Interface shear transfer of precast concrete dry joints in segmental columns

Engineering Structures 175 (2018) 257–272

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Interface shear transfer of precast concrete dry joints in segmental columns ⁎

T

Zhan-Yu Bu , Xu Zhang, Han-Hui Ye, Kang Xi, Wei-Ye Wu Faculty of Architectural, Civil Engineering and Environment, Ningbo University, Ningbo 315211, China

A R T I C LE I N FO

A B S T R A C T

Keywords: Precast segmental column Dry joint Shear test Direct shear loading Failure pattern Shear strength formula

Precast dry joints in segmental columns are important to ensure the structural integrity and continuous to transfer both shear and flexural forces. Though dry joints have the concern of durability problems, they are the trend for convenience in site construction and less climate influenced. Ten precast column joint specimens were fabricated and tested under direct shear loading. The failure patterns of solid section keyed specimens were shearing off of male shear keys and diagonal cracks in female shear keys. The failure sequence was from top to lower shear keys. But the failure pattern of hollow section specimens was flexural failure of upper male or lower female shear keys, shearing off of vertical shear keys, and diagonal cracks in vertical female shear keys. The shear force-relative slippage relations of specimens exhibited two or three stages. Higher confinement stress, solid section (or enough key cover concrete thickness), and enough number of keys were essential to maintain high shear strength and ductile responses. Two formulation methods of shear strength prediction formula were deduced. One is based on the failure envelope of uniaxial compression and tension strength of concrete, the other one is based on tension strength Mohr circle only. The proposed formula has comparable prediction precision for the joint shear test database. Shear key shear stress and shear strength contribution were analyzed with the test data. The specimen shear stress transfer and distribution were qualitatively investigated with a two dimensional FE model.

1. Introduction Precast substructures are prevailing nowadays, from Europe to America and also in Asia areas. In Europe, water-proof layer is used on top of girder, so epoxy-free joint is permitted in practice. But in the U.S., for durability considerations, dry joints in precast components are forbidden. In the present stage of China, the most popular precast construction techniques of substructures evolve from cast-in-place joint to grout joint, and dry joint is the trend to reducing manual work in site. The present design method of precast joint is to prevent the joint opening at service limit state (SLS). So under the SLS, the joint is compressed with a minimum of 0.5 MPa confinement stress remained in all joints. Under the act of higher load, the column reaches ultimate limit state (ULS), the interfaces at critical joints will open, and the reinforcement across the joint will prevent the joint opening. For the columns located in low seismicity area, mild steel deformed bar is used. For the columns in moderate to high seismicity areas, prestressed finished deformed bar or the combination of mild steel deformed bar and prestressing strand are preferred. Usually, in the SLS stage, the crack width controls the design, but in the ULS stage, the tensile or compressive stress of cover concrete and tensile stress of reinforcement will play the decisive role. ⁎

The shear transfer mechanisms of precast joint in columns or girders consists of two parts, the interlocking resistance provided by castellated or prismatic shear keys, and the friction force under the confinement pressure of flat interface provided by asperities. Several shear failure mechanisms are identified in castellated joints, such as S (Single curvilinear) cracks and M (Multiple diagonal) cracks. Many researchers have conducted push off test of precast joint, the experiment parameters include confining stress, dry or epoxy joint, epoxy thickness, flat or keyed joint and number of keys, key depth, distance between keys, monolithic joint and steel fiber joint [17,4,7,6,8]. The main findings include: (1) Failure sequence and pattern of keys, mostly in shear failure, part of three-keyed high pressure epoxied joint failed in bending, failure pattern of keyed dry joint is ductile, while keyed epoxied joint fails more brittle; (2) The epoxied joint strength was consistently higher than that of dry joints; (3) Sequential failure of multiple keyed joints was observed from test and was verified by finite element model, a reduce factor of 0.7 was proposed for the shear strength of three-keyed joint; (4) AASHTO [1] and Rombach and Specker’s [13] formula both underestimated the shear strength of single-keyed dry joint, but overestimated the shear strength of threekeyed joints. Other researchers studied the response of segmental concrete beams with external prestressing under shear loading

Corresponding author. E-mail address: [email protected] (Z.-Y. Bu).

https://doi.org/10.1016/j.engstruct.2018.08.037 Received 1 December 2017; Received in revised form 25 June 2018; Accepted 11 August 2018 0141-0296/ © 2018 Elsevier Ltd. All rights reserved.

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Fig. 1. Specimens configuration.

researchers. Finally a planar finite element model was developed to study the stress distribution and direction qualitatively.

[15,16,11]. The main conclusions are: (1) The cracking that led to failure of structure arised from the joint, the shear failure of the beams was spatial, shear strength of the beams once the joint was open couldn’t be predicted by the present formula for shear strength; (2) A simplified failure mode to such beams was suggested, and formulas to evaluate the resistance of the joints were proposed. Researchers also used finite element model to simulate response of push off test [14]. Theoretical mechanical model by combination of fracture mechanics and truss model theory was used to analyze the shear loading test of precast joints [9]. But the study on prismatic shear key is very limited, and the contribution of various mechanisms still has controversies.

1.2. Specimen fabrication In order to study the performance of specimens with different section and shear key configurations under direct shear loading, five batches of specimens were designed. Each batch had two same specimens which have 1 MPa or 2 MPa confinement stress, respectively. In reference of Anagnostopoulou et al. [3] for the design of shake table test specimen of precast column and girder bridge and practical design of common monolithic concrete bridges, a superstructure total load ratio of live load plus dead load was approximately N / fc′ A = 0.10. For concrete grade C30, the cubic compressive strength is fcu, k = 30 MPa, the cylinder compressive strength is approximately fc′ = 0.79 fcu, k = 23.7 MPa. The compressive stress on the column section is approximately σ = 0.1 fc′ = 2.37 MPa. So in the test, the stress on interface chose 1 MPa and 2 MPa. The specimen configurations are shown in Fig. 1. The typical specimen side view after assembled in laboratory is illustrated in Fig. 1(a). One specimen is composed with two L type parts. If the specimen has shear keys, the male shear key part is always put underneath the female part in test. Fig. 1(b)–(f) designates the interface configuration. Fig. 1(g) presents the section of shear key. The specimen properties are listed in Table 1. The specimen identifier is represented as R(H)-D (Q,N)-S-n, where R represents section shape of rectangular, H means section shape of hollow, D indicates double (two) shear keys, Q is for quard (four) shear keys, N dictates no (without) shear key, S designates specimen, the number following S indicates the confinement stress (MPa) on interface.

