Noy . 19, 1886. THE ENGINEER. 399 bending moments from the unit of the forces S. The These are the conditions of the statically undetermined THE NATIONAL AGRICULTURAL HALL AT latter expressions would simply be added as so much structure, and it is now to be made statically determined. KENSINGTON* additional work done.J In the first place, the whole of the [right half, which The first step is to decide which are the additional bars and forms an elastic support for the left half on the line C C, is CALCULATION OF STRESSES IN THE ROOF PRINCIPAL supports ; their removal would make the structure statically removed, and in place of it are put the two forces H and V AS AN ELASTIC STRUCTURE ACCORDING TO THE determined. In most cases this can be decided in various and a bending moment M. This is always admissible when PRINCIPLE OF WORK. ways. we cut through a solid beam. It will be observed that under It is now generally known that the principle of work can all circumstances these forces and the corresponding dis ­ be applied to the calculation of strains in elastic structures tances which the left half yields under them—viz., A*£, Ay which are statically undetermined, and it will therefore be and the torsional angle A (f>, will be identical with or oppo ­ sufficient to state in the following paper the theory broadly, site to those of the right half, and that from this fact three referring for its full development to the numerous works equations can be derived, from which H, Y, and M can eventu ­ on the subject. If a force S be made to act upon any ally be calculated. But the removal of the right half alone point of an elastic structure, and if it yields to this force does not make the left half statically determined. It would a distance A l, the force has performed work to the be necessary, besides, to remove five bars ; but in order to amount of £ S A l- At the same time the structure B, keep the calculation within practical dimensions, bearing resists, and while yielding some of its parts are elon ­ in mind that one more bar nearly doubles its length, one gated, others compressed, and still others bent. Each B\ of the five had to be selected to remain in the statically of these deformations represents an amount of work which determined structure. The bar which seemed to be of in each single case can be expressed by means of the least influence upon the stresses of the structure, if it were following data (1) The length of the part; (2) its sec- ir left to remain, is No. 16, Fig. 2; and it was assumed that tional area—moment of inertia in case of bending; (3) the this bar would under all circumstances have an equal and stress—bending moment —which is in it; and (4) the opposite stress with No. 17, although this is only approxi- modulus of elasticity of the material. The principle of The present structure contains two kinds of parts, viz., mately the case. The four bars taken out and replaced by work requires that the sum of all work represented (1) such with a close and more or less complicated lattice- forces were then Nos. 22, 24, 25, 26, and the stresses in by these single deformations should equal £ S A l- The factor £ can be omitted, as it occurs in all expressions. In a statically deter ­ mined structure the stresses are found without kV having regard to the elastic deformation of the parts, and an equation of the nature here Fig. Z . LojtfL. described would only be used for calculating A l, i.e.y the deflection of the structure under the f<£, 4. 67 T. ' force S in the direction of S. But in a statically undetermined structure fo — there are more parts or more points of support than are absolutely necessary for its stability; .3 APT. Let v3~Xoii8 and inasmuch as these parts receive stresses and -4 '*s *4.13 T. the supports pressures, as well as the necessary parts and supports, the stresses and pressures y>4-30T. in the latter are modified by their presence, and 2 ^........ the extent of the modification will largely depend 3-70 T. on the extensibility and compressibility of those ■—I additional parts or supports. The mode of $-i*- treatment of such a structure then prescribes Y 3-70T. that forces S', S®, . .—unknown quan ­ 6-21T, tities—be introduced in place of the additional parts or supports ; that the stresses from these forces in the remaining determined structure be calculated in the usual way. Further, that the for expressions S1 A ll, S® A . • be formed, ?