wsdot 04 06(v stiffness) apresentacao

8/13/2019 WSDOT 04 06(v Stiffness) Apresentacao

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Prepared by

J. P. Singh & Associates

in association withMohamed Ashour, Ph.D., PE

West Virginia University Techand

Gary Norris Ph.D., PE

University of Nevada, Reno

APRIL 3/4, 2006

Computer Program DFSAP

Deep Foundation System Analysis ProgramBased on Strain Wedge Method

Washington StateDepartment of Transportation


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Pile and Pile Group Stiffnesseswith/without Pile Cap


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SESSION I

STIFFNESS MATRIX FOR BRIDGE

FOUNDATION AND SIGN CONVENTIONS

How to Build the Stiffness Matrix of Bridge Pile

Foundations (linear and nonlinear stiff. matrix)?

How to Assess the Pile/Shaft Response Based on

Soil-Pile-Interaction with/without Soil Liquefaction

(i.e. Displacement & Rotational Stiffnesses)?


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Y

X X

Z

Z

Y

Foundation Springs in

the Longitudinal Direction

K11

K22K66

Column Nodes

Longitudinal


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Loads and Axis

F1

F2

F3

M1M2

M3 X

Z

Y

F1

F2

F3

M1

M2

M3X

Z

Y


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K11 0 0 0 -K15 0

0 K22 0 K24 0 0

0 0 K33 0 0 0

0 K42 0 K44 0 0-K51 0 0 0 K55 0

0 0 0 0 0 K66

x y z x y z

Force Vector for x = 1 unit

Full Pile Head Stiffness Matrix

Lam and Martin (1986)

FHWA/RD/86-102


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= 0

Applied P

Applied M

Applied P

Induced M

A. Free-Head Conditions B. Fixed-Head Conditions

= 0

Applied P

= 0Induced PApplied M

Induced M

C. Zero Shaft-Head Rotation, = 0 D. Zero Shaft-Head Deflection, = 0

Shaft/Pile-Head Conditions in the DFSAP Program

Special Conditions forLinear Stiffness Matrix


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Y

X X

Z

Z

Y

Foundation Springs inthe Longitudinal Direction

K11

K22K66

Column Nodes

Loading in the Longitudinal

Direction (Axis 1 or X Axis )

Single Shaft

K22

Y

P2

K11

K66

P1

M3

Y

X X

P2

K22

K33

K44

P3

M1

Y

Y

Z Z

Loading in the Transverse

Direction (Axis 3 or Z Axis)


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Steps of Analysis

Using SEISAB (STRUDL), calculate the forces at the

base of the fixed column (Po, Mo, Pv) (both directions)

Use DFSAP with special shaft head conditions to

calculate the stiffness elements of the required

(linear) stiffness matrix.

K11 0 0 0 0 -K16

0 K22 0 0 0 0

0 0 K33 K34 0 0

0 0 K43 K44 0 0

0 0 0 0 K55 0

-K61 0 0 0 0 K66

F1 F2 F3 M1 M2 M3

1

2

3

1

2

3


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Steps of Analysis

Using SEISAB and the above spring stiffnesses at the

base of the column, determine the modified reactions

(Po, Mo, Pv) at the base of the column (shaft head)

K11 0 0 0 0 -K16

0 K22 0 0 0 0

0 0 K33 K34 0 0

0 0 K43 K44 0 0

0 0 0 0 K55 0

-K61 0 0 0 0 K66

1 2 3 1 2 3


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Steps of Analysis Keep refining the elements of the stiffness matrix used

with SEISAB until reaching the identified tolerance forthe forces at the base of the column

Why KF3M1KM1F3 ?

KF3M1 = K34 =F3 /1 and KM1F3 = K43= M1/3Does the linear stiffness matrix represent the actual

behavior of the shaft-soil interaction?

KF1F1 0 0 0 0 -KF1M3

0 KF2F2 0 0 0 0

0 0 KF3F3 KF3M1 0 0

0 0 KM1F3 KM1M1 0 0

0 0 0 0 KM2M2 0

-KM3F1 0 0 0 0 KM3M3

F1

F2

F3

M1M2

M3

1 2 3 1 2 3


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y

p

(Es)1

Po

(Es)3

(Es)4

(Es)2p

p

p

y

y

y

(Es)5

p

y

Laterally Loaded Pile as a Beam

on Elastic Foundation (BEF)