1.1. Research significance The preceding research on shear response of precast segmental bridge dry or epoxied joints greatly contributed to the fast application of this kind of bridge types. But the research still has inadequacy to be overcome, (1) Most of the research objectives are joints on the web or flange of girder in precast superstructures, shear response of joints in precast column is still debatable; (2) Most of the research is about castellated joints, the failure mechanism of prismatic shear key joints is seldom reported; (3) The shear capacity of precast joint predicted by current code is not well recognized. In order to adapt the fast development of precast substructures application in moderate to high seismicity areas, a series of experiments were carried out to study the shear performance of dry joints in precast segmental bridge columns. Five groups of shearing off test specimens were fabricated for different confining stresses. Among them, two groups are flat joints with solid or hollow interface, three groups are keyed joints with different shear key geometries, numbers and solid or hollow interface. The failure sequence and pattern of cracking or push off of shear keys of the five group precast dry joint specimens were discussed firstly. Then the shear force versus relative slippage curves of test results were presented and explained. To revise the current code for shear response design of precast dry joints, shear capacity formulas deduced from failure plane of tensile and compressive envelope of concrete, or from the tensile failure plane of concrete were presented. The proposed formula was validated for shear capacity estimation of precast dry joint by the test results of this contribution and other

2. Experiment setup and instrumentation The test setup and instrumentation is shown in Fig. 2. The measurement instrumentation can be divided into three categories, namely strain, displacement, and force measurement. The shear force is applied as direct shear through the interface plane, and the interface has no flexural moment theoretically. The shear force is applied on top of the female part by a universal testing machine. 258

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Table 1 Specimen properties. Specimen

Joint interface (mm)

Hollow area (mm)

Shear key dimension (mm)

Shear key number

Nominal confinement stress (MPa)

RDS-1 RDS-2 RQS-1 RQS-2 RNS-1 RNS-2 HQS-1 HQS-2 HNS-1 HNS-2

450 × 450

D70

200 × 60

2

150 × 60

4

–

0

150 × 60

4

–

0

1.0 2.0 1.0 2.0 1.0 2.0 1.0 2.0 1.0 2.0

210 × 210

2.1. Specimen failure pattern

The loading protocol is displacement controlled monotonic loading. The loading velocity is about 0.50 mm/minute. The data acquisition system includes shear force on top of specimen, interface lateral and vertical displacement, concrete surface strain and confinement stress, which are measured by load cell-2, displacement gauge, strain gauge, and load cell-1, respectively. The readings of the instrumentations are recorded by a static-dynamic strain indicator. The test setup of specimen HQS-1 is shown in Fig. 3. The front face of specimen is instrumented with hydraulic jack to produce prestressing force, and the back face of the specimen is installed with steel plate and anchorage of prestressing strands.

The shear failure mode after test of specimen RDS-1 is shown in Fig. 4(a). The failure of specimen under 1 MPa confinement stress is nearly totally shearing off of male shear keys, and diagonal crack on left female shear key toward the corner. The failure characteristics of specimen RDS-2 is similar to that of specimen RDS-1, completely shearing off of male shear key and diagonal crack in female shear key upper corner (Fig. 4(b)). The difference lies in the shearing off extent of male shear keys. Specimen RQS-1 exhibits shearing off of four male shear keys, while

Fig. 2. Specimen instrumentation. 259

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(a) Front view

(b) Back view

Fig. 3. Test setup of specimen HQS-1.

larger interface area and smaller key base area compared with HQS-1. The same reason can be applied to RDS-2 and HQS-2. RQS-1 has same interface area, larger key area, and less confinement stress compared with RDS-2, thus the resulting shear capacities are similar. And RQS-2 has largest shear capacity for the relatively larger interface area, key area and higher confinement pressure. Another feature of shear resistant force-slippage curve is the post peak shear capacity of specimens. The shear keys are totally or partially sheared off after the peak strength in the force-displacement curve. So theoretically, the shear capacities of specimens are contributed by the friction only in this loading stage. The post peak shear capacity of HQS1 is the lowest, RQS-1 and RDS-1 have similar post peak shear strength in the 1 MPa confinement stress batch. HQS-2 has the relatively lower shear capacity of after peak, RQS-2 and RDS-2 have analogous post peak shear capacities in the 2 MPa confinement stress batch. The shear resistant force-relative slippage curves of specimen RDS-1 and HQS-2 present three stages, linear-nonlinear ascending stage, leveling stage, and descending stage. The shear force-slippage curves of specimen HQS-1, RQS-1, and RQS-2 exhibit three stages, namely, ascending stage, abrupt descending stage, and gently descending stage. The shear force-slippage curves of specimen RDS-2 shows two stages, linear-nonlinear ascending stage, and descending stage. It is believed that the sudden drop before summit is the point where partial shear keys (top male and bottom female) are sheared off, the summit is the point where all shear keys are sheared off, and the abrupt descending after peak is the point where large slip initiates. The shear capacity versus slippage relations of four flat specimens are depicted in Fig. 6. The shear resistance of four flat specimens was divided by 250 kN to make the shear capacity dimensionless, as shown in Fig. 6. All shear capacity versus slippage curves presented two stages, namely, linear-nonlinear ascending stage and leveling stage. The shear capacity of RNS-1 is a little higher than HNS-1, but the shear capacity of RNS-2 is much higher than HNS-2. The shear capacity of flat joint specimens is contributed by the small asperities in the interface. As the shear loading increases to some extent, the interlocking force by small asperities interlocking is overcame, large relative slip between the two

the female shear keys remain intact (Fig. 4(c)). Under 2 MPa confinement stress, the specimen RQS-2 presents failure mode of four male shear keys shearing off and diagonal cracks in upper left female shear key (Fig. 4(d)). Specimen RNS-1 has no shear key on the interface, the main failure mode is scratches on interface and diagonal cracks of 45 degrees inclined angle from the horizontal direction through the center hole (Fig. 4(e)). When the confinement pressure increases from 1 MPa to 2 MPa, the failure pattern of specimen RNS-2 under direct shear is the scratches on interface and cross cracks through the center hole (Fig. 4(f)). The failure mode of specimen HQS-1 is shearing off of upper male shear key, lower female shear key, and complete or partial of middle male shear keys (Fig. 4(g)). Concrete cracking near outer edge of middle right female shear key is observed. After the confining stress increases from 1 MPa to 2 MPa, the specimen HQS-2 exhibits totally shearing off of three upper male shear keys and one lower female shear keys. Diagonal cracks on both middle female shear keys are also evident (Fig. 4(h)). Specimen HNS-1 and HNS-2 only has scratches on the interface, no obvious cracks are observed (Fig. 4(i) and (j)).

