Project 1

To determine the loading on the beam indicated, we need to know the distance between adjacent beams and the distance between the column and the first beam.

Assuming the beams are equally spaced, we can calculate the distance between adjacent beams as follows:

Distance between adjacent beams = (Total length of floor - 2 x column width) / (Number of beams - 1) = (24 m - 2 x 0.45 m) / (5 - 1) = 5.775 m

Assuming the distance between the column and the first beam is also 0.45 m, the distance between the column center and the center of the indicated beam is 5.775 m x 2 + 0.45 m = 12 m.

To analyze the beam, we can use the slope deflection method. The first step is to determine the fixed-end moments due to the applied loads. The loading on the beam consists of dead load, live load, and floor finish and mech loads. The total load on the beam can be calculated as follows:

Total load = Dead load + Live load + Floor finish and mech loads = (120 mm x 24 kN/m³) / 1000 m³ + 3 kPa + 5 kPa = 3.072 kN/m + 5 kPa

The load per unit length on the beam is equal to the total load multiplied by the distance between adjacent beams:

Load per unit length = Total load x Distance between adjacent beams = (3.072 kN/m + 5 kPa) x 5.775 m = 17.733 kN/m

The fixed-end moments due to the applied loads can be calculated as follows:

Fixed-end moment at A = Fixed-end moment at B = (17.733 kN/m x 12 m)² / 16 = 3158.07 kNm

The next step is to calculate the slope deflection coefficients for each span. The moment of inertia of the beam can be calculated as follows:

Moment of inertia = (1/12) x Width x Height³ = (1/12) x 0.5 m x 0.75 m³ = 0.02344 m⁴

The slope deflection coefficients for each span can be calculated using the following formulas:

a = 5EI / (L² + 4EI) b = 6EI / (L² + 4EI) c = EI / (L + 2EI / K)

where E is the modulus of elasticity of concrete, I is the moment of inertia of the beam, L is the span length, and K is the stiffness of the support.

Assuming the stiffness of the support is infinite, the slope deflection coefficients can be calculated as follows:

a = 5 x 10⁹ Pa x 0.02344 m⁴ / (12 m)² = 2.385 x 10⁻⁴ b = 6 x 10⁹ Pa x 0.02344 m⁴ / (12 m)² = 2.862 x 10⁻⁴

The span length of the beam is equal to the distance between adjacent beams, which is 5.775 m. The slope deflection coefficient for the mid-span is:

c = 10⁹ Pa x 0.02344 m⁴ / (5.775 m + 2 x 10⁹ Pa x 0.75 m x 0.5 m³) = 4.756 x 10⁻⁴