Block shear failure in welded gusset plates under combined loading

Journal of Constructional Steel Research 170 (2020) 106079

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Journal of Constructional Steel Research

Block shear failure in welded gusset plates under combined loading Shervin Maleki ⁎, Majid Ghaderi-Garekani Dept. of Civil Engineering, Sharif University of Technology, Azadi Ave., Tehran, Iran

a r t i c l e

i n f o

Article history: Received 2 February 2020 Received in revised form 18 March 2020 Available online xxxx Keywords: Block shear Base metal strength Inclined loading Combined loading Welded connections

a b s t r a c t Block shear failure in the base metal of welded steel connections is a potential failure mode affecting many steel structures. However, there are only a few studies on the block shear failure of welded connections under combined shear and axial loading. Combined loading is defined as a simultaneous loading parallel and perpendicular to the weld lines (or an inclined loading) in the plane of the connecting plate. In this research, a nonlinear finite element model is used to study the effect of connection geometry and weld group configuration on the block shear strength of welded connections under combined loading. In current design standards, the block shear failure planes are assumed to consist of shear and tension planes, parallel and perpendicular to the applied load, respectively. The results obtained indicate that for combined loading, inclined failure planes are possible and the state of stress in such planes is a combination of tensile and shear stresses, the extent of which is proportional to the angle of loading. The results are used to describe the behavior of block shear rupture planes in such cases. Finally, to provide a better estimate of block shear strength in welded connections subjected to combined loading, a new equation is proposed that considers the actual shape of displaced block. © 2020 Elsevier Ltd. All rights reserved.

1. Introduction 1.1. Background Block shear failure is one of the probable failure modes for welded structural steel connections in which a block of base metal material surrounding the welded region is detached from the connecting element. The most remarkable feature of this failure mode is a variable contribution of two stress components: tensile stresses in the planes perpendicular to the loading direction and shear stresses in the planes parallel to the loading direction. The block shear design strength equations in current North American design standards have been mainly based on research results of steel members with bolted connections. The structural behavior and the block shear capacity of gusset plates with bolted end connections have been studied by a number of researchers [1–7]. However, for welded connections, research studies are scarce. In current design standards, a combination of shear and tensile strength on failure planes controls the block shear capacity. The ANSI/AISC 360–16 specification [8]

⁎ Corresponding author. E-mail address: [email protected] (S. Maleki).

https://doi.org/10.1016/j.jcsr.2020.106079 0143-974X/© 2020 Elsevier Ltd. All rights reserved.

suggests the block shear nominal strength, Rn, as the lesser of the following two equations: Rn ¼ 0:6F u Anv þ U bs F u Ant

ð1Þ

Rn ¼ 0:6F y Agv þ U bs F u Ant

ð2Þ

where Fy and Fu are the yield and tensile strengths of the steel material, respectively, Anv and Agv are the net and gross areas subjected to shear, respectively, Ant is the net area subjected to tension and Ubs is a reduction coefficient for nonuniform tensile stresses. Where the tension stress is uniform, Ubs = 1; where the tension stress is nonuniform, as in the case of multi-row bolted clip angle connections, Ubs = 0.5. In this respect, it should be noted that for welded connections the net and gross areas are identical. Therefore, Eq. (2) yields a lower capacity and governs. Based on reliability analyses using a large experimental database gathered by Kulak and Grondin [1] for bolted connections, Driver et al. [9] recommended a “unified equation” which provides a more consistent level of safety than Eqs. (1) and (2), as follows:   Fy þ Fu pffiffiffi Rn ¼ Rt Ant F u þ Rv Agv ð3Þ 2 3 where Rt and Rv are tension area and shear area mean stress correction factors, respectively. Experimental and numerical studies on the block

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shear failure of welded gusset plates conducted by Topkaya [10], indicated that the mechanics of block shear in welded connections is distinct from bolted connections. Also, a more precise nominal resistance equation was proposed for block shear failure of gusset plates with uniaxial concentrically loaded welded connections considering stress triaxiality as: Fu Rn ¼ 1:25 F u Agt þ pffiffiffi Agv 3

ð4Þ

in which Agt is the gross area subject to tension, and all other variables have been defined previously. Oosterhof and Driver [11] conducted numerical and experimental research on concentrically loaded welded lap plate connections. The results confirmed the accuracy of the unified equation (Eq. (3)), with a value of Rt = 1.25 for the design of these connections. They showed that the effects of connection geometry and weld arrangement on block shear capacity are small, and there is no need for modification in the unified equation (Eq. (3)). Experimental and numerical investigations on block shear failure of coped beams with welded end connections have been conducted by Yam et al. [12,13], based on which, a new block shear equation was proposed. In addition, providing extended tests data by Wei et al. [14] and further finite element analyses by Yam et al. [15], the proposed design strength equation was modified in order to take into account the effect of connection rotational stiffness. When an applied load on a connection consists of an inclined load with two perpendicular components with respect to the weld lines, the resultant force acting on each weld line is also inclined. This type of loading occurs frequently in steel structures particularly, in welded column and beam splice connections, in welded beam to beam/column

Fig. 2. Block shear failure in bolted single plate connection subjected to combined axial and shear forces [16].

double-angle or single plate shear connections, as shown in Fig. 1. In these cases, the stresses in the base metals adjacent to the welds are in a combination of shear and tensile stresses and are not clearly investigated.

Fig. 1. Examples of combined loading in welded connections: (a) welded double-angle shear connection, (b) welded column splice and (c) welded beam splice.

