B) Nonlinear Force – Based Timoshenko Beam Element

To include shear deformations in the element, the element flexibility must be established. To express clearly the modifications to the existing Bernoulli formulation, both the Timoshenko and Bernoulli formulations will be carried out.

The beam elements added by Spacone et al [15] have five degrees of freedom, without rigid body modes, in three dimensions at the element level. They are two bending moments, Q1 – Q4, or rotations, q1 – q4, at each end and one axial force, Q5, or displacement, q5 (figure IV-4).

Figure IV-4: Three Dimensional Flexibility Based Element

The capital X, Y, and Z in figure IV-4 are the global degrees of freedom while the lowercase x, y, and z are the local degrees of freedom for the element. The element is divided into transverse sections along its length at each one of the classic Guass or Guass – Lobatto integration points. Each of these transverse sections is further divided into longitudinal fibers. This configuration is shown in figure IV-5.

Figure IV-5: Element Divided into Transverse Sections and Longitudinal Fibers x

y

z

X Z

Y

Q

, q

Q

, q

Q

, q

Q

, q

Q

, q

To obtain the flexural and axial response along the element, the fibers are analyzed and the stresses, σ, and moduli, E, are summed to compute the section response. The section responses (mainly forces and flexibility) are then summed according to weight factors depending on the location of the section and the integration scheme used to compute the total axial and flexural response along the element. The fiber section model yields interaction between flexural and axial responses. The shear response for each element is calculated at the section level. This response is then summed over all the sections, again according to the weight factors depending on the location of the section and the integration scheme used, to get the total element response due to shear. In the proposed approach shear response is independent of the axial and flexural responses at the section level. However, since force equilibrium is satisfied pointwise along the element, interaction among the axial – flexural and shear responses is enforced at the element level. This will be clarified in the following formulation.

There are three major steps in the force – based formulation. In the first step, equilibrium, the force fields are expressed as functions of the nodal forces:

Q x b x

D ( ) = ( )

Where D(x) are the section forces, b(x) contain the element force shape functions, and Q contain the nodal forces as illustrated in figure IV-4.

The second step is to write the section constitutive law in which the section deformations, d(x), are related to the section forces, D(x), through the section flexibility matrix, f(x):

)

The third step is to satisfy compatibility (in an average sense). Starting from a compatible state of deformations, the element relation between forces, Q, and corresponding deformations, q, is obtained with the application of the principle of virtual forces:

∫

Putting together IV.4, IV.5 and IV.6, along with noting the arbitrariness of δQ, one obtains:

Q

is the element flexibility.

The above equations apply to the force – based formulation of both Bernoulli and Timoshenko beams. However there are differences in the individual components which enter into the equations. These differences will be illustrated below. As shown in figure IV-4, the three dimensional element has five degrees of freedom, two moments at each end and an axial load, irrespective of whether shear deformations are included or not. However, on the section level, the number of non – zero deformations depend on whether or not shear deformations are included. The section degrees of freedom are illustrated in figure IV-6.

Figure IV-6: Three Dimensional Section Degrees of Freedom – Bernoulli and Timoshenko Beams ,ε

As can be seen from figure IV-6, the Bernoulli beam section has three non – zero deformations while the Timoshenko beam section has five deformations. Because the shear forces are required in the section formulation, the equilibrium statement must include these forces and therefore the shape functions for the Timoshenko beam must include the relation between shear force applied at a section to the nodal forces of the element. Adding these to the shape functions included in the Bernoulli beam formulation, which assumes constant axial force and linear moment along the element, gives the constant shear force distributions of the Timoshenko beam. The constitutive relation must also be modified because the sections now include shear forces and deformations. The section flexibility, which is obtained by inverting the section stiffness, contains the shear flexibility of the section decoupled from the axial and bending terms. Weighted integration of f(x), however, couples shear, axial and bending responses on the element level. These relations are detailed thoroughly in figure IV-7.

Referring to figure IV-7, it is interesting to note that while the constitutive equation for the Bernoulli beam exhibits a full 3x3 section flexibility matrix, the same relation for the Timoshenko beam is a full 3x3 matrix for the flexural and axial responses but is merely diagonal for the shear response resulting in a 5x5 section flexibility matrix. This is due to the fact that, as mentioned earlier, the fiber responses are added over the section to obtain the response at each section. So, at the section level, the axial and flexural responses are naturally coupled. The shear response is not calculated at the fiber level, but is determined for the entire section as an average response. However, the flexibility equation for both beam types is a full 5x5 matrix. This is due to the fact that the element response is found by summing the responses of each section weighted through the shape functions, b(x), that impose linear moments and constant shear in equilibrium with the end moments. So, the formulation used here for the Timoshenko beam includes coupling of axial and flexural responses on the section level, but shear responses are independent on the section level. However, shear and axial – flexural responses are coupled at the element level. In other words, because the shear forces are related to the end moments through equilibrium, the shear and bending forces are coupled through equilibrium.

Figure IV-7: Comparison of Components in Bernoulli and Timoshenko Element Formulations Element Level – 5 dof’s

Section Level – 3 dof’s Element Level – 5 dof’s Section Level – 5 dof’s

Bernoulli Beam Timoshenko Beam



f(x) is section flexibility from fiber section and Nonlinear V-γ relation.

Axial and Bending Deformations Coupled Shear Deformations Uncoupled f(x) is section flexibility from fiber section.

Axial and Bending Deformations Coupled

3) Compatibility

q = FQ

[ ] 5x 5 F =

F is full 5x5 Element Flexibility Matrix With Coupling Among Axial and Flexural Deformations

F is full 5x5 Element Flexibility Matrix With Coupling Among Axial, Flexural

and Shear Deformations

[ ] 5 x 5 F =

Note: a through h are values yet undefined. They are included to represent non – zero flexibility determined values