Foundations and SSI aspects of FEMA 356 and 440

EERI TechnicalSeminar Series

Impact of Soil-Structure Interaction on Response of StructuresSeminar 1: Practical Applications to Shallow Foundations

Performance-based guidelines for practitioners

Foundations and SSI aspects of FEMA 356 and 440

Craig D. Comartin



Outline

Example buildingFoundation modelingInertial effectsKinematic effectsResponse spectrum for analysisBearing capacity



Example building plan

160’-0”

100’

-0”

Plan

8” R/C wall – 20’Ltypical



Example building wall elevation and section

Elevation @ wall Section @ wall

Roof

2nd

1st

20’-0”

Footing 26’L x 3’B x 1.5’d

3’D

10’-0”typical

Elevation @ wall Section @ wall

Roof

2nd

1st

20’-0”


3’D

10’-0”typical



Example building description

• Two story concrete structure: FEMA model building Type C2• Story to story heights 10 ft: total building height of 20 ft. • Plan dimensions: 100 ft. by 160 ft. • Floors and roof construction: two-way reinforced concrete flat slab

• Roof DL = 140 psf• Floor DL = 160 psf

• Vertical support: concrete columns and interior and exterior reinforced concretebearing walls

• Lateral system: six shear walls in each direction, 12 total – L=20 ft., t=8 in. • Foundations: spread footings bearing 3 ft. below grade and reinforced concrete

slab on grade• Soils conditions: very stiff alluvium, u= 1200 fps, NEHRP Site Class C• Ground motion: shaking with a 10% chance of being exceed in 50 yrs.• Analysis objective: Maximum global displacement for specified ground motion

NOTE: See FEMA 440 for complete example.



Generic Foundation Element Models

Fy

Fx

Mz

SθSx

Sy

Sx

Siy i=1 to 5

c. Winkler Component Model

b. Uncoupled ComponentModel

a. Foundation Actions

Structural Component Geotechnical Components



Flexible base properties

Flexible base model

20’-0”

10’-0”typical

Mroof

Mfloor


3’D

Flexible base model

20’-0”

10’-0”typical

Mroof

Mfloor


3’D

Flexible base model

20’-0”

10’-0”typical

Mroof

Mfloor


3’D

Flexible base model

20’-0”

10’-0”typical

Mroof

Mfloor


3’D



Gazetas’ equations (stiffness)



Gazetas’ equations (embedment)




Soil properties Foundation dimensions

initial shear modulus 31 ksi length, L= 26 ftwidth, B= 3 ft

unit wt. of soil, 100 pcf thickness, d 1.5 ft shear wave velocity, 1200 fps depth, D= 3 ft

effective shear modulus 0.75

23 ksi

Poisson's ratio 0.3

0/G G =

γ =

sν =

20 sG

gγ ν= =

G=

υ=

ATC 40Sect. 10.4.1.2

FEMA 356Table 4.7



Soil properties

shear wave velocity = 1200 ft/secsoil unit weight = 100 pcf

gravity = 32.2 ft/sec2

G0 = 31 ksi

G = 0.75 G0

= 23 ksi




Rotational stiffness

at surface 780,342,531 k-in/rad

embedment factor 1.39

at depth 1,081,161,315 k-in/rad

θ υ

⎡ ⎤⎛ ⎞= + =⎢ ⎥⎜ ⎟− ⎝ ⎠⎢ ⎥⎣ ⎦

2.43

, 0.47 0.034 61surGB LK X walls

B0.6 1.9 0.6

1 1.4 1.5 3.7d d dL L Dθβ

−⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + + =⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦

,surK Kθ θ θβ= =

FEMA 356Fig. 4.4ATC 40

Table 10.2&3

7.8 x 108

1.1 x 109




Translational stiffness

at surface 44,505 k/in

embedment factor 1.86

at depth 82,607 k/in

υ

⎡ ⎤⎛ ⎞= + =⎢ ⎥⎜ ⎟− ⎝ ⎠⎢ ⎥⎣ ⎦

0.65

, 3.4 1.2 62x surGB LK X walls

B

( ) 0.4

21 0.21 1 1.6x

Dd B LDB BL

β⎡ ⎤⎛ ⎞ ⎛ ⎞+⎢ ⎥= + + =⎜ ⎟ ⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎝ ⎠⎣ ⎦

,x x sur xK K β= =

FEMA 356Fig. 4.4ATC 40

Table 10.2&3



Linear periodsEvaluate the linear periods for the structural model

assuming a fixed base, , and a flexible base, , using appropriate foundation modeling assumptions.

