pull-out and shear strength equations for headed …
TRANSCRIPT
69(69s)
PULL-OUT AND SHEAR STRENGTH EQUATIONS
FOR HEADED STUDS CONSIDERING EDGE DISTANCE
Hirokazu HIRAGI1, Shigeyuki MATSUI2, Takashi SATO3, Abubaker AL-SAKKAF4, Shigeru ISHIZAKI5 and Yasuhiro ISHIHARA6
1Member of JSCE, Dr. Eng., Assoc. Professor, Dept. of Civil Eng., Setsunann University
(17-8 Ikeda-Nakamachi, Neyagawa, Osaka 572-8508, Japan) 2Fellow of JSCE, Dr. Eng., Professor, Dept. of Civil Eng., Osaka University
(2-1, Yamadaoka, Suita, Osaka 565-0871, Japan) 3 Student Member of JSCE, Graduate Student, Dept. of Civil Eng., Osaka University
(2-1, Yamadaoka, Suita, Osaka 565-0871, Japan) 4Student Member of JSCE, M. Eng., Graduate Student, Dept. of Civil Eng., Osaka University
(2-1, Yamadaoka, Suita, Osaka 565-0871, Japan) 5Member of JSCE, Engineer, Bridge Design Dept., Sakai Iron Works Co. Ltd.
(3-1, Dezima-Nishimachi, Sakai, Osaka 590-0831, Japan) 6Member of JSCE, Engineer, Bridge Design Dept., Katayama Strutech Corp.
(6-2-21, Minamiokazima,Taisyouku, Osaka 551-0021, Japan)
Studs are often used as shear connectors or anchors between concrete and steel members at various composite structures. As the studs are welded in a finite range of the steel members and each stud generally has different distance from the edge of structural concrete, it can be expected that they do not show the same pull-out and shear strength. In this paper, therefore, the test data of previous investigations including new data obtained by authors and existing formulae for pull-out and shear strength of the stud are reevaluated, and the strength equations were revised in the case of independent of the edge distance. Then for the studs near the concrete edge, new influence factors were found and composed into the revised equations as coefficients. Key Words : stud, shear connector, anchor, ultimate capacity, hybrid structure
1. INTRODUCTION
Recently in Japan, due to intending to simplify the highway bridge system, several new types of hybrid structures composed of steel and concrete members have been actively developed. In order to realize such new type of hybrid structures, it is necessary not only to satisfy the required performance of joint part and/or connection members, but also to derive reliable formulae to evaluate the strength of connection members and establish an accurate design method.
The composite joints of hybrid structures are designated as the jointing part between steel girder and RC pier of a hybrid rigid frame bridge, the jointing part of steel and concrete members at a mixed bridge, anchoring part of steel pier to concrete footing and contact parts between flange plates of main girders and RC-slab at a plate girder bridge system, etc.
In these structural parts, stud shear connecters have been used frequently as jointing elements between steel and concrete. However, when the studs are used in narrow jointing space, it can be seen that the expected load carrying capacity cannot be obtained because of insufficient distance from the edge of concrete. And, in the field of bridge engineering, previous researches on the strength of the stud are limited to the one concerning to composite girder, and almost all of them are on studs welded on the steel girder with sufficient edge distance1). But, there are a few studies on the stud strength considering edge distance, and these are only in the field of architectural engineering2). Therefore, for the design of composite joints mentioned above, it is important to clarify the effects of edge distance on the pull-out and shear strength of the stud.
Under these circumstances, firstly in this research, testing methods adopted in the previous
70(70s)
researches and existing proposals of pull-out and shear strength of the stud are reevaluated, and both the domestic and foreign tested data including new results obtained by authors are collected and rearranged.
And then, based on the statistical analysis of these data, the influence factors on the strength of the stud are clarified, and the estimating equations of pull-out and shear strength of the stud are proposed both in the case of ignoring edge distance and considering them.
However, the estimating equations for pull-out and shear strength of the stud with 6mm diameter are already reported in reference 3). 2. LITERATURE REVIEW (1) In the case being independent of edge distance a) Pull-out strength
Existing equations to estimate the pull-out strength of headed stud are shown in Table 14)-12). These equations are derived by assuming that a stud subjected to pull-out load would fracture with conical shear surface, and can be basically classified into 2-types of which employs either the gross surface area or the effective horizontal projective area of the cone as a parameters for evaluating the strength. And either the shear strength ( '
cf ) or the tensile strength ( 3/2'
cf ) is employed as a concrete strength. Though, basically, the shear strength should be used for the gross surface area of the cone, and the tensile strength should be used for the effective horizontal projective area, such treatments are not done so, in practice.
Photo 1 shows a conical pull-out fracture mode of the concrete. A pull-out strength of the stud is generally derived under the assumption that the conical fracture surface inclination is 45 degrees. However, it can be seen as the photo. that the concrete just under the stud head shows a pure shear mode with the sharp inclination, and as the fracture
face get closer to the concrete surface, the inclination of the cone gets to gradually flat. While the photo shows a result of a specimen with no reinforcement, it can be expected that the fracture face of a specimen with reinforcing mesh will show more complicate. In any case, the inclination angle of the cone does not show 45 degrees as assumed generally, so far.
