imperfect manner. To make the frame into a true beam, this tendency of the loaded frame to slip through between the others must be counteracted by tongues, T, T 1? T 2 , projecting from one frame and working in a groove in the neighbouring frame (vide 19). Each tongue should be made so that it does not abut against the bottom of the groove, and is thus incapable of resisting any horizontal force it must neither prevent the whole beam nor any
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Fig. 11.
The structure will now be found capable of carrying weights as a beam. It deflects or bends as in fig. 11 under the action of a load on A l ; all the pieces of india-rubber above the centre of the beam are then com pressed ; all those below the centre are then extended ; and at the sections between the frames the horizontal internal forces are wholly met by the elastic reaction due to these horizontal extensions and compressions.
At any one section, say at a distance x l from the support Q, the pieces c x and d^ are just as much compressed as the pieces b l a x are extended ; the equal and opposite parallel resultants of these forces consequently constitute a couple, and the moment of this couple must be equal to the moment of the couple tending to bend the beam at this section, or to what is called the bending moment ; now the bending moment in this simple case is due to the up ward vertical reaction P l at N and the equal downward force with which the frame A x bears on the tongue T x ; for it is clear that, neglecting the weight of the frame, if a weight W on frame A l is borne by two forces P and Pj at the two piers, the tongue next N must also bear a vertical force P x and the tongue next Q a vertical force P. The stresses borne by the india-rubber pieces are exactly the same as if the frame A x were firmly held, and a vertical force P x applied to pull up the right hand part of the beam, while the tongue T a acted as a hinge ; the moment tending to turn the right hand part round in a left-handed direction would be P x (L x-^). This moment is resisted by the elasticity of the india-rubber, which must exert an equal and opposite moment round the same point. Calling s lt s 2 , s s the sectional areas of the pieces of rubber, and y v y%, y^ their distances from the axis where the section is neither extended nor compressed, and^ 1 ,p 2 ,^ 3 the inten sity with which each piece is strained, the moment due to the elasticity of the pieces of rubber tending to turn back the left hand part of the beam in a right-handed direction will be Sjosy. Now, if the modulus of elasticity of the rubber is constant, the stresses p v p 2 ,2) 3 will be propor tional to their distance from the unstrained axis ; thus if b l is 18 inches from the axis and c x only 9 inches, & x being shortened twice as much as c 1? the stress on b-^ will be twice that on c v and calling a the stress at unit distance from the axis we have p l ? ay l and ji z ay^ p n ? ay n , so that we may write ^psy ? a^sy z as the expression for the moment of the elastic forces. Hence equating the bending moment and the moment of the "elastic forces, we have—
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From this equation when the load is given we may determine a, and hence the intensity of the stress p ? ay at any distance y from the axis. If this intensity is less than the safe stress for the material, the beam is, at the section considered, strong enough to bear the load so far as the horizontal extending and compressing forces are concerned.
Thus if the dimensions of the beam be those given above and it be supported so that L may be 6 feet and the distance x l 2 feet 3 inches, the distance y of the outermost piece of rubber from the unstrained axis 8 inches, the weight 50 lb, and the section of the rubber in each row 2 inches (two cylinders side by side, each with a section of 1 inch) we shall have as the numerical values in the above equation, neglecting the weight of the frame itself (P : being nearly ? 187), 187 x 45 in. ? a (4 x 8 2 + 4 x 4 2 ) from which a ? 2 64; then the force supported by each inch of either of the rows of rubber a l or d l will be p ? 8 x 2 64 ? 21 12 H> ; the stress on each of the inner rows will be half this amount ; the same equations give the load which (so far as that particular section is concerned) can safely be placed on the frame A x consistently with a given stress per square inch on the rubber. The strength required in the tongue T : is still more easily found, the stress tending to shear it off is P 1; and it must have a suffi cient cross section to bear that shearing stress. Similar reasoning would allow us to calculate the strength of our beam at either of the two other sections where the india- rubber pieces and tongues are placed. The general relation between the external and internal forces in any beam is similar to that illustrated by the model ; at any section the moment due to the elastic forces must balance the moment due to the external forces tending to bend the beam at that section. The problem, therefore, of determining the strength of a beam at any section resolves itself, so far as the horizontal forces are concerned, into finding expres sions for these two moments and equating them. The equation thus stated will give the maximum horizontal stress thrown on the material of a given girder by a given load, or it will give the maximum load on a given girder consistent with a safe stress on the material, or if, as is generally the case in bridges, the load and maximum safe stress on the material are given, the equation will allow us to fix the dimensions required for the cross section of the beam so far as the horizontal forces are concerned. The provision to be made for resisting the shearing stress for which the tongues are required in the model will be explained in 19.
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Fig. 12.
In any solid beam the stresses do not divide themselves into horizontal and vertical components. This division .is made by the engineer to simplify his calculations. In the beam the actual stress at any point will be the resultant of the horizontal stress (borne by the india-rubber in the model) and the vertical stress (borne by the tongue in the model). The diagram, fig. 12, shows the direction of )g> > the resultant stress at each point of a beam of rectangular cross section. The curves are called the lines of principal stress.
of a given beam under a given distribution and magnitude of load is the sum of the moments, taken relatively to the section, of all the external forces acting on one of the two segments into which the section divides the beam. It is u matter of indifference which segment is considered, but the moment on one segment will be positive and that on the other negative. Let x be the distance of the section from the left hand abutment Q of any beam of span L (fig. 1 3). Let P
be the load borne by abutment Q, and P x the load borne by