ShaftWidth

x x

Longitudinal

Steel

Steel Shell


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Linear Stiffness Matrix

K11 0 0 0 0 -K160 K22 0 0 0 0

0 0 K33 K34 0 0

0 0 K43 K44 0 0

0 0 0 0 K55 0-K61 0 0 0 0 K66

F1 F2 F3 M1 M2 M3

Linear Stiffness Matrix is based on

Linear p-y curve (Constant Es), which is not the case Linear elastic shaft material (Constant EI), which is not

the actual behavior

Therefore,

P, M= P+ M and P, M= P+ M

1

2

3

1

2

3

A t l S i


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Shaft Deflection, y

LineLoad,p

yP, M> yP+ yM

yM

yPyP, M

y

p

(Es)1

(Es)3

(Es)4

(Es)2p

p

p

y

y

y

(Es)5

p

y

MoPo

Pv

Nonlinear p-y curve

As a result, the linear analysis

(i.e. the superposition technique )

can not be employed

Actual Scenario


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Applied P

Applied M

A. Free-Head Conditions

K11 or K33= PApplied/

K66 or K44 = MApplied/

Nonlinear (Equivalent) Stiffness Matrix


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Nonlinear (Equivalent) Stiffness Matrix

K11 0 0 0 0 00 K22 0 0 0 0

0 0 K33 0 0 0

0 0 0 K44 0 0

0 0 0 0 K55 00 0 0 0 0 K66

F1 F2 F3 M1 M2 M3

Nonlinear Stiffness Matrix is based on

Nonlinear p-y curve Nonlinear shaft material (Varying EI)

P, M> P+ M K11 = Papplied/ P, MP, M> P+ M K66 = Mapplied/ P, M

1

2

3

1

2

3


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Pile Load-Stiffness Curve

Linear Analysis

Pile-He

adStiffness,K

11,

K33,

K44,K

66

Pile-Head Load, Po, M, Pv

P1,

M1

P2,

M2

Non-Linear Analysis


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Linear Stiffness Matrixand

the Signs of the Off-Diagonal Elements

KF1F1 0 0 0 0 -KF1M3

0 KF2F2 0 0 0 00 0 KF3F3 KF3M1 0 0

0 0 KM1F3 KM1M1 0 0

0 0 0 0 KM2M2 0

-KM3F1 0 0 0 0 KM3M3

F1 F2 F3 M1 M2 M3

1

2

3

1

2

3

Next Slide


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F1 X or 1

Z or 3

Y or 2

Induced M3

1

K11= F1/1K61 =-M3/1

X or 1

Z or 3

Y or 2

M3

3

K66= M3/3K16 =-F1/3

Induced F1

Elements of the Stiffness Matrix

Next SlideLongitudinal Direction X-X


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F3X or 1

Z or 3

Y or 2

K33 = F3/3K43 =M1/3

X or 1

Z or 3

Y or 2

1

K44= M1/1K34 =F3/1

Transverse Direction Z-Z

Linear Stiffness Matrix for Pile group


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(Lam and Martin, FHWA/RD/86-102)

Linear Stiffness Matrix for Pile group


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Pile Load-Stiffness Curve

Linear Analysis

Pile-HeadStiffness,K

11,

K33,

K44,K

66

Pile-Head Load, Po, M, Pv

P1,

M1

P2,

M2

Non-Linear Analysis

P


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(KL)1(KL)2(KL)3

(Kv)2 (Kv)1(Kv)3

(KL)C

(Kv)G(KL)G

(KR)G

(KL)G = (KL)i+ (KL)C= PL / L Ldue to lateral/axial loads

(Kv)G= Pv / v vdue to axial load (Pv)

(KR)G= M / due to moment (M)

PL

Pv

M

Rotational angle

Lateral deflection L

Axial settlement v

P P


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PL

Pv

M

Pv

(pv)M(pv)M

(pv)Pv(pv)Pv

(pL)PL

PL

Pv

M

Pile Cap with Free-Head Piles

xx

z

z

(pv)M(pv)M

(pv)Pv(pv)Pv

PLM

(pL)PL

Pile Cap with Fixed-Head Piles

(Fixed End Moment)

P


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PL

Pv

M

Rotational angle

Lateral deflection L

Axial settlement v

Axial Rotational Stiffness

of a Pile Group

K55 = GJ/L WSDOT

MT= (3.14 D i) D/2 (Li)= zT

/ L

K55= MT/

P


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PL

Pv

M

(K22)(K11)

(K66) xx

K11 0 0 0 0 00 K22 0 0 0 0

0 0 K33 0 0 0

0 0 0 K44 0 0

0 0 0 0 K55 0

0 0 0 0 0 K66

1

2

3

1

2

3

(K11) = PL/ 1

(K22) = Pv/ 2

(K33) = M3

Group Stiffness Matrix

(pv)M(pv)M

(pv)Pv(pv)Pv

PL

Pv (1)

M

(pL)PL

(Fixed End Moment)


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wsdot 04 06(v stiffness) apresentacao

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