Modelling and Analysis of Cable Stayed Structures

Modeling and Analysis of Cable-Stayed Structures Dr. Michael H. Swanger Senior Research Engineer CASE Center March 2001

Basic Cable Behavior W T

u

L

Tinitial

The stiffness of the cable is a function of its state of stress and its state of deformation. “u” is the displacement at the end of the cable. u

Basic Cable Behavior (cont.) Force per unit projected length

Parabolic Shape

Force per unit arc length of cable

Cable shape and length are functions of applied loads and state of stress Catenary Shape

Concentrated force

Harped (Kinked) Shape

Basic Cable Behavior (cont.) The Principal Cable Analysis Problem:






Computation of cable stress depends on a known cable length and shape geometry. In turn, cable length and shape geometry depend on a known state of stress! Therefore, an iterative solution is required.

General Cable Analysis Procedure Cable Prestress Analysis Do a nonlinear systematic search to determine the cable shape geometry and cable length that satisfies user specified initial cable tension or initial cable geometry constraints (i.e., initial construction state)

Service Load Analysis With initial prestress conditions satisfied, perform nonlinear analysis for applied service loads

Prestress Analysis Procedure 1. Model entire structure -- define initial cable geometry (usually a straight cable state)

2. Define cable prestress loading – usually self weight only

Prestress Analysis Procedure (cont.)

3. Define cable prestressing constraints -- any

one cable may have an initial tension or geometric constraint specified. For example:

• • • •

Initial stress in cables Deformed sag condition in main bridge suspension cables Bridge deck deformed profile conditions Guyed tower plumbed conditions

Prestress Analysis Procedure (Cont.)

4. Perform nonlinear analysis on full structure for prestress load condition

5. Check compliance of prestress conditions

(i.e., are prestress initial conditions satisfied). If satisfied, end prestress analysis; if not, make adjustments to cable lengths and go back to Step 4.

Cable Element Characteristics

• Two to five nodes per cable element – using an isoparametric finite element formulation 2

1 3

Example of a 3-node cable element (Note the cable node incidence order)

• Cross-section area – AX • Self weight and its direction – force/arc length • Length factor (initial strain specification) • Number of quadrature (integration) points • Applied concentrated and uniform and/or linear

(per unit arc or projected length) member loads; joint temperature loads

Example

Simple Suspension Bridge Suspension cable Elev. = 541.2 FT Elev. = 230.0 FT

x

x

Hangers (truss members in this example) x

3280.00 FT 656.00 FT

Deck Suspension Cable Elements: AX = 300 sq. inches E = 24500 ksi SW = 1800 plf (self weight) Fu = 40000 kips (breaking tension) Prestress geometry constraint is the elevation value of 230 ft. shown at the midpoint of the suspension cable under Load 1

Elev. = 162.5 FT

Hanger Truss Members: AX = 100 sq. inches E = 24500 ksi Deck Frame Members: AX = 6000 sq. inches IZ = 750000 in4 E = 29000 ksi

Example

Simple Suspension Bridge (cont.) Geometry Highlights UNITS FEET JOINT COORDINATES 'S1' 0.0 0.0 0.0 S; 'S2' 4592.0 0.0 0.0 S 'C1' 656.0 541.2 0.0 S; 'C33' 3936.0 541.2 0.0 S 'D1' 656.0 0.0 0.0 S; 'D17' 3936.0 0.0 0.0 S JOINT RELEASES 'C1' 'C33' FORCE X 'D1' 'D17' FORCE Y $* ** $* ** Cable nodes $* ** GENERATE BETWEEN 'C1' 'C33' XDIR 32 ID INC 1

Example

Simple Suspension Bridge (cont.) Geometry Highlights $* ** $* ** Generate suspension cable elements $* ** GENERATE 16 ELEMENTS ID 'CABLE1' 1 FROM 'C1' 2 TO 'C3' 2 TO 'C2' 2

$* ** $* ** Main suspension cable element properties $* ** ELEMENT PROPERTIES 'CABLE1' TO 'CABLE16' TYPE 'IPCABLE' AX 300.0 SW 0.150 DIR -Y LF 0.999

Example

Simple Suspension Bridge (cont.) Geometry Highlights Initial Input Geometry Before Prestress Analysis

CABLE3 C1 C33 xxxxxx xxxxxxxxx xxxxxx xxxxxx xxxxxx

D1 S1 x

x

D17 x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

S2 x

Example

Simple Suspension Bridge (cont.) Prestress Analysis Highlights

$* ** $* ** Define initial stress loading condition: includes SW cable element property $* ** UNITS KIPS FEET LOAD 1 'Full traffic + SW' MEMBER LOADS 'DECK1' TO 'DECK16' FORCE Y GLOBAL UNI FR W -2.8

Example

Simple Suspension Bridge (cont.) Prestress Analysis Highlights $* ** $* ** Describes prestressing conditions for the suspension cable $* ** UNITS FEET DEFINE CABLE NETWORK 1 INCLUDE 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 joint ADJUST COORDINATES Y $ of the free joints along cable END

