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**Seismic Site Response Analysis**

Soil-Structure Interaction: Basic Concepts Steve Kramer University of Washington Steve Kramer University of Washington EERI Technical Seminar Series Impact of Soil-Structure Interaction on Response of Structures Seminar 2: Practical Applications to Deep Foundations

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**Outline Soil-structure interaction basics Kinematic interaction**

Inertial interaction Analysis of soil-structure interaction Effects of soil-structure interaction Soil-pile-structure interaction basics Deep foundations Single pile/shaft behavior Analysis of single piles Forms of loading Pile groups Dynamic response Methods of analysis Conclusions

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**Soil-Structure Interaction**

How does the presence of soil affect the response of a structure? Soil Rock Rock Does the structure founded on rock respond differently than when founded on soil?

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**Soil-Structure Interaction**

How does the presence of a structure affect the response of the soil? Foundation input motion Free-field motion Soil Rock How does the motion at the base of the structure differ from the free-field motion?

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**Soil-Structure Interaction**

In reality, the response of the soil affects the response of the structure, and the response of the structure affects the response of the soil Soil-Structure Interaction Two components: Kinematic interaction Presence of stiff foundation elements on or in soil cause foundation motions to deviate from free-field motions. Inertial interaction Inertial response of structure causes base shear and moments which cause displacements of foundation relative to free-field.

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**Soil-Structure Interaction**

Kinematic SSI has three primary causes: Base slab averaging – results from stiffness of foundation Embedment – variation of ground motion with depth Wave scattering – scattering off corners and edges Base slab averaging – stiffness of foundation prevents it from matching free-field deformations. Exists even for massless foundation.

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**Outline Soil-structure interaction basics Kinematic interaction**

Inertial interaction Analysis of soil-structure interaction Effects of soil-structure interaction Soil-pile-structure interaction basics Deep foundations Single pile/shaft behavior Analysis of single piles Forms of loading Pile groups Dynamic response Methods of analysis Conclusions

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**Soil-Structure Interaction**

Kinematic SSI has three primary causes: Base slab averaging – results from stiffness of foundation Embedment – variation of ground motion with depth Wave scattering – scattering off corners and edges Flexible pile Deformation matches free-field deformation Surface motion = free-field motion No rotation at surface

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**Soil-Structure Interaction**

Kinematic SSI has three primary causes: Base slab averaging – results from stiffness of foundation Embedment – variation of ground motion with depth Wave scattering – scattering off corners and edges Rigid pile Deformation different than free-field deformation – can translate and rotate Surface motion = free-field motion Rotation and displacement at surface

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**Soil-Structure Interaction**

Kinematic SSI has three primary causes: Base slab averaging – results from stiffness of foundation Embedment – reduction of ground motion with depth Wave scattering – scattering off corners and edges Ground motion amplitude decreases with depth

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**Soil-Structure Interaction**

Kinematic SSI has three primary causes: Base slab averaging – results from stiffness of foundation Embedment – reduction of ground motion with depth Wave scattering – scattering off corners and edges Vertically propagating shear waves can cause rocking as well as translation

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**Soil-Structure Interaction**

Kinematic SSI has three primary causes: Base slab averaging – results from stiffness of foundation Embedment – reduction of ground motion with depth Wave scattering – scattering off corners and edges Wave scattering reduces amplitude of high frequency components

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**Outline Soil-structure interaction basics Kinematic interaction**

Inertial interaction Analysis of soil-structure interaction Effects of soil-structure interaction Soil-pile-structure interaction basics Deep foundations Single pile/shaft behavior Analysis of single piles Forms of loading Pile groups Dynamic response Methods of analysis Conclusions

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**Soil-Structure Interaction**

Inertial SSI results from compliance of soil Soil is not rigid – will deform due to loads from structure Translation in three directions

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**Soil-Structure Interaction**

Inertial SSI results from compliance of soil Soil is not rigid – will deform due to loads from structure Rotation about three axes 6 x 6 stiffness matrix to describe foundation compliance

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**Soil-Structure Interaction**

Inertial SSI results from compliance of soil Soil is not rigid – will deform due to loads from structure Displacement in two directions Rocking about one axis 3 x 3 stiffness matrix to describe foundation compliance

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**Soil-Structure Interaction**

Inertial SSI results from compliance of soil Soil is not rigid – will deform due to loads from structure Deformations resulting from structural forces will propagate away from structure Energy “removed” from structure – radiation damping