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As shown in Fig. 2, a similar situation exists in beam to column bolted single plate (shear tab) connections when beam axial load is also present due to seismic collector forces (or from other sources). The block shear failure pattern and strength in these cases are affected by the simultaneous presence of the two perpendicular loads. Using the von Mises yield criterion, Astaneh-Asl [16] suggested a method to check the block shear limit state for a bolted single plate subjected to combined loading. Block shear strength under combined axial and shear forces also appears in the design examples presented by AISC Seismic Design Manual (SDM) [17]. In particular, in the example problem 5.2.4, which deals with the Brace-to-Beam/Column connection design using the Uniform Force Method (UFM), as illustrated in Fig. 3. In this example, using elliptical interaction, SDM applies the AISC specification equations (i.e., Eqs. (1) and (2)) to check the block shear limit state for the bolted single plate at the column interface under combined shear and axial force as follows: 

V ϕRnV

2

 þ

H ϕRnH

2 ≤1

ð5Þ

where Vand H are the shear and axial components of the applied load, respectively. RnV and RnH are the block shear nominal capacities relative to the applied shear and axial loads, respectively, as calculated by Eqs. (1) and (2) and ϕ is the resistance factor. In Eq. (5), the block shear capacity is calculated separately once for the horizontal and once for the vertical direction assuming no interactions, and then combined via the interaction equation given above. However, under real combined loading, the stress state on tensile and shear rupture planes would be coupled and the “no interaction” assumption might cause serious errors. In this paper,

3

this equation is later examined for welded connections under combined loading. Although block shear failure may occur in both bolted and welded steel connections, a review of the relevant literature indicates that there has been scant attention to the block shear failure of welded connections. More specifically, the failure mechanism and strength calculation of this limit state under combined loading has neither been addressed in the current design standards nor has been researched before. 1.2. Problem statement The lap splice of two thick plates welded to a thin gusset plate subjected to an inclined loading (R) is depicted in Fig. 4. The reason for using two thick plates as a member is to ensure that the base metal failure occurs in the gusset plate and load eccentricity and shear lag effects are minimized. The applied inclined load can be resolved into combined loads consisting of a horizontal component (N) and a vertical component (V). Probable block shear failure path in the gusset plate is shown by the dashed lines in the same figure. As shown in Fig. 4, planes No. 1 to No. 3 are the failure planes of the base metal adjacent to the weld, and (β) is the angle between the failure plane No. 3 and the direction of pure shear (i.e., ‘x’ axis). The ratio of the axial (N)and shear (V) components of the load defines the angle of loading (θ) as follows: tanðθÞ ¼

N V

ð6Þ

Accordingly, when the load R is applied at angle θ = 0° the loading is considered as a pure shear load (V), and forθ = 90°, the loading is all tensile (N) with respect to the gusset plate. In pure tensile loading, the

Fig. 3. Connection details for example 5.2.4 of SDM [17].

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Fig. 4. Typical block shear failure path under combined loading.

block shear failure path in the gusset is assumed to be U-shaped such that, planes No. 1 and No. 2 are considered as shear planes and plane No. 3 as tensile plane and angle β is zero. This is a common case of block shear failure in welded connections. As indicated in ANSI/AISC 360-16 [8] specification, block shear limit state must be checked around the periphery of welded connections using Eq. (2) by assuming a failure path comprised of one or two shear planes and one tensile plane. Therefore, as shown in Fig. 5(a), U-shaped failure path (path 1) with its associated tensile and shear rupture planes should be considered. It is also possible that L-shaped failure path (path 5 in Fig. 5(b)) takes place when edge distance of the gusset is short. By varying θ from 90° to 0°, the loading turns from pure tension to pure shear and the block shear failure path occurs in L-shaped form (path 5 in Fig. 5(b)) and angle β equals to 90°. For (0° b θ b 90°), the loading is a combined axial and shear load. In this situation, as shown in Fig. 5(b), there are several possible failure paths (e.g., paths 2–4) in addition to the previous ones. Obtaining the true failure path with the aid of current design specifications is not possible. Furthermore, the shear or tensile stress states on the failure planes are not distinguishable from each other as they are coupled. This paper discusses the numerical investigations that have been carried out to assess the block shear capacity of welded plates subjected to combined shear and axial loads. A nonlinear finite element (FE) model is utilized to execute a parametric study in order to examine the effects of various parameters such as angle of the inclined load,

connection geometry and weld arrangement on the block shear capacity and failure path of welded plates. Moreover, the main objective of this research is to develop a new block shear strength prediction equation under combined loading for welded connections. 2. Finite element analysis The general purpose finite element (FE) software ABAQUS [18] is utilized to model the welded lap plate connection depicted in Fig. 4 for large deformation nonlinear analyses. The analyses are primarily intended to scrutinize the block shear failure mechanism of a welded plate under combined loading with an emphasis on the influence of weld arrangement. The FE models are composed of three parts: lap plates, gusset plate, and weld. Two types of welding are taken into account as shown in Fig. 6. Weld group Type (A) have only longitudinal welds and weld group Type (B) have both longitudinal and transverse welds. 2.1. Elements and meshing Fig. 7 shows a typical FE model and the density of the mesh adopted. All components of the model were meshed by three dimensional 8node solid elements with reduced integration (C3D8R). To get rid of hourglass effects, the number of elements in the y-direction (thru thickness) for the gusset plate, lap plate and welds were selected as 1, 3 and

Fig. 5. Block shear failure under (a) uniaxial loading (b) combined loading.

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Fig. 6. Weld group configuration.

Fig. 7. Typical meshing and boundary condition of FE models.

