Finite element model based on displacement method

The most widely spread finite element method is based on the motion method; the commer-cial programs mostly apply this basic method. The fundamentals of the method are the fol-lowings: the body must be divided into elements, and then kinematically admissible dis-placement fields must be considered on the elements by approximate functions. After that, by applying the geometric and constitution equation alongside with the boundary conditions a linear algebraic equation system is created. The solution of this equation system is the approx-imate displacement field. The stress field, calculated from the displacement field, will particu-larly satisfy the equilibrium equations. In the description, vectors (column matrixes) will be used instead of tensors.

3.5.1. Introduction of vector fields

Vector of stress components (column matrix): the vector, including the stress tensor

compo-nents is described in a spatial system as:

 

 

Strain vector (column matrix): vector, including the stress tensor components is described

If the displacement method is used, then the geometric and constitution equations are also required. These equations have to be reformulated to vector equations. Let us define the scalar components of the geometric equation 



uu



2

 1 in a Descartes coordinate system:

x

and substitute them into the deformation vector. Let us convert them into a product form:

u (in-cluding the differential rules) differential operator matrix. The proper elements are substituted into the stress vector by the use of the constitution equation 





x y z



x y z

Then, let us convert them into product form:



Thus the stress vector is derived as a product of the  deformation vector and the C ma-trix which includes the material constants. Introducing the vector fields, both the geometric equation:

u

  (3.18)

and the constitution equation:



 C . (3.19)

are obtained as single products. Substituting (3.18) into (3.19):  Cu, thus the displace-ment field is the unknown function, while the stress and deformation can be directly calcu-lated.

3.5.2. Elasticity problem and the method of solution

The finite element method is presented on an elasticity problem. The general elasticity prob-lem is the following:

p0

q

u0

V

Au

x y

z

Ap

r P

Figure 3.4.: Elasticity problem According to Figure 3.4, the following data are given:

 The geometry of the body,

 The material constants of the body,

 loads,

 Constraints.

Demanded functions: u

 

r , 

 

r , 

 

r . Steps to solution:

 Firstly, the body is divided to finite domains so-called elements. Special points, nodes are appointed on these elements. The elements cover the total volume of the body, and their geometric representation is a mesh. The single elements are connected to each other by the nodes.

 The displacement field is approximated element by element, generally with polyno-mials which are fit to the nodes. The displacement fields of the nearby elements are fit to each other through the nodes, and they describe a continuous function on the body.

 The approximate stress- and deformation field can be derived from the displacement field by the use of the geometric- and constitution equation. Then, by the applying the principle of Lagrange variation, a linear, algebraic equation system can be derived with respect to the nodes. This is the so-called ‘stiffness equation’. The algebraic

sys-tem of equation is solvable, if a load or displacement parameter – derived from the ki-nematic or dynamic boundary conditions – is specified to each nodes on the surface.

Thus the unknown values are the displacements of the nodes.

 By solving the system of equation, the approximate nodal displacement field is ob-tained, thus the approximate stress- and deformation fields can be calculated as well.

3.5.3. Finite element, approximate displacement field

The body is divided to arbitrary shaped and sized finite domains, finite elements. Naturally, it is taken into account that the basis functions have to fit to the element.

Figure 3.5.: Discretization, finite element

The displacement field of e element is approximated by a continuously differentiable func-tion. The type of the function is determined, and according to this function, the demanded numbers of nodes (2 points in case of linear function, 3 points in case of quadratic function) are appointed on the element. Then the displacement field is described by the nodal displace-ment. The displacement of element node i on element:



















ei ei ei ei

w v u

u ,

The displacement vector of element e derived from the displacement of i,j,k,,n nodes:

 element is derived from the interpolation of u_e nodal displacement vector:

 

e

 

e

e r N r u

u   , (3.20)

where ^N_e

 

^r is the approximate matrix (matrix of the interpolation functions). This matrix is built up by (3x3) blocks, and each block includes the interpolation function of each node.

The displacement of the element can be derived from the nodal displacement of iwith re-spect to e element:

ei are the interpolation functions. Definition of the indexes:

The N_eixz

 

r function defines the displacement along x direction of element e related to any arbitrary r location due to the displacement of z direction of node i, while the other compo-nents of the nodal displacement vector of element e are zero.

The functions must to satisfy the following conditions:

 The functions must be continuously differentiable,

 Nei

 

ri E, the function must provide unit value of displacement in node i,

By the approximation of the element’s displacement field, the deformation field can be ob-tained by substituting (3.20) into (3.18):

 

e

 

e

 

e

e r u r N r u

 ,

Introducing B_e

 

r nodal-deformation matrix as a product of the differential operator and the approximate matrix:

 

_e

 

_e

e r B r u

 . (3.21)

The stress field of the element:

 

e

 

e

 

e

e r C r CB r u

 . (3.22)

The potential energy of the element according to (3.6):

   

r r dV u

 

r qdV u

 

r pdA

ep e

e A

e V

e V

e

e ^



e ^{ }



^{ }



^

  

2

1 .

Rewriting the formula by forming the scalar and double scalar products into matrix prod-ucts (the constants of the internal energy are replaced) and introducing them as vectors instead of tensors:

  

r

  

r dV



u

 

r



qdV



u

 

r



pdA

ep e

e A

T e V

T e V

e T e

e^



^



^



  

2

1 .

Substituting (3.20), (3.21), (3.22) and separating the constants out of the integrals:

 

^u



^B

 

^r



^CB

 

^{r dVu}

 

^u



^N

 

^r



^qdV

 

^u



^N

 

^r



^pdA

ep e

e A

T e T e V

T e T e e V

e T e T

e ^ e



^



^



 2

1 .

Let us introduce the stiffness matrix:



B

 

r



CB

 

r dV K

Ve

e T

e ^



e ^{, (3.23)}

And the nodal load vectors with respect to the volume and surface forces:



^N

 

^r



^qdV

F

Ve

T

qe^



e ^{, (3.24)}



^N

 

^r



^pdA

F

Aep

T

pe^



e ^{, (3.25)}

pe qe

e F F

F :  .

Thus the potential energy of the element is:

   

e

T e e e

T

e ue K u  u F

 2

1 .

The energy theorems can only be applied on the whole body; they are not valid on indi-vidual elements. If the body has Q number of elements, the potential energy of the body is derived from the sum of all elements’ potential energy.

 

^{U TKU}

 

^{U TF}

Q

In document Finite element methode (Pldal 48-55)