Question 23.7: A column of hollow rectangular cross-section and made of ste...

\text { Given } S_{y t}=400 N / mm ^{2} \quad E=207000 N / mm ^{2} .

l = 1 m n = 1.5 (for bending about long axis)
n = 1 (for bending about short axis)
Step I Slenderness ratio about XX and YY axis
A = 30 × 20 – 25 × 15 = 225 mm².

I_{x x}=\frac{1}{12}\left[30(20)^{3}-25(15)^{3}\right]=12968.75 mm ^{4} .

I_{y y}=\frac{1}{12}\left[20(30)^{3}-15(25)^{3}\right]=25468.75 mm ^{4} .

k_{x x}=\sqrt{\left(\frac{I_{x x}}{A}\right)}=\sqrt{\frac{12968.75}{225}}=7.59 mm .

k_{y y}=\sqrt{\left(\frac{I_{y y}}{A}\right)}=\sqrt{\frac{25468.75}{225}}=10.64 mm .

\left(\frac{l}{k_{x x}}\right)=\frac{1000}{7.59}=131.75            (i).

\left(\frac{l}{k_{y y}}\right)=\frac{1000}{10.64}=93.98          (ii).

Step II Critical load along XX-axis

\frac{S_{y t}}{2}=\frac{n \pi^{2} E}{(l / k)^{2}} \text { or } \frac{400}{2}=\frac{(1.5) \pi^{2}(20701}{\left(\frac{l}{k_{x x}}\right)^{2}} .

\therefore \quad\left(\frac{l}{k_{x x}}\right)=123.78                   (iii).

From (i) and (iii), the column is treated as a long column. Using Euler’s equation,

P_{c r}=\frac{n \pi^{2} E A}{\left(l / k_{x x}\right)^{2}}=\frac{(1.5) \pi^{2}(207000)(225)}{(131.75)^{2}} .

=39723.05 N               (a)
Step III Critical load along YY-axis

\frac{S_{y t}}{2}=\frac{n \pi^{2} E}{\left(l / k_{y y}\right)^{2}} \quad \text { or } \quad \frac{400}{2}=\frac{(1) \pi^{2}(207000)}{\left(\frac{l}{k_{y y}}\right)^{2}} .

\therefore \quad\left(\frac{l}{k_{y y}}\right)=101.07               ( iv)
From (ii) and (iv), the column is treated as a short column. Using Johnson’s equation,

P_{c r}=S_{y t} A\left[1-\frac{S_{y t}}{4 n \pi^{2} E}\left(\frac{l}{k_{y y}}\right)^{2}\right] .

=(400)(225)\left[1-\frac{400}{4(1) \pi^{2}(207000)}(93.98)^{2}\right] .

or                P_{c r}=51091.61 N               (b)
Step IV Load carrying capacity
From (a) and (b), the load carrying capacity of the column is 39 723.05 N.