If the option to incorporate the effects of soil-structure interaction
is exercised, the requirements of this section are permitted
to be used in the determination of the design earthquake forces
and the corresponding displacements of the structure if the
model used for structural response analysis does not directly
incorporate the effects of foundation flexibility (i.e., the model
corresponds to a fixed-based condition with no foundation
springs). The provisions in this section shall not be used if a
flexible-base foundation is included in the structural response
model.

The provisions for use with the equivalent lateral force procedure are given in Section 19.2, and those for use with the modal analysis procedure are given in Section 19.3.

The provisions for use with the equivalent lateral force procedure are given in Section 19.2, and those for use with the modal analysis procedure are given in Section 19.3.

The following requirements are supplementary to those presented
in Section 12.8.

To account for the effects of soil-structure
interaction, the base shear (

The reduction (Δ

*V*) determined from Eq. 12.8-1 shall be reduced to*Ṽ*=

*V*- Δ

*V*(19.2-1)

The reduction (Δ

*V*) shall be computed as follows and shall not exceed 0.3*V*:
(19.2-2)

where
C_{S} |
= |
the seismic design coefficient computed from Eqs. 12.8-2,
12.8-3, and 12.8-4 using the fundamental natural period of
the fixed-base structure (T or T) as specified in Section
12.8.2_{a} |

C̃ |
= |
the value of C computed from Eqs. 12.8-2, 12.8-3, and
12.8-4 using the fundamental natural period of the flexibly supported structure (_{S}T̃) defined in Section 19.2.1.1 |

β̃ |
= |
the fraction of critical damping for the structure-foundation system determined in Section 19.2.1.2 |

W̅ |
= |
the effective seismic weight of the structure, which shall be
taken as 0.7W, except for structures where the effective
seismic weight is concentrated at a single level, it shall be
taken as equal to W |

The effective period (

The foundation stiffnesses (

Alternatively, for structures supported on mat foundations that
rest at or near the ground surface or are embedded in such a way
that the side wall contact with the soil is not considered to remain
effective during the design ground motion, the effective period
of the structure is permitted to be determined from

Note: Use straight-line interpolation for intermediate values of

*T̃*) shall be determined as follows:
(19.2-3)

where
T |
= | the fundamental period of the structure as determined in Section 12.8.2 |

k̅ |
= | the stiffness of the structure where fixed at the base, defined by the following: |

(19.2-4)

where
h̅ |
= |
the effective height of the structure, which shall be taken
as 0.7 times the structural height (h), except for structures
where the gravity load is effectively concentrated at a
single level, the effective height of the structure shall be
taken as the height to that level_{n} |

K_{y} |
= |
the lateral stiffness of the foundation defined as the horizontal force at the level of the foundation necessary to produce a unit deflection at that level, the force and the deflection being measured in the direction in which the structure is analyzed |

K_{θ} |
= |
the rocking stiffness of the foundation defined as the moment necessary to produce a unit average rotation of the foundation, the moment and rotation being measured in the direction in which the structure is analyzed |

g |
= | the acceleration of gravity |

*K*and_{y}*K*_{θ}) shall be computed by established principles of foundation mechanics using soil properties that are compatible with the soil strain levels associated with the design earthquake motion. The average shear modulus (*G*) for the soils beneath the foundation at large strain levels and the associated shear wave velocity (*v*) needed in these computations shall be determined from Table 19.2-1 where_{s}v_{so} |
= |
the average shear wave velocity for the soils beneath the
foundation at small strain levels (10^{-3} percent or less) |

G_{o} |
= | γv^{2}/_{so}g = the average shear modulus for the soils beneath
the foundation at small strain levels |

γ | = | the average unit weight of the soils |

(19.2-5)

where
a |
= | the relative weight density of the structure and the soil defined by |

(19.2-6)

**Table 19.2-1 Values of***v*_{s}*v*and_{so}*G*/*G*_{o} Value of v/_{s}v_{so} |
Value of G/G_{o} |
|||||
---|---|---|---|---|---|---|

S/2.5_{DS} |
S/2.5_{DS} |
|||||

Site Class | ≤0.1 | 0.4 | ≥0.8 | ≤0.1 | 0.4 | ≥0.8 |

A B C D E F |
1.00 1.00 0.97 0.95 0.77 α |
1.00 0.97 0.87 0.71 0.22 α |
1.00 0.95 0.77 0.32 αα |
1.00 1.00 0.95 0.90 0.60 α |
1.00 0.95 0.75 0.50 0.05 α |
1.00 0.90 0.60 0.10 αα |

*S*/2.5._{DS}^{α}Should be evaluated from site-specific analysis.**Table 19.2-2 Values of α**

_{θ}r_{m}/vT_{s} |
α_{θ} |
---|---|

<0.05 0.15 0.35 0.5 |
1.0 0.85 0.7 0.6 |

*r*and_{a}*r*= characteristic foundation lengths defined by_{m}
(19.2-7)

and
(19.2-8)

where
A_{o} |
= | the area of the load-carrying foundation |

I_{o} |
= | the static moment of inertia of the load-carrying foundation about a horizontal centroidal axis normal to the direction in which the structure is analyzed |

α_{θ} |
= | dynamic foundation stiffness modifier for rocking as determined from Table 19.2-2 |

v_{s} |
= | shear wave velocity |

T |
= | fundamental period as determined in Section 12.8.2 |

The effective damping factor for
the structure-foundation system (β̃) shall be computed as follows:

