Effect and evaluation of prying action for top- and seat-angle connections

Nonlinear finite element (FE) static analyses of top- and seat-angle connections were performed using ABAQUS standard to investigate the influence of connection parameters and its properties on prying action developed due to the interaction between column flange and top angle’s vertical leg. Contact phenomenon between two interfaces with finite sliding and bolt pretension in the initial step of analysis was considered in the FE model. FE analysis results were compared with the experimental ones to examine the applicability of the FE model. Then, the location of plastic hinges in connection assemblages was investigated at the ultimate state of the connection and a parametric study was performed varying connection parameters, material properties of connection assemblages, and magnitude of bolt pretension to visualize their effects on prying force and on the position of prying force on top angle’s vertical leg. Current study shows that plastic hinges not only develop at top angle’s heel and bolt hole region but also in the bolt shank that differs from some assumptions of power model (Kishi and Chen 1990): (1) top angle thickness and gage distance from angle heel to bolt hole center line have an distinct effect on prying action; and (2) distributed prying force developed near the region of the top edge of tension angle’s leg adjacent to column flange can contribute to the failure of the connection. Finally, a mathematical formulation to identify the location of prying force action point is proposed.


Introduction
Nowadays semi-rigid connections are of considerable interest to researchers and design offices because they are easy to install in steel frames and have considerable energy dissipating capacity under seismic excitation. Recognizing the importance of these connections in practice, AISC (2001AISC ( , 2005 specifications recommend inclusion of this group of connections in the Partially Restrained (PR) construction category. However, detail design guidelines for this group of connections, especially for angle type of connections, are not developed yet.
In the AISC-ASD specifications, top-and seat-angle connections are treated as connections intended only for transferring beam shear force. However, experimental evidence and the present study on the connections show that besides transferring beam shear force, this type of connection transfers fairly considerable beam end moment to the column. When transferring this moment through the top angle to the column, an increase in tensile force is incurred in the bolts due to local deformation of the top angle's vertical leg. This additional tensile force is commonly known as the prying force (Q in Fig. 4). Recently, researchers have paid serious attention to this additional force in their mathematical representation of connection behavior. Initially, the T-stub modeling concept has been used for designing extended end plate connections and modified for use to design angle type of connections (Faella et al. 1996;CEN 1997). However, the T-stub model is not accurate enough to represent actual deformation pattern of true angle type of connections at failure, and may not correctly estimate the connection strength.
Accordingly, in the 80s and 90s, available moment-rotation models were modified and improved by incorporating more realistic representation of prying action Chen 1986, 1990;Faella et al. 1996;CEN 1997). Kishi and Chen's (1990) connection model (named as power model) considered bending and shear deformations of the angle, but disregarded deformation of bolts for deriving the ultimate moment capacity of angle type of connections. To establish a rational prediction model for precise representation of the M-h r curves of the connections, the effect of prying action on connection behavior should not be ignored. In this context, FE analyses were performed to investigate the contribution of prying action in momentrotation behavior (Ahmed 2002;Komuro et al. 2006). The places of plastic zones, effect of connection parameters on prying action and the magnitude and location of prying action are also examined for tracing a failure mechanism of the connection. Finally, based on the FE analysis and experimental results (Azizinamini et al. 1985;Harper 1990), an equation determining the location of prying force at the final load level is suggested for general application of estimating the ultimate moment capacity of the connection.

FE analysis of connections
A 3D FE mesh of the steel top-and seat-angle connection is constructed employing ABAQUS/CAE (2005) structural analysis modeling tool to execute a nonlinear static analysis. A complete list of connections with geometric details which are used for FE analysis to study the connection parameter and property effects on prying action is provided in Table 1. This list includes connections used by the first author in his previous study (Ahmed et al. 2001) and an additional connection designated as ''Test 3'' which was tested by Harper in the University of South Carolina (Harper 1990).