Within the whole tested data of 294 collected in this investigation as shown in Table 3, 93 data except the data affected by edge distance and broken at the stud shank, are arranged based on the relation proposed by the research group of Lehigh-University as shown in Fig. 1. The proposal by Lehigh- University was adopted entirely in the PCI design hand book. In the figure, the ordinate shows a experimental value and the abscissa shows a calculated value given by the estimating equation. It can be seen that the pull-out strength of the stud with conical fracture surface is accurately expressed with high correlation coefficient of 0.9632. However, the estimating equation is inconsistent because the pull-out strength is estimated by shear strength of the concrete despite of using the effective horizontal projective area of the cone.
Table 1 Existing proposals for pull-out strength of headed stud
Photo 1 Cone fracture mode by pull-out
Fig.1 Comparison of measured pull-out loads with the equation proposed by Lehigh-University
( ) 'fhhd cssh + )mmN10( 3 ⋅×
0
20
40
60
80
100
120
140
160
0 20 40 60 80 100 120 140
Mea
sure
d pu
ll-ou
t loa
d
P
(×1
03 N)
Correlation coefficient(R ) ：0.9632
cssh fhhdP ')(207.1 +⋅=
Proposed by Estimating equations F-value
4) Leigh-University 1.207
5) Sattler 0.935
6) Utescher 0.946
7),8) CEB-ECCS 1.384
9) PCI Design data book 1.207
10) Roik,Bode,Hanenkamp 11.3
11) McMackin 0.272
12) Ohtani 11.3
Ref.Evaluation methods for pull-out strength (<A s ・ f sy )
F :Coefficient，ｈ：Overall hight of the stud(mm)，h s :Length of the stud(mm)d h :Diameter of the stud head(mm)，f' c :Compressive strength of concrete(N/mm2)
cshs fhdhF ')( +⋅
chs fdhF '⋅32
'2cs fhF ⋅
cfhF '927.0 2⋅
cshs fhdhF ')( +⋅
cshs fhdhF ')( +⋅
cshs fhdhF ')(2 +⋅ π
cshs fhdhF ')( +⋅
71(71s)
b) Shear strength based on push-out tests for single and double shear surfaces In general, shear resistance of a stud has been
obtained experimentally with the testing method as shown in Fig. 2, so far.
The most widely used testing method is a push-out test with the composite steel section having outer double shear surfaces, and it is just called “Push-out shear test”. The method can be loaded with a pure shear state until ultimate limit, and is hardly influenced by the edge distance because the concrete resists by bearing strength of the concrete under the stud shank. In addition, it is based on the assumption that the steel plate at the stud root does not deform until ultimate state as shown in the right side of the figure. Therefore, no rapid reduction of shear resistance is recognized due to confined effect under the tri-axial compression of the concrete at the front part of the stud, even if the concrete around the root of the stud crushes locally.
Push-out test for the sandwiched concrete specimen with two cover plates is developed to determine the strength of shear connectors used in sandwich structures such as submerge tunnel, harbor caisson block etc. In this method, due to the relatively thin plate thickness of the specimen, the plate at root of the stud deforms locally in the early
stage of the load. This rotational deformation of the plate increases with the increase of loading, and the concrete at front of the stud crashes locally. So that, the large deformation of the plate caused by the crush of concrete induces the decrease of shear resistance.
Shear test by tensile loading for single shear surface is used frequently to obtain the shear strength at anchoring part between steel pier and concrete footings mainly in architectural engineering field. In this testing method, since the steel plate with welded studs is pulled horizontally by high strength steel bar, the shear force is loaded eccentrically, and the plate will be bent out of the shear surface. Consequently, the concrete at front of the first stud locally crushes in early stage due to a complex stress state in the direction of stud axis, and then the shear resistance of the stud decreases. (2) In the case of considering edge distance a) Pull-out strength
When the distance e from the edge of concrete to the center of stud is shorter than the length of stud shank hs, a cone surface formed by pull-out load becomes a partially lacked one cut by the concrete end. Therefore, as specified by Architectural Institute of Japan (AIJ), the pull-out strength is reduced by considering the lack of the effective horizontal
Fig. 2 Testing methods of the stud for shear1),13),25)
Testing method
Push-out test for double shear
surface
(outer concrete)
Push-out test for double shear
surface (inner concrete)
Pull-out test for single shear
surface
Configurations of specimens Deformation and distributing stress of stud’s bottom
High strength bolts
Base plate Bedding
Stud
Concrete
Loading PlateLoad cell
machine Head of testing
Flange plate
Web plate
Crushing zone
Stud Concrete
Steel plate Loading plate Stud
Load cellHydraulic jack Concrete Crushing zone
Stud Concrete
Steel plate
Hydraulic jack(center-hole
Load
cell
Concrete
High-strength-steel bar
Steel plate
Stud
Steel plate
Crushing zone Stud Concrete
Stud reactionCompressive
reactionCompressive
Free
Channel Steel
72(72s)
projective area as shown in Fig. 3. For reference, the general fracture model ignoring edge distance is also shown in this figure.
For the whole collected data of 294 shown in Table 3, the relationship between measured pull-out load and edge distance is analyzed as shown in Fig. 4. Though the data widely scattering, the envelope enclosing upper bound of pull-out strength shows a tendency that the tested values decrease with the decrease of edge distance. So a more detailed
analysis for the effect of edge distance seems to be necessary for the design of stud by the pull-out strength.
