GATE Civil Engineering (CE) 2019 Shift 1 Solved Paper

Question : 54 of 65

Marks: +1, -0

The rigid-joined plane frame QRS shown in the figure is subjected to a load

{P}

at the joint R. Let the axial deformation in the frame be neglected. If the support S undergoes a settlement of

∆=\;{P L^{2}}/{β EI}

, the vertical reaction at the support

S

will become zero when

β

is equal to

Solution:

Using slope deflection method,

M_{\QR}=\;{A I}/{ℓ}(θ_{R}-\;{3 ∆}/{ℓ})

M_{\RQ}=\;{2 G}/{ℓ}(2 θ_{R}-\;{3 ∆}/{ℓ})

M_{\RS}=\;{2 E I}/{ℓ}(2 θ_{R})

If reaction at S is equal to zero

M_{RQ}+M_{QR}+Pℓ=0

\;{6 {EI} θ_{ {R}}}/{ℓ}-\;{12 {EI} ∆}/{ℓ}^{2}+ {Pℓ}=0

\;{6 E l θ_{R}}/{ℓ}-\;{12EI}/{ℓ^2}+{Pℓ^{2}}/{βEI}+Pℓ=0

\;{6 {EI} θ_{ {R}}}/{ℓ}-\;{12 {Pℓ}}/{β}+ {Pℓ}

.....(i)
From equilibrium of joint

M_{RQ}+M_{RS}=0

\;{8 EIθ_{R}}/{ℓ}-\;{6 E I ∆}/{ℓ^{2}}=0

\;{6 {EI} θ_{ {R}}}/{ℓ}-\;{6}/{8}(\;{6 {EI}}/{ℓ^{2}} × \;{ {P} ℓ^{3}}/{β {EI}})

\;{6 {EI} θ_{ {R}}}/{ℓ}=\;{36 {Pℓ} ℓ}/{8 {β}}

...(ii)
⇒ From (i) & (ii)

\;{36 {P} ℓ}/{8 {β}}-\;{96 {Pℓ}}/{8 {β}}+ {Pℓ}=0

-\;{60 P ℓ}/{8 β}+Pℓ=0

⇒ \;\; 8 β=60

β=\;{60}/{8}=7.5

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