COMPUTERS AND STRUCTURES, INC., FEBRUARY 2016 TECHNICAL NOTE AUTOMATIC GENERATION OF POINT SPRING SUPPORTS BASED ON DEFINED SOIL PROFILES AND COLUMN-FOOTING PROPERTIES Introduction This technical note describes the capability whereby foundation springs and dashpots can automatically be generated to represent the flexibility and damping associated with soil-foundation interaction in translational and rotational vibration modes for shallow foundations (i.e., isolated column footings). The methodology used for calculation of the shallow foundation translational stiffness and damping coefficients is based on a NIST report (NISTGCR 12-917-21) titled Soil-Structure Interaction for Building Structures, September 2012. Chapter 2 of this report presents the detailed procedures for computing foundation stiffness and damping. The input required for automatic generation of soil springs and dashpots at column bases include geotechnical and shear wave velocity profiles, soil shear strength parameters and their variation with depth, Poisson s ratios, soil hysteretic damping ratios, column footing geometries, and a building period. Procedural Overview The following steps are typically involved when automatically generating column base point springs and dashpots from soil profiles and column footing geometry. 1. Define soil profiles. This includes soil layering information for the site in terms of shear wave velocity profiles and material profiles. 2. Define isolated column footings. Required information consists of gross footing dimensions and footing embedment depth. 3. Define point spring properties using the soil profiles and isolated column footing definitions as described above. This definition also requires the first mode of vibration period for the building. 1
Calculating Soil-Structure Interaction 2 4. Assign the point springs defined above to column bases. Based on the procedure described above, soil springs and dashpots will automatically be created for the selected column bases. Calculating Soil-Structure Interaction The soil and foundation parameters input by the user are used to generate a nonlinear link element at the column base with translational and rotational stiffness and damping. An outline of the steps necessary to determine the stiffness and damping is presented below. The notations used in the calculations are as follows: Notations for Soil Profile Parameters nn = Number of soil layers in the soil profile TT ii = Thickness of i th layer of the soil profile ww ii = Unit weight of i th layer of the soil profile GG ii = Shear modulus of i th layer of the soil profile νν ii = Poisson s ratio of i th layer of the soil profile cc ii = Cohesion of i th layer of the soil profile φφ ii = Friction angle of i th layer of the soil profile VV ii = Shear wave velocity of i th layer of the soil profile ββ ss = Soil hysteretic damping ratio Notations for Isolated Column Footing
Calculating Soil-Structure Interaction 3 LL = Half the dimension of isolated footing along column major axis BB = Half the dimension of isolated footing along column minor axis DD = Isolated footing thickness DD ee = Depth of embedment, measured from the base of the column Notations for General and Intermediate Variables ρρ ss = Soil mass density (average of all soil layers) gg = Acceleration due to gravity νν = Average Poisson s ration (average of all soil layers) TT = First-mode time period of the structure ωω = Circular frequency corresponding to First-mode time period of the structure zz pppp = Depth interval to compute average effective profile velocity for lateral translational stiffness in X direction zz pppp = Depth interval to compute average effective profile velocity for lateral translational stiffness in Y direction
