Page 375 - statbility for masters and mates

Let
Bending and shear stresses 363
bEdy dA The force acting on dA fEdA
M1 yEdA I1
Let A1 be the cross-sectional area at ab, then the total force (P1) acting on the area A1 is given by:
P1 M1 S yEdA I1
M1 y A1 I1
Let P2 be the force acting on the same area (A2) at section cd.
Then
where and
Since dx is small then A1 A2 and I1 I2. Therefore let A area and I second moment of the section about the neutral axis.
P2 M2 y A2 I2
M2 Bending moment at this section,
I2 Second moment of the section about the neutral axis.
Shearing force at A P1 P2
M1 M2 y A
I
M1 M2 dx y A
M M dxI
1 2 is equal to the vertical shearing force at this section of the
; P1 P2 F y A dx I
Now let `q' be the shearing stress per unit area at A and let `t' be the thickness of the beam, then the shearing force is equal to qEtEdx.
; qEtEdx F y A dx I
But
beam. dx