Test: Determinacy And Indeterminacy - 3

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QUESTION: 1

Neglecting axial deformation, the kinematic indeterminacy of the structure shown in the figure below is:

A.
12
B.
14
C.
20
D.
22

Solution:

Kinematic indeterminacy means degree of freedom of structure at various joints.
No rotation or translation is possible at A so degree of freedom at A is zero. There is a possibility of rotation at C but no translation so degree of freedom is one. At G both rotation and translation is possible so. degree of freedom is 2. At J no rotation but translation so d.o.f. is 1 . At B, D, H and K there are 4 rotations and 1 translation so d.o.f. is 5. At E, F and I three rotations and two translations so d.o.f., is 5.
So kinematic indeterminacy,
= 0 + 1 + 2 + 1 + 5 + 5 = 14
Alternate:
From direct formula
External reactions r_e = 3 + 2 + 1 + 2 = 8
Number of members (m) = 11
Number of rigid joints (j) = 9
Number of hinged joints (j') = 2
There are no internal hinges so number of releases is zero.
r_r = 0
Degree of kinematic indeterminacy,
D_k = 3(j + j') - r_e + r_r- m
= 3 x (9 + 2) - 8 - 11
= 33 - 19 = 14

QUESTION: 2

What is the statical indeterminacy for the frame shown below?

A.
12
B.
15
C.
11
D.
14

Solution:

Static indeterminacy D_s = 3 m + r_e- r_r - 3(j + j') Where,
m = total members = 12
r_e = total external reactions = 12
j = total number of rigid joints = 11
j' = total number of hybrid joints = 1
r_r = total number of released reactions

∴ D_s = (3 x 12) + 12 - 1 - 3 (11 + 1)
= 36 + 11 - 36 = 11

QUESTION: 3

The statical indeterminacy for the given 3D frame is

A.
8
B.
6
C.
9
D.
12

Solution:

D_s = D_se + D_si
D_se = r_E - 6
r_E = 6 + 3 + 3 + 6 = 18
D_se = 12
D_si = 6_c - r_R = 6 x 1 - 3
= 6 - 3 [(2 - 1) + (3 - 1)]
= 6 - 9 = -3
∴ D_s = 12 - 3 = 9

QUESTION: 4

The statical indeterminacy for the given 2D frame is

A.
8
B.
6
C.
9
D.
11

Solution:

D_s = D_se + D_si
D_se = r_E - 3
r_E = 2 + 3 + 3 + 3 = 11
D_se = 8
D_si = 3C - r_R

= 6 - (2 - 1) + (2 - 1) + (2 - 1) = 3
∴ D_s = 8 + 3 = 11

QUESTION: 5

Consider the following pin-jointed plane frames

Which of these frames are stable?

A.
1, 2 and 3
B.
Both 3 and 4
C.
2, 3 and 4
D.
Both 2 and 4

Solution:

QUESTION: 6

The kinematics indeterminacy of the beam as shown in figure is

A.
5
B.
9
C.
14
D.
15

Solution:

The kinematic indeterminacy of the beam is given by,
D_k= 3j - r_e + r_r
Now, j = 5
r_e = 7
r_r = 1
∴ D_k = 3 x 5 - 7 + 1 = 9

QUESTION: 7

The degree of static indeterminacy of the hybrid plane frame as shown in figure is

A.
10
B.
11
C.
12
D.
13

Solution:

D_s = D_se + D_si
D_se = r_e - 3 = 14 - 3 = 11
D_si = 3c - r_r

= 18 - ( 2 + 2 + 3 + 2 + 3 + 2 + 2)
= 18 - (16) = 2
∴ D_s = 11 + 2 = 13

QUESTION: 8

A plane frame ABCDEFGH shown in figure has clamp support at A hinge supports at G and H, axial force release at C and moment release (hinge) at E. The static (d_s) and kinematic (α_k) indeterminacies respectively are

A.
4, 9
B.
3, 11
C.
2, 13
D.
1, 14

Solution:

QUESTION: 9

The total degree of kinematic indeterminacy of the plane frame shown in the given figure considering columns to be axially rigid is

A.
20
B.
37
C.
44
D.
28

Solution:

D_k= [3(j + j) - r_e] + r_r - m
where,
j = total number of rigid joints = 12
j' = total number of hybrid joints = 2
r_e = total number of external reactions
= 3+ 2 + 2 = 7
r_r = = (2 - 1) + (2 - 1) = 2
m = total number of axially rigid member = 9
∴ D_k = [3(12 + 2) - 7] + 2 - 9 = 35 - 7 = 28