Question 1.4.16: Develop specific Euler equations for the sizes of columns ha...

(a) Using A = πd2/4 and k =√I/A = {\pi d^2}/{4} \ and \ k=\sqrt{{l}/{A}} =\left[\left({\pi d^2}/{64}\right)\left({\pi d^2}/{4}\right) \right]^{{1}/{2}} ={d}/{4} with Eq. (4–44) gives

\frac{P_{cr}} {A} = \frac{C\pi ^2E} {\left({l}/{k}\right)^2 }       (4–44)

d=\left(\frac{64P_{cr}l^2}{\pi ^3CE} \right) ^{{1}/{4}}       (4-51)

(b)- For the rectangular column, we specify a cross section h \times b with the restriction that h \leq b. If the end conditions are the same for buckling in both directions,then buckling will occur in the direction of the least thickness. Therefore

A=bh k^2={I}/{A}=\frac{h^2}{12}

Substituting these in Eq. (4–44) gives

\frac{P_{cr}} {A} = \frac{C\pi ^2E} {\left({l}/{k}\right)^2 }       (4–44)

b=\frac{12 P_{cr}l^2}{\pi ^3CEh^3} h \leq b      (4-52)

Note, however, that rectangular columns do not generally have the same end conditions in both directions.