Method description
Contenido
El método matricial requiere asignar a cada barra elástica de la estructura una matriz de rigidez, llamada matriz de rigidez elemental que dependerá de sus condiciones de enlace extremo (articulación, nudo rígido,...), la forma de la barra (recta, curvada, ...) y las constantes elásticas del material de la barra (módulo de elasticidad longitudinal y módulo de elasticidad transversal). A partir del conjunto de matrices elementales mediante un algoritmo conocido como acoplamiento que tiene en cuenta la conectividad de unas barras con otras se obtiene una matriz de rigidez global, que relaciona los desplazamientos de los nudos con las fuerzas equivalentes sobre los mismos.
Igualmente a partir de las fuerzas aplicadas sobre cada barra se construye el llamado vector de fuerzas nodales equivalentes que dependen de las acciones exteriores sobre la estructura. Junto con estas fuerzas anteriores deben considerarse las posibles reacciones sobre la estructura en sus apoyos o enlaces exteriores (cuyos valores son incógnitas).
Finalmente se construye un sistema lineal de ecuaciones, para los desplazamientos y las incógnitas. El número de reacciones incógnita y desplazamientos incógnita depende del número de nodos: es igual a 3N para problemas bidimensionales, e igual a 6N para un problema tridimensional. Este sistema siempre puede ser dividido en dos subsistemas de ecuaciones desacoplados que cumplen:.
Una vez resuelto el subsistema 1 que da los desplazamientos, se substituye el valor de estos en el subsistema 2 que es trivial de resolver. Finalmente a partir de las reacciones, fuerzas nodales equivalentes y desplazamientos se encuentran los esfuerzos en los nudos o uniones de las barras a partir de los cuales pueden conocerse los esfuerzos en cualquier punto de la estructura y por tanto sus tensiones máximas, que permiten dimensionar adecuadamente todas las secciones de la estructura.
Elementary stiffness matrices
To construct the stiffness matrix of the structure, it is necessary to previously assign an elemental stiffness matrix to each individual bar (element). This matrix depends exclusively on:
The elemental matrix relates the nodal forces equivalent to the forces applied on the bar with the displacements and rotations suffered by the ends of the bar (which in turn determines the deformation of the bar).
A knot where two bars are joined is called rigid or embedded if the angle formed by the two bars after deformation does not change with respect to the angle they formed before deformation. Even if the two bars as a whole are unable to change the angle between bars, they can rotate with respect to the node, but maintaining the angle they form at their end. In reality, rigid welded or rigidly bolted joints can be treated as rigid knots.
For a bar rigidly joined at both ends, the elemental stiffness matrix that adequately represents its behavior is given by:.
Where:
are the geometric magnitudes (length, area and moment of inertia).
the longitudinal elasticity constant (Young's modulus).
Alternatively, the stiffness matrix of a straight two-embedded bar can be written more abbreviated, introducing the characteristic mechanical slenderness:.
Where:
It is the characteristic mechanical slenderness.
In this case, when turns are imposed on the articulated node, no forces are transmitted to the non-articulated node. In that case the stiffness matrix, using the same notation as in the previous section, is given by:.
Where it has been assumed that the articulated knot is the second. If it were the first, the elements of the previous matrix would have to be permuted to obtain:.
Since a straight bar with hinged nodes can only transmit forces along its axis, the corresponding stiffness matrix of that bar only has different components for the longitudinal degrees of freedom. In that case the stiffness matrix, using the same notation as in the previous section, is given by:.
A three-dimensional straight bar has 6 degrees of freedom per node (3 translational and 3 orientation "Orientation (geometry)"), as the bar has two nodes the rigidity matrix is a 12 submatrices:
Where the submatrices are:.
And the geometric and mechanical quantities associated with the bar are:
nodal forces
For each bar, an elementary vector of generalized nodal forces is defined, which is statically equivalent to the forces applied on the bar. The size of the nodal force vector depends on the dimensionality of the bar:
The components of this vector make up a system of forces and moments of force, such that the resulting force and the resulting moment coincide with the force and moment of the original system of forces on the bar.
For the loads shown in the attached figure on a two-dimensional bar or beam, the nodal force vector consists of two vertical forces (F, F) applied at each of the two ends, two horizontal forces (F, F) applied at each of the ends and two moments of force (M, M) applied at each of the ends. These six components form the nodal force vector. It is easy to verify that the force and moment resulting from these six components are statically equivalent to the original system of forces formed by P and q if the following values are taken:.
Displacement calculation
Once the global stiffness matrix and the global nodal force vector have been found, a system of equations like () is constructed. This system has the property that it can be decomposed into two subsystems of equations:
By solving the first determined compatible subsystem, the unknown displacements of all the nodes of the structure are known. Inserting the solution of the first subsystem into the second results in the reactions.
We can illustrate the calculation of displacements with an example. For example, if we consider the bending in the
Rows 3 and 6 contain the unknown turns (displacements) of the ends of the beam and taken together they make up the first subsystem for the displacements. Ignoring the null terms and rewriting in matrix form the subsystem of equations for the displacements is simply:
Whose solution gives us the value of the angle rotated by the right and left end of the beam under those loads:
Once these values are known and inserted into the matrix, rows 1, 2, 4 and 5 provide us with the value of the four previously unknown hyperstatic reactions.
Calculation of reactions
Once the displacements are calculated by solving a system of equations, the calculation of the reactions is simple. From equation () we simply have:
Taking the same example as in the last section, the calculation of reactions on the two-hinged beam with load P and q would be:
By introducing the values of the rotations at the ends and multiplying the stiffness matrix by the displacement vector, we finally have:
This completes the reaction calculation.
Stress calculation
The stress calculation is carried out by examining in local coordinates of the bars the axial stress, the shear stresses, the bending moments and the torsional moment generated in each of the bars, knowing the displacements of all the nodes of the structure. This can be done using the stiffness matrices expressed in local coordinates and the nodal displacements also expressed in local coordinates.
Dynamic analysis
The static analysis discussed above can be generalized to find the dynamic response of a structure. To do this, it is necessary to represent the inertial behavior of the structure through a mass matrix, model the dissipative forces through a damping matrix, which together with the stiffness matrix allow us to propose a system of second-order equations of the type:
The solution of the previous system involves a calculation of the eigenfrequencies and eigenmodes. Admitting that the dissipative forces are unimportant, the natural frequencies can be determined by solving the following polynomial equation in:.
These magnitudes allow a modal analysis to be carried out that reproduces the behavior of the structure under different types of situations.