=80. Deflection.=—It is frequently necessary, both in the design of beams and framed bridges, to ascertain how much the given loading will cause the beam or truss to sag, or, in engineering language, to deflect below the position occupied when unloaded. The deflection is determined by the sagging in the vertical plane of the neutral line below its position when the structure carries no load. In Fig. 10 the curved line _AB_ is the neutral line of the beam when supporting loads. If the loads should be removed, the line _AB_ would return to a horizontal position. The line drawn horizontally through _A_ and indicated by _x_ is the position of the centre line of the beam before being bent. The vertical distance _w_ below this horizontal line shows the amount by which the point at the end of the line _x_ is dropped in consequence of the flexure of the beam. The vertical distance _w_ is therefore called the deflection. Evidently the deflection varies with the amount of loading and with the distance from the end of the beam. The curved line _AB_ in one special case only is a circle. The general character of that curve is determined by the loading and the length of span. In order that the deflection may be properly considered it is necessary that the relation between _x_ and _w_ shall be established for all conditions of loading and length of span. If the value of _u_ from equation (2) be placed in equation (3), there will result _E_ _a_ = ---. (6) ρ If the value of _a_ from equation (6) be substituted in the last member of equation (1), there will at once result _EI_ _M_ = ----. (7) ρ It is established by a very simple process in differential calculus that _I d²w_ ---- = ------. (8) ρ dx² Hence, substituting from equation (8) in equation (7), _d²w_ _M = EI_ -----. (9) _dx²_ Equation (9) may be used by means of some very simple operations in integral calculus to determine the value of _w_ in terms of _x_ and the loads on the beam when the value of the bending moment _M_ is known, and the procedures for determining that quantity will presently be given. Using the processes of the calculus, the two following equations will immediately be found: _dw_ 1 ----- = ---- ⌠ _Mdx_; (10) _dx EI_ ⌡ 1 _w = ----- ⌠⌠ Mdx²_ (11) _EI_ ⌡⌡