CAAD of Bridges Handouts

8/13/2019 CAAD of Bridges Handouts

1/12

Computer Aided AnalysisComputer Aided Analysis

& Design& Design

General Concepts ofGeneral Concepts of CAADCAADErEr.. SarojSaroj BhattaraiBhattarai

DORDOR

Conventional Practices in bridgeConventional Practices in bridge

analysis/designanalysis/design Different elements of the structure considered individually;Different elements of the structure considered individually;

Interaction with other elements are covered mostly by empiricalInteraction with other elements are covered mostly by empirical

relationship and assumptionsrelationship and assumptions

e.g. factors for con tinuity of slabs, load distribution among th e girders,e.g. factors for con tinuity of slabs, load distribution among th e girders,

act ion and behav ior of c ross i rders etcact ion and behav ior of c ross i rders etc..

Deck slabs are analyzed with the help of empirical graphs (Deck slabs are analyzed with the help of empirical graphs (Pigaud'sPigaud's

curves) or effective width methodcurves) or effective width method

Loads distribution withLoads distribution with Courbon'sCourbon's oror MorriceMorrice & Little's methods& Little's methods

Analytical or graphical analysis of forces in the truss members in aAnalytical or graphical analysis of forces in the truss members in a

2D plane2D plane

Conventional Practices in bridge analysis/designConventional Practices in bridge analysis/design

Optimization of design is time consuming. A small change in any

parameter of a structure will cause repetition of the whole process.

As the degree of indeterminacy increases, it becomes more and

more cumbersome to analyze the structure.

True analysis of a real structure in a 3D space considering the

Optimization of design is time consuming. A small change in any

parameter of a structure will cause repetition of the whole process.

As the degree of indeterminacy increases, it becomes more and

more cumbersome to analyze the structure.

True analysis of a real structure in a 3D space considering theeffects of all connected elements with conventional methods is very

difficult, if not impossible.

effects of all connected elements with conventional methods is very

difficult, if not impossible.


2/12

Use of Computers for EngineersUse of Computers for Engineers

There is no instant solution

Use computers for repetitive and complicated tasks

Use of Spreadsheets for customized solutions Customized solution through programming

General purpose software based on Finite Element Analysis

SAP2000, STAAD, ANSYS, etc.

Structural software for specific solutions

SAFE, ETABS, etc.

Other engineering and General Purpose software

GIS, DTM software, Hydrological analysis programs, Road design software,etc.

Drafting software, e.g. AutoCAD

Simplified Structural AnalysisSimplified Structural AnalysisReal Structure is governed byReal Structure is governed by

Partial DifferentialPartial Differential

Equations of various orderEquations of various order

P

A - cons tantE - cons tant

L

dl

dl = PL/AE L

dl

P

A - variesE - may vary

dl = ?

a) b)

dl is obta ined byby s olvi ng differential equation

c)

xx yy zz

vxx y z p+ + + =0

P

L1 L2 L3 L4

L5

dl1 dl2

dl3 dl4

dl5

The tapered bar subs tituted by a numberof s tra ight bars (finite e lements ).

The tota l e longation = vector sum ofindividual e longations ..As the number of e lements increase,the result becomes more accurate .

A and E are cons tant within the lengthof each e lement.dl1 = PL1/A1 E1, dl2 = PL2/A2 E2 . . .an d s o o n .

Direct solution is only possible for:Direct solution is only possible for:

Simple geometrySimple geometry

Simple BoundarySimple Boundary

Simple Loading.Simple Loading.