3, respectively. The preliminary FE analyses revealed that the results depend on the mesh density of the gusset plate rather than the lap plate. Therefore, prior to modeling, a convergence study was conducted in order to provide a better estimate of the mesh size in the gusset plate. It was observed that there is not a significant change in the block shear strength when the mesh size decreases below 5 mm. Thereby, based on the models dimensions a 5 mm mesh size was adopted.

2.2. Boundary conditions Taking advantage of symmetry, only one of the lap plates and half the thickness of gusset plate were modeled, which reduces the computational time. Symmetry boundary condition was applied to the nodes which lie in the plane of symmetry. The end of gusset plate (i.e., edge AD) was fully fixed against translation and rotation. The edge AB of the gusset plate which is parallel to z-axis was fixed against translation in the x- and y- directions. Also, in order to let the gusset plate fail in an L-shape block shear, the edge DC was only fixed against translation in the y-direction. Furthermore, the findings showed that changing the rotational boundary condition of the edges AB and DC had no remarkable

effect on the connection block shear capacity and its associated failure pattern. 2.3. Constraints and interactions Adjacent surfaces of the weld with the lap plate and gusset plate were tied together via surface-based tie constraints which make displacement and rotation at the two adjacent nodes equal. In this way, the weld nodes were considered as master nodes and the gusset nodes as slave nodes, since they are more flexible. In order to avoid surface penetration of the components into each other, the interaction between the lap plate and the gusset plate was modeled as contact surfaces. In particular, the “hard contact” and “sliding without friction” were used in the normal and tangential directions, respectively. 2.4. Material modeling Stress-strain data were taken directly from the results of coupon test of CSA G40.21300 W structural steel [19]. The modulus of elasticity of 207 GPa and Poisson's ratio of 0.3 were used in accordance with the same coupon test data. The yield and ultimate strengths obtained

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from the test were 328 MPa and 503 MPa, respectively. In addition, true stress and strain values were obtained from the engineering stress and strain results using the following equations:   σ true ¼ σ eng 1 þ εeng

ð7Þ

  σ true ε ptrue ¼ ln 1 þ εeng − E

ð8Þ

in which σtrue is the true stress, σeng is the engineering stress, εptrue is the true plastic strain, εeng is the engineering strain and E is the modulus of elasticity. Plastic deformation properties were not considered in modeling the lap plates and welds, for these members were designed to remain elastic before rupture takes place in the gusset plate. The plasticity model of the gusset plate was based on the von Mises yield criterion and its associated flow rule. 2.5. Damage criteria With the aim of an accurate examination of block shear failure path in a welded gusset plate, material strength degradation due to initiation and evolution of cracks, was taken into account using “Damage for Ductile Metals” module available in ABAQUS [18]. A stress triaxiality dependent fracture locus calibrated for industrial aluminum and steel was invoked [20], in which, the damage is supposed to initiate when the equivalent plastic strain, (ε−pl), reaches the failure strain,(ε−pl 0 ). In order to capture the residual load-carrying capability of the material (i.e., a post-peak softening branch of a stress-strain curve), progressive damage evolution option in ABAQUS was activated, and also, among various damage evolution models, a linear softening law was utilized. Thereupon, damage parameters were determined and calibrated through trial and error to match the coupon test data of the material. 2.6. Loading protocol Combined loading was applied by enacting its two perpendicular components at the weld group centroid, in which, the ratio of the two load components is representative of the angle of loading (θ) as shown by Eq. (11). A static-general step was defined for loading, during which, load was applied through displacement control by a ramp function. 2.7. Verification of the FE modeling In order to validate the discussed FE modeling assumptions and procedure, two instances of tested gusset plate with welded connections, available in the literature, were modeled with similar techniques discussed above: (I) –specimen #11, tested by Topkaya [10], and (II) – specimen W3, tested by Oosterhof and Driver [21]. 2.7.1. Specimen #11 by Topkaya [10] Specimen #11 consists of flat bars, 15 mm thick and 150 mm wide, welded on both faces of gusset plate which has dimensions of 500 mm length, 750 mm width and 4 mm thickness. The weld group configuration consists of both longitudinal and transverse welding similar to Type (B) shown in Fig. 6 . Material properties used in the FE model had yield strength Fy = 309 MPa and ultimate strength Fu = 402 MPa, as reported by Topkaya [10]. Also, minimum specified strength of weld, (FEXX), were taken as 550 MPa. The connection details and the dimensions of the modeled specimen are shown in Fig. 8 and Table 1, respectively. The specimen was modeled in ABAQUS according to the previously described procedure and the load-displacement response curve of the model is compared with the test results of Topkaya [10] in Fig. 9. As can be seen, an excellent correspondence is observed. Moreover, failure

Fig. 8. Specimen #11 configuration [10].

Table 1 Dimensions of specimen #11 [10]. Dimensions (mm) Weld length (L) Weld separation (W) Edge distance Plate thickness Gusset thickness

150 99.9 300 15 4

pattern of the tested specimen matches well with that obtained from the FE analysis (Fig. 10). It can be seen that the FE analysis, particularly, the damage model implemented, predicts fracture initiation and propagation in block shear failure of the welded plate accurately. 2.7.2. Specimen W3 by Oosterhof and Driver [21] In order to assess the capability of the ABAQUS model in predicting the block shear capacity of welded lap joints, specimen W3 tested by Oosterhof and Driver [21] was modeled in the software. As shown in Fig. 11, the specimen consists of a wide flange member connected to the thick narrower plates using longitudinal and transverse welds. The thickness of the web plate and lap plates were 6.2 mm and 8 mm, respectively. The modulus of elasticity and Poisson's ratio were 206,800 MPa and 0.3, respectively. The yield stress and ultimate strength of the material used in the FE model was 379 MPa and472 MPa, respectively. Dimensions of the modeled specimen and material properties are given in Table 2. Representative load-displacement response curve of the modeled assembly is compared with the corresponding results obtained from test in Fig. 12. As can be seen, the test data and the results of the FE analyses are in a very good agreement. 3. FE analysis results and discussion The efficiency of the FE modeling to investigate the block shear failure in welded connections has been verified. Using this validated FE modeling technique, a comprehensive parametric study was conducted to further investigate the influence of various parameters on the block shear strength of welded plates subjected to combined axial and shear forces. Table 3 shows a summary of the geometric parameters involved

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Fig. 9. Comparison of the FE model and test results of specimen #11 tested by Topkaya [10].

