Abstract
Thermal effects can be one of the most harmful conditions that any steel structure should expect throughout its service life. To counteract this effect, a new beam, with a capability to dissipate thermally induced axial force by slanting of endplate connection at both ends, is proposed. The beam was examined in terms of its elastic mechanical behavior under symmetric transverse load in presence of an elevated temperature by means of direct stiffness finite element model. The performance of such connection is defined under two resisting mechanisms; by friction force dissipation between faces of slant connection and by small upward crawling on slant plane. The presented numerical method is relatively easy and useful to evaluate the behavior of the proposed beam of various dimensions at different temperatures. Its applicability is evident through satisfactory results verification with those from experimental, analytical and commercially available finite element software. Based on the good agreement between theoretical and experimental methods, a series of design curves were developed as a safepractical range for the slant endplate connections which are depend on the conditions of the connection.
Keywords:
Slanted endplate connection; elevated temperature; symmetric transverse load; direct stiffness finite element model; analytical model
1 INTRODUCTION
The search for an economical resistance method to improve the performance of steel structures due to elevated temperature remains a challenging task that captures the interest of structural engineers in recent decades. Commonly employed existing engineering solutions to address temperature related concerns are in the forms of increasing section area, provision of lateral supports (Usmani et al., 2001Usmani A.S., Rotter J.M., Lamont S., Sanad A.M., Gillie M. (2001). Fundamental principles of structural behaviour under thermal effects. Fire Safety Journal 36(8): 721744.), cooling action by airconditioning and watering (Bailey et al., 1996Bailey C.G., Burgess I.W., Plank R.J. (1996). Analyses of the effects of cooling and fire spread on steelframed buildings. Fire Safety Journal 26(4): 273293.), and thermal break (Larson and Van Geem, 1987Larson S.C., Van Geem M.G. (1987). Structural thermal break systems for buildings: Feasibility study: Final report. Other Information: Portions of this document are illegible in microfiche products. Original copy available until stock is exhausted. p. Medium: X; Size: Pages: 102.). For better thermal resistance, structural members and connections behaviors at elevated temperature remain the principal subject of study of numerous papers exploring on various issues of structural matters. From 168 tests on compressed steelhinged bars with elastic thermal elongation restrain, (Rodrigues et al., 2000Rodrigues J.P.C., Cabrita Neves I., Valente J.C. (2000). Experimental research on the critical temperature of compressed steel elements with restrained thermal elongation. Fire Safety Journal 35(2): 7798.) concluded that thermal axial restraint is essential in the determination of fire resistance. This implies the importance of the consideration of connection or restraining component in structures under thermal loads. At elevated temperatures, the designs used for steel structures can be largely deviated from the normal thermalfree condition (Bradford, 2006Bradford M. (2006). Elastic Analysis of Straight Members at Elevated Temperatures. Advances in Structural Engineering 9(5): 611618.). To characterize the behaviors of steels such that a full utilization of their special feature can be executed, (Chen and Young, 2006Chen J., Young B. (2006). Stressstrain curves for stainless steel at elevated temperatures. Engineering Structures 28(2): 229239.) had proposed a stressstrain model that unifies the yield strength, elastic modulus, ultimate strength, and ultimate strain of steel at elevated temperatures for general use. (Mourão and E Silva, 2007Mourão H.D.R., E Silva V.P. (2007). On the behaviour of singlespan steel beams under uniform heating. Journal of the Brazilian Society of Mechanical Sciences and Engineering 29(1): 115122.) had pointed out that it is insufficient to rely only on the consideration of reduction in strength and modulus of steel structure in the thermal load environment analyses, even when the heat is uniformly distributed. They suggested that the behavior of the structures including the realistic boundary conditions at high temperatures is required under such condition, indicating the significance of design details for the elevated temperature loading resistance. (Takagi and Deierlin, 2007Takagi J., Deierlein G.G. (2007). Strength design criteria for steel members at elevated temperatures. Journal of Constructional Steel Research 63(8): 10361050.), making comparisons using nonlinear finite element simulations, evaluated design equations from the American Institute of Steel Construction (AISC, 2013AISC (2013). AISC Design Guide 27: Structural Stainless Steel. Institute of Steel Construction) and European Committee for Standardization (EC3, 2005EC3 (2005). Eurocode 3 (EC3), BS EN 199312: Design of Steel Structures, Part 12: General RulesStructural Fire Design.. British Standards Institution London, UK.) for steel members at elevated temperatures. The (EC3, 2005EC3 (2005). Eurocode 3 (EC3), BS EN 199312: Design of Steel Structures, Part 12: General RulesStructural Fire Design.. British Standards Institution London, UK.) provisions were exhibited to