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Chapter: Civil - Structural Analysis - Flexibility Method

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Important Questions and Answers:Flexibility Matrix Method For Indeterminate Structures

Civil - Structural Analysis - Flexibility Method - Flexibility Matrix Method For Indeterminate Structures

 

FLEXIBILITYMATRIXMETHOD FORINDETERMINATE STRUCTURES

 

1. What is meant by indeterminate structures?

 

Structures that do not satisfy the conditions of equilibrium are called indeterminate structure. These structures cannot be solved by ordinary analysis techniques.

 

2. What are the conditions of equilibrium?

 

The three conditions of equilibrium are the sum of horizontal forces, vertical forces and moment sat any joint should be equal to zero.

 

i.e. ?H=0;?V=0;?M=0

 

3. Differentiate between determinate and indeterminate structures.

Determinate structures can be solving using conditions of equilibrium alone (?H=0;?V =0;?M=0). No other conditions are required. Indeterminate structures cannot be solved using conditions of equilibrium because (?H? 0;?V?0;?M?0). Additional conditions are required for solving such structures. Usually matrix methods are adopted.

 

4. Define degree of indeterminacy (i).

 

The excess number of reactions that make a structure indeterminate is called degree of indeterminacy, and is denoted by (i).Indeterminacy is also called degree of redundancy. Indeterminacy consists of internal and external indeterminacies.

i  =II+EI

where II=internal indeterminacy

and EI=external indeterminacy.

 

5. Define internal and external indeterminacies.

Internal indeterminacy(II) is the excess no of internal forces present in a member that make a structure indeterminate.

External indeterminacy(EI) is excess no of external reactions in the member that make the structure indeterminate.

i  =II+EI;

 

EI=r- e;

Where r=no of support reactions and

e=equilibrium conditions II=i -EI

e=3 (plane frames) and

e=6 (space frames)

 

6. Write the formulae for degree of indeterminacy for:

 

(a)Two dimensional pin jointed truss(2D Truss)

 

i = (m+r) - 2j            where m=no of members

 

r=no of reactions j

=no of joints

(b)Two dimensional rigid frames/plane rigid frames (2DFrames)

 

i =(3m+r)- 3j           

where

m=no ofmembers

r=no of reactions

j =no ofjoints


 

 

(c)Three dimensional space truss (3D Truss)      

i = (m+r)- 3j         where

m=no of members

          r=no of reactions j

          =no of joints

(d)Three dimensional space frames (3DFrame)   

i =(6m+r)- 6j       

where m=no         of members

          r=no of reactions j

          =no of joints

 

7. Determine the degree of indeterminacy for the following 2D truss. i =(m+r)-2j

Where m=19

r=4

 

j =10 e=3

?i =(19+4)-2x10=3

 

External indeterminacy EI=r-e=4-3=1

?Internal indeterminacy II=i-EI=3-1=2


 

8. Determine the total, internal and external degree of indeterminacy for the plane rigid frame below.

 

i =(3m + r)- 3j

where m=7

r=4 j =6 e=3

?i =(3x7+ 4)- (3x6) =7

 

External indeterminacy EI=r-e=4-3=1

?Internal indeterminacy II=i-EI=7-1=6

 

9. Determine i, EI, II for the given plane truss. i =(m + r)- 2j

 

Where

m=3 r=4 j =3 e=3

 

?i =(3+ 4)-(2x3) =1

 

External indeterminacy EI=r-e=4-3=1

?Internal indeterminacy II=i-EI=1-1=0


 

10. Find the indeterminacy for the beams given below.

 

For beams degree of indeterminacy is given by i =r-e


i =r-e

where

r=no of reactions, e=no of equilibrium conditions r=4 and e=3

?i =4-3=1

 


i =r-e

where

r=5 and e=3

 

?i =5-3=2

 

11. Find the indeterminacy for the given rigid plane frame. i =(3m + r)- 3j

Where

m=3

r=4 j =4

?i =(3x3+ 4)- (3x4) =1

External indeterminacy EI=r-e=4-3=1

?Internal indeterminacy II=i-EI=1-1=0

 

12. Find the indeterminacy of the space rigid frame. i =(6m + r)- 6j

Where

m=8

r=24 (i. e. 6persupportx4) j =8 e=6

?i =(6x8+24)- (6x 8) =24

 

External indeterminacy EI=r-e=24-6=18 ?Internal indeterminacy II=i-EI=24-18=6

13. Find the indeterminacy for the given space truss. i =m +r-3j

Where m=3

r=18 (i. e. 6reactions persupport x3) j =4

?i =(3+18)- (3x4) =9

External indeterminacy EI=r-e=18-6=12 ?Internal indeterminacy II=i-EI=9-12=-3

 

14.        What are the different methods of analysis of indeterminate structures.

 

The various methods adopted for the analysis of indeterminate structures include:

(a) Flexibility matrix method.

(b) Stiffness matrix method

(c)Finite Element method

 

15. Briefly mention the two types of matrix methods of analysis of indeterminate structures.

The two matrix methods of analysis of indeterminate structures are: (a) Flexibility matrix method- This method is also called the force method in which the forces in the structure are treated as unknowns. Then o of equations involved is equal to the degree of static indeterminacy of the structure.

 

(b)Stiffness matrix method- This is also called the displacement method in which the displacements that occur in the structure are treated as unknowns. Then o of displacements involved is equal to then o of degrees of freedom of the structure.

 

16.        Define a primary structure.

 

A structure formed by the removing the excess or redundant restraints from an indeterminate structure making it statically determinate is called primary structure. This is required for solving indeterminate structures by flexibility matrix method.

 

17.        Give the primary structures for the following indeterminate structures.

 

Indeterminate structure         Primary Structure


 

18. Define kinematic indeterminacy (Dk) or Degree of Freedom (DOF)

 

Degrees of freedom is defined as the least no of independent displacements required to define the deformed shape of a structure. There are two types of DOF: (a)Nodal type DOF and (b)Joint type DOF.

 

19. Briefly explain the two types of DOF.

 

(a)Nodal type DOF- This includes the DOF at the point of application of concentrated load or moment, at a section where moment of inertia changes, hinge support, roller support and junction of two or more members.

 

(b)Joint type DOF- This includes the DOF at the point where moment of inertia changes, hinge and roller support, and junction of two or more members.

 

20. For the various support conditions shown below give the DOFs.

(a)     No DOF

(b)     1- DOF

(c)      2- DOF

(d)     1- DOF

 

21. For the truss shown below, what is the DOF?


Pin jointed plane frame/truss

 

DOF/ Dk = 2j-r

where r=no of reactions

j = no of joints

 

22. Define compatibility in force method of analysis.

 

Compatibility is defined as the continuity condition on the displacements of the structure after external loads are applied to the structure.

 

23.        Define the Force Transformation Matrix.

 

The connectivity matrix which relates the internal forces Q and the external forces R is known as the force transformation matrix. Writing it in a matrix form,

 

{Q} =[b]{R}

where Q=member force matrix/vector

b= force transformation matrix

 

R = external force/load matrix/ vector

 

24.        What are the requirements to be satisfied while analyzing a structure?

 

The three conditions to be satisfied are:

(a)Equilibrium condition

(b)Compatibility condition

(c)Force displacement condition

25. Define flexibility influence coefficient(fij)

Flexibility influence coefficient (fij) is defined as the displacement at joint 'i' due to a unit load at joint 'j', while all other joints are not load.

 

26.        Write theelementflexibility matrix(f)fora truss member.

 

The element flexibility matrix(f) for a truss member is given




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