2.2. Shear resistant force-relative slippage relation The shear resistant force versus relative vertical displacement curves of six keyed specimens are shown in Fig. 5. In order to compare the relative value of shear resistance, the shear resistance of six keyed specimens was divided by 400 kN to make the shear capacity dimensionless, as shown in Fig. 5. The first characteristic is the peak of force-displacement curve, i.e. shear capacity. The shear capacities of specimens RDS, RQS and HQS increase notably as the confinement stress increasing from 1 MPa to 2 MPa. Among all specimens, RDS-1 and HQS-1 have similar shear capacities, RDS-2, RQS-1 and HQS-2 have comparable shear capacities, and RQS-2 has largest shear capacity. The results can be explained from the different interface areas, key number and geometries of specimens. Specimen RDS-1 has 260

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(i) Male key part

(ii) Female key part

(i) Male key part

(b) RDS-2

(a) RDS-1

(i) Male key part

(i) Male key part

(ii) Female key part

(ii) Female key part

(d) RQS-2

(c) RQS-1

(i) Lower part

(ii) Female key part

(i) Lower part

(ii) Upper part

(ii) Upper part (f) RNS-2

(e) RNS-1 Fig. 4. Specimen failure pattern.

element of concrete taken from the shear plane are shown in Fig. 7(b). There are three kinds of stresses on the tiny element. The shear stress τ due to restraint provided by confinement force N , the vertical normal stress σy due to the applied shear force Q , and the lateral normal stress σx due to the applied confinement force N . The averaged stress can be used for the unknown of exact stress distribution.

interfaces initiates, and the shear capacity curve exhibits leveling off. The Mohr-Coulomb relation can be used in the friction of interface. So the friction coefficient can be calculated with the average shear stress divided by normal stress. The friction coefficients are 0.769, 0.864, 0.851 and 0.766 for specimen RNS-1, RNS-2, HNS-1 and HNS-2, respectively. And the average friction coefficient of the preceding specimens is 0.813.

τ= 3. Shear capacity formulae for precast to precast dry joint

Q bd

(1)

σx =

N bd

(2)

3.1. Proposed shear failure plane tangent with envelope of tensile and compressive damage surface

σy =

Q bw

(3)

Consider a direct shear off specimen with width w , thicknessb , and shear contact plane depth d Fig. 7(a)). The stresses acting on a tiny

The aim is to determine the state of these stresses which will cause in failure of concrete in the shear plane. The failure envelope proposed 261

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(i) Male key part

(ii) Female key part (g) HQS-1

(i) Male key part

(i) Male key part (ii) Female key part (h) HQS-2

(ii) Female key part (i) HNS-1

(i) Male key part (ii) Female key part (j) HNS-2

Fig. 4. (continued)

1.0

RDS-2 RQS-2 HQS-2

0.8 0.6

RQS-1

0.4

HQS-1

RDS-1

0.2

1.0

HNS-2

0.8 RNS-1

0.6 0.4

-0.2 -2

0

2

4

6

8

10

12

14

16

18

HNS-1

RNS-1(Peak159.4kN) RNS-2(Peak 342.2kN) HNS-1(Peak 145.3kN) HNS-2(Peak 243.0kN)

0.2 0.0

0.0 -0.2

RNS-2

1.2

Relative shear capacity

1.2

Relative shear capacity

1.4

RDS-1(Peak 282.0kN) RDS-2(Peak 413.3kN) RQS-1(Peak 404.7kN) RQS-2(Peak 494.5kN) HQS-1(Peak 255.5kN) HQS-2(Peak 389.1kN)

1.4

-2

Relative vertical displacement (mm)

0

2

4

6

8

10 12 14 16 18 20 22 24

Relative vertical displacement (mm)

Fig. 5. Shear capacity of keyed specimens.

Fig. 6. Shear capacity of flat specimens.

by Hofbeck et al. [5] is used. The failure envelope shown in Fig. 8 consists of two parts: (1) A line L1 inclined at α = 37 degree to the confining stress axis and tangent to the Mohr circle representing failure in uniaxial compression; (2) A line L2 drawn from the point of intersection of the first line with the ordinate, and tangent to the Mohr circle denoting failure in uniaxial tension. In building the failure envelope for some strength of concrete, the uniaxial compression strength fc is taken as 0.85fc′ and the tensile strength ft is taken as 0.23 fc′ ksi(0.604 fc′ MPa) [2]. Based on former experimental observation, the shear resistant mechanism of the test specimen shown in Fig. 1 has two parts, the first part

is the shear resistance provided by shear keys, and the second part is the frictional force provided by interlocking of small particle on shear plane under confining stress. So the expression of shear resistant force can be written as:

Vj = Akey fc′ (C1 + C2 σn ) + 0.6Asm σn

(4)

The inclined angle of line L2 relative to X axis is θ .

θ = 2β−90°

(5)

Using the relationship of compression strength, fc′ = 0.79fcu, k , the angles θ and β of different concrete strength grades are listed in Table 2. The detailed deduction of the coefficients C1 and C2 can refer to Appendix A. The coefficient C1 of shear strength provided by shear keys 262

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7

C10 C30 C50 C70 C1=0.165

6

Coefficient C1

5

C20 C40 C60 C80

4 3 2 1 0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

w/(2d)

(a) Push-off specimen

Fig. 9. Coefficient C1 variation with concrete compressive strength and shear test specimen geometry in the proposed formula.

(b) Stress state

Fig. 7. Direct shear off specimen and stress state.

where A, B and C are coefficients defined in Appendix A. There are three parameters in Eq. (7), for convenience, the specimen geometry parameter w /2d = 0.5 is used. Then the variation of coefficient C2 with concrete strength and confinement stress was depicted in Fig. 10. It can be observed from Fig. 10, the coefficient C2 varies with confinement stress σn (σx = σn in Eq. (7)). As the concrete compressive strength increases, the influence of confinement stress on the coefficient C2 decreases. The relation between concrete compressive strength fc′ and the coefficient C2 was linear interpolated for the calculation results by Eq. (7) at σx = 10 MPa. The third times of standard deviation offset lines of the linear interpolation were also shown in Fig. 11. The linear interpolation minus third times of standard deviation line equation was shown in Eq. (8).

C2 = −0.0006fc′ + 0.04133

(8)

Substitute C1 = 0.165 and Eq. (8) into Eq. (4), the shear strength of precast joint can be written as,

Vj = Akey fc′ [0.165 + (−0.0006fc′ + 0.04133)·σn] + 0.6Asm σn 4. Degree diagonal compressive shear failure plane with fixed principal tensile stress (AASHTO 1999 method)

Fig. 8. Failure envelope under combined normal stresses and shear stresses.

Roberts [12] deduced the direct shear capacity of dry precast shear key joint assuming 45 degrees compression strut angle using the Mohr stress circle theory. The procedure assumed the tensile strength of concrete is fixed when the shear force loading increased. This procedure later was adopted by AASHTO [1] code for shear strength estimation of precast dry or epoxied joints. The shear strength of keyed dry joint in push off can be written in a function of the tensile strength of concrete and the compressive stress on the shear plane. The equation can be written in the form,

Table 2 Angle parameter variation with concrete compressive strength. Concrete grade fcu, k

C10

C20

C30

C40

C50

C60

C70

C80

β (Degree) θ (Degree)

63.06 36.11

70.23 50.46

73.65 57.29

75.74 61.48

77.19 64.39

78.28 66.56

79.13 68.25

79.81 69.63

without confinement can be written as,

0.2125cosθ C1 = w (0.447−sinθ) 2d + 0.894

Vj = Akey fc′ (C1 + C2 σn ) + 0.6Asm σn

C2 =

−B + 2Aσx fc′

−

Vj = Akey fc′ (12 + 0.017σn ) + 0.6Asm σn (lb, σn and fc′ in psi)