The pull-out strength equations proposed by Fisher et al. and specified in PCI design handbook are shown in Table 2. For reference, the table also shows the equation proposed by AIJ which is presented by lower bound of tested value. As it is clear from the table, those equations are giving a simple reduction factor for the edge effect. The procedure seems to be
Fig. 3 Evaluation method for pull-out strength of headedstud provided in AIJ Standard
Edge distance
PCU : Pull-out strength of headed stud P'CU : Reduced Pull-out strength of the studf'C : Compressive strength of concrete AC : Effective horizontal projective areaAC : Effective horizontal projective area(e=∞) (Considering the edge distance e)hS : Length of stud shank e : edge distancedh : Diameter of the stud headdS : Diameter of the stud shank
Estimatingequations by
AIJ
Fracturemodels
Non-Considered Considered
concrete
Conical fracture
(Shaded portion)
0
50
100
150
200
250
0 100 200 300 400 500 600Edge distance e (mm)
Mea
sure
d pu
ll-ou
t loa
d P
(×
103 N
)
(Reference diagram)
Pull-out strength
P'CU : Reduced pull-out strength of the stud considering edge distance e : Edge distancePCU : Pull-out strength of the stud ignoring edge distance ds : Diameter of the stud shank
dh : Diameter of the stud headQ'u : Reduced shear strength of the stud considering edge distance hs : Length of the stud shankQu : Shear strength of the stud ignoring edge distance f'c : Compressive strength of concrete
K : Coefficientλ : Material factor
λ=1.0 for nomal concrete λ=0.85 for light-weight aggregate concreteλ=0.75 for light-weight concrete
As : Cross section area of the stud shank Ec : Young's modulus of the concretePmax : Measured lower bound of pull-out strength of the stud considering edge distanceAcr : Effective horizontal projective area
Qmax : Measured lower bound of shear strength of the stud considering edge distanceAcr : Effective projective area formed at end verticl surface of the concrete
(Refer to right side illustration)
Shear strength
Proposed by Fisher et al. (1973)
AIJ Standard (1985)
PCI design handbook(1978)
)'f)dh(hK( chss +⋅λ⋅=
( ) )E'fA5.0( 5.0ccs ⋅⋅=
( ) { }���
��� +⋅+−−+⋅++π= − )dh(ecos)dh(e)dh(edhh hs
12hs
22hshss
( ){ }2e 2⋅π=
cucu P'P ⋅α= γ
( )5000'f)1e(3250'Q cu λ−=
sd9e2=αγ
uQu Q'Q ⋅α=
sQ d8
1e −=α
crmax A2256.0P ⋅= CQcmax A'f6.0Q ⋅=
cucu PC'P ⋅= γ
1heCs
≤=γ
(Reduction factor)
(Reduction factor) (Reduction factor)
(Pound and inch)
Table 2 Estimating equations for pull-out and shear strength considering edge distance (Previous research)
Fig. 4 Relationship between measured pull-out load andedge distance
73(73s)
too simple to express the effect of edge distance. And the equation by AIJ seems to be too much safety side, because it is given by the lower bound data. b) Shear strength based on push-out test for
single and double shear surfaces Because the push-out test for double shear surfaces
in standing type is hardly influenced by edge distance as mentioned above, the test data would be excluded from the object of discussions.
The strength equations for shear resistance of the stud for the specimen with single shear surface proposed by Fisher et al. and specified in PCI design handbook are also shown in Table 2. The table also shows the AIJ equation obtained by the same manner as pull-out strength, for reference. As it is clear from the table, in the same manner as pull-out strength, Fisher et al. expresses the shear strength of the stud considering edge distance by only multiplying the equation being independent of edge distance by a reduction factor. And, PCI equation also is a mere experimental equation, and has no clear basis. In the same manner, the AIJ standard gives an estimating equation for shear resistance considering edge distance using the vertical effective projective area obtained by assuming the conical fracture surface starting from the bearing portion at the stud shank surface as shown in the right side illustration of Table 2. And the equation is also irrational due to adoption of the lower bound of tested data. When a shear test is carried out by tensile loading through reinforcing bars in concrete block, the concrete surface of pull side is free and the studs nearest to the concrete surface will be affected by edge distance. In the same manner as pull-out strength, the whole 118 data shown in Table 5 for single shear tests are collected and arranged in this study, and the relationship between measured shear capacity and edge distance is shown in Fig. 5. From the figure, though the tested data are widely scattering, the upper bound of tested values decrease linearly with the decrease of edge distance, and the influence of the edge distance on the shear strength of the stud is clear.
3. PROPOSAL OF PULL-OUT AND
SHEAR STRENGTH EQUATIONS FOR HEADED STUD
(1) Influence factors on pull-out and shear
strengths and basic statistical data of previous tests
a) Pull-out strength From the strength equations proposed so far as
shown in Table 1, it can be supposed that the significant influence factors on the pull-out strength
of the stud are shank diameter (ds), length of stud shank (hs), diameter of stud head (dh), compressive strength of the concrete (f'c), effective horizontal projective area parameter of the cone (ds+hs) and edge distance (e).
The basic statistical data found so far both in the domestic and foreign papers for pull-out strength including new data obtained by authors are 294 as shown in Table 35),10),12),15),18). b) Push-out shear strength
As for the shear strength based on the push-out test for the specimen with double shear surfaces, the shaft diameter (ds), overall length (h), tensile strength (fsu) of the stud and compressive strength of the concrete (f'c) would be the significant factors. Because, for the push-out shear test, it can be considered that the shear strength is hardly influenced by edge distance, since the force acting on H-shaped steel is resisted by bearing strength of the concrete under the stud shank.