For values of between 0.10 and 0.20 the values of β

The quantity

For intermediate values of , the value of

(19.2-9)

where
β_{o} |
= | the foundation damping factor as specified in Fig. 19.2-1 |

_{o}shall be determined by linear interpolation between the solid lines and the dashed lines of Fig. 19.2-1.The quantity

*r*in Fig. 19.2-1 is a characteristic foundation length that shall be determined as follows:
(19.2-10)

(19.2-11)

where
L_{o} |
= | the overall length of the side of the foundation in the direction being analyzed |

r and _{a}r_{m} |
= | characteristic foundation lengths defined in Eqs. 19.2-7 and 19.2-8, respectively |

*r*shall be determined by linear interpolation.**EXCEPTION:**For structures supported on point-bearing piles and in all other cases where the foundation soil consists of a soft stratum of reasonably uniform properties underlain by a much stiffer, rock-like deposit with an abrupt increase in stiffness, the factor β_{o}in Eq. 19.2-9 shall be replaced by β*'*_{o}if < 1 where*D*is the total depth of the stratum. β_{s}*'*_{o}shall be determined as follows:
(19.2-12)

The value of β̃ computed from Eq. 19.2-9, both with or without
the adjustment represented by Eq. 19.2-12. shall in no case be
taken as less than β̃ = 0.05 or greater than β̃ = 0.20.The distribution
over the height of the structure of the reduced total seismic
force (

*Ṽ*) shall be considered to be the same as for the structure without interaction.The modified story shears, overturning
moments, and torsional effects about a vertical axis shall be
determined as for structures without interaction using the reduced
lateral forces.

The modified deflections (δ̃) shall be determined as follows:

The modified story drifts and P-delta effects shall be evaluated
in accordance with the provisions of Sections 12.8.6 and 12.8.7
using the modified story shears and deflections determined in
this section.

The modified deflections (δ̃) shall be determined as follows:

(19.2-13)

where
M_{o} |
= | the overturning moment at the base using the unmodified seismic forces and not including the reduction permitted in the design of the foundation |

h_{x} |
= | the height above the base to the level under consideration |

δ_{x} |
= | the deflections of the fixed-base structure as determined in Section 12.8.6 using the unmodified seismic forces |

The following provisions are supplementary to those presented
in Section 12.9.

To account for the effects of soil-structure
interaction, the base shear corresponding to the fundamental
mode of vibration (

The reduction (Δ

The period

where

The preceding designated values of

*V*_{1}) shall be reduced to*Ṽ*

_{1}=

*V*

_{1}- Δ

*V*

_{1}(19.3-1)

The reduction (Δ

*V*_{1}) shall be computed in accordance with Eq. 19.2-2 with*W̅*taken as equal to the effective seismic weight of the fundamental period of vibration,*W̅*, and*C*computed in accordance with Eq. 12.8-1, except that_{s}*S*shall be replaced by design spectral response acceleration of the design response spectra at the fundamental period of the fixed-base structure (_{DS}*T*_{1}).The period

*T̃*shall be determined from Eq. 19.2-3 or from Eq. 19.2-5 where applicable, taking*T*=*T*_{1}, evaluating*k̅*from Eq. 19.2-4 with*W̅*=*W̅*_{1}, and computing*h̅*as follows:
(19.3-2)

where

w_{i} |
= | the portion of the total gravity load of the structure at
level i |

φ_{il} |
= | the displacement amplitude at the i^{th} level of the structure
when vibrating in its fundamental mode |

h_{i} |
= | the height above the base to level i |

*W̅*,*h̅*,*T*, and*T̃*also shall be used to evaluate the factor α from Eq. 19.2-6 and the factor β_{}*o*from Fig. 19.2-1. No reduction shall be made in the shear components contributed by the higher modes of vibration. The reduced base shear (*Ṽ*i) shall in no case be taken less than 0.7*V*_{1}.The modified modal seismic
forces, story shears, and overturning moments shall be determined
as for structures without interaction using the modified
base shear (

The modified modal drift in a story (Δ̃

*Ṽ*_{1}) instead of*V*_{1}. The modified modal deflections (δ̃_{xm}) shall be determined as follows:
(19.3-3)

and
δ̃

where
_{xm}= δ_{xm}for*m*= 2, 3, ... (19.3-4)M_{o1} |
= |
the overturning base moment for the fundamental mode
of the fixed-base structure using the unmodified modal
base shear V_{1} |

δ_{xm} |
= |
the modal deflections at level x of the fixed-base structure
using the unmodified modal shears, V_{m} |

*) shall be computed as the difference of the deflections (δ*_{m}_{xm}) at the top and bottom of the story under consideration.The design values of the modified
shears, moments, deflections, and story drifts shall be determined
as for structures without interaction by taking the square
root of the sum of the squares (SRSS) of the respective modal
contributions. In the design of the foundation, it is permitted to
reduce the overturning moment at the foundation-soil interface
determined in this manner by 10% as for structures without
interaction.

The effects of torsion about a vertical axis shall be evaluated in accordance with the provisions of Section 12.8.4, and the P-delta effects shall be evaluated in accordance with the provisions of Section 12.8.7 using the story shears and drifts determined in Section 19.3.2.

The effects of torsion about a vertical axis shall be evaluated in accordance with the provisions of Section 12.8.4, and the P-delta effects shall be evaluated in accordance with the provisions of Section 12.8.7 using the story shears and drifts determined in Section 19.3.2.