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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.
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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
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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
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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
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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
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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
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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
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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
=
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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
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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
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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