The Need for Structural ModelThe Need for Structural Model

EXCITATIONEXCITATION

LoadsLoads

RESPONSESRESPONSESDisplacementsDisplacements

StrainsStrains

STRUCTURESTRUCTURE

StructuralStructural

ModelModel

rat onsrat ons

SettlementsSettlements

Thermal ChangesThermal Changes

StressStress

Stress ResultantsStress Resultantsppvv


3/12

From Classical to FEMFrom Classical to FEM

Ass ump tio ns

Equilibrium

Stress-Strain Law

Classical

Actual Structure

FEM

Structural Model

xx yy zz

vxx y zp+ + + =0

t

v

t

s

t

v

dV p u dV p u ds_ _ _

= +

Compatibility

(Principle of Virtual Work)

Partial Differential

Equations

r=

Algebraic Equations

K = Stiffness

r = Response

R = Loads

Simplified Structural SystemSimplified Structural System

Loads (F)Loads (F) Deformations (u)Deformations (u)

FvFv

K (Stiffness)K (Stiffness)uu

F = K uF = K u

FF

Equilibrium Equation

Special Analysis TypesSpecial Analysis Types NonNon--linear Analysislinear Analysis

PP--Delta AnalysisDelta Analysis

Buckling AnalysisBuckling Analysis

Static Pushover AnalysisStatic Pushover Analysis

Fast NonFast Non--Linear Anal sis FNALinear Anal sis FNA Large Displacement AnalysisLarge Displacement Analysis

Dynamic AnalysisDynamic Analysis

Free Vibration and Modal AnalysisFree Vibration and Modal Analysis

Response Spectrum AnalysisResponse Spectrum Analysis

Steady State Dynamic AnalysisSteady State Dynamic Analysis


4/12

Role of EngineerRole of EngineerEvaluate Real Structure

Create Structural Model

Discretize Model in FE

EngineerEngineer

Solve FE Model

Interpret FEA Results

Physical significance of Results

SoftwareSoftware

EngineerEngineer

Concept of modelingConcept of modeling

Discretization3DSOLIDS

(Governed by partial

differential equations)

mp cat on

(geometric)

CONTINUOUS MODEL

OF STRUCTURE

(Governed by either

partial or total dif-ferential equations)

DISCRETE MODEL

OF STRUCTURE

(Governed by algebraic

equations)

3D-CONTINUM

MODEL

Global Modeling of Structural GeometryGlobal Modeling of Structural Geometry

(a) Real Structure

(b) Solid Model (c) 3D Plate-Frame (d) 3D Frame

(e) 2D Frame

Fig. 1 Various Ways to Model a Real Struture

(f) Grid-Plate


5/12

Physical Categorization ofPhysical Categorization of

StructuresStructures

Structures can be categorized in many ways.Structures can be categorized in many ways.

For modeling and analysis purposes, the overallFor modeling and analysis purposes, the overallphysical behavior can be used as basis ofphysical behavior can be used as basis of

ca egor za onca egor za on

Cable or Tension StructuresCable or Tension Structures

Skeletal or Framed StructuresSkeletal or Framed Structures

Surface or Spatial StructuresSurface or Spatial Structures

Solid StructuresSolid Structures

Mixed StructuresMixed Structures

Structural MembersStructural MembersContinuum

Regular Solid

(3D)

Plate/Shell (2D)

x zz

y

Dimensional Hierarchy of Structural Members

eam (1

b h

L>>(b,h)

b

ht

z

t


6/12

The Basic Structural QuantitiesThe Basic Structural Quantities

LoadsLoads

ActionsActions

DeformationsDeformations

The main focus of Structural

Mechanics is to develop

relationships between thesequantities

StrainsStrains

StressesStresses

Stress ResultantsStress Resultants

e ma n ocus o s

solve these relationships

numerically

Mechanics RelationshipsMechanics RelationshipsLoad

Act ion Defor matio n

StrainStressStress Resultant

Primary RelationshipsPrimary Relationships

LoadLoad Action RelationshipAction Relationship

ActionAction Deformation RelationshipDeformation Relationship

DeformationDeformation Strain RelationshipStrain Relationship StrainStrain Stress RelationshipStress Relationship

StressStress Stress Resultant RelationshipStress Resultant Relationship

Stress ResultantStress Resultant Action RelationshipAction Relationship

Most of these relationships can definedMost of these relationships can defined

mathematically, numerically and by testingmathematically, numerically and by testing


