コンクリートの圧縮強度と引張強度の確率分布と寸法効果に関する研究

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コンクリートの圧縮強度と引張強度

の確率分布と寸法効果に関する研究

小池扶千朗 @ 加藤善之助

In the case of calculating the probability distribution of strength of r巴inforcedconcrete members by Mont巴Carlosimulation by the computer， it is very important to sample the probability distribution of mechanical properties of cach materials. This study巴xaminedthe probability distribution and size e百ectof compr巴ssivestrength of threεkinds of concrete cylinders and four kinds of concrete prisms and direct tensile strength of four kinds of concr巴te pris;ns and splitting tensile strength of three kinds of concrete cylinders， using four kinds of concrete mix proportions having each di妊εrentmaximum sizes of aggregates

The experimental valuεof strength shows the probability distribution quite close to the straight line when plotted eith巴ron w巴ibullprobability pap巴ror on normal ones， but some values are slightly apart from the straight line near the maximum and minimum巴xperimentalvalues But， experimental values of coe伍cientof variation CV of strength of concrete show the lower

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is gamma function. Therefore， the probability distribution of its strength cannot always be expressed by the w巴ibulldistribution. Compressive strength of concr巴tedecreases with decrease in size of specimen， both in prism and cylinder specimen， and with increase in size of aggregate. On the other hands， tensile strength of concrete reaches the top at specimen size of 10cm both in prism and cylinder specimens， but it rath巴rdecreases when specimen size becomes smaller than 10cm

203 1. INTRODUCTION Study on valiability of streogth and deformation of reinforced concrete members is particularly import -ant to discuss the structural safety. Recently， papers simulated these variabilities by Monte Carlo method have been reported by many research巴rs CRef. 1 -7J . In the case of calculating size巴ffectsand the probability distribution of flexural and shear strength of reinforced concrete members by Monte Carlo simulation by the computer， sampling the probability distribution of compressive and tensile strength of concrete being one of the materials such as rein forcing bars and concrete composing reinforced con crete members has a very important m己amng

well known that the probability distribution of con crete strength follows a Weibull distribution when co町 reteshows a perfectly brittle fracture mode [19， 20，15 and 21J， but Tanigawa， Yamada and Yokoyama showed in their日xperiments that the probability distribution of concrete strength could not always be expressed by a Weibull distribution [Ref.22J . Recent -ly， making allowance for these facts， some of the failure probability models are proposed for the materials which can not be expr巴ssedby Weibull distribution [Ref. 23， 21， 18， 24 and 25J， and it is expected to invεstigate the validity of these failure models.

Tests and theories on size effect of compressive strength of concrete have been examined by many researchers since 1925 [8-18J ， while no generally accepted theories and experimental equations for predicting size effects exist at pr芭sent.G邑nerally，it is On the other hand， tensible strength of cor町 etehas a significant influence on several important physical properties such as flexural and shear cracking load and creack patt巴rns，shear strength且ndbond strength of deformed bars in reinforced concrete members [Ref. 18J . Many researchers have purs田d their studi邑son tensile strength of concrete using many

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kinds of test arrangment [Ref. 26-33， 34-39， 40-45J But， resεarches refering to size effect呂swell as to the prob呂bilitydistribution of tensile strength of concrete in direct or indirect tensile t巴stare not sufficient at present [Ref 3. 4-39J It is necessary to pursue the reseaches on thes己probabilitydisrtibution in order to simulate the prob旦bilitydistribution of flexural and shear cracking load and ultimate shear strength of reinforced concrete members by Monte Carlo tech mque

This study examined size effects and the probabili -ty distribution of direct tensilεstrength of concrete with four kinds of maximum size of aggr巴gateby lazy

Table 1 Outline of experime孔 lOAg.series 15Ag.series 20Ag.series 25Ag.series (1) Test Specimen Prism tens日巴 specimensare plain concrete prisms reducεd central parallel section with enlarged ends T

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5 D:::4.46 PR-1-15.0 PR-T-9朗 PR~下.7，'t5PR+4絹 tongs grips method [Ref. 40J and indirect t巴nsile strength by cylinder splitting test， fabricating four kinds of prism specimens for direct tensile test and three kinds of concrete cylinders for splitting tensile test， and examined those of compressive strength of concrete by prism and cylinder specim巴ns，and offer ed data to simulate occurence of the probability distribution of concrete strength used

2. EXPERIMENTAL PROCEDURE

The experiment was carried out in accordance with thεtest program as shown in Table 1

m 附肝油 Cubic specimen Compressive test

亘日画

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7.5 x 15，φ10x20 and

φ15 x 30 cm were cast芭d for compressiv巴test and splitting tentile test， where G占isdiameter of cylinder.

