Problem 7.1. Slenderness of a column

Slenderness, λ=

Buckling in x-z plane.

Step 1.  Determine the buckling length in x-z plane.

Check buckling lengthHide buckling length

See Figure 7.19.

Bottom of the column is built-in, the top is free, thus the buckling length is:
$l_{\mathrm{o}}=\nu l=2l=2\times 300=600 \mathrm{cm}$

Step 2.  Determine the slenderness in x-z plane.

Check slenderness in x-z planeHide slenderness

Eq.(7-18)

${\lambda }_y=\frac{l_{\mathrm{o}}}{i_y}=\frac{600}{11.23}=53.43$

Buckling in x-y plane.

Step 3.  Determine the buckling length in x-y plane.

Check buckling lengthHide buckling length

See Figure 7.19.

Bottom of the column is built-in, the top is hinged, thus the buckling length is:
$l_{\mathrm{o}}=\nu l=0.7l=0.7\times 300=210 \mathrm{cm}$

Step 4.  Determine the slenderness in x-z plane.

Check slenderness in x-y planeHide slenderness

Eq.(7-18)

${\lambda }_z=\frac{l_{\mathrm{o}}}{i_z}=\frac{210}{3.02}=69.54$

Step 5.  Choose the relevant slenderness.

Check relevant slenderness in x-y planeHide relevant slenderness

Buckling in x-z plane.

Bottom of the column is built-in, the top is free, thus the buckling length in the x-z plane is:
$l_{\mathrm{o}}=\nu l=2l=2\times 300=600 \mathrm{cm}$

See Figure 7.19.

The slenderness in x-z plane is calculated as:

Eq.(7-18)

${\lambda }_y=\frac{l_{\mathrm{o}}}{i_y}=\frac{600}{11.23}=53.43$

Buckling in x-y plane.

Bottom of the column is built-in, the top is hinged, thus the buckling length in the x-y plane is:
$l_{\mathrm{o}}=\nu l=0.7l=0.7\times 300=210 \mathrm{cm}$

See Figure 7.19.

The slenderness in x-z plane results in:

Eq.(7-18)

${\lambda }_z=\frac{l_{\mathrm{o}}}{i_z}=\frac{210}{3.02}=69.54$

Buckling occurs in the x-y plane in the direction of the higher slenderness. The relevant slenderness is 

${\mathit{\lambda }}_{\mathbf{z}}\mathbf{=}\mathbf{69}\mathbf{.}\mathbf{54}>{\lambda }_y=53.43$