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In stiffness matrix method of structure analysis, the quantity taken as redundant is- a)Deflection
- b)Rotation
- c)Both (A) and (B)
- d)None of the above
Correct answer is option 'C'. Can you explain this answer?
In stiffness matrix method of structure analysis, the quantity taken as redundant is
a)
Deflection
b)
Rotation
c)
Both (A) and (B)
d)
None of the above
|
Tanvi Shah answered |
Stiffness Matrix Method:
- The Stiffness method provides a very systematic way of analyzing determinate and indeterminate structures.
- These local (member) force-displacement relationships can be easily established for all the members in the truss, simply by using the given material and geometric properties of the different members.
- In the stiffness matrix method, nodal displacements are treated as the basic unknowns for the solution of indeterminate structures.
The properties of the stiffness matrix are
- It is a symmetric matrix
- The sum of elements in any column must be equal to zero.
- It is an unstable element therefore the determinant is equal to zero.
- The joint displacements are treated as basic unknowns
- The number of displacements involved is equal to the no of degrees of freedom of the structure
- The method is the generalization of the slope deflection method.
- The same procedure is used for both determinate and indeterminate structures.
- Deflection and Rotation are the quantity taken as redundant.
A stiffness matrix is to be generated for beam AB as a horizontal flexural member. As per the method adopted for calculation of stiffness matrix, if end A is given translational displacement in a vertically upward direction, the end forces generated at the end ‘B’ are- a)No force at end B
- b)-12EI/L3 vertical force and 6EI/L2moment
- c)-6EI/L3 vertical force and 2EI/L moment
- d)-6EI/L2 vertical force and 4EI/L moment
Correct answer is option 'B'. Can you explain this answer?
A stiffness matrix is to be generated for beam AB as a horizontal flexural member. As per the method adopted for calculation of stiffness matrix, if end A is given translational displacement in a vertically upward direction, the end forces generated at the end ‘B’ are
a)
No force at end B
b)
-12EI/L3 vertical force and 6EI/L2moment
c)
-6EI/L3 vertical force and 2EI/L moment
d)
-6EI/L2 vertical force and 4EI/L moment
|
|
Tanvi Shah answered |
Concept:
Stiffness Matrix:
The elements of the stiffness matrix are the forces that are required to hold the restrained structure with a unit displacement at one of the coordinates and with zero displacements at all other coordinates are determined.
Hence,
For the fixed beam shown below -
Let, the end A is given translational displacement in an upward direction keeping all other coordinates fixed,
Let, the end A is given translational displacement in an upward direction keeping all other coordinates fixed,
The following figure shows the forces required to hold the structure in the given position -
Hence, from the above figure, the end forces generated at B are -
- Vertical force = 12EI/L3 (Downward) = - 12EI/L3
Negative sign because opposite to the direction of displacement.
- Moment = 6EI/L2 (Anticlockwise)
Stiffness matrix with reference to coordinates 1, 2 and 3 as shown in Fig. below is:- a)
- b)
- c)
- d)
Correct answer is option 'A'. Can you explain this answer?
Stiffness matrix with reference to coordinates 1, 2 and 3 as shown in Fig. below is:
a)
b)
c)
d)
|
Pioneer Academy answered |
Concept:
Development of stiffness matrix:
To develop jth column of a stiffness matrix – unit displacements will be given in jth coordinate direction without giving displacement in any other coordinate directions and forces developed in all the coordinate direction are found out.
Ist Column:
Ist Column:
K13 = 0
K23 = 0
Important Point:
- Stiffness Matrix is a displacement method of analysis in which Degree of Kinematic Indeterminacy is found and number of coordinate direction are chosen accordingly. The structure is analysed by developing stiffness matrix.
- Flexibility Matrix is a force method of analysis in which Degree of Static Indeterminacy is found and number of coordinate direction are chosen accordingly. The structure is analysed by developing flexibility matrix.
The flexibility matrix for the simply supported beam with reference to the coordinates, as shown below, is
- a)
- b)
- c)
- d)
Correct answer is option 'A'. Can you explain this answer?
The flexibility matrix for the simply supported beam with reference to the coordinates, as shown below, is
a)
b)
c)
d)
|
|
Pioneer Academy answered |
Concept:
Flexibility matrix:
- The redundant forces are identified and assign one co-ordinate to each redundant.
- The redundant are released to obtained to obtain the primary structure.
- Due to external load, the displacements in the direction of various co-ordinate and at the location of co-ordinate are found out.
- Positive sign is taken when the direction of slope/deflection is the same as the original moment/force.
- Negative sign taken when the direction of slope/deflection is opposite as of original moment/force
Slope at the point A due to the moment at A is given by
θα = ML/3EI
Slope at point B due to moment at point A is given by
θb = ML/6EI
Calculation:
Given: EI = Constant
First, apply unit moment at A and draw its deflected shape
First, apply unit moment at A and draw its deflected shape
δ11 = L/3EI (Same as of original moment) and δ21 = L/6EI (same as of original moment)
Now apply unit moment at B and draw its deflected shape
δ21 = L/6EI(Same as of original moment) and δ22 = L/3EI(Same as of original moment)
So, The flexibility matrix will be
∴ The flexibility matrix is given by 
For stable structures, one of the important properties of flexibility and stiffness matrices is that the element on the main diagonal(i) of a stiffness matrix must be negative.