Calculating Soil-Structure Interaction 4 zz pppp = Depth interval to compute average effective profile velocity for vertical translational stiffness in Z direction zz pppppp = Depth interval to compute average effective profile velocity for rotational stiffness about X axis zz pppppp = Depth interval to compute average effective profile velocity for rotational stiffness about Y axis zz pppppp = Depth interval to compute average effective profile velocity for rotational stiffness about Z axis VV ssss = Average effective profile velocity for lateral translational stiffness in X direction VV ssss = Average effective profile velocity for lateral translational stiffness in Y direction VV ssss = Average effective profile velocity for vertical translational stiffness in Z direction VV ssssss = Average effective profile velocity for rotational stiffness about X axis VV ssssss = Average effective profile velocity for rotational stiffness about Y axis VV ssssss = Average effective profile velocity for rotational stiffness about Z axis GG xx = Effective shear modulus for lateral translational stiffness in X direction
Calculating Soil-Structure Interaction 5 GG yy = Effective shear modulus for lateral translational stiffness in Y direction GG zz = Effective shear modulus for vertical translational stiffness in Z direction GG xxxx = Effective shear modulus for rotational stiffness about X axis GG yyyy = Effective shear modulus for rotational stiffness about Y axis GG zzzz = Effective shear modulus for rotational stiffness about Z axis ηη xx = Embedment correction factor for lateral translational stiffness in X direction ηη yy = Embedment correction factor for lateral translational stiffness in Y direction ηη zz = Embedment correction factor for vertical translational stiffness in Z direction ηη xxxx = Embedment correction factor for rotational stiffness about X axis ηη yyyy = Embedment correction factor for rotational stiffness about Y axis ηη zzzz = Embedment correction factor for rotational stiffness about Z axis aa 0xx = Dimensionless frequency for lateral translational stiffness in X direction
Calculating Soil-Structure Interaction 6 aa 0yy = Dimensionless frequency for lateral translational stiffness in Y direction aa 0zz = Dimensionless frequency for vertical translational stiffness in Z direction aa 0xxxx = Dimensionless frequency for rotational stiffness about X axis aa 0yyyy = Dimensionless frequency for rotational stiffness about Y axis aa 0zzzz = Dimensionless frequency for rotational stiffness about Z axis αα xx = Dynamic stiffness modifier for translational stiffness in X direction αα yy = Dynamic stiffness modifier for translational stiffness in Y direction αα zz = Dynamic stiffness modifier for translational stiffness in Z direction αα xxxx = Dynamic stiffness modifier for rotational stiffness about X axis αα yyyy = Dynamic stiffness modifier for rotational stiffness about Y axis αα zzzz = Dynamic stiffness modifier for rotational stiffness about Z axis
