the beam has a maximum design moment of 272 kn-m. the beam is limited to a cross-section of 250mm x 460mm, having an effective depth of 400mm. fc' = 20.7 mpa, fy = 344.7 mpa. use a steel covering of 60mm on both the top and bottom. a) design the reinforcement steel rebar at the bottom of the beam. b) design the reinforcement steel rebar at the top of the beam. c) is the tensile steel yields? show your solution

a) To design the reinforcement steel rebar at the bottom of the beam, we need to calculate the required area of steel reinforcement.

Step 1: Calculate the effective depth of the beam after deducting the steel covering from the overall depth. Effective depth = Overall depth - Steel covering Effective depth = 400mm - 60mm - 60mm Effective depth = 280mm

Step 2: Calculate the moment capacity of the beam using the given maximum design moment and the properties of the cross-section. Moment capacity = Maximum design moment / (Effective depth * Width of the beam) Moment capacity = 272 kN-m / (280mm * 250mm) Moment capacity = 0.0389 kN/mm

Step 3: Calculate the required area of steel reinforcement using the moment capacity and the properties of the steel. Required area of steel reinforcement = Moment capacity / (0.87 * fy) Required area of steel reinforcement = 0.0389 kN/mm / (0.87 * 344.7 MPa) Required area of steel reinforcement = 0.0389 kN/mm / 299.769 MPa Required area of steel reinforcement = 0.0001296 mm^2

Step 4: Select an appropriate size and number of steel reinforcement bars to meet the required area. Let's assume we use 16mm diameter steel reinforcement bars. Area of one 16mm diameter bar = π * (16mm/2)^2 Area of one 16mm diameter bar = 201.06 mm^2

Number of bars required = Required area of steel reinforcement / Area of one 16mm diameter bar Number of bars required = 0.0001296 mm^2 / 201.06 mm^2 Number of bars required = 0.000000644 bars

Since we cannot have a fraction of a bar, we need to round up the number of bars required to the nearest whole number. Therefore, we would need at least 1 bar of 16mm diameter steel reinforcement at the bottom of the beam.

b) To design the reinforcement steel rebar at the top of the beam, we follow the same steps as above.

Step 1: Calculate the effective depth of the beam after deducting the steel covering from the overall depth. (Same as in part a)

Step 2: Calculate the moment capacity of the beam using the given maximum design moment and the properties of the cross-section. (Same as in part a)

Step 3: Calculate the required area of steel reinforcement using the moment capacity and the properties of the steel. (Same as in part a)

Step 4: Select an appropriate size and number of steel reinforcement bars to meet the required area. (Same as in part a)

Therefore, we would also need at least 1 bar of 16mm diameter steel reinforcement at the top of the beam.

c) To determine if the tensile steel yields, we need to calculate the maximum moment capacity of the beam and compare it to the given maximum design moment.

Maximum moment capacity = 0.87 * fy * (Area of steel reinforcement at the bottom + Area of steel reinforcement at the top) Maximum moment capacity = 0.87 * 344.7 MPa * (1 * 201.06 mm^2) Maximum moment capacity = 0.87 * 344.7 MPa * 201.06 mm^2 Maximum moment capacity = 0.0597 kN/mm

Since the maximum moment capacity (0.0597 kN/mm) is greater than the given maximum design moment (0.0389 kN/mm), the tensile steel does not yield.