Question

A rectangular singly reinforced concrete beam has a width of 300 mm and an effective depth of 550 mm. The grades of steel and concrete are Fe500 and M25, respectively. The area of steel in tension is 1963 \(\text{mm}^2\). The depth of neutral axis is 302 mm. The ratio of the limiting depth of the neutral axis (\(x_{u,max}\)) to the effective depth (\(d\)) of the beam is 0.456.
The most probable mode of failure of the beam is

• A. compression failure
• B. bond failure
• C. shear failure
• D. ductile failure

Hint:

Compare the actual depth of neutral axis \(x_u\) with the limiting depth \(x_{u,max}\); if \(x_u > x_{u,max}\), the section is over-reinforced and fails in compression.

Answer 1

The Correct Option is A

Step 1: Recall the idea of a limiting (balanced) neutral axis.
In limit-state design of a singly reinforced beam, the limiting neutral axis depth \(x_{u,max}\) is the depth at which the concrete reaches its ultimate compressive strain of 0.0035 at exactly the same moment the tension steel reaches its design yield strain. If the actual neutral axis \(x_u\) sits deeper than \(x_{u,max}\), there is too much steel for the concrete to let it yield first, this is an over-reinforced section. If \(x_u\) is shallower than \(x_{u,max}\), the steel yields first, giving a ductile, under-reinforced section.

Step 2: Compute the limiting neutral axis depth for this beam.
We are given \(\dfrac{x_{u,max}}{d}=0.456\) and \(d=550\) mm, so:
\[ x_{u,max}=0.456\times 550=250.8 \text{ mm} \]

Step 3: Compare the actual neutral axis depth to this limit.
The given actual depth of neutral axis is \(x_u=302\) mm. Compare:
\[ x_u=302\text{ mm} \;>\; x_{u,max}=250.8\text{ mm} \]
The actual neutral axis lies well deeper than the limiting one.

Step 4: Interpret what this means physically.
Since \(x_u>x_{u,max}\), the beam has more tension steel than a balanced section would need. As the load increases, the concrete on the compression side reaches its crushing strain of 0.0035 before the steel on the tension side reaches its yield strain. The concrete crushes suddenly while the steel is still behaving elastically, this is a compression failure (also called an over-reinforced or brittle failure), and it gives little to no warning before collapse, unlike a ductile, under-reinforced beam.

Step 5: Rule out the other options.
Bond failure and shear failure are governed by anchorage length and shear reinforcement respectively, neither of which is being discussed here, no bond or shear data is even given. Ductile failure is exactly the opposite of what happens when \(x_u>x_{u,max}\), ductile behaviour needs the steel to yield first, which is not the case here.

Final Answer:
Since the actual neutral axis depth exceeds the limiting value, the beam is over-reinforced and fails by concrete crushing before the steel yields, a compression failure. \[ \boxed{\text{Compression failure}} \]