Design an economical square wood column that will support an axial load of 350 kN. The unsupported length is 4m and c =5 MPa. Use E = 2 GPa

Step 1: Determine the required cross-sectional area

The first step in designing the column is to determine the required cross-sectional area. This can be done using the formula for the axial stress in a column, which is given by:

σc = P/A

where: σc is the allowable stress in the column (5 MPa), P is the axial load (350 kN), and A is the cross-sectional area of the column.

Rearranging the formula to solve for A gives:

A = P/σc

Substituting the given values gives:

A = 350 kN / 5 MPa = 70,000 mm^2

Step 2: Determine the required side length of the square column

Since the column is square, its cross-sectional area is given by the square of its side length. Therefore, we can find the required side length by taking the square root of the required cross-sectional area:

side length = sqrt(A) = sqrt(70,000 mm^2) = 264.575 mm

So, the side length of the square column should be approximately 265 mm.

Step 3: Check the slenderness ratio

The slenderness ratio of a column is given by the formula:

Slenderness ratio = unsupported length / (radius of gyration)

For a square column, the radius of gyration is equal to the side length divided by the square root of 12. Therefore, the slenderness ratio is:

Slenderness ratio = 4000 mm / (265 mm / sqrt(12)) = 87.18

This is less than the typical maximum allowable slenderness ratio of 120 for wooden columns, so the column is not too slender.

Step 4: Check the buckling load

The buckling load for a column is given by the formula:

Pcr = π^2EI / L^2

where: E is the modulus of elasticity (2 GPa), I is the moment of inertia (for a square column, I = side length^4 / 12), and L is the unsupported length (4000 mm).

Substituting the given values gives:

Pcr = π^22 GPa(265 mm)^4 / 12 / (4000 mm)^2 = 516.3 kN

Since the buckling load is greater than the axial load, the column is safe against buckling.

Therefore, a square wooden column with a side length of approximately 265 mm should be sufficient to support the given axial load.