predict to a small disagreement of within 10%20% of the simulated finite element results. Conversely, (AISC, 2013AISC (2013). AISC Design Guide 27: Structural Stainless Steel. Institute of Steel Construction) specification predicts strengths that are up to twice as large, from which modifications had been suggested for improvement. (Yin and Wang, 2004Yin Y.Z., Wang Y.C. (2004). A numerical study of large deflection behaviour of restrained steel beams at elevated temperatures. Journal of Constructional Steel Research 60(7): 10291047.) presented the numerical modeling of the large deflection behavior of steel beams at elevated temperatures with different elastic axial and rotational restraints at the ends, focusing in particular the response in catenary action. They stressed that high temperature engineering design must take into account the effect of axial forces in the beam on the adjacent structure, including the connections, since the level of axial restraint is of the greatest importance due to its influence in the beam deflection and the catenary force under catenary action. Much effort was therefore devoted also to the investigation of the effect of connecting element on the global steel structural response when exposed to temperature change (AlJabri, 2011AlJabri K.S. (2011). Modelling and simulation of beamtocolumn joints at elevated temperature: A review. Journal of the Franklin Institute 348(7): 16951716.). (Dai et al., 2010Dai X.H., Wang Y.C., Bailey C.G. (2010). Numerical modelling of structural fire behaviour of restrained steel beamcolumn assemblies using typical joint types. Engineering Structures 32(8): 23372351.) witnessed various failure modes in the results of fire tests simulations of restrained steel beamcolumn assemblies adopting five different types of joints, signifying the importance of connection type details in thermal load resistance. (Simões da Silva et al., 2001Simões da Silva L., Santiago A., Vila Real P. (2001). A component model for the behaviour of steel joints at elevated temperatures. Journal of Constructional Steel Research 57(11): 11691195.) had extended the commonly practiced component method to the prediction of the response of steel joints under fire loading, focusing on a cruciform flush endplate beamtocolumn connection and comparing it to the experimental results obtained under various loading conditions. Realizing that restraint against thermal expansion of the beams can produce large axial forces, (Heidarpour and Bradford, 2009Heidarpour A., Bradford M.A. (2009). Elastic behaviour of panel zone in steel moment resisting frames at elevated temperatures. Journal of Constructional Steel Research 65(2): 489496.) presented an elastic analysis of a panel zone in a steel frame at elevated temperatures, joined in a rigid or semirigid fashion. Using a simple equilibrium premise, they developed an analytical model that takes into account the shear and flexural deformations of the panel zone. It was found that the main characteristics of the panel zone depend extensively on the thermal regime and the large compressive axial load induced in the connection, which is builtup from the restraint to thermal expansion at initial stages of the thermal loading. In much the same vein, numerical study of beamcolumn joints under thermal load using numerous types of connections had been carried out by (Yang and Tan, 2012Yang B., Tan K.H. (2012). Numerical analyses of steel beamcolumn joints subjected to catenary action. Journal of Constructional Steel Research 70: 111.), from which the static solver remains a more accurate tool compared to that of explicit in obtaining structural response although nonconvergence can be an issue when dealing with fracture simulation in the former. They commented that changes in the arrangement of bolts in flush end plate connections can improve the structural performance, and that current acceptance criteria of rotation capacities for steel joints are somewhat too conservative as they only consider pure flexural resistance. Accordingly, new connection acceptance criteria were proposed. In addition to available numerical works, (Qian et al., 2008Qian Z.H., Tan K.H., Burgess I.W. (2008). Behavior of steel beamtocolumn joints at elevated temperature: Experimental investigation. Journal of Structural Engineering 134(5): 713726.) conducted an experimental investigation of steel extended endplate beamtocolumn joints at elevated temperatures at 400, 550, and 700°C with different axial compression forces applied to the beams to simulate restraint effects. It was observed that axial restraint force plays a principal role on the steel joint moment capacity, and that there was a clear beam web tension field action near the column in thermally unrestrained cruciform tests. The fundamental development of an analytical model of such a connection when subjected to a nonsymmetric gravity load and a temperature increase has been formulated by (Zahmatkesh et al., 2014bZahmatkesh F., Osman M.H., Talebi E., Kueh A.B.H. (2014b). Analytical study of slant endplate connection subjected to elevated temperatures. Steel and Composite Structures 17(1): 4767.). It was demonstrated from the study that an increase in temperature in the beam with a general vertical bolted connection induces a huge extra axial force that decreases the capability in carrying external gravity loads. Also, this study showed that by changing the connection to that of slanting type, this additional axial force can be damped efficiently.
