RCC Numerical – Find limiting moment of resistance and steel required for a beam

Find limiting moment of resistance and steel required for a beam 300 × 600 mm (effective), if concrete M 25 and Fe 415 steel are used.

To find the limiting moment of resistance and the amount of steel required for a beam with the given dimensions and materials, follow these steps:

Given:

• Beam dimensions: Width 𝑏=300 mm, Effective depth 𝑑=600 mm
• Concrete grade: M25
• Steel grade: Fe415

1. Limiting Moment of Resistance:

The limiting moment of resistance occurs when the concrete reaches its maximum strain (0.0035) and the steel reaches its yield strain. This is usually calculated using the concept of limiting balanced condition where the beam is on the verge of failure.

For M25 concrete and Fe415 steel, we use the following standard formulas.

Step 1: Calculate the Effective Depth of the Beam

Effective depth 𝑑 = 600 mm (already given).

Step 2: Assume a Depth for the Neutral Axis

Assume that the neutral axis (na) is at 𝑥 mm from the top of the beam. For a balanced section, this neutral axis depth can be approximated using:

𝑥=0.48⋅𝑑

where 𝑑=600 mm, so:

𝑥≈0.48⋅600=288 mm

Step 3: Calculate the Moment of Resistance

The design parameters for concrete and steel are:

• 𝑓𝑐𝑘 = 25 MPa (Concrete compressive strength)
• 𝑓𝑦 = 415 MPa (Yield strength of steel)
• 𝜙 = 0.87 (Strength reduction factor for steel)
• 𝜆 = 0.87 (Strength reduction factor for concrete)

The limiting moment of resistance 𝑀𝑢 is given by:

𝑀𝑢=0.36𝑓𝑐𝑘𝑏𝑥(𝑑−𝑥2)

where 𝑓𝑐𝑘 is in MPa, 𝑏 is in mm, and 𝑥 is the depth of the neutral axis in mm. Here,

𝑀𝑢=0.36×25×300×288(600−2882)

𝑀𝑢=0.36×25×300×288×(600−144)

𝑀𝑢=0.36×25×300×288×456

𝑀𝑢≈1,584,092,800 Nmm

𝑀𝑢≈1,584.1 kNm

2. Steel Required:

The area of steel required 𝐴𝑠 can be found by using the formula for the design moment and the assumed depth of neutral axis.

Step 1: Determine the Moment Capacity

Use the formula for the moment of resistance considering the effective depth:

𝑀𝑢=0.87𝑓𝑦𝐴𝑠(𝑑−𝑥2)

Rearrange to solve for 𝐴𝑠:

𝐴𝑠=𝑀𝑢0.87𝑓𝑦(𝑑−𝑥2)

Substitute the values:

𝐴𝑠=1,584,092,8000.87×415×(600−2882)

𝐴𝑠=1,584,092,8000.87×415×456

𝐴𝑠≈10,237 mm2

Summary

• Limiting Moment of Resistance: Approximately 1,584.1 kNm
• Steel Required: Approximately 10,237 mm2

These calculations are approximations and should be verified with detailed structural analysis and design codes as needed.