Tensile behaviour of adhesive anchors under different strain rates

Engineering Structures 192 (2019) 113–125

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Tensile behaviour of adhesive anchors under different strain rates ⁎

T

Lenda T. Ahmed , Abass Braimah Department of Civil and Environmental Engineering, Carleton University, 1125 Colonel By Drive, Ottawa, ON K1S 5B6, Canada

A R T I C LE I N FO

A B S T R A C T

Keywords: Adhesive anchor Concrete Finite element analysis Tensile load Strain rate Blast loading Impact loading

The use of post-installed anchors in connecting window frames to concrete elements has seen a dramatic increase in the post September-11 era. However, the behaviour of post-installed anchors under high strain rates of loading corresponding to blast loading transferred from retrofitted windows has not been extensively investigated. This paper presents the results of a finite element analysis to investigate the tensile behaviour of adhesive anchor embedded into concrete at strain rates ranging from 10−5 s−1 to 103 s−1. The adhesive anchor to concrete numerical models were validated with results from a drop-mass experimental test and then used to investigate the tensile capacity and the corresponding dynamic increase factor (DIF) at the different strain rates (ε )̇ and for different anchor diameters (d) and embedment depths (hef). Failure modes and ultimate capacity under different rates of loading of the adhesive anchor were examined. The results show that the increase in the strain rate increased the tensile capacity and the DIF. The adhesive anchors exhibited maximum DIF of 3.77 under tensile load for the 19.1-mm anchor diameter with 76.2 mm embedment depth at the high strain rate of 103 s−1. The strain rate affects the failure mode for the adhesive anchor; the failure mode transitions from concrete cone failure to combined cone-bond failure and then to steel failure as the strain rate increased.

1. Introduction Post-installed anchors are increasingly used in construction due to their flexibility of installation and the increasing demand for shorter construction times. Post-installed anchors have found use in a variety of applications including: new construction, retrofit and rehabilitation of concrete and masonry structures [1]. Post-installed anchors have also found use in blast retrofit applications to resist, minimize or mitigate the influence of blast load where they are subjected to high strain rate loading. Post-installed anchors are classified according to the load transfer mechanism into mechanical anchors and bonded anchors [2]. The bonded anchors can be either adhesive anchors or grouted anchors. The use of adhesive anchors has gained popularity in the past decades due to their fast curing time in comparison with grouted anchors and their associated superior cost-effectiveness [3]. Anchorage to concrete systems can be exposed to static or dynamic loading conditions resulting from use and occupancy or from the environment or climate. The behaviour of adhesive anchors embedded into concrete and subjected to static tensile load has been widely investigated experimentally [4–9]. Many of the researchers have investigated the effect of parameters such as embedment depth, anchor

⁎

diameter, anchor spacing in multi-anchor configurations, concrete compressive strength, and use of fibre reinforced concrete on the capacity of the anchorage to concrete system. Braimah et al. [10] investigated the behaviour of adhesive anchors embedded into concrete substrate and subjected to impulsive loading. Steel anchor failure was observed. The authors investigated the influence of anchor penetration angle on the dynamic increase factor (DIF) and recommended DIFs of 3.2 and 1.2 for the anchors embedded in concrete substrate with penetration angles of 45° and 90° respectively [10]. Very little information is available on the dynamic behaviour of adhesive steel anchors embedded in concrete substrates; especially under high strain rate effects arising from blast and impact loading. It is therefore important to investigate the behaviour of steel anchors under dynamic loads in order to minimize or prevent anchorage failure under these loads. It has been experimentally established that both concrete and steel experience an increase in strength under high strain rate of loading. The combined effect of the strength increase in concrete and steel has not been adequately investigated under strain rates experienced under blast and impact. Predicting the failure mode of anchorage to concrete system, relationship between the anchorage system strength increase and strain rate will provide designers with the requisite information to

Corresponding author. E-mail address: [email protected] (L.T. Ahmed).

https://doi.org/10.1016/j.engstruct.2019.04.072 Received 20 June 2018; Received in revised form 18 April 2019; Accepted 23 April 2019 Available online 09 May 2019 0141-0296/ © 2019 Elsevier Ltd. All rights reserved.

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CONTACT_AUTOMATIC_ONE_WAY_SURFACE_TO_SURFACE_TIEBREAK. The tiebreak contact is a penalty-based contact which allows transmission of the forces in normal and tangential directions and has the capability to model the failure [11]. Before failure, the tiebreak contact prevents separation between slave (adhesive) and the master (anchor) segments. After failure, the coupling in normal direction is eliminated and the contact behaves as surface to surface contact with thickness offset [11]. CONTACT_ERODING_SINGLE_SURFACE was used to remove the distorted elements of the adhesive material. ENMASS parameter was set equal to 1 in the CONTROL_CONTACT key card to retain the mass of these removed distorted elements and they remain active in contact [11]. BOUNDARY_PRESCRIBED_MOTION_SET in zdirection was applied to the top surface of the anchor to represent the tensile load. Fig. 2 shows the geometric configuration and boundary condition of the adhesive anchor model under tensile load. One-quarter of the concrete block and adhesive anchor was modelled for the tensile load specimen to take advantage of symmetry. The concrete block was fixed at the bottom surface (XY plane) to prevent translational and rotational motion during the load application. Symmetry plane boundary conditions were applied in XZ and YZ planes of the adhesive anchor model to investigate the tensile behaviour as shown in Fig. 2. The concrete block was modeled using continuous surface cap model (MAT_159) while the steel anchor was modeled using piecewise linear plasticity model (MAT_024). MAT_ARUP_ADHESIVE (MAT_169) was used to model the adhesive material. The properties of the steel anchor used in the analysis were in accordance with ASTM A354 specification with nominal yield and ultimate tensile strengths of 896 N/ mm2 and 1034 N/mm2 respectively, density of 0.00785 g/mm3, Young’s modulus of 200000 N/mm2, failure strain of 14% and Poisson’s ratio of 0.3. The concrete material on the other hand was modeled with a density of 0.0024 g/mm3 and nominal compressive strength of 30 N/ mm2. Tensile strength of the concrete is not required as an input parameter for the concrete material model (MAT_159). The concrete material is assumed free from cracking or damage, and this is implemented in the concrete material model (MAT_159) through the preexisting damage parameter PRED equal to zero. In addition, the MAT_159 material model has built-in damage criterion to represent the damage of the concrete when subjected to compression or tension load [11]. The value of the damage parameter (d) ranges from zero to one and allows for the prediction of crack initiation and propagation in concrete. The MAT_159 material model has been reported to be more suitable for modelling concrete compared to the other material models available in LS-DYNA [16]. Also, MAT_159 has built-in erosion criterion that can represent concrete failure [17]. Furthermore, MAT_159 has a built-in parameter (IRATE) to represent the strain rate effect on the concrete strength. By activating the IRATE formulation, increase in the strain rate increases the strength of the concrete material [18]. Also, the adhesive material model (MAT-169) has parameters; EDOT0 and EDOT2 to represent the strain rate effect [11] of the adhesive material. The adhesive material was modeled with a density of 0.0012 g/mm3, Elastic modulus of 3034 N/mm2, Poisson’s ratio of 0.4, tensile strength of 56 N/mm2 and shear strength of 44 N/mm2 [19].