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**Outline Soil-structure interaction basics Kinematic interaction**

Inertial interaction Analysis of soil-structure interaction Effects of soil-structure interaction Soil-pile-structure interaction basics Deep foundations Single pile/shaft behavior Analysis of single piles Forms of loading Pile groups Dynamic response Methods of analysis Conclusions

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**Soil-Structure Interaction**

Analysis of soil-structure interaction Two approaches Direct approach – model soil and structure together Requires detailed model of structure and soil in one computer program Can handle nonlinear soil and structural response

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**Soil-Structure Interaction**

Analysis of soil-structure interaction Two approaches Direct approach – model soil and structure together Substructure approach – model separately and combine Kinematic SSI Inertial SSI Can use different codes for soil and structural response Superposition requires linearity

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**Soil-Structure Interaction**

Analysis of kinematic soil-structure interaction

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**Soil-Structure Interaction**

Analysis of kinematic soil-structure interaction Influenced by stiffness and geometry of soil and foundation uFIM qFIM Model foundation as massless but with actual stiffness Compute foundation input motions, uFIM and qFIM

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**Soil-Structure Interaction**

Impedance function – foundation stiffness and damping kv cv Qv kh kq ch cq M Qh Kv = kv + icvw 6 x 6 matrix of complex impedance coefficients 3 translational coefficients 3 rotational coefficients Cross-coupling (off-diagonal) coefficients

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**Soil-Structure Interaction**

Analysis of structure on compliant base subjected to FIM Based on principle of superposition – assumed linearity Frequently performed using equivalent linear approach kh kq uFIM cq ch cv kv qFIM

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**Outline Soil-structure interaction basics Kinematic interaction**

Inertial interaction Analysis of soil-structure interaction Effects of soil-structure interaction Soil-pile-structure interaction basics Deep foundations Single pile/shaft behavior Analysis of single piles Forms of loading Pile groups Dynamic response Methods of analysis Conclusions

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**Soil-Structure Interaction**

Effects of soil-structure interaction Consider simplified model Base of structure can translate and rotate SDOF system on compliant base Horizontal translation Rocking

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**Soil-Structure Interaction**

Effects of soil-structure interaction Consider simplified model Displacement due to distortion of structure Displacement due to rocking Displacement of ground Displacement due to horizontal translation SDOF system on compliant base Horizontal translation Rocking

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**Soil-Structure Interaction**

Effects of soil-structure interaction Consider simplified model

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**Soil-Structure Interaction**

Structure on circular footing of radius, r Soft soil Stiff structure Period lengthening is negligible for a soft structure on stiff soil – it’s effects increase with increasing structure/soil relative stiffness. SSI effects are small for flexible structure on stiff site, but significant for stiff structure on soft soil. ~ T/T Stiff soil Flexible structure h/(VsT)

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**Soil-Structure Interaction**

Soft soil Stiff structure Radiation damping is negligible for a soft structure on stiff soil – it’s effects increase with increasing structure/soil relative stiffness. Relative importance of radiation damping decreases with increasing h/r (increasing rocking response). SSI effects are small for flexible structure on stiff site, but significant for stiff structure on soft soil. Stiff soil Flexible structure h/(VsT)

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**Soil-Structure Interaction**

Effects of soil-structure interaction Consider simplified model SSI can decrease structural deformations, loads With increasing foundation flexibility, Period lengthens Damping increases

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**Soil-Structure Interaction**

Effects of soil-structure interaction Consider simplified model SSI can increase total displacements With increasing foundation flexibility, Period lengthens Damping increases

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**Soil-Structure Interaction Basics**

Summary SSI is not significant for cases of flexible structures on stiff soil deposits SSI can be quite significant for stiff structures founded on soft soils Fundamental period of soil-structure system is longer than that of fixed-base structure Effective damping of soil-structure system is higher than damping of structure alone Total displacements can be increased by SSI – can be important for closely-spaced tall structures Neglecting SSI is equivalent to assuming the structure is supported on rigid materials

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**Outline Soil-structure interaction basics Kinematic interaction**

Inertial interaction Analysis of soil-structure interaction Effects of soil-structure interaction Soil-pile-structure interaction basics Deep foundations Single pile/shaft behavior Analysis of single piles Forms of loading Pile groups Dynamic response Methods of analysis Conclusions

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**Deep Foundations Why? Poor soil conditions Soft, weak, compressible**