0.2125cosθ

(11a)

Vj = Akey fc′ (1.008 + 0.205σn) + 0.6Asm σn (N, σn and fc′ in MPa)

w

(0.447−sinθ) 2d + 0.894⎤ σx ⎡ ⎣ ⎦

(10)

The coefficient C1 = 12.14 (in psi) and C1 = 1.008 (in MPa) are derived using the 45 degree diagonal compressive shear failure plane assumption. The variation of coefficient C2 with the confinement stress and concrete compressive strength is depicted in Fig. 12. A mean value of C2 = 0.017 (σn in psi) and C2 = 0.205 (σn in MPa) are used in AASHTO [1] specification, the shear strength of keyed joint can be written as,

(6)

Considering more common cases of concrete strength varying from C10 to C80 and specimen geometry w/2d = 0.1–2.0, the coefficient C1 variation with w /2d and concrete strength was depicted in Fig. 9. It can be observed that C1 = 0.165 is conservative for most of the cases considered. The coefficient C2 can be written as,

B2−4AC

(9)

(11b)

(7) 263

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0.08

C10 C30 C50 C70

0.04

0.02

0.00

0

2

4

6 8 σn(MPa)

10

0.2

0.0 0

10

20

30

40

50

60

70

fc'(MPa)

0.06

Calculated Linear interpolation Interpolation+3σ Interpolation-3σ

Fig. 13. Coefficient C2 interpolation with varying concrete strength at confinement stress σn = 10 MPa in 45 degree compressive strut failure formula.

30 Shear key shear stress resistance (MPa)

0.05

0.04

0.03

0.02

0.01

0.00 0

10

20

30

40

50

60

70

Fig. 11. Coefficient C2 interpolation with varying concrete strength at confinement stress σn = 10 MPa in the proposed formula.

0.6

C10 C30 C50 C70 C2=0.205

0.5 0.4

C10 C30 C50 C70 C10 C30 C50 C70

25 20

C20 C40 C60 C80 by new formula C20 C40 C60 C80 by AASHTO

15 10 5 0

fc'(MPa)

Coefficient C2

0.3

0.1

12

Fig. 10. Coefficient C2 variation with concrete compressive strength and confinement stress in the proposed formula.

Coefficient C2

Calculated Linear interpolation Interpolation+3σ Interpolation-3σ

0.4

Coefficient C2

Coefficient C2

0.06

0.5

C20 C40 C60 C80

0

2

4

6 σn(MPa)

8

10

Fig. 14. Shear key shear stress resistance comparison.

Fig. 13. The equation of linear interpolation line minus three times of standard deviation line can be written as,

C20 C40 C60 C80

C2 = −0.0037fc′ + 0.23916

(12)

Thus Eq. (11b) can rewritten as,

Vj = Akey fc′ [1.008 + (−0.0037fc′ + 0.23916)·σn] + 0.6Asm σn 0.3

(13)

The comparison of shear stress capacity provided by shear keys estimated by first part of Eq. (9) and that by first part of Eq. (11b) is shown in Fig. 14. The new formula calculation results by Eq. (9) are lower than those of AASHTO [1] estimation by Eq. (11b), especially at high confinement stress cases.

0.2 0.1

12

0

2

4

6 8 σn(MPa)

10

12

5. 30 and 60 degree compressive strut failure plane with fixed principal tensile stress

Fig. 12. Coefficient C2 variation with concrete compressive strength and confinement stress for 45 degree compressive strut failure.

Similar procedure as 45 degree diagonal compression shear failure plane shown in the preceding section can be applied to the α1 degree diagonal compression shear failure plane case. The coefficients of Eq. (10) can be derived in the following procedure. The shear stress for no confinement stress case,

From Fig. 12, it is evident that the coefficient C2 was in relation with confinement stress and concrete compressive strength. In order to calculate the coefficient C2 precisely, the influence of concrete strength was considered. The coefficient C2 at 10 MPa confinement stress was calculated for different concrete compressive strength. Linear interpolation of relation between C2 and fc′ was shown in the middle line of

τ1 =

264

tanα1 +

4 + tan2 α1 ft 2

(14)

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0.40

C10 C30 C50 C70

0.35

C10 C30 C50 C70

0.7 Coefficient C2

Coefficient C2

0.30

0.8

C20 C40 C60 C80

0.25 0.20 0.15

0.6

C20 C40 C60 C80

0.5 0.4 0.3

0.10 0.05

0.2 0

2

4

6

8

10

12

0

2

4

6

σn(MPa)

Fig. 15. Coefficient C2 variation with concrete compressive strength and confinement stress for 30 degree compressive strut failure.

τ2 =

(σx + ft )tanα1 +

(σx +

+

4(ft2

(15)

The increase of shear stress with confinement relative to the no confinement case,

σx tanα1 +

(σx + ft )2tan2 α1 + 4(ft2 + ft σx ) − 4 + tan2α1 ft

The shear stress coefficient can be written as,

C2 =

τ1 fc′

= 0.3114(tanα1 +

4 + tan2 α1 ) (17)

σx tanα1 + τin = fc′ σx

(σx + ft )2tan2 α1 + 4(ft2 + ft σx ) − 4 + tan2α1 ft 2 fc′ σx (18)

Substitute the angle α1 = 30 degrees into Eqs. (17) and (18), then coefficient C1 = 0.828 is derived. Coefficient C2 variation with concrete grade and confinement stress is depicted in Fig. 15. Using similar procedure with 45 degree diagonal failure plane method, the influence of concrete compressive strength on coefficient C2 was interpolated as

(19)

Vj = Akey fc′ [0.828 + (−0.0023fc′ + 0.15837)·σn] + 0.6Asm σn

(20)

C2 = −0.0063fc′ + 0.39394

(21)

Vj = Akey fc′ [1.363 + (−0.0063fc′ + 0.39394)·σn] + 0.6Asm σn

(22)

5.1. Precast joint direct shear loading test results and validation of the proposed formula The comparison between direct shear test carried out by the 0.7

Calculated Linear interpolation Interpolation+3σ Interpolation-3σ

Calculated Linear interpolation Interpolation+3σ Interpolation-3σ

0.6 0.5

0.2

Coefficient C2

Coefficient C2

0.3

C2 = −0.0023fc′ + 0.15837

For α = 60 degrees, the coefficient C1 = 1.363 is derived. Coefficient C2 variation with confinement stress is depicted in Fig. 17. The interpolation of coefficient C2 variation with concrete compressive strength was shown in Fig. 18. The equation of linear interpolation minus three times standard deviation can be written as that of Eq. (21). The shear strength estimation of precast dry joints by 60 degree diagonal failure plane assumption was shown in Eq. (22).

2 (16)

C1 =

12

shown in Fig. 16. The linear interpolation relation minus three times standard deviation was depicted in Eq. (19). The shear strength estimation of precast dry joints by 30 degree diagonal failure plane assumption can be written in form of Eq. (20).

