Page 174 - Aircraft Stuctures for Engineering Student

158 Structural instability

changes take place outside the elastic limit of the material, we see, from our remarks
in the previous paragraph, that the modulus of elasticity of the material in the area
Al is E while that in A2 is Et. The homogeneous column now behaves as if it were
non-homogeneous, with the result that the stress distribution is changed to the
form shown in Fig. 6.7(b); the linearity of the distribution follows from an assump-
tion that plane sections remain plane.
As the axial load is unchanged by the disturbance

Also, P is applied through the centroid of each end section a distance e from nn so
that

/: cx(yl +e) dA + r uv(y2 - e) dA = -Pv (6.10)

From Fig. 6.7(b)

ff1 ff2
ffx = -Y1, ffv = -Y2 (6.1 1)
4 d2
The angle between two close, initially parallel, sections of the column is equal to the
change in slope d2v/dz2 of the column between the two sections. This, in turn, must be
equal to the angle 64 in the strain diagram of Fig. 6.7(c). Hence

(6.12)

and Eq. (6.9) becomes, from Eqs (6.1 1) and (6.12)

(6.13)

Further, in a similar manner, from Eq. (6.10)
d2
$ (. y: dA + Et r yf dA) + e 2 (. y1 dA - Et J” y2 d A) = - Pv (6.14)
0
The second term on the left-hand side of Eq. (6.14) is zero from Eq. (6.13). Therefore
we have

(6.15)

in which

Il =Jd’y;dA and I2,sd,ygdA
0
0
the second moments of area about nn of the convex and concave sides of the column
respectively. Putting

EJ = EI1 + EtIz