Stiffness method

Stiffness Method

Introduction

Fig. 1 Frame structure

Stiffness method is an efficient way to solve complex determinant or indeterminant structures (Fig. 1). It is also called finite element method, which is a powerful engineering method and has been applied in numerous engineering fields such as solid mechanics and fluid mechanics. The idea of stiffness method is as following:

Subdividing the structures into a series of discrete elements

Formulating the stiffness matrix for each of the elements

Assembling the global matrix

Applying the boundary conditions to obtain the reduced matrix

Inverting the reduced matrix

Multiplying the inverted reduced matrix with the forces to get the displacements of the nodes

Post-processing to obtain the stresses and strains of elements

Truss Stiffness Matrix

Fig. 2 Truss element for stiffness matrix

For truss structures, each member is an element. In order to assemble the matrices in a systematic way, all the truss member stiffness matrices are formed in global coordinates. Every member's stiffness matrix is formulated in the same way which is as following:

Where
            A = Cross section area of the member
            E = Elasticity modulus
            L = Length of the member
            C = cos(q)
            S = sin(q)
            k = Member stiffness method

Frame Stiffness Matrix

Fig. 3 Plane frame element

For frame structures, the stiffness matrix is more complex than truss structures since the rotation is also considered. Here only plane frame is considered. In order to derive the frame stiffness matrix, the arbitrarily oriented beam element's matrix is formed and then the axial effects are included. Combining both, the frame stiffness matrix is formulated as following:

(2)

Where
            A = Cross section area of the element
            E = Elasticity modulus
            I = Moment of inertia
            L = Length of the element
            C = cos(q)
            S = sin(q)
            k = Element stiffness method

Assembling the Stiffness Matrices and Force Matrices

The global stiffness matrix can be obtained by summing the stiffness matrix for each element, the formulation is

Where
            K = Global stiffness matrix
            k = Local element matrix
            N = Total number of element
            e = Index

Similarly, the force can be assembled as following:

Where
            F = Global force matrix
            f = Local force matrix
            N = Total number of element
            e = Index

Formulation of a System of Equations

To determine the displacements, the boundary conditions are imposed and the following equation is solved.

Where
            d = Displacement matrix

By post processing, the stresses, strains and nodal forces can be obtained.

Stiffness Method

		Introduction
Fig. 1 Frame structure		Stiffness method is an efficient way to solve complex determinant or indeterminant structures (Fig. 1). It is also called finite element method, which is a powerful engineering method and has been applied in numerous engineering fields such as solid mechanics and fluid mechanics. The idea of stiffness method is as following: Subdividing the structures into a series of discrete elements Formulating the stiffness matrix for each of the elements Assembling the global matrix Applying the boundary conditions to obtain the reduced matrix Inverting the reduced matrix Multiplying the inverted reduced matrix with the forces to get the displacements of the nodes Post-processing to obtain the stresses and strains of elements




		Truss Stiffness Matrix
Fig. 2 Truss element for stiffness matrix		For truss structures, each member is an element. In order to assemble the matrices in a systematic way, all the truss member stiffness matrices are formed in global coordinates. Every member's stiffness matrix is formulated in the same way which is as following: Where A = Cross section area of the member E = Elasticity modulus L = Length of the member C = cos(q) S = sin(q) k = Member stiffness method

		Frame Stiffness Matrix
Fig. 3 Plane frame element		For frame structures, the stiffness matrix is more complex than truss structures since the rotation is also considered. Here only plane frame is considered. In order to derive the frame stiffness matrix, the arbitrarily oriented beam element's matrix is formed and then the axial effects are included. Combining both, the frame stiffness matrix is formulated as following:
(2)
		Where A = Cross section area of the element E = Elasticity modulus I = Moment of inertia L = Length of the element C = cos(q) S = sin(q) k = Element stiffness method

		Assembling the Stiffness Matrices and Force Matrices
		The global stiffness matrix can be obtained by summing the stiffness matrix for each element, the formulation is Where K = Global stiffness matrix k = Local element matrix N = Total number of element e = Index Similarly, the force can be assembled as following: Where F = Global force matrix f = Local force matrix N = Total number of element e = Index

		Formulation of a System of Equations
		To determine the displacements, the boundary conditions are imposed and the following equation is solved. Where d = Displacement matrix By post processing, the stresses, strains and nodal forces can be obtained.