SESSION # 3

STIFFNESS MATRIX FOR BRIDGE FOUNDATION AND SIGN CONVETIONS

Loads and Axis

F1

F2

F3

M1M2

M3 X

Z

Y

F1

F2

F3

M1

M2

M3

X

Z

Y

Y

X X

Z

Z

Y

Foundation Springs in the Longitudinal Direction

K11

K22K66

Column Nodes

Loading in the Longitudinal Direction (Axis 1 or X Axis )

Single Shaft

P2

K22

K11

K66

P1

M3

Y

Y

X X

P2

K22

K33

K44

P3

M1

Y

Y

Z Z

Loading in the Transverse Direction (Axis 3 or Z Axis )

Steps of Analysis

• Using SEISAB, calculate the forces at the base of the fixed column (Po, Mo, Pv)

• Use S-SHAFT with special shaft head conditions to calculate the stiffness elements of the required stiffness matrix • Longitudinal (X-X)• KF1F1 = K11 = Po / (fixed-head, = 0)• KM3F1 = K61 = MInduced /

• KM3M3 = K66 = Mo / (free-head, = 0)

• KF1M3 = K16 = PInduced /

K11 = PApplied / K66 = MApplied/

K61 = MInduced / K16 = PInduced/

B. Zero Shaft-Head Deflection, = 0

= 0

Applied P

= 0Induced P

Applied MInduced M

A. Zero Shaft-Head Rotation, = 0

X-Axis X-Axis

Linear Stiffness Matrix

Steps of Analysis

KF1F1 0 0 0 0 -KF1M3

0 KF2F2 0 0 0 00 0 KF3F3 KF3M1 0 00 0 KM1F3 KM1M1 0 00 0 0 0 KM2M2 0-KM3F1 0 0 0 0 KM3M3

F1 F2 F3 M1 M2 M3

• 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)

1

2

3

1

2

3

Steps of Analysis

• Keep refining the elements of the stiffness matrix used with SEISAB until reaching the identified tolerance for the forces at the base of the column

Why KF3M1 KM1F3 ?KF3M1 = K34 = F3 /1 and KM1F3 = K43 = M1 /3

Does the linear stiffness matrix represent the actual behavior of the shaft-soil interaction?

Linear Stiffness Matrix

K11 0 0 0 0 -K16

0 K22 0 0 0 00 0 K33 K34 0 00 0 K43 K44 0 00 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 behaviorTherefore,

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

1

2

3

1

2

3

Shaft Deflection, y

Lin

e L

oad

, 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

Mo

Po

Pv

Nonlinear p-y curve

As a result, the linear analysis (i.e. the superposition technique ) can not be employed

Actual Scenario

Applied P

Applied M

A. Free-Head Conditions

K11 or K33 = PApplied /

K66 or K44 = MApplied/

Nonlinear (Equivalent) Stiffness Matrix

Nonlinear (Equivalent) Stiffness Matrix

K11 0 0 0 0 00 K22 0 0 0 00 0 K33 0 0 00 0 0 K44 0 00 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