Fig. 10. Fractured profiles of specimen #11: (a) FE prediction and (b) experimental observation.

Fig. 11. Specimen W3 configuration [11].

(as defined in Fig. 4), for twenty welded lap plate connections used in the FE models. These parameters consist of weld length, weld separation distance, gusset plate thickness and edge distance. Also, the last

five models (Type B) consider the effect of added transverse weld on the block shear strength. In all models, the lap plates are 12 mm in thickness. The effect of edge distance is considered by defining angle (α)

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S. Maleki, M. Ghaderi-Garekani / Journal of Constructional Steel Research 170 (2020) 106079 Table 2 Summary of specimen dimensions [21].

Table 3 Finite element models geometries.

Dimensions (mm) Weld length (L) Weld separation (W) Edge distance Lap plate thickness Web plate thickness

201.4 103.7 NA 8 6.2

shown in Fig. 4, herein called the ‘diagonal angle’, which is proportional to the ratio of weld length (lw) to the edge distance (e). Varying (θ) from (0°) to (90°) in (15°) increments, each model was analyzed under seven angles of loading. Therefore, the ultimate failure loads of the 140 FE models, all of which failed in block shear mode, are summarized in Table 4.

3.1. Block shear failure path All models were failed in block shear as was expected. However, different failure paths were observed. In pure tensile loading, when θ = 90°, the failure occurred generally in U-shaped form such that, the angle β is 0° and tension rupture in plane No. 2 preceded shear ruptures in planes No. 1 and No. 3 (see Fig. 4). Block shear limit state under this common loading condition (pure tensile) has been studied previously and is confirmed here as well [10,11]. Fig. 13 illustrates the failure initiation from the tensile plane and its propagation to the shear planes for this loading. In pure shear loading condition, which corresponds to angle of loading θ = 0°, connection failure was in L-shaped form as shown in Fig. 14. This failure pattern consists of a tensile plane perpendicular, and a shear plane parallel to the load direction. In this case, referring to Fig. 4, the FE findings revealed that the angle β is 90° and so, the plane No. 1 is tension plane which is perpendicular to the loading direction and planes No. 2 and No. 3 are shear planes which are parallel to the loading direction. Also, it was observed that tension rupture in plane No. 1 preceded shear rupture in planes No. 2 and No. 3. Fig. 14 illustrates failure initiation from the tensile plane and its propagation to the shear planes. By varying θ from 0° to 90°, it was observed that β also changes from 0° to 90° and the order of rupture planes changes. In the mid-range for θ, (e.g., θ = 45°), the stress state in all failure planes is a combination of tensile and shear stresses. Fig. 15 shows initiation and propagation of failure in Model #6 for angle of loading equal to45°. In this case, the

Model #

Type

(lw) (mm)

(s) (mm)

(e) (mm)

(t) (mm)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

A A A A A A A A A A A A A A A B B B B B

100 100 100 100 50 100 100 100 100 50 100 100 100 100 50 100 100 100 100 50

100 100 100 50 100 100 100 100 50 100 100 100 100 50 100 100 100 100 50 100

150 100 50 50 100 150 100 50 50 100 150 100 50 50 100 150 100 50 50 100

4 4 4 4 4 5 5 5 5 5 6 6 6 6 6 4 4 4 4 4

Table 4 Finite element analysis results summary. Model #

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Ultimate load (R) (kN) θ = 0°

θ = 15°

θ = 30°

θ = 45°

θ = 60°

θ = 75°

θ = 90°

568.1 507.2 439.5 375.8 381.2 711.8 632.1 550.5 477.5 477.8 866.8 770.1 655.6 599.3 561.2 571.5 510.5 438.0 384.3 378.9

583.2 516.2 461.2 393.5 401.2 752.3 647.1 568.2 475.7 506.6 869.9 786.4 682.0 569.5 598.7 600.4 529.3 443.5 381.6 408.3

669.6 574.4 475.4 394.3 424.3 842.4 698.3 583.7 498.8 532.5 987.9 843.9 718.0 602.9 641.2 667.6 582.6 481.5 392.4 439.2

673.4 632.1 517.8 412.2 425.6 800.2 805.6 642.6 524.8 524.2 985.1 968.2 767.4 612.2 629.4 640.2 641.6 521.6 412.5 423.1

612.3 609.4 580.2 466.1 425.9 747.6 765.3 719.0 589.0 516.5 920.2 939.4 871.0 685.9 625.3 601.5 612.5 571.2 460.0 416.0

569.1 575.2 560.9 446.8 412.0 700.3 733.7 719.5 560.5 527.7 876.2 845.6 857.3 674.1 623.9 571.6 580.5 571.2 443.8 414.6

525.2 518.9 527.4 385.6 391.5 650.3 642.5 624.1 473.6 484.9 763.2 748.6 778.5 570.0 572.1 504.6 508.1 518.9 389.0 385.6

Fig. 12. Comparison of the FE model and test results of specimen W3, tested by Oosterhof and Driver [21].

