Determine the ratio of strenghts of a solid steel column to that a hollow column of the same material and having the same cross sectional area. The internal diameter of the hollow column is 1/2 of its external diameter. Both the columns are same length and are pinned at their both ends

To determine the ratio of strengths between a solid steel column and a hollow column with the same material and cross-sectional area, you'll need to understand the principles of structural mechanics.

1. Calculate the area of the cross-section for both columns:
- For the solid steel column, the cross-sectional area will be equal to the area of a circle with a diameter equal to the external diameter of the hollow column.
- For the hollow column, the cross-sectional area will be equal to the difference between the area of a circle with a diameter equal to the external diameter and the area of a circle with a diameter equal to the internal diameter.

2. The strength of a column is typically proportional to its cross-sectional area. Therefore, the ratio of strengths between the solid steel column and the hollow column can be determined by comparing their cross-sectional areas.

3. Substitute the calculated areas into the following formula:
Ratio of strengths = Area of solid steel column / Area of hollow column

4. Simplify the equation to obtain the ratio of strengths.

Note: It is important to consider that there are additional factors, such as the material properties, column length, and boundary conditions, which may also influence the strength of the columns.

To determine the ratio of strengths between a solid steel column and a hollow column with the same material, cross-sectional area, and length, we need to consider the moment of inertia.

The moment of inertia is a measure of how resistant an object is to changes in rotational motion. For a column, it represents the resistance to bending.

Let's denote the external diameter of the hollow column as D, and the internal diameter of the hollow column as d (which is 1/2 of D). The cross-sectional area of both columns is the same.

The moment of inertia for a solid column is given by:

I_solid = (π/4) * D^4

The moment of inertia for a hollow column is given by:

I_hollow = (π/4) * (D^4 - d^4)

Since both columns are made of the same material and have the same cross-sectional area, we can compare their strengths using the moment of inertia.

The ratio of strengths is given by:

Strength_ratio = I_solid / I_hollow

Substituting in the equations for the moment of inertia, we get:

Strength_ratio = (π/4) * D^4 / [(π/4) * (D^4 - d^4)]

Simplifying, we can cancel out the common terms:

Strength_ratio = (D^4) / (D^4 - d^4)

Since d = D/2, we can substitute d with D/2:

Strength_ratio = (D^4) / (D^4 - (D/2)^4)

Strength_ratio = (D^4) / (D^4 - (D^2/16))

To further simplify, we can multiply the numerator and denominator by 16:

Strength_ratio = (16 * D^4) / (16 * D^4 - D^4 + (D^2)/16)

Strength_ratio = (16 * D^4) / (15 * D^4 + (D^2)/16)

Therefore, the ratio of strengths of the solid steel column to the hollow column is:

Strength_ratio = (16 * D^4) / (15 * D^4 + (D^2)/16)

Hints:

Axial stress is the same because the same load is distributed over the same area.
However, buckling strength is widely different.

Calculate the ratio of the area moments of inertia, and compare buckling their strengths using Euler's buckling formula:
F=π²2EI/(KL)²
where E,I are Young's modulus, I=area moment of inertia, K=end condition constant, equals 1 when pinned, and L=length of columns.

See also for more explanation on buckling:
http://en.wikipedia.org/wiki/Buckling
or other textbooks on strength of materials.