Finite-Element Methods

From the basic principles given in preceding articles, systematic procedures have been developed for determining the behavior of a structure from a knowledge of the behavior under load of its components. In these methods, called finite-element methods, a structural system is considered an assembly of a finite number of finitesize
components, or elements. These are assumed to be connected to each other only at discrete points, called nodes. From the characteristics of the elements, such as their stiffness or flexibility, the characteristics of the whole system can be derived.
With these known, internal stresses and strains throughout can be computed.
Choice of elements to be used depends on the type of structure. For example, for a truss with joints considered hinged, a natural choice of element would be a bar, subjected only to axial forces. For a rigid frame, the elements might be beams subjected to bending and axial forces, or to bending, axial forces, and torsion. For a thin plate or shell, elements might be triangles or rectangles, connected at vertices.

For three-dimensional structures, elements might be beams, bars, tetrahedrons, cubes, or rings.
For many structures, because of the number of finite elements and nodes, analysis by a finite-element method requires mathematical treatment of large amounts of data and solution of numerous simultaneous equations. For this purpose, the use of computers is advisable. The mathematics of such analyses is usually simpler and more compact when the data are handled in matrix for. (See also Art. 5.10.7.)

Force and Displacement Methods

The methods used for analyzing structures generally may be classified as force (flexibility) or displacement (stiffness) methods.
In analysis of statically indeterminate structures by force methods, forces are chosen as redundants, or unknowns. The choice is made in such a way that equilibrium is satisfied. These forces are then determined from the solution of equations that ensure compatibility of all displacements of elements at each node. After the redundants have been computed, stresses and strains throughout the structure can be found from equilibrium equations and stress-strain relations.
In displacement methods, displacements are chosen as unknowns. The choice is made in such a way that geometric compatibility is satisfied. These displacements are then determined from the solution of equations that ensure that forces acting at each node are in equilibrium. After the unknowns have been computed, stresses and stains throughout the structure can be found from equilibrium equations and stress-strain relations.
In choosing a method

the following should be kept in mind: In force methods, the number of unknowns equals the degree of indeterminacy. In displacement methods, the number of unknowns equals the degrees of freedom of displacement at nodes. The fewer the unknowns, the fewer the calculations required.
Both methods are based on the force-displacement relations and utilize the stiffness and flexibility matrices described in Art. 5.10.7. In these methods, displacements and external forces are resolved into components usually horizontal, vertical, and rotational at nodes, or points of connection of the finite elements. In accordance with Eq. (5.103a), the stiffness matrix transforms displacements into forces. Similarly, in accordance with Eq. (5.103b), the flexibility matrix transforms forces into displacements. To accomplish the transformation, the nodal forces and displacements must be assembled into correspondingly positioned elements of force and displacement vectors. Depending on whether the displacement or the force method is chosen, stiffness or flexibility matrices are then established for each of the finite elements and these matrices are assembled to form a square matrix, from which the stiffness or flexibility matrix for the structure as a whole is derived. With that matrix known and substituted into equilibrium and compatibility equations for the structure, all nodal forces and displacements of the finite elements can be determined from the solution of the equations. Internal stresses and strains in the elements can be computed from the now known nodal forces and displacements.

Element Flexibility and Stiffness Matrices

The relationship between independent forces and displacements at nodes of finite elements comprising a structure is determined by flexibility matrices for stiffness matrices k of the elements. In some cases, the components of these matrices can be developed from the defining equations:

The jth column of a flexibility matrix of a finite element contains all the nodal displacements of the element when one force Sj is set equal to unity and all other independent forces are set equal to zero.
The jth column of a stiffness matrix of a finite element consists of the forces acting at the nodes of the element to produce a unit displacement of the node at which displacement Sj occurs and in the direction of Sj but no other nodal displacements of the element.

Beams with Bending Only. As another example of the use of the definition to determine element flexibility and stiffness matrices, consider the simple case of an elastic prismatic beam in bending applied by moments Mi and Mj at nodes i and j, respectively (Fig. 5.92a). The beam might be a finite element of a rigid frame.
Connections to other members are made at nodes i and j, which can transmit moments and forces normal to the beam.

Displacement (Stiffness) Method

With the stiffness or flexibility matrix of each finite element of a structure known, the stiffness or flexibility matrix for the whole structure can be determined, and with that matrix, forces and displacements throughout the structure can be computed (Art. 5.13.2). To illustrate the procedure, the steps in the displacement, or stiffness, method are described in the following. The steps in the flexibility method are similar.
For the stiffness method:
Step 1. Divide the structure into interconnected elements and assign a number, for identification purposes, to every node (intersection and terminal of elements).
It may also be useful to assign an identifying number to each element.
Step 2. Assume a right-handed cartesian coordinate system, with axes x, y, z.
Assume also at each node of a structure to be analyzed a system of base unit vectors, e1 in the direction of the x axis, e2 in the direction of the y axis, and e3 in the direction of the z axis. Forces and moments acting at a node are resolved into components in the directions of the base vectors. Then, the forces and moments at the node may be represented by the vector Piei, where Pi is the magnitude of the force or moment acting in the direction of ei. This vector, in turn, may be conveniently represented by a column matrix P. Similarly, the displacements translations and rotation of the node may be represented by the vector iei, where i is the magnitude of the displacement acting in the direction of ei. This vector, in turn, may be represented by a column matrix .
For compactness, and because, in structural analysis, similar operations are performed
on all nodal forces, all the loads, including moments, acting on all the nodes may be combined into a single column matrix P. Similarly, all the nodal displacements may be represented by a single column matrix .
When loads act along a beam, they should be replaced by equivalent forces at the nodes simple-beam reactions and fixed-end moments, both with signs reversed from those induced by the loads. The final element forces are then determined by adding these moments and reactions to those obtained from the solution with only the nodal forces.
Step 3. Develop a stiffness matrix ki for each element i of the structure (see Art. 5.13.2). By definition of stiffness matrix, nodal displacements and forces for the i the element are related by



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