The elasticity and resistance of the materials of engineering . epresent 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 each single segment ; it Is evi-dently equal to R for RP and R for RP. It becomes zero atP, where is found the greatest bending moment. Case II, Again, let Fig. 2 represent the same beam shown in Fig. i,but let the load be one of uniform intensity, /, extending fromend to end of the beam. Let C be placed at the centre of thespan, and let R and Ry as be- a fore, represent the two reac-ti
The elasticity and resistance of the materials of engineering . epresent 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 each single segment ; it Is evi-dently equal to R for RP and R for RP. It becomes zero atP, where is found the greatest bending moment. Case II, Again, let Fig. 2 represent the same beam shown in Fig. i,but let the load be one of uniform intensity, /, extending fromend to end of the beam. Let C be placed at the centre of thespan, and let R and Ry as be- a fore, represent the two reac-tions. Since the load is sym-metrical in reference to C, R = R. For the same reason the mo- b c> ments and shears in one half of the beam will be exactly likethose in the other ; consequently, reference will be made toone half of the beam only. Let x and Xj then be measuredfrom R toward C. The forces acting upon the beam are Rand /, the latter being uniformly continuous. Applying theformulae of the preceding Art., the bending moment at anysection x will be :. 140 THEORY OF FLEXURE. [Art. 21. M = Rx — p {x — X,) dx,. /. M=Rx^^ (3) If / is the span, at C, M becomes : 2 8 But because the load is uniform : R^^. ^.^^-<^ (4) Hence ; ^/.=%---^ ...... (5) if W is put for the total load. Placingin Eq. (3) : M =i- {Ix - X) o (6) 2 The moments M, therefore, are proportional to the abscis-sae of a parabola whose vertex is over C, and which passesthrough the origin of co-ordinates 7^. Let y^^T, then, normalto RR, be taken equal to M^, and let the parabola RAR be Art. 21.] SPECIAL MOMENTS AND SHEARS. I4I drawn. Intercepts, as FH, parallel to AC^ will represent bend-ing moments in the sections, as //, at their shear at any section is : . = « = .-,„,£_.) .. .<, or, it is equal to the load covering that portion of the beam be-tween the section i7i question and the centre. Eq. (7) shows that the shear at the centre is zero; it also showsthat S — R 2X the ends of the bea
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