These literatures have clearly identified that the chief concern in steel structures under thermal load lies in the connection details and restraining method. It is worthwhile to note that most existing studies on steel beams have only considered the structural response in the use of conventional vertical endplate connection due to applied thermal axial forces. So far, there is little evidence that suggests a structural method to reduce the amount of axial force in restrained beams when subjected to elevated temperature. Also, there has been no controlled study comparing differences in the thermal responses of vertical (conventional) and slanted endplate connections for steel beams.
In the present work, we therefore explore further by means of direct stiffness method the thermal behavior of the slanted endplate connections in offering the structural solution to protect the beam and connections against primary failure due to elevated temperature. In the spirit of the finding by reference (Zahmatkesh et al., 2014aZahmatkesh F., Osman M.H., Talebi E. (2014a). Thermal Behaviour of Beams with Slant EndPlate Connection Subjected to Nonsymmetric Gravity Load. The Scientific World Journal 2014.), a 1D direct stiffness finite element method is presented in the current work to examine the behaviors of beams with a slant endplate connection in the presence of an elevated temperature when subjected to a symmetric transverse load in the elastic regime. In addition to complement the existing analytical solution, this numerical model offers potential to be extended to structures of continuous system with multiple degrees of freedom. In many fire accidents, axially restrained beams are partially or fully yielded before temperature goes to higher than 100° C (elastic) simply due to thermal expansion and axially exerted forces. The current research proposed the slant endplate connections instead of conventional connections to increase safety of structures for extended heating exposure time for fire fighting. On this basis, the findings from the current paper are useful and applicable for the early occurrence of fire incidents.
2 VERTICAL AND SLANT ENDPLATE CONNECTION CHARACTERISTICS OF BEAM SUBJECTED TO TEMPERATURE INCREASE
Figures 1 and 2 show the structural arrangements of vertical and test scale of slant endplate connection systems respectively. The popularity of bolted endplate connections is largely due to its simplicity in fabrication and installation. In general, these connections consist of endplates that are welded to the ends of a beam in the workshop and then bolted to the flange of columns on site. They are usually used in industrial structures, residential towers in terms of knee connections with endplate and beam connections in gable roofs. The maximum angle of slanting endplate connection (Figure 2) has been set to 60°, because a higher angle is not practical for crossing bolts into the holes at the top and bottom of endplate for fastening (Zahmatkesh et al., 2014cZahmatkesh F., Osman M.H., Talebi E., Kueh A.B.H. (2014c). Direct stiffness model of slant connection under thermal and nonsymmetric gravity load. Journal of Constructional Steel Research 102(0): 2443.).
The beam with restrained supports tends to have expansion when it is subjected to temperature increase. So, if the supports do not allow enough tolerance for motion, their reactions will apply huge axial forces in the beam. Most of the times, elongation of members in the elastic regime of material is small. Therefore, the introduction of the slant endplate connection can damp the existing axial force by allowing the ends of beam to crawl on the connection surface due to elongation of the beam. Hypotheses about the stages of the performance and the reaction of endplate connections due to increase in temperature are shown in Figures 3 and 4, for vertical and slant types, respectively. As shown in Figure 3, after an increase in temperature, the beam generally tends to buckle due to increase in axial load (P_{cr} ), because vertical endplate connection does not allow the beam to have expansion. On the other hand, as shown in Figure 4, the supports reactions apply axial forces after temperature increase to the beam through member expansion such that the slant surface allows the beam to damp axial force and expansion by means of linear crawling on the slant surface. Although there is a vertical motion tolerance between the surfaces in vertical endplate connections, it is unable to absorb the expansion of two ends of the beam in the horizontal direction because the direction of expansion is perpendicular to the direction of moving surface. In the slant endplate connection, there is a slanting tolerance between the surfaces such that it can absorb the expansion of two ends of the beam using crawling mechanism over the slanting faces, because the direction of horizontal expansion can be propelled to the slanting plane of connection.