The basic statistical data for the push-out shear strength by the tests for double shear surfaces including by authors results (whole number : 197) are shown in Table 419). c) Shear strength for single shear surface
For the shear strength based on the push-out test for the specimens with single shear surfaces, the shaft diameter (ds), overall length (h), diameter of stud head (dh), compressive strength of the concrete (f'c) and edge distance (e) would be the significant factors. And basic statistical data for push-out shear strength by single shear test including by authors (whole number : 118) are shown in Table 511),20),22). (2) Derivation of strength equation for pull-out
strength of the stud a) Studs embedded deeply from concrete edge
The appropriate experimental data of 231 was statistically analyzed by logarithmic multiple regression analysis.
Based on the parameters in the existing strength
Fig. 5 Relationship between measured shear capacity and edge distance
0
50
100
150
200
250
300
0 50 100 150 200 250 300 350Edge distance e (mm)
Mea
sure
d sh
ear c
apac
ity
Q
(×10
3 N
)
74(74s)
equations and the authors judgment, the pull-out strength of the stud finally could be composed as the following exponential multivariate regression model. Namely,
b
ca
shu fhdP ')( ⋅+⋅= α (1)
Where, Pu : Pull-out strength (N), hs : Length of the stud shank (mm), dh : Diameter of the stud head (mm), f'c : Compressive strength of the
concrete (N/mm2),
α: Coefficient.
Then, Eq.(2) can be derived by the multiple regression analysis.
316.0967.1 ')(61.1 cshu fhdP ⋅+⋅= (2) ｔ value：63.7 for 1.967, and 3.75 for 0.316
Where, the significance of the exponential part in
the regression equation is examined by the t value, and it is judged that a bigger t-value gives a higher significance. In addition, the multiple correlation coefficient was 0.974.
Then, from taking the previous strength equations into account, each exponential part can be rounded off for practical usage as Eq.(3).
cshu fhdP ')( 2 ⋅+⋅= α (3)
Then, the 93 data showing the high correlation were rearranged using the relation of Eq.(3) as shown in Fig. 6. From the figure, it was clarified that the regression equation gives the medium value (median) of tested data in all range.
The validity of Eq.(3) was confirmed by the Fig. 6. However, the equation form of PCI design handbook was also investigated in order to keep the consistency as a pull-out strength equation.
The equations of PCI design handbook are derived under the assumption of two fracture modes that the stud shank are broken itself and the concrete cone fracture around the stud. The tensile strength of the stud shank is the upper bound. Then PCI equations are given by Eq. (4).
suscsshu
suscconeu
fAfhhdFP
fAfAFP
⋅≤⋅+⋅=
⋅≤⋅⋅=
')(2
'
π (4)
Where,
F : Coefficient, Acone : Area of an assumed conical fracture
surface of the concrete (mm2),
As : Cross-sectional area of stud the shank (mm
2),
fsu : The tensile strength of the stud (N/mm2).
Here, in Eq.(4) making the acting direction of the
external force coincides with the resisting direction, and by assuming that the strength equation is given in the form of the product of projected area of the shear cone on a concrete face (Ac) and the tensile strength of the concrete (ft), the form of basic equation is expressed as follows.
sustsshu fAfhhdFP ⋅≤⋅+⋅= )(π (5)
0
20
40
60
80
100
120
140
160
0 30 60 90 120 150 180 210
Mea
sure
d pu
ll-ou
t loa
d P
(×10
3 N)
)mmN10(')( 32 ⋅×+ csh fhd
R=0.9678
csh fhdP ')(897.0 2+⋅=
Fig. 6 Relation obtained by multiple regression analysis
Table 3 Basic statistical data of studs for pull-out test
Mean Standarddeviation Min Max
Diameter of stud shank ds(mm) 11.88 6.18 6.00 22.00Length of the stud shank hs(mm) 79.98 49.85 29.90 200.00
Diameter of stud head dh(mm) 21.44 9.97 12.00 44.50Compressive strength of concrete f'c(N/mm2) 26.58 4.70 11.18 37.66
Projective aream parameter hs+dh(mm) 101.41 57.99 41.90 232.00edge distance e(mm) 181.02 129.95 30.00 500.00
Number of data : 294
Table 4 Basic statistical data of studs for push-out shear test of double shear faces
Mean Standarddeviation
Min Max
Diameter of stud shank ds(mm) 17.90 3.70 6.00 32.00Overall length h(mm) 86.89 20.50 35.00 214.00
Tensile strength of stud fsu(N/mm2) 492.36 59.26 349.12 620.47Compressive strength of concrete f'c(N/mm2) 30.95 7.83 13.63 61.98
Number of data : 197
Table 5 Basic statistical data of studs for push-out test of the stud of single shear face
Mean Standarddeviation Min Max
Diameter of stud shank ds(mm) 14.78 9.77 6.00 51.00Overall length h(mm) 131.86 107.16 35.00 508.00
Diameter of stud head dh(mm) 28.41 22.53 12.00 114.00Compressive strength of concrete f'c(N/mm2) 28.72 4.70 18.73 36.28
Edge distance e(mm) 151.06 67.18 40.00 305.00
Number of data : 118
75(75s)
Then, for the tensile strength of concrete, the strength equation shown in the Ref. 24) seems to be useful. So, Eq.(5) can be written as
sus
csshu
fAfhhdFP
⋅≤⋅⋅+⋅=
)'267.0()( 3
2
π (6)
Fig. 7 shows rearranging the data of Fig. 6 using
Eq.(6). As is obvious from the figure, the results of Fig. 7 shows an equivalent evaluation in comparison with the result by Eq.(3) derived from multiple regression analysis with the equivalent correlation coefficient (0.9678→0.9673). Therefore, it can be considered that the Eq.(6) using the tensile strength of the concrete gives an appropriate estimation for the pull-out strength of the stud, too.