This is the standard master slide for presentations – hopefully, we will all look quite similar (same font, etc.) if we use this

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**Deep Foundations Why? Poor soil conditions Soft, weak, compressible**

This is the standard master slide for presentations – hopefully, we will all look quite similar (same font, etc.) if we use this

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**Deep Foundations Why? Poor soil conditions Soft, weak, compressible**

This is the standard master slide for presentations – hopefully, we will all look quite similar (same font, etc.) if we use this

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**Deep Foundations Why? Poor soil conditions Soft, weak, compressible**

This is the standard master slide for presentations – hopefully, we will all look quite similar (same font, etc.) if we use this

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**Deep Foundations Pile Foundations**

All deep foundation photos courtesy of Geo-Photo album (Ross Boulanger and Mike Duncan)

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Deep Foundations Pile Foundations Note excavation outside of forms

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Deep Foundations Pile Foundations Note excavation outside of forms

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Deep Foundations Drilled Shaft Foundations

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Deep Foundations Drilled Shaft Foundations

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Deep Foundations Drilled Shaft Foundations

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Deep Foundations Drilled Shaft Foundations

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**Outline Soil-structure interaction basics Kinematic interaction**

Inertial interaction Analysis of soil-structure interaction Effects of soil-structure interaction Soil-pile-structure interaction basics Deep foundations Single pile/shaft behavior Analysis of single piles Forms of loading Pile groups Dynamic response Methods of analysis Conclusions This is the standard master slide for presentations – hopefully, we will all look quite similar (same font, etc.) if we use this

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**Deep Foundations Single pile/shaft behavior – vertical loading**

Applied load High in clays (adhesion) Low in sands (friction) Skin resistance Low in clays (cohesive) High in sands (frictional) Tip resistance

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**Deep Foundations Single pile/shaft behavior – vertical loading**

Applied load High in clays Low in sands Skin resistance Possible momentary suction Tip resistance Zero* in clays Zero in sands

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**How do we measure vertical load resistance?**

Deep Foundations Single pile/shaft behavior – vertical loading Applied load How do we measure vertical load resistance? Skin resistance Tip resistance

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Deep Foundations Pile load test

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**Deep Foundations Single pile/shaft behavior – vertical loading**

Applied load, Q Qult Q Strain gauges Skin resistance Tip Skin Tip resistance

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**Nonlinear soil response means pile stiffness is not constant**

Deep Foundations Single pile/shaft behavior – vertical loading Pile head load displacement Q Applied load, Q Qult Q Qult Strain gauges Nonlinear soil response means pile stiffness is not constant Skin resistance d Tip Skin Tip resistance

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**Deep Foundations Single pile/shaft behavior – lateral loading**

Applied load No lateral load Horizontal plane Lateral load

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**Deep Foundations Single pile/shaft behavior – lateral loading**

Applied load No lateral load Horizontal plane p Lateral load y

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**Deep Foundations Single pile/shaft behavior – lateral loading Strength**

Applied load p pult Stiffness y Horizontal plane p Lateral load y

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**Deep Foundations Single pile/shaft behavior – lateral loading**

At large depths, p pult y Pile moves through soil Soil appears to flow around pile Soil movement in horizontal plane Solutions for pult available p Lateral load y

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**Deep Foundations Single pile/shaft behavior – lateral loading**

At shallow depths, p pult y p Lateral load Wedge of soil is pushed up and out Vertical and horizontal components of movement y

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**Same resistance in both directions**

Deep Foundations Single pile/shaft behavior – lateral loading At shallow depths, Same resistance in both directions Wedge of soil is pushed up and out Vertical and horizontal components of movement

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**Symmetric loading leads to generally symmetric response**

Deep Foundations Single pile/shaft behavior – lateral loading At shallow depths, Symmetric loading leads to generally symmetric response Wedge of soil is pushed up and out Vertical and horizontal components of movement

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**Much greater resistance to loading in upslope than downslope direction**

Deep Foundations Single pile/shaft behavior – lateral loading At shallow depths, Much greater resistance to loading in upslope than downslope direction Wedge of soil is pushed up and out Vertical and horizontal components of movement

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**Symmetric loading leads to asymmetric response**

Deep Foundations Single pile/shaft behavior – lateral loading At shallow depths, Symmetric loading leads to asymmetric response Wedge of soil is pushed up and out Vertical and horizontal components of movement

