Bars and Beams FEM Linear Static Analysis. Stiffness matrix formulation: bar element. Uniform prismatic elastic bar of length L, elastic modululus E and area A ( 2 dof ) Only axial direct displacement are allowed.
Forces that must be applied to the nodes in order to maintain the displacement state
Nodal displacement vector
Bstrain displacement matrix
E elastic modulus
dVincrement of the element volume
Nshape function matrix
The definition of k guarantees that it will be a Symmetric matrix
Aarea of the cross section
E elastic modulus
Icentroidal moment of inertia
Nodal displacements consist of lateral translations v1 and v2 and rotations qz1 and qz2 about the z axis
The beam centerline has lateral displacement is v=v(x)
v=v(x) is cubic in x for a uniform prismatic beam loaded only at its ends
We will ignore transverse shear deformation, although commercial software account for it
One d.o.f. has unit value and all other d.o.f. are zero
kij are nodal forces and moments that must be applied to sustain the assigned deformation state, positive direction are upward for forces and counterclokwise for moment
To solve for the first column of k, the column vector
the following conditions are used
which operates on the vector of nodal d.o.f.
where B is a matrix that yields curvature d2n/dx2 of the beam element from the product B d.
In each case the expression: represents the strain
energy in an element under nodal displacement d.
In bars, strain energy depends on axial strain; in beams, strain energy depends on curvature.
In terms of generalized coordinates the lateral displacement is a cubic function in x:
The bj terms can be stated in terms of nodal d.o.f. making the substitution at beam extremities, like, for example:
Thus an alternative form for the lateral displacement uses shape function Ni
where each Ni states the deflected shape associated with a particular end translation of rotation
The curvature of the beam element is expressed in matrix form using the derivative of the shape function and the nodal displacements
Where strain-displacement matrix B is:
Bending moment and flexural stresses are computed from curvature, which in turn depends on nodal d.o.f. d.
If y is the distance from the neutral axis, the following holds:
Flexural moment M caused by nodal displacement d varies linearly with x in each element
3D beam element is a cubic function in x:
Web of beam section
Local reference frame
global reference frame
A beam element in a general purpose FE program has three-dimensional capability and may also be called a “space beam” element.
A global reference frame XYZ is introduced, while the longitudinal axis of the beam lies along a local x axis, defined by the coordinates of nodes 1 and 2, of a local reference frame xyz.
In order to orient the local frame xyz, a node 3 is introduced, whose coordinates serve to orient the xy plane in XYZ space. In the example the xy plane is placed along the web of the beam section. No d.o.f. are associated to node 3.
d.o.f. of 3D beam element is a cubic function in x:
6 dof for each node
12 dof for each element
At each node the element has six d.o.f. (three displacements and three rotations)
The matrix k is formulated in the local reference frame, then kis transformed so that global d.o.f. replace local d.o.f. at each node.
The element resists forces in any direction and moments about any axis.
3D beam element: some remarks is a cubic function in x:
Additional data are needed for stress computation, such as the appropriate distance y in the flexure formula.
If the section is noncircular, note that J is not the polar moment of inertia of the cross sectional area A. J is a property of the cross section, such that the correct relative rotation of nodes a and 2 under the torque T is given by TL/GJ.
Often (especially for open sections, like the one of the example), is much lower than the polar moment of A
Properties of k (element) and K (structure) is a cubic function in x:
Avoiding singularities for K is a cubic function in x:
K D = R
- if unsupported, it can have two rigid-body motions in the xy plane lateral translation and rotation about a point
- is adequately supported if at least two d.o.f. are prescribed, expect that the two rotations are prescribed, in such case a rigid translation is possible.
Maximum value not well approximated