Problem 10.5. Local buckling of an I beam

Problem a)

Buckling load of the flange, N_x,cr[kN/m]=

Problem b)

Buckling load of the web, N_x,cr[kN/m]=

Critical load of the total cross section, P_cr [kN]=

Buckling loads of long plates are given in Table 10.4.

Problem a)

Step 1.  Determine the buckling load of the flange with one hinged and one free edges.

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8^th row of Table 10.4.

${\mathit{N}}_{\mathbf{x}\mathbf{,}\mathbf{cr}}=6\frac{D}{L_y^2}\left(1–\nu \right)=6\times \frac{2.4}{0.{075}^2}\left(1–0.3\right)\mathbf{=}\mathbf{1792}\mathbf{ }\mathbf{kN}\mathbf{/}\mathbf{m}$

where L_y in the formula is the half width of the flange as it is shown in the Figure.

Step 2.  Give the value of the load, P acting on the cross section which results the critical load in the flange.

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Problem b)

Step 1.  Determine the buckling load of the hinged web.

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1^st row of Table 10.4.

${\mathit{N}}_{\mathbf{x}\mathbf{,}\mathbf{cr}}=\frac{4{\mathrm{\pi }}^2}{L_y^2}D=\frac{4{\mathrm{\pi }}^2}{0.{225}^2}2.4\mathbf{=}\mathbf{1872}\mathbf{ }\mathbf{kN}\mathbf{/}\mathbf{m}$

where L_y in the formula is the height of the web as it is shown in the Figure.

Step 2.  Give the value of the load, P acting on the cross section which results the critical load in the flange.

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Step 3.  Choose the relevant buckling load.

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Buckling loads of long plates are given in Table 10.4.

Problem a)

The buckling load of the flange with one hinged and one free edges is

8^th row of Table 10.4.

where L_y in the formula is the half width of the flange as it is shown in the Figure.

The critical load, P acting on the cross section which causes the buckling of the flange is 

${\mathit{P}}_{\mathbf{c}\mathbf{r}}=\min \left(973.2, 931.8\right)\mathbf{=}\mathbf{931}\mathbf{.}\mathbf{8}\mathbf{ }\mathbf{kN}$

Problem b)

The buckling load of the hinged web is

1^st row of Table 10.4.

${\mathit{N}}_{\mathbf{x}\mathbf{,}\mathbf{cr}}=\frac{4{\mathrm{\pi }}^2}{L_y^2}D=\frac{4{\mathrm{\pi }}^2}{0.{225}^2}2.4\mathbf{=}\mathbf{1872}\mathbf{ }\mathbf{kN}\mathbf{/}\mathbf{m}$

where L_y in the formula is the height of the web as it is shown in the Figure.

The critical value of the load, P acting on the cross section which cause web buckling is

${\mathit{P}}_{\mathbf{c}\mathbf{r}}=\min \left(973.2, 931.8\right)\mathbf{=}\mathbf{931}\mathbf{.}\mathbf{8}\mathbf{ }\mathbf{kN}$

Flange loses its stability first, thus flange buckling results the relevant critical load of the cross section.

${\mathit{P}}_{\mathbf{c}\mathbf{r}}=\min \left(973.2, 931.8\right)\mathbf{=}\mathbf{931}\mathbf{.}\mathbf{8}\mathbf{ }\mathbf{kN}$