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**Deep Foundations Determination of p-y behavior – lateral load test**

Applied load Strain gauge pairs

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**Deep Foundations Determination of p-y behavior – lateral load test**

lateral soil resistance shear force bending moment p y

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**Deep Foundations Determination of p-y behavior lateral soil resistance**

shear force bending moment

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**Nonlinear soil response means lateral stiffness is not constant**

Deep Foundations Determination of p-y behavior lateral soil resistance shear force bending moment Nonlinear soil response means lateral stiffness is not constant p y

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**Deep Foundations static cyclic sand**

Rate-dependence has been observed in some fine-grained soils (increases with increasing plasticity) May provide ~ 10% increase in stiffness/strength for 10-fold increase in strain rate Implies frequency-dependence in dynamic stiffness stiff clay soft clay

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**Outline Soil-structure interaction basics Kinematic interaction**

Inertial interaction Analysis of soil-structure interaction Effects of soil-structure interaction Soil-pile-structure interaction basics Deep foundations Single pile/shaft behavior Analysis of single piles Forms of loading Pile groups Dynamic response Methods of analysis Conclusions

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Deep Foundations Analysis of deep foundation response – lateral loading, single foundation Dynamic beam on nonlinear Winkler foundation

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**Deep Foundations NF FF NF FF NF FF NF FF NF FF NF FF NF FF NF FF**

Analysis of deep foundation response – lateral loading, single foundation Dynamic beam on nonlinear Winkler foundation NF FF NF FF NF FF NF FF Free-field displacement NF FF NF FF NF FF NF FF

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Deep Foundations Analysis of deep foundation response – lateral loading, single foundation Near-field element Nonlinear, inelastic behavior close to pile p y D y . m (1,1) m (1,2) nf nf m (2,1) m (2,2) nf nf

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Deep Foundations Analysis of deep foundation response – lateral loading, single foundation Far-field element Frequency-dependent radiation damping

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Deep Foundations Analysis of deep foundation response – lateral loading, single foundation Far-field element Frequency-dependent radiation damping y k k k ff 1 2 3 m ff c c c 1 2 3

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**Dimensionless Frequency**

Deep Foundations Analysis of deep foundation response – lateral loading, single foundation Far-field element Frequency-dependent radiation damping c 2 3 Dimensionless Frequency Stiffness Real part Imaginary part y k k k ff 1 2 3 m ff c c c 1 2 3

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Deep Foundations Analysis of deep foundation response – vertical loading, single foundation Discretize pile, represent nonlinear skin resistance using t-z curves t-z t z Tip resistance mobilized at larger displacements Skin resistance generally mobilized quickly Q z What about tip resistance? Q-z

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**Outline Soil-structure interaction basics Kinematic interaction**

Inertial interaction Analysis of soil-structure interaction Effects of soil-structure interaction Soil-pile-structure interaction basics Deep foundations Single pile/shaft behavior Analysis of single piles Forms of loading Pile groups Dynamic response Methods of analysis Conclusions

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Deep Foundations All forms of loading p-y t-z Q-z

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Deep Foundations Vertical loading

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Deep Foundations Vertical loading

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**Vertical stiffness influenced by entire soil profile**

Deep Foundations Vertical loading Adhesion/frictional resistance of soil and interface strength mobilized along length of pile Tip resistance mobilized in bulb beneath base of foundation Vertical stiffness influenced by entire soil profile

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Deep Foundations Lateral loading at pile head

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**Deep Foundations Lateral loading at pile head**

Resistance (stiffness) dominated by near-surface soils Deeper soils don’t contribute much to lateral resistance (stiffness)

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**Homogeneous soil modulus:**

Single Pile Stiffness – Static Loading KMM KMH KHM Es KHH 1 1 d Es* z Homogeneous soil modulus: Rocking and swaying are coupled KHH ≅ Es d ( Ep / Es* )0.21 KMM ≅ Es d 3 ( Ep / Es* )0.75 KMH = KHM ≅ Es d 2 ( Ep / Es* )0.50

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**“Gibson soil” modulus:**

Single Pile Stiffness – Static Loading KMM KMH KHM Es KHH 1 1 d Es* z “Gibson soil” modulus: KHH ≅ Es d ( Ep / Es* )0.35 KMM ≅ Es d 3 ( Ep / Es* )0.80 KMH = KHM ≅ Es d 2 ( Ep / Es* )0.60

