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

1
Review Beam Theory 2 (Contd)
IV. Successive Integration Method / Shear and
Bending Moment Diagrams q(x)
(q-dV/dx) q loading function V(x)
-?q(x)dxC1 (VdM/dx) Vshear force M(x)
?V(x)dxC1x C2 (Mdq/dx) Mbending
moment q(x) 1/EI?(M(x)dx1/2C1x2 C2 xC3
(qEI(dv/dx)) qcurvatureslope of
y-displacement curve y(x) ?q(x)dx 1/6C1x3
1/2C2 x2C3x C4 yvertical displacement V.
Sign Conventions VI. Stresses and
Strains in Beams FLEXURE FORMULA sx
-My/I where yvertical distance from
NA sx(max)-Mmaxymax/I, (rectangular)
ymaxh/2 Imoment of inertia of cross
section about NA Irectangularbh3/12,
Icircularpr4/4 PARALLEL AXIS THEORUM
IAAIooAd2 STRAIN FORMULA
ex-yM/EI SHEAR FORMULA txy VQ/Ib,
txy(max)VmaxQmax/Ib (at NA) Qfirst moment
of the area above y about the NA?Aiyi
(areamoment arm) Qrectangularb/2(h2/4-y2)?tx
y(max)3Vmax/2bh
V
tension
compression
M
M
M
M
(-)
()
V
compression
tension
sx (y)
txy (y)
y
sy 0
sx (max)c
txy (max)
M
c
x
NA
M
o
h
o

d
b
sx (max)T
A
A
( moment)