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Fig. 13. Progression of block shear failure path in model #6 with (θ = 90°).

failure plane No. 3 is inclined as depicted in Fig. 15. Also, as can be seen, the block shear failure path is neither U-shaped and nor L-shaped. It appears that, under combined loading, the stress state on all failure planes is a combination of tensile and shear stresses rather than a pure tension or shear.

planes take place, whether transverse weld is present or not. So, the presence of transverse weld has no significant effect on block shear capacity or its failure path.

3.2. Weld arrangement effects

In the conventional method used by the AISC SDM [17] (i.e., Eq. (10)), the inclined direction of failure plane No. 3 is not considered. In other words, failure planes are supposed to be either in tension or in shear, rather than a combination of these two stress states. In order to focus more on the varying stress states on block shear failure planes under combined loading, average stresses normalized with respect to ultimate strength (Fu) on each failure plane for all angles of loading were obtained and shown in Table 5. In pure tensile loading (θ = 90°), planes No. 1 and No. 3 are shear planes, and plane No. 3 is in tension. Referring to Table 5, the average shear and tensile stresses are 0.66Fu and 1.33Fu for this case, respectively. This finding reveals that increasing the tensile plane capacity to a value higher than Fu due to presence of stress triaxiality is a reasonable assumption as was confirmed through tests by Topkaya [10]. Stress triaxiality is defined as the ratio of maximum principal stress to the von Mises equivalent stress. By varying the loading direction, the tensile and shear behavior of the failure planes switch from one to the other. So, failure planes are in a combination of tensile and shear stress states, such that, the extent of participation of each stress component is proportional to the angle of loading, (θ). More specifically, according to Table 5, it can be stated that for small values of (θ), (β) is large and

To understand the effects of weld arrangement, or in particular the presence of transverse weld in addition to longitudinal welds, the results of Models #16 to #20 with transverse weld were compared to Models #1 to #5 without the transverse weld (see Table 3). The findings indicate that change in the block shear strength is less than 5% as illustrated in Table 4. So, in the case of combined loading, transverse welding does not affect the results significantly for all angle of loading, which is similar to the results obtained in the previous study on block shear strength of uniaxially loaded (i.e., θ = 90°) welded gusset plate connections [11]. Representative load-displacement response curves comparing the connection behavior of Model #2 and Model #17 are shown in Fig. 16, for different loading angles. As can be seen, the change in load-displacement curve due to the presence of transverse weld is negligible. In addition, it should be noted that the failure path was not changed dramatically by the presence of transverse welding. Fig. 17 depicts the peak equivalent plastic strain (PEEQ) comparison between Model #2 and #17 which are about the same. The reason for this is that the separation of the base metal does not occur until the failure of all three

3.3. Determination of failure planes

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Fig. 14. Progression of block shear failure in model #6 with (θ = 0°).

plane No. 1 would fail in tension, and planes No. 2 and No. 3 would mostly fail in shear. However, increasing (θ) would result in smaller (β) which increases the shear portion of plane No. 1 and tensile portion of plane No. 2.

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 1 Rn ¼ u u    ≥ Ru =ϕ t sinθ 2 cosθ 2 þ RnN RnV

4. Proposed design strength equations 4.1. Conventional equation As stated in the introduction, in order to calculate the block shear capacity under combined loading, AISC SDM [17] applied an elliptical interaction between block shear strength relative to two perpendicular directions. In this method which was utilized for bolted connections, the strength of block shear limit state is calculated relative to applied axial and shear forces separately and then combined together as an interaction equation as follows:     Ru cosθ 2 Ru sinθ 2 þ ≤1 ϕRnv ϕRnN

and (RnN), respectively. Then these values are combined together using Eq. (9). Using simple algebra the above equation can be transformed to the nominal block shear strength of a welded plate subjected to an inclined load (Ru) as follows: ð10Þ

The above equation is herein called the conventional equation for combined loading of welded connections. It predicts the capacity by calculating the tensile and shear strengths of the base metal separately and combining them in the form of an elliptical interaction. In this approach, the real shape of the displaced block which corresponds to failure path under combined loading is not considered. Note that, the resistance factor, (ϕ), equal to 0.75 is considered according to AISC specification [8] for design purposes.

ð9Þ

In this study, the application of this equation is examined for welded connections. In this way, for a welded connection shown in Fig. 4, the block shear strength is first calculated relative to axial and shear loads (via Eq. (2) which governs) to obtain the block shear capacities (RnV)

4.2. Proposed equation based on FE analysis The aim of this study is to develop an equation to predict the block shear capacity of welded connections subjected to combined shear and axial loads. As stated in the previous section, in the conventional method the block shear capacity is calculated with respect to each

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Fig. 15. Progression of block shear failure in model #6 with (θ = 45°).

load component separately. Therefore, a modified block shear design equation is needed to take into account the real stress state due to combined loading on the tension and shear planes, as well as, stress triaxiality which is available in welded connections. Based on the FE analyses findings, the vector resultants of the block shear capacities with respect to the pure tension and pure shear components of the applied load can be utilized as a basis of the formulation. In the proposed approach, the shear and tensile capacities relative to shear load V and axial load N are calculated such that the contribution of each stress component on the failure planes is proportional to the angle of loading. Furthermore, as the FE analyses revealed, it is reasonable to assume that the inclined failure plane (i.e., plane No. 3) would rupture parallel to the direction of the applied load. Fig. 18(a) and (b) illustrate the failure planes when the angle of loading θ is greater and smaller than diagonal connection angle α, respectively. Referring to Fig. 18, the diagonal angle (α) the load angle (θ) and the length of failure plane No. 3 (lr) can be found as follows:   lw ð11Þ α ¼ tan−1 e θ ¼ tan−1

