Finite Element Analysis

Aerospace Application

Course Plan

• Day 1: Introduction to FEA• Day 1: Direct Stiffness Method • Day 2: Direct Stiffness Method (2D Structures)• Day 2: Structural Applications (Beam Elements)• Day 3: Solid Elements and Shell Elements• Day 3: FEM Check• Day 4: FEA Analysis Process• Day 5: Design of Experiments (DOE)• Day 6 and 7: Project: Propeller Design• Day 8 and 9: Project: Wing design• Day 10: Project Presentations

Tools we’ll use!!!

Excel

Solidworks

Course Plan

• Day 1: Introduction to FEA• Day 1: Direct Stiffness Method • Day 2: Direct Stiffness Method (2D Structures)• Day 2: Structural Applications (Beam Elements)• Day 3: Solid Elements and Shell Elements• Day 3: FEM Check• Day 4: FEA Analysis Process• Day 5: Design of Experiments (DOE)• Day 6 and 7: Project: Propeller Design• Day 8 and 9: Project: Wing design• Day 10: Project Presentations

Introduction to FEA

• What is the Finite Elements Analysis (FEA)?– A numerical method.– Traditionally, a branch of Solid Mechanics.– Nowadays, a commonly used method for multiphysics

problems.• What areas can FEA be applied?

– Structure analysis: a cantilever, a bridge, landing train…– Solid Mechanics: a gear, a automotive power train…– Dynamics: bullet impact…– Thermal analysis: thermal stress brake disc…– Electrical analysis: Piezo actuator, MEMS…

What is the Finite ElementsAnalysis (FEA)?

What is the Finite ElementsAnalysis (FEA)?

What is the Finite ElementsAnalysis (FEA)?

How does FEA work?Example: Find the perimeter of the circle

Exact solution: P = π * (3 in) = 9.4247779608 in

Easy because we know pi (π), thanks Archimedes!!!

What if pi does not exist????

3 in

We can solve this problem considering the circle as an polygon, we will start with the square.

r

r

r

rθ L

N = 4r = 1.5 inθ = 360⁰ / N = 90⁰L = sqrt(2*r^2*(1-cos(θ))) = 2.12132P = N * L = 8.4852813742Error = 9.97%

How does FEA work?Let’s solve it in excel for different number of sides

Approximate solution = 9.42 in

Convergence

How does FEA work?

How does FEA work?

Course Plan

• Day 1: Introduction to FEA• Day 1: Direct Stiffness Method • Day 2: Direct Stiffness Method (2D Structures)• Day 2: Structural Applications (Beam Elements)• Day 3: Solid Elements and Shell Elements• Day 3: FEM Check• Day 4: FEA Analysis Process• Day 5: Design of Experiments (DOE)• Day 6 and 7: Project: Propeller Design• Day 8 and 9: Project: Wing design• Day 10: Project Presentations

Direct Stiffness Method

• The direct stiffness method is used mostly for Linear Static Analysis

• The development of the direct stiffness method originated in the 1940s and is generally considered the fundamental of finite elemental analysis.

• Linear Static analysis is appropriate if deflections are small and vary only slowly.

• Linear Static analysis omits time as variable.• It also exclude plastic action and deflections that change the

way loads are applied.• The direct stiffness method for Linear Static analysis follows the

laws of Statics and the laws of Strength of Materials.

Direct Stiffness Method

Direct Stiffness Method

• One-dimensional Truss Element– The simplest type of engineering structure is the truss structure– A truss member is a slender (the length is much larger than the cross

section dimensions) two-force member.– Members are joined by pins and only have the capability to support

tensile and compressive loads axially along the length.

Direct Stiffness Method• For the analysis, we will establish the following sign conventions:

– Forces and displacements are defined as positive when they are acting in the positive X direction.

– The position of a node in the undeformed condition is the finite element position of than node.

• If equal and opposite forces of magnitude F are applied to the end nodes, the member will undergo a change in length according to the equation:

• This equation can also be written as , which is similar to Hooke’s Law used in linear spring.

• K is called the spring constant or stiffness of the spring

Direct Stiffness Method• We will use the general equations of a single one-dimensional truss element to illustrate the formulation of the stiffness

matrix method:

• By using the relative Motion Analysis method, we can derive the general expressions of the applied forces (F1 and F2) in terms of the displacements of the nodes (X1 and X2) and the stiffness constant (K)

Direct Stiffness Method

• Example: Find the nodal displacements and reaction forces.

Free body diagram

Direct Stiffness Method

Direct Stiffness Method

Direct Stiffness Method

Inverse Matrix

Direct Stiffness Method

ExercisesExercise 1: Find the nodal displacements and reaction forces.

Exercise 2: Find the nodal displacements and reaction forces.

Homework

1.Reading lecture: Truss Elements in Two Dimensional Spaces

2.Make sure you have installed Solidworks, Solidworks Simulation and Solidworks Flow Simulation in your laptop.