Short column of rectangular cross section 160mm by 60mm carrying point load of 40 KN at max eccentricity position along the longer axis so that no tensile stress is applied.
I worked out max eccentricity as 26.66 mm from RH and LH, and compressive stress as 4.167 MPa.
The column is to be replaced by an I beam having at least same section modulus (Z) as the rectangular column.
Not sure how to work out Z ??
To determine the section modulus (Z) of the I-beam that would have the same section modulus as the rectangular column, you need to follow a few steps:
Step 1: Calculate the section modulus (Z) of the rectangular column.
The section modulus (Z) can be calculated using the formula: Z = (b * h^2) / 6, where b is the base width and h is the height of the rectangular column.
Given that the base width (b) is 160mm and the height (h) is 60mm, plug these values into the formula to calculate the section modulus (Z) for the rectangular column.
Z = (160mm * (60mm)^2) / 6
Z = (160mm * 3600mm^2) / 6
Z = 96,000mm^3
Step 2: Find an appropriate I-beam that has a section modulus (Z) equal to or greater than the calculated value for the rectangular column.
You can refer to standard beam tables or manufacturer's documentation to find an I-beam that meets the required section modulus.
Keep in mind that the section modulus (Z) for an I-beam depends on the dimensions and shape of the beam's cross-section. It is typically given in units of millimeters cubed (mm^3).
By comparing the calculated section modulus of the rectangular column (96,000mm^3) to the listed section modulus of various I-beams in the table, you can find an I-beam that has a section modulus equal to or greater than 96,000mm^3.
Ensure that the I-beam you select is capable of withstanding the applied loads and meets other design requirements. Consulting a structural engineer or referring to appropriate design codes and standards can be helpful in this process.
To calculate the section modulus (Z) of an I-beam, you need to know the dimensions of the I-beam's cross-section, specifically the moment of inertia (I) and the distance from the neutral axis to the outermost fibre (c).
The section modulus (Z) can be calculated using the formula:
Z = I / c
However, before we can proceed with the calculations, we need to determine the dimensions of the I-beam that will provide the same section modulus as the rectangular column.
Given:
Width of rectangular column (b) = 160 mm
Height of rectangular column (h) = 60 mm
Eccentricity (e) = 26.66 mm
Compressive stress (σ) = 4.167 MPa
First, let's determine the moment of inertia (I) of the rectangular column:
I(rectangular) = (b * h^3) / 12
Substituting the known values:
I(rectangular) = (160 * 60^3) / 12
Now we can calculate the distance to the outermost fiber (c) for the rectangular column. Since the rectangular column carries the load at its maximum eccentricity position along the longer axis, the distance to the outermost fiber would be half the height of the rectangle.
c(rectangular) = h / 2 = 60 / 2 = 30 mm
Now, to calculate the section modulus (Z) for the rectangular column:
Z(rectangular) = I(rectangular) / c(rectangular)
Next, let's calculate the section modulus (Z) for the I-beam. Since the I-beam has a different cross-sectional shape, its moment of inertia and distance to the outermost fiber will be different.
The moment of inertia for an I-beam can be calculated as the sum of the moments of inertia of its individual components. The formula for the moment of inertia (I) of an I-beam is given by:
I(beam) = 2 * I(f) + A(f) * d(f)^2
Where:
I(f) = Moment of inertia of the flange
A(f) = Area of the flange
d(f) = Distance from the centroid of the flange to the neutral axis
Since we need to have the same section modulus (Z) as the rectangular column, we can set the Z's of both shapes equal to each other:
Z(rectangular) = Z(beam)
This implies that:
I(rectangular) / c(rectangular) = I(beam) / c(beam)
Simplifying, we get:
I(beam) = (I(rectangular) / c(rectangular)) * c(beam)
Now we need to determine the moment of inertia of the flange and the distance from the centroid of the flange to the neutral axis for the I-beam. These parameters depend on the specific dimensions and design of the I-beam.