Loading...
No Permit Sculpture (MISC)City of La Quinta Arts Society - Granite Monolith Sculpture by Roger Hopkins ' Outside of Omri & Bonnie's Restaurant Washington Street - La Quinta, California Engineer: Dennis S. Wish, PE ' 54-625 Avenida Bermudas, La Quinta CA. 92253 (760) 564-0884 (Phone and Fax) Artist: Roger Hopkins Sculptures 44489 Town Center Way — Suite D, Palm Desert CA. 92260 . Q�OFESS/0, ~ DENNIS S. WISH r NO.C-041250 m , .. IA C� I l a t Project La Quinta Arts Society - Stone Monoliths Job Ref. 0314 Section Sheet nolrev. California Professional Engineer 54-625 Avenida Bermudas Installation of Monoliths S-1 La Quinta, California 92253 Catc. by Date • Chck'd by Date App'dby Date 760-564-0884 (phone 8: fax) DSW 5/30/03 DSW 5/30/03 DSW ' 5/30/03 City of La Quinta Arts Society - Granite Monolith Sculpture by Roger Hopkins ' Outside of Omri & Bonnie's Restaurant Washington Street - La Quinta, California Engineer: Dennis S. Wish, PE ' 54-625 Avenida Bermudas, La Quinta CA. 92253 (760) 564-0884 (Phone and Fax) Artist: Roger Hopkins Sculptures 44489 Town Center Way — Suite D, Palm Desert CA. 92260 . Q�OFESS/0, ~ DENNIS S. WISH r NO.C-041250 m , .. IA C� I l a t _ . I e� Project La Quinta Arts So aety -Stone Monoliths Job Ref. 0314 Section Sheet no./rev. California Professional Engineer 5x-625 Avenida Bermudas Installation of Monoliths S-2 La Quinta, California 92253 Calc. by Date' Chck'd by Date App'd by Date 760-564-0884 (phone & fax) DSW 5/30/03 DSW 5/30/03 DSW 1 5/30/03 DESIGN ASSUMPTIONS AND METHODS f The Stone Monoliths started as a project for Omri and Bonni's restaurant under construction on Washington Street in the city of La Quinta. The project was funded by the La Quinta Arts Society and consisted of three granite. monoliths weighing approximately 6,500 -pounds each with a roof or"Cap-stone" that weighed 13,000 -pounds each. The vertical stones are pyramidal in shape and the base is approximately 5 -feet x 2 -feet with each stone varying in shape within. these dimensions. They taper to a near point — often no more than 4 or 6 -inches in width and a 2 to 3 inches in depth. They extend approximately 6.5 -feet above the finished grade (although, again, each stone varies by nature).. The stones are relatively thin in the weak direction. The shape used in this analysis package for the design of the foundation is assumed to be an Obelisk that represents the approximate limits of the exterior dimensions. They are as they would appear in nature although the stones are intended to provide the viewer with the impression that they exist naturally and are protruding from deep in the ground. In reality, the stones are cut with a flat base and sit upon a foundation to distribute the weight of the stone to soil below and also upon a Pile or Cantilevered Column. foundation intentionally designed to help control the motion of the stone in a moderate earthquake. Life Safety Issues The sculpture is intended to draw the public to the artwork and allow them to interact with it — move around it, sit on some of the stones, consider the possibility of climbing on the smaller surrounding stones, pushing against the monoliths etc. The interaction of the work with the public raised great concerns for me as to how I was going to design the structural system so as to protect those who would