Th色variablesin the expεriment are as follows: four different sizes of tension prism specimens (d=4 46， 7.25， 9.68 and 15.0 cm) and three different sizes of splitting and compressive cylinder specimens (φ=7.5，

10.0 and 15.0 cm) were also prepared to obtain the properties of concrete used.

(2) Fablication and Curing of Specimen

Ordinary portland cement， Yahagi river sand and Tεnryu river gravel wεre used for concrete. The propξrties of aggreg抗告sused are shown in Table 2

Mix proportions of four kinds of concrete are shown Table 2 Properties of aggregate Kind 01J Kind 01 IAgg元ateSpecific Water Fineness concrete! aggregateI Slze gravity absorption modulus (醐) 24hrs.(%) river gravel 1O ~2.5 2.65I 0.99 5.57 lOAg.series_r_i_v_e_r_s_a_n_d ₂_.₅₈ I 1.56 1.2~ 2.95 river gravel 15~ 5 2.65 0.93 6.25 15Ag.se口出 river sand 1. 2~ 2.58 1.56 2.95 river gravel 20~ 5 2.66 0.90 6.57 20Ag.series river sand 2.5~ 2.58 1.50 2.95 river gravel 20~5 2.66 0.90 7.00 25Ag.series river sand 2.5~ 2.51 1.80 2.58

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コンクリトの圧縮強度と引張強度の確率分布と寸法効果に関する研究 205 Tabl巴3 Mix proportion of concret巴 Kind of Size of gravel Water

一

Sand concrete (mm) (kg/m') Ckg/m') I (kg/mづ 10Ag.series 10~2. 5 230 383 659 15Ag.series 15~ 220 I 367 708 20Ag.series 20~ 210 350 759 25Ag.series 25~ 350 739 in Table 3 and water-cement ratio (w/c) of concrete was 60% by weight. Four kins of maximum sizes of aggregate (sieve dimension= 10， 15， 20 and 25 mm) were prepared as inclusion， respectively

Prism specimens for compressive test having steel mold at both ends were cast in wood mold holizontal ly. Each concrete specimens werεfabricated most carefull， so as to placεthe aggregates as inclusion in concrete molds with巳qualdensity. Cylinder speci mens stored in a labolatory during 48 hours after casting， th巴n they were remolded and cured in m01sture room at a t巴mperatureof 20"士l"C and a relative humidity of over 80% until just b巴forethe test during six weeks

(3) Method of Loading and M田surement

The loadings and supports were accomplished with the s旦mesize of plates as specimens both prisms and cylinders and spherical seats molded to the same scale as the test specimen used in comperessive test Direct tensile test technique was used for specimens with enlarged ends to which load was applied purely by friction using four kinds of lazy tongs grips shown in Fig. 2 [Ref.40-45]. Generaly， it is more suitable for testing a large number of di妊erentsiz巴sof speci -m巴ns.T otal of 240 prism specimens wεre tested in compressive test， 210 cylinders in compressive， 480 prisms in direct tensile and 270 cylinders in splitting，

respectively

Longitudinal strain (ε) wasm四suredby two strain gauge type deformation transformers attached to the specimen (meansured length= 1.8D) in compr巴sSlve test and was measured by wire resistance strain gauges (gauge length= 60 mm) in direct tensile test 3. TEST RESULT AND DISCUSSION

Table 4.1 and 4.2 show the actual dimensions of specimen after removing mold and show the test results， where電'size of specimen" indicates mean

value (1) Fracture Distribution Direct tensile test specimens have a hight of three times its depth (d) in central parallel t芭stlength司The incidence of fracture was very greater in the top parts of test length in the case of PR-T -15.0 seriεs sp巴ci -mens，and was vεry greater in the central 2d part in the case of another sizes of specimens.J ohnston and Gravel Ckg/m') 1015 1004 996 996 山 j Design Measured C /vl) I AirC /vl) SI泊a叩n州1

