Question 8.4: A thin-walled pin-ended column is 2 m long and has the cross...

Since the cross-section of the column is doubly symmetrical, the shear center coincides with the centroid of area and x_{S}=y_{S}=0; Eqs. (8.74), (8.75), and (8.76) therefore apply. Further, the boundary conditions are those of the column whose buckled shape is defined by Eqs. (8.71), so that the buckling load of the column is the lowest of the three values given by Eqs. (8.77).

E I_{x x} \frac{ d ^{2} v}{ d z^{2}}=-P v  (8.74)

E I_{y y} \frac{ d ^{2} u}{ d z^{2}}=-P u  (8.75)

E \Gamma \frac{ d ^{4} \theta}{ d z^{4}}-\left(G J-I_{0} \frac{P}{A}\right) \frac{ d ^{2} \theta}{ d z^{2}}=0  (8.76)

u=A_{1} \sin \frac{\pi z}{L}, \quad v=A_{2} \sin \frac{\pi z }{L} \quad \theta=A_{3} \sin \frac{\pi z}{L}  (8.71)

P_{ CR (x x)}=\frac{\pi^{2} E I_{x x}}{L^{2}} P_{ CR (y y)}=\frac{\pi^{2} E I_{y y}}{L^{2}} P_{ CR (\theta)}=\frac{A}{I_{0}}\left(G J+\frac{\pi^{2} E \Gamma}{L^{2}}\right)  (8.77)

The cross-sectional area A of the column is

The second moments of area of the cross-section about the centroidal axes Cxy are (see Chapter 15), respectively,

The polar second moment of area I_{0} is

T(z)=P\left(x_{ S } \frac{ d ^{2} v}{ d z^{2}}-y_{ S } \frac{ d ^{2} u}{ d z^{2}}\right)-I_{0} \frac{P}{A} \frac{ d ^{2} \theta}{ d z^{2}}  (8.69)

that is

The torsion constant J is obtained using Eq. (17.11), which gives

J=\sum \frac{s t^{3}}{3} \quad \text { or } \quad J=\frac{1}{3} \int_{\text {sect }} t^{3} d s  (17.11)

Finally, \Gamma is found using the method of Section 27.2 of Ref. 3 and is

Substituting these values in Eqs. (8.77), we obtain

The column therefore buckles in bending about the Cy axis when subjected to an axial load of 0.41 \times 10^{4} N.