+ ft σx )

2

τin = τ2−τ1 =

10

Fig. 17. Coefficient C2 variation with concrete compressive strength and confinement stress for 60 degree compressive strut failure.

The shear stress for with confinement stress case,

ft )2tan2 α1

8

σn(MPa)

0.1

0.4 0.3 0.2 0.1

0.0

0.0 0

10

20

30

40

50

60

70

0

fc'(MPa)

10

20

30

40

50

60

70

fc'(MPa)

Fig. 16. Coefficient C2 interpolation with varying concrete strength at confinement stress σn = 10 MPa in 30 degree compressive strut failure formula.

Fig. 18. Coefficient C2 interpolation with varying concrete strength at confinement stress σn = 10 MPa in 60 degree compressive strut failure formula. 265

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Table 3 Interface shear strength comparison between formula and test results by the author. Specimen

RDS-1 RDS-2 RQS-1 RQS-2 RNS-1 RNS-2 HQS-1 HQS-2 HNS-1 HNS-2 Mean Standard deviation

fc′ (MPa)

22.83 19.20 23.78 19.28 24.02 18.57 24.10 18.72 23.15 18.57 – –

σn (MPa)

Compressive displacement (mm)

Beginning

Max

Min

1.091 2.014 0.983 2.022 1.033 2.003 1.257 2.038 1.075 1.997 – –

– 2.021 1.154 2.065 1.079 – 1.283 2.096 1.203 2.011 – –

– 2.005 1.119 2.008 1.046 – 1.282 2.043 1.088 2.003 – –

Test (kN)

0.616 0.674 1.221 0.453 0.163 0.128 1.837 1.105 1.093 1.256 – –

282.0 413.3 404.7 494.5 159.4 342.2 255.5 389.1 145.3 243.0 – –

AASHTO

Eq. (8)

−45 deg strut

−45 deg −3σ

−30 deg −3σ

−60 deg −3σ

0.905 0.870 0.761 0.852 0.780 0.694 1.238 0.957 0.705 0.783 0.855 0.221

0.883 0.851 0.738 0.828 0.780 0.694 1.190 0.928 0.705 0.783 0.838 0.224

0.788 0.778 0.639 0.736 0.780 0.694 1.025 0.811 0.705 0.783 0.774 0.260

1.070 0.995 0.931 1.009 0.780 0.694 1.516 1.157 0.705 0.783 0.964 0.253

0.784 0.760 0.641 0.714 0.780 0.694 1.038 0.778 0.705 0.783 0.768 0.267

Table 4 Precast joint shear strength test results by other researchers. Test author

Test by Zhou et al. [17]

Test by Buyukozturk et al. [4]

Test by Jiang et al. [7]

Test by Koseki and Breen [10]

Mean Standard deviation

Specimen

M1-D-F M2-D-F M3-D-F M1-D-K1 M2-D-K1 M3-D-K1 M4-D-K1 M4.5-D-K1 M0.5-D-K3 M1-D-K3 M1.5-D-K3 M2-D-K3 F-D-100 F-D-300 F-D-500 K-D-100 K-D-300 K-D-500 F-01 F-02 K1-01 K1-02 K1-01-S K2-01 K2-02 K3-01 K3-02 NK SK MK NKE SKE MKE – –

fc′ (MPa)

52.2 52.8 53.5 44.4 57.9 64.5 36.9 37.7 41.9 37.7 59.1 63.7 47.4 47.4 47.4 48.4 40.7 49.4 37.64 37.64 38.16 38.58 43.06 39.27 38.90 38.87 38.87 48.82 46.23 49.78 43.79 42.06 44.44 – –

σn (MPa)

1.00 2.00 3.00 1.00 2.00 3.00 4.00 4.50 0.50 1.00 1.50 2.00 0.69 2.07 3.45 0.69 2.07 3.45 1.00 2.00 1.00 2.00 1.00 1.00 2.00 1.00 2.00 2.91 2.99 2.85 2.95 2.88 2.97 – –

Test (kN)

34.0 73.5 110.0 202.0 336.0 404.0 373.0 375.0 339.5 451.3 661.0 740.0 3.2 8.4 10.0 65.5 84.2 111.2 12.4 23.4 88.3 113.9 112.3 135.4 180.5 181.3 235.6 162.2 197.8 213.4 257.8 297.8 275.6 – –

AASHTO

Eq. (8)

45 deg strut

45 deg −3σ

30 deg −3σ

60 deg −3σ

0.905 0.870 0.761 0.852 0.780 0.694 1.238 0.957 0.705 0.783 0.882 0.816 0.818 1.074 0.892 0.918 0.905 0.970 1.631 1.304 1.216 1.228 0.750 0.857 1.200 0.942 0.874 0.890 0.969 1.026 0.916 0.879 0.763 0.930 0.254

0.883 0.851 0.738 0.828 0.780 0.694 1.190 0.928 0.705 0.783 0.882 0.816 0.818 0.967 0.688 0.613 0.738 0.776 1.545 1.197 0.974 0.902 0.750 0.857 1.200 0.862 0.737 0.646 0.969 1.026 0.841 0.760 0.689 0.842 0.278

0.788 0.778 0.639 0.736 0.780 0.694 1.025 0.811 0.705 0.783 0.882 0.816 0.818 0.803 0.587 0.538 0.618 0.651 1.272 0.984 0.820 0.770 0.750 0.857 1.200 0.712 0.607 0.542 0.969 1.026 0.696 0.633 0.572 0.752 0.320

1.070 0.995 0.931 1.009 0.780 0.694 1.516 1.157 0.705 0.783 0.882 0.816 0.818 1.292 0.894 0.770 0.979 1.026 2.085 1.617 1.285 1.172 0.750 0.857 1.200 1.161 0.997 0.858 0.969 1.026 1.129 1.013 0.922 1.021 0.331

0.784 0.760 0.641 0.714 0.780 0.694 1.038 0.778 0.705 0.783 0.882 0.816 0.818 1.061 0.857 0.801 0.759 0.806 1.645 1.217 1.234 1.186 0.750 0.857 1.200 0.988 0.785 0.761 0.969 1.026 0.860 0.788 0.746 0.872 0.274

during the loading procedure. The increment of tendon stress of specimen RQS-1 is 0.174 MPa, and that of specimen HNS-1 is 0.119 MPa. The joint compression of specimens RDS, RQS and RNS is normally less than 1.0 mm. But the joint compression of specimens HQS and HNS is normally larger than 1.0 mm. To justify the validity of the different form of shear capacity formulae derived and introduced in this contribution, the calculation results utilizing the AASHTO [1] specification and its derivatives and the proposed formula are compared with test data by researchers (Table 4). Generally, the proposed formula gives relatively accurate estimation for the shear capacity of precast joint as the 45 degree strut AASHTO

authors, and the formulae estimation with AASHTO equation and its derivatives (45, 30 and 60 degree compressive strut model), and author proposed formula (Eq. (9)) was listed in Table 3 column 7 to column 12. The confinement stress variation on the interface and the lateral relative displacement between two parts test results were also presented in Table 3 column 3 to column 6. The AASHTO 45 degree strut formula predicted well for both keyed and flat dry joint shear capacity. The AASHTO 30 degree strut formula gave similar results with the proposed formula of Eq. (9), both underestimated the shear strength of keyed dry joints. The AASHTO 60 degree strut formula overestimated the shear strength of hollow keyed dry joints. The stress of tendon increased 266

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10

RDS shear key RQS shear key HQS shear key Proposed formula shear key AASHTO formula shear key RNS HNS Proposed and AASHTO smooth

Interface shear stress (MPa)

9 8 7 6 5 4 3 2 1 0 1

Confinement stress (MPa)

2

Proportion of shear capacity assumed by shear keys

Fig. 19. Shear stress variation with confinement stress.