Finally, by rounding off the coefficient of Eq.(6), Eq.(7) can be proposed, as an appropriate pull-out strength of the stud.
)'267.0()(85.0 32
csshu fhhdP ⋅⋅+⋅= π (7) with a limitation of {(e-ds/2)/hs ≥ 2.0} as shown in the following paragraph. b) Considering edge distance
As mentioned above, there are many unsolved problems for the influence of edge distance on the pull-out strength of the stud. However, in order to estimate the reduction of pull-out strength due to the smaller distance from the concrete edge to the stud, an edge distance parameter was introduced and investigated. In the PCI equation, the value of the edge distance e (distance from the concrete edge to the central axis of the stud) divided by the length of
stud shank hs is defined as the edge distance parameter. In this case, the pure cover changes depend on the diameter of the stud, even if the edge distances are same.
Therefore, a pure cover ratio {(e-ds/2)/hs} was employed as the edge distance parameter in order to eliminate the difference.
Moreover, on formulation of pull-out strength of the stud considering edge distance, the data gotten by break at stud shank were excluded, and the tested data for the specimen with reinforcements around the stud were also omitted, because the pull-out strength seems to increase by restraint of the reinforcements.
Taking the pure covering ratio to the abscissa as the new edge distance parameter, and the normalized force which is divided the experimental pull-out strength by the calculated value given by Eq.(7) to the ordinate, tested data were plotted as shown in Fig. 8. From the figure, it is obvious that the tested data are remarkably scattering. However, from general view of the figure, it is judged that the normalized force of ordinate converges roughly to 1.0 when the edge distance parameter exceeds 2.0. Namely, it can be said that the pull-out strength of the stud is hardly influenced by edge distance for the range of the edge distance parameter above 2.0.
Therefore, by only the data of the abscissa value under 2.0, the regression analysis by linear and exponential type were carried out again. As a result, it was clarified that the results obtained by the exponential regression analysis shows higher correlation, though the tested data were scattered (See Fig. 9). From this fact, it was considered that the strength equation for the range of {(e-ds/2)/hs ≤ 2.0} should be given as the exponential function, and the following reduction factor (αp) could be introduced in order to simplify the expression.
Fig. 7 Comparison of measured pull-out load with the estimated value with Eq.(6) independing on edge distance
0
20
40
60
80
100
120
140
160
0 30 60 90 120 150 180 210
Mea
sure
d pu
ll-ou
t loa
d
P (×
103 N
)
R=0.9673
)'267.0()(8453.0 32
cssh fhhdP ⋅+⋅= π
)'267.0()( 32
cssh fhhd ⋅+π )mmN10( 23 ⋅×
Fig. 8 Relationship between P/Pu and edge distance parameter
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 2 4 6 8 10 12Edge distance parameter e
P/P u
giv
en b
y eq
.(7)
a
76(76s)
Fig. 9 Fitting of estimating equation based on multiple regression analysis (e ≤ 2)
s
d
s
d
up
he
he
PP
s
s
2
2
21
712.0
−⋅
−=
≒
＝α (8)
As shown in Fig. 10, the tested data was plotted
with the pure covering as abscissa and the loading ratio (P/Pu) as ordinate, in a similar manner as Fig. 8, and an applicability of Eq.(9) which is incorporated a coefficient given by Eq.(8) into Eq.(7) was examined (whole data of 73 with the concrete strength over 25N/mm2).
s
d
csshu he
fhhdPs
2
21)'267.0()(85.0 3
2 −⋅⋅⋅⋅+⋅= π (9)
with a limitation of {(e-ds/2)/hs ≤ 2.0}.
From Fig. 10, it can be observed that the Eq.(7) is linked to Eq.(9) at the edge distance parameter equal to 2.0, but at that point, the relation between the edge distance parameter and the loading ratio changes from the exponential function to the constant value.
For the data same as Fig. 10, taking the edge distance parameter into account, the tested pull-out strength were compared with the calculated ones as shown in Fig. 11. As is clear from the figure, though the calculated value gives the medium of experimental one in the range of less than 100kN, in the range of more than 100kN, the experimental value tends to take a less value than the calculated one. However, because of the high correlation coefficient of 0.9578, it was judged the strength equation for pull-out strength of the stud can be given by Eq.(7) and Eq.(9). 70% and/or 80% reduction value of Eq.(7) and Eq.(9) are also shown in Fig. 10 and Fig. 11 . From these figures, it can be found that the 70% reduction curve of Fig. 10 well expresses the lower bound of the tested data. Therefore, the reduction curve can be employed as the design strength formulae which take the scattering of tested data into account.
(3) Derivation of shear strength equations of the
stud a) In the case being independent of edge distance ① Push-out shear test
The shear strength of the stud obtained by the ordinary push-out test method is considered to be hardly influenced by edge distance due to the characteristics of testing method. From the statistical arrangements of the previous tested data, Hiragi and Matsui proposed the strength equations for shear strength of the stud19). In this study, whole test data including the new data obtained by the authors were rearranged according to the above evaluating method.