7/12

Simplified Examples of ActionSimplified Examples of Action--

DeformationDeformation

P

V

M

V

v

M

P

V

M

P

V

M

V

v

M

V

v

M

=

L

MV

EI

Lv

32

6

3

P

M

V

P

P

M

V

M

V

EA

LP=

= V

L

M

EI

L 2

2

2

The Basic Six DOFThe Basic Six DOF

Three Translations along theThree Translations along the

reference axisreference axis

Dx, Dy, DzDx, Dy, Dz

ree otat ons a out t eree otat ons a out t e

reference axisreference axis

Rx, Ry, RzRx, Ry, Rz

Constraints and RestraintsConstraints and Restraints

Restraints:Restraints:

Direct limits on the DOFDirect limits on the DOF

External Boundary ConditionsExternal Boundary Conditions

Fixed Su ort , Su ort SettlementFixed Su ort , Su ort Settlement

ConstraintsConstraints

Linked or dependent limits on DOFLinked or dependent limits on DOF

Internal linkages within the structure, in addition to orInternal linkages within the structure, in addition to orin place of normal connectionsin place of normal connections

Rigid Diaphragm, MasterRigid Diaphragm, Master--Slave DOFSlave DOF


8/12

Body ConstraintsBody Constraints

A Body Constraint causes all of its constrained joints toA Body Constraint causes all of its constrained joints to

move together as a threemove together as a three--dimensional rigid body.dimensional rigid body. All constrained joints are connected to each other by rigidAll constrained joints are connected to each other by rigid

links and cannot displace relative to each other.links and cannot displace relative to each other.

This Constraint can be used to:This Constraint can be used to:

Model rigid connections, such as where several beams and/orModel rigid connections, such as where several beams and/or

columns frame togethercolumns frame together

Connect together different parts of the structural model thatConnect together different parts of the structural model that

were defined using separate mesheswere defined using separate meshes

Connect Frame elements that are acting as eccentric stiffenersConnect Frame elements that are acting as eccentric stiffeners

to Shell elementsto Shell elements

What is Stiffness ?What is Stiffness ?

In structural terms, stiffness mayIn structural terms, stiffness maybe defined as Resistance tobe defined as Resistance toDeformationDeformation

So for each type of deformation,So for each type of deformation, FKu

Fu

=

For Linear Response

ere s a correspon n s nessere s a correspon n s ness

Stiffness can be considered orStiffness can be considered orevaluated at various levelsevaluated at various levels

Stiffness is also the constant inStiffness is also the constant inthe Actionthe Action--DeformationDeformationRelationshipRelationship

u

FK=

The Structure StiffnessThe Structure StiffnessMaterial Stiffness

Section Stiffness

Material Stiffness

Cross-section Geometry

Member Stiffness

Structure Stiffness

Member Geometry

Structure Geometry


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Global Structure StiffnessGlobal Structure Stiffness

The overall resistance of theThe overall resistance of thestructures to over all loads,structures to over all loads,

called thecalled the Global StructureGlobal StructureStiffness.Stiffness. This is derivedThis is derivedfrom the sum of stiffness offrom the sum of stiffness of

,,connectivity and theconnectivity and theboundary or the restrainingboundary or the restrainingconditions.conditions.

Advantage of FEM is to analyze the structure as a

whole!

Advantage of FEM is to analyze the structure as a

whole!