1.0 RDS-Test RQS-Test HQS-Test RDS-Proposed RQS-Proposed HQS-Proposed RDS-AASHTO RQS-AASHTO HQS-AASHTO

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 1.0

Confinement stress (MPa)

2.0

Fig. 21. Mesh grid of the planar finite element model of joint.

Fig. 20. Shear capacity assumed by shear keys.

The reason was on this kind of solid section shear key contribution increased very slowly compared with that by smooth area with increasing of confinement pressure, and the shear key had enough cover thickness to prevent premature failure under shear force. So as the confinement stress increased, the shear stress on shear keys decreased. But shear stress on specimen HQS increased with the increasing of confinement, because on this specimen the shear capacity contribution of shear key was higher than 50%, and the shear keys had relatively high stiffness than the nearby smooth area. The AASHTO formula (Eq. (11b)) gives higher prediction to shear stress on shear keys with test results of specimen HQS. The proposed formula (Eq. (9)) provides reasonable prediction to test results of specimen HQS. For both equations, stress on shear keys increases with increasing of confinement stress. The stress on smooth area of specimen RNS and HNS increased as the confinement stress increased, which showed same trend with proposed or AASHTO formula. The proportion of shear capacity assumed by shear keys in test, proposed formula and AASHTO formula were presented in Fig. 20. Specimen RQS-1 has the highest shear capacity assumed by shear keys (0.678), the second is HQS-1 (0.561), and the third is RDS-1 (0.503), in the 1 MPa confinement stress case. The ratio of shear capacity assumed by shear keys decreased as the confinement increased, because the shear strength provided by friction increased very fast, but the shear strength offered by shear keys increased very slowly. The proportion of shear capacity assuming by shear keys became 0.434, 0.517 and 0.272 for specimen RQS-2, HQS-2 and RDS-2 under 2 MPa confinement stress, respectively. The shear capacity undertaken by shear keys of specimen

specification formula does. The 30 and 60 degree diagonal shear failure plane formulas provide lower bound and upper bound estimation of test data, respectively. The mean value of non-dimensional calculation results by 45 degree failure plane formula is 0.952. The 30 degree and 60 degree failure plane formulae give 0.807 and 1.196 prediction relative to test data, respectively. The proposed formula gives best estimation of 0.994 in average. The proposed formula gives the best estimation of mean shear capacity. The standard deviation of 45, 30, 60 degree failure plane mechanism, and proposed formula is 0.267, 0.289, 0.450, and 0.294, respectively. The standard deviation of the proposed formula is a little higher than 45 degree strut formula. 5.2. Shear key shear stress and shear strength contribution analysis Assume the shear stresses on smooth area are the same for specimens with same total interface area and confinement stress, then utilizing the tested results, the shear stress on shear keys can be calculated. This may provide an insight into the role the shear key plays on the total shear capacity and its stress variation rules. For the specimens tested in this contribution, RNS-1 was used to calculate the shear stress of RDS-1 and RQS-1, HNS-1 was referred when calculating HQS-1. The shear stress on shear keys and smooth area of interface are depicted in Fig. 19 utilizing the assumption afore mentioned. The shear stress on shear keys is between 4.0 and 7.6 MPa, which is higher than that on smooth area of 0.8–1.7 MPa. The shear stress on specimen RDS and RQS shear key decreased as the increasing of confinement stress. 267

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(a) Contour at start

(c) Contour at middle

(b) Vector at start

(d) Vector at middle

Fig. 22. Minimum principal stress.

Referring to Figs. 2 and 3, the 2D FE model was composed with six components, lower male concrete part, upper female concrete part, left steel plate, right steel plate, upper steel plate, and prestressing tendon. All materials are assumed to be elastic, which means the model simulation is applicable to the before cracking stage of joint specimen. The dimension of the specimens was same to specimen RDS, except the shear key was changed to castellated type for simplicity of modeling. Central located hole was not modeled in the FE model. The section area of tendon is 420 mm2, and the prestressing stress is set as 471.6 MPa in the initial loading stage, which becomes 602.7 MPa after releasing. The corresponding average confinement stress on the interface is 1.250 MPa at starting of shear loading. The boundary condition is the tendon embedded in 40 mm thick steel plate on left and right side of two parts, and the bottom of male shear key part fixed on the floor. The shear loading was exerted on top of upper steel plate. The interface of two concrete parts is defined as surface-to-surface contact, the normal direction is hard contact, and the

HQS changed very slightly as the confinement stress increased. There are two reasons, first is the ratio of key to smooth area of specimen HQS is relatively higher, and the second is precast quality is the male-female shear key friction is fully developed. The proposed formula gave better estimation than AASHTO formula on the proportion of shear capacity undertaken by shear keys. Both equations used real concrete strength in calculation.

6. Qualitative investigation of stress transfer and distribution For the complexity of the interface configuration studied in this contribution, it needs three dimensional (3D) finite element (FE) model to investigate the loading performance of the joint. But the 3D FE model usually requires more calculation efforts, which is not good choice for bridge design engineers. To study the general stress transfer mechanism of keyed dry joints, a two dimensional (2D) FE model was utilized, as shown in Fig. 21. 268

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Fig. 22. (continued)

author and other researchers. A 2D FE model was calibrated to qualitatively study the stress transfer and distribution of the joint specimens. Some useful results were concluded.

tangent direction is penalty type friction slip, with a friction coefficient of 0.6. The interact of left and right plates to concrete and upper steel plate to concrete are defined as tie constraint, which means the two areas of master and slave are fixed together, even the mesh of two areas are not consistent. The shear capacity of the joint in Fig. 21 was estimated with AASHTO 45 degree strut formula Eq. (11b), which is about 618.9 kN. Three stages results at beginning (0%), middle (50%) and end (100%) of shear strength, are extracted from the simulation results. The minimum compressive principal stress contour and vector are shown in Fig. 22. Small arrow represents the vector of the stress component, big arrow designates the resultant of the vector. Before the vertical shear loading is exerted, the minimum principal stress and vector are shown in Fig. 22(a) and (b), respectively. The compressive stress is transferred near the center of male part interface axis, and spreads through two corner of shear key to the female part, and to the right side of female part at last. The direction of resultant compressive principal stress in the beginning is normal to the short edge of shear key. Then as the shear force acted on the joint, compressive principal stress in vertical direction is observed. The maximum stress is concentrated at the upper corner of shear key (Fig. 22(c)). The resultant stress illustrated the compressive stress flow direction and intensity (Fig. 22(d)). As the shear force increased to shear capacity, the compressive stress in the concrete part increased. The vertical stress became larger than lateral confinement stress (Fig. 22(e)). The vector intensity increased, and the resultant force was more concentrated in the upper corner of shear key (Fig. 22(f)). Thus a compressive inclined strut near the joint interface was formed, which showed tendency of upper compression and lower opening in the shear key interlocking location. This phenomenon was also observed in the test procedure.