design safe and cost-effective adhesive anchorage systems. Also, the effect of adhesive anchor diameter and embedment depth on the anchorage system capacity at different strain rates will be invaluable to designers. This paper reports on an investigation on the tensile behaviour of adhesive anchors subjected to strain rate ranging from static strain rate of 10−5 s−1 to high strain rate of 103 s−1 using LS-DYNA software. Different design parameters (anchor diameter and embedment depth) were considered in the analyses. The numerical model was validated by comparing the finite element results with experimental tests results reported in the literature and presented in Section 3. Also, comparison was made between the results obtained from the finite element analysis under static rates of loading and the predictions from analytical methods (ACI and CCD). Level of damage and failure modes at different strain rates were also investigated. Tensile load-displacement relation for the adhesive anchors at low and high strain rates was drawn and analyzed. DIF was determined and analytical equations to relate DIF and strain rate for the adhesive anchors were developed.

2. Finite element modelling Finite element models were developed using LS-DYNA software [11] to investigate the tensile behaviour of adhesive anchors embedded into concrete. Concrete block size of (4hef + 125) × (4hef + 125) × (2hef ) mm was used to investigate the tensile behaviour, where hef is the embedment depth of the adhesive anchor. A single adhesive anchor was placed in the center of each concrete block. The size of the concrete blocks was chosen to ensure that the edge distance was sufficient for concrete cone formation without edge effects. Three standard steel anchor diameters of 12.7 mm, 15.9 mm and 19.1 mm with embedment depths of 76.2 mm, 101.6 mm, 127 mm and 152.4 mm were investigated. A schematic view of the adhesive anchorage to concrete system is shown in Fig. 1. The steel anchor and adhesive material were modelled using eightnoded solid elements. The concrete was modeled using four-noded tetrahedron solid elements which are capable of transitioning from coarse mesh to fine mesh as required [12]. The mesh of concrete elements was refined and biased towards the adhesive anchor along the embedment depth to improve the accuracy of the analysis and to minimize discretization errors. CONTACT_TIED_SURFACE_TO_SURFACE key card in LS-DYNA was used to model the bond between adhesive and the concrete. The tied contact is a constrained-based contact used to ensure that the adhesive material is tied to the concrete material with no gap [11]. The tied contact allows for mesh transition [13] and because of the difference in the mesh sizes and element types of the concrete and adhesive to assure that the slave nodes (adhesive) lie on the master surface (concrete) without gap [11]. LS-DYNA updates the coordinates of the slave nodes to be identical on the master surface and neglect the slave nodes that are located far from the master surface [14]. When the failure of the elements in contact occurs, the elements in the tied contact are automatically deleted [15]. The adhesive contact with steel anchor was modeled using

3. Validation of the finite element model To ensure the material models chosen for the concrete, adhesive and steel elements are adequate for the research program reported in this paper, an adhesive anchorage to concrete system representing experimental program conducted by Braimah et al. [10,20] was developed in LS-DYNA to validate the numerical model. The numerical model consisted of a 6.4-mm diameter steel anchor with embedment depth of 114 mm and embedded into 35 N/mm2 compressive strength concrete. The adhesive anchor was modeled with a yield strength of 874 N/mm2 and ultimate tensile strength of 1138 N/mm2. The epoxy adhesive material was 1.5 mm thick with 30 N/mm2 tensile strength, 23 N/mm2 shear strength and 4900 N/mm2 modulus of elasticity. A mesh

Fig. 1. A schematic view of the adhesive anchorage to concrete system. 114

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Fig. 2. Geometric configuration and boundary condition of adhesive anchor model; (a) concrete mesh, (b) adhesive anchor mesh. Table 1 Effect of mesh size refinement on the convergence of ultimate tensile load. Model No.

Mesh size (mm) Anchor

1 2 3 4

0.75 0.75 0.75 0.50

Adhesive

0.75 0.75 0.50 0.50

Concrete Min.

Max.

3 1 1 1

8 8 8 8

Ultimate tensile load (kN)

Displacement (mm)

FEA

EXP.

FEA

EXP.

29.50 32.11 33.14 33.28

34.37

0.83 0.86 0.89 0.89

0.99

sensitivity analysis was carried out to achieve an optimum mesh size that resulted in high accuracy in comparison with experimental results and minimized computational effort. Table 1 presents the results of the mesh sensitivity analysis. The experimental displacement profile from a drop-mass test was applied to the steel anchor in the numerical model by using BOUNDARY_PRESCRIBED_MOTION_SET. The drop mass test frame is shown in Fig. 3. The drop mass was guided by a steel rod of 50 mm diameter. A ring sensor (compression load cell) was placed at the bottom of the steel guide rod to measure the impact loading while a tension load cell was placed at the top of the guide rod to measure the tensile load on the anchor from the drop mass impact. The adhesive anchor specimen was placed on supports at the top of the drop mass test frame and clamped down. The 50-mm diameter guide rod was connected to the steel anchor by a steel coupler through which the load was applied to the anchor along its axis by a drop mass. During the testing, the drop mass was lifted to a specified height by an electromagnet and then released to free-fall onto an anvil that consists of compression load cell. Upon impact on the anvil, the steel guide rod transmits the energy applied to the steel anchor and concrete. A linear variable displacement transformer (LVDT) was connected to the guide rod and used to measure the displacement of the steel anchor [20]. In the mesh sensitivity analyses, the mesh sizes of models No. 3 and 4 resulted in converged results to the experimental results obtained by Braimah et al. [10,20] (Table 1). Fig. 4 presents a comparison of loaddisplacement response of the experimental tests and the finite element analysis. The figure shows that as the mesh size decreases the loaddisplacement response approaches the experimental. As shown in Fig. 5, concrete cracking in a small area around the anchor was observed on the top surface of concrete where a small amount of anchor pullout occurred followed by steel anchor failure. The same failure mode (steel anchor failure) was observed for both the experimental test and finite element analysis. The numerical model with 0.75-mm steel, 0.50-mm adhesive, and a biased 1.0-mm to 8.0-mm concrete elements

Fig. 3. Drop mass test frame [20]

115

Failure mode

Steel Steel Steel Steel

failure failure failure failure

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40

Mesh size (0.75x0.75x3) mm Mesh size (0.75x0.75x1) mm Mesh size (0.75x0.5x1) mm Mesh size (0.5x0.5x1) mm Experimental

35

Tensile load (kN)

30 25 20 15 10 5 0 0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

Displacement (mm) Fig. 4. Comparison of tensile load-displacement relation between FEA and experimental results obtained by Braimah et al. [20].

