Allows to be the vector of nodal displacements of a typical element. The displacements at some point of the element must be found by Interpolation functions, symbolically:.

where.

Equation (6) gives rise to other large interest amounts:

Contenido

Para un elemento típico de volumen , el trabajo virtual interno debido a los desplazamientos virtuales es obtenido por sustitución de (5) y (9) dentro de (1):.

Element Arrays

Primarily for the convenience of references, the following matrices pertain to typical elements that must be defined:

These matrices are usually evaluated numerically using Gaussian Quadrature for Numerical Integration.

Its use simplifies (10) to the following:.

Virtual work of elements in terms of the nodal displacements of the system

Since the nodal displacement vector q is a subject of the nodal displacement system r (for compatibility with adjacent elements), we can replace q with r by expanding the size of the element matrices with new columns and rows of zeros:

Where, for simplicity, we use the same symbols for the element arrays, which have now expanded their size as well as properly reordered their rows and columns.

Summing the internal virtual jobs (14) for all the elements given on the right side of (1):.

Now considering the left side of (1), the external virtual work of the system consists of:.

Adding (16), (17b) and equating the sum to (15) gives:.

Since virtual displacements are arbitrary, the previous equality reduces to:

Comparing with (2) shows that:.

In practice, arrays of elements are not expanded or reordered. Instead, the system stiffness matrix is ​​assembled by adding individual coefficients to where the subscripts ij, kl mean that the nodal displacements of the elements respectively coincide with the nodal displacements of the system. Similarly, it is assembled by adding individual coefficients to where matches . This direct addition of within gives the procedure the name Direct rigidity method (Matrix rigidity method).

Allows to be the vector of nodal displacements of a typical element. The displacements at some point of the element must be found by Interpolation functions, symbolically:.

where.

Equation (6) gives rise to other large interest amounts:

Contenido

Para un elemento típico de volumen , el trabajo virtual interno debido a los desplazamientos virtuales es obtenido por sustitución de (5) y (9) dentro de (1):.

Element Arrays

Primarily for the convenience of references, the following matrices pertain to typical elements that must be defined:

These matrices are usually evaluated numerically using Gaussian Quadrature for Numerical Integration.

Its use simplifies (10) to the following:.

Virtual work of elements in terms of the nodal displacements of the system

Since the nodal displacement vector q is a subject of the nodal displacement system r (for compatibility with adjacent elements), we can replace q with r by expanding the size of the element matrices with new columns and rows of zeros:

Where, for simplicity, we use the same symbols for the element arrays, which have now expanded their size as well as properly reordered their rows and columns.

Summing the internal virtual jobs (14) for all the elements given on the right side of (1):.

Now considering the left side of (1), the external virtual work of the system consists of:.

Adding (16), (17b) and equating the sum to (15) gives:.

Since virtual displacements are arbitrary, the previous equality reduces to:

Comparing with (2) shows that:.

In practice, arrays of elements are not expanded or reordered. Instead, the system stiffness matrix is ​​assembled by adding individual coefficients to where the subscripts ij, kl mean that the nodal displacements of the elements respectively coincide with the nodal displacements of the system. Similarly, it is assembled by adding individual coefficients to where matches . This direct addition of within gives the procedure the name Direct rigidity method (Matrix rigidity method).