Fixed base model

20’-0”

10’-0”typical

Mroof

Mfloor

T

0.20sec=%T

T%

0.14sec=TFlexible base model

20’-0”

10’-0”typical

Mroof

Mfloor


3’D

Flexible base model

20’-0”

10’-0”typical

Mroof

Mfloor


3’D

xK Kθ

Flexible base model

20’-0”

10’-0”typical

Mroof

Mfloor


3’D

Flexible base model

20’-0”

10’-0”typical

Mroof

Mfloor


3’D

xK Kθ



Initial spectrum for analysis

Select ground motion spectrum

Site class C and shear wave velocity,

Acceleration parameters for MCE shaking Damping coefficients for initial short period 1.5 glong period 0.6 g

Adjustment for site class Cshort periodlong period

short period 1.0 g

long period 0.5 g

To reduce to design level motions (e.g.10% chance of being exceeded in 50 years), multiply accelerations by 2/3

sv =1200fps

sS =5%β =

1.0sB =

1 1.0B =

FEMA 356Sect. 1.6ATC 40

Sect. 4.4

S1 =

( )xs a sS F S g= = 11.5=1.5( )1 1= = 1.3 0.6 =0.78x vS FS g

1.0aF =

=1.3vF

DS XSS S2= =3

D XS S1 12= =3



Acceleration vs. period

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.50 1.00 1.50

Period, T (sec)

Sa

free field motion (FFM) @5% damping

Initial design spectrum for β =5%

T=0.2 sec



Foundation (inertial) damping

2

1 21 1eff efff

eff eff

T Ta a

T Tβ

⎛ ⎞ ⎛ ⎞= − + −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

% %

( )*exp . . /1 4 7 1 6ea c h rθ= −

( )*ln /2 25 16ea c h rθ⎡ ⎤= −⎣ ⎦( )1.5 / 1e xc e r= +

Fixed to flexible base period shift

Ratio of effective height to foundation radius for rotation

Ratio of effective height to foundation radiusfor translation



Foundation (inertial) dampingParameters necessary to compute:

%eff

eff

TT

*h

rθ

xr

e

Effective period shift

Effective building height

Equivalent foundation radius for rotation

Embedment of structure

Equivalent foundation radius for translation



Effective period shift

0.5211 1μ

⎧ ⎫⎡ ⎤⎛ ⎞⎪ ⎪⎢ ⎥= + −⎨ ⎬⎜ ⎟⎢ ⎥⎝ ⎠⎪ ⎪⎣ ⎦⎩ ⎭

% %eff

eff

T TT T

μ is the expected ductility demand for the system (i.e. including structure and soil effects). This is an approximation that assumes all inelastic behavior is concentrated in the structure.

Fixed to flexible base period shift

FEMA 440Eqn. 8-8



Example building

Effective period lengthening

Assume μ = 3

1.2 FEMA 440Eqn. 8-8

0.5211 1eff

eff

T TT Tμ

⎧ ⎫⎡ ⎤⎛ ⎞⎪ ⎪⎢ ⎥= + − =⎨ ⎜ ⎟ ⎬⎢ ⎥⎝ ⎠⎪ ⎪⎣ ⎦⎩ ⎭

% %



Effective building height

174 in.* 1

1

N

i iiN

ii

h Mh

M

ϕ=

=

= =∑

∑FEMA 440Sect. 8-3

This is a simple dynamic property of the structure.



Foundation radius for rotation

( )133 1

8K

rG

θθ

υ⎛ ⎞−= ⎜ ⎟⎝ ⎠

FEMA 440Eqn. 8-7




( )133 1

8K

rG

θθ

υ⎛ ⎞−= ⎜ ⎟⎝ ⎠

91.1 x 10θ =K

231in.=

From previous calculation



Alternative approximation

( )2* *

2 *

1

fixed

fixed

x

K hK

KTT K

θ =⎛ ⎞

− −⎜ ⎟⎝ ⎠

%FEMA 440

Eqn.8-6



Fixed base stiffness

2* * 2fixedK M

Tπ⎛ ⎞= ⎜ ⎟

⎝ ⎠*M is the effective mass for the first mode calculated as

the total mass times the effective mass coefficient (see ATC 40 Eqn. 8-21).

