Proposal of shear strength formula including proportion effects of shear planes YAMAGUCHI Kazuhiro1, KOBAYASHI Yoshihiro2, INAYAMA Masahiro3 ABSTRACT: In this research, we try to estimate shear strength including proportion effects with our proposing empirical formula of shear strength. For practical structural calculation, we made shear strength formula simple. We replaced stress concentration effects with proportion effects between width and height of a shear plane in the formula. Shear tests were examined in four methods. First tests are horizontal loading tests on beam-column joints with buried tensile bolts. Second tests are monotonic loading tests on joints with buried tensile bolts. Third tests are shear tests specified in the Japanese Industrial Standards (JIS Z2101). Fourth tests are double shear tests with parameters including the proportion of a shear plane. We proposed empirical formula of shear strength, based on results of fourth tests.

KEYWORDS: Shear strength, Proportion effects, Size effects, Empirical formula, Tensile bolts 1 INTRODUCTION 1 2 3

In this research, we try to estimate shear strength including proportion effects with our proposing empirical formula of shear strength. In our previous research [1–6], we had verified analytical model of joints with buried tensile bolts, and had proposed its design formula. Beam-column joints tests, column-base joints tests and portal frame tests had been examined. It has shown that the design formula has estimated stiffness and yield strength well. Meanwhile, the design formula has not estimated ultimate strength of shear failure mode, because the formula did not have parameters including stress concentration effects. For practical structural calculation, we made shear strength formula simple. We replaced stress concentration effects with proportion effects between width and height of a shear plane in the formula. The formula could be applied to design various joints of wood structure (bevelled housing etc.) (Figure1). The formula and the previously proposed design formula could estimate elasto-plastic behaviour of joints with buried tensile bolts (Figure 2).

Figure 1: Bevelled housing 1 YAMAGUCHI Kazuhiro, Graduate School of Agricultural and Life Sciences, The University of Tokyo, 1-1-1,Yayoi, Bunkyo-ku, Tokyo, Japan.113-8657 Email: yamakm5@yahoo.co.jp

Column Depth

Bea

m D

epth

d

Center of tensile bolts

Neutral Axis

C

T T

xp

Moment

Dis

tanc

e bet

wee

n C

ente

rs o

fTe

nsio

n an

d Co

mpr

essio

n

2 KOBAYASHI Yoshihiro, Graduate School of Agricultural and Life Sciences, The University of Tokyo, 1-1-1,Yayoi, Bunkyo-ku, Tokyo, Japan.113-8657 Email: arch@archkoba.com 3 INAYAMA Masahiro, Graduate School of Agricultural and Life Sciences, The University of Tokyo, 1-1-1, Yayoi, Bunkyo-ku, Tokyo, Japan.113-8657 Email: ainayama@mail.ecc.u-tokyo.ac.jp Figure 2: Beam-Column Joints with buried tensile bolts

2 EXPERIMENT 2.1 HORIZONTAL LOADING TESTS ON BEAM-

COLUMN JOINTS WITH BURIED TENSILE BOLTS

Table 1 shows specification of test specimens used in this experiment. In each specification, six replications were prepared. For column and beam members, glulam having a grade of JAS E120-F330 (MOE=12kN/mm2, Fb=32.4N/mm2) Scotch Pine. Figure 3 shows a test set-up of beam-column joint. This configuration represents the upper half and lateral half portion of portal frame subjected to a horizontal load. Loading protocol applied was ± 1/450, ± 1/300, ±1/200, ±1/150, ±1/100, ±1/75,±1/50, ±1/30 rad. and finally Pmax then returns back to zero. In this experiment, three cycle repeat was adopted within each peak deformation.

Table 1: Specification of test specimens

SpecimenCode Name

Length ofShear Plane

Width ofShear Plane

Depth ofShear Plane

H-L180*W54*D54-1-2-3-4-5-6

180mm 54mm 54mm

H-L240*W54*D54-1-2-3-4-5-6

240mm 54mm 54mm

225

3320

390

2930

Beam : Scotch Pine Glulam E120-F330 105*450*2950

Nut F16mm(S45C)

28 672

1482

.514

47.5

Loading Point

*10

*13

*20

Washer: 54*54*9‚”(SPHC)

Tensle Bolt: 16mm(SNR400B)

*14

*2

*4

*7

*11*12

*19

*15*16

*1

*3

*6 *5

*8 *9

20

2950

50

105

24001425

180

or 2

40

Cent

er o

fBe

am D

epth

20

180

or 2

40

Joint using forSecond Test

30558012030 80

12

104

60

55

60

Tensle Bolt: 16mm (S45C)

