A (15×15×60)cm concrete beam is subjected to three-point bending. If the load at failure is (1950+202)kgf, calculate the modulus of rupture (flexural strength) of the concrete.
The correct answer and explanation is :
To calculate the modulus of rupture (flexural strength) of the concrete beam, we use the formula:
[
f_r = \frac{P L}{b d^2}
]
where:
- ( f_r ) = modulus of rupture (MPa)
- ( P ) = load at failure (kgf)
- ( L ) = span length (cm)
- ( b ) = width of beam (cm)
- ( d ) = depth of beam (cm)
Step 1: Convert Load to kgf
Given failure load:
[
P = 1950 + 202 = 2152 \text{ kgf}
]
Step 2: Given Dimensions
- Beam width ( b = 15 ) cm
- Beam depth ( d = 15 ) cm
- Span length ( L = 60 ) cm
Step 3: Calculate Modulus of Rupture
[
f_r = \frac{2152 \times 60}{15 \times 15^2}
]
[
f_r = \frac{129120}{3375}
]
[
f_r = 38.27 \text{ kgf/cm²}
]
Step 4: Convert to MPa
Since 1 kgf/cm² = 0.0981 MPa,
[
f_r = 38.27 \times 0.0981
]
[
f_r = 3.75 \text{ MPa}
]
Final Answer:
The modulus of rupture (flexural strength) of the concrete is 3.75 MPa.
Explanation (300 words)
The modulus of rupture (MOR), also known as flexural strength, is a measure of a material’s ability to resist bending failure. It is crucial in determining the performance of concrete beams under load. In this case, we analyze a simply supported (15 × 15 × 60 cm) concrete beam subjected to three-point bending.
To determine the MOR, we use the standard formula for three-point bending. The failure load (P), which is the total load applied at the center of the beam before failure, is given as 2152 kgf. The span length (L = 60 cm) is the distance between the two supports.
The beam’s cross-sectional dimensions are 15 cm in width (b) and 15 cm in depth (d). These values are substituted into the formula to compute the modulus of rupture.
Once we obtain the value in kgf/cm² (38.27 kgf/cm²), we convert it to MPa using the conversion factor 1 kgf/cm² = 0.0981 MPa. This results in a final modulus of rupture of 3.75 MPa.
The modulus of rupture is an essential parameter in concrete design, as it helps engineers predict how a beam will behave under bending stress. A higher MOR indicates better flexural performance. This value is influenced by factors such as concrete composition, curing conditions, and reinforcement. In practical applications, MOR is used to ensure that beams can support expected loads without failure, maintaining structural integrity and safety.