And the result is shown in Fig. 12 (whole data of 197 shown in Table 4). The validity of the equation by Hiragi and Matsui can be certified from the figure. Therefore, it was judged that this equation can be adopted as the strength equation for push-out shear
Fig. 11 Comparison of the estimated loads and measured pull-out load by ignoring edge distance
Fig. 10 Distribution of P/Pu for full range and fractal value
0
50
100
150
200
250
300
350
0 50 100 150 200 250 300 350Calculated pull-out load P u (×103N)
Mea
sure
d pu
ll-ou
t loa
d
P
(×10
3 N)
R =0.9784
Median
Reduced 80％
Reduced 70％
(e -d s /2)/h s≦2(e -d s /2)/h s＞2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 2 4 6 8 10 12Edge distance parameter e
P/P u
giv
en b
y eq
.(7)
af' c≧25N/mm2
Reduced 80%
Reduced 70%
eq.(9)
eq.(7)
0
0.5
1
1.5
2
0 0.5 1 1.5 2Edge distance parameter e
P/P u
giv
en b
y eq
.(7)
a
)7125.0(7124.05466.0
2 =��
�
�
��
�
� −⋅== R
he
PP
s
d
up
S
α
)5143.0(6586.0 2 =��
�
�
��
�
� −⋅== R
he
PP
s
d
up
S
α
22 ≤−
s
d
he S
77(77s)
strength in the case of ignoring edge effect. The equation is given by
9800'3.31 += cs
su fdhAQ (10)
Where,
Qu : Push-out shear strength of a stud (N), As :Cross-sectional area of the stud shank (mm
2),
h : Overall height of the stud (mm), ds : Diameter of the stud shank (mm), f'c : Compressive strength of concrete (N/mm
2).
For reference, both the shear test data obtained by
double shear surface specimens for inner concrete and by the single shear surface push-out test (Refer to Fig. 2) are also shown in Fig. 12. From this figure, those test results seem to be obviously a little bit lower, in comparison with the ordinary push-out test results. However, the gradients of both results seem to be similar. That is, though the regression line for the ordinary push-out shear test results has the intercept at y-axis, the regression lines for shear strength of double shear surfaces by inner concrete and for single shear test seem to be expressed without the intercept.
Hereafter, the data collected by the new testing method which can represent the stress transfer mechanism of actual structures is strongly expected. ② Shear test of single shear surface by tensile load
The shear test results of single shear surface by tension type loading are shown with the regression line in Fig. 13. The data were a few data (whole 50 data). As is clear from the figure, the intercept of the
regression equation becomes zero, and the correlation coefficient (R) shows a high value of 0.9943. And the gradient of regression line shows a little higher value of 32.9 in comparison with 31.3 which was obtained by the push-out shear test .
From the results, as the strength equation for single shear strength of the stud independent of edge distance, the following expression can be proposed considering the same parameter of the ordinary push-out shear strength equation.
'3.31 c
ssu f
dhAQ = (11)
In the shear test by tension type loading for single
shear surface, the stud welded on steel plate has a tendency to develop local bending of the plate due to the eccentricity of acting force. And the strength becomes a little bit smaller than the push-out results. Therefore, it can be said that the Eq.(11) is a shear strength equation for the studs welded on thin steel plate. Moreover, the equation form was determined under the engineering judgment to make the gradient of the equation coincide with push-out shear strength one of Eq.(10) and to put the intercept of the equation zero. b) Considering edge distance ① Shear test for single shear surface by tensile load
As noted earlier, generally in the push-out shear test, the shear strength is hardly affected by edge distance of the concrete. However, in the shear test by tension type loading for single shear surface, the shear strength of the stud seems to be influenced by edge distance. The ratio of the pure cover to the length of stud shank {(e-ds/2)/hs} is employed as a edge distance parameter as well as the pull-out
Fig. 12 Comparison between measured push-out shear loadsand estimated loads proposed herein by ignoring edgedistance
0
50
100
150
200
250
0 2000 4000 6000 8000
Mea
sure
d pu
sh-o
ut sh
ear l
oad
Q
(×10
3 N）
9800'3.31 +⋅⋅⋅= cs
su fdhAQ
R= 0.9161
○ Double shear (Outer concrete)
▲ Double shear (Inner concrete)
● Single shear
( ) )mmN(' ⋅⋅ css fdhA Fig.13 Comparison between measured single shear capacityand estimated loads proposed herein by ignoring edgedistance
0
40
80
120
160
0 1000 2000 3000 4000
Mea
sure
d si
ngle
shea
r loa
d
Q (×
103 N
)
R =0.9943
cs
su fdhAQ '918.32 ⋅⋅=
( ) )mmN(' ⋅⋅ css fdhA
78(78s)
strength. Fig. 14 shows the relationship between the shear
force ratio [experimental value/ calculated value of Eq.(11)] and edge distance parameter (whole data is 118 shown in Table 5). It is recognized from this figure that the shear force ratio of ordinate converges roughly to 1.0 when the edge distance parameter exceeds 2.0. This means the shear strength of the stud is hardly influenced by edge distance for the range of edge distance parameter over 2.0 as well as the pull-out strength.
