Subproject Name

f4f0dfef-c158-4a95-ac32-acd89c363b0f 59 Subproject Name GFDIntano Philipps Technical Consultants Corporation 4 5 decimal

&[DATE] &[TIME] - &[FILENAME]

REFERENCES:

1. ...

SEISMIC PARAMETER

seismic zone coefficient

Project Title...

Z 0.40 :

(Reference or Note to the Reader)

near-source factor

N.a 1.50 :

(Reference or Note to the Reader)

\[TITLE]\

N.v 2.00 :

(Reference or Note to the Reader)

seismic coefficient, Soil Profile D

C.a 0.40 N.a * :

(Reference or Note to the Reader)

C.a 0.6

\[AUTHOR]\

\[DATE[YYYY\002D\MM\002D\DD]]\

C.v 0.56 N.v * :

(Reference or Note to the Reader)

C.v 1.12

DESIGN SPECTRAL RESPONSE ACCELERATION

T.s C.v 2.5 C.a * / :

T.o 0.20 T.s * :

T.s 0.7467

T.o 0.1493

T 03 0.01 range :

T 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2 0.21 0.22 231 mat

T.o 0.149333333333333

T 0 4.149333333333333 T.o range :

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

T s.A 2.50 C.a * 0.40 0.60 T T.o / * + (* T T.o ≤ 2.5 C.a * T T.o T.s lele C.v T / cases :

T s.A vectorize 0.6 1.5 1.5 1.5 1.5 1.5 1.25 1.0714 0.9375 0.8333 0.75 0.6818 0.625 0.5769 0.5357 0.5 0.4688 0.4412 0.4167 0.3947 0.375 0.3571 0.3409 0.3261 0.3125 0.3 0.2885 0.2778 281 mat

Spectrum T T s.A vectorize augment :

spectrum Spectrum . 14 red plot :

Spectrum T.s 2 T.s 0.5 - 2 2 mat spectrum 0.75 1.875 ◄ Ts ... 12 black 1 5 mat 0 1.5 + 200 green 1 5 mat 5 1 sys

basic  Bolean &
Bolean (,)*(,)
'lele' compact T.o T ≤ (T T.s ≤ (& T.o T ≤ (T T.s ≤ (* T T.o T.s lele 31 sys

data Spectrum :

data Σ Finite differences Cumul Integration Y 0 : vx data 1 col : vy data 2 col : Y 1 el 0 : i 2 vx rows range Δx i el vx i el vx i 1 - el - : Y i el Y i 1 - el 12 / vy i 1 - el vy i el + (Δx i el * (* + : 21 line for vx Y augment 71 line :

Σ data Σ :

Σ

Σ Σ rows row 4.032 2.9452 12 mat

Adapt 'backsubstitution' data data 2 data rows range 12 range el :

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

data Deriv data Derivative BackSubstitution X data 1 col : Y data 2 col : 12 mat n X rows : deriv 1 el 0 : deriv n el 0 : 12 mat i 2 n 1 - (range deriv i el 0.5 Y i 1 + el Y i el - X i 1 + el X i el - / Y i el Y i 1 - el - X i el X i 1 - el - / + (* : for deriv 1 el deriv 2 el : deriv n el deriv deriv rows el : X 1 deriv * augment 81 line :

D data Deriv :

Deriv D o 6 black plot :

Deriv D 2 1 sys

For this particular application, FD [Finiite Diferences] is inexact. Usable for more conventional projects

XY Spectrum :

Apply . . . Thiele interpolation

Thiele interpolation is offered for concluding the project.

Generally, Thiele is exceptional extrapolation.

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XY XY 619 range 12 range el :

<p> Thiele modules </p>

X Y Δ Thiele ε 1018 -^: n X rows : i 1 n range M i 1 el Y i el : for j 2 n range i j n range dM M i j 1 - el M j 1 - j 1 - el - : dM dM abs ε < ε M i j el X i el X j 1 - el - dM / : if : 21 line for for i 1 n range c i 1 el X i el : c i 2 el M i i el : 21 line for Δ 0 ≡ c M if 61 line :

X XY 1 col : Y XY 2 col : 12 mat

length Thiele j XY rows : 14

Thiele on length of support points 'j' K X x Cfr N j 1 + el K j 1 + el : c j el N j el : a j el K j el : jj j 1 - (0 range c jj el a jj 1 + el : a jj el K jj el x X jj el - c jj el / + : 21 line for a 1 el 51 line :

recast the Cf on the system  [K,X,x] x Cf K X x Cfr :

Thiele K's coefficients @ support points X X Y 0 Thiele 0.746667 1.5 0.896 0.597333 - 1.045333 1.5 - 1.194667 2.072678 1013^* 1.344 2.538003 1015 -^* - 1.493333 1.231426 1014^* - 1.642667 1.674701 1015 -^* 1.792 3.076374 1012^* - 1.941333 3.227557 1015 -^* 2.090667 1.138182 1013^* 2.24 9.718632 1015 -^* - 2.389333 2.219349 1013^* 2.538667 8.797922 1015 -^* - 2.688 1.197324 1013^* - 142 mat

Thiele K's coefficients. K X Y 0 Thiele 2 col :

K K 3 round vectorize eval :

x λ 1 X 1 el x ≤ (x 5 ≤ (& cases :

Very slow plot x D x Cf x diff :

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

X X eval :

DXY i 1 X rows range DY i el X i el D eval : for X DY augment 21 line :

support XY . 12 black plot :