Nut: 16mm(S45C)16mm Anchor Bolt

Joint using for Second Test

Column : Scotch Pine Glulam E120-F330 105*390*2400

Inflection Pointof Beam

Infle

ctio

n Po

int

of C

olum

n

M16 Anchor Bolt*17*18

Figure 3: Test set-up for horizontal loading tests on beam-column joints specimen

2.2 MONOTONIC LOADING TESTS ON JOINTS WITH BURIED TENSILE BOLTS

Figure 4 shows sampling of test specimens. The specimens were sampled from beams used for horizontal loading tests on beam-column joints. Table 2 shows specification of test specimens used in this experiment. Figure 5 shows a test set-up of joints with buried tensile bolts. The specimens were fixed to the testing machine with six 12m diameter bolts. Relative displacements between joints were measured using two transducers.

Beams used for horizontal loading testson beam-column joints

Monotonic loading test specimens

Figure 4: Sampling of test specimens

Table 2: Specification of test specimens

Specimen Code NameSpecimen Code Name

(HorizontalLoading Tests)

Length ofShear Plane

Width ofShear Plane

Depth ofShear Plane

M-L180*W54*D54-1 H-L180*W54*D54-1

M-L180*W54*D54-2 H-L180*W54*D54-2

M-L180*W54*D54-3 H-L180*W54*D54-3

180mm 54mm 54mm

550

6018

0

6022.5

30

60

105 210

310

22.5

Scotch Pine GlulamE120-F330

Nut : 16mm (S45C)Washer: 54*54*9‚” (SPHC)

Tensle Bolt: 16mm (S45C)

Monotonic Load

Nut : 16mm (S45C) Load Cell

Figure 5: Test set-up for monotonic loading test

2.3 SHEAR TESTS SPECIFIED IN THE JAPANESE INDUSTRIAL STANDARDS (JIS Z2101)

Figure 6 shows sampling of test specimens. The specimens were sampled from beams used for horizontal loading tests on beam-column joints. Figure 7 shows dimensions of JIS block test specimens. Table 3 shows specification of test specimens used in this experiment. In each specification, three replications were prepared. Figure 8 shows a test set-up of JIS block test specimens. A universal test machine (Shimadzu co.) was used for the test. Loading speed was 1mm/min. The push-out displacement was measured by transducers.

Beams used for horizontal loading testson beam-column joints

JIS block test specimens

First outer layerof glulam

Second outer layerof glulam

30 30

Figure 6: JIS Block test specimens

30

20 20

10

20

90 45 0

Figure 7: Dimensions of JIS block test specimens

Table 3: Specification of test specimens

Test Length ofShearPlane

Width ofShearPlane

Specimen Code Name(Horizontal

Loading Tests)

Layerof glulam

Demention

JIS blocktest

20mm 20mmH-L180*W54*D54-1H-L180*W54*D54-3H-L180*W54*D54-2

First outerSecond outer

90450

Specimen Code Name (e.g. J-L20*W20-1F90-1)

Figure 8: Test set-up for JIS block test specimens

2.4 DOUBLE SHEAR TESTS WITH PARAMETERS INCLUDING THE PROPORTION OF A SHEAR PLANE

Figure 9 shows size of test specimens. Each specimen was made of Douglas fir with MOE of 10.6kN/mm2, mean density of 490kg/m3 and average moisture content of 14.1%. Table 4 shows specification of test specimens used in this experiment. We estimate effects of the length of shear plane by tests of L30W30-L540W30, and effects of the width of shear plane by tests of L180W30-L180W120. In each specification, three replications were prepared. Figure 10 shows that specimens of L30W30-L540W30 are sampled from a lumber. Figure 11 shows a test set-up of specimens with parameters including the proportion of a shear plane. The specimens were fixed to the testing machine with six 12m diameter bolts. Relative displacements between joints were measured using two displacement transducers.

Table 4: Specification of test specimens

L W 30 60 90 12030 L30W3060 L60W30

120 L120W30180 L180W30 L180W60 L180W90 L180W120240 L240W30300 L300W30360 L360W30420 L420W30480 L480W30540 L540W30

900

60

60

L

W150

Figure 9: Size of test specimens

Figure 10: Sampling of test specimens

Figure 11: Test set-up for double shear tests

3 RESULTS AND DISCUSSION 3.1 HORIZONTAL LOADING TESTS ON

BEAM-COLUMN JOINTS WITH BURIED TENSILE BOLTS

Figure 12 shows the relationship between moment and rotational angle. Figure 13 shows shear strength and shear failure behaviour. Figure 14 shows failure behaviour. Table 6 shows Shear strength and characteristics value in horizontal loading tests.

