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IV. Successive Integration Method / Shear and Bending Moment Diagrams :

V. (+). (-). V. Review : Beam Theory 2 (Cont’d). IV. Successive Integration Method / Shear and Bending Moment Diagrams : q(x) (q=-dV/dx) q= loading function V(x) =-  q(x)dx+C 1 ( V=dM/dx) V=shear force M(x) =  V(x)dx+C 1 x +C 2 ( M=d q /dx) M=bending moment

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IV. Successive Integration Method / Shear and Bending Moment Diagrams :

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  1. V (+) (-) V Review : Beam Theory 2 (Cont’d) IV. Successive Integration Method / Shear and Bending Moment Diagrams : q(x)(q=-dV/dx) q= loading function V(x)=-q(x)dx+C1 (V=dM/dx) V=shear force M(x)=V(x)dx+C1x +C2 (M=dq/dx) M=bending moment q(x)=1/EI(M(x)dx+1/2C1x2 +C2 x+C3(q=EI(dv/dx)) q=curvature=slope of y-displacement curve y(x)=q(x)dx +1/6C1x3 +1/2C2 x2+C3x +C4y=vertical displacement V. Sign Conventions : VI. Stresses and Strains in Beams : FLEXURE FORMULA :sx=-My/I where : y=vertical distance from NA sx(max)=-Mmaxymax/I, (rectangular) ymax=h/2 I=moment of inertia of cross section about NA Irectangular=bh3/12, Icircular=pr4/4 PARALLEL AXIS THEORUM :IAA=Ioo+Ad2 STRAIN FORMULA:ex=-yM/EI SHEAR FORMULA :txy=VQ/Ib, txy(max)=VmaxQmax/Ib (at NA) Q=first moment of the area above y about the NA=Aiyi(area•moment arm) Qrectangular=b/2(h2/4-y2)txy(max)=3Vmax/2bh V tension compression M M M M (-) (+) compression V tension sx (y) txy (y) y sy =0 sx (max)c txy (max) M c x M NA o h o • d b sx (max)T A A (+ moment)

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