For the steel anchor failure, the ultimate tensile load can be determined using Eq. (2). The minimum value of the ultimate tensile load obtained from Eqs. (1) and (2) is considered for comparison with the finite element results and included in Table 2.

sizes (Fig. 4) was deemed adequate for modelling the adhesive anchorage system. The tensile load-displacement response was almost linear up to the peak load followed by a sharp drop in the load where the failure occurred at the post peak load. Load fluctuation about the residual tensile load was observed. The load fluctuation can be attributed to the pullout of the anchor to a small displacement associated with concrete cracking followed by steel anchor failure.

Nu = Ase fut where

Nu = ultimate tensile force (N); f cc' = compressive strength of concrete cube; hef = effective embedment depth; do = hole diameter; Ase = effective cross-sectional area of the anchor; and fut = ultimate tensile strength of the steel where ( fut = 1034 N/mm2). A concrete block of 37 N/mm2 cube compressive strength (f cc' ) that is equivalent to 30 N/mm2 cylinder compressive strength was used in Table 2 [22,23].

4. Comparison of numerical ultimate tensile anchor capacity with the design methods Ultimate capacity of the adhesive anchors obtained from the finite element analysis (FEA) at the static strain rate of 10−5 s−1 was compared with the American Concrete Institute design method (ACI 349) and Concrete Capacity Design method (CCD) as shown in Table 2. According to ACI method [21], the ultimate tensile load of post-installed anchors failing by the concrete cone failure mode can be determined by Eq. (1).

d Nu = 0.96 f cc' hef2 ⎜⎛1 + 0 ⎞⎟ h ef ⎠ ⎝

(2)

The CCD design method [21,22] on the other hand, proposed Equation (3) for calculating the ultimate tensile capacity of post-installed adhesive anchors based on concrete cone failure.

Nu = K f cc' hef1.5 (1)

where K = 13.5 for post installed anchors.

Fig. 5. Failure mode obtained from:(a) finite element analysis and (b) experimental results obtained by Braimah et al. [20]. 116

(3)

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Table 2 Comparison of ultimate tensile loads obtained from FEA with ACI and CCD methods. d (mm)

hef (mm)

Ultimate tensile load (kN) ACI

CCD

FEA

ACI/FEA

CCD/FEA

Failure mode (FEA)

12.7

76.2 101.6 127.0 152.4

40.45 69.00 105.09 130.98

54.62 84.10 117.53 130.98

57.12 80.45 95.72 96.79

0.71 0.86 1.10 1.35

0.96 1.05 1.23 1.35

CC CC S S

15.9

76.2 101.6 127.0 152.4

41.92 70.96 107.53 151.64

54.62 84.10 117.53 154.49

65.69 87.31 126.18 159.72

0.64 0.81 0.85 0.95

0.83 0.96 0.93 0.97

CC CC CCB S

19.1

76.2 101.6 127.0 152.4

43.70 73.33 110.50 155.20

54.62 84.10 117.53 154.49

69.62 105.49 157.28 179.40

0.63 0.70 0.70 0.87

0.78 0.80 0.75 0.86

CC CC CC CC

observed for the 15.9-mm diameter at embedment depth of 127 mm. Steel anchor failure was observed for the 12.7-mm diameter adhesive anchor at embedment depths of 127 mm and 152.4 mm. Also, steel anchor failure was observed for the 15.9-mm anchor diameter at embedment depth of 152.4 mm. For long embedment depths and smaller anchor diameters of 12.7-mm and 15.9-mm, the steel anchor is not capable to withstand the applied load and resulted in steel fracture. Under the tensile loading, concrete cracking started to appear on the top surface of the concrete around the anchor. Also, the cracks generated at the bottom of the anchor, propagated diagonally forming concrete cone and leading to failure of the anchorage system. It can also be observed that the level of concrete damage increased with the increase in the anchor diameter from 12.7-mm to 19.1-mm for the same embedment depth. The increase in the embedment depth resulted in increase in the concrete cone diameter and the failure surface area of the concrete. The concrete cone diameter increased from 225 mm to 268 mm and from 236 mm to 291 mm for increase in embedment depth from 76.2 mm to 101.6 mm for the 12.7-mm and 15.9-mm diameter adhesive anchors respectively. Further increase in the anchor embedment depth to 152.4 mm resulted in steel anchor failure as shown in Fig. 6. The increase in the concrete cone diameter was from 244 mm to 421 mm for the 19.1-mm diameter adhesive anchor when the embedment depth increased from 76.2 mm to 152.4 mm.

From Table 2 it can be seen that the ACI and CCD methods underpredict the ultimate tensile load in comparison with the FEA results for most of the adhesive anchors, except for the 12.7-mm diameter at embedment depth of 127 mm and 152.4 mm, as the design methods incorporate more conservatism than the FEA. In general, the CCD method gives a better agreement with the finite element method. Similar observation was reported by Fuchs et al. (1995) where the ACI method underpredict the failure load for shallow embedment depths and unconservative for the deep embedment depths [21]. This is attributed to the fact that the ACI method disregards the size effect [21]. Same material properties were used for the finite element analysis and the design codes. However, in the finite element analysis assumptions such as materials model, boundary conditions and contact formulation, were considered in developing the numerical model for the adhesive anchorage to concrete system. These assumptions may have an influence on the variation between the FEA results and the design methods results. 5. Parametric analysis of adhesive anchor concrete systems After the adhesive anchor system was validated against the experimental results to establish the adequacy of the mesh sizes and the material models for concrete, adhesive and steel, the numerical (finite element) models were used in a parametric analysis to investigate the effect of various anchorage to concrete design parameters (anchor diameter and embedment depth) on the behaviour of the anchorage system. Twelve models were developed with 12.7-mm, 15.9-mm and 19.1-mm diameter steel anchors and embedment depths of 76.2 mm, 101.6 mm, 127.0 mm and 152.4 mm. Effect of the design parameters on the capacity of the anchorage system was investigated when subjected to various strain rates of loading ranging from the static strain rate of 10−5 s−1 to a higher strain rate of 103 s−1.