(1) Higher confinement pressure can ensure complete function of shear keys under shearing force. Large shear key number and effective shear key base area are very important to maintain higher shear strength of joint. Shear off failure of male shear keys were observed in solid section specimen and vertical shear keys of hollow section specimens. But for the hollow sections, the male and female shear key cover concrete thickness can be the control parameter. Some of the shear keys exhibited flexural failure due to the insufficient cover thickness, both in upper male and lower female shear keys of hollow section specimens. Diagonal cracks toward the outer face in female shear key are very common failure type in solid section specimens. (2) The relative slippage-relative shear resistant force curve of ten specimens gave insight into the outcome of shear strength by various combinations of flat area, key number or key base area, and confinement pressure. The shear strength of keyed specimens can be divided into three grades for specimens tested. The first grade is solid section specimen with four keys and 2 MPa pressure. The third grade specimens are solid section specimen with two keys and hollow section specimen under 1 MPa confinement stress. The second grade includes the rest of keyed specimens. The shear strength provided by the shear keys increased slowly compared with the shear strength provided by smooth area friction when the confinement stress increased. So as the confinement stress increased, the ratio of keyed and flat joint shear strength decreased. (3) Two formulation methods of shear strength prediction formula were presented, one is based on the failure envelope of both uniaxial compressive and tensile strength, the other one is inclined failure strut model. The latter was adopted in AASHTO [1] specification. A total of four formulae are deducted including the AASHTO equation. For the test conducted by the author, the AASHTO 45 degree strut formula predicted well for joint shear capacity. The AASHTO 30 degree strut formula and the proposed formula underestimated the shear strength of specimens. The AASHTO 60 degree strut formula overestimated the shear strength of hollow keyed dry joints. The tested specimens showed

7. Conclusions Ten dry joint precast column shearing off specimens were fabricated and tested to failure in this contribution. The failure pattern and sequence were investigated. Relative slippage and relative shear resistant force relations were recorded and analyzed. Two failure mechanism and envelope formation methods were utilized to construct the shear strength formulae for precast concrete dry joints. Then the formulae were used to predict the shear strength of joint specimens tested by the 269

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proportion of shear capacity assumed by shear keys than the AASHTO formula did. (5) The stress transfer and distribution properties on the precast joint specimens were investigated through a planar FE model. Under the biaxial loading of axial compression and vertical shear, each specimen part can be looked as an eccentrically compressed column. The stress distribution on the interface transferred from uniform compression to compression and flexure as the vertical shear load increased. The vector intensity increased, and the resultant force was more concentrated in the upper corner of shear key. Thus a compressive inclined strut near the joint interface was formed, which showed tendency of upper compression and lower opening at joint.

compression in upper corner of joint, and the prestressing tendon stress increased in different extent. The proposed formula predicted well for the tests by other four researchers than AASHTO 45 degree strut formula did. While the 30 degree strut formula underestimated the shear strength of joints, and the 60 degree strut formula overestimated the test results. (4) Assume the shear stresses on smooth area are the same for specimens with same total interface area and confinement stress, the shear stress on shear keys of keyed dry joints were calculated and analyzed. The shear stress on shear keys of tested specimen by author is between 4.0 and 7.6 MPa, and that on smooth area is between 0.8 and 1.7 MPa. The shear stress on shear key of solid section specimens decreased with increasing of confinement stress. The shear stress on shear keys of hollow section specimens increased with increasing of confinement stress. The AASHTO formula overestimated the shear stress on shear keys of hollow section specimens, but the proposed formula gave lower bound prediction to test results. The stress on smooth area of flat section specimens increased as the confinement stress increased, which showed same trend with AASHTO and proposed formula. The shear capacity assumed by shear keys decreased as the increasing of confinement stress. The proposed formula gave reasonable prediction of the

Acknowledgment This work was financially supported by the National Key R&D Program of China under Grant No. 2017YFC0702900, Zhejiang Provincial Public Welfare Technology Research Industrial Project under Grant No. LGG18E080004. This work was also sponsored by the K.C. Wong Magna Fund in Ningbo University. Their support is gratefully acknowledged.

Appendix A A. Proposed shear failure plane tangent with envelope of tensile and compressive damage surface The shear resistant mechanism of the test specimen shown in Fig. 1 has two parts, the first part is the shear resistance provided by shear keys, and the second part is the frictional force provided by interlocking of small particle on shear plane under confining stress. So the expression of shear resistant force can be written as:

Vj = Akey fc′ (C1 + C2 σn ) + 0.6Asm σn

(A-1)

The following steps are used to find the coefficient C1 and C2 in Eq. (A-1). (i) To find the equations and inclined angles of the lines that defines the failure envelope The coordinate of the center of uniaxial compression failure circle is (x 0 , y0 ) = (fc /2, 0) Fig. 8. The line that inclines at 37° with X axis and is tangent with the uniaxial compression strength circle is defined as line L1. The tangent point of L1 with uniaxial compression circle is (x1, y1) . The coordinate of (x1, y1) can be written as

x1 = x 0−R1sinα = fc /2−fc /2·sinα = fc /2(1−sinα )

(A-2a)

y1 = y0 + R1cosα = 0 + fc /2·cosα = fc /2·cosα

(A-2b)

where α = 37°, R1 is radius of uniaxial compression strength circle. The intersection point coordinate of line L1 with Y axis is (x2 , y2 ) .

x2 = 0

(A-3a)

y2 = fc /2·(−tanα + sinα tanα + cosα ) ≈ fc /4

(A-3b)

The center coordinate of uniaxial tensile strength circle is (x3 , y3 ) = (−ft /2, 0) . The inclined angle of line connecting the point (x2 , y2 ) and (x3 , y3 ) relative to X axis is β .

tanβ =

f /4 f y2 = c = 0.5 c x3 ft /2 ft

(A-4a)

f β = tan−1 ⎛⎜0.5 c ⎞⎟ ⎝ ft ⎠

(A-4b)

Using ft = 0.604 fc′ and fc = 0.85fc′ unit in MPa, substitute into Eq. (A-4b),

β = tan−1 (0.7 fc′ )

(A5)

The line drawn from (x2 , y2 ) and tangent with uniaxial tensile strength circle is defined as L2. The inclined angle of line L2 relative to X axis is θ .

