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.

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