# Determinacy and Indeterminacy

 Question 1
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 1 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 2
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 2 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 3
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 3 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}
 Question 4
The kinematic indeterminacy of the plane truss shown in the figure is A 11 B 8 C 3 D 0
GATE CE 2016 SET-2   Structural Analysis
Question 4 Explanation:
Kinematic indeterminacy,
\begin{aligned} D_{k}&=2 j-r_{e} \\ &=2 \times 7-3=11 \end{aligned}
 Question 5
A guided support as shown in the figure below is represented by three springs (horizontal, vertical and rotational) with stiffness $k_{x},k_{y}\: and \: k_{\theta }$ respectively. The limiting values of $k_{x},k_{y}\: and \: k_{\theta }$ are: A $\infty ,0,\infty$ B $\infty ,\infty ,\infty$ C $0 ,\infty ,\infty$ D $\infty ,\infty ,0$
GATE CE 2015 SET-2   Structural Analysis
 Question 6
The static indeterminacy of the two-span continuous beam with an internal hinge,shown below,is_________ A 0 B 1 C 2 D -1
GATE CE 2014 SET-2   Structural Analysis
Question 6 Explanation:
Number of member,
$m=4$
Number of external reaction,
$r_{e}=4$
Number of joint,
$j=5$
Number of reaction released,
$r_{r}=1$
Degree of static -indeterminacy
\begin{aligned} D_{s} &=3 m+r_{e}-3 j-r_{r} \\ &=3 \times 4+4-3 \times 5-1 \\ &=0 \end{aligned}
 Question 7
The degree of static indeterminacy of a rigid jointed frame PQR supported as shown in the figure is A zero B one C two D unstable
GATE CE 2014 SET-1   Structural Analysis
Question 7 Explanation:
\begin{aligned} D_{s} &=D_{s_{e}}+D_{s_{i}} \\ &=\left(r_{e}-3\right)+3 C-r_{r} \\ &=(4-3)+3 \times 0-1 \\ &=0 \end{aligned}
Since cable is provided, it implies that there is only vertical loading.
Hence the structure is stable.
 Question 8
The degree of static indeterminacy of rigidity jointed frame in a horizontal plane and subjected to vertical load only, as shown in figure below, is A 6 B 4 C 3 D 1
GATE CE 2009   Structural Analysis
Question 8 Explanation:
In general, total number of equilibrium equations for a 3-D rigidly jointed frame are 6. i.e.
$\Sigma F_{x}=0, \Sigma F_{y}=0, \Sigma F_{2}=0, \Sigma M_{x}=0, \Sigma M_{y}=0$
$\Sigma M_{z}=0$
But for vertical loading, these equilibrium equations reduces to 3.
I.e. $\Sigma F_{y}=0, \Sigma M_{x}=0\text{ and }\Sigma M_{z}=0$
Also, at fixed support in a 3 -D rigidly jointed frame
total number of external reactions are 6 i.e..
$R_{x^{\prime}}, R_{y^{\prime}} R_{z^{\prime}} M_{x^{\prime}} M_{y^{\prime}} M_{z}$
But for vertical loading, these external reactions reduce to 3 i.e.,
$R_{y^{\prime}} M_{x^{\prime}} M_{z}$
Thus for two fixed supports total external reactions will be 6
$\therefore$ Degree of static indeterminacy
= Total external reactions - Total number of available equilibrium equations
$=6-3=3$
 Question 9
The degree of static indeterminacy of the rigid frame having two internal hinges as shown in the figure below, is A 8 B 7 C 6 D 5
GATE CE 2008   Structural Analysis
Question 9 Explanation:
The degree of static indeterminacy for a rigid hybrid frame is given by,
\begin{aligned} D_{s} &=3 m+r_{e}-r_{r}-3(j+7) \\ \text { Where, } m &=\text { total number of members }=9 \\ r_{e} &=\text { total number of external reactions } \\ &=2+1+1=4\\ r_{r}&= \text{total number of released reactions at }\\ &\text{hybrid joint}\\ &=\Sigma\left(m_{j}-1\right)=(2-1)+(2-1)=2 \\ j &=\text { total number of rigid joints }=6 \\ j &=\text { total number of hybrid joints }=2 \\ \therefore \quad D_{S} &=(3 \times 9)+4-2-3(6+2) \\ &=27+4-2-24=31-26=5 \end{aligned}
 Question 10
Considering beam as axially rigid, the degree of freedom of a plane frame shown below is A 9 B 8 C 7 D 6
GATE CE 2005   Structural Analysis
Question 10 Explanation:
Degree of kinematic indeterminacy or degree of freedom,
\begin{aligned} D_{k} &=3 j-r_{e}-m \\ &=3 \times 4-3-1 \\ &=8 \end{aligned}
There are 10 questions to complete.