be interacting with it while a seismic event may occur. The bottom line is that the stones are unreinforced monoliths as they appear in nature and will crack, fracture or fail in the event of a strong motion earthquake. There is no practical means to tie the stones together or internally reinforce them without impairing the aesthetic value of the sculpture. Furthermore, the Art Society and the Artist are adamant that the public be allowed access to interact with the art. The only solution possible, given the constraints of the project is to attempt to design the foundation and connection of the monoliths to the foundation in such a manner as to control the movement of the stones and attempt to keep them rocking without "shearing"- or fracturing long enough to allow people to get safely away. Controlled Rocking Mode The retrofit of Unreinforced Masonry taught us that there are three modes of motion during an earthquake on URM masonry piers — shear strength, rocking mode and failure. The shear resisting elements of URM materials are allowed to rock *in the hopes that rocking modes will not lead to overturning of the piers and that in cyclical dynamic motion the subsidence of the gradual ground motion will bring the piers back to stability. Shear design attempts to balance rocking modes with strength of the masonry at the grouted joints (head and bed joints as well as the keying of the wythes of masonry as a unit. Shear failures lead to total failure, but generally occur where the piers } e� -L(JCll�dr�.�l�/G¢�L���i. Project La Quinta Arts Society -Stone Monoliths Job Ref. 0314 Section Sheet no./rev. Califomia Professional Engineer 54-625 Avenida Bermudas Installation of Monoliths, S-3 La Quinta, California 92253 Calc. by Date Chck'd by Date App'd by Date 760-564-0884 (phone & fax) DSW ' 5/30/03 . DSW 5/30/03 1 DSW 5/30/03 ° are incapable of rocking because they are either too long in relationship to their height or are too narrow (such'as a column. Analysis in strong and weak direction for stones Weight of stone; Wd, =6500 lbs Height of stone; H.= 6.5 ft Lateral weight of stone; Fim = 0.46xWd, = 2990.00 lbs Applied at approx. 1/3 of H; H,at = H/3 = 2.17 ft Assumed base of stone; L= 5 ft; D = 2 ft; (Rocking on edges of stone) Overturning in strong direction;. Mot = H,atxF,at = 6478 Ib—ft �cJ 001 �� � ,J#� Y• Resisting Moment; Mras = Wd,xU2 =16250 Ib ft Therefore — rock is stable in strong direction: Check weak direction; Mres2 = Wd,xD/2 = 6500 Ib ft There is not a sufficient factor of safety (1.5 times the Overturning moment) to prevent the stones from falling over in the event _ of a strong motion earthquake, however, if the stones are not narrower than 24 -inches at the base they will most likely rock without failing. Prevention of Overturning t' Should the stones overturn in the weak direction we have taken precautions to prevent or minimize the collapse by pinning the stones to the foundation at the center of gravity where shear would be equal to zero force. The.reason for this is that if we attempt to prevent the stones from rocking, the uplift would be sufficient to disintegrate the ' stones by pulling out the Holddown device. In the weak direction, the uplift for would be too close to the center of mass to be of any value in resisting uplift. A2 -inch diameter SAE 1018 