A 25 × 50 mm bar of rectangular cross-section is made of plain carbon steel 40C8 \left(S_{y t}=380 N / mm ^{2} \text { and } E=207000 N / mm ^{2}\right) . The length of the bar is 500 mm. The two ends of the bar are hinged and the factor of safety is 2.5. The bar is subjected to axial compressive force.
(i) Determine the slenderness ratio;
(ii) Which of the two equations—Euler’s or Johnson’s—will you apply to the bar?
(iii) What is the safe compressive force for the bar?
Given For bar, cross-section = 25 × 50 mm.
l=500 mm \quad S_{y t}=380 N / mm ^{2} .
E = 207 000 N/mm² (fs) = 2.5
Step I Slenderness ratio
I=\frac{(50)(25)^{3}}{12} mm ^{4} .
k=\sqrt{\frac{I}{A}}=\sqrt{\left[\frac{(50)(25)^{3}}{12(50 \times 25)}\right]}=7.22 mm .
\left(\frac{l}{k}\right)=\frac{500}{7.22}=69.25 (i).
Step II Selection of equation
The boundary line between Johnson’s and Euler’s equations is given by
\frac{S_{y t}}{2}=\frac{n \pi^{2} E}{(l / k)^{2}} \quad \text { or } \quad \frac{380}{2}=\frac{(1) \pi^{2}(207000)}{(l / k)^{2}} .
\therefore \quad\left(\frac{l}{k}\right)=103.7 (ii).
Since the slenderness ratio of the bar (69.25) is less than 103.7, the bar is treated as a short column and Johnson’s equation is applicable.
Step III Safe compressive force on bar
P_{c r}=S_{y t} A\left[1-\frac{S_{y t}}{4 n \pi^{2} E}\left(\frac{l}{k}\right)^{2}\right] .
=(380)(25 \times 50)\left[1-\frac{380}{4(1) \pi^{2}(207000)(69.25)^{2}}\right] .
\text { or } \quad P_{c r}=369077.88 N .
The safe compressive force is given by
P=\frac{P_{c r}}{(f s)}=\frac{369077.88}{2.5}=147631.15 N (iii).