Calculating Soil-Structure Interaction 7 kk xx,ssssss = kk yy,ssssss = kk zz,ssssss = kk xxxx,ssssss = kk yyyy,ssssss = kk zzzz,ssssss = Elastic static stiffness at ground surface for translation in X direction Elastic static stiffness at ground surface for translation in Y direction Elastic static stiffness at ground surface for translation in Z direction Elastic static stiffness at ground surface for rotation about X axis Elastic static stiffness at ground surface for rotation about Y axis Elastic static stiffness at ground surface for rotation about Z axis kk xx = Dynamic stiffness of embedded footing for translation in X direction kk yy = Dynamic stiffness of embedded footing for translation in Y direction kk zz = Dynamic stiffness of embedded footing for translation in Z direction kk xxxx = Dynamic stiffness of embedded footing for rotation about X axis kk yyyy = Dynamic stiffness of embedded footing for rotation about Y axis kk zzzz = Dynamic stiffness of embedded footing for rotation about Z axis
Calculating Point Springs & Dashpots 8 ψψ = Dimensionless factor for computing radiation damping ββ xx = Damping ratio for translation along X axis ββ yy = Damping ratio for translation along Y axis ββ zz = Damping ratio for translation along Z axis ββ xxxx = Damping ratio for rotation about X axis ββ yyyy = Damping ratio for rotation about Y axis ββ zzzz = Damping ratio for rotation about Z axis cc xx = Damping coefficient for translation along X axis cc yy = Damping coefficient for translation along Y axis cc zz = Damping coefficient for translation along Z axis cc xxxx = Damping coefficient for rotation about X axis cc yyyy = Damping coefficient for rotation about Y axis cc zzzz = Damping coefficient for rotation about Z axis Calculating Point Springs & Dashpots Step 1: Calculate Depth Interval The depth interval necessary for computing an average effective profile velocity can be taken as the half-dimension of an equivalent square
Calculating Point Springs & Dashpots 9 foundation matching the area of the actual foundation or the half-dimension of an equivalent square foundation matching the moment of inertia of the actual foundation. The coordinate axes are oriented as shown below such that L B. zz pppp = BBBB ; zz pppp = zz pppp zz pppp = BBBB 4 zz pppppp = BB 3 LL 4 zz pppppp = BBLL 3 4 zz pppppp = BB 3 LL + BBLL 3
Calculating Point Springs & Dashpots 10 Step 2: Calculate Average Soil Mass density and Average Poisson s Ratio nn ii=0 ρρ ss = 1 nn ww ii gg νν = 11 nn (νν nn ii=00 ii) Step 3: Calculate Average Profile Velocity For calculating the average profile velocity, layers that are partially or fully within the depth interval are used. For checking the soil layers that are effective for computing average profile velocity, the following parameters are used: ZZ mmmmmm = ZZ DD ee ZZ pp ZZ mmmmmm = ZZ DD ee Where, ZZ mmmmmm = Global Z coordinate of the top of depth range ZZ mmmmmm = Global Z coordinate of the bottom of depth range ZZ = Global Z coordinate of the column base ZZ pp = Depth interval The average effective profile velocity is calculated as follows: VV ssss = ZZ pppp nn2 zz ii ii=nn1 VVii
Calculating Point Springs & Dashpots 11 Where, nn 1 = First layer, partially or fully within depth interval nn 2 = Last layer, partially or fully within depth interval VV ssss = VV ssss VV ssss = ZZ pppp nn2 zz ii ii=nn1 VVii VV ssssss = ZZ pppppp nn2 zz ii ii=nn1 VVii VV ssssss = ZZ pppppp nn2 zz ii ii=nn1 VVii VV ssssss = ZZ pppppp nn2 zz ii ii=nn1 VVii Step 4: Calculate Effective Shear Modulus GG xx = ρρ ss (VV ssss ) 2 GG yy = ρρ ss VV ssss 2 GG zz = ρρ ss (VV ssss ) 2 GG xxxx = ρρ ss (VV ssssss ) 2 GG yyyy = ρρ ss VV ssssss 2 GG zzzz = ρρ ss (VV ssssss ) 2