<wycrv where A l', Als, . would be known quanti ­ 4.72 T. ties for the unit of S—they would be = 0 ______ ..-•**•* _______ _____ 3 85 T. 14-72 T, 3 13 T. _y FiocacL JjooloL xtv Tons »> 0-61 T. jy OooousicnjCfJL JjoouLh » 1170 T. » M 28-39 T 18-41 T- aJb 30Tbs -p.f* siip. »> jy ypuicb Treasure. >» »> ^/•843^ ¥ iyZSi DistajtjC& ocpccrb of J\FocLn JFtCbj~-3£- 0 *T /' i y7T>' ' '0-41 T.*f \ * jo-70 T. 83'- 0- \ \ Z4 \ , ■-$ $ £ -V \ W \ \ 20. \ 13 Floor Ixjte 4? 8? 2 <1 £ 18 14 work —cross-lined in Fig. 1—and (2) single bars. The them were denoted by S2S, S* 4, S®-*, S* fl. The parts former can be treated as solid beams, that is to say— treated as solid beams were divided into a number of sec- although this is not quite accurate—forces acting upon tions —viz., Nos. 1 to 9 and 28 to 39. It was assumed that in case of a perfectly firm support, but not = 0 in them will bend them, and compress or extend their in each the bending moment and the axial stress was case of an elastic bar or an elastic support — and, neutral fibres as if they were solid, the latter will be only equal from end to end. Fig. 2 is then a diagram of the finally, that for each S an equation be constructed of the compressed or extended according to the compressive or statically determined structure, which also contains the form referred to above. It would then be possible, by tensile stresses which are produced in them by the forces. final results of the calculation illustrated by funicular solving these equations, to determine the unknown quan ­ A A are the principal abutments of the structure, B B are polygons. Upon it act ten different forces —yiz., H, Y, M, tities S, and with these all the stresses and deflections that flexible connections which can be regarded as hinges, D the four forces S, the fixed load, the wind pressure, and may be required. The general form of these equations is and E are ball pivots, of which the lower ones are fixed the occasional load on the gallery floor. Accordingly there as follows : points, but the frames A E are free to move horizontally are ten _ diagrams of stresses, and their contents are em- X rx n X e* n s b at E. The concrete in which the frames are enclosed can bodied in the Table A on next page. - m* Sz = S1 2 z <r* ' + S® 2 rFf\>x 0"x * ^ « The first column contains the expression m for the exten ­ « 1 resist such movement only to a small extent, even if the * 50 r=s n sibility of the parts, viz., length divided by product of w + adhesion to the iron parts were very great, because the + . + in x 2 )® area and modulus of elasticity, 10,000 tons ; the ninth Ini modulus of elasticity of concrete is much smaller than Itan x « n that of iron. In the present case the concrete enclosure column contains the expression A e for the flexibility of + S“ 2 ™x (Tx * <TX u + 2 Wlx&x2 Sz • . (1)1* has no greater density than is necessary to keep off the sections of beams, viz. : Length divided by product of I *- 1 1*1 moisture, and while there were no means of ascertaining moment of inertia and modulus of elasticity. Both ex­ where S2 is any one of the forces S', S*, . Su, its elastic resistance, it had to be left out of the calcula­ pressions are for convenience multiplied with 107 . Columns which replace the u additional bars; <rx z is the stress due tion altogether. The same is the case with the side walls, 2 to 8 contain the stresses from the units of the forces to the unit of this force in any one, the Xth, bar of a series which have no very great power of resisting laterally. H, Y, M and the forces S ; columns 10 to 16 the bending of n necessary bars. A similar meaning have cr* *, <rx 4, moments from the same units and columns 17 to 22 the t M. am Ende .—“ Stresses in Statically Undetermined Systems.”— stresses and moments from the fixed load, the wind . <rx u; mx is an expression for the extensibility of Engineering, 1883, p. 509. Further references: —Maxwell .—“On the any one of the n necessary bars, and m* the same for any Calculation of the Equilibrium and Strength of Frames.”—Philosophical pressure, and the occasional load. Where no figures are one of the u additional bars. S* is the stress from any given Magazine, 1864, xxvii., p. 204. Mohr .—“ Beitrag zur Theorie der Holz marked there are no stresses.