Mesh generation of FE model
In the FE analysis modeling, top and seat angles, bolts, beam and stub column of the tested connection are idealized by homogeneous steel material and modeled with eight-node solid (brick) elements, which are identified as C3D8R elements in ABAQUS. These elements with reduced integration are generated considering precise constitutive law of integration and are very suitable for nonlinear static analysis allowing finite strain and rotation in large-displacement analysis (ABAQUS 2005). Moderately fine mesh is used to construct the mesh of the connection model to obtain close representations to the experimental specimens. The mesh of the connection model strictly followed the geometries of the tested connection. The measurements of all connections including Test 3 are shown in Table 1. The connection parameters shown with symbols in Fig. 1 are enlisted in Table 1. The other connection parameters shown with numeric values in the same figure are kept unchanged in all connection models. Because of symmetry, one quarter of the connection is used for mesh generation of all FE model connections with half of span length (such as, 1.829 m for Test 3 is illustrated in Fig. 2). Bolts in the mesh are divided into shank, head and nut elements to consider their individual affections on connection behavior. The bolt hole diameter is supplied 1.6 mm (1/16 in.) larger than the diameter of the bolt in accordance with Azizinamini et al. (1985) and Harper's (1990) test data. The total number of elements and nodes defined in the FE mesh of model Test 3 are 8652 and 16,421, respectively. As an example, the mesh patterns

Contact surface modeling
To precisely analyze the behavior of connecting members, contact definition with finite sliding option is applied for the contact surfaces between the vertical legs of angles and column flange, between the horizontal legs of angles and corresponding beam flanges, and between the bolts and bolthole elements during the analysis. Moreover, to consider the frictional resistance between contact surfaces, the penalty frictional coefficient is assumed as 0.1.

Boundary condition
FE models are analyzed consistent with the experimental setup of Azizinamini et al. (1985) and Harper (1990), in which (1) two beams are symmetrically connected to the column flanges in a cruciform shape (Fig. 1); (2) the ends of these beams are simply supported; and (3) letting the center of top plate of stub column move upward so that the forces corresponding to the prescribed bending moment can be distributed among the connection assemblages. In addition, rollers are used to correct any movement of the stub column due to slippage and asymmetric movement. Based on such experimental setup and considering structural symmetry, one-quarter model of connection composed of stub column, beam, top and seat angles and bolts is used for FE analysis modeling. Figure 3 demonstrates the boundary conditions applied to FE model. To enforce connection symmetry, plane Z-X in the middle of the stub column is constrained in direction of Y and plane Y-Z in the middle of the beams and angles in length and stub column in width is constrained in direction X (Fig. 3). To produce only vertical reaction forces, the beam end support is assumed to be pin.

Loading method
The analysis was carried out with a monotonically incremental loading scheme for all connection models imitating experimental loading procedure. The connection models are analyzed in three loading steps. In the first step, a prescribed force, 40 % of minimum tensile strength of bolt, is applied to the pretension node of a predefined section of the bolt shank. As a result, the length of bolt shank at the pretension section changes by the amount necessary to carry the prescribed load. In the second step, the prescribed bolt load is replaced by changing the length of pretension section back to the initial length.  Table 2 is analyzed considering zero pretension force of bolts. In the third step, bending moment is introduced to the beam-to-column connection by employing vertical displacement of the middle section of plane Z-X of the stub column (Fig. 3). Automatic load increment scheme is preferred because ABAQUS can select the precise incremental rate to attain the optimum computational efficiency.