  N V

when θ N α:

ð12Þ

lr ¼

lw sinθ

and when θbα lr ¼

ð13Þ e cosθ

ð14Þ

Based on the above-mentioned assumptions the nominal block shear capacity of a welded plate subjected to a combined loading can be estimated as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rn ¼ ððSN þ T N Þ  sinθÞ2 þ ððSV þ T V Þ  cosθÞ2 ð15Þ SV ¼ 0:6  F u  t  ðs þ lr cosθÞ

ð16Þ

T V ¼ 1:2 F u  t  lw

ð17Þ

SN ¼ 0:6  F u  t  ðlw þ lr sinθÞ

ð18Þ

T N ¼ 1:2 F u  t  ðs þ lr cosθÞ

ð19Þ

where Rn is the nominal block shear strength and t is the thickness of the base metal (e.g., gusset plate). SV and SN are the shear capacities relative to shear and axial component of the applied load, respectively. TV and TN

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Fig. 16. Load-displacement curves of Models #2 and #17 with (a) (θ = 0°), (b) (θ = 45°) and (c) (θ = 90°).

Fig. 17. Displaced block for (a) Model #2 and (b) Model #17.

are tensile capacities relative to shear and axial component of the applied load, respectively. In the proposed equation, it is assumed that the tension plane has a 20% higher capacity in excess of the ultimate strength (Fu) due to stress triaxiality effects. Note that, the above equation considers a ruptured state for the block shear phenomenon and therefore, the yield stress is not used. In addition, the resistance factor equal to 0.75 can be applied for design purposes [8].

4.3. Discussion and comparison A comprehensive comparative study was performed in order to assess the accuracy of the conventional and proposed equations. Accordingly, the block shear strength was calculated through Eqs. (10) and (15) for all angles of loading for the twenty models. The percent differences between the predicted nominal capacities obtained by each approach and those obtained by the FE analyses are given in Tables 6

S. Maleki, M. Ghaderi-Garekani / Journal of Constructional Steel Research 170 (2020) 106079 Table 5 Normalized average stresses on failure planes. Normal stresses

13

Table 6 Comparison of the predictions of the conventional equation with FE analyses. Shear stresses

Model #

θ (deg.)

σ xx in Fu plane No. 1

σ zz in Fu plane No. 2

σ xx in Fu plane No. 3

τ zx in Fu plane No. 1

τ zx in Fu plane No. 2

τ zx in Fu plane No. 3

0 15 30 45 60 75 90

1.31 1.19 0.98 0.83 0.51 0.29 0.00

0.00 0.32 0.54 1.10 1.16 1.24 1.33

0.00 0.38 0.56 0.79 0.61 0.33 0.00

0.00 0.09 0.20 0.25 0.39 0.51 0.63

0.67 0.54 0.39 0.30 0.24 0.13 0.00

0.59 0.56 0.52 0.46 0.56 0.61 0.66

and 7, respectively. Referring to these tables, the key observations are as follows: 1. The conventional equation is highly conservative. It predicts the nominal block shear capacities that are on the average 34% less than the numerical findings. Note that, the resistance understrength factor ϕ was not applied in this comparison, which will add to its conservatism. The reason for this discrepancy is that in the conventional method the failure planes are considered to be either in tension or shear rather than a combination of the two. In addition, inclined failure plane is not considered. 2. The proposed equation (Eq. (15)) shows an excellent agreement with the numerical findings. The average difference is about 5% on the conservative side for all cases considered. In the proposed equation, the extent of contribution of each component of the stress, i.e., shear or tensile, is taken into account, through which, the true failure path and consequently block shear capacity is predicted. This leads to a more accurate estimation of base metal capacity and more economical design. To examine the accuracy of the two approaches in detail, the block shear strength calculated by the resistance equations are compared with the failure loads obtained by FE analyses in Fig. 19. In these figures, data points above the diagonal line, illustrate conservative load predictions. The data points closer to the diagonal line indicate more accurate predicted capacities. Referring to Fig. 19(a), it can be deduced that the nominal resistance predicted by the conventional equation is very conservative. Furthermore, a closer inspection of Fig. 19(b) indicates that the proposed equation predicts the block shear strength with an excellent accuracy.

Rn FEM −Rn #1 Rn FEM

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Mean Unbiased standard deviation

(%)

θ= 0°

θ= 15°

θ= 30°

θ= 45°

θ= 60°

θ= 75°

θ= 90°

29.9 29.3 27.4 25.5 32.3 30.1 29.1 27.5 26.7 32.5 31.1 30.1 26.9 29.9 31.0 30.4 29.7 27.1 27.2 31.9 29.3 2.0

32.3 30.5 30.3 29.3 35.4 34.4 30.7 29.3 26.9 36.0 31.9 31.6 29.3 26.7 35.0 34.2 32.2 27.5 27.1 36.5 31.3 3.1

42.2 37.6 31.0 30.5 38.0 42.6 35.8 29.8 31.4 38.3 41.2 36.3 31.5 31.8 38.5 42.0 38.4 31.9 30.2 40.1 36.0 4.5

44.0 43.3 34.9 34.9 37.0 41.1 44.4 34.4 36.1 36.0 42.6 44.4 34.1 34.3 36.1 41.1 44.1 35.3 35.0 36.6 38.5 4.1

40.0 41.1 40.1 43.6 35.7 38.6 41.4 39.6 44.2 33.7 40.1 42.7 40.2 42.5 34.3 38.9 41.4 39.2 42.8 34.2 39.7 3.1

36.6 37.6 36.6 42.0 32.5 35.6 38.9 38.2 42.2 34.1 38.2 36.4 37.8 42.3 33.1 36.9 38.2 37.8 41.6 32.9 37.5 3.0

31.7 30.9 32.0 33.1 28.5 31.1 30.2 28.2 31.9 27.8 29.5 28.1 30.9 32.1 26.6 28.9 29.4 30.9 33.7 27.4 30.1 2.0

Notes: Rn FEM - capacity obtained from FE analyses; Rn #1 - capacity calculated by Eq. (10).

