Operasi Matrix Di Staad

Purpose of the Course

Lecturer: ƒ Lars Andersen, MSc, PhD, Associate Professor ƒ Department of Civil Engineering, Division of Structural Mechanics A lb Aalborg U University, i it S Sohngaardsholmsvej h d h l j 57 57, DK DK-9000 9000 A Aalborg lb ƒ Phone: 9940 8455 | E-mail: [email protected] Homepage of the course: ƒ www.wind.civil.aau.dk → Teaching Activities → Finite Element Design After the course, the student must be able to: ƒ Understand the basic concepts in Finite Element Analysis (FEA) ƒ Use a FEA program (STAAD.Pro 2007) ƒ Know K the th basic b i tterms and d algorithms l ith b behind hi d th the analysis l i ƒ Be able to analyse large complex structures ƒ Come up with realistic dimensions of structural elements.

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Contents of the lecture

Introduction to the Finite-Element Method (FEM) ‫ ﻩ‬What is Finite-Element Analysis (FEA)? ‫ ﻩ‬Historic overview ‫ ﻩ‬Why use FEA? p and output p from an FEA ‫ ﻩ‬Input ‫ ﻩ‬How does FEM work? ‫ ﻩ‬Example in STAAD.Pro ‫ ﻩ‬Exercise: Learn to use STAAD.Pro (plane-frame problem)

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Short Historic Overview of FEM

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 integrated, and CAD programs are used for modelling complex structures g y the method was developed p for the analysis y of Originally, stresses in structures – but today FEM is used to analyse heat transfer, fluid flow, electric and magnetic fields etc.

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Why use FEA? Simple Statically Determinate System

Freeway-crossing north of Aalborg ...

... approximated i t db by statically t ti ll d determinate t i t system t

5 reaction forces determined by 5 equilibrium equations

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Why use FEA? Simple Statically Indeterminate System

Freeway-crossing north of Aalborg ...

... approximated i t db by statically t ti ll iindeterminate d t i t system t

6 reaction forces but only 3 equilibrium equations

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Why use FEA? Complex Structure

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David Fay Custom Chair

Golden Gate Bridge, San Francisco

Antwerp Railway Station, Belgium

Kandahar Airport, Afghanistan

Why use FEA? Load combinations

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Discrete Versus Continuous System

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Prototype = reality

Continuous system

Discrete system = FE model

Input and Output from an FEA

Input: ‫ ﻩ‬Dimensions Di i off the h structure ‫ ﻩ‬Cross-section types (circular, rectangular, I-profile, ...) ‫ ﻩ‬Material properties (wood, steel, concrete, glass, ...) ‫ ﻩ‬Supports (fixed, free, moving, ...) ‫ ﻩ‬Loads (concentrated, line, surface, combinations) Output: ‫ ﻩ‬Deformation components (translation, rotation) ‫ ﻩ‬Section force curves, reactions (shear, normal, moment) ‫ ﻩ‬Strains and stresses (shear, normal) ‫ ﻩ‬Fulfilment of design criteria (Eurocode, ...) ‫ ﻩ‬Eigenmodes (dynamic resonance risk, ...)

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Eigenmodes/Eigenfrequencies

Tacoma Narrows Bridge, 1940

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Eigenmodes/Eigenfrequencies

Millenium Bridge London, 2000/2002

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Eigenmodes/Eigenfrequencies

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FEA Programs

Commercial Finite-Element Programs: ‫ ﻩ‬ABAQUS (www.simulia.com) ‫ ﻩ‬COSMOSWorks (SolidWorks) (www.cosmosm.com) ‫ ﻩ‬FEMLAB (www.comsol.dk) ‫ ﻩ‬STAAD.Pro (www.bentley.com) ‫ ﻩ‬ANSYS Structural (www.ansys.com) ‫ ﻩ‬etc. etc

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How Does FEA work? Dividing the structure into elements

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‫ ﻩ‬From the user input, a given structure is divided into small elements (finite elements) (done partly by user and partly by program) ‫ ﻩ‬Each element is assigned material properties (done by user) ‫ ﻩ‬Each element’s element s mechanical behaviour is defined by a set of differential equations from the choice of element type and material properties (done by program)

How Does FEA work? Matrix Equations for the Elements are Found

‫ ﻩ‬The differential equations for each element are solved and arranged d iinto t a matrix t i fformulation l ti suitable it bl ffor computert aided solutions (done by program)

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How Does FEA work? Matrix Equation for the Global System is Assembled

‫ ﻩ‬The element matrices are combined into a global system of equations ti d defined fi d ffrom th the placement l t off th the respective ti elements (done by program) ‫ ﻩ‬From this the global structural equation is obtained (done by program)

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How Does FEA work? Load and Boundary Conditions are Applied

‫ ﻩ‬The boundary conditions (loads and supports) are specified (d (done b by user)) ‫ ﻩ‬The boundary conditions are incorporated into the system of differential equations (done by program)

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How Does FEA work? The Structural Matrix Equation is Solved

‫ ﻩ‬The displacement (and rotations) of all nodes are found from solving th system the t off equations ti (done (d b by program)) ‫ ﻩ‬Displacements at intermediate points are found from interpolation of nodal values (done by program)

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How Does FEA work? Stresses and Strains are found

‫ ﻩ‬The strains are found from the displacements (done by program) ‫ ﻩ‬Stresses are found from a constitutive relation (done by program)

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How Does FEA work? The design criteria are checked

‫ ﻩ‬The design criteria are checked (done by user/program) ‫ ﻩ‬The structure is modified to fulfil criteria and a new analysis is made (done by user)

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Example – Analytical Solution

Structural system:

Analytical solution:

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STAAD.Pro – Overview

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Editor

Add beam Mark beam

Hold down ctrl to move the starting point of a beam

Menu

S Snap node d

STAAD.Pro – Overview

General, General property, support, l d load

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D t b Database

STAAD.Pro – Overview

Analyse/print All

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STAAD.Pro – Script file

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Node coordinates Member definition M t i ld Material definition fi iti Section assignment Material assignment Support assignment Load assignment Analysis definition

STAAD.Pro – Analysis

Analyze

Mode

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STAAD.Pro – Results

Double click gives Double-click section displacement

Node displacement Reactions Section forces

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Example – Numerical Result from STAAD.Pro

Analytical solution:

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Today's Problem

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Plane bridge

Determine ete e tthe ep profile o e types from o tthe e de deformation o at o ccriteria te a ((1/200 / 00 o of spa span). ) Change the supports and determine the profile types in the same manner. Get familiar with the p program. g ((Use the menu Geometry/Split y p Beam to divide the vertical beam into 3 for easy applying the load)

Today's Problem

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Support types

Profile types

Maximum deformation

(F)ixed, (P)ined, (F)ixed (B)ut (direction)

(Global deformation), local (x,y)-coordinates (x,y) coordinates

+ FB (Fx,Mz)

P

F

FB (Fx,Mz)

P

F

F

F

F

P

P

P

50 mm

IPE160

HE200B

y-dir di 139.2 mm x=4.167

y-dir di 165.6 mm x=5.833

x-dir di 33.8 mm y=2.85+1.45

50 mm 50 mm

Maximum allowed deformation 1/200 of span = 50 mm