Example

Simple Suspension Bridge (cont.) Prestress Analysis Highlights $* ** $* ** Specify prestress analysis control parameters $* ** CABLE ANALYSIS DATA CONVERGENCE TOLERANCE GEOMETRY 0.01 CONVERGENCE TOLERANCE DISPLACMENT 0.001 MAXIMUM NUMBER OF GEOMETRY ITERATIONS 10 MAXIMUM NUMBER OF EQUILIBRIUM ITERATIONS 50 LOAD 1 $ Prestress Loading END

Example

Simple Suspension Bridge (cont.) Prestress Analysis Highlights

$* ** $* ** Perform the prestress analysis $* ** PERFORM CABLE PRESTRESS ANALYSIS

Example

Simple Suspension Bridge (cont.) Prestress Analysis Highlights New undeformed and unstressed Geometry computed during the Prestress Analysis of a weightless cable required to achieve the desired prestressed position of Y = 230 ft. at joint 'C17’ under the application of initial Load 1 (see next slide).

X 2296.00 FT Y 257.21 FT

Joint 'C17'

Z 0.00 FT x x x x x xx x x xx x xx xx xx x x x xxx xx xx xxx xx x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

Example

Simple Suspension Bridge (cont.) Prestress Analysis Highlights Deformed Geometry After Prestress Analysis Under Initial Load 1 Joint ‘C17‘ displacements: X 1.625E-06 Y -2.619E+01 Z 0.000E+00

x

New undeformed and unstressed geometry (at Y = 257.2 ft.)

xx x xx xx xx x xx x xxx x xx xx xxx x xxxxxxxx x x x x xxxxxxxxx xx xx xxx x x xxxxxxxxx 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

New deformed and prestressed geometry after prestress analysis: Y = 257.2 – 26.19 = 231.01 ft (approx. 230)

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 FEET KIP DEG CABLE GEOMETRY DATA =================== ELEMENT ------CABLE1 CABLE2 CABLE3 CABLE4

TEMPERATURE DEGF

UNSTRESSED LENGTH ----------

STRESSED LENGTH --------

217.454 212.702 211.173 208.887

217.840 213.071 211.536 209.242

217.454 ==========

217.840 ========

3347.45

3353.15

. . . CABLE16

TOTAL LENGTHS = CABLE NODAL STRESSES ==================== ELEMENT -------

NODE ----

CABLE1

C1 C3 C2

SXX ----6264.81 6264.74 6264.77

TIME SEC

Example

Simple Suspension Bridge (cont.) Continue with Service Load Analysis & Construction Sequence Simulation Phases 1. Main suspension cable only in place 2. Partial deck in place

Example

Simple Suspension Bridge (cont.) Phase I Highlights $* ** $* ** Phase I of construction sequence $* ** Suspension cable only in place $* ** $* ** Remove deck and hangers $* ** INACTIVE MEMBERS 'DECK1' TO 'DECK16' 'H1' TO 'H15' INACTIVE JOINTS 'D1' TO 'D17' $* ** $* ** Perform nonlinear analysis continuation $* ** CONVERGENCE TOLERANCE DISPLACMENT 0.001 MAXIMUM NUMBER OF CYCLES 25 NONLINEAR ANALYSIS CONTINUE UNITS INCHES LIST DISPLACEMENTS JOINT 'C17'

Example

Simple Suspension Bridge (cont.) Sequential Analysis Phase I Highlights Initial Deformed Geometry After Phase I Under Self Weight of Suspension Cable Alone Joint ‘C17‘ displacements: X -2.873E-06 Y -7.332E+00

New undeformed and prestressed geometry during prestress analysis (Y = 257.2 ft.)

Z 0.000E+00 xx xx

xx xx x x xx x xx x x x xxx xxxxx xxx xxxx xxxxxxxxx xxxxxx x x xxx x xxxxxxxxxx

xx

x

x

Y = 249.9 ft. (257.2 – 7.3)

Example

Simple Suspension Bridge (cont.) 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 FEET CHANGES LOAD 1 ADDITIONS MEMBER LOADS 'DECK1' TO 'DECK16' FORCE Y GLOBAL UNI FR W 0.8 $* ** $* ** Perform nonlinear analysis continuation $* ** NONLINEAR ANALYSIS CONTINUE UNITS INCHES KIPS' LIST DISPLACEMENTS JOINT 'C17'

Example

Simple Suspension Bridge (cont.) Phase II Highlights Deformed Geometry After Phase II X -9.493E+01 Y -3.988E+02 Z 0.000E+00 xx xx x x x x x x x xx x x x xx xx xx x x x xx xx xx x xx x x xx x x xxx x x x xx xx xx xx xx xx xx x x x

x x

x x

x x

x x

x x

x x

x x

x x

x X -8.797E+01 Y 2.049E+02 Z 0.000E+00

Example

Simple Suspension Bridge (cont.) Continuation of Analysis Sequence





Continue sequential construction simulation cable bridge analysis by adding additional portions of the deck structure and continuing the nonlinear analysis. After completion of the sequential analysis, the final position of joint ‘C17’ is 230 ft.