bar stock is being used to connect the stone to a 24° diameter Sonotube filled with a ; ' light slurry concrete mix. The other end of the rod is secured T-0" minimum into a clean cored hole (3°) up through the center of the stone base. To prevent the bar stock from shattering the stone, the bar will only be held in place by a non-drying, non -shrink elastomeric type product (Deco-SealTm) The.end of the stock in the Sonotube is intended to break up the slurry mix in the event of an earthquake and dampen the acceleration of ground motion to prevent the shearing or fracturing of the stone. - The goal is to allow the stones.to rock while accommodating their failure in the bending of the bar stock (as it pulls out of the Deco-Seal"m and slowly lower the stone toward the ground allowing for people to get safely away, The stones themselves will be 2 -inches below grade and will'bear on a 12 -inch thick pad that will distribute their ` weight to the soil below. 7„ `r 4' e1� Project La Quinta Arts Society - Stone Monoliths Job Ref. 0314 Section Sheet, no./rev. California Professional Engineer 54-625 Avenida Bermudas Installation of Monoliths S-4 La Quinta, California 92253 Calc. by Date- Chdc'd by Date App'd by Date .760-564-0884 (phone & fax) DSW 5/30/03 DSW 5/30/03 DSW 5/30/03 CONCLUSION: It is my professional opinion that the design set forth for this sculpture has considered the life safety of the public while accommodating the forces of nature in compliance with the 97 UBC (R _value of 2.2). The stones were treated like columns or flag poles'and the overturning calculations were done based upon conservative -values. The Stones connections to their foundations are designed to provide time for people to get safely away from the. structures but not to prevent their breakage.or collapse. The use of bar -stock steel is intended to dampen and slow ^the rate of decent of the rock should an earthquake cause it to overturn. There is no guarantee or means to anticipate the effect of the sculpture on life safety without resorting to elaborate base isolation systems similar to those used in museums. The concept of base isolation, motion dampening to accommodate movement so as to avoid fracturing the stone and causing failure was fully. considered in this design in order to protect the safety of the public. r Furthermore, the original design required a 13,000 pound cap stone which was negotiated out of the artwork due to i its impracticality of maintaining a useful connection of the vertical stones to the cap stone in the event of an earthquake (similar to Stonehenge). The Commission and artist were advised by this office to create a barrier that would prevent the public from interactingwith the art - yet this was considered an important part of the project and was finally rejected. Based upon the full constraints of this project, the attached design represents, in my professional opinion, the most practical solution to the problem of insuring the public safety while providing a foundation (hidden) for these Granite monoliths. - r 1 ; ^ Title: Job # " Dsgnr: Date: 9:21 PM, . 29 MAY 03 Description.: Scope Rev: 560100 Ser Kw -060390, Ver 5.6.1. 