Then, for only the data of the edge distance parameter under 2.0 in Fig. 14, the linear regression equation was obtained with higher correlation with the correlation coefficient of 0.9257 (See Fig. 15). Therefore, the strength equation for the affected range of edge distance {(e-ds/2)/hs ≤ 2.0} is given by linear function, and the following reduction factor (αp) can be introduced.
s
s
uq h
de
QQ �
�
���
� −= 2
21
≒α (12)
As shown in Fig. 16, the tested data were plotted
with the pure cover of edge distance parameter as abscissa and the shear force ratio as ordinate, and the Eq.(13) with the coefficient given by Eq.(12) and Eq.(11) for the range of {(e-ds/2)/hs ≥ 2.0}.was drawn on the figure (same data number as Fig. 14).
��
�
�
��
�
� −=
s
d
cs
su he
fdhAQ
s2
21'3.31 (13)
From Fig. 16, it is observed that the Eq.(11) is
linked to Eq.(13) at the edge distance parameter equal to 2.0, and the relation between the edge distance parameter and the shear force ratio can be expressed as a bi-linear curve. Furthermore, 70% and/or 80% reduction value of Eq.(11) and Eq.(13) are drawn in Fig. 16. From the figure, the 70% reduction curve of Fig. 16 seems to be the lower bound of tested data.
As by all the test data of 118 and taking the edge distance parameter into consideration, the test values
Fig. 14 Relationship between Q/Qu and edge distance parameter
Fig.15 Fitting of an estimating equation based onmultiple regression analysis (e≦2)
Fig. 16 Comparison between measured single shear capacity and estimated loads by Eq.(13) and Eq.(11)
Fig.17 Comparison between measured single shear capacity and estimated loads by Eq.(13) considering edge distance
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 1 2 3 4 5 6Edge distance parameter e
Q/Q
u
ss h)2de( −=
(e -d s /2)/h s≦2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0.0 0.5 1.0 1.5 2.0 2.5Edge distance parameter e
Q/Q
u a
)9257.0(4919.0 2 =−
⋅== Rh
eQQ
s
d
uq
S
α
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 1 2 3 4 5 6Edge distance parameter e
Q/Q
u gi
ven
by e
q.(1
1) a
Reduced 80%
Reduced 70%
Eq.(13) Eq.(11)
0
50
100
150
200
250
300
350
400
0 50 100 150 200 250 300 350 400Calculated single shear capacity Q u (×103N)
Mea
sure
d si
ngle
shea
r cap
acity
Q
(×10
3 N)
Median
Reduced 80%
Reduced 70%
R =0.9562
79(79s)
were compared with the calculated ones as shown in Fig. 17. As is clear from the figure, though the calculated value gives the medium of experimental one in the range of less than 200kN, the experimental value in the range of more than 200kN tends to take a less value than the calculated one. However, because of the high correlation coefficient of 0.9562, it was judged that the strength equation for shear strength of the stud with single shear surface can be given by Eq.(11) and Eq.(13). Moreover, it can be seen the 70% reduction values of Eq.(11) and Eq.(13) well expresses the lower bound of scattering data. Therefore, the reduction curves can be employed as the design strength formulae which take the scattering of tested data into account. 4. CONCLUSIONS
Firstly, in this paper, the strength equations of the pull-out and shear strength of the stud, proposed by previous researches and specified in design handbook were introduced, and by presenting a problem from the degree of conformity for the existing tested data, the necessity of presenting the strength equation of the stud was emphasized. Secondly, the statistical processing was carried out for the existing tested data including the data obtained by authors, and then, the strength equations for both the pull-out and shear strength of the stud were derived rationally. Finally, the design strength formulae which take the scattering of tested data into account were also proposed by introducing the reduction factor. This study is summarized as follows. (1) The pull-out strength equation of the stud was derived by revising of strength equation specified in PCI. For the pull-out strength of the stud ignoring edge distance, Eq.(7) was proposed by formulating in the form of the product of projected area of the shear cone and the tensile strength of the concrete.
Then, for the pull-out strength equation of the stud considering edge distance, the value dividing the pure cover thickness by the length of stud shank {(e-ds/2)/hs} was newly introduced as an edge distance parameter, and the Eq.(9) was proposed for the edge distance parameter under 2.0. In addition, the Eq.(9) is linked to the strength equation of Eq.(7) which is unaffected by edge distance at {(e-ds/2)/hs}=2.0. (2) The validity of the push-out shear strength equation of Eq.(10) proposed by Hiragi and Matsui was certified statistically, and adopted herein as the push-out shear strength equation of the stud. Especially, as for the push-out shear strength, the data including fine diameter stud (φmm) tested in
this study could be accurately rearranged. (3) Shear strengths for double shear surfaces of inner concrete and for single shear surface show a little lower values in comparison with push-out shear strength. As the shear strength equation of these cases, Eq.(11) that put the intercept of the push-out shear strength equation to zero can be proposed in the case of ignoring edge distance.
In addition, as the shear strength equation for single shear surface of the stud considering edge distance, the Eq.(13) was proposed by using a new edge distance parameter {(e-ds/2)/hs} for the parameter under 2.0 as well as pull-out strength equation. In addition, the Eq.(13) is linked to the Eq.(11) which is unaffected by edge distance at {(e-ds/2)/hs}=2.0. (4) The 70% reduction values of pull-out and shear strength equations of the stud proposed herein can be adopted as design values considering the scattering of tested data. ACKNOWLEDGEMENT ： This study was conducted as a joint research of the RCJ (Research group of Composite Joints). The authors wish to express their gratitude to the precious contribution of the research members of Nippon Stud Welding Co., Matsuo Bridge Construction Inc. and Japan Information Processing Co. And, the authors would like to acknowledge students of Osaka Univ. and Setsunan Univ. in the laboratory tests. REFERENCES 1) Maeda, Y., Matsui, S. and Hiragi, H.: Effects of Concrete-
Placing Direction on Static and Fatigue Strengths of Stud Shear Connectors, Technology Reports of the Osaka University, Vol.33, No.1733, pp.397-406, 1983.
