three pinned-end columns of the same material have the same length and the same cross-sectional area (see figure). the columns are free to buckle in any direction. the columns have cross sections as: (a) a circle, (b) a square, and (c) an equilateral triangle. determine the ratios pa : pb : pc of the critical loads for these columns.

Question

12-12-2022
Mathematics

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three pinned-end columns of the same material have the same length and the same cross-sectional area (see figure). the columns are free to buckle in any direction. the columns have cross sections as: (a) a circle, (b) a square, and (c) an equilateral triangle. determine the ratios pa : pb : pc of the critical loads for these columns.

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contexto1028 contexto1028 · Answer 1 · 2022-12-19T08:02:08-05:00

The maximum load that a beam can withstand before breaking is known as the critical load. The SI unit of a critical load is the Newton, which is the same as the unit of a load.

The articulation for the snapshot of dormancy of a circle is,

IC=πd432=0.098174 d4

Here the measurement of the circle is d.

As a result of the materials' identical cross-sectional area, A C = A S, substitute the given values.

I S = 1 6 A 4 = 1 6 (0.88622 d) 4 = 0.102805 d 4 The materials have the same cross-sectional area, so A C = A T 4 d 2 = 3 4 B 2 B = 1.34677 d Here, the side of the triangle is B. I S = 1 6 A 4 = 1 6 (0.88622 d) 4 = 0.102805 d 4

The expression for the rectangle's moment of inertia is I T = 1 12 ( 4 d 2 ) (1.34677 d ) 2 = 0.11872 d 4. The expression for the critical load is P C = 2 E I L 2. It is evident from the aforementioned expression that the critical load is directly propositional to the moment of inertia.

The ratio of moment of inertia can be expressed as P 1:

P2:P3=IC:IS:

I T Replace the given values.

P1:P2: P3=0.098174:0.102805:

As a result, the critical load-to-weight ratio for these columns is 0.11872:

0.102805: 0.11872

To learn more about critical load here

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