25 -Oct -2002 General Footing Analysis & Design T Page -1 (c)1983-2002 ENERCALC Engineering Software . c:tprogram filesker-55X0314.ecw:Calculations �i Description Gravity load support for 6,500 lbs granite stone,(traiangular Oblesk) General IrlfOrrttatlOn Calculations are designed to ACI 318-95 and 1997 UBC Requirements Allowable Soil Bearing 1,500.0 psf - - r: Dimensions... Short Term Increase 1.330 , ; Width along X -X Axis - ; ` 5.000 ft Seismic Zone 4 ; Length along Y -Y Axis ? `• 4.000 ft ' ►• Biaxial Applied Loads Footing Thickness `d '' 12.00 in a " V Live & Short Term Combined Col Dim. Along X -X Axis 0.00 in "Pc ` .4 ,"' 2,000.0 psi ' Col Dim. Along Y -Y Axis.0.00 in Fy -60,000.0 psi r Base Pedestal Height ;,' 0.000 in 'Concrete Weight 145.00 pcf • o + Min Steel /o Overburden Weight 20.00 psf 0.0014 Rebar Center To Edge Distance 3.50 in 3 Y xLoads - Applied vertical Load... t Dead Load 6.500 k r e ...ecc along X -X Axis. 0.000 in t r 7 . Live Load k, .ecc along Y -Y Axis .. ..0.000 in .�4 ` " `g .Short Term Load k :.�' I r .: Creates Rotation about Y -Y Axis : r' Creates Rotation about X -X Axis- ' Applied Moments... (pressures @ left & right) "F Y• . (pressures @ top & bot) + ,f 'Dead Load ,Live Load k -ft , , k -ft ' k -ft , "k -ft .. ;` Short Term. L5 .* k -ft ' ., 5 , k -ft' Creates Rotation about Y -Y Axis , , , Creates Rotation about X -X Axis 6 { Applied Shears.. ; w' (pressures @ left & right) „s (pressures @ top & bot) +' ` i Dead Load .� k Live Load k F ..- k ` Short Term k ''4 k - Summary. -' Footing Design OK '12.Oin '. 5.06ft x 4.00ft Footing, Thick, w/ Column,Support 0.00 x 0 OOin x O.Oin high ` " 5.00ft x 4.00ft Footing, .12.Oin Thick, w/ Column Suppo • - ... ' ' - DL+LL DL+LL+ST `' * > Actual ' • Allowable , Max Soil Pressure' - 490.0 490.0 psf Max Mu 1'.422 k -ft per ft " Allowable 1,500.0 1,995.0 psf £ Required Steel Area 0:143 in2per ft '71 J t, 1 'X' Ecc!of Resultant ".0.000 in • 0.000 in t �: s ' - ` "Y' Et cof Resultant +� 0.000 in 0.000 in _Shear Stresses.... Vu rF . Vn Phi tY F • 3 _ f� 1-Wa -� y .. 7.952. 76.026 psi X -X Min. Stability Ratio No Overturning t" 1:500 `' ; 2 -Way '`'. 30.738.' 152.053 psi ' + -1 +Y -Y Min. Stability Ratio No Overturning - Footing Design Shear Forces ACI 9-1 ACI 9-2 , , * ACI 9-3 Vn ' Phi , " . I Two -Way Shear y 26.22 psi 30.74 psi 19.76 psi 152.05 psi ' One -Way Shears... f Vu @Left 7.95 psi 7.95 psi '-'" ,� a 5.11 psi .. . 76.03 psi ' Vu @Right w 7.95 psi 7.95 psi 5.11 psi 76.03 psi , Vu @',Top 6.04 psi 6.04 psi 3.88 psi 76.03 psi -`� •� + Vu @,Bottom 6.04 psi 6.04 psi fX 3.88 psi 76.03 psi . ` - • y ' . ' Moments ACI 9-1 ACI 9-2 ACI 9-3 Ru / Phi As Read Mu @' Left 1.42 k -ft ,'• 1.42 k -ft , 0.91 k=ft 21.9 psi ' . -0.14 in2 per ft ' L Mu @ Right .1.42 k -ft p 1.42 k -ft "., r 0.91 k -ft 21.9 psi 0.14 in2 per ft ' Mu @ Top ' 0.91 k -ft 0.91 k -ft ' 0.58 k -ft " _ 14.0 psi 'y ` 0.14 in2 perft. . Mu @' Bottom 0.91 k -ft . •-0.91 k -ft , .. '; 0.58 k -ft .14.0 psi" , ^ F 0.14.in2 , per ft � ' . Y J$ ..t .. t • r - ti4f i .� s i � � r •c y .. ^ K � '1:.• , G- .. us t A : F ? �"L:F Y x +• -.r ��, ih � '?, t �'i � r' :. { {� . w Title : }: • } Job # ' Dsgnr: Date: 9:21 PM, 29 MAY 03 •. ". ,, Description Scope Rev: 560100 User: KW -060390, Ver 5.6.1, 25 -Oct -2002 General Footing Analysis Pae 2 &Design 9 • (c)1983 2002 ENERCALC Engineering Software c:kprogram fileskec55\0314.ecw:Calculations ' Description Gravity load support for 6,500 lbs granite stone (traiangular Oblesk) . Soil Pressure Summary Service Load Soil Pressures Bottom- Top -Rig Bottom- • . Top-Lef ;�' i• ,�' DL + LL 490.00 490.00 ? , , 490.00 ,"° 490.00 psf • DL + LL + ST 490.00 490.00 : 490.00 490.00 psf Factored Load Soil Pressures a ACI Eq. 9-1 _ 686.00 686.00 ..686.00 686.00 psf - ACI Eq, 9-2 686.00 686.00 686.00' 686.00 psf ACI Eq. 9-3 441.00 441.00 441.00 `�. 