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.