5.2. Tensile load-displacement behaviour at low strain rate In order to investigate the tensile behaviour of the adhesive anchors, numerical models of adhesive anchorage to concrete systems were developed. Strain rates of 10−5 s−1, 10−3 s−1, 10−1 s−1, 10 s−1, 102 s−1 and 103 s−1 were selected for the analysis to investigate the effect of strain rates on the behaviour of adhesive anchor systems. Load-displacement graphs for the 12.7-mm and 19.1-mm diameter adhesive anchors at a low strain rate of 10−5 s−1 (static strain rate) are presented in Figs. 7 and 8 respectively. Figs. 7 and 8 show that the tensile load increased with the displacement until the ultimate value. This is attributed to the concrete resistance to the applied load where the tensile load transfers from the anchor to the concrete through the adhesive material. The post-peak response shows a reduction in the load with further increase in displacement until failure. The failure of the anchorage system is due to initiation and propagation of concrete cracking and resulting in concrete cone breakout failure. However, for the 12.7-mm diameter anchor with embedment depths of 127 mm and 152.4 mm, the reduction in the load after the ultimate value is due to steel anchor failure. Hence the failure load is the same for both anchors. As shown in Figs. 7 and 8, the displacement (δ) at the ultimate load decreases with increase in anchor

5.1. Failure mode of adhesive anchors under tensile load Fig. 6 shows the crack patterns for the 12.7-mm, 15.9-mm and 19.1mm diameter adhesive anchors with embedment depths of 76.2 mm, 101.6 mm, 127 mm and 152.4 mm at the low strain rate of 10−5 s−1. The crack patterns are presented with the plastic strain contour fringe plots representing damage to concrete and varying between 0 and 1.0. Concrete with no damage is represented by fringe value of 0 while fringe values higher than zero represent damaged concrete. Maximum damage occurs at the fringe value of 1 where the damaged concrete with no useable strength [24]. The cracks are along areas with high fringe levels between 0.6 and 1. As shown in the figure, concrete cone breakout failure was observed at low strain rate of 10−5 s−1 for most of the adhesive anchors investigated. Combined cone-bond failure was 117

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d=12.7 mm

d=15.9 mm

d=19.1 mm

hef=76.2 mm

hef=101.6 mm

hef=127 mm

hef=152.4 mm Fig. 6. Plastic strain contours of adhesive anchors at strain rate of 10−5 s−1. 200

200

hef=76.2 mm

180

hef=101.6 mm hef=127 mm

160

hef=152.4 mm

150

Tensile load (kN)

140

Tensile load (kN)

hef=76.2 mm hef=101.6 mm hef=127 mm hef=152.4 mm

175

120 100 80

125 100 75

60

50

40

25

20

0 0

0 0

1

2

3

4

5

6

1

2

3

4

Displacement (mm)

5

6

7

7

Fig. 8. Tensile load-displacement response of 19.1-mm diameter adhesive anchor at strain rate of 10−5 s−1.

Displacement (mm) Fig. 7. Tensile load-displacement response of 12.7-mm diameter adhesive anchor at strain rate of 10−5 s−1.

is the dominant failure mode.

diameter from 12.7-mm to 19.1-mm, at the same embedment depths. The increase in anchor diameter from 12.7-mm to 19.1-mm increases the contact area between the adhesive anchor and concrete and hence increase the tensile capacity of the anchorage. The tensile load was also observed to increase with increase in the embedment depth from 76.2 mm to 152.4 mm. The increase in the anchor embedment depth increased the displacement at the ultimate tensile load. The embedment depth was observed to have a greater effect on the ultimate tensile load at the same strain rate when concrete cone breakout failure is the dominant failure mode. However, the increase in the embedment depth has no influence on the ultimate tensile load when steel anchor failure

5.3. Strain rate effect on the failure mode of the adhesive anchors under tensile load Figs. 9 and 10 show the failure mode of 12.7-mm and 19.1-mm diameter adhesive anchors at strain rates ranging from 10−3 s−1 to 103 s−1. For the 12.7-mm diameter adhesive anchor (Fig. 9), combined cone bond failure was observed for the 76.2 mm and 101.6 mm embedment depths at strain rates of 10−3 s−1 to 102 s−1; wherein a shallow cone was observed at the top of the concrete accompanied by adhesive bond failure at the remaining part of the embedment depth below the shallow concrete cone. Steel anchor failure was observed for 118

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=10-3 s-1

hef (mm) 76.2

=10-1 s-1

=10 s-1

=102 s-1

=103 s-1

101.6

127.0

152.4

Fig. 9. Failure mode of 12.7 mm diameter adhesive anchor at different strain rates.

=10-3 s-1

hef (mm) 76.2

=10-1 s-1

=10 s-1

=102 s-1

=103 s-1

101.6

127.0

152.4

Fig. 10. Failure mode of 19.1 mm diameter adhesive anchor at different strain rates. 140

the 127 mm and 152.4 mm embedment depths at all the strain rates investigated. Fig. 10 presents the failure mode of 19.1-mm diameter adhesive anchor. The damage profiles show that for the embedment depth of 76.2 mm, concrete cone failure was observed at strain rate of 10−3 s−1. The increase in the strain rate to 102 s−1 resulted in combined cone-bond failure. For anchor embedment depths of 101.6 mm, 127 mm and 152.4 mm, combined concrete cone-bond failure mode was observed at strain rates of 10−3 s−1 to 102 s−1. Anchors investigated at the highest strain rate of 103 s−1 exhibited steel anchor failure mode for all the anchor diameters and embedment depths. It is clear from Figs. 9 and 10 that the strain rate has an influence on the failure mode. The failure mode is observed to transition from concrete cone or combined cone-bond failure mode to steel anchor failure mode with increase in strain rate. This behaviour can be attributed to the increase in concrete and steel capacity with increase in strain rate. The increase in the tensile capacity of the concrete is higher than the increase in the steel capacity [25,26]. Hence, the concrete resistance to the tensile load at high strain rate increase results in steel anchor failure.

hef=76.2 mm hef=101.6 mm

120

hef=127 mm hef=152.4 mm

Tensile load (kN)

100 80 60 40 20 0 0

1

2

3

4

5

6

7

8

9

Displacement (mm)

Fig. 11. Tensile load-displacement graph for 12.7-mm diameter adhesive anchor at strain rate of 10 s−1.

119

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L.T. Ahmed and A. Braimah 140

300

hef=76.2 mm

120

hef=101.6 mm

hef=127 mm

250

hef=127 mm

hef=152.4 mm

hef=152.4 mm

225

Tensile load (kN)

100

Tensile load (kN)

hef=76.2 mm

hef=101.6 mm

275

80

60

40

200 175 150 125 100 75 50

20

25 0

0 0

1

2

3

4

5

6

7

8

0

9

1

2

3

4

5

6

7

8

9

Displacement (mm)

Displacement (mm)

Fig. 13. Tensile load-displacement graph for 19.1-mm diameter adhesive anchor at strain rate of 10 s−1.