Element Stiffness MatrixElement Stiffness Matrix

R K K K K K K r

Node1 Node2

r1

r2r3

r4

r5 r6

1 11 12 13 14 15 16 1

R2

K21

K22

K23

K24

K25

K26

r2

R3

K31

K32

K33

K34

K35

K36

r3

R4

K41

K42

K43

K44

K45

K46

r4

R5

K51

K52

K53

K54

K55

K56

r5

R6

K61

K62

K63

K64

K65

K66

r6

=

A 2D Frame Element StiffnessA 2D Frame Element Stiffness

Node1 Node2

U1

U2U3

U1

U2U3

E ,A ,I ,L

( )1 - ( )1

(P2)1 0 12EI/L3 6EI/L2 0 -12EI/L3 6EI/L2 (U2)1

(P3)1 0 6EI/L2 4EI/L 0 -6EI/L2 2EI/L (U3)1

(P1)2 -EA/L 0 0 EA/L 0 0 (U1)2

(P2)2 0 -12EI/L3 -6EI/L2 0 12EI/L3 -6EI/L2 (U2)2

(P3)2 0 6EI/L2 2EI/L 0 -6EI/L2 4EI/L (U3)2

(U1)1 (U2)1 (U3)1 (U1)2 (U2)2 (U3)3

=


10/12

Some Sample Finite ElementsSome Sample Finite Elements

Truss and Beam Elements (1D,2D,3D)

Plane Stress, Plane Strain, Axisymmetri c, Plate and Shell Elements (2D,3D)

Brick Elements

Usage of 1D ElementsUsage of 1D Elements

3D Frame

2D Grid

2D Frame

Shell ElementShell Element

GeneralGeneral Total DOF per Node = 6 (or 5)Total DOF per Node = 6 (or 5)

Total Displacements per NodeTotal Displacements per Node

= 3= 3

Total Rotations per Node = 3Total Rotations per Node = 323

U1, R1

Node 3

U3, R3

U2, R2

U1, R1

Node 4

U3, R3

U2, R2

U 3 R 3

ApplicationApplication For Modeling surface elementsFor Modeling surface elements

carrying general loadscarrying general loads

1

U1, R1

Node 1

U3, R3 U2, R2

U1, R1

Node 2

,

U2, R2

Shell


11/12

Load CasesLoad Cases

Load cases are defined by the user and used forLoad cases are defined by the user and used for

analysis purpose onlyanalysis purpose only

Static Load CasesStatic Load Cases

Dead LoadDead Load

Live Load, Moving LoadLive Load, Moving Load

Wind Load, Water current load, etcWind Load, Water current load, etc

Earthquake Load CasesEarthquake Load Cases

Response Spectrum Load CasesResponse Spectrum Load Cases

Time History Load CasesTime History Load Cases

What Results Can We Get ?What Results Can We Get ?(in SAP2000)(in SAP2000)

At JointsAt Joints

Joint DisplacementsJoint Displacements

Spring ReactionsSpring Reactions

Restrained ReactionsRestrained Reactions

Constrained ForcesConstrained Forces

For all Available DOFFor all Available DOF

Given on the Local Joint CoordinatesGiven on the Local Joint Coordinates

Given for all Load Case, Mode Shapes,ResponseGiven for all Load Case, Mode Shapes,ResponseSpectrums, Time Histories, Moving Loads, and LoadSpectrums, Time Histories, Moving Loads, and LoadCombinationsCombinations


12/12

For Frame ElementsFor Frame Elements The Actions Corresponding to Six DOF atThe Actions Corresponding to Six DOF at

Both Ends, in Local Coordinate SystemBoth Ends, in Local Coordinate System

1122

+V2+V2+P+P

1122

+M2+M2+T+T

33

33

22

+P+P+V+V22

+V+V33

+V3+V3

33

33

22

+T+T+M2+M2

+M3+M3

+M3+M3

For Shell ElementFor Shell Element

The Shell element internal forces (also called stress resultants)The Shell element internal forces (also called stress resultants)

are the forces and moments that result from integrating theare the forces and moments that result from integrating the

stresses over the element thickness.stresses over the element thickness.

The results include the Membrane Results (inThe results include the Membrane Results (in--plane forces)plane forces)

The results are given for Element Local AxisThe results are given for Element Local Axis

It is very important to note that these stress resultants are forcesIt is very important to note that these stress resultants are forces

and momentsand momentsper unit of inper unit of in--plane lengtplane lengthh

Obtaining Envelop ResultsObtaining Envelop Results

Comb1 Comb2 Comb3 Comb N

Load Case -1

Load Case - 2

TotalMax, P

Min, P

o a a se -

Load Case - M

Envelop Results

P1 P2 P3 P N

CAAD of Bridges Handouts

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