GATE 2011 AEROSPACE ENGINEERING (Q 31-40)

1. A rocket engine is tested on a test bed under the ideal condition of fully expanded jet. The exhaust velocity is 2 km/s through a nozzle of area 2.5 m². The mass flow rate is 200 kg/s. The specific impulse of the propellant and the thrust developed respectively are (assume g = 9.81 m/s²)
   1. 175.87 s and 200 kN
   2. 203.87 s and 400 kN
   3. 231.87 s and 200 kN
   4. 280.87 s and 400 kN

   Answer:- 280.87 s and 400 kN

2. A body undergoes deformation under plane strain conditions when subjected to the following stresses (in MPa): σ_xx=450, σ_yy=450, τ_xy=75, τ_xz=0, τ_yz=0. What are the remaining components of stresses (in MPa) and strains? Assume the material to be isotropic and linear-elastic with Young’s modulus E=200 GPa and Poisson’s ratio ν=1/3.
   1. σ_zz=0, ε_xx=0.00225, ε_yy=0.00225, γ_xy=0.002, γ_xz=0, γ_yz=0
   2. σ_zz=300, ε_xx=0.001, ε_yy=0.001, γ_xy=0.001, γ_xz=0, γ_yz=0
   3. σ_zz=300, ε_xx=0.00225, ε_yy=0.00225, γ_xy=0.001, γ_xz=0, γ_yz=0
   4. σ_zz=0, ε_xx=0.001, ε_yy=0.001, γ_xy=0.002, γ_xz=0, γ_yz=0

   Answer:- σ_zz=300, ε_xx=0.001, ε_yy=0.001, γ_xy=0.001, γ_xz=0, γ_yz=0
   For plain strain condition, ε_zz=0
   ⇒ (1/E)[σ_zz-ν(σ_xx+σ_yy)]=0
   ⇒ σ_zz=ν(σ_xx+σ_yy)
   ⇒ σ_zz=(1/3)(450+450)=300 MPa
   ε_xx=(1/E)[σ_xx-ν(σ_yy+σ_zz)]
   ⇒ ε_xx=(1/200000)[450-(1/3)(450+300)]
   ⇒ ε_xx=0.001
   ε_yy=(1/E)[σ_yy-ν(σ_zz+σ_xx)]
   ⇒ ε_yy=(1/200000)[450-(1/3)(300+450)]
   ⇒ ε_yy=0.001
   Shear Modulus G=E/[2(1+ν)]
   γ_xy=τ_xy/G
   ⇒ γ_xy=2τ_xy(1+ν)/E
   ⇒ γ_xy=(2)(75)[1+(1/3)]/200000=0.001
   For plain strain condition, γ_xz=0, γ_yz=0