Fig. 12. Tensile load-displacement graph for 12.7-mm diameter adhesive anchor at strain rate of 103 s−1.

concrete and steel. Concrete has been reported to have increased tensile and compressive strengths with increase in the strain rate [29]. Hence, the concrete resistance increase results in increase of the ultimate load capacity of the anchorage system when failure is by either cone or combined cone-bond. Also, increasing the strain rate increases the modulus of elasticity of the concrete [30] and its energy absorption capacity [31]. Moreover, the yield and ultimate strength of steel increases with increase in strain rate [32]. This is due to the increase in the deformations and dislocations of steel at high strain rate. Figs. 13 and 14 show the tensile load-displacement graphs for the 19.1-mm diameter adhesive anchor at strain rates of 10 s−1 and 103 s−1 respectively. As shown in Fig. 13 the ultimate tensile load increased with the increase in the embedment depth from 76.2 mm to 152.4 mm where combined cone-bond failure was observed at all the embedment depths. Transition in the failure mode from combined cone-bond failure to steel anchor failure was observed at high strain rate of 103 s−1. The embedment depth has no effect of on the tensile load at high strain rate of 103 s−1 where steel failure is the dominant failure mode.

5.4. Tensile load-displacement behaviour at high strain rates Figs. 11 and 12 show the tensile load-displacement graphs for the 12.7-mm diameter adhesive anchor at strain rates of 10 s−1 and 103 s−1 respectively. The tensile load increased with displacement up to the peak tensile load. At intermediate strain rate of 10 s−1, the peak tensile load increased as the embedment depth increased from 76.2 mm to 127 mm where the failure mode changed from combined cone-bond failure to steel anchor failure. Further increase in the embedment depth to 152.4 mm shows no increase of the peak tensile load as steel failure is observed. At embedment depths of 76.2 mm and 101.6 mm, the post peak behaviour at strain rate of 10 s−1 shows a decrease in the tensile load due to bond failure at the lower part of the anchor accompanied by crack initiation and propagation in the concrete to the top surface of the concrete. The crack initiation and propagation lead to fracturing of the concrete and results in combined cone-bond failure as shown in Fig. 9. As the strain rate increased from 10 s−1 to 103 s−1, the tensile load increased with the displacement until the maximum load. The post peak behaviour shows a sharp decrease in the tensile load due to steel anchor failure. The failure mode transitions from combined cone-bond failure to steel anchor failure for the embedment depths of 76.2 mm and 101.6 mm. At embedment depths of 127 mm and 152.4 mm steel anchor failure was the dominant failure mode (Fig. 9). At the high embedment depths, the ultimate failure load of the anchorage systems is the same as failure is dependent on steel anchor resistance. It can be seen from Figs. 7, 11 and 12 that the higher strain rate of 103 s−1 exhibited higher tensile load than that observed at strain rates of 10−5 s−1 and 10 s−1. Also, it can be seen that the displacement at the ultimate tensile load at high strain rate of 103 s−1 for the 12.7-mm anchor diameter at embedment depths of 127 mm and 152.4 mm (Fig. 12) is less than that obtained at static strain rate of 10−5 s−1 (Fig. 7). Steel material behaviour is inherently brittle without a pronounced yield plateau under high strain rates of loading compared to its behaviour under static loading. This can be attributed to fact that metals subjected to static load exhibit ductile behaviour which can change to brittle failure at higher loading rates [27]. The results agree with those obtained by Albertini and Montagnani who observed that the strain for the Austenitic steel decreased with increase in the strain rate ranging from 10−2 s−1 to 5 × 102 s−1. Increasing the strain rate resulted in increasing stress flow while decreasing the fracture strain (elongation) [28]. However, at enhanced strain rates, the ductility behaviour could be influenced by steel material type. The increase in the ultimate load with increase in strain rate is attributed to the fact that strain rate affects the mechanical properties of

5.5. Strain rate effect on the ultimate tensile load and dynamic increase factor The ratio of the dynamic to static strength of the adhesive anchor is defined as the dynamic increase factor (DIF). The effect of strain rate on concrete strength (compressive and tensile), yield and ultimate strength

Tensile load (kN)

300

hef=76.2 mm

275

hef=101.6 mm

250

hef=127 mm

225

hef=152.4 mm

200 175 150 125 100 75 50 25 0 0

1

2

3

4

5

6

7

8

9

Displacement (mm) Fig. 14. Tensile load-displacement graph for 19.1-mm diameter adhesive anchor at strain rate of 103 s−1. 120

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Ultimate tensile load, Fu (kN)

300

DIF, d=12.7 mm

DIF, d=15.9 mm

DIF, d=19.1 mm

Fu, d=12.7 mm

Fu, d=15.9 mm

Fu, d=19.1 mm

4.0

3.0

250

2.5 200 2.0 150 1.5 100

1.0

50

Also, it can be seen that the DIF is higher for the shallow embedment depth of 76.2 mm compared to the deep embedment depth of 152.4 mm. Maximum DIFs of 1.88, 2.64 and 3.77 were obtained for the 12.7-mm, 15.9-mm and 19.1-mm diameter adhesive anchors with 76.2mm embedment depth at the highest strain rate of 103 s−1 where steel anchor failure is observed for all the adhesive anchors. Maximum DIFs of 1.12, 1.12 and 1.49 were obtained for the 12.7-mm, 15.9-mm and 19.1-mm diameter adhesive anchors with 152.4 mm embedment depth at the highest strain rate of 103 s−1. The increase in the strain rate increased the ultimate load and the DIF for the concrete and steel. The DIF for concrete tensile strength is much higher than that for steel [25,26]. Hence as the strain rate increases, the increase in the anchor capacity for concrete cone and combined concrete cone-bond failure exceeds the increase attributed to steel anchor failure where the concrete cone failure is observed at strain rates of 10−5 s−1 for most of the adhesive anchors (Fig. 6). Combined cone-bond failure is observed at strain rates ranging from 10−3 s−1 to 102 s−1 for most of the adhesive anchors while steel failure is the dominant failure mode for all the adhesive anchors at high strain rate of 103 s−1 as shown in Figs. 9 and 10. Table 3 presents the ultimate tensile load and failure mode of adhesive anchorage to concrete system with different anchor diameters and embedment depths at strain rates varying from 10−5 s−1 to 103 s−1. The increase in anchor embedment depth resulted in increase in the ultimate tensile load at all strain rates investigated and when concrete cone or combined cone-bond failure mode was observed. Maximum tensile loads of 108.85 kN, 178.59 kN and 266.96 kN were obtained at high strain rate of 103 s−1 and embedment depth of 152.4 mm for the 12.7-mm, 15.9-mm and 19.1-mm diameter adhesive anchors, respectively.

3.5

Dynamic increase factor (DIF)

350

0.5

0

0.0 1.E-05

1.E-03

1.E-01

1.E+01

Strain rate

1.E+02

1.E+03

(s-1)

Fig. 15. Ultimate tensile load and DIF versus strain rate for the 76.2 mm embedment depth adhesive anchor.