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**Parabolic soil modulus:**

Single Pile Stiffness – Static Loading KMM KMH KHM Es KHH 1 1 d Es* z Parabolic soil modulus: KHH ≅ Es d ( Ep / Es* )0.28 KMM ≅ Es d 3 ( Ep / Es* )0.77 KMH = KHM ≅ Es d 2 ( Ep / Es* )0.53

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**Layered nonlinear soil:**

Single Pile Stiffness – Static Loading KMM KMH KHM Es KHH 1 1 d z Layered nonlinear soil: Use p-y analysis

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Deep Foundations Lateral loading from ground shaking – uniform soil profile Very flexible pile High curvatures, low bending moments Flexural demands can be as high at depth as near the surface Pile head motion is same as free-field ground surface motion

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Deep Foundations Lateral loading from ground shaking – uniform soil profile Stiff pile Low curvatures, high bending moments Flexural demands can be as high at depth as near the surface Pile head motion is different than free-field ground surface motion – reflects distribution of motions along length of pile

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Deep Foundations Lateral loading from ground shaking – uniform soil profile High free-field curvature at boundary Soft Stiff Flexural demands can be much higher at depth than near surface

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Deep Foundations Lateral loading from ground shaking – uniform soil profile Stiff High free-field curvature at both boundaries Soft Stiff Flexural demands can be high at both locations

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**Deep Foundations - Lateral loading summary**

Inertial Kinematic Loading from cyclic soil deformation Crust Lateral spreading Loading from permanent soil deformation Loading from superstructure

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Deep Foundations All forms of loading p-y t-z Q-z

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**Deep Foundations P P d Lateral loading p y Liquefiable soils**

p-y curves are “softened” as pore pressures increase – shape remains the same Actual p-y behavior of liquefied soil is more complicated – stiffness and shape both change

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**Outline Soil-structure interaction basics Kinematic interaction**

Inertial interaction Analysis of soil-structure interaction Effects of soil-structure interaction Soil-pile-structure interaction basics Deep foundations Single pile/shaft behavior Analysis of single piles Forms of loading Pile groups Dynamic response Methods of analysis Conclusions

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Deep Foundations Pile Groups

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**Deep Foundations Pile Groups**

Increased axial load – potential for bearing failure Decreased axial load – potential for pullout Lateral loading of pile groups mobilizes axial resistance (t-z, Q-z springs) as well as lateral resistance (p-y springs)

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**Deep Foundations Pile Groups**

Axial resistance of piles generally sufficient to greatly reduce pile cap rotation

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**Deep Foundations Pile Groups**

Axial resistance of piles generally sufficient to greatly reduce pile cap rotation

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**Deep Foundations Pile Groups**

Axial resistance of piles generally sufficient to greatly reduce pile cap rotation

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**Deep Foundations Pile Groups**

Axial resistance of piles generally sufficient to greatly reduce pile cap rotation

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Deep Foundations Pile Groups 3 x 3 group

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Deep Foundations Pile Groups 4 x 4 group

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**Deep Foundations Pile Groups**

Zones of influence overlap with each other and with other piles Not all piles produce same resistance Pile-soil-pile interaction can affect group capacity and stiffness Zones of influence Piles interact at spacings less than 7-8 diameters

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**Deep Foundations Pile Groups Single pile Row 4 Row 3 Row 2 Row 1 Row 1**

Rows 3-5 Leading row takes greatest load Trailing rows take less load Leading row Trailing rows Group effects handled by p-multipliers Multiple cycles can diminish row effects

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**Deep Foundations Pile Groups Leading row takes greatest load**

Trailing rows take less load Group effects handled by p-multipliers Mokwa, 1999 Multiple cycles can diminish row effects

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**Deep Foundations Pile Groups Single pile has greater bending moment**

Leading row piles have largest M in group Trailing row moments stabilize after Row 3 Rollins et al., 2005

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**Passive resistance on pile cap**

Deep Foundations Pile Groups – Embedded pile cap Passive resistance on pile cap Pile cap can provide substantial contribution to lateral resistance (stiffness) Effectiveness can be affected by compaction of backfill soils

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Deep Foundations Pile Groups – High overturning moment M High axial demands placed on outer rows of piles – upward and downward Can lead to yielding of these piles – plastic deformation of soils M q

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Deep Foundations Pile Groups – High overturning moment M Dissipated energy High axial demands placed on outer rows of piles – upward and downward Can lead to yielding of these piles – plastic deformation of soils M q

