# Plastic Analysis

 Question 1
For the square steel beam cross-section shown in the figure, the shape factor about $z- z$ axis is $S$ and the plastic moment capacity is $M_P$. Consider yield stress $f_y = 250 MPa$ and $a = 100 mm.$ The values of $S$ and $M_P$ (rounded-off to one decimal place) are
 A $S = 2.0, M_P= 58.9 kN-m$ B $S = 2.0, M_P=100.2 kN-m$ C $S = 1.5, M_P= 58.9 kN-m$ D $S = 1.5, M_P=100.2 kN-m$
GATE CE 2022 SET-2   Design of Steel Structures
Question 1 Explanation:
Shape factor for diamond shaped section = 2
\begin{aligned} S&=\frac{M_P}{M_y}=\frac{f_yZ_P}{f_yZ_e}\\ M_P&=S.M_y=S.f_y.Z_e \end{aligned} \begin{aligned} I_{ZZ}&=\frac{a^4}{12}\\ Z_{ZZ}&=\frac{a^4 \times \sqrt{2}}{12 \times a }=\frac{\sqrt{2}a^3}{12}mm^3\\ M_P&=\left [ 2 \times 250 \times \frac{\sqrt{2} \times (100)^3}{12} \right ] \times 10^{-6} kNm\\ &=58.93kNm \end{aligned}
 Question 2
A prismatic steel beam is shown in the figure. The plastic moment, $M_{p}$ calculated for the collapse mechanism using static method and kinematic method is
 A $M_{P, \text { static }} \gt \frac{2 P L}{9}=M_{P, \text { kinematic }}$ B $M_{P, \text { static }}=\frac{2 P L}{9} \neq M_{P, \text { kinematic }}$ C $M_{P, \text { static }}=\frac{2 P L}{9}=M_{P, \text { kinematic }}$ D $M_{P, \text { static }} \lt \frac{2 P L}{9}=M_{P, \text { kinematic }}$
GATE CE 2021 SET-2   Design of Steel Structures
Question 2 Explanation: \begin{aligned} \text{At collapse,} \quad M_{p} \theta+M_{p} \phi&=P \Delta\\ \Rightarrow \quad \quad 3 M_{P} \frac{\Delta}{l}+\frac{3 M_{P} \Delta}{2 l} &=P \Delta \\ M_{P} &=\frac{2 P l}{9}\\ \text{Also,} \qquad\qquad \quad M_{P ,\text { static }}&=M_{P,\text{ kinematic}} \end{aligned}

 Question 3
The ratio of the plastic moment capacity of a beam section to its yield moment capacity is termed as
 A aspect ratio B load factor C shape factor D slenderness ratio
GATE CE 2020 SET-2   Design of Steel Structures
Question 3 Explanation:
Ratio of $\frac{M_p}{M_y}=$ Shape factor
 Question 4
If the section shown in the figure turns from fully-elastic to fully-plastic, the depth of neutral axis (N.A.), $\bar{y}$, decreases by A 10.75 mm B 12.25 mm C 13.75 mm D 15.25 mm
GATE CE 2019 SET-1   Design of Steel Structures
Question 4 Explanation: $\bar{y}=\frac{A_1y_1+A_2y_2}{A_1+A_2}=\frac{300\times 2.5+300 \times 35}{300+300}=18.75m$ Question 5
A prismatic propped cantilever beam of span L and plastic moment capacity $M_{p}$ is subjected to a concentrated load at its mid-span. If the collapse load of the beam is $\alpha \frac{M_{p}}{L}$, the value of $\alpha$ is ______
 A 2 B 4 C 6 D 8
GATE CE 2018 SET-2   Design of Steel Structures
Question 5 Explanation: $P_{u}=\frac{6M_{P}}{l}$ So, $\alpha =6$

There are 5 questions to complete.