Lecturer: Lars Andersen, MSc, PhD, Associate Professor Department of Civil Engineering, Division of Structural Mechanics

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1 Purpose of the Course 1 Lecturer: Lars Andersen, MSc, PhD, Associate Professor Department of Civil Engineering, Division of Structural Mechanics Aalborg University, it Sohngaardsholmsvej h 57, DK-9000 Aalborg Phone: la@civil.aau.dk Homepage of the course: Teaching Activities Finite Element Design After the course, the student must be able to: Understand the basic concepts in Finite Element Analysis (FEA) Use a FEA program (STAAD.Pro 2007) Know the basic terms and algorithms behind the analysis Be able to analyse large complex structures Come up with realistic dimensions of structural elements.

2 Contents of the lecture 2 Introduction to the Finite-Element Method (FEM) What is Finite-Element Analysis (FEA)? Historic overview Why use FEA? Input and output from an FEA How does FEM work? Example in STAAD.Pro Exercise: Learn to use STAAD.Pro (plane-frame problem)

3 Short Historic Overview of FEM 3 FEM is tied with the development of computer technology Approximately 40 years old NASA developed NASTRAN in the 1960s First College Course in FEM was offered in 1970 In the 1970s, FEM was limited to large corporations with expensive mainframe computers In the 1980s, powerful desktop computers made FEM an indispensable engineering tool In the 1990s, more complex elements are introduced, optimization capabilities are integrated, and CAD programs are used for modelling complex structures Originally, the method was developed for the analysis of g y p y stresses in structures but today FEM is used to analyse heat transfer, fluid flow, electric and magnetic fields etc.

4 Why use FEA? Simple Statically Determinate System 4 Freeway-crossing north of Aalborg approximated by statically ti determinate t system 5 reaction forces determined by 5 equilibrium equations

5 Why use FEA? Simple Statically Indeterminate System 5 Freeway-crossing north of Aalborg approximated by statically ti indeterminate i t system 6 reaction forces but only 3 equilibrium equations

6 Why use FEA? Complex Structure 6 David Fay Custom Chair Golden Gate Bridge, San Francisco Antwerp Railway Station, Belgium Kandahar Airport, Afghanistan

7 Why use FEA? Load combinations 7

8 Discrete Versus Continuous System 8 Prototype = reality Continuous system Discrete system = FE model

9 Input and Output from an FEA 9 Input: Dimensions of the structure Cross-section types (circular, rectangular, I-profile,...) Material properties (wood, steel, concrete, glass,...) Supports (fixed, free, moving,...) Loads (concentrated, line, surface, combinations) Output: Deformation components (translation, rotation) Section force curves, reactions (shear, normal, moment) Strains and stresses (shear, normal) Fulfilment of design criteria (Eurocode,...) Eigenmodes (dynamic resonance risk,...)

10 Eigenmodes/Eigenfrequencies 10 Tacoma Narrows Bridge, 1940

11 Eigenmodes/Eigenfrequencies 11 Millenium Bridge London, 2000/2002

12 Eigenmodes/Eigenfrequencies 12

13 FEA Programs 13 Commercial Finite-Element Programs: ABAQUS ( COSMOSWorks (SolidWorks) ( FEMLAB ( STAAD.Pro ( ANSYS Structural ( etc.

14 How Does FEA work? Dividing the structure into elements 14 From the user input, a given structure is divided into small elements (finite elements) (done partly by user and partly by program) Each element is assigned material properties (done by user) Each element s mechanical behaviour is defined by a set of differential equations Each element s mechanical behaviour is defined by a set of differential equations from the choice of element type and material properties (done by program)

15 How Does FEA work? Matrix Equations for the Elements are Found 15 The differential equations for each element are solved and arranged dinto a matrix ti formulation suitable for computeraided solutions (done by program)

16 How Does FEA work? Matrix Equation for the Global System is Assembled 16 The element matrices are combined into a global system of equations defined d from the placement of the respective elements (done by program) From this the global structural equation is obtained From this the global structural equation is obtained (done by program)

17 How Does FEA work? Load and Boundary Conditions are Applied 17 The boundary conditions (loads and supports) are specified (done by user) ) The boundary conditions are incorporated into the system of differential equations (done by program)

18 How Does FEA work? The Structural Matrix Equation is Solved 18 The displacement (and rotations) of all nodes are found from solving the system of equations (done by program) ) Displacements at intermediate points are found from interpolation of nodal values (done by program)

19 How Does FEA work? Stresses and Strains are found 19 The strains are found from the displacements (done by program) Stresses are found from a constitutive relation (done by program)

20 How Does FEA work? The design criteria are checked 20 The design criteria are checked (done by user/program) The structure is modified to fulfil criteria and a new analysis is made (done by user)

21 Example Analytical Solution 21 Structural system: Analytical solution:

22 STAAD.Pro Overview Editor 22 Mark beam Add beam Hold down ctrl to move the starting point of a beam Menu Snap node

23 STAAD.Pro Overview 23 General, property, support, load Database

24 STAAD.Pro Overview 24 Analyse/print All

25 STAAD.Pro Script file 25 Node coordinates Member definition Material definition iti Section assignment Material assignment Support assignment Load assignment Analysis definition

26 STAAD.Pro Analysis Analyze Mode 26

27 STAAD.Pro Results 27 Double-click gives section displacement Node displacement Reactions Section forces

28 Example Numerical Result from STAAD.Pro 28 Analytical solution:

29 Today's Problem 29 Plane bridge Determine e the profile types from the deformation o criteria (1/200 of span). Change the supports and determine the profile types in the same manner. Get familiar with the program. (Use the menu Geometry/Split Beam to p g ( y p divide the vertical beam into 3 for easy applying the load)

30 Today's Problem 30 Support types (F)ixed, (P)ined, (F)ixed (B)ut (direction) Profile types + Maximum deformation (Global deformation), local (x,y)-coordinates FB P F IPE160 HE200B di di di (Fx,Mz) P F IPE160 HE200B y-dir mm x=4.167 FB P F (Fx,Mz) y-dir mm x=5.833 x-dir 33.8 mm y= F F F P P P 50 mm 50 mm 50 mm Maximum allowed deformation 1/200 of span = 50 mm

Lecture #9 Matrix methods

Lecture #9 Matrix methods METHODS TO SOLVE INDETERMINATE PROBLEM Small degree of statical indeterminacy Force method Displacement methods Displacement method in matrix formulation Large degree of statical