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**Outline Soil-structure interaction basics Kinematic interaction**

Inertial interaction Analysis of soil-structure interaction Effects of soil-structure interaction Soil-pile-structure interaction basics Deep foundations Single pile/shaft behavior Analysis of single piles Forms of loading Pile groups Dynamic response Methods of analysis Conclusions

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**Single Pile Stiffness – Dynamic Loading**

Under harmonic loading, pile will respond (deform) harmonically Both amplitude and phase of response will vary with frequency Can model resistance (pile impedance) as having two parts Elastic resistance – K(w) Viscous resistance – C(w) In phase 90o out of phase Can model dynamic stiffness using stiffness multiplier K(w) = k(w)Kstatic

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**Deep Foundations – Dynamic Amplification Factors**

αo = ωd / VS Frequency dependence for single piles not that strong Not uncommon to assume k(w) ~ 1

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**Outline Soil-structure interaction basics Kinematic interaction**

Inertial interaction Analysis of soil-structure interaction Effects of soil-structure interaction Soil-pile-structure interaction basics Deep foundations Single pile/shaft behavior Analysis of single piles Forms of loading Pile groups Dynamic response Methods of analysis Conclusions

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**Methods of Analysis Direct Analysis**

Model entire soil-pile-structure system Compute response in single analysis

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**Direct Analysis p-y t-z Q-z Attach p-y curves to all nodes**

Attach t-z curves to all nodes p-y t-z Attach Q-z curves to pile tip nodes Q-z

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Direct Analysis Apply depth-varying free-field motions to free ends of p-y, t-z, and Q-z elements Compute resulting response Coupled analysis of soil-pile-structure system p-y t-z Q-z

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**Substructure Modeling**

Cut piles at mudline and replace with springs/dashpots Apply kinematic pile motions at mudline to free ends of horizontal, vertical, and rotational springs Compute resulting response

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**Substructure Modeling**

Cut column at pile cap and replace with springs/dashpots Apply kinematic pile cap motions to horizontal, vertical, and rotational springs at centroid of pile cap Compute resulting response Substructure modeling can provide exact solution for linear system Can iterate to approximate nonlinear effects using equivalent linearization No direct way to handle nonlinear systems

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**Outline Soil-structure interaction basics Kinematic interaction**

Inertial interaction Analysis of soil-structure interaction Effects of soil-structure interaction Soil-pile-structure interaction basics Deep foundations Single pile/shaft behavior Analysis of single piles Forms of loading Pile groups Dynamic response Methods of analysis Conclusions

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**Conclusions Physical behavior of deep foundations is complicated**

For buildings … … kinematic interaction can affect foundation input motions … inertial interaction effects are more significant For bridges, wharves, etc. … kinematic interaction effects can be very significant … kinematic effects due to permanent deformations can be critical … inertial interaction effects can still be important For inertial interaction, estimation of stiffness becomes important … single piles – resistance mostly flexural … pile groups – resistance provided by flexural and axial components … pile cap stiffness can be significant – backfill characteristics important Approximations to actual behavior frequently required – requires communication between structural and geotechnical engineers

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Thank you

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References Gazetas, G. (), Chapter 15 in Foundation Engineering Handbook, Winterkorn and Fang, Van Nostrand Reinhold. Lam, I.P. and Law, H. (2000). “Soil-structure interaction of bridges for seimsic analysis,” Technical Report MCEER , 119 pp. Mokwa, R.L. (1999). “Investigation of the Resistance of Pile Caps to Lateral Loading,” Ph.D. Dissertation, Virginia Tech. Rollins, K.M., Lane, J.D., Gerber, T. M. (2005). "Measured and Computed Lateral Response of a Pile Group in Sand," J. Geotechnical and Geoenvironmental Engrg., ASCE Vol. 131, No. 1, p Stewart, J.P. (2000). “Variations between foundation-level and free-field earthquake ground motions” Earthquake Spectra, 16 (2), Stewart, J.P., Fenves, G.L. and Seed, R.B. (1999). “Seismic soil-structure interaction in buildings. II: Analytical aspects,” J. Geotech. & Geoenv. Engrg., ASCE, 125 (1), Stewart, J.P., Seed, R.B., and Fenves, G.L. (1999). “Seismic soil-structure interaction in buildings. II: Empirical findings,” J. Geotech. & Geoenv. Engrg., ASCE, 125 (1),

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