θ = 2β−90°

(A6)

(ii) To find the coefficient C1 of shear strength provided by shear keys without confinement The coefficient C1 is the shear strength of shear keys without confining stress in Eq. (A-1). Assume the Mohr circle at the non-confinement state is tangent with envelope L2. As shown in Fig. 8, the intersection of initial Mohr circle with shear stress axis is (0, τ ) . A line is drawn between (0, τ ) and center of that circle, the line cuts that circle diametrically opposite from (0, τ ) , that is (σy, −τ ) . The center of that circle is point (σy /2, 0) . The distance from point (σy /2, 0) to L2 is, 270

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f wτ sinθ + c cosθ 2d 4

R=

(A7)

where R is radius of present Mohr stress circle. Above equation uses the relation below from Eqs. (1) and (3).

σy τ

=

w d

(A8)

The radius R of the present stress circle can also be written as,

σ 2 w 2 ⎛ y ⎞ + τ 2 = ⎛ ⎛ ⎞ + 1 ⎞·τ ⎜ ⎟ 2d ⎝2⎠ ⎝ ⎝ ⎠ ⎠

R=

(A9)

Using Eqs. (A-8) and (A-9), the shear stress τ on shear plane can be written as, fc 4

τ=

w 2 2d

( )

cosθ

+

=

w 1 − 2d sinθ

0.2125fc′ cosθ w

(0.447−sinθ) 2d + 0.894

(A10)

The right equation in Eq. (A-10) is derived by Taylor expansion of the square root calculations in the denominator of the left equation in Eq. (A10). Compare Eq. (A-10) with Eq. (A-1), the coefficient C1 of shear strength provided by shear keys without confinement can be written as,

0.2125cosθ w (0.447−sinθ) 2d + 0.894

C1 =

(A11)

(iii) To find the coefficient C2 of shear strength provided by shear keys with confinement When the confining stress σx exists, the stress circle will enlarge. The enlarged stress circle still is tangent with the strength envelope line L1 or L2. The stresses on two normal planes are (σx , τ ) and (σy, −τ ), respectively. The line connects these two points diametrically. The center of the stress circle is

(

σx + σy 2

)

, 0 . The radius of the stress circle calculated from (σx , τ ) to

(

σx + σy 2

)

, 0 is,

2

σ σy τ2 + ⎛ x − ⎞ ⎝2 2⎠

R=

(A12)

Assume the stress circle is tangent with line L2, then the radius also equals to the distance from the circle center with line L2.

σy f σ R = ⎛ x + ⎞ sinθ + c cosθ 2 2 4 ⎝ ⎠

(

σx + σy 2

)

, 0 to the tangent point

(A13)

The relation in Eq. (A-8) still works when the stress circle enlarges, the radius of Eq. (A-12) equals to that of Eq. (A-13), then the quadratic equation of shear stress τ can be written as, (A14)

Aτ 2 + Bτ + C = 0 where the coefficient A , B and C are in the form of,

w 2 A = 1 + ⎛ ⎞ cos2 θ ⎝ 2d ⎠

(A-15a)

f w B = −⎡σx (1 + sin2 θ) + c sinθcosθ⎤· ⎢ ⎥ 2d 2 ⎣ ⎦

(A-15b)

f2 f σx2 cos2 θ−σx c sinθcosθ− c cos2 θ 4 4 16

C=

(A-15c)

The root of Eq. (A-14) can be written as,

τ=

−B +

B2−4AC 2A

(A-16)

The increase of shear stress can be written as,

τin =

−B +

0.2125fc′ cosθ B2−4AC − w 2A (0.447−sinθ) 2d + 0.894

(A-17)

Referring to Eq. (A-1), the coefficient C2 can be written as,

C2 =

−B + B2−4AC 0.2125cosθ − w 2Aσx fc′ (0.447−sinθ) 2d + 0.894⎤ σx ⎡ ⎣ ⎦

(A-18)

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References

failure of concrete shear key. I: Theory. J Eng Mech ASCE 1993;119(4):681–700. [10] Koseki K, Breen JE. Exploratory study of shear strength of joints for precast segmental bridges. Research Report 248-1. Center for Transportation Research. The University of Texas at Austin, Austin, Texas; 1983. [11] Li GP, Yang DH, Lei Y. Combined shear and bending behavior of joints in precast concrete segmental beams with external tendons. J Bridge Eng 2013;18(10):1042–52. [12] Roberts CL. Measurement-based revisions for segmental bridge design and construction criteria. Dissertation for the Degree of Doctor of Philosophy. the University of Texas at Austin; 1993. [13] Rombach G, Specker A. Design of joints in segmental hollow box girder bridges. 1st FIB kongress, Osaka, Japan, Session 2, Paper E-73; 2002, 1–6. [14] Shamass R, Zhou X, Alfano G. Finite-element analysis of shear-off failure of keyed dry joints in precast concrete segmental bridges. J Bridge Eng, ASCE 2015;20(6):04014084. [15] Turmo J, Ramos G, Aparicio AC. Shear strength of dry joints of concrete panels with and without steel fibres: application to precast segmental bridges. Eng Struct 2006;28(1):23–33. [16] Turmo J, Ramos G, Aparicio AC. Shear behavior of unbounded post-tensioned segmental beams with dry joints. ACI Struct J 2006;103(3):409–17. [17] Zhou XM, Mickleborough N, Li ZJ. Shear strength of joints in precast concrete segmental bridges. ACI Struct J 2005;102(1):3–11.

[1] AASHTO. Guide specifications for the design and construction of segmental concrete bridges. Washington, D.C.: American Association of State Highway and Transportation Officials; 1999. [2] AASHTO. AASHTO LRFD Bridge Design Specifications, Customary U.S. Units. American Association of State Highway and Transportation Officials, Washington, D.C.; 2012. [3] Anagnostopoulou M, Filiatrault A, Aref A. Seismic design and analysis of a precast segmental concrete bridge model. Technical Rep. MCEER-11-0002, Univ. at Buffalo, State Univ. of New York; 2011. [4] Buyukozturk O, Bakhoum MM, Beattie SM. Shear behavior of joints in precast concrete segmental bridges. J Struct Eng ASCE 1990;116(12):3380–401. [5] Hofbeck JA, Ibrahim IO, Mattock AH. Shear transfer in reinforced concrete. ACI J 1969;66(2):119–28. [6] Issa MA, Abdalla HA. Structural behavior of single key joints in precast concrete segmental bridges. J Bridge Eng ASCE 2007;12(3):315–24. [7] Jiang HB, Chen L, Ma ZJ, Feng W. Shear behavior of dry joints with castellated keys in precast concrete segmental bridges. J Bridge Eng, ASCE 2015;20(2):04014062. [8] Jiang HB, Wei RB, Ma ZJ, Li YH, Jing Y. Shear strength of steel fiber-reinforced concrete dry joints in precast segmental bridges. J Bridge Eng 2016;ASCE:04016085. [9] Kaneko Y, Connor JJ, Triantafillou TC, Leung CK. Fracture mechanics approach for

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