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Some beam and shell element formulations use Hermite polynomials instead of Lagrange interpolating polynomials for the displacement approximation functions.

Why is this different approach required, if it is required?

In lecture, we discussed three approaches to beam formulations: Euler-Bernoulli, Timoshenko, and continuum mechanics.

Which type requires this approach, and which doesn’t?

Answer :

Hermite polynomials are used in some beam and shell element formulations to include nodal derivative information important for capturing bending and curvature behaviors; essential for Euler-Bernoulli beam theories, but not mandatory for Timoshenko beam theories or continuum mechanics approaches.

Some beam and shell element formulations utilize Hermite polynomials for displacement approximation functions because they can naturally include derivative information at the nodes, which is essential for capturing the bending behavior of beams and the curvature of shells. For instance, in the Euler-Bernoulli beam theory, the displacement field is assumed to be a cubic polynomial, analogous to the cubic Hermite polynomial, enabling the precise capture of the beam's slope at the nodes.

On the other hand, the Timoshenko beam theory, which accounts for shear deformation, does not necessarily require Hermite polynomials, and Lagrange interpolating polynomials can be sufficient depending on the formulation. The inclusion of derivative continuity is particularly important in higher-order theories or when the solution gradients are as important as the values themselves, such as in curvature or bending moments in beams and shells.

The continuum mechanics approach to beam formulation, which provides a more general framework, can leverage either Hermite or Lagrange polynomials, depending on the desired accuracy and complexity of the model.

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