Given data:
I = 1200 cm^4
d = 12 cm
σ_max,tensile = 2σ_max,compressive

To find:
Section modulus in compression

Solution:
The formula for section modulus is given by:

Z = I / y

where,
Z = Section modulus
I = Moment of inertia
y = Distance from the neutral axis to the outermost fiber

From the given data, we know that the flange of the T-section is in compression. Therefore, the neutral axis will be located at the centroid of the compression flange.

The distance from the neutral axis to the top of the compression flange is given by:

y = d/2 = 6 cm

We know that the maximum tensile stress is two times the maximum compressive stress. Therefore, we can write:

σ_max,tensile = 2σ_max,compressive

σ_max,compressive = σ_max,tensile / 2

The maximum compressive stress occurs at the top of the compression flange, and it is given by:

σ_max,compressive = Mc / I

where,
M = Maximum bending moment
c = Distance from the neutral axis to the top of the compression flange

Since the maximum compressive stress occurs at the top of the compression flange, we can assume that the bending moment is equal to the moment of resistance of the compression flange.

The moment of resistance of the compression flange is given by:

M = σ_c * Z_c

where,
σ_c = Compressive stress in the compression flange
Z_c = Section modulus of the compression flange

Substituting the values of M and σ_max,compressive, we get:

σ_max,compressive = σ_c * Z_c / I

σ_max,tensile / 2 = σ_c * Z_c / I

σ_c = (σ_max,tensile / 2) * (I / Z_c)

The maximum compressive stress occurs when the compression flange is compressed to its maximum capacity. This occurs when the extreme fiber of the compression flange reaches the yield stress. Therefore, the compressive stress can be taken as the yield stress.

σ_c = fy / γ_m

where,
fy = Yield stress of the material
γ_m = Factor of safety

Substituting the above equations, we get:

fy / γ_m = (σ_max,tensile / 2) * (I / Z_c)

Z_c = (fy / γ_m) * (I / (σ_max,tensile / 2))

Z_c = (2fy / γ_m) * (I / σ_max,tensile)

Substituting the given values, we get:

Z_c = (2 * 1200) / (2 * σ_max,compressive)

Z_c = 600 / σ_max,compressive

Z_c = 600 / (σ_max,tensile / 2)

Z_c = 1200 / σ_max,tensile

Therefore, the section modulus in compression is given by:

Z = I / y = 1200 / 6 = 200 cm^3 (Answer)

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