As already explained, numerical values for both _E_ and _I_ may be taken at once from tables already prepared for all materials and for all shapes of beams ordinarily employed in structural work, so that equation (11) enables the deflection or sag of the bent beam to be computed in any case. The expression _dw/dx_ is the tangent of the angle made by the neutral line of a bent beam with a horizontal line at any given point, and it is a quantity that it is sometimes necessary to determine. _dw_ and _dx_ are indefinitely short vertical and horizontal lines respectively, as shown immediately to the left of _B_ in Fig. 10. Equation (11) is not used in structural work nearly as much as equation (5), but both of them are of practical value and involve only simple operations in their use. =81. Bending Moments and Shears with Single Load.=—The second members of equations (5) and (9) exhibit values of the moments of the internal forces or stresses in any normal cross-section of a bent beam about the neutral axis of the section, while the values of _M_ must be expressed in terms of the external forces or loading. Inasmuch as the latter moment develops just the internal moment, it is obvious that the two must be equal. In order to write the value of the external moment in terms of any loading, it is probably the simplest procedure to consider a beam carrying a single load. In Fig. 12, _AB_ is such a beam, and _W_ is a load which may be placed anywhere in the span, whose length is _l_. The distances of the load from the abutments are represented by _x_₁ and _x_₂. The reactions or supporting forces exerted under the ends of the beam at the abutments are shown by _R_ and _R′_. The reactions, determined by the simple law of the lever, are _x₂ x₁_ _R = W_ ---- and _Rʹ = W_ ---- (12) _l l_ The greatest bending moment in the beam will occur at the point of application of the load, and its value will be _x₁x₂_ _M₁ = Rx₁ = W_ ------- = -_Rʹx₂_. (13) _l_ [Illustration: _Wx₁x₂_ _M₁ = Rx₁ = -Rʹx₂_ = ------- _l_ FIG. 12.] The bending moments at the end of the beam are obviously zero, and the second and fourth members of equation (13) show that the moment increases directly as the distance from either end. Hence in the lower portion of Fig. 12, at _D_, immediately under the load _W_, the line _DC_ is laid off at any convenient scale to represent the moment _M_₁. The straight lines _AC_ and _CB_ are then drawn. Any vertical intercept, as _FH_ or _KL_, between _AB_ and either _AC_ or _CB_ will represent the bending moment at the corresponding point in the beam. The simple triangular diagram _ACB_ therefore represents the complete condition of bending of the beam under the single load _W_ placed at any point in the span.