Modelling and Analysis of Cable Stayed Structures

8/3/2019 Modelling and Analysis of Cable Stayed Structures

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Modeling and Analysisof

Cable-Stayed StructuresDr. Michael H. Swanger

Senior Research EngineerCASE Center

March 2001


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Basic Cable Behavior

u

u

T

The stiffness of the cable is a function of its state of stress

and its state of deformation. u is the displacement at theend of the cable.

W

L

Tinitial


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Basic Cable Behavior (cont.)

Parabolic Shape

Catenary Shape

Harped (Kinked) Shape

Cable shape and length are functions

of applied loads and state of stress

Force per unit projected length

Force per unit arc length of cable

Concentrated force


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Basic Cable Behavior (cont.)

The Principal Cable Analysis Problem:

Computation of cable stress depends

on a known cable length and shapegeometry.

In turn, cable length and shape geometrydepend on a known state of stress!

Therefore, an iterative solution is required.


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General Cable Analysis ProcedureCable Prestress Analysis

Do a nonlinear systematic search to determinethe cable shape geometry and cable length thatsatisfies user specified initial cable tension or initial

cable geometry constraints (i.e., initial constructionstate)

Service Load Analysis

With initial prestress conditions satisfied, perform

nonlinear analysis for applied service loads


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Prestress Analysis Procedure

1. Model entire structure -- define initialcable geometry (usually a straight

cable state)

2. Define cable prestress loading usually self weight only


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Prestress Analysis Procedure

(Cont.)

4. Perform nonlinear analysis on full structurefor prestress load condition

5. Check compliance of prestress conditions(i.e., are prestress initial conditionssatisfied). If satisfied, end prestress

analysis; if not, make adjustments to cablelengths and go back to Step 4.


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Example

Simple Suspension Bridge

xxxx xxxx

656.00 FT

xxxx xxxx

3280.00 FT

Elev. = 162.5 FT

Elev. = 541.2 FT

Elev. = 230.0 FT

Suspension Cable Elements:AX = 300 sq. inches

E = 24500 ksiSW = 1800 plf (self weight)Fu = 40000 kips (breaking tension)

Hanger Truss Members:AX = 100 sq. inches

E = 24500 ksi

Deck Frame Members:AX = 6000 sq. inchesIZ = 750000 in4

E = 29000 ksi

Suspension cable

Hangers(truss membersin this example)

Deck

Prestress geometry constraint is theelevation value of 230 ft. shown at themidpoint of the suspension cableunder Load 1


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Example

Simple Suspension Bridge (cont.)

Geometry Highlights

InitialInput Geometry Before Prestress Analysis

x x

x x

x x

x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x

x x x x x x x x x x x x x x xxxS1 xx

S2

xx

D1

xx

D17

xx

C1

xx

C33

xx

CABLE3


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Example


Prestress Analysis Highlights

$* **$* ** Define initial stress loading condition: includes SW cable element property

$* **UNITS KIPS FEETLOAD 1 'Full traffic + SW'MEMBER LOADS'DECK1' TO 'DECK16' FORCE Y GLOBAL UNI FR W -2.8


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Example



$* **$* ** Describes prestressing conditions for the suspension cable$* **

UNITS FEETDEFINE CABLE NETWORK 1INCLUDE ELEMENTS 'CABLE1' TO 'CABLE16'ATTACH JOINTS 'C1' 'C33' $ At extreme fixed joints of cable network

SAG COORDINATE Y 230.0 JOINT 'C17' $ desired Y position of jointADJUST COORDINATES Y $ of the free joints along cable

END


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Example



$* **$* ** Specify prestress analysis control parameters

$* **CABLE ANALYSIS DATACONVERGENCE TOLERANCE GEOMETRY 0.01CONVERGENCE TOLERANCE DISPLACMENT 0.001

MAXIMUM NUMBER OF GEOMETRY ITERATIONS 10MAXIMUM NUMBER OF EQUILIBRIUM ITERATIONS 50LOAD 1 $ Prestress Loading

END


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Example



New undeformed and unstressedGeometrycomputed during the Prestress Analysis of aweightless cable required to achieve the

desired prestressed position of Y = 230 ft. atjoint 'C17 under the application of initial Load 1(see next slide).

x x

x x

x x

x x x x x x x x x x x x x x x x x x x x xx x x

x xx x

x xx

x x x x x x x x x x x x x x x

xxxx

X 2296.00 FT

Y 257. 21 F T

Z 0.00 FTJoint 'C17'


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Example



Deformed Geometry After Prestress AnalysisUnder Initial Load 1

x x

x x

x x

x x x x x x x x xx x x x x x x x x x x x x

x x xx x

x xx x

x x x x x x x x x x x x x x xx x

x x

x x

xx

xx x

x x x x x x x x x x x x x x xx x

x xx x

x xx

xx

x x x x x x x x x x x xx x

x

xxxx

X 1.625E-06

Y -2.619E+01

Z 0.000E+00

Joint C17 displacements:

New undeformed and unstressed

geometry (at Y = 257.2 ft.)

New deformed and prestressedgeometry after

prestress analysis:Y = 257.2 26.19 = 231.01 ft (approx. 230)


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Example

Simple Suspension Bridge (cont.){ 122} > list cable analysis results elements 'CABLE1' TO 'CABLE16'

*********************************************

* RESULTS OF LATEST CABLE GEOMETRY ANALYSIS *

*********************************************

ACTIVE UNITS (UNLESS INDICATED OTHERWISE):

LENGTH WEIGHT ANGLE TEMPERATURE TIME

FEET KIP DEG DEGF SEC

CABLE GEOMETRY DATA

===================

ELEMENT UNSTRESSED STRESSED

LENGTH LENGTH

------- ---------- --------

CABLE1 217.454 217.840

CABLE2 212.702 213.071

CABLE3 211.173 211.536

CABLE4 208.887 209.242

.

.

.

CABLE16 217.454 217.840

========== ========

TOTAL LENGTHS = 3347.45 3353.15

CABLE NODAL STRESSES

====================

ELEMENT NODE SXX

------- ---- -----

CABLE1 C1 6264.81

C3 6264.74C2 6264.77


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Example


Phase II Highlights

$* **$* ** Phase II: partial deck with hangers$* **ACTIVE MEMBERS 'DECK1' TO 'DECK7' 'H1' TO 'H7'ACTIVE JOINTS 'D1' TO 'D8'

$* **$* ** Remove traffic load for construction sequence$* **UNITS KIPS FEETCHANGESLOAD 1ADDITIONS

MEMBER LOADS'DECK1' TO 'DECK16' FORCE Y GLOBAL UNI FR W 0.8

$* **$* ** Perform nonlinear analysis continuation$* **NONLINEAR ANALYSIS CONTINUEUNITS INCHES KIPS'LIST DISPLACEMENTS JOINT 'C17'


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Example


Continuat ion of Analysis Sequence

Continue sequential construction simulation

cable bridge analysis by adding additionalportions of the deck structure andcontinuing the nonlinear analysis.

After completion of the sequential analysis,the final position of joint C17 is 230 ft.

Modelling and Analysis of Cable Stayed Structures

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