Results of analysis and discussion
Connection moment M is determined by multiplying reaction force at the pin support of the beam and minimum distance between the point of pin support of beam and the instantaneous center of rotation. The instantaneous center of rotation for the connection is located at the cross-point of the horizontal middle plane and the vertical section at the top of the heel of angle leg adjacent to the compression beam flange (Fig. 4). Relative rotation of the connection h r calculated from the results of FE analysis is estimated using the following equation: where d t and d b are the horizontal displacements at the upper and lower edges of beam flanges, respectively, and h is the beam section height. In the FE analysis, connection moment is considered as the ultimate moment capacity  Therefore, the FE analysis method is precise enough to investigate the magnitude of prying force and locations of the prying action and the yielding zones of the connection.
The stress-deformation results of the FE analyses show that yielding occurs initially at the top of the tension angle's filet (Fig. 6), and is then propagated in the area around the tension bolt head. Simultaneously, it is observed from FE analysis results that stresses higher than the yield point are generated in the bolt shank near the bolt head (Fig. 7), which indicates that the nonlinear behavior of the connection differs from some assumptions of the power model (Kishi and Chen 1990). With this observation, the place of formation of the upper plastic hinge in the top angle's vertical leg should be revised and an additional plastic hinge need to be added in the tension bolts to match with the results of FE analyses for the prediction model determining ultimate connection moment capacity. ABAQUS/standard is able to estimate the pressure and its resultant force of a contact surface. In this study, the distributed pressure of the contact surface on the back of the top angle's vertical leg and its resultant which represents the relative prying force of the connection (illustrated in Fig. 4, Detail A) are investigated to determine the magnitude and location of prying force at the ultimate state of connections. Figure 8 illustrates nodal force distributions due to the contact pressure caused on the column-facing surface (back) of top angle's vertical leg. The illustrations in Fig. 8 demonstrate the numerical results for three loading stages: (a) the initial step before the connection moment is introduced (Fig. 8a); (b) near the elastic limit (Fig. 8b); (c) in the middle of the plastic range (Fig. 8c). In addition, contact surface pressure distribution on the back of the top angle's vertical leg and its location of the resultant at the ultimate state of connection are demonstrated in Fig. 9. Figure 8a shows that contact pressure is almost uniformly developed on the back of the top angle's vertical leg under the bolt head area by the bolt tightening force and the contact nodal forces are moved to the upper area of the left-hand side of vertical leg as the connection moment increased up to the plastic range (Fig. 8b, c). The contact pressure distribution on the column surface facing top angle's vertical leg is accumulated at the upper area of the left-hand side of vertical leg. The resultant force due to the contact pressure is determined by summing up nodal forces developed on the contact surface that was defined as slave surface in contact definition between the two surfaces. This resultant is located in the contact area at a distance b from the centerline of bolt hole at the ultimate state of the connection (Fig. 9). It should be noted that this contact pressure resultant at the ultimate state of a connection should be considered as the prying force. Figure 10a shows the distribution of internal forces in tension bolt for model FE12 Po in which initial bolt pretension is ignored. The total tensile force in bolt, which is shown by solid curvilinear line in Fig. 10a, is the summation of shear force (developed in top angle's vertical leg at the top of filet) illustrated with solid straight line in Fig. 10a and prying force. It can be observed from the figure and Table 3 that bolt tensile force (V t ? Q) for model FE12 is 1.87 times higher than shear force (V t ) at the ultimate state. So, it is obvious that the prying force (Q), which is caused by the deformation of top angle's vertical leg, adds extra tensile force in bolt.

Distribution of forces in tension bolt (Ahmed 2002)
In Fig. 10b, solid straight, broken and solid curvilinear lines represent top angle's vertical leg shear (beam tension flange force), summation of shear force at top angle's vertical leg and contact force due to bolt pretension, and bolt tensile force, respectively. Thus, the vertical difference in between broken and solid curvilinear lines demonstrates the prying force corresponding to the shear force (Fig. 10b). It is apparent from Fig. 10 that prying force is also introduced in the tension bolt in application of pretension force in bolts of model FE12. However, it is difficult to estimate the prying force working on tension bolts because bolt tensile force consists of three components: shear force at top angle's vertical leg, contact force due to bolt pretension, and prying force. Shear force is determined as the transferred force from beam flange; whereas, the other two forces can be determined as contact force that is the summation of contact force due to bolt pretension and prying force. To get the individual forces from the contact force, it is assumed that the distribution of prying force corresponding to the shear force for the case considering bolt pretension is similar to that for the case ignoring pretension forces in bolts because distribution of prying is the same in both cases (considering and ignoring pretension forces in bolts) at the initial and ultimate level of loading (Figs. 10a, b, 11a). Even though, for the case considering pretension forces in bolts, the prying force may be little higher in the medium level of loading than in the case ignoring bolt pretension, it may not exceed 5 % of the total tensile force of bolt and may be ignored to avoid complexity of calculation for determining prying force of the connection considering pretension forces in bolts. It is also confirmed later in Fig. 11a that distribution of prying force in both cases considering with/without pretension forces in bolts is almost the same according to the results obtained from the post-analysis investigation of the influence of bolt pretension on prying action. Figure 10b shows the distribution of forces in tension bolt of model FE12, which considers pretension forces in bolts. It is seen that three components of bolt force can be properly estimated. From this figure, it is observed that the component of contact force caused by pretensioning of bolts is decreased nonlinearly with increasing shear force of connection due to the bending effects of top angle's vertical leg.