3 DIRECT STIFFNESS FINITE ELEMENT FORMULATION
3.1 Case Study Description
In the current formulation, three case studies are simulated in two common categories; before thermal effect and after uniform elevated temperature. In the first case study, it is assumed that the two ends of the beam are supported with rollers on a fixed slanting plane that can carry the moment. The second case study is similar to the first case but its difference is there exists now a friction force between the two endplates. In the third case study, we consider the use of the friction bolts in the beam with slant endplate connection. The effects of various types of friction factors and slanting angles are to be investigated. The results from the direct stiffness formulation are first validated by an experimental test for selected cases. To expand study to all cases, results from the direct stiffness method and the modeling using the commercial finite element package, ABAQUS 6.9, are compared with those of analytical equations given by (Zahmatkesh et al., 2014bZahmatkesh F., Osman M.H., Talebi E., Kueh A.B.H. (2014b). Analytical study of slant endplate connection subjected to elevated temperatures. Steel and Composite Structures 17(1): 4767.).
3.2 The Connection with Roller Supports
For convenience, the joints of endplate connections in the current case are assumed to be rigid. In addition, the ends of the lower part of slanting line are prevented from further downward movement while the other ends are free as shown in Figure 5. The supports are rollers on slanted lines, so that the beam is in static equilibrium before thermal effect. In the static equilibrium, the uniform symmetric transverse load, W, causes the roller endplates to move downward, but the initial axial load in the beam, Pi, resists against downward sliding. Therefore, in this case study, if any compression axial force due to elevated temperature is applied, the endplates tend to move upward without any resistive force from support reaction such as friction. The boundary condition (B.C.) of roller supports can channel the total elongation of beam by free sliding upward over the slanting surface. As a result, the amount of axial force due to elevated temperature goes to zero and is damped completely. For model description, the beam member in Figure 5 is divided into two elements and three joints (nodes) such that the length of each element is equal to L with the same properties of section (crosssectional area, A, and second moment of area, I) and material (Young's modulus, E). For convenience of displacements and reaction forces computations, local axes definitions that follow the slant plane are adopted: X1Y1 at the left and X3Y3 at the right hand sides and X2Y2 local axes at node 2 that is coincided with the global axes. From the standard stiffness matrix of general frame element, K_{element} , and the transformation matrices, T1 and T2, the stiffness matrices of elements, K1 and K2, at local axes can be obtained from Equations 1 and 2. Note that local axes are used instead of global axes to obtain force and displacement at the slanting direction. The angle of beam element with the left support in the X1Y1 local axes is equal to θ_{1}, and it is equal to θ_{3} at the right support in the X3Y3 local axes. In the model, the angle of endplate with the vertical on the left hand side is considered similar to that of the right hand side (θ).
Where
Force diagram of beam with slant endplate connections due to symmetric transverse load and increase in temperature (frictionless support).
Standard stiffness matrix of general frame element:
So that (Equations 1) and (2) can now be written as:
Where
Hence, the total equilibrium equation for the beam takes the following form.
Where
are the displacement and the force vectors, respectively.
In the current case, the B.C. are:
Substituting B.C. in Equation 9 and solving, we get:
The amount of axial force (Pi) in the beam due to symmetric transverse load and elevated temperature when frictionless supports are used can be obtained from:
3.3 The Connection with Friction Support
3.3.1 Before Elevated Temperature
In this case study, it is assumed that there is a friction force between two faces of slant endplates at connection. So, an elastic spring stiffness, Kg, is used instead of friction reaction in supports at the local axes directions. As shown in Figure 6, the stiffness of this spring depends on the amount of friction factor and direction of sliding. Before any thermal effect, the beam is subjected only to symmetric transverse load. Therefore, the two endplates with rollers tend to slide downward but the friction force that is replaced by spring, Kg, resists against sliding on the slant plane. In calculating the stiffness of spring, the displacements of the endplate on slanting lines (X1 and X3) due to shortening of beam by initial axial load, Pi, are required.
Force diagram of beam with slant endplate connections due to symmetric transverse load (friction supports, before elevated temperature).
The total equilibrium relationship of such a beam can be expressed as:
From Equation 14, it can be found that the total stiffness matrix is similar to the first case study (frictionless support). But when the friction support is used, the reaction force of spring, K _{g} d _{xi}, should be considered. After the substitution of the reaction force of spring into the reaction vector, the displacement and initial axial force due to the symmetric transverse load before thermal effect can be obtained with the following B.C.
and the following corresponding expressions.
Elongation of beam in the X_{1} direction (frictionless case):
Free displacement in the X_{1} direction depending on friction:
Friction force in the slanting direction:
Stiffness of spring in the slanting direction:
From Equation 16, the free displacement due to contraction on the slant direction at roller support can be obtained. Equation 17 shows that when there is a friction force between endplates, the amount of such displacement will be decreased. Hence, we can use a spring to replace and cover the free downward displacement where the friction force tends to resist sliding. The stiffness of spring before thermal effect can be obtained using Equations 18 and 19.
3.3.2 After Elevated Temperature
After elevated temperature, the primary length of the beam increases but two ends of the member are axially semirestrained. Therefore, the reaction of support induces an axial force in the beam. This axial force depends on the elevated temperature and the boundary condition at the supports. The slanting direction of supports allows the endplates to crawl upward when the amount of initial axial load due to elevated temperature, Pt, is higher than the axial force only due to symmetric transverse load and friction force on the slant surface. Here, the upward resistance force (friction) is simulated by a spring stiffness, Kt (Figure 7).
Force diagram of beam with slant endplate connections due to symmetric transverse load and increase in temperature (friction support, after elevated temperature).
As shown in Figure 7, the direction of movement is upward, and the spring resists against this sliding. The total equilibrium relationship of such a beam can be expressed as:
With the following corresponding important parameters.
Elongation of beam in the X_{1} direction (frictionless case):
Free displacement in the X_{1} direction depending on friction:
Friction force in the slanting direction:
Stiffness of spring in the slanting direction:
From Equation 20, it can be observed that the total stiffness matrix is similar to the case of before elevated temperature, but now the reaction force of spring, K_{tdxi} , should be considered. Also, the amount of pure axial force due to elevated temperature should be added in terms of forces in the matrix equation. In calculating the stiffness of spring, Kt, the difference of the amount of shortening in the beam due to only transverse load, ΔL_{x1} , and the amount of elongation in the beam due to elevated temperature (Equation 21) should be obtained. From Equation 21, the free displacement of the endplate on slanting plane after elevated temperature, Δ_{dx1} (Equation 22), can be computed. The stiffness of spring after uniform elevated temperature can be determined from Equations 23 and 24. After the substitution of the reaction force of spring in the reaction vector with the consideration of boundary conditions, the displacement and initial axial force due to an increase in temperature can be accordingly evaluated.
3.4 The Connection with Friction Bolt Support
3.4.1 Before Elevated Temperature
In many general endplate connections, friction bolts are used instead of general bolts to increase the ability of connections due to sliding forces. By applying the tightening force, Pb, perpendicular to the slant endplate, the friction force of bolts can be increased. This means there exists an additional resisting force against downward sliding when the beam is subjected to only symmetric transverse load. The total equilibrium relationship of such a beam can be expressed as:
with the following expressions.
Elongation of beam in the X_{1} direction (frictionless case):
Free displacement in the X_{1} direction depending on friction:
Friction force in the slanting direction:
Stiffness of spring in the slanting direction:
From Equation 28, the decrease in initial axial force in the beam before any thermal effect due to the friction bolts can be determined. Here, an elastic spring, Kg, is used to simulate the free displacement of endplate on the slant plane due to the transverse load. As shown in Figure 8, initiating from equilibrium, (Equation 25), the stiffness of simulated spring, Kg, can be obtained from (Equation 29).
Force diagram of beam with slant endplate connections due to symmetric transverse load (with friction bolts, before elevated temperature).
3.4.2 After Elevated Temperature
When the friction bolts are used in the case of after elevated temperature, the two endplate connections tend to resist against upward movement compared with friction support with general bolts, because the amount of friction force is increased by perpendicular force that is resulted from the tightening of friction bolts. In the thermal case, the friction force changes its direction downward to set equilibrium. The total equilibrium relationship of such a beam can be expressed as Equations 3034. As shown in Figure 9, the friction and free displacement due to elevated temperature are simulated by spring, Kt, from which the reaction of this spring would tend to resist against upward sliding. The minimum required initial axial force, Pt, to initiate the sliding with the amount of stiffness simulated by the spring, Kt, in this case is resulted from Equation 34.
Force diagram of beam with slant endplate connections due to symmetric transverse load and increase in temperature (with friction bolts, after elevated temperature).
along with the following expressions.
Elongation of beam in the X_{1} direction (frictionless case):
Free displacement in the X_{1} direction depending on friction:
Friction force in the slanting direction:
Stiffness of spring in the slanting direction:
4 EXPERIMENTAL TESTS AND COMPARISON WITH DIRECT STIFFNESS FORMULATION RESULTS
The structural responses computed with the direct stiffness method are first validated with experimental results. Experimental data was taken from small scale steel beam and frame tests carried out at the Faculty of Civil Engineering of Universiti Teknologi Malaysia (UTM). The geometrical details for the considered structure are shown in Figure 10. H sections with a thickness of 1.5 mm were employed for beam and columns. The height of columns and beam's length of 235 mm and 450 mm were considered, respectively.
Geometrical details of the slant endplate connection in a beamtocolumn system for experimental study (all units are in mm).
Figure 11 shows the composition of components used in the experimental study. The structure consists of five separated parts; two columns, two short side beams and a middle beam with two inclined ends. These five parts were assembled by normal bolts and nuts. For the tests, the side beams and middle beam were detached as separate components from the main frame, resembling those of the analytical assumptions. The slant endplate connection can damp thermal axial force by motion on slant surface, so in the experimental model, the holes on slant endplate connection were constructed such that they are of oval shape to allow this deformation. According to direct stiffness equations, the friction factor, µs, between two faces of slant endplate connection joint is desired. So, a basic test was conducted to determine this as shown in Figure 12. In the friction test the selfweight of beam is measured 6.67 N with a connection slanting angle of 45° (θ = 45°).
Basic test for finding the static friction factor (µs) between two faces of slant endplate connection.
A 39.22 N attachment weight was used on the top of side beam in the vertical position. To measure the amount of sliding force, a load cell with 50 kA 5 kN capability was located at the side of the beam. An automated data logger (UCAM70A) was used to measure the corresponding axial force. The basic test measured a magnitude of 0.37 (µs = tan φ) for the static friction factor of the inclined surfaces. After the determination of the friction factor, two additional tests; before and after elevated temperature, were carried out for measuring the structural performance of the current connection system. The test setup can be seen in Figure 13 where the beam was tested with seven symmetric gravity load intensities (6.67 N, 26.29 N, 45.91 N, 65.53 N, 85.15 N, 104.77 and 124.39 N) including beam selfweight. Similar to the basic test, the axial force is recorded using the data logger with a load cell located at the end of the beam. For controlling the temperature under 100° (elastic zone) (EC3, 2005EC3 (2005). Eurocode 3 (EC3), BS EN 199312: Design of Steel Structures, Part 12: General RulesStructural Fire Design.. British Standards Institution London, UK.), ten thermocouples were attached along the beam with the heat source coming from a hot air blower placed at the middle of the beam. Note that (Equations 14) and (20) from the model with the friction support assumption are specifically employed to produce the numerical results, to be compared to experimental responses.
The relationships between applied load and the amount of axial force in the beam for the investigated two cases are shown in Figure 14. It is evident that the direct stiffness model conforms acceptably to the outcomes produced by experiments, both before and after temperature increase, and thus validating the applicability of the current numerical method.
The relationship between symmetric load and axial force in the beam (friction factor, μs = tan φ = 0.37, θ = 45°).
5 NUMERIC PARAMETER STUDY
After a good agreement with experimental outcomes, we demonstrate next using the model approach the evaluation of the minimum requirement of initial axial force for equilibrium that initiates the motion at the two ends of the beam for several parametric changes. Also, the corresponding minimum elevated temperatures for endplate crawling are computed. The structural model described in the previous section has been used to analyze a steel beam section at elevated temperature, in which an Isection (buildup) beam is connected at its ends to slant fixed endplate supports (Figure 15). For demonstration purpose we have set for the beam: cross section area (A) = 14