The elasticity and resistance of the materials of engineering . f bending at any given section, the greatestpossible amount of load must be brought on the beam for thegreatest bending moment at any section. Hence, tJie greatest bendijig moment, at the specified section,will exist when the load covers the whole span. It also follows that all sections will suffer their greatestbending moments with the same position of load. The principles involved in these results find important ap-plications in the theory of truss bridges. 138 THEORY OF FLEXURE. [Art. 21. Art. 21.—Moments and Shears in Special
The elasticity and resistance of the materials of engineering . f bending at any given section, the greatestpossible amount of load must be brought on the beam for thegreatest bending moment at any section. Hence, tJie greatest bendijig moment, at the specified section,will exist when the load covers the whole span. It also follows that all sections will suffer their greatestbending moments with the same position of load. The principles involved in these results find important ap-plications in the theory of truss bridges. 138 THEORY OF FLEXURE. [Art. 21. Art. 21.—Moments and Shears in Special Cases. Certain special cases of beams are of such common occur-rence, and consequently of such importance, that a somewhatmore detailed treatment than that already given may bedeemed desirable. The following cases are of this character. Case I. Let a non-continuous beam, supporting a single weight P at any point, be considered,and let such a beam be rep-resented in Fig. I. If thespan RR is represented by I ^ a -^ b = RP-^ RP, the reactions R and R will be :. i? = ^ P, and R = y P (I) Consequently, if x represents the distance of any section inRP from R, while x represents the distance of any section ofRP from R, the general values of the bending moments forthe two segments a and b of the beam will be : M = Rx, and M = Rx (2) These two moments become equal to each other and repre-sent f/ie greatest bendmg moment in the beam when X ?=^ a and x = b, Art. 21.] SPECIAL MOMENTS AND SHEARS. 139 or, wJien the section is taken at the point of application of theload P. Eq. (2) shows that the moments vary directly as the dis-tances from the ends of the beam. Hence, if ^/(normal toRR^ is taken by any convenient scale to represent the greatest moment, — P, and if RAR is drawn, any intercept parallel to AP 2iVidi lying between RAR and RR will represent the bend-ing moment for the section at its foot, by the same scale. Inthis manner CD is the bending moment at D. The shear is uniform for
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