MECH3300 Finite element Methods. LECTURE 2 - Commercial packages and how they work. Commercial finite element packages . History - initially punched card input and printer output. The original codes are now the “solver”, but have graphical preprocessing and post-processing added.
- Commercial packages and how they work
Output can be displayed as scaled deflected shapes, contour plots of stress, animation of deformation, xy plots etc.
Researchers continue to push the limits of what problems can be solved. Eg at Northwestern University, Professor Belytschko has recently been growing cracks through finite element meshes.
What happens when you press the “solve” option on a menu….
1. THE DIRECT STIFFNESS METHOD
2. ASSEMBLY OF ELEMENT MATRICES
3. DEGREES OF FREEDOM IN A STRUCTURE
4. RESTRAINTS [or constraints or freedom conditions]
5. THE SOLUTION PROCESS
Most structures are statically indeterminate, so that loads are shared between members in accordance with their stiffness. Hence we can’t just use statics.
The direct stiffness method refers to writing a large stiffness matrix to describe a structure. It was invented as a method of hand calculation, but only became popular once computers could solve the equations.
Structural equations can be written as displacement = f(loads) giving “compliance” or “flexibility” terms or as loads = f(displacements) giving stiffness terms.
The advantage of the latter is that it leads to a sparse stiffness matrix, with non-zero terms only where local interactions between members or elements occur.
Let spring 1 have axial displacements u1 at node 1 and u2at node 2. It’s element equations are
Ie the force in the spring is its stiffness k times the difference in displacement over its length. To add another spring in series, add another node with F3and u3associated with it. The equations for spring 1 are now
Spring 1 Spring 2
For the second spring, its expanded stiffness matrix is that in the eqns.
Adding the forces at nodes
Hence the assembled stiffness matrix is
Note this summation assumes at all elements at a node have the same displacement - for beams and plates rotations are matched as well, so this means a joint is always treated as a rigid joint by default, not as a pin joint.
The process of expanding element matrices with zeroes and adding them can be continued. For instance, for 5 equal springs end to end, [K] is
Note this matrix is banded - the non-zero terms cluster around the diagonal and it is symmetric. These properties typically apply to matrices of structural problems and are exploited in equation solvers.
Consider the 3 by 3 set of equations below
If u2= 0 then column 2 of [K] is multiplied by zero, and F2becomes an unknown reaction. Hence we can solve for u1and u3by solving
Having found u1and u3we can then find F2 = k2 u1 + k5 u3 if the reaction is of interest. Finding a reaction may need to be done to check that the reaction is zero, in a case where the restraint is an artificial one, applied just to stop rigid body motion.
1.Find element stiffness matrix for each member, linking forces/moments to displacements/rotations at joints. Write these in the global coordinate system.
2. Add (or assemble) these to give a large matrix [K], by expanding each member matrix with rows/columns of zeroes.
3. Remove rows and columns of [K] corresponding to any constrained displacements or rotations. This must be done to at least prevent any rigid body motion or mechanism.
4. Solve [K] u = F for joint displacements or rotations u.
5. Use each element matrix times relevant displacements to find the forces/moments on nodes of that element. These imply internal loadings (eg bending moment), and internal loadings enable stresses to be found.
A beam element loaded at each end only will have linear variation of bending moment over its length in each principal plane.
Gravity or inertia loading can give a uniform distributed loading. This leads to quadratic variations of bending moment.
The sign of a moment at a node may be different to that of the bending moment in the beam there.
Results must be reported in local axes, aligned with the principal axes of a beam, not in the global axes used to describe the whole model, in order to distinguish loading causing bending from that causing twisting or axial loading.
Local axis directions are determined by the reference node position.