# Determinacy and Indeterminacy

 Question 1
Consider the following three structures: Structure I: Beam with hinge support at A, roller at $C$, guided roller at $E$, and internal hinges at $B$ and $D$. Structure II: Pin-jointed truss, with hinge support at $A$, and rollers at $B$ and $D$. Structure III: Pin-jointed truss, with hinge support at $A$ and roller at $C$.

Which of the following statements is/are TRUE?
 A Structure I is unstable B Structure II is unstable C Structure III is unstable D All three structures are stable
GATE CE 2023 SET-1   Structural Analysis
Question 1 Explanation:
Unstability in beam can be checked

(i) If support reactions are not enough.
(ii) If reactions are concurrent.
(iii) If reactions are parallel.
(iv) If there is mechanism.

where as for truss also along with the above given points the triangular panels are generally stable.
But, a virtual inspection must be conducted to understand the stability (I)
It is unstable as it shows mechanisms.
Also to understand if we apply a vertical force at slider side, the deflected shape will be as follows. (II) The frame is internally stable but all the reactions are concurrent and meeting at hinge A, and the frame can rotate about A. Hence, it is unstable. (III) The frame is unstable because it cannot resist shear in DE and AB since DE and AB are slender members. Question 2
Consider a beam PQ fixed at P, hinged at Q, and subjected to a load F as shown in figure (not drawn to scale). The static and kinematic degrees of indeterminacy, respectively, are A 2 and 1 B 2 and 0 C 1 and 2 D 2 and 2
GATE CE 2022 SET-2   Structural Analysis
Question 2 Explanation: Static indeterminacy, SI=r-3=(3+2)-3=2
Kinematic indeterminacy=0+1=1

 Question 3
The degree of static indeterminacy of the plane frame as shown in the figure is ___ A 10 B 15 C 20 D 25
GATE CE 2019 SET-2   Structural Analysis
Question 3 Explanation:
Static indeterminacy $D_s = D_{se} + D_{si}-\text{Force releases}$
External indeterminacy, $D_{se} = r - s$
No. of support reactions, r = 7
Number of equilibrium equations, s = 3
$D_{se} = 7-3=4$
Internal indeterminacy $D_{si}= 3\times \text{No of closed boxes}$
$= 3\times 4 = 12$
Force releases = 1 [At the internal hinge]
$D_S= 4 + 12 - 1 = 15$
 Question 4
Consider the frame shown in the figure: If the axial and Shear deformation in different members of the frame are assumed to be negligible. The reduction in the degree of Kinematical indeterminacy would be equal to
 A 5 B 6 C 7 D 8
GATE CE 2017 SET-2   Structural Analysis
Question 4 Explanation: $D_{k}$ (when extensible) =14
$D_{k}$( when inextensible )=$D_{k}$( when extensible) - No. of axially rigid member
$=14-6=8$
Shear deformation are already neglected. They have no significance on $D_{k}$. So, reduction in $D_{k}$ is 6.
 Question 5
A Planar truss tower structure is shown in the figure. Consider the following statements about the external and internal determinacies of the truss.
(P) Externally Determinate
(Q) External Static Indeterminacy = 1
(R) External Static Indeterminacy = 2
(S) Internally Determinate
(T) Internal Static Indeterminacy = 1
(U) Internal Static Indeterminacy = 2
Which one of the following options is correct?
 A P-False; Q-True; R-False; S-False; T-False; U-True B P-False; Q-True; R-False; S-False; T-True; U-False C P-False; Q-False; R-True; S-False; T-False; U-True D P-True; Q-True; R-False; S-True; T-False; U-True
GATE CE 2017 SET-1   Structural Analysis
Question 5 Explanation:
\begin{aligned} \text{External Indeterminacy}&=r-s=4-3=1 \text { degree } \\ \text{Internal indeterminacy}&=m-(2 j-3)=15-(2 \times 8-3)\\ & =2 \text{ degrees} \end{aligned}

There are 5 questions to complete.