P, M > P + M

1

2

3

1

2

3

Shaf

t-H

ead

Stif

fnes

s, K

11, K

33, K

44, K

66Load Stiffness Curve

Shaft-Head Load, Po, M, Pv

P 1, M

1

P 2, M

2

Linear Stiffness Matrix and

the Signs of the Off-Diagonal Elements

KF1F1 0 0 0 0 -KF1M3

0 KF2F2 0 0 0 00 0 KF3F3 KF3M1 0 00 0 KM1F3 KM1M1 0 00 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

F1X or 1

Z or 3

Y or 2

Induced M3

1

K11 = F1/1

K61 = -M3/1

X or 1

Z or 3

Y or 2

M3

3

K66 = M3/3

K16 = -F1/3

Induced F1

Elements of the Stiffness Matrix

Next SlideLongitudinal Direction X-X

F3X or 1

Z or 3

Y or 2

Indu

ced M

1

3

K33 = F3/3

K43 = M1/3

X or 1

Z or 3

Y or 2

1

K44 = M1/1

K34 = F3/1

M 1

Induced F 3

Transverse Direction Z-Z

MODELING OF INDIVIDUAL SHAFTS AND

SHAFT GROUPS WITH/WITHOUT SHAFT CAP

K33 = F3/3

K44 = M1/1

K22 = F2/ 2

F2

F3

M1

Y

Y

Z Z

F2 F3

F2 F3

K22

K33

K44

Y

Y

Z Z

Single shaft

Pv

Po

Mo y

Cap Passive Wedge

Shaft Passive Wedge

Shaft Group with Cap

Ground Surface Shaft Group (Transverse Loading)(with/without Cap Resistance)

With CapPaxial = Pv/ n + Pfrom Mo

Po = Pg = PCap + Ph * n

PCap

Paxial

Ph

Kaxial

KLateral

Krot. (free/fixed)

n piles

Kgaxial

KgLateral

Kgrot.

No CapPaxial = Pv/ n Po = Pg = Ph * nMshaft = Mo/n

Pv

PoMo

Ground Surface Shaft Group (Longitudinal Loading)(with/without Cap Resistance)

With Cap (always free) Paxial = Pv/ n Po = Pg = PCap + Ph * nMshaft = Mo/n

PCap

Paxial

Ph

Kaxial

KLateral

Krot. (free)

n piles

Kgaxial

KgLateral

Kgrot.

No CapPaxial = Pv/ n Po = Pg = Ph * nMshaft = Mo/n

Pv

PoMo

SHAFT GROUP EXAMPLE PROBLEM EXAMPLE PROBLEMS

M o

Axia l Load

D iam eter o f Shaft Segm ent #2 = 6.0 ft

Seg

men

t # 1

= 4

3 ft

Seg

men

t # 2

=

17

ft

Layer # 1 Clay

Layer # 3 Rock

W ater Table

SAND

G round Surface

Shaft Section # 2N onlinear ana lys isSegm ent length =17 ftShaft d iam ete r =6.0 ftfc o f concrete = 5 Ksi

fy o f the stee l bars = 60 Ksi

R atio o f S tee l bars (A s/A c)= 3%R atio o f Transversestee l (A 's/A c)= 0 .5%C oncrete C over = 4.0 in

S haft W idth

x x

Long itudina l S teel

Shaft Section # 1N onlinear ana lysisSegm ent length = 43 ftShaft d iam eter = 10 ftfc o f concrete =5 K si

fy o f the stee l bars = 60 Ks i

R atio o f S tee l bars (A s/A c)= 1 .1%R atio o f Transversestee l (A 's/A c)= 0 .5%C oncrete cover = 6 .0 in

P o

D iam eter of S haft Segm ent #1 = 10 ft

S haft W id th

x x

Longitud ina l S tee l

37 ft

25 ft

3 ft= 42 O, = 74 pcf, 50= 0.0025

S u= 400 psf

= 64 pcf

50= 0.02

q u= 30000 psf

= 75 pcf

50= 0.0004

Single Shaft with Two Different Diameter

Example 3, Shaft Group (WSDOT)(Longitudinal Loading)

Shaft Group LoadsShaft PropertiesS haft length = 52 ftS haft d iam eter = 8.0 ftfc of concre te = 5 kips

R atio o f steel rebars (As/Ac)= 1.5 %

Shaft W idth

x x

Longitudinal S tee l

Ground Surface

60 f

t

20 f

t20

ft 8 ft

52 f

t

6 ft

Pv

PoMo

Average Shaft (????)

Shaft Group

Example 3, Shaft Group (WSDOT)Longitudinal Loading)

Example 3, Shaft Group (WSDOT)(Transverse Loading)

Shaft Group Loads

Ground Surface

60 ft

20 ft20 ft

8 ft

52 f

t

6 ft

10 ft

Pv

PoMo

Average Shaft

Shaft Group

Example 3, Shaft Group (WSDOT)(Transverse Loading)

K1 K2

FH

FV

KH

P- EFFECT

The moment developed at the column base is a function of Fv, FH, and