441.00 psf ' ACI Factors (pe(ACI, applied internally to entered loads) ACI 9-1V& 9-2 DL 1.400 ACI 9-2 Group Factor 0.750 UBC 1921.2.7 "1A" Factor' 1.400 ACI 9-1!& 9-2 LL 1.700 ACI 9.3 Dead Load Factor 0.900- UBC 1921.2.7 "0.9" Factor 0.900 'ACI 9-2 ST 1,700 ACI 9-3 Short Term Factor 1.300' + J - i .;.:seismic= ST a : 1.100 p, ''� .. Jr r` '� .. _ � {F.. 'r..',.- kms.. ,� ' •f `'( • a"FJr' $, r { - i. � - � .. ,. 1. > • , . . r 'rte J� Ft 2 ,.-�. ,, ••,.t 4.t+a� .Y - -s J. „V v ' t '., .. - ` j .. '+- �.'a. «L ,4' and � / -`14 '�-',y , . • •' .. , C` 1� w � �` H ',� i. .� r Y• •.e �• f i a •k, - .. of ' r' e;� ..+ ' r,•• + t'• �,; ' { � �+ �'1� 'r ,� '.�� - � .. ,.. t .. .. f, n ... •r • *+ _' -,• �e;� � _ r s..el' f ° i' .� .a'y�.''�y Y •<a,,t t ' `• `, � f• - r,� a i• • •, - Rev: 560100 User: KW -0603 ,] (c)1983-2002 E rincn w•i n4i Title t Job # V, Dsgnr: ' : Date: 9:22PM, 29 MAY 03 ` Description Scope. er5.6.1,25-opt-2002 Pole Embedment in Soil Page 1 .ALC Engineering Software c:iprogram files\ec5510314.ecw:Calculations foundation support for. 6,500.lbs - consider 46% for R=2.2 to resist lateral support in the side to side motion General Information I E AllowjPassive 250.00 pcf Applied Loads... , Max Passive 1,500.00 psf a Point Load 3,000.00 lbs .• Load duration factor , 1.330 , distance from base .2.500 ft , t Pole is Circular t Diameter - 24.000 in Distributed Load . k 16.00 #/ft 1 Restrained @ Surface , distance to top 3.000 It distance to bottom 6.500 ft , Summary Moments @ Surface.,,. Point 7,500.00 ft-# Total Moment 7,234.00 ft-# r Distributed load,, 266.00 "� t ,,Total _ Lateral > 2,944.00 lbs ;, n 1 P. '' - With Surface Restraint... ,` „r . _ Req'd Depth 3.589 ft - Pressure @ Base.. y Actual 1,193.37 psf r Allowable 1,193.37 psf "' : �.. tAV Surface Restraint Force 5,673.41 lbs . a . �' - tit., � _. '. ',a 1 .:y , �• ;�• # , ' . . 1 e : 4. 3 (� ` E i. s t. - " .. :. xi ' ` - a ,.' � „ � d W �. ` • .. .. ,r• � . . i a YY 1 t h rh,, ' ,4 •+i •• .. .f } , ` Y r , �' lir. 9 3 �.. - 1.7 •?..' ] 1H�{:�•. •,+ '• t••� - `R+- -. � r� i �r. R.. tom. ., REVISIONS REV DESCRfiMON DATE -. APPROVED 4 01 01 F 1'. FROM ACTUAL EDGE OF STONE - 'TYP. 15 RE84R O 12' O.C. BOTH WAYS F -T. -T 2 APPROXIMATE SHAPE OF STONE MONOLITH OLa 2" DIA. SAE1018 SOLID BAR W STOCK 0 I 2' � Q a 24" DIA. SONOTUBE W/ SLURRY CONCRETE MIX TO HELP DAMPEN MOTION OF BAR STOCK DURING a' SEISMIC EVENT - TO 4'-0" BELOW GRADE. SEE SECTIONS -11"vi n.,l L ALL, CONCRETE TO BE 2,000 4APPROXIMATE PSI CONCRETE TOP OF CONCRETE TO BE 2 -INCHES r BELOW TOP OF GRADE SO AS TO BE HIDDEN FROM VIEW. Dennis S. Wish, PE - Structural Engineering. } 54-625 Avenkla Bermudas, La Qubdo CalHomia 92253 - 760.5"OSU OFFICE — - 20&361.5447 E—Fax f PROJECT La Quinta Art's Society . Granite Oblesk foundation for Roger Hopkins DATE DWG NO. REV, NO. SCALE: ENG: OSIV SHEET:.1 OF.: - ,• s + . , r . REVISIONS r1 w. .' ° ` REY ; DESCRIPRON tr DATE APPROVED al AA NOTE. SHAPE OF STONE. VARIES BUT r _ u SHOULD NOT EXCEED THE SIZES SHOWN t ,� t 2 r ON THIS PLAN' p ' �, ! rA' L 84" DIA. 