5. Design example With the aim of showing how to apply the conventional and proposed equations, block shear capacity of a welded gusset plate connection which is assumed to be identical to Model #2 is considered and shown in Fig. 20. The connection is subjected to axial and shear compopffiffiffi nent of an inclined force such that, ratio (N/V) equals 3. The gusset plate thickness (t) equals 4 mm. The weld length (lw), the weld separation (s) and the edge distance (e) are all 100 mm. The yield strength Fy is 328 MPa and the ultimate tensile strength Fu is 503 MPa. Therefore for this connection, the load angle, (θ), and the diagonal angle, (α), are: pffiffiffi θ ¼ tan−1 3 ¼ 60∘ : α ¼ tan−1 ð1Þ ¼ 45∘ :

Fig. 18. Failure planes assumed in the proposed equation when (a) θ N αand (b) θ b α.

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Table 7 Comparison of the predictions by the proposed equation with FE analyses. Model #

Rn FEM −Rn #2 Rn FEM

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Mean Unbiased standard deviation

(%)

θ= 0°

θ= 15°

θ= 30°

θ= 45°

θ= 60°

θ= 75°

θ= 90°

4.4 4.8 3.9 3.6 5.0 4.6 4.5 4.1 5.2 5.3 6.0 5.9 3.3 9.4 3.2 4.9 5.4 3.5 5.8 4.4 4.9 1.4

3.7 4.2 7.2 7.7 5.2 6.7 4.5 5.8 4.5 6.2 3.2 5.7 5.8 4.3 4.7 6.5 6.6 3.5 4.8 6.9 5.4 1.3

6.4 6.5 5.7 5.9 2.4 7.0 3.9 4.0 7.0 2.8 4.8 4.6 6.4 7.7 3.2 6.1 7.9 6.9 5.4 5.7 5.5 1.6

8.6 2.6 6.0 4.5 5.4 3.8 4.5 5.3 6.3 4.0 6.3 4.6 4.9 3.6 4.0 3.8 4.1 6.7 4.6 4.8 4.9 1.3

5.2 4.7 5.1 5.0 6.4 2.9 5.2 4.2 6.0 3.5 5.4 7.3 5.1 3.1 4.4 3.5 5.2 3.6 3.7 4.2 4.7 1.1

5.3 6.3 4.0 5.7 6.1 3.8 8.2 6.4 6.0 8.4 7.8 4.4 5.7 6.3 7.0 5.8 7.2 5.7 5.1 6.7 6.1 1.2

8.1 6.9 8.4 6.1 7.5 7.2 6.1 3.3 4.4 6.6 5.1 3.2 7.0 4.7 5.0 4.3 5.0 6.9 6.9 6.1 5.9 1.5

Notes: Rn FEM - capacity obtained from FE analyses; Rn #2 - capacity calculated based on Eq. (15).

Given that (θ N α), the shear capacities, SV and SN, due to shear and tensile component of the applied load, are:   100 SV ¼ 0:6  503  4  100 þ pffiffiffi ¼ 190:5 kN: 3 SN ¼ 0:6  503  4  ð2  100Þ ¼ 241:4 kN: Also, the tensile capacities, TV and TN, due to shear and tensile component of the applied load, are as follows: T V ¼ 1:2  503  4  100 ¼ 221:3 kN:

Fig. 20. Design example for the block shear of a welded gusset plate subjected to combined loading.

  100 T N ¼ 1:2  503  4  100 þ pffiffiffi ¼ 381 kN: 3 Hence, the block shear strength can be obtained by using Eq. (15) as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rn ¼ ðð241:4 þ 381Þ  sinð60∘ ÞÞ2 þ ðð190:5 þ 221:3Þ  cosð60∘ ÞÞ2 ¼ 580:6 kN: The block shear capacity using the conventional equation is also shown below. In this approach, the strength is calculated independently relative to each load component. Considering the U-shaped and Lshaped failure paths for the axial (N) and shear (V) components of the applied load, the capacities are: RnN ¼ 503  4  100 þ 0:6  328  2  100  4 ¼ 358:6 kN: RnV ¼ 503  4  100 þ 0:6  328  2  100  4 ¼ 358:6 kN: The resulting capacities are combined via an elliptical interaction as follows: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 1 Rn ¼ u u    ¼ 358:6 kN: t sin60∘ 2 cos60∘ 2 þ 358:6 358:6

Fig. 19. Comparison of FE results with predicted capacities by (a) the conventional equation and (b) the proposed equation.