Influence of bolt pretension on prying force (Ahmed 2002)
Initial pretension forces applied on models A1 Po , FE5, A1m, and FE6 are 0, 20, 40, and 60 % of the ultimate strength of bolt, respectively, and all the other properties are kept similar. In Fig. 11, the vertical difference in between two similar lines such as short dash and short dash with solid circle defines the magnitude of prying force corresponding to the shear force (tension flange force) for each bolt. It can be observed in Fig. 11a that bolt tensile forces of corresponding models starting from different levels of pretensioning get closer as external load increases, and ultimately meet in the same point. Thus, prying force near the ultimate level of loading is not affected by the magnitude of pretension force. At the higher level of loading, contact force due to pretensioning of bolts is entirely neutralized, and only prying force is retained.

Bolt diameter
To investigate the effect of bending stiffness of bolt on prying force, comparison has been made between FE4 and A1m connection models in which bolt diameters (d b ) are taken to be 19.05 mm (3/4 in.) and 22.23 mm (7/8 in.), respectively, and pretension force is taken to be 40 % of the ultimate strength of bolt keeping the other parameters and material properties the same. It is observed from Fig. 11b that prying force develops very similarly in both connection models at the beginning of loading, but near the ultimate state, connection model with larger bolt diameter develops higher prying forces.

Angle thickness
Angle thicknesses (t t ) of connection models FE9, FE10 and FE11 are assigned to be 9.5 mm (3/8 in.), 12.7 mm (1/2 in.), and 19.1 mm (3/4 in.), respectively, and their other   properties are kept the same. The effect of angle thickness on bolt force is depicted in Fig. 11c. Bolt tensile force in model FE9 increases most rapidly among the three models. The analysis results also show that prying force develops faster for thinner angle, and angle thickness is one of the most influential parameters to determine the contribution of prying action to the ultimate moment capacity of the connection.

Gage distance from the angle heel to the centerline of bolt hole
The gage distances from the heel to the centerline of bolt hole on top angle's vertical leg g 0 t are taken to be 51 mm (2 in), 64 mm (2.5 in) and 114 mm (4.5 in) for connection models FE12, FE9, FE13, respectively, when the other properties of those connection models are the same. It is evident from Fig. 11d that prying force develops rapidly in connection with larger gage distance g 0 t which also shows significant effect on prying action in determining connection's ultimate moment capacity.
Influence of mechanical properties of connection assemblages on prying force (Ahmed 2002) Angle's material All the measurements and material properties between FEA models FE1 and FE7 are the same except angle's material properties. The yield stress f y,t and ultimate strength f u,t of angles are 365 and 550 MPa, respectively, for model FE1 and 250 and 400 MPa, respectively, for model FE7. It is observed from Fig. 11e that connection model FE7 with thinner angles (f y,t = 250 MPa and f u,t = 400 MPa) develops a little greater prying force than that for connection model FE1 with thicker angles (f y,t = 365 MPa and f u,t = 550 MPa) at the final stages of loading.

Bolt's material
All the respective measurements and properties of models A2 m and FE8 are the same except bolt mechanical properties. The yield stress f y,b and ultimate strength f u,b of bolts are assumed to be 635 and 830 MPa for model A2 and 830 and 1035 MPa for model FE8, respectively. It is observed in Fig. 10f that mechanical properties of bolt have no effect on prying action throughout the early and late middle loading ranges. But connection model FE8 with stiffer bolts (f y,b = 830 MPa and f u,b = 1035 MPa) exhibits a little increased prying force than the connection model A2m with weaker ones (f y,b = 635 MPa and f u,b = 830 MPa) at the ultimate state.

Influence of connection parameters on distance b
The location of the resultant of contact surface pressure is automatically generated by ABAQUS with respect to global axis. This resultant is known as prying force at the ultimate stage of loading. Later this location is determined from the centerline of bolt hole as shown in Table 3 by b. The table shows that the distance b mostly depends on the thickness of top angle t t and gage distance g 0 t from angle heel to the centerline of bolt hole. Other parameters, which influence this distance to some extent, are: bolt diameter, and material properties of angle and bolts (Table 3).

Prediction of distance b and assessment
It is observed that the distance b related to the location of prying force (i.e., resultant of contact pressure as shown in Fig. 9) action point from the centerline of bolt hole can be related to the angle thickness and can be expressed as The distance b of the prying force action point from the centerline of bolt hole can also be linked to the angle thickness in combination with gage distance from angle heel to the centerline of bolt hole and can be determined as where a is the distance from the centerline of bolt hole to the top edge of tension angle's leg adjacent to column flange (Fig. 4), t t is thickness of top angle and g 0 t is gage distance from angle heel to the centerline of bolt hole.
A brief summary of the FE analysis results and the proposed distance of the prying force location b are listed in Table 3. This table shows that prying force Q can be of as much as 2.4 times the shear force V t (i.e., beam flange force), and it must be considered in the calculation of the design resistance of the tension fasteners and of the ultimate connection moment. This table also shows that the performance of Eq. (3) over Eq. (2) estimating b distance demonstrates superior in eleven cases, equal in three cases and inferior in only two cases. In addition, the maximum and minimum discrepancies in b distance calculation by Eq. (3) are 11.4 and -5.3 %, respectively. Corresponding discrepancies obtained using Eq. (2) are 13.8 and -5.4 %, respectively. However, Eq.
(2) is simpler as it uses only one parameter: tension angle thickness. But Eq. (3) shows superior results because of inclusion of more connection parameters to predict the distance of prying force from the center line of bolt hole of top angle's vertical leg. However, overestimation of the value of b using Eq. (3) may be as high as 11.4 %. Overestimation of distance b (in other words leads to conservative design) by the proposed equations could slightly decrease the predicted strength of the connections.

Conclusions
Confirming about the applicability of FE analysis model, the locations of plastic hinges at the ultimate state and the effects of prying force developing on the back of the top angle's vertical leg is investigated at the different loading levels of top-and seat-angle connection. Following conclusions are reached from this investigation: 1. The plastic hinges develop at the top of the tension angle's filet and in the area around the tension bolt head of the vertical leg. Simultaneously, the other plastic hinge is developed in the bolt shank near the bolt head of the tension bolt, which indicates that the nonlinear behavior of the connection differs from some assumptions of Kishi and Chen's power model. 2. Prying force moves from the bolt head area to the upper area of the left-hand side of vertical leg as the connection moment increases from zero to the ultimate level. 3. Reduction of flange angle thickness or increment of gage distance from the angle heel to the centerline of bolt hole can increase the magnitude of prying force. 4. The location of prying force can be determined using the proposed expressions (Eqs. 2, 3). But Eq. (3) is recommended because of superior results. 5. Prying force can be of as much as 2.4 times the shear force at the vertical leg of top angle (i.e., beam flange force), and it must be considered in the calculation of the design resistance of the tension fasteners and the ultimate moment capacity of the connection.
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