Calculating Point Springs & Dashpots 12 Step 5: Calculate correction factors for embedment depth ηη xx = 1.0 + 0.33 + 1.34 LL BB DD ee BB 0.8 ηη yy = ηη xx ηη zz = 1.0 + 0.25 + 0.25 1+LL BB DD ee BB 0.8 ηη xxxx = 1.0 + DD ee + 1.6 BB 0.35+LL BB DD ee BB 2 ηη yyyy = 1.0 + DD ee + 1.6 BB 0.35+(LL BB) 4 DD ee BB 2 ηη zzzz = 1.0 + 1.3 + 1.32 LL BB DD ee BB 0.9 Step 6: Calculate dimensionless frequency parameter ωω = aa 0xx = aa 0yy = aa 0zz = aa 0xxxx = 2ππ TT ωωωω VV ssss ωωωω VV ssss ωωωω VV ssss ωωωω VV ssssss
Calculating Point Springs & Dashpots 13 aa 0yyyy = aa 0zzzz = ωωωω VV ssssss ωωωω VV ssssss Step 7: Calculate Dynamic Stiffness Modifiers αα xx = 1.0 αα yy = 1.0 αα zz = 1.0 αα xxxx = 1.0 0.2 0.4+ LL BB aa 0zz 2 10 1+3(LL BB 1) +aa 2 0zz 0.55+0.01LL BB 1 aa 0xxxx 2 0.4 2.4 (LL BB) 3+aa 0xxxx 2 αα yyyy = 1.0 αα zzzz = 1.0 0.55aa 2 0yyyy 1.4 0.6+ 2 (LL BB) 3+aa 0yyyy 0.33+0.03LL BB 1 aa 0zzzz 2 0.8 1+0.33(LL BB 1) +aa 0zzzz 2
Calculating Point Springs & Dashpots 14 Step 8: Calculate Translational and Rotational Stiffness The elastic static stiffness at the ground surface is calculated as follows: kk xx,ssssss = GG xx BB 2 νν 6.8 LL BB 0.65 + 2.4 kk yy,ssssss = GG yy BB 6.8 2 νν LL BB 0.65 + 0.8 LL + 1.6 BB kk zz,ssssss = kk xxxx,ssssss = kk yyyy,ssssss = GG zz BB 1 νν 3.1 LL BB 0.75 + 1.6 GG xxxx BB 3 1 νν GG yyyy BB 3 1 νν 3.2 LL + 0.8 BB 3.73 LL BB 2.4 + 0.27 kk zzzz,ssssss = GG zzzz BB 3 4.25 LL BB 2.45 + 4.06 The dynamic stiffness of the embedded footing is calculated as follows: kk xx = αα xx ηη xx kk xx,ssssss kk yy = αα yy ηη yy kk yy,ssssss kk zz = αα zz ηη zz kk zz,ssssss kk xxxx = αα xxxx ηη xxxx kk xxxx,ssssss kk yyyy = αα yyyy ηη yyyy kk yyyy,ssssss
Calculating Point Springs & Dashpots 15 kk zzzz = αα zzzz ηη zzzz kk zzzz,ssssss Step 9: Calculate radiation damping ratios and damping coefficients ψψ = 2 (1 νν) (1 2νν) ; ψψ 2.5 Damping ratios are calculated as follows: 4[ψψ(LL BB ββ zz = )+(DD ee BB)(1+LL BB (kk zz GG zz BB) 4[LL BB+ ββ yy = (DD ee BB)(1+ψψLL BB kk yy GG yy BB 4[LL BB+ ββ xx = (DD ee BB)(ψψ+LL BB ββ zzzz = kk yy GG yy BB )] )] )] aa 0zz aa 0yy aa 0xx 2αα zz 2αα yy 2αα xx 4 3 3LL BB DDee BB +ψψll BB 3 DD ee BB +3LL BB 2 DD ee BB +ψψdd ee BB +LL BB 3 + LL BB aa 0zzzz 2 kk zzzz GGzzzzBB 1.4 1+3(LL BB 1) 0.7+aa 0zzzz 2 aa 0zzzz 2αα zzzz ββ yyyy = 4 3 LL BB 3 DDee BB +ψψdd ee BB 3 LL BB +3DD ee BB LL BB 2 +ψψ LL BB 3 aa 2 0yyyy 4 3 LL BB +ψψdd ee BB 3 kkyyyy GGyyyyBB 3 aa0yyyy 2αα yyyy kk yyyy GGyyyyBB 3 1.8 1+1.75(LL BB 1) +aa 0yyyy 2 +
References 16 ββ xxxx = 4 3 DDee BB +DD ee BB 3 +ψψ LL BB DD ee BB 3 +3 DD ee BB LL BB +ψψll BB aa 0xxxx 2 4 3 ψψll BB +1DD ee BB 3 kkxxxx GGxxxxBB 3 aa0xxxx 2αα xxxx kk xxxx GGxxxxBB 3 1.8 1+1.75(LL BB 1) +aa 0xxxx 2 The damping coefficients are calculated as follows: + cc xx = cc yy = cc zz = cc xxxx = cc yyyy = 2(ββ ss +ββ xx )kk xx ωω 2ββ ss +ββ yy kk yy ωω 2(ββ ss +ββ zz )kk zz ωω 2(ββ ss +ββ xxxx )kk xxxx ωω 2ββ ss +ββ yyyy kk yyyy ωω cc zzzz = References 2(ββ ss +ββ zzzz )kk zzzz ωω National Institute of Standards and Technology. Soil-Structure Interaction for Building Structures, NISTGCR-12-917-21 (Gaithersburg, MD, 2012).