. Area of Section.
_BC_ ½_W₂_ tan _α_ 360 inches = _l_ 140 square inches _DC_ ½_W₂_ sec _α_ 204.5 ” 4.14 ” ” _DG_ _W₂_ 96 ” 80 ” ”
10 × 14³ 27440 _I_ = -------- = ------- = 2286.7. 12 12
The substitution of those quantities in the first term of the second member of equation (52) will give
1 ⎲ _S²L_ 1 ( _W₂²_ tan² _α_.360 _W₂²_.96 ) -----⎳ ----- = ---------- (------------------ + ----------) 2_E A_ 2,000,000 ( 4 × 140 80 )
2 _W₂²_ sec² _α_ .204.5 + ----------- --------------------- = .000,003,73 _W₂²_. 56,000,000 4 × 4.14
The substitution of numerical quantities in equation (54) gives
1 _W₁²l³_ ---- -------- = .000,213_W₁²_. _EI_ 96
Or, since _W - W₂ = W₁_,
_e_ = .000,003,73_W₂²_ + .000,213(_W - W₂)²_. (55)
Hence
_de_ ---- = .000,007,46_W₂_ - .000,426(_W - W₂_) = 0. (56) _dW₂_
The solution of this equation gives
_W₂_ = .893_W_ = 19,660 pounds.
_W₁_ = 340 ”
It is interesting to observe that the first term of the second member of equation (56) is the deflection of the point of application of _W₂_ as a point in the frame, while the second term is the deflection of the point of application of _W₁_ considered as a point of the beam. In other words, the condition resulting from the application of the principle of least work is equivalent to making the elastic deflections by _W₁_ and _W₂_ equal. Indeed equation (53) expresses the equivalence of deflections whenever the features of the problem are such as to involve concurrent deflections of two different parts of the structure.
=116. Removal of Indetermination by Methods of Least Work and Deflection.=—The indetermination existing in connection with the computations for such trusses as those shown in Fig. 22 and Fig. 23 can be removed by finding equations of condition by the aid of the method of least work or of deflections. It is evident that the component systems of bracing of which such trusses are composed must all deflect equally. Hence expressions may be found for the deflections of those component trusses, each under its own load. Since these deflections must be equal, equations of condition at once result. A sufficient number of such equations, taken with those required by statical equilibrium, can be found to solve completely the problem. Such methods, however, are laborious, and the ordinary assumption of each system carrying wholly the loads resting at its panel-points is sufficiently near for all ordinary purposes.
The method of least work can be very conveniently used for the solution of a great number of simple problems, like that which requires the determination of the four reactions under the four legs of a table, carrying a single weight or a number of weights, and many others of the same character.