DIF, d=12.7 mm Fu, d=12.7 mm

DIF, d=15.9 mm Fu, d=15.9 mm

DIF, d=19.1 mm Fu, d=19.1 mm

Ultimate tensile load, Fu (kN)

300

4.0 3.5 3.0

250

2.5 200 2.0 150 1.5 100

1.0

50

0.5

0

0.0

Dynamic increase factor (DIF)

350

6. Regression analysis 1.E-05

1.E-03

1.E-01

1.E+01

1.E+02

1.E+03

To develop an accurate predictive formula for determining the DIF of the adhesive anchorage to concrete systems based on the finite element results, regression analysis was performed. As shown in Table 3, most of the adhesive anchors investigated at strain rates ranging from 10−3 s−1 to 103 s−1 exhibited combined cone-bond and steel anchor failure under tensile load. Hence regression analysis was performed for these failure modes. Average value of the DIF for the adhesive anchorage systems with anchor diameters of 12.7 mm, 15.9 mm and 19.1 mm was calculated to adjust the DIF for the effect of anchor diameter. Figs. 17 and 18 present the relation between the DIF and the strain rate ratio (εḋ / εṡ ) for the combined cone-bond and steel anchor failure modes respectively. Various statistical models were used to predict the relation between the DIF of the adhesive anchors and strain rate ratio: exponential, linear, logarithmic and power. The power and exponential models exhibit the highest coefficient of determination of 77% and 91% for the combined cone-bond and steel anchor failure modes respectively. The predicted formula for the DIF for the adhesive anchors can be presented as in Eqs. (4) and (5) for the combined cone-bond failure and steel failure modes respectively.

Strain rate (s-1)

Fig. 16. Ultimate tensile load and DIF versus strain rate for the 152.4 mm embedment depth adhesive anchor.

of steel are often expressed by the DIF [18,25,33,34]. In this paper, in order to predict the increase in strength of anchorage system due to increase in the steel and concrete strength with the increase in the strain rate, DIF for the anchorage to concrete system was investigated. The lowest strain rate of 10−5 s−1 is representative of static loading rate and was used as the baseline for comparison with adhesive anchor capacity at the higher strain rates. Figs. 15 and 16, show the effect of strain rate on the ultimate tensile load and DIF for the 12.7-mm, 15.9mm and 19.1-mm diameter adhesive anchors at embedment depths of 76.2 mm and 152.4 mm, respectively. It can be seen from Figs. 15 and 16 that the ultimate tensile load for the adhesive anchor increased with the increase in the strain rate. As shown in Fig. 15 for anchor embedment depth of 76.2 mm, the relationship between the ultimate tensile load and the strain rate appears to be bilinear with a change in slope at about 101 s−1. This is similar to the reported relationship between concrete strength and strain rate [33,35,36]. As shown in Fig. 16 for anchor embedment depth of 152.4 mm, bilinear relationship was obtained for the 19.1-mm diameter adhesive anchor where concrete cone breakout and combined cone-bond failure was observed at strain rates up to 102 s−1. The failure mode transitioned to steel anchor failure at high strain rate of 103 s−1. While, linear relation was obtained for the 12.7-mm and 15.9-mm diameter adhesive anchors where steel anchor failure was the dominant failure mode at all the strain rates investigated. As shown in Figs. 15 and 16, the DIF and ultimate tensile capacity increases with the increase in the strain rate. The 19.1-mm diameter anchor exhibited higher DIF than that obtained for anchor diameters of 12.7-mm and 15.9-mm for all the strain rates investigated.

ε̇ DIF = 0.9689 ⎛ d ⎞ ⎝ εṡ ⎠ ⎜

DIF = 1.0215e

0.0198

⎟

ε̇ 3E − 9 ⎛ d ⎞ ⎝ εṡ ⎠

(4) (5)

The adequacy of the predicted model is verified according to the following tests: calculating coefficient of determination R2 and residual analysis [37]. A good probability distribution of the results is obtained when the coefficient of determination (R2) is closer to one [38]. Residual analysis has been performed to measure the difference between the results obtained from finite element analysis and fitted results of DIF obtained from Eqs. (4) and (5). Figs. 19 and 20 present 121

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Table 3 Ultimate tensile load and failure mode for the adhesive anchors at different strain rates. hef (mm)

ε ̇ (s−1)

Fu (kN)

−5

Failure mode

d = 12.7 mm

d = 15.9 mm

d = 19.1 mm

d = 12.7 mm

d = 15.9 mm

d = 19.1 mm

76.2 101.6 127 152.4

10

57.12 80.45 95.72 96.79

65.69 87.31 126.18 159.72

69.62 105.49 157.28 179.40

CC CC S S

CC CC CCB S

CC CC CC CC

76.2 101.6 127 152.4

10−3

63.33 88.59 96.27 97.74

69.70 94.88 135.36 162.02

78.68 113.94 161.79 187.48

CCB CCB S S

CCB CCB CCB S

CC CCB CCB CCB

76.2 101.6 127 152.4

10−1

67.39 92.31 97.29 98.82

74.94 101.48 143.56 164.08

89.32 120.78 169.02 196.47

CCB CCB S S

CCB CCB CCB S

CCB CCB CCB CCB

76.2 101.6 127 152.4

10

71.14 95.49 99.73 99.95

87.36 108.31 151.14 168.41

114.01 139.14 185.58 211.45

CCB CCB S S

CCB CCB CCB S

CCB CCB CCB CCB

76.2 101.6 127 152.4

102

88.11 98.60 102.19 103.38

118.21 124.28 162.82 171.08

180.25 196.53 217.96 234.18

CCB CCB S S

CCB CCB CCB S

CCB CCB CCB CCB

76.2 101.6 127 152.4

103

107.24 107.93 108.43 108.85

173.63 175.89 177.57 178.59

262.69 264.59 266.13 266.96

S S S S

S S S S

S S S S

3.0

3.0 Steel failure

2.5

2.5 Dynamic increase factor (DIF)

Dynamic increase factor (DIF)

Combined cone bond

2.0

0.9689x0.0198

y= R² = 0.7708

1.5

1.0

0.5

0.0 1.E-02

2.0 y = 1.0215e3E-09x R² = 0.9125

1.5 1.0 0.5

1.E+00

1.E+02

1.E+04

1.E+06

1.E+08

0.0 1.E-02

1.E+10

1.E+00

Strain rate ratio

1.E+02 1.E+04 Strain rate ratio

1.E+06

1.E+08

1.E+10

Fig. 17. Effect of strain rate ratio on DIF for adhesive anchors exhibited combined cone-bond failure.