Consequently, the predictions according to the proposed and the conventional methods are 580.6 kN and 358.6 kN, respectively. The corresponding FE result of the block shear strength for this connection is 609.4 kN (see Table 4). Accordingly, the proposed equation is within 4.7% of the FE result while the conventional equation is 41% on the conservative side. As a second design example, the block shear capacity of a beam web in a welded/welded double-angle shear connection as shown in Fig. 21 is calculated by the two methods. The connection is subjected to V = 300 kN shear force and N = 400 kN axial force due to gravity and seismic loads, respectively. In such shear connections, when the pure shear force is applied, the limit state of block shear is not a governing failure mode unless the beam is coped. Albeit, when the axial load is applied in addition to the shear force, block shear failure in the beam web is

S. Maleki, M. Ghaderi-Garekani / Journal of Constructional Steel Research 170 (2020) 106079

15

Fig. 21. Design example for the block shear failure of beam web under combined shear and axial force.

possible due to a combination of shear and axial loads [16]. Referring to Fig. 21, the double-angle shear connection consists of IPE 400 beam connected to a column by double equal leg 100 × 100 × 10 angle with 200 mm length. The beam has a clear distance of 20 mm from column. The members are constructed from St-37 steel with a nominal yield strength, Fy, of 240 MPa and nominal ultimate tensile strength, Fu, of 370 MPa. Note that, the beam depth is d = 400 mm and its flange thickness is tf = 13.5 mm and web thickness is tw = 8.6 mm. Therefore, the clear vertical distance between the angle and inside face of the flange is 86.5 mm. The loading angle, (θ), and the diagonal angle, (α), are calculated as: θ ¼ tan−1 ð400=300Þ ¼ 53∘ : α ¼ tan−1 ð80=86:5Þ ¼ 42:8∘ : Given that (θ N α), the shear capacities, SV and SN, due to shear and tensile component of the applied load, are:   80 ¼ 496:6 kN: SV ¼ 0:6  370  8:6  200 þ 1:33 SN ¼ 0:6  370  8:6  2  80 ¼ 305:5 kN: Similarly the tensile capacities, TV and TN, due to shear and tensile component of the applied load, are: T V ¼ 1:2  370  8:6  80 ¼ 305:5 kN:   80 ¼ 993:4 kN: T N ¼ 1:2  370  8:6  200 þ 1:33 Therefore, the block shear strength is obtained by Eq. (15) and checked against the applied load as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 Rn ¼ ðð305:5 þ 993:4Þ  sinð53∘ ÞÞ þ ðð496:6 þ 305:5Þ  cosð53∘ ÞÞ ¼ 1144:6 kN:

ϕRn ¼ 0:75  1144:6 ¼ 858:1 kNNRu ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3002 þ 4002 ¼ 500 kN O:K:

Also, using the conventional equation the design strength is calculated as follows: RnN ¼ 370  8:6  200 þ 0:6  240  8:6  2  80 ¼ 834:5 kN: RnV ¼ 370  8:6  80 þ 0:6  240  8:6  ð200 þ 86:5Þ ¼ 609:4 kN: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 1 Rn ¼ u u   2 ¼ 1012:6 kN: t sinð53∘ Þ 2 cosð53∘ Þ þ 834:5 609:4 ϕRn ¼ 0:75  1012:6 ¼ 759:5kNNRu ¼ 500 kN O:K: So, the strength prediction according to the conventional method is about 11% less than the capacity obtained by the proposed equation. 6. Conclusions The block shear limit state in welded gusset plates subjected to simultaneous axial and shear forces (called combined loading in this paper) was investigated using a finite element analysis model. The modeling and assumptions involved were validated through comparisons with similar tested specimens available in the literature. In order to capture the block shear failure path, ductile damage module in ABAQUS was utilized as a material degradation model in the numerical simulations. Subsequently, the verified finite element modeling was employed to conduct a parametric study. Applying the load at different angles, (θ), ranging from (0°) to (90°) in (15°) increments, the failure path and the ultimate strength of the models with various geometries and weld arrangements were obtained. From the analytical and numerical investigations detailed herein, the following conclusions can be drawn: • Block shear strength and the shape of a displaced block is influenced by the angle of loading. For (θ) equal to zero which corresponds to pure shear loading, block shear failure occurs in L-shaped form,

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while for pure tensile loading (θ = 90°), the failure occurs in U-shaped form. In both cases, the failure planes include one or two shear planes and one tension plane. For loading angles in between these cases (i.e., 0° b θ b 90°) which represent the case of combined loading, one failure plane has an inclined direction. In addition, the state of stress on failure planes is a combination of tensile and shear stresses, the contribution of each is proportional to the angle of loading (θ). • The presence of transverse weld in addition to the longitudinal welds does not significantly affect the block shear failure path or its ultimate strength under combined loading. This finding is similar to the findings obtained by Oosterhof and Driver [11] for the case of uniaxial loading. • The conventional equation suggested by the AISC Seismic Design Manual for block shear capacity of bolted connections subjected to combined loading was also examined for welded connections. This equation predicted the capacities on the average 34% lower than the FE method, which can yield uneconomical designs. Moreover, it cannot predict the true failure path in block shear rupture. • Based on the FE analyses results, an equation (Eq. (15)) was proposed to estimate the block shear strength of welded plates under combined loading. In this equation, attention to the real shape of the displaced block, true stress state on the failure planes, as well as stress triaxiality effects available in welded connections have been included. Comparing the results of the FE analyses with the calculated capacities by the proposed equation indicated that the estimated capacities are on the average about 5% on the conservative side. Using a resistance factor of ϕ = 0.75 in design along with this equation should ensure the safety needed for this limit state. References [1] G.L. Kulak, G.Y. Grondin, AISC LRFD rules for block shear in bolted connections- a review, Eng. J. 38 (2001) 199–203. [2] C. Topkaya, A finite element parametric study on block shear failure of steel tension members, J. Constr. Steel Res. 60 (2004) 1615–1635, https://doi.org/10.1016/j.jcsr. 2004.03.006.

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