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Page 2
The load vector corresponding to unconstrained degrees of freedom is,
{}
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9
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p
p
p
p
p
p
p
k
(5)
In the given frame, constraint (known) degrees of freedom are
. Eliminating rows and columns corresponding to constrained
degrees of freedom from global stiffness matrix and writing load displacement
relationship,
12 11 10 3 2 1
, , , , , u u u u u u
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33 . 5 1 1 33 . 1 1 0
1 5 . 500 0 1 5 . 0 0
1 0 5 . 500 0 0 500
33 . 1 0 . 1 0 33 . 5 0 . 1 0 . 1
0 . 1 5 . 0 0 0 . 1 5 . 500 0
0 0 500 0 . 1 0 5 . 500
10
24
24
0
24
24
10
u
u
u
u
u
u
(6)
Solving we get,
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3 -
5 -
2 -
3 -
5 -
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9
8
7
6
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10 3.85
10 5.65 -
10 1.43
10 8.14 -
10 3.84 -
10 43 . 1
u
u
u
u
u
u
(7)
Page 3
The load vector corresponding to unconstrained degrees of freedom is,
{}
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=
24
24
0
24
24
10
9
8
7
6
5
4
p
p
p
p
p
p
p
k
(5)
In the given frame, constraint (known) degrees of freedom are
. Eliminating rows and columns corresponding to constrained
degrees of freedom from global stiffness matrix and writing load displacement
relationship,
12 11 10 3 2 1
, , , , , u u u u u u
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3
33 . 5 1 1 33 . 1 1 0
1 5 . 500 0 1 5 . 0 0
1 0 5 . 500 0 0 500
33 . 1 0 . 1 0 33 . 5 0 . 1 0 . 1
0 . 1 5 . 0 0 0 . 1 5 . 500 0
0 0 500 0 . 1 0 5 . 500
10
24
24
0
24
24
10
u
u
u
u
u
u
(6)
Solving we get,
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3 -
5 -
2 -
3 -
5 -
-2
9
8
7
6
5
4
10 3.85
10 5.65 -
10 1.43
10 8.14 -
10 3.84 -
10 43 . 1
u
u
u
u
u
u
(7)
Let be the support reactions along degrees of freedom
respectively. Support reactions are calculated by
12 11 10 3 2 1
, , , , , R R R R R R
12 , 11 , 10 , 3 , 2 , 1
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12
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12
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3
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1
33 . 1 0 0 . 1 0 0 0
0 500 0 0 0 0
0 . 1 0 5 . 0 0 0 0
0 0 0 33 . 1 0 0 . 1
0 0 0 0 500 0
0 0 0 0 . 1 0 5 . 0
10
9 8 7 6 5 4
u
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u
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p
p
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p
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R
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42 . 19
28 . 28
99 . 10
43 . 3
71 . 19
99 . 0
42 . 19
28 . 28
99 . 10
43 . 3
71 . 19
99 . 0
0
0
0
0
0
0
12
11
10
3
2
1
R
R
R
R
R
R
(8)
Summary
In this lesson, the analysis of plane frame by the direct stiffness matrix method is
discussed. Initially, the stiffness matrix of the plane frame member is derived in
its local co-ordinate axes and then it is transformed to global co-ordinate system.
In the case of plane frames, members are oriented in different directions and
hence before forming the global stiffness matrix it is necessary to refer all the
member stiffness matrices to the same set of axes. This is achieved by
transformation of forces and displacements to global co-ordinate system. In the
end, a few problems are solved to illustrate the methodology.
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FAQs on The Direct Stiffness Method: Plane Frames - 4 - Structural Analysis - Civil Engineering (CE)
| 1. What is the direct stiffness method in plane frames? |  |
Ans. The direct stiffness method is a structural analysis technique used to determine the displacements, forces, and moments in a plane frame structure. It involves dividing the structure into smaller elements, applying the principles of equilibrium and compatibility, and solving a system of equations to obtain the unknowns.
| 2. How is the direct stiffness method used in analyzing plane frames? | |
Ans. In analyzing plane frames using the direct stiffness method, the structure is divided into individual elements such as beams or trusses. Each element is characterized by its stiffness matrix, which relates the displacements and forces at its ends. By assembling the stiffness matrices of all elements and considering the boundary conditions, a global stiffness matrix is formed. By applying equilibrium equations and solving the system of equations, the displacements and forces in the plane frame can be determined.
| 3. What are the advantages of using the direct stiffness method in analyzing plane frames? | |
Ans. The direct stiffness method offers several advantages in analyzing plane frames. It allows for the accurate determination of displacements, forces, and moments in complex structural systems. It can handle both static and dynamic loads, making it applicable to a wide range of engineering problems. Additionally, the method is computationally efficient, allowing for quick analysis of large and complex structures.
| 4. Are there any limitations or assumptions associated with the direct stiffness method? | |
Ans. Yes, there are certain limitations and assumptions associated with the direct stiffness method. It assumes linear behavior of the structural elements, neglecting any nonlinear effects. It also assumes small deformations and linear elastic material properties. Additionally, the method requires the structure to be divided into finite elements, which may introduce errors depending on the complexity of the structure and the size of the elements used.
| 5. Can the direct stiffness method be used for three-dimensional structures? | |
Ans. While the direct stiffness method is primarily used for analyzing plane frames, it can be extended to three-dimensional structures. However, the complexity and computational requirements increase significantly in three dimensions, making it more challenging to apply. In such cases, specialized software and techniques are often employed to handle the increased complexity and calculations involved in the direct stiffness method for three-dimensional structures.
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