Fig. 18. Effect of strain rate ratio on DIF for adhesive anchors exhibited steel failure.

the residual plots. As shown in Fig. 19, for the combined cone-bond failure the variance in the residual increases with the increase in the strain rate ratio. For the steel anchor failure (Fig. 20), the residual exhibits horizontal trend line at strain rate ratio up to 107 then the residual increases with the increase in the strain rate ratio of 108. In addition, to evaluate the adequacy of the proposed equations, new adhesive anchor models with diameters of 9.5-mm, 12.7-mm, 15.9mm and 19.1-mm and embedment depths of 89 mm, 114 mm and 140 mm were developed. The relation between the DIF obtained from the finite element analysis of the new developed models and the regression models (Eqs. (4) and (5)) are presented as shown in Figs. 21 and 22 for the combined cone-bond failure and steel failure modes respectively. As shown in the figures, the DIF is observed distributed around the equality line. However, some divergence was observed for the higher values of the DIF where the residual increased at higher strain rates. Fujikake et al. (2003) proposed an equation to determine the

ultimate dynamic cone resistance for the shallow embedment depths [39] as follows:

α hef

(6)

Ae = π·hef ·tanθ (d + hef ·tanθ)

(7)

Fcd = Ae ·Ftd ·

where Ftd is determined according to the proposed equation by Ross et al. [40] as follows: 3.373

Ftd εḋ ⎞ ⎛ = exp ⎡ ⎢0.00126 log10 εṡ Fs ⎝ ⎠ ⎣ ⎜

Fs = 0.23(f c' )2/3

⎟

⎤ ⎥ ⎦

(8) (9)

where Fcd is the ultimate dynamic concrete cone breakout strength, Ae is the projected area of concrete cone failure, α = 3.48 × 10−3, θ is the crack propagation angle (θ = 60o ). Fs and Ftd are the tensile static and 122

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L.T. Ahmed and A. Braimah 3.0

1 Combined cone bond

Steel failure

2.7

0.6

2.4

0.4

2.1

DIF (Predicted)

Residuals

0.8

0.2 0 -0.2 -0.4

1.8 1.5 1.2 0.9

-0.6

0.6

-0.8

0.3

-1 1.E-1

0.0

1.E+0 1.E+1 1.E+2 1.E+3 1.E+4 1.E+5 1.E+6 1.E+7 1.E+8 1.E+9

0.0

0.3

0.6

0.9

1.2

Strain rate ratio

2.1

2.4

Steel failure

0.8

ε̇ Fb = π·dh·hb·τbs·⎛ d ⎞ ⎝ εṡ ⎠ ⎜

0.6 0.4

3.0

0.013

⎟

(11)

where Fccb is the ultimate tensile load for the combined cone-bond failure, Fb is the ultimate tensile load for the bond failure mode, hb is the bond failure depth (hb = hef − hc ), hc is the failure cone depth in the combined cone-bond failure mode (hc = 35 mm), τbs is the static bond strength (τbs = 19 N/mm2). In order to verify the results obtained from the finite element analysis, a comparison has been made between the ultimate load obtained from the finite element analysis and the proposed equations by Fujikake et al. (Eqs. (6) and (10)) for the concrete cone breakout failure and the combined cone-bond failure modes respectively [39]. Concrete cone breakout failure was observed at low strain rate of 10−5 s−1. The embedment depth (hef) substituted in Eq. (10) is equal to the cone failure depth (hc) for the combined cone-bond failure mode. Tables 4 and 5 show a comparison of the ultimate load obtained from the finite element analysis for the concrete cone breakout failure and combined cone-bond failure modes and the proposed equations by Fujikake et al. [39]. It can be seen from Tables 4 and 5 that the ultimate loads obtained from the finite element analysis agree well with the proposed equations by Fujikake et al. [39] with average values of 1.15 and 1.03 for the concrete cone breakout failure and combined cone bond failure modes respectively.

0.2 0 -0.2 -0.4 -0.6 -0.8 -1 1.E-1 1.E+0 1.E+1 1.E+2 1.E+3 1.E+4 1.E+5 1.E+6 1.E+7 1.E+8 1.E+9

Strain rate ratio Fig. 20. Residual versus strain rate ratio for the adhesive anchor exhibited steel failure. 2.4 Combined cone bond

2.2

2.7

(10)

Fccb = Fcd + Fb

1

Residuals

1.8

Fig. 22. DIF obtained from the finite element analysis versus the predicted DIF for the adhesive anchors exhibited steel failure.

Fig. 19. Residual versus strain rate ratio for the adhesive anchor exhibited combined cone-bond failure.

2.0 1.8 1.6

DIF (Predicted)

1.5

DIF (FEA)

1.4

7. Conclusions

1.2 1.0

The tensile behaviour of adhesive anchors subjected to strain rates in the range from 10−5 s−1 to 103 s−1 was investigated using LS-DYNA – a multi-physics based finite element analysis program. Adhesive anchor diameters of 12.7 mm, 15.9 mm and 19.1 mm with embedment depths of 76.2 mm, 101.6 mm, 127 mm and 152.4 mm were

0.8 0.6 0.4 0.2 0.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

Table 4 Comparison between ultimate load obtained from the FEA and the proposed equation by Fujikake et al. (2003) for concrete cone breakout failure mode.

DIF (FEA)

Fig. 21. DIF obtained from the finite element analysis versus the predicted DIF for the adhesive anchors exhibited combined cone-bond failure.

dynamic strength of concrete respectively. Also, Fujikake et al. (2003) proposed equations to determine the ultimate dynamic load for the combined cone-bond failure mode [39] as follows:

123

d (mm)

hef (mm)

Fus FEA (kN)

Fcd Fujikake (kN)

Fus FEA/Fcd Fujikake

12.7 12.7 15.9 15.9 19.1 19.1

76.2 101.6 76.2 101.6 76.2 101.6

57.12 80.45 65.69 87.31 69.62 105.49

53.10 79.96 54.27 81.31 55.45 82.67

1.08 1.01 1.21 1.07 1.26 1.28

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Table 5 Comparison between ultimate dynamic load obtained from the FEA and the proposed equations by Fujikake et al. [39] for combined cone-bond failure mode. d (mm)

12.7 12.7 12.7 12.7 12.7 12.7 12.7 12.7 15.9 15.9 15.9 15.9 15.9 15.9 15.9 15.9 19.1 19.1 19.1 19.1 19.1 19.1 19.1 19.1 19.1 19.1 19.1 19.1 19.1 19.1 *

hef (mm)

76.2 76.2 76.2 76.2 101.6 101.6 101.6 101.6 76.2 76.2 76.2 76.2 101.6 101.6 101.6 101.6 76.2 76.2 101.6 101.6 101.6 101.6 127.0 127.0 127.0 127.0 152.4 152.4 152.4 152.4

εṡ (s−1)

−5

10 10−5 10−5 10−5 10−5 10−5 10−5 10−5 10−5 10−5 10−5 10−5 10−5 10−5 10−5 10−5 10−5 10−5 10−5 10−5 10−5 10−5 10−5 10−5 10−5 10−5 10−5 10−5 10−5 10−5

εḋ (s−1)

–3

10 10–1 10 102 10–3 10–1 10 102 10−3 10−1 10 102 10−3 10−1 10 102 10−1 10 10−3 10−1 10 102 10−3 10−1 10 102 10−3 10−1 10 102

εḋ / εṡ

2

10 104 106 107 102 104 106 107 102 104 106 107 102 104 106 107 104 106 102 104 106 107 102 104 106 107 102 104 106 107