5 Cf 5 D 21 mat 0.2239 0.0448 - 21 mat

d DXY o 6 black plot :

x Cf x λ * x D x λ * support d 4 1 sys

DXY 0.7467 2.0103 - 0.896 1.3957 - 1.0453 1.0253 - 1.1947 0.7849 - 1.344 0.6201 - 1.4933 0.5022 - 1.6427 0.415 - 1.792 0.3487 - 1.9413 0.2971 - 2.0907 0.2562 - 2.24 0.2232 - 2.3893 0.1961 - 2.5387 0.1737 - 2.688 0.155 - 142 mat

X Clear 1

L H N xd U 0 : dx H L - N 1 - / : 12 mat i 1 N range U i el L dx - (dx i * + : for U 31 line :

L 0.7467 : H 5 : N 40 : 13 mat X L H N xd : 21 line

Export i 1 X rows range expY i el X i el D eval : for X expY augment 21 line :

t0 1 time :

OPTION ... 1. Populate 'X' as convenient 2. Vectorize D(X)... Export.

Export Export eval :

1 time t0 - min 1.3

Export Export rows row 5 0.0448 - 12 mat

Export 0.7467 2.0101 - 0.8558 1.5301 - 0.9648 1.2036 - 1.0739 0.9715 - 1.1829 0.8005 - 1.292 0.671 - 1.4011 0.5706 - 1.5101 0.4911 - 1.6192 0.4272 - 1.7282 0.375 - 1.8373 0.3317 - 1.9463 0.2956 - 122 mat

Recover Thiele ... Export

X 0 0.1493 0.2987 0.448 0.5973 0.7467 0.896 1.0453 1.1947 1.344 1.4933 1.6427 1.792 1.9413 2.0907 2.24 2.3893 2.5387 2.688 2.8373 2.9867 3.136 3.2853 231 mat :

K 1.5 0.597 - 1.5 - 2.0727 1013^* 0 1.2314 1014^* - 0 3.0764 1012^* - 0 1.1382 1013^* 0 2.2193 1013^* 0 1.1973 1013^* - 141 mat :

K X x Thiele K 1 el x X 6 el - K 2 el x X 7 el - K 3 el x X 8 el - K 4 el x X 9 el - K 5 el x X 10 el - K 6 el x X 11 el - K 7 el x X 12 el - K 8 el x X 13 el - K 9 el x X 14 el - K 10 el x X 15 el - K 11 el x X 16 el - K 12 el x X 17 el - K 13 el x X 18 el - K 14 el / + / + / + / + / + / + / + / + / + / + / + / + / + :

x λ 1 X 6 el x ≤ (x 5 ≤ (& cases :

Sanity check K X 0.7467 Thiele K X 5 Thiele 21 sys 1.5 0.2239 21 sys

K X x Thiele x λ * 0 0.75 3 0.75 2 2 mat 2 1 sys K X x Thiele 0 0.75 3 0.75 2 2 mat 1.493 0.75 1 2 mat 3 1 sys

½ magnitude K X x Thiele 0.75 - x 0.75 2 solve 1.4929

X K x expr K 1 el x X 6 el - K 2 el x X 7 el - K 3 el x X 8 el - K 4 el x X 9 el - K 5 el x X 10 el - K 6 el x X 11 el - K 7 el x X 12 el - K 8 el x X 13 el - K 9 el x X 14 el - K 10 el x X 15 el - K 11 el x X 16 el - K 12 el x X 17 el - K 13 el x X 18 el - K 14 el / + / + / + / + / + / + / + / + / + / + / + / + / + :

X K x expr x diff 5580375000000008290800168 - 125 x * + (* 16427 - 10000 x * + (* 19413 - 10000 x * + (* 56 - 25 x * + (* 25387 - 10000 x * + (* 11947 - 10000 x * + (61570016427 - 10000 x * + (* 19413 - 10000 x * + (* 56 - 25 x * + (* 25387 - 10000 x * + (* - 14933 - 10000 x * + (1538219413 - 10000 x * + (* 56 - 25 x * + (* 25387 - 10000 x * + (* - 224 - 125 x * + (455280056 - 25 x * + (* 25387 - 10000 x * + (* 20907 - 10000 x * + (2219325387 - 10000 x * + (* 1197323893 - 10000 x * + (* - (* + (* + (* + (* + (* - 10453 - 10000 x * + (168 - 125 x * + (* 16427 - 10000 x * + (* 19413 - 10000 x * + (* 56 - 25 x * + (* 25387 - 10000 x * + (* + (* 359725387 - 10000 x * + (556 - 25 x * + (* 19413 - 10000 x * + (16427 - 10000 x * + (10453 - 16084 - 125 x * + (* + (* 8010453 - 10000 x * + (* 168 - 125 x * + (* + (* 8010453 - 10000 x * + (* 168 - 125 x * + (* 16427 - 10000 x * + (* + (* 10453 - 10000 x * + (168 - 125 x * + (* 16427 - 10000 x * + (* 19413 - 10000 x * + (* + (* 40010453 - 10000 x * + (168 - 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Some coefficients can't be reduced ...

x expr confrac x convert maple 100083887400000000597 / - 660749291464500567613306956915016297245850980000000590323676819904227438718859249694673039800212776173 x 2^4619045009476457376167912678623213308559817430447294212875551 x * 4619045009399961615244294193695966939819573147031787981086255000 / - 426711534721194256470769210190620791795680515675872528636826793811171684396440406654053819383592723299295277490808432999717068461439090153189225152092725516431325126860929966447264329778757847471280882267485760268031341962664408084487799940020000000 / + 136504217434446420414391811650584114914985739986165413291752249407694963268878287613802979285889700887637558916451989887414732608473362929808563363237715694306208313648718265480206171069107861721333005123169274624428611887387551937891394468507336732258763671875000 x 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