High School

Locate the centroid and calculate the moments of inertia for a "T" shaped beam that consists of a stem and a flange.

- The stem (upright piece) is a "2x8" with actual dimensions of 1-1/2" x 7-1/2".
- The flange (piece across the top) is a "2x6" with actual dimensions of 1-1/2" x 5-1/2", centered on the stem.

Determine the centroid location and compute the moments of inertia for this configuration.

Answer :

Final answer:

The centroid of the T-shaped beam is located at (5.625", 2.625"). The moments of inertia for the stem and flange of the beam are 5 square inches and 2.1667 square inches, respectively, with a total moment of inertia of 7.1667 square inches.

Explanation:

The centroid of a T-shaped beam can be located by finding the center of mass of the entire beam. Since the stem is a 2x8 with actual dimensions of 1-1/2"x7-1/2" and the flange is a 2x6 with actual dimensions of 1-1/2"x5-1/2", we can calculate the centroid as follows:

  1. Calculate the area of the stem: A_stem = length x width = 7.5" x 2" = 15 square inches.
  2. Calculate the area of the flange: A_flange = length x width = 5.5" x 2" = 11 square inches.
  3. Calculate the total area: A_total = A_stem + A_flange = 15 square inches + 11 square inches = 26 square inches.
  4. Calculate the x-coordinate of the centroid: x = ((A_stem x x_stem) + (A_flange x x_flange)) / A_total = ((15 square inches x (7.5"/2)) + (11 square inches x (7.5" + (5.5"/2)))) / 26 square inches = 5.625".
  5. Calculate the y-coordinate of the centroid: y = ((A_stem x y_stem) + (A_flange x y_flange)) / A_total = ((15 square inches x (2.75" + (2"/2))) + (11 square inches x (2.75" + (5.5"/2))))) / 26 square inches = 2.625".

The centroid of the T-shaped beam is located at (5.625", 2.625").

To calculate the moments of inertia, we need to consider the stem and the flange separately:

  1. Moment of inertia of the stem: I_stem = (1/12) x (height x width^3) = (1/12) x (7.5" x 2^3) = (1/12) x 7.5" x 8" = 5 square inches.
  2. Moment of inertia of the flange: I_flange = (1/12) x (height x width^3) = (1/12) x (5.5" x 2^3) = (1/12) x 5.5" x 8" = 2.1667 square inches.
  3. The total moment of inertia can be found by summing the individual moments of inertia: I_total = I_stem + I_flange = 5 square inches + 2.1667 square inches = 7.1667 square inches.

The moments of inertia of the T-shaped beam are 5 square inches for the stem and 2.1667 square inches for the flange, with a total moment of inertia of 7.1667 square inches.

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