FEA

Regression Fu (kN)

Fus (kN)

Fud (kN)

57.12 57.12 57.12 57.12 80.45 80.45 80.45 80.45 65.69 65.69 65.69 65.69 87.31 87.31 87.31 87.31 69.62 69.62 105.49 105.49 105.49 105.49 157.28 157.28 157.28 157.28 179.40 179.40 179.40 179.40

63.33 67.39 71.14 88.11 88.59 92.31 95.49 98.60 69.70 74.94 87.36 118.21 94.88 101.48 108.31 124.28 89.32 114.01 113.94 120.78 139.14 196.53 161.79 169.02 185.58 217.96 187.48 196.47 211.45 234.18

60.63 66.42 72.76 76.15 85.39 93.54 102.47 107.25 69.72 76.38 83.67 87.57 92.67 101.52 111.21 116.40 80.95 88.68 111.97 122.66 134.37 140.63 166.94 182.87 200.33 209.68 190.42 208.59 228.51 239.17

Fujikake Eq. (10)

Fud FEA/Fccb Fujikake

Fcd (kN)

Fb (kN)

Fccb (kN)

18.48 20.88 31.01 44.55 18.48 20.88 31.01 44.55 19.28 21.79 32.37 46.49 19.28 21.79 32.37 46.49 22.70 33.72 20.09 22.70 33.72 48.44 20.09 22.70 33.72 48.44 20.09 22.70 33.72 48.44

38.38 40.75 43.26 44.58 62.04 65.87 69.93 72.06 47.00 49.90 52.97 54.58 75.97 80.66 85.63 88.23 60.98 64.75 92.85 98.58 104.66 107.84 128.26 136.18 144.58 148.97 163.67 173.77 184.49 190.10

56.86 61.63 74.28 89.12 80.52 86.75 100.95 116.61 66.28 71.69 85.34 101.07 95.25 102.45 118.00 134.73 83.68 98.47 112.94 121.28 138.38 156.28 148.35 158.88 178.30 197.41 183.76 196.47 218.22 238.54

1.11 1.09 0.96 0.99 1.10 1.06 0.95 0.85 1.05 1.05 1.02 1.17 1.00 0.99 0.92 0.92 1.07 1.16 1.01 1.00 1.01 1.26 1.09 1.06 1.04 1.10 1.02 1.00 0.97 0.98

Fus: ultimate static load obtained from FEA, Fud: ultimate dynamic load obtained from FEA.

investigated. The main conclusions obtained from the finite element analyses on the adhesive anchors can be summarized as follows:

References [1] Cook RA. Behavior of chemically bonded anchors. J Struct Eng 1993;119(9):2744–62. [2] McMullin PW, Price JS, Persellin EH. Concrete design. New York: Routledge Taylor and Francis Group; 2016. [3] Upadhyaya P, Kumar S. Pull-out capacity of adhesive anchors: an analytical solution. Int J Adhes Adhes 2015;60:54–62. [4] Epackachi S, Esmaili O, Mirghaderi SR, Behbahani AAT. Behavior of adhesive bonded anchors under tension and shear loads. J Constr Steel Res 2015;114:269–80. [5] Cattaneo S, Muciaccia G. Adhesive anchors in high performance concrete. Mater Struct 2015:1–12. [6] Yilmaz S, Özen MA, Yardim Y. Tensile behavior of post-installed chemical anchors embedded to low strength concrete. Constr Build Mater 2013;47:861–6. [7] Cook RA, Doerr GT, Klingner RE. Bond stress model for design of adhesive anchors. ACI Struct J 1993;90(5):514–24. [8] Zamora NA, Cook RA, Konz RC, Consolazio GR. Behavior and design of single, headed and unheaded, grouted anchors under tensile load. ACI Struct J 2003;100(2):222–30. [9] Cook RA, Kunz J, Fuchs W, Konz RC. Behavior and design of single adhesive anchors under tensile load in uncracked concrete. ACI Struct 1998;95(1):9–26. [10] Braimah A, Contestabile E, Guilbeault R. Behaviour of adhesive steel anchors under impulse-type loading. Can J Civ Eng 2009;36(11):1835–47. [11] Livermore Software Technology Corporation, LSTC. LS-DYNA keyword user’s manual volume II material models. vol. II, Livermore California; 2014. [12] Livermore Software Technology Corporation, LSTC. LS DYNA theory manual; 2015. [13] Hallquist JO. LS-DYNA theory manual. Livermore Software Technology Corporation; 2006. [14] Bala S. Tolerance used for tying slave nodes in TIED contacts. LSTC: Livermore Software Technology Corporation; 2008. [15] Chen S, Zang M, Wang D, Yoshimura S, Yamada T. Numerical analysis of impact failure of automotive laminated glass: a review. Compos Part B 2017;122:47–60. [16] Jaime MC. Numerical modeling of rock cutting and its associated fragmentation process using the finite element method; 2011. [17] Bermejo M, Goicolea JM, Gabald́ on F, Santos A. Impact and explosive loads on concrete buildings using shell and beam type elements. 3rd ECCOMAS themat conf comput methods struct dyn earthq eng. 2011. p. 1–14. [18] Murray Y. Users manual for LS-DYNA concrete material model 159; 2007. [19] Dogan F, Hadavinia H, Donchev T, Bhonge PS. Delamination of impacted composite structures by cohesive zone interface elements and tiebreak contact. Cent Eur J Eng 2012;2(4):612–26. [20] Braimah A, Guilbeault R, Contestabile E. High strain rate behaviour of chemically

• The ultimate tensile load and DIF of adhesive anchors increases with increase in the strain rate from 10 s to 10 s . The failure mode of adhesive anchorage systems is affected by the • −5 −1

• • • • •

3 −1

strain rate; at the static strain rate of 10−5 s−1, most of the adhesive anchors under tensile load exhibited concrete cone failure, at strain rates of 10−3 s−1, 10−1 s−1 and 10 s−1 combined cone-bond failure was observed while at the highest strain rate of 103 s−1, steel anchor failure was observed. Maximum DIF of 3.77 was obtained for the 19.1 mm diameter adhesive anchor with embedment depth of 76.2 mm at high strain rate of 103 s−1 where steel anchor failure mode is observed. The ultimate tensile load increased with the increase in the embedment depth at the same strain rate when the concrete cone breakout failure is the dominant failure mode. The increase in the anchor diameter from 12.7 mm to 19.1 mm increased the concrete cracking and level of damage sustained by the concrete substrate. The concrete cone breakout diameter increased with the increase in the anchor diameter. Analytical equations were developed to relate the DIF and the strain rate for the adhesive anchorage to concrete systems under tensile load and for use in design of adhesive anchorage systems.

Acknowledgements The authors would like to thank Department of Civil and Environmental Engineering, Carleton University for providing the facilities and the software necessary to accomplish this research.

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L.T. Ahmed and A. Braimah

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