0

5

10

15

20

25

30

35

40

0 50 100 150

Mom

ent

(kNm

)

Rotational Angle (*E-3rad)200

H-L240*W54*D54-1H-L240*W54*D54-2H-L240*W54*D54-3H-L240*W54*D54-4H-L240*W54*D54-5H-L240*W54*D54-6H-L180*W54*D54-1H-L180*W54*D54-2H-L180*W54*D54-3H-L180*W54*D54-4H-L180*W54*D54-5

H-L180*W54*D54

-1

H-L180*W54*D54

-2

H-L180*W54*D54

-3

H-L180*W54*D54

-4

H-L180*W54*D54

-5

shear strength

101.23 kN 72.58 kN 72.12 kN 85.28 kN 63.41 kN

Failure behaviour

Shear plane

Shear plane

Figure 13: Shear strength and Shear failure behaviour Figure 12: Relationship between moment and rotational

angle

Shear Failure of beam Fracture of tensile bolt (Tensile bolt : S45C) (Tensile bolt : SNR400B)

Push out Push out Pull out Pull out

Figure 14: Failure behaviour

Table 6: Shear strength and characteristics value in horizontal loading tests

Push outPmax5)

Pull outPmax4) K M1/120 Mu Mmax θy θv θu θmax μ Ds S

(kN) (kN) (kNm/rad) (kNm) (kNm) (kNm) (*E-3rad) (*E-3rad) (*E-3rad) (*E-3rad) ── ── (*E-3kNrad)H-L240*W54*D54-1 82.77 76.12 1724.95 18.40 33.11 37.70 15.12 19.19 173.65 173.65 9.05 0.24 5431.46

-3 89.23 78.33 1796.18 19.13 33.28 38.09 14.70 18.53 156.90 156.90 8.47 0.25 4913.76-4 92.23 78.12 1641.03 17.80 33.22 37.56 16.14 20.24 147.27 145.40 7.27 0.27 4556.06-5 79.32 75.12 1458.03 16.72 33.43 37.53 17.66 22.93 184.58 184.58 8.05 0.26 5787.11-6 93.48 81.94 1499.57 16.64 33.50 37.96 17.54 22.34 174.75 174.75 7.82 0.26 5479.57-7 94.45 82.37 1570.12 17.63 33.58 38.59 16.82 21.39 176.13 176.13 8.24 0.25 5554.96

ave1) 88.58 78.67 1614.98 17.72 33.35 37.91 16.33 20.77 168.88 168.57 8.15 0.26 5287.15

SD2) 6.19 2.96 130.82 0.96 0.18 0.40 1.23 1.75 13.91 14.50 0.60 0.01 459.01

5 percent TL3) 69.06 69.33 1202.63 14.69 32.79 36.64 12.44 15.26 125.05 122.88 6.26 0.22 3840.36H-L180*W54*D54-1 101.23 71.50 974.17 12.99 25.32 29.34 18.27 25.99 142.08 142.08 5.47 0.32 3268.29

-2 72.58 75.58 1215.03 14.20 25.41 28.87 14.75 20.92 121.55 121.55 5.81 0.31 2823.29-3 75.12 71.10 1024.31 12.58 25.16 29.20 17.14 24.56 146.68 146.68 5.97 0.30 3381.63-4 85.28 71.19 1287.23 13.98 25.88 28.34 14.61 20.11 114.94 114.94 5.72 0.31 2714.50-5 63.41 68.31 1029.58 11.04 23.09 26.58 15.63 22.43 113.04 113.04 5.04 0.33 2351.17

ave 79.52 71.54 1106.06 12.96 24.97 28.47 16.08 22.80 127.66 127.66 5.60 0.31 2907.78SD 14.42 2.60 136.59 1.26 1.09 1.12 1.59 2.46 15.67 15.67 0.36 0.01 420.95

5 percent TL 45.84 65.46 787.00 10.00 22.44 25.84 12.37 17.05 91.05 91.05 4.75 0.29 1924.431) ave : average, 2) SD : standard deviation, 3) 5 percent TL : 95 percent lower limit in 75 percent confidence level, 4) shear failure of beam, 5) fracture of tensle bolt

Specimen Code Name

3.2 MONOTONIC LOADING TESTS ON JOINTS WITH BURIED TENSILE BOLTS

Figure 15 shows Failure mode. Figure 16 shows Shear strength and shear failure behaviour.

