Nonlinear finite elements/Homework 6

Problem 1: Continuum-based Nonlinear Beam Bending
You have seen that the derivation of the finite element equations for beams can get quite complicated.

One alternative is to use the continuum-based approach given in Belytschko, Liu, and Moran. Your task for this problem is to read the portion on beams from the chapter on Beams and Shells from that book. After you have completed that task, answer the following questions.


 * 1) What is the continuum-based approach for the finite element analysis of beams? Draw figures to elucidate.
 * 2) Why is the continuum-based approach preferred in nonlinear finite element analysis?
 * 3) The rate of deformation is defined as

\boldsymbol{D} = \text{sym}(\boldsymbol{\nabla} \mathbf{v}) = \frac{1}{2}\left[\boldsymbol{\nabla} \mathbf{v} + (\boldsymbol{\nabla} \mathbf{v})^T\right] $$ where $$\mathbf{v}$$ is the velocity.In index notation, we write

D_{ij} = \text{sym}(v_{i,j}) = \frac{1}{2}\left(v_{i,j} + v_{j,i}\right) = \frac{1}{2}\left(\frac{\partial v_i}{\partial x_j} + \frac{\partial v_j}{\partial x_i}\right), \qquad i,j=1,2,3~. $$ Given the above definition, derive equations (9.2.1) through (9.2.7) from the book.
 * 4) Why is the Euler-Bernoulli beam theory $$C^1$$? Why is the Timoshenko beam theory $$C^0$$?
 * 5) Why is a $$C^0$$ beam theory preferable?
 * 6) What are the major assumptions in the Timoshenko beam formulation described in the handout (the authors call these the modified Mindlin-Reissner assumptions)?
 * 7) Derive the equation (9.3.13) of the book starting from equation (9.3.3).

Problem 1a: Curved laminated beam
To enforce the plane stress assumption, we have to rotate the rate of deformation so that its components are aligned with a local coordinate system aligned with the laminae. This exercise is meant to improve your understanding of the process.

Consider the curved composite beam shown in Figure 1.

Assume that the beam has been shaped into an arc of a circle. The material of the beam is a transversely isotropic fiber composite material with the fibers running along the length of the beam. The rate constitutive relation of the material is given by

\cfrac{D}{Dt} \begin{bmatrix} \sigma_{11} \\ \sigma_{22} \\ \sigma_{33} \\ \sigma_{23} \\ \sigma_{31} \\ \sigma_{12} \end{bmatrix} = \begin{bmatrix} C_{11} & C_{12} & C_{13} & 0 & 0 & 0 \\ C_{12} & C_{11} & C_{13} & 0 & 0 & 0 \\ C_{13} & C_{13} & C_{33} & 0 & 0 & 0 \\ 0 & 0 & 0 & C_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & C_{44} & 0 \\ 0 & 0 & 0 & 0 & 0 & (C_{11}-C_{12})/2 \end{bmatrix} \begin{bmatrix} D_{11} \\ D_{22} \\ D_{33} \\ D_{23} \\ D_{31} \\ D_{12} \end{bmatrix}~. $$


 * 1) Construct a laminar coordinate system at the blue point.  (Use equations 9.3.16 and 9.3.17 from the book).  Assume that the blue point is at the center of element 5.
 * 2) Assume that the velocities at nodes 5 and 6 are $$\mathbf{v}_5$$ and $$\mathbf{v}_6$$, respectively.  Compute the velocity gradient at the blue point.
 * 3) Compute the rate of deformation at the blue point.
 * 4) Rotate the rate of deformation so that its components are with respect to the laminar coordinate system.
 * 5) Compute the stress rate at the blue point.  This stress rate is expressed in terms of the laminar coordinate system.
 * 6) Apply the plane stress condition and solve for $$D_{yy}$$.
 * 7) Express the stress rate and the modified rate of deformation in the global coordinate system.

You may use numbers instead of symbols to make the above process more concrete in your mind.

Problem 1b: ( Optional: Extra Credit)
Show how we obtain equation 9.3.22 of the book. Recall that, in matrix notation, the nodal internal force is given by

\mathbf{f}^{\text{int}} = \int_{\Omega} \mathbf{B}^T \boldsymbol{\sigma}~d\Omega $$ where the $$\mathbf{B}$$ matrix relates the rate of deformation to the nodal velocities ($$\mathbf{v}$$) by

\mathbf{D} = \mathbf{B} \mathbf{v} ~. $$

Problem 1c: ( Optional: Extra Credit)
As you have noticed, not all the steps on the finite element solution process have been described by the authors. Write down the extra steps that you think you will need to complete the solution process.

Problem 2: Exploring commercial software
Track down the different types of beam and shell elements that ANSYS and LS-DYNA provide.For each element that you find, answer the following
 * 1) What is the name of the element?
 * 2) How many nodes does the element have?
 * 3) How many displacement type (displacements, rotations) degrees of freedom are there at each node?  List them.
 * 4) How many force type (forces, moments) boundary conditions can be applied at each node?List them.
 * 5) What is the form of the shape functions used by the element?  (Alternatively, you may write down the approximate solutions assumed by the element.)
 * 6) Is the beam element a continuum-based one or one derived from the equations of beam theory?
 * 7) Does the element allow for full and reduced integration?