FEMA 440Eqn. 8-3

23,250 k/in2

* * 2fixed mK M M

Tπ α⎛ ⎞= = =⎜ ⎟

⎝ ⎠

where0.77

2

1

2

1 1

/

/ /

N

i imi

m N N

i i imi i

w g

w g w g

φα

φ

=

= =

⎡ ⎤⎢ ⎥⎣ ⎦= =

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦

∑

∑ ∑

ATC 40Eqn 8-21



Translational stiffness of the foundation

This can be evaluated using the procedures in FEMA 356 (Chap. 4) or ATC 40 (Chap. 10). For many applications, it can be estimated as

82x xK Gr

υ=

−

where G = effective, strain-degraded soil shear modulus and υ = soil Poisson’s ratio (∼0.3 for sand, ∼0.45 for clay).

FEMA 440Eqn. 8-5



Foundation radius for translation

/x fr A π=

fA is the area of the foundation footprint if the foundation components are inter-connected laterally.

FEMA 440Eqn. 8-4

71 ft. = 856 in./x fr A π= =



Translational stiffness of the foundation

This can be evaluated using the procedures in FEMA 356 (Chap. 4) or ATC 40 (Chap. 10). For many applications, it can be estimated as

82x xK Gr

υ=

−

where G = effective, strain-degraded soil shear modulus and υ = soil Poisson’s ratio (∼0.3 for sand, ∼0.45 for clay).

FEMA 440Eqn. 8-5

93,867 k/in82x xK Gr

υ= =

−

83,000=xKFrom previous calculation



Foundation rotational stiffness

6.0E+08 k-in/rad( )2* *

2 *

1

fixed

fixed

x

K hK

KTT K

θ = =⎛ ⎞

− −⎜ ⎟⎝ ⎠

%

88.4 x 10θ =K

91.1 x 10θ =KFrom previous calculation

Adjust for embedment




189 in( ) θ

θυ−⎛ ⎞

= =⎜ ⎟⎝ ⎠

133 1

8K

rG

86.0 x 10θ =KNeglecting embedment and using



Foundation (inertial) damping

Determine the basement embedment, e, if applicable and calculate

x

er

Relative translational stiffness index



Example building

No basement so e = 0



Example buildingCalculate foundation damping

3.73 %2

1 21 1eff efff

eff eff

T Ta aT T

β⎛ ⎞ ⎛ ⎞

= − + − =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

% % FEMA 440Eqn. 8-9a

⎩ ⎭

where25.19

-18.06

1.00

( )1 exp 4.7 1.6 /ea c h rθ= − =

( )2 25ln / 16ea c h rθ⎡ ⎤= − =⎣ ⎦

( )1.5 / 1e xc e r= + =

FEMA 440Eqn. 8-9b

FEMA 440Eqn. 8-9c

FEMA 440Eqn 8-9d



Foundation dampingGraphical approach

1 1.5 2Period Lengthening

0

10

20

30

Foun

dati o

nD

amp i

ng,β

f( %

)= 0

(radiation damping only)

= 0.5

1.0

2.0 /h

/e r

1 1.5 2Period Lengthening

0

10

20

30

Foun

dati o

nD

amp i

ng,β

f( %

)= 0

(radiation damping only)

= 0.5

1.0

2.0 /h

/e r

4%

1.2



Flexible base damping ratio

where 5 %iβ =

6.9 %( )0 3

if

eff effT T

ββ β= + =%

FEMA 440Eqn. 8-10



Reduce spectrum for increased damping

Evaluate the effect on spectral ordinates of the change in damping ratio accordance with Section 6.3.

1.09

( ) ( ) ( )0

0 0

5%

( ) ( )

a a FIMa

S SS

B Bββ β

= =

00

4(5.6 ln (in %))

Bβ β= =

−FEMA 440Eqn.6-17

FEMA 440Eqn.6-16




0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.50 1.00 1.50

Period, T (sec)

Sa

free field motion (FFM) @5% damping

free field motion includingfoundation damping

Spectrum adjusted for foundation damping

T=0.2 sec



Kinematic effectsEvaluate the spectral reduction from base slab

averaging (RRSbsa) as a function of period.