2) Matsuzaki, I.: The Earthquake-proof Installation Method of the Equipment -Type of the Anchor and the Support Load Capacity-, The Construction Technology of Architecture, pp.31-51, 1980. (In Japanese)
3) Al-sakkaf, A., Hiragi, H. , Takabayashi, K. and Matsui, S.: Investigation on Pull-out and Shear Strength of φ6mm Studs, Proceedings of the Japan Concrete Institute, Vol.23, No1, pp.757-762, 2001.
4) TRW, Nelson-Division: Embedment Properties of Head Studs, TRW-Inc., USA, p.47, 1974.
5) Sattler, K.: Betrachtungen über neue Verdubelungen im Verbundbau, Der Bauingnieur,37,H1, pp.1-8, 1962.
6) Utescher, G.: Beurteilungsgrundlagen für Fassadenveran -kerungen, Berlin, Ernst & Sohn, 1978.
7) European Convention for Constructional Steelwork: Composite Structures, The Construction Press, 1981.
8) Daft European Recommendations on Composite Structures: CEB-ECCS-FIP- IABSE, Joint Committee on Composite Structures 1977.
9) PCI Design Handbook: Precast Prestressed Concrete, Prestressed Concrete Institute, Chicago, Illinois, 1978.
10) Bode, H. and Roik, K.: Headed Studs -Embedded in Concrete and Loaded in Tension-, Paper presented at the ACI Annual Convention, Los Angels, pp.61-88, 1983.
11) McMackin, P. J., Slutter, R. G. and Fisher, J. W.: Headed
80(80s)
Steel Anchors under Combined Loading, AISC Engineering Journal, pp.43-52, 1973.
12) Ohtani, Y. and Fukumoto, Y.: Failure Behavior of Stud Anchor due to Pullout Tension, Technology Reports of the Osaka University, Vol.39, No.1981, pp.297-305, 1989.
13) Sonoda, K., Kitou, H., Nakajima, K. and Uenaka, K.: Systematic Research on Shear Transfer Characteristic of Projecting Steel Plate, Journal of Structural mechanics and earthquake engineering, JSCE, No.598/I-44, pp.183-202, 1998. (In Japanese)
14) AIJ: Design Guidelines and Commentary for Composite Structure, Part 4; Design Guidelines and Commentary for Anchor Bolts, 1985. (In Japanese)
15) TRW, Nelson-Division: Nelson Stud Project No.802, Report No.1966-5, 1966.
16) TRW, Nelson-Division: Design Data, Nelson Concrete Anchor Studs, Manual, No.21, 1961.
17) Shirasaka, Y., Matsuzaki, I., Abe, Y. and Usami, S.: Experimental Study on Loading Capacity of Support Structure (Embedding Metallic Material) for Piping Equipment, Part-1. Pull-out Strength of Stud Bolt Embedded in Concrete -, The AIJ Convention Scientific Lecture Outline, pp.1375-1376, 1979. (In Japanese)
18) Shirasaka,Y., Matsuzaki,I., Abe,Y. and Usami, S.: Experimental Study on Loading Capacity of Support Structure (Embedding Metallic Material) for Piping Equipment - 2. Tensile and Shear Fatigue Strength of Headed
Stud with 19mm diameter, The AIJ Convention Scientific Lecture Outline, pp.1377-1378, 1979. (In Japanese)
19) Hiragi,H., Matsui,S. and Fukumoto,Y.: Derivation of Strength Equations of Headed Stud Shear Connectors – Fatigue Strength, Proceedings of Structural Engineering, JSCE, Vol.35A, pp.1221-1232, 1989. (In Japanese)
20) Klingner, R. E. and Mendonca, J. A. : Shear Capacity of Short Anchor Bolts and Welded Studs: A Literature Review, ACI Journal, Proceedings Vol.79, No.34, 1982.
21) Bailey, J. W. and Burdette, E. G.: Edge Effects on Anchorage to Concrete, Civil Engineering Research Series,No.31, The University of Tennessee, Knoxville, 1977.
22) Swirsky, R. A., et al.: Lateral Resistance of Anchor Bolts Installed in Concrete, Report No.FHWA-CA-ST-4167-12, California, Department of Transportation, Sacramento,1977.
23) Kawaguchi, C.: An introduction to Multi-variable Linear Regression Analysis, pp.3-33, 1973. (In Japanese)
24) Okamura, H.: Limit State Design Method for Concrete Structures, Concrete seminar 4, Kyouritsu-Shuppann, pp.17, 1979. (In Japanese)
25) Takabayashi, K., Al-sakkaf, A., Matsui, S., Kawasaki, T., Hiragi, H. and Ishizaki, S.: Research on Shear and Pull-out Strength of the Stud with 6mm Diameter, Proc. of the 54th Annual meeting of JSCE, I-A142, 1999. (In Japanese)
(Received August 15, 2001)