Question

As per IS:800-2007, design of a cantilever steel beam section for its moment capacity requires fulfilment of an upper bound, expressed as:
\[ M_d \le 1.5 Z_e \frac{f_y}{\gamma_{m0}} \]
The reason for such upper bound is to

• A. control deflection
• B. restrain lateral-torsional buckling
• C. avoid plastic deformation under working load
• D. avoid yielding at ultimate load

Hint:

Compare the plastic-modulus based \(M_d\) with the first-yield moment \(Z_e f_y/\gamma_{m0}\); the 1.5 factor is a working-load serviceability safeguard, not a buckling or deflection check.

Answer 1

The Correct Option is C

Step 1: Understand the code clause.
IS:800-2007 gives the design bending strength of a laterally supported beam as \(M_d = \beta_b Z_p \dfrac{f_y}{\gamma_{m0}}\), where \(Z_p\) is the plastic section modulus. For a cantilever, the clause also fixes an upper ceiling on \(M_d\) in terms of the elastic section modulus \(Z_e\): \[ M_d \le 1.5 Z_e \frac{f_y}{\gamma_{m0}} \]

Step 2: Ask what this ceiling is doing.
The plastic modulus \(Z_p\) is always larger than the elastic modulus \(Z_e\) (the shape factor \(Z_p/Z_e\) is typically 1.12 to 1.5 for standard rolled sections). If the code let \(M_d\) grow purely off \(Z_p\) with no check against \(Z_e\), a section with a high shape factor could be assigned a design moment far above its first-yield moment \(M_y = Z_e f_y/\gamma_{m0}\), meaning the section would already be yielding and picking up plastic deformation well before the design working loads are even reached.

Step 3: Connect the limit to working load behaviour.
\(Z_e\) corresponds to the moment at which the extreme fibre just reaches yield stress \(f_y\), the first-yield moment. By capping \(M_d\) at 1.5 times this first-yield capacity, the code keeps the working-load stresses within a range where the member does not undergo large plastic deformation while still under service load. This is a serviceability-linked safeguard, not a buckling or deflection check.

Step 4: Eliminate the other options.
Deflection control (A) is checked separately through span/deflection serviceability clauses, not this formula. Lateral-torsional buckling (B) is governed by the reduction factor \(\chi_{LT}\) applied to \(\beta_b Z_p f_y\), a different clause entirely, and this bound applies even to fully restrained beams where buckling is not a concern. Avoiding yielding at ultimate load (D) is not the intent either, since limit state design accepts some yielding and moment redistribution at the factored ultimate stage; the 1.5\(Z_e\) bound is about restraining plastic deformation while the beam still resists working loads.

Final Answer:
The upper bound exists to avoid plastic deformation under working load. \[ \boxed{\text{Option (C)}} \]