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10Ag 10_10-CY-SP-~1Oo。-ccCYy y-sSsyPp--44d7ll 51l_{1 2}25750 1₅ ₁₉_.₉₉₁₂₄5_.0₀₆619.5~38.11 _{24.0~33 .4 1} ₂2₉6_..₈7_'₁1 4₂_._₅2₉4_I115₈._.8₇9₀_o 0_9₉5₈4₈1 6₁₁₂₁悩₅_XX_--₄2₁3_..₇2₁4₁16₁₂_.₁₅似₁₁₁5_.₁似₅₁_-₀0962268496098131176_.₉₇₂₁₂₉₈₁₃₀₀₁_-₀_.₉₆₁₁₃_.₃₉_D_.₁₀₆ senes 唱10-CY-SP-oI5 25 114.98130.0618.5~28 01 24.01 5_731 9.97 0.983110 .44X-33. 65110 .441 9.441-0.966124.012.791-0.95213.1710.124 15 Ag 15-CY-SP-~7 5 24 7.50 15.11 17.4~35. 9 26.5 5_17 19.52 0.977 5.56X-18.62 5.56 4.56 0.987 26.4 5.84 0.992 3 _ 25:0 222 15-CY-SFFd-41l 0 25 9.99 20.1323.3~36.8 30.0 3.87 12.88 0.986 8. 38X-28. 95 83817.38 0_ 993 30.1 38 0.990 3.40 149 senes _15-CY-SP-~15 ₂₅ ₁₁₅₀₃₃₀_.₁₅ 2 1.1 ~33.3 27.0 3.01 11.15 0.978 9. 73X-32. 55 9.731 8_73 0.985包7.13.44 0.986 3.29 128 20 Ag 20-CY-SP廿7_5 22.0~3 1.3 26.31 2.50 9.48 0.986 11. 15X-36 . 94 11.15110_15 0.989 26.42 r 8 3.27 111 20-CY-SPP--4a10 199991mω 25.0~36.3 29.5 I 3.36 11.40 0.944 9 _ 20X-31.62 9 201 8.20 0_970 29.6 3 _ 971-0 . 980 3.38 130 senes ₂₀_-_C_Y_-_S_P -! l i15 19 114.96 !30 0422.4~29.9 25_6 i 2.05 8.00 0.965 13 . 19X-43 _ 25 13_19112.19 0.986 25.6 2.381-0.991 3.24 092 25 Ag. ₂2₅5-_-C_CY_Y-_-S_S_PP-_-O_O7.5 19I749lM 20 .4~33.3 7.39 0.995 27.0 lO 19 1 9.9712005246~34.4 29.81 2.88 9651l00 191094X3761 1094 9.94 0.987 29.8 3 . 341-0 . 988 3.39D 112 senes ₂₅_-_C_Y_-_S_P_φ₁₅ ₂₀ ₁₁₄₉₇₁₃₀₁₄_17.4~27.7 ₂₃_.₇₁₃_.₀₀₁₂_.₆₆₁₀₉₈₅₁₈_.₁₉_X_-₂₆_.₃₉₁₈_.₁₉ 7.19 0_ 982 23.7 3491-0.974 3 16D.154

(5)

コンクりートの圧縮強度と引張強度の確率分布と寸法効果に関する研究 207 Table 4.2 Test result (2)

Kind of Notation of Size of Bending concr巴te SpeClmei1 speClm巴n(cm) span(cm) 10Ag lO-PR-B-IO.O 10.17 x 9.97 30 senes 10-PR -B-15.0 15.12 x 15.03 45 15Ag 15-PR-B-IO.O 10.21 x 9.96 30 senes 15-PR←B-15.0 15.10 x 15.03 45 20Ag 20-PR-B-I0.0 10.18x 10.00 30 S巴nes 20-PR-B-15.0 15.15 x 15.05 45 25Ag 25-PR-B-I0.0 10 . 00 x 10. 04 30 senes 25-PR守B-15.0 15.02x 15.13 45 lN(ーLN(トP)) lNトLN(1-P)) 10司PR-C 15-PR-C 2か PR-C Z5-P件〈 lN(ーlN(l-P)) lN←lN(トP)) 10-PR-T 15-PR-i 20-PR-T 25-PR-T LN(-LN(いP)) 1 .0 ー1.0 -2.0 3.0 Fig.3・1 Relation between non-failure probability ln(-ln(l-P)) and strength ln(F) (Weibull distribution)

Modulus of Cubic compressive rupture(kg/cm') strength(kg / cm') 32.9 35.9 30.7 34.1 33.1 24.6 34.0 31.2 315 324 303 324 326 303 318 296

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(6)

208

P=-l

一一ー…E・....一・・・ (1) N+1 where， P: failure probability N : the total number of specimens When a material shows perfectly bittle fracture mode， the probability distribution of strangth follows a Weibull distribution [Ref. 19， 20， 15 and 21J，the relation between non-failure probability 1n(ー1n(1-P)) and the strength 1n(F) is expressed by the straight line expression (Weibull distribution). In(ーIn(l-P)) = In(A)+

(

s

+

1)・In(F) ・・ー(2) where， P: failure probability βparameter related to properties of con crete A: a constant determined by kind of materials， circumstances， size of speci -men， etc. F : strength of specimen The slope of straight line (β+ 1) does not reated to the quantity of tdetects.In this study， it was assumed that the experimental errors were small and distributions of that errors were uniform in the large range of D/d The straight lines obtained by the least square approximation of experimental values are shown in Fig. 3.1 and 3.2， and the expressions of these straight lines and correlation coefficients are shown in Table 4.1 The results of tests show correlation coefficient of 0.85-0.99 in prism compressive specimens， 0.94-0.99 in cylinder comperssive， 0.96-0.996 in prism direct tensile and 0.94-0.99 in cylinder splitting. In the case of prism direct tensile test， correlation coe伍cientis nearest to1.Fig. 3.2 shows the probability distribu -tion of test results protted on Normal probability paper.Fig. 3.1 and 3.2 show that distributions of experimental values are quite close to the straight line， but some values are slightly apart from the straight line near the maximum and minimum values This matter requires futher examinations to discuss the structural safety. In order to propose the pro -bability distribution for adequate indication of distri -bution of exerimental values， it is necessary to accumulate more experimental data. 1 ) Effect of size of specimen

Fig. 4 and Table 4.1 show the relation between material constant (β) and spacimen size (S).βwas calculated by the following formula form "a" (slope of the straight line drawn by the probability distribu -tion of strength calculated by the method of reast squares on the weibull probability paper. β = a-l 一一…・・…・・・・・(3) Fig. 4 shows the tendency that the value of βincrease with increse of S (size of spacimen; prism spec. : S= prism width， cyclinder spec. : S=cylinder diamater). N agamatsu stated that the value ofβwas constant [Ref. 15J . However， Fig. 4 shows that βis greatly

affected by the size of aggregate in concrete or by the size of specimen， and not constant. Hoshino and Tomeji showed thatβwas 5.25 for the direct tensile strength of mortar [Ref. 38Jー 2) Effect of Size of Aggregates Fig. 5 shows the relation between material constant (β) and width (diameter) (D) of spacimen to size of aggregate (d) ratio (D/d)， in the tensile strength of concrete. In the case of PR -T specimen (prism speci men in direct tensile test)，仕levalue ofβincreases straight up to D/d = 10， but after D/d exceeds 10， it shows tendency to decrease In the case of CY -SP specimen (cylinder splitting specimen)， the value ofβhas the tendency to increase up to D/d= 10 while showing considerable variabia bility， but after D/d exceeds 10， it shows the tendency to decrease same as PR-T specimen. (3) Coe伍cientof Variation of Strength Coe伍cientof variation of strength (CV) was calcu -lated by the following formula CV=

去

where， CV: coeflicient of variation Fi: measured value of strength F : mean value of strength N: total number of specimens 1) E妊ectof size of specimen -・・(4) Fig. 6 shows the relation between coefficient of variation of strength (CV : %) and size of specimen (S). The value of CV of compressive strength of prism specimen decreases greatly with increase in prism width S in the range whear S is 4.46cmー7.25cm，butin the rang where S is larger than 7.25cm， the tendency of decrease suddenly becomes small.This tendency coincides well with the result of experiments reported by the auther [Ref. 7J . The value of CV of compressive strength of cylinder specimen in the range where s is "'7.5-φ15 decrease continuously with increase in S. The value of CV of the direct tensile strength of prism specimen d巴creasesgreatly with increase in S in the rang where the prism width (S) is 4.46ー7.25，but after the value of S exceeds 7.25 each series of concrete show very little decrease and show constant value. However.the specimen of which maximum size of aggregates is smaller， shows the smaller values of CV. On the other hand， the value of CV of the spling tensile strength of cylinder specimen decreases greatly with increase in S in the range where cylinder diameter (S) is 7.5cm 10cm， but after exceeding this rang the value of CV decreases very little and shows almost constant value. Hoshino reported that the variability of direct tensile strength was larger than the variablility of splitting tensile strength [Ref. 37J . However， according to this experimental results， the value of

(7)

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7.510 IS Cylinder diameter S(cm) Fig. 4 Relation between mat巴rialconstant(s)and

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(8)

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(9)

コンクリートの圧縮強度と引張強度の確率分布と寸法効果に関する研究 211 cuFc ( 均叫 Cu:lic叩 P閣 slve ゐ∞ιstrer唱th Fb 均加

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15X15X53 10 15 20 25 内謁regatesize d (mm) Fig. 13 Cubic compressive strength and modulus of rupture of prism specimen CV of direct tensile and splitting showed similar tendency and the variability of splitting strength showed slightly larger value. 2) Effect of Size of Aggregeate

Hoshino， Johnston and sidewell， Sabnis and Mirza， et al reported that the specimen with aggregate of large size showed larger variability of tensile strang th in the splitting tensile test [Ref. 37，43，18，46]. However， the experimental result of this time showed that specimens of 20 Ag. series and 25 Ag. series had smaller variablility than specinens of 15 Ag. series and 10 Ag. series. More researches are needed in this area. Johnston reported that the specimen with aggregate of large size showed larger value of CV of direct tensile strength in the direct tensile test， which concides with the experimental results of this time [Ref. 43J. Fig. 7 shows the relation between values of CV of direct tensile strength and Djd of prism specimen The value of CV shows tendency to decrease with increase inDjd in the hyperbolic shape， but the rela -tion between CV and Djd can not be indicated in one formula because it is a任ectedby size of aggregate.

3) E百ectof Value of β

Fig. 8 shows the relation between CV and β. The value of cofficient of variation (CV) can be calculated by the following theoretical equation [Ref. 15，21J

/

r

(1十一三一} ""''[T _-.j/

、

担キl' r- ~._1 …… "'(5)

V r

，、

2(1+ーニー}_β+1' where，

r

:

garr盟国function Nagamatsu reported that coefficient of variation CV of strength was a任ectedonly by material constantβ，

not a妊ectedby size of specimen and proporsed eq. 5. However， it is obvious in Fig. 8 that the experimental values are larger than the theoretical value in both prism compressive strength and prism direct tensile strength. Therefore， the probability distribution of its

strength can not always be expected by a Weibull distribution [Ref. 22Jー

(4) Mean Value of strength

Fig. 9 shows the relation between strangth (F) and size of specimens (S)ーCompressivestrength of prism specimens decreases straight when size of specimen becomes smaller， and strangth decreases in parallel with increase in size of aggregate in concrete. These tendencies were already confirmed by the auther's previous study [Ref. 7J The direct tensile strength of prism specimen shows considerab!e increase with decrease in size of speci -mens S in the rang where S is 15.0-9.68cm， but this increase of direct tensile strength reaches the top when S is smaller than 9.68 and it rather decreases when S is smaller than 7.25.

These strengths of specimens of aggregate size 20 Ag. series and 25 Ag. series showed tendency to become lower than these strangth of 10 Ag. and 15 Ag. series when the value of S become smaller.On the other hand， as shown in Fig. 9， splitting tensile strangth of cylinder specimen showed a little di任er

-ence according to the size of aggregate in case of φ15cm cylinder-the specimen with aggregate of large size showing slightly low strength， butin .case ofφ10 and φ7.5cm cylinders， any di百erenceof strangth by size of aggregate was not recognized. Besides， the value of splitting tensile strength increases greatly with decrease in the value of diameter S when S is 15 -10cm， but with S = 10cm as border ，when S decreases from 10cm to 7.5cm strangth rather falls. On the other hand， Subnis and Mariza repored "Mirza [Ref. 46J tested series of cylinders， cast form the same model concrete， ranging from lin.X 2in. (25mm X 50mm) to 6in.X 12in. (150古田1X 300mm) in splitting tensile tests，

and the mean strength and the standard deviation were found to decrease with an increase in size of specimen [Ref. 20J. This difierence will be studied here after.

Fig. 10 shows the relation between strength (F) and the value ofDjd.Compressive strength (F) of prism specimen and cylinder specimen shows tendency to increase with increase inDjd and in the case of same value ofDjd， the larger the size of specimen iS，the slightly larger value concrete shows.

On the other hand， with same Djd value， direct tensile strength of prism specimen showed that the specimens of 20 Ag. and 25 Ag. series had consider -ably lower value than the specimens of 10 Ag. and 15 Ag. series. The experiment of this time shows that tensile strangth of concrete is not determined only by Djd， but also affected greatly by size of aggregate. Fig. 11 show the relation between relative concrete strangth (lnFjF15) and relative volume of specimen (1n V 15

!

V

)

plotted on the logarithmic graph (both co -ordinates) where F15and V15 are the strangth and volume of S=15.0cm series of prism specimen， res

(10)

-pectively. According to th邑Weibull'sweakest statis. tical theory and the stochastic theory for the perfect -ry brittle fracture mode， size effect of strength can be writtl巴nas follows ~ =

，

Y

(

o)l!(β十1) ...・(6) Fo ¥ V!

where， F 0 and Vo are strength and volum of standard speClmen.βis material constant

According to the above formula， relation between 1n(F!F 0) and 1n(Vo!V) is indicated on the straight line wlth the slope -1!(β十1)

In Fig. 11， relation between comprεssive strength of prism and volume of specimen is shown nearly by the formula of straihit line regardless of aggregate size except secimen of S=4.46cm， and the above formula is almost a妊ected，but in the case of direct tensile strength， it can not be considered as straight line. The value of direct tensile strength shows tendency to increase on the contrary when the value of specimen size becomes larger than the certain value Fig. 12. shows relationship between prFt/F5P ratio (pr Ft=direct tensile strength of prism， F5p=splitting tensile strength of cylinder) and size of aggregate. In Fig. 12， values of PτFt7日!F5P</>

，

.5' prF t9.68!F 5Pφ10 and pr Ftl5.0/FSP</>15 were calculated， supporsing the size of prism S and diameter of cylinder φwere equal for convenience' sake. Fig. 13 shows the cubic compres sive strength calculated from the mean value of巴ach six specimens and the modulus of rupture calculated from the mean value of each three specimens， for referrence 4. CONCLUSION The following are the conclution of study on probability distribution and size effect， with tests of compressive strength， direct tensile strength and splitting tensile strength using different sizes of aggregate and specimen which are main factors to determine strength of concrete

1) The experim巴ntalvalue of strangth shows the probability distribution quite close to the strainght line when plotted either on Weibull probability papers or on Normal ones， but some values are slightly apart from the straight line near the maximum and minimum experimental values 2) The value of material constant βshows tendency to increase with increase in size of specimen S. The value of βis largely affectεd by thεsize of aggregate in concrete and by the size of specimen， and cannot be considered as constant value 3) Coefficient of variation for strength in compres sive test showed gradual increase with decrease of sp巴cimensize in the range of 15cm to 10ー7cm，but it showed greatly increase when the specimen size becomes smaller than the range of 10-7cm 4) The value of coe伍cientof variation (CV) for tensile strength showed same tendency as compres sive strength， in both cases of direct tensile test and splitting test 5) The value of coe伍cientof variation (CV) for tensile strength of concrete in direct t巴nsiletest shows larger v旦lueswhen the size of aggregate in the specimen is larger 6) Experimental values of coefficient of variation (CV) of strength show the lower values than theoretical ones indicated by the formula (5). Ther邑forethe probability distribution of its streng -th cannot always be expressed by the Weibull distribution. 7) Compressive strength decreases with decrease in size of specimen， both in prism and cylinder speci -mens， and with incr巴asein size of aggregates 8) Tesile strength of concrete reaches the top at specimen size of 10cm both in prism and cylinder specimens， but it rather decreases when specimen size becomes small巴rthan 10cm. 9) The formula (6) is almost effected in compressive test of prism specimen， but the formula (6) is not effected in direct tensile test AC区NOWLEDGEMENT The author is greatly indebted to Messrs.M.Asai，

Y.Kato， K.Mizuno， M.Iwata， A.Kokuanin， K Nakanishi， N.Ito， K.Ueda， A.Hasegawa，孔 S.Uek王awa孔

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