Matrix Method of Structural Analysis:
Matrix method of structural analysis is a matrix approach to solve the analysis problems of the structures. In this method the behavior of the structures written in the form of the forces and displacement equations, and these equations are then solved with matrix method of solving a system of simultaneous equations. This method is laborious when done manually but this has vast applications with the computers. So read on:
Here we divide the structure into a numbers of co-ordinates and then we define the forces and displacements at the co-ordinates in terms of their stiffness and their flexibility.
[P] = [k].[D]....(1) Here, [k] is the stiffness matrix.
Above equation shows that we can write the force in terms of the stiffness of the joint and the amount of the displacement required. In this method, displacements are considered redundant and so this method is known as displacement method. You can find out the displacement by taking the inverse of the stiffness matrix on both sides and you can find out displacement matrix as the final result.
[D] = [f].[P]......(2)
here [f] is the flexibility matrix,
you can pre-multiply the above equation with inverse of flexibility matrix and you will get the force matrix. So this method is known as the force method. You have to treat the forces as the redundant and you can find them as the final result.
So in the force method and displacement method, the first step is to find the redundant forces and redundant displacements respectively. Then you have to form the flexibility matrix by producing unit force at the chosen co-ordinates. Then pre-multiply the inverse of the flexibility matrix on both sides of equation(2) and you will get the force matrix.
In displacement method, you have to form the stiffness matrix and pre-multiply its inverse on both sides of the equation(1), and you will get the displacements at the respective co-ordinates.
That is a small introduction to the matrix approach, if you want further assistance, please feel free to leave a comment.
Thank you,
:)
Matrix method of structural analysis is a matrix approach to solve the analysis problems of the structures. In this method the behavior of the structures written in the form of the forces and displacement equations, and these equations are then solved with matrix method of solving a system of simultaneous equations. This method is laborious when done manually but this has vast applications with the computers. So read on:
Here we divide the structure into a numbers of co-ordinates and then we define the forces and displacements at the co-ordinates in terms of their stiffness and their flexibility.
[P] = [k].[D]....(1) Here, [k] is the stiffness matrix.
Above equation shows that we can write the force in terms of the stiffness of the joint and the amount of the displacement required. In this method, displacements are considered redundant and so this method is known as displacement method. You can find out the displacement by taking the inverse of the stiffness matrix on both sides and you can find out displacement matrix as the final result.
[D] = [f].[P]......(2)
here [f] is the flexibility matrix,
you can pre-multiply the above equation with inverse of flexibility matrix and you will get the force matrix. So this method is known as the force method. You have to treat the forces as the redundant and you can find them as the final result.
So in the force method and displacement method, the first step is to find the redundant forces and redundant displacements respectively. Then you have to form the flexibility matrix by producing unit force at the chosen co-ordinates. Then pre-multiply the inverse of the flexibility matrix on both sides of equation(2) and you will get the force matrix.
In displacement method, you have to form the stiffness matrix and pre-multiply its inverse on both sides of the equation(1), and you will get the displacements at the respective co-ordinates.
That is a small introduction to the matrix approach, if you want further assistance, please feel free to leave a comment.
Thank you,
:)
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