'SONOTUBE W/ CONCRETE SLURRY MIX. • BRACE STONE IN PROPER t Q ` I • ` 4' ORIENTATION ON CONCRETE PIER AT BASE OF. SONOTUBE ' ! AND PLACE BEARING'.` c3 . FOUNDATION AT SAME TIME' \ 12'- THICK CONCRETE Z, •� , { '{> ; ; '' ,FOUNDATION W/ 15 (Fs=60 KSI) REBAR AT 12" O.C. IN r EACH DIRECTION •gym 1<71`< 17177 48 4 // \�/\/\\/j �j\j% �i\j/\\/j\%/\\//\ .°. %\\�j\�j\�/\\//\\�\�\\�j\��\�/ 'Kt. * •fix 2 _ +�' \ \ \ \ \ \ \ \ \ \ 7 t lv �! •'\\\\\/\\\\\\\\\\\\\\/\\\\�\\\\\\\4..r. �' ! V°; i•`F,_+1 is \/\ \ \ \ \ \ \ \.\ \ faE r / y •• ; .. A f• \//\//\\//\//\//\//\//\// \r \/� _ \//\//\//\//\//\//\//\//\// i' i '� /. ,jai M f s'. ry ),y ' ! •' •- ,• 7. e i 4 \\/\\/\/\\/\\/\\/\\/\\/\\/\\/\ .. \/\\/\\/\\/\\/\\/\\/\\/\\/ F f •�� Y .. F`'4'' gi •,tip ,.r•' -. 15 /�\////\// //. // ti �, f r. r> i V r� ✓�? i ` .. . 1 �' I 'c' x NIX `PRE -MADE 18—INCH SQUARE" ` CONCRETE PIER TO TEMPORARY SUPPORT OF THE," , � • , .. ,. ,. - B CK AND STONE UNTIL.' � .� . AR STO A TI SLURRY AND SURROUNDING BEARING PAD ARE PLACED ` NOTE: THE PURPOSE _ OF THE CONNECTION OF,THE MONOLITH TO THE SONOTUBE Dennis S. Wish, PE Structural Engineering FOUNDATION WITH TWO-INCH BAR STOCK 54-625 Avenida Bermudos, La , Qu1Mo Collfomlo 92253,1 IS TO ALLOW THE STONE TO ROCK IN 760.564,0664 OFFICE_ - 20&361.5447 E—fox 1, { �" f THE EVENT OF AN EARTHQUAKE. THE USE r OF A SLURRY MIX AND THE DECO=SEAL' PROJECT • CONNECTION OF 2—INCH BAR STOCK IS r j INTENDED TO DAMPEN THE ENERGY ' La Quinta Art s SOLI®! CAUSING THE ROCKING SO THAT THE ;GronRe.06lesk foundation ,for Roger Hopkins ! y "' hg� • STONE MAY ROCK AND NOT SHATTER AT �• REy .NO THE BAR CONNECTION. DAZE DWG No. ' r + K SCALE. .wry i ENG: DSW J ? SHEET 2 F . OF: ..x i& .. '} m �' r :1 ., ! .'ti .ate t ,� '�, 'r._. r s-., 'a '.h• �'. ..' REVISIONS . s L l REV DESCRIPTFON DATE APPROVED • GRANITE OBLESK SHAPE 'VARIES i . •„ ' WITH EACH STONE AS DOES THE , "+ ,E RELATIVE HEIGHT. - ,- 2—INCH DIAMETER SOLID BAR STOCK ' ` z • 4i �, (SAE 1018 OR BETTER) IN 3—INCH m DIAMETER HOLD. CENTER ROD IN HOLE WITH RUBBER. GASKET AT TOP AND BOTTOM . AND FILL CAVl11' WITH DECO—SEAL NON -SHRINK ADHESIVE. a z ; DECO SEAL ALLOWS SOME DAMPENING ; . . „ OF THE BAR STOCK TO PREVENT I 9 SHATTERING OF STONE /N LATERAL ' „ .: + MOVEMENT.` r f 4 x' 7 x 12" THICK 2" F /SHEDy GRADE OVER BEARING PAD'(SEE (SEE PLAN :. r" _'CONCRETE FNDT. - VIEW) W/ 14 REBAR AT 12 O.C.. EACH WAY, y 41i "e. XX y/may/xxx may/may/\° i \j/\\j/\\j/\\j/\\� ., j .,/ „/ . ; 24 DIAMETER SONO—TUBE V/V/s t wa \\ \\ \ FILLED WITH SLURRY MIX c. . r Y /x\�X\\//\�// : _ CONCRETE TO ALLOW FOR /\ /\ /\ / // // //' `' ; THE CONCRETE TO BREAK UP /N A 'SEISMIC EVENT — �• � :- + SEE STRUCTURAL , ANALYSIS "{ , .AND RTATTACHED, REPO t PREMADE 18 SQUARE. CONCRETE PIER TO T 2, ` TEMPORARILY SUPPORT THE�` 2—INCH BAR STOCK WITHIN ` k SONOTURE ,,. Dennis S. Wish, PE,.,. Structural Engineering'. 54-625 Avenida Bsimudos, La Quinfa CalHomia 92253, c { i 780.564.0864 OFFICE.' -,.,'20&361.5447 E—fax '"� �. '•1:f PROJECT La 'Quinta Art's SOCletyr { «{ >�`r 'x �OmnRe ObInk foundation for Roger. Hopkins ` t= + �i •DATE, OWG N0. -* �a REV. NO .r # SCALE: 'OSW SHEET. 3 - rOF. , ,1f. C �r. rgr 4 �, + r r K � L, t y• .j � '�"; •,m;4 '{' ,f * va•. , _ y 1 v fir. y v1 . 2v +