(ii) of a stiffness matrix must be positive
(iii) of a flexibility matrix must be positive
(iv) of a flexibility matrix must be negative- a)(i) and (iii)
- b)(i) and (iv)
- c)(ii) and (iii)
- d)(ii) and (iv)
Correct answer is option 'C'. Can you explain this answer?
For stable structures, one of the important properties of flexibility and stiffness matrices is that the element on the main diagonal
(i) of a stiffness matrix must be negative.
(ii) of a stiffness matrix must be positive
(iii) of a flexibility matrix must be positive
(iv) of a flexibility matrix must be negative
(ii) of a stiffness matrix must be positive
(iii) of a flexibility matrix must be positive
(iv) of a flexibility matrix must be negative
a)
(i) and (iii)
b)
(i) and (iv)
c)
(ii) and (iii)
d)
(ii) and (iv)
|
|
Tanvi Shah answered |
Flexibility and stiffness matrix:
To generate the jth column of the flexibility matrix- a)a unit force is applied at coordinate j and the displacements are calculated at all coordinates
- b)a unit displacement is applied at coordinate j and forces are calculated at all coordinates
- c)a unit force is applied at coordinate j and the forces are calculated at all coordinates
- d)a unit displacement is applied at coordinate j and the displacements are calculated at all coordinates
Correct answer is option 'A'. Can you explain this answer?
To generate the jth column of the flexibility matrix
a)
a unit force is applied at coordinate j and the displacements are calculated at all coordinates
b)
a unit displacement is applied at coordinate j and forces are calculated at all coordinates
c)
a unit force is applied at coordinate j and the forces are calculated at all coordinates
d)
a unit displacement is applied at coordinate j and the displacements are calculated at all coordinates
|
|
Tanvi Shah answered |
Stiffness Matrix Method
1. This method is also known as displacement method or equilibrium method.
2. It is suitable if static indeterminacy (Ds) > kinematic indeterminacy (Dk) for the structure.
Stiffness is defined as Force required to produce unit displacement, k = P/Δ
K12 → Force at 2 due to unit displacement at 1
Flexibility Matrix Method
1. This method is also known as force method or compatibility method.
2. It is suitable if kinematic indeterminacy (Dk) > static indeterminacy (Ds) for the structure.
Flexibility is defined as Displacement caused due to unit Force.
δ12 → Displacement at 2 due to Force at 1
∴ To generate a jth column of flexibility matrix a unit force is applied at coordinate j and the displacement are calculated at all coordinates.
The stiffness coefficients Kij indicate- a)Force at i due to a unit deformation at j
- b)Deformation at j due to a unit force at i
- c)Deformation at i due to a unit force at j
- d)Force at j due to a unit deformation at i
Correct answer is option 'A'. Can you explain this answer?
The stiffness coefficients Kij indicate
a)
Force at i due to a unit deformation at j
b)
Deformation at j due to a unit force at i
c)
Deformation at i due to a unit force at j
d)
Force at j due to a unit deformation at i
|
|
Tanvi Shah answered |
The difference between flexibility matrix and stiffness matrix is as follows:
The stiffness matrix for a beam element is 
The corresponding flexibility matrix is- a)
- b)
- c)
- d)
Correct answer is option 'B'. Can you explain this answer?
The stiffness matrix for a beam element is The corresponding flexibility matrix is
The corresponding flexibility matrix isa)
b)
c)
d)
|
|
Tanvi Shah answered |
Concept:
Stiffness matrix (K) is inverse of a flexible matrix (F) or vice versa.
K = [F]-1 or F = [K]-1
Calculation:
The stiffness matrix of a beam is given as 
Calculate the flexibility matrix. Flexibility matrix will be ______.- a)
- b)
- c)
- d)
Correct answer is option 'D'. Can you explain this answer?
The stiffness matrix of a beam is given as Calculate the flexibility matrix. Flexibility matrix will be ______.
Calculate the flexibility matrix. Flexibility matrix will be ______.a)
b)
c)
d)
|
|
Tanvi Shah answered |
If we denote stiffness matrix as M and flexibility matrix as Δ
It is stiffness matrix, and then flexibility matrix is: Δ = K-1
Calculation:
If the stiffness matrix of the beam element is given as 
then the flexibility matrix is- a)
- b)
- c)
- d)
Correct answer is option 'A'. Can you explain this answer?
If the stiffness matrix of the beam element is given as then the flexibility matrix is
then the flexibility matrix isa)
b)
c)
d)
|
|
Tanvi Shah answered |
Given:
Flexibility matrix is inverse of the stiffness matrix
So, A-1 is the stiffness matrix of A
So, A-1 is the stiffness matrix of A
∴The flexibility matrix of given stiffness matrix is 
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