0 0.2 0.4 0.6 0.8 1 1.2

Period (s)

0.4

0.5

0.6

0.7

0.8

0.9

1

Foun

datio

n/fre

e-fie

ld R

RS

fro

m b

ase

slab

ave

ragi

ng (R

RS

bsa)

Simplified Modelbe = 65 ft

be = 130 ft

be = 200 ft

be = 330 ft



Kinematic effects

An approximation to curves is given by:

≥ the valuefor T= 0.2 sec.

1.21114100

ebsa

bRRST

⎛ ⎞= − ⎜ ⎟⎝ ⎠

FEMA 440Eqn. 8-1



Kinematic effects

Evaluate effective foundation size where aand b are the full footprint dimensions (in feet) of the building foundation in plan view.

abbe =



Kinematic effectsIf the structure has a basement embedded a depth e

from the ground surface, evaluate an additional spectral reduction from embedment (RRSe) as a function of period.

0 0.4 0.8 1.2 1.6 2

Period (s)

0

0.2

0.4

0.6

0.8

1

1.2Fo

unda

tion/

Free

-Fie

ld R

RS

e = 30 ftVs = 2500 ft/sVs = 1200 ft/sVs = 600 ft/s



Kinematic effectsAn approximation to curves is given by:

vs = shear wave velocity n = shear wave velocity

reduction factor for the expected PGA

cos

.

2

the larger of 0.453or the RRS value efor T=0.2sec

es

eRRST nvπ⎛ ⎞

= ≥⎜ ⎟⎝ ⎠ FEMA 440

Eqn. 8-2



Adjust for kinematic interactionModify kinematic soil-structure interaction

Effective foundation size: Embedment:a = 100 ft. e = 0b = 160 ft.

126 ft.

Ratio of response spectra for base slab averaging

Foundation input motion (FIM)

no basement

eb ab= =

1.211 the value for = 0.2 sec.14100

ebsa

bRRS TT⎛ ⎞= − ≥⎜ ⎟⎝ ⎠

FEMA 440Eqn. 8-1

( ) ( )a FIM bsa e a FFMS RRS RRS S=

eRRS =1.0




0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.50 1.00 1.50

Period, T (sec)

Sa free field motion (FFM) @5% damping

free field motion includingfoundation damping

foundation input (FIM) withfoundation damping

Final spectrum

T=0.2 sec



0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.5 1 1.5

Roof displacement, D

Base shear,V/W

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.5 1 1.5

Roof displacement, D

Base shear,V/W

Displacement demand and base shear

0.4 gVW

=

.0.4 inroofΔ =



Elasto-Plastic Behavior

Possible Stress States

1

l

Elastic prior touplift

2Elastic at uplift

3Elastic after uplift

4

l

Yield prior touplift

5Yield afteruplift

6 Inelastic limit

q< qu

q< qu

q< qu qu

qu

qu



Elasto-Plastic Behavior

Moment,M

Rotation,θ

3

1

2

6

5

4

Infinitely strong soil

P

M

l

BStress (q) distribution

Ultimate soil capacity = q

u

55

Pl2

Pl2 - P2

2quB

P l6



Check soil bearing capacity

( ) ( )13 23 3( ) 41932OTM floor roof floor roof slab slabLM F F F F M V ft k⎛ ⎞= + − + − + =⎜ ⎟

⎝ ⎠∑ ∑

2OTM DLL aM P −⎛ ⎞= ⎜ ⎟

⎝ ⎠

149yroof

Vm ft k

W= =

171yfloor

Vm ft k

W= =

20’-0”

10’-0”typical

F floor

20’-0”

10’-0”typical

F roof

PDL

PDL

qu

−2

L a

Mslab,Vslab

F total

20’-0”

10’-0”typical

F floorF floor

20’-0”

10’-0”typical

F roofF roof

PDL

PDL

qu

−2

L a

Mslab,Vslab

F totalF total

540k≈

17.2DLu

Pq ksfBa

= =2 10.4OTM

DL

Ma L ftP

→ = − =

5002slab slabLM V ft k+ ≈∑ ∑



Ultimate capacities for shallow foundations

Presumptive values tabulated for classes of materialsVertical bearing pressures increase with depth and widthLateral passive pressures increase with depth

Prescriptive values based on original design data3.0 times specified "allowable" dead plus live loads1.5 times all required "working loads"

Site-specific values based on geotechnical investigation



Documents

Foundations and SSI aspects of FEMA 356 and 440