Figure 15: Failure modes

Figure 16: Shear strength and Shear failure behaviour

3.3 SHEAR TESTS SPECIFIED IN THE JAPANESE INDUSTRIAL STANDARDS (JIS)

Table 7 shows results of .JIS block test. Figure 17 shows the relationship between shear strength and density.

y = 1.7134x + 1.0333R² = 0.2983

0

2

4

6

8

10

12

14

16

0 2 4 6 8

Shear stren

gth  τ(N/m

m2 )

Density (kN/m3)

Side (Shear failure) Center (Splitting failure)

Figure 17: Relationship between shear strength and density

M-L180*W54*D54

-1

H-L180*W54*D54

-1

M-L180*W54*D54

-2

H-L180*W54*D54

-2

M-L180*W54*D54

-3

H-L180*W54*D5

-3

shear strength

101.23 kN 65.67 kN 72.58 kN 70.90 kN 72.12 kN 74.10 kN

Failure mode

Side(Shear failure)

Center(Splitting failure)

Side(Shear failure)

Side(Shear failure)

Center(Splitting failure)

Center(Splitting failure)

Failure behavior

Shear plane

Shear plane

Table 7: Test results 3.4 DOUBLE SHEAR TESTS WITH PARAMETERS INCLUDING THE PROPORTION OF A SHEAR PLANE

Pmaxarea of

shear plane density

(kN) (mm2) (kN/m3)

J-L20*W20-1F90-1 13.43 5.47 407 5.84

J-L20*W20-1F90-2 11.17 4.39 393 5.23

J-L20*W20-1F90-3 12.41 4.81 388 5.74

J-L20*W20-1F45-1 8.88 3.47 391 5.36

J-L20*W20-1F45-2 10.23 4.08 399 6.01

J-L20*W20-1F45-3 9.54 3.63 380 5.23

J-L20*W20-1F 0-1 11.89 4.76 400 6.06

J-L20*W20-1F 0-2 10.91 4.36 400 5.73

J-L20*W20-1F 0-3 10.09 4.05 402 5.38

J-L20*W20-1S90-1 9.39 3.57 380 4.92

J-L20*W20-1S90-2 9.67 3.92 406 5.60

J-L20*W20-1F90-3 9.71 3.83 394 5.38

J-L20*W20-1F45-1 7.97 3.05 383 4.54

J-L20*W20-1F45-2 8.45 3.30 391 4.61

J-L20*W20-1F45-3 8.46 3.34 395 4.58

J-L20*W20-1F 0-1 8.18 3.25 397 5.74

J-L20*W20-1F 0-2 10.08 3.96 393 5.57

J-L20*W20-1F 0-3 9.22 3.60 390 4.98

J-L20*W20-2F90-1 11.01 4.34 394 4.96

J-L20*W20-2F90-2 9.94 3.90 392 5.01

J-L20*W20-2F90-3 10.83 4.23 391 5.01

J-L20*W20-2F45-1 9.91 3.90 394 4.91

J-L20*W20-2F45-2 8.30 3.34 402 4.86

J-L20*W20-2F45-3 8.95 3.53 395 4.88

J-L20*W20-2F 0-1 7.77 3.07 395 4.96

J-L20*W20-2F 0-2 8.67 3.28 378 4.63

J-L20*W20-2F 0-3 8.54 3.19 374 5.07

J-L20*W20-2S90-1 9.85 3.91 397 5.16

J-L20*W20-2S90-2 10.62 4.23 398 5.21

J-L20*W20-2F90-3 9.18 3.60 392 4.60

J-L20*W20-2F45-1 9.59 3.69 385 4.56

J-L20*W20-1F45-2 8.65 3.35 387 5.01

J-L20*W20-2F45-3 9.73 3.87 398 5.34

J-L20*W20-2F 0-1 8.28 3.23 390 5.12

J-L20*W20-2F 0-2 9.98 3.83 384 5.12

J-L20*W20-2F 0-3 10.35 3.92 379 4.79

J-L20*W20-3F90-1 12.47 4.92 395 5.12

J-L20*W20-3F90-2 10.58 4.30 407 4.99

J-L20*W20-3F90-3 10.19 4.02 395 4.96

J-L20*W20-3F45-1 9.17 3.75 409 4.81

J-L20*W20-3F45-2 9.28 3.73 402 4.81

J-L20*W20-3F45-3 11.31 4.47 395 5.02

J-L20*W20-3F 0-1 9.09 3.79 417 4.42

J-L20*W20-3F 0-2 9.06 3.65 403 5.06

J-L20*W20-3F 0-3 9.60 3.94 410 5.13

J-L20*W20-3S90-1 7.44 2.98 401 4.50

J-L20*W20-3S90-2 11.62 4.76 410 5.33

J-L20*W20-3F90-3 11.34 4.70 415 5.55

J-L20*W20-3F45-1 9.07 3.67 405 4.97

J-L20*W20-3F45-2 9.47 3.90 412 4.97

J-L20*W20-3F45-3 8.64 3.54 410 5.04

J-L20*W20-3F 0-1 9.60 3.80 396 5.31

J-L20*W20-3F 0-2 9.25 3.54 382 4.54

J-L20*W20-3F 0-3 9.49 3.89 410 4.46

Specimen Code Name

10.95

12.34

9.01

9.59

10.96

shear strengthτ(N/mm2)

