module 1 introduction to fem

Introduction to FEM

Finite Element Analysis (ENGR 455)

Dr. Andreas SchifferAssistant Professor, Mechanical Engineering

Tel: +971‐(0)2‐4018204

andreas.schiffer@kustar.ac.ae

2

The Finite Element Method• 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

The Finite Element Method

• 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

General Steps of an FE analysisStep 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

General Steps of an FE analysisStep 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

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

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)

1d K F

There are 3 categories of steps in an FE analysis:

F K d

8

Application of the FE method

Impact analysis of an ice deflector ramp for the railway industry

Experimental investigation

9

Application of the FE method

Impact analysis of an ice deflector ramp for the railway industry

Plastic strains induced in the ramp

10

Application of the FE 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

Application of the FE 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

Application of the FE 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

Application of the FE method

Fluid-structure interaction in underwater blast loading

Free‐standing rigid plate

14

Application of the FE method

Crack propagation in a random heterogeneous material

Crack propagation is modeled by using the extended FE method (XFEM)

Experiments:

FE simulations: