8. Beams and Plates#

If one spatial dimension of a domain $\mathrm{\Omega }=[0,1{]}^2\times [-t/2,t/2]$ is small compared to the other directions, $t\ll 1$, we have the following options

• Use a uniform mesh with an enormous amount of elements

• Use anisotropic elements

• Make a dimension reduction and only mesh the two-dimensional domain $[0,1{]}^2\times \{0\}$

The first option is inefficient as we would waste a lot of elements. The second option can easily lead to locking problems if the thickness $t$ is small. Therefore, the third option is commonly considered, where the 3D elasticity problem is reduced to a two-dimensional plate problem.

If two directions are small compared to the third one, we can apply a dimension reduction to one-dimensional beams.

In this chapter we first derive the Reissner-Mindlin and Kirchhoff-Love plate equations. Then, we present the Timschenko and Euler-Bernoulli beam and discuss the arising problems of shear locking or how to handle a fourth order problem as a mixed problem. After this, locking-free and stable formulations of the Kirchhoff-Love plate via Hellan-Herrmann-Johnson (HHJ) elements and Reissner-Mindlin plates with the tangential-displacement-normal-normal-stress continuous (TDNNS) elements are considered:

• Derivation plates

• Timoschenko and Euler-Bernoulli beam

• Kirchhoff-Love plates

• Reissner-Mindlin plates