average

9.55

9.16

9.32

9.54

9.55

10.13

9.06

9.45

9.88

11.08

9.92

9.25

8.29

10.08

9.32

10.59

9.05

8.32

9.58

Figure 18 shows relationship between shear strength and length of shear plane. Figure 19 shows relationship between shear strength and width of shear plane. Figure 20 shows shear failure behaviour. Empirical formula (1) of shear strength, based on linear regression in Figure 18

Length of shear plane: under 300mm 2.51P L W= × 2.51A= (N)

Length of shear plane: over 300mm (0.442 662)P L W= + × 0.442 662LW W= + 0.442 662A W= + (N)

(1)

P(N): shear strength L(mm): length of shear plane W(mm): width of shear plane A: (mm2): area of shear plane

y = 2.5187xR² = 0.8424

y = 0.4426x + 662.67R² = 0.0958

0

200

400

600

800

1000

1200

0 60 120 180 240 300 360 420 480 540

Pmax

/ W

idth

of s

hear

pla

ne (N

/mm

)

Length of shear plane (mm)

experiment(ave)experiment(L30W30-L540W30)

Figure 18: Relationship between shear strength and length of shear plane

y = 2.4599xR² = 0.9369

0

50

100

150

200

250

300

350

0 30 60 90

Pmax

/ L

engt

h of

shea

r pl

ane

(N/m

m)

Width of shear plane (mm)

120

experiment(ave)experiment(L180W30-L180W120)

Figure 19: Relationship between shear strength and width of shear plane

4 CONCLUSIONS

We proposed empirical formula of shear strength, based on results of double shear tests with parameters including the proportion of a shear plane. REFERENCES [1] Inayama M, Yamaguchi K, Miyaya Y : “Study on

structural design method of timber semi-rigid joints drawn with tensile bolts”, Summaries of technical papers of Annual Meeting Architectural Institute of Japan. C-1, Structures III, Timber structures steel structures steel reinforced concrete structures, pp.355-356, 2008

[2] Miyaya Y , Yamaguchi K, Kawahara S, Inayama M, : “Study on Wooden Portal Frame Structure with Precut Systems (Part1) Development and Analysis ” , Summaries of technical papers of Annual Meeting Architectural Institute of Japan. C-1, Structures III, Timber structures steel structures steel reinforced concrete structures, pp.389-390, 2008

[3] Yamaguchi K, Miyaya Y , Kawahara S, Inayama M, : “Study on Wooden Portal Frame Structure with Precut Systems (Part2) Lateral Loading Test”, Summaries of technical papers of Annual Meeting Architectural Institute of Japan. C-1, Structures III, Timber structures steel structures steel reinforced concrete structures, pp.391-392, 2008

[4] Iezumi R, Yamaguchi K, Nakamura M, Kawahara S, Inayama M, : “Analysis and Tests on Timber Semi-rigid Joints Drawn with Tensile Bolts (Part1) Tests on Beam-Column Joints” , Summaries of technical papers of Annual Meeting Architectural Institute of Japan. C-1, Structures III, Timber structures steel structures steel reinforced concrete structures, pp.97-98, 2009

[5] Nakamura M, Yamaguchi K, Iezumi R, Kawahara S, Inayama M, : “Analysis and Tests on Timber Semi-rigid Joints Drawn with Tensile Bolts (Part2) Elemental Tests”, Summaries of technical papers of Annual Meeting Architectural Institute of Japan. C-1, Structures III, Timber structures steel structures steel reinforced concrete structures, pp.99-100, 2009

[6] Yamaguchi K, Nakamura M, Iezumi R, Kawahara S, Inayama M, : “Analysis and Tests on Timber Semi-rigid Joints Drawn with Tensile Bolts (Part3) Analysis ” , Summaries of technical papers of Annual Meeting Architectural Institute of Japan. C-1, Structures III, Timber structures steel structures steel reinforced concrete structures, pp.99-100, 2009 Figure 20: Shear failure behaviour