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![6
General Steps of an FE analysis
Step 5: Assemble the Global System Equation
• All individual elements are assembled using the method of
superposition (or direct stiffness method) to produce the global or
total system of equations of the problem.
Here, {F} is the vector of global nodal forces, [K] is the structure
global or total stiffness matrix, {d} is now the vector of known and
unknown structure nodal degrees of freedom (displacements).
• It can be shown that at this stage, the global stiffness matrix [K] is
a singular square matrix because its determinant is equal to zero.
• To remove this singularity, we must invoke certain boundary
conditions (or constraints or supports) so that the structure
remains in place instead of moving as a rigid body.
• Step 6: Apply Boundary Conditions and Loading
• Prescribe forces and displacements at nodes.
F K d
F K d
](https://image.slidesharecdn.com/282165203-module-1-introduction-to-fem-250809232933-1709c3e2/75/282165203-Module-1-Introduction-to-FEM-pdf-6-2048.jpg)
![7
General Steps of an FE analysis
• Step 7: Solve for the Unknown Degrees of Freedom
• Involves finding the inverse of the global stiffness matrix [K]-1 .
• Then the structure’s unknown nodal degrees of freedom {d} can
be calculated via
• Step 8: Solve for the Element Strains and Stresses
• Strains can be directly expressed in terms of the displacements
determined in Step 7.
• Stresses are obtained from the strain solutions through the
constitutive law (e.g. Hooke’s law).
• Step 9: Interpret the Results (Post-processing)
Determination of locations in the structure where large deformations
and large stresses occur is generally important in making design
decisions.
1. Pre-processing (Step 1-6)
2. Solution (Step 7-8)
3. Post-processing (Step 9)
1
d K F
There are 3 categories of
steps in an FE analysis:
F K d
](https://image.slidesharecdn.com/282165203-module-1-introduction-to-fem-250809232933-1709c3e2/75/282165203-Module-1-Introduction-to-FEM-pdf-7-2048.jpg)
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282165203-Module-1-Introduction-to-FEM.pdf
- 1. Introduction to FEM FiniteElement Analysis (ENGR 455) Dr. Andreas Schiffer Assistant Professor, Mechanical Engineering Tel: +971‐(0)2‐4018204 andreas.schiffer@kustar.ac.ae
- 2. 2 The Finite ElementMethod • The Finite Element Method (FEM) is a numerical method for solving problems of engineering and mathematical physics. • In this method, the partial differential equations of a mathematical model are discretized to obtain a set of simultaneous algebraic equations. • The discretization is achieved by dividing the solution domain into an equivalent system of smaller bodies or units (= finite elements). Elements Nodes The elements are interconnected at points common to two or more elements (nodal points or nodes)
- 3. 3 The Finite ElementMethod • The FE solutions yields approximate values for the unknowns at discrete points in space. • The FE method is important because for problems involving complicated geometries, loadings and material properties, it is generally not possible to obtain analytical mathematical solutions. • The FE method can be applied to many engineering problems, inculding structural analysis, heat transfer, fluid flow, mass transport and electromagnetic potential.
- 4. 4 General Steps ofan FE analysis Step 1: Discretize the Problem and Select the Element Types • Divide the structure into small pieces, usually called meshing • Type of element depends on the nature of the problem (structural) and the dimension (1D, 2D or 3D)
- 5. 5 General Steps ofan FE analysis Step 2: Select a Displacement Function • Choosing a displacement function within each element connecting the nodes. • Linear, quadratic and cubic polynomials are most common. Step 3: Define the Stress-Strain Relationships • Describe the constitutive law relating stresses to strains in each element • The simplest constitutive relation is Hooke’s law, σx = E εx (in 1D). Step 4: Derive the Element Stiffness Matrix and Equations • Write the system of equations describing the structural behavior of an element. • Relating nodal forces to nodal displacements of the element through an element stiffness matrix: f k d
- 6. 6 General Steps ofan FE analysis Step 5: Assemble the Global System Equation • All individual elements are assembled using the method of superposition (or direct stiffness method) to produce the global or total system of equations of the problem. Here, {F} is the vector of global nodal forces, [K] is the structure global or total stiffness matrix, {d} is now the vector of known and unknown structure nodal degrees of freedom (displacements). • It can be shown that at this stage, the global stiffness matrix [K] is a singular square matrix because its determinant is equal to zero. • To remove this singularity, we must invoke certain boundary conditions (or constraints or supports) so that the structure remains in place instead of moving as a rigid body. • Step 6: Apply Boundary Conditions and Loading • Prescribe forces and displacements at nodes. F K d F K d
- 7. 7 General Steps ofan FE analysis • Step 7: Solve for the Unknown Degrees of Freedom • Involves finding the inverse of the global stiffness matrix [K]-1 . • Then the structure’s unknown nodal degrees of freedom {d} can be calculated via • Step 8: Solve for the Element Strains and Stresses • Strains can be directly expressed in terms of the displacements determined in Step 7. • Stresses are obtained from the strain solutions through the constitutive law (e.g. Hooke’s law). • Step 9: Interpret the Results (Post-processing) Determination of locations in the structure where large deformations and large stresses occur is generally important in making design decisions. 1. Pre-processing (Step 1-6) 2. Solution (Step 7-8) 3. Post-processing (Step 9) 1 d K F There are 3 categories of steps in an FE analysis: F K d
- 8. 8 Application of theFE method Impact analysis of an ice deflector ramp for the railway industry Experimental investigation
- 9. 9 Application of theFE method Impact analysis of an ice deflector ramp for the railway industry Plastic strains induced in the ramp
- 10. 10 Application of theFE method Side-crash simulation of a road car v0 Deformation of the B‐frame after the impact Modelling of damage and failure by element deletion
- 11. 11 Application of theFE method Deformation of a sandwich plate under blast loading (Source: ABAQUS) TNT after 1 kg TNT: after 2 kg TNT: Deformation of the honeycomb core after 1kg TNT
- 12. 12 Application of theFE method Structural dynamics analysis of a steel bridge 1st mode (bending mode) f = 19.6 Hz 2nd mode (bending mode) f = 51.4 Hz 3rd mode (swaying mode) f = 69.1 Hz
- 13. 13 Application of theFE method Fluid-structure interaction in underwater blast loading Free‐standing rigid plate
- 14. 14 Application of theFE method Crack propagation in a random heterogeneous material Crack propagation is modeled by using the extended FE method (XFEM) Experiments: FE simulations: