Problem 1-3
Determine the critical load (buckling Load) and critical stress of the 4 m long column having its cross-section as shown in the following figure. The column has both ends fixed and the value of E for the column material is 200 GPa. .
Figure 1-3(a)
Figure 1-3(b)
Solution:
Critical load of a column is the maximum axial load at which the column will start buckling. The critical load or buckling load of column is calculated by using Euler formula as given below.
Pcr = π2 E I / (KL)2
where E = modulus of elascity
I = Least Moment of inertia
KL = Effective length of column where K is the factor for effective length which depends on the end conditions of column
Critical Stress = Critical Load / Area
= {π2 E I / (KL)2}/ A
substituting I = A r2 , where r = radius of gyration
= {π2 E A r2 / (KL)2}/ A
Critical Stress = π2 E / (KL/r)2
KL/r is known as slenderness ratio of the column
The buckling of column depends on slenderness ratio which is defined as the ratio of effective length and the least radius of gyration. The radius of gyration is calculated as (I/A)0.5.
Lower the slenderness ratio, higher the critical load. It should be kept in mind that the critical stress can not be more than yield stress of the material of column.
The moment of inertia is calculated as shown in solved problem 1-1 .
You can also use our online calculator for moment of inertia of plane sections
We get least moment of inertia (Imin) = 94.582 cm4. Please refer to the solved problem 1-1 for moment of inertia calculation.
Area of cross-section = 5 x 1 + 10 x 1 + 10 x 1 = 25 cm2 = 25 x 10 -4 m2
Minimum radius of gyration (rmin) = (Imin/A)0.5 = (98.582/25)0.5= 1.94 cm = 0.0194 m
Effective length of column depends on the end conditions as given in the following figure.
The value of effective length factor (K) is 0.5 when both ends are fixed.
Therefore effective length Leff = 0.5 x L = 0.5 x 4 = 2 m
Slenderness ratio = KL/rmin = 0.5 x 4 /0.0194 = 102.82
Pcr = {π2 x 200 x 106 kN/m2 x 94.582 x 10-8 m4}/{22 m2}
Pcr = π2 x 2 x 94.582/4 = 466.27 kN
Critical Stress = Critical Load/Area
Critical Stress = 466.27/0.0025 = 186508 kN/m2 = 186.508 x 106 N/m2
1 MPa = 106 N/m2
Critical Stress = 186.508 MPa
You can also use our online Calculator for Buckling Load Capacity of Column with I section
To get knowledge about calculation of shear force and bending moment you can visit the following related links of solved examples
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