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Both concentrated and distributed loads f(ξ,η,ζ) have to be transformed into equivalent concentrated loads according to the following integration.

This way, the load vector {F} is being assembled.

Boundary Conditions     
Equilibrium                      
Pseudo Displacements  
Displacement Field         
Strain Field                       
Stress Field                      

Boundary Conditions

Certain degrees of freedom are fixed, in that their displacements are considered being zero. These are called stationary and the corresponding rows and columns are deleted from the stiffness matrix and the load vector. This leaves a stiffness matrix and a load vector having only free degrees of freedom, [Kff ] and  {Ff }  respectively.

Equilibrium

Pseudo Displacements

The (zero) displacements for the bounded degrees of freedom are considered to creat the displacement vector {D}.

Displacement Field

The equation's solution represents the displacements of the control points, the so-called pseudo-displacements. Unlike in classical FEM, control points aren't material points, means that they don't belong to the structure. In general, displacements of the model differ from the displacements of the corresponding control points. Pseudo-displacements play an auxiliary role in calculating the model's displacement field. The displacement field is field is calculated via shape functions [2].

Displacements of interpolatory control points are physical model’s displacements as well.

Strain Field 

The strain vector can be evaluated at any point in the field with the help of control point displacements and the deformation matrix [B] [2]:

Stress Field

Applying Hooke’s constitutive law leads to:

Note that stress and strain vectors are evaluated via the derivatives of the shape functions. This means that their distribution is going to be one order less than the displacement distribution. This is why stress and strain continuity cannot be achieved in FEM models, where shape functions are always C-1  continuous. This problem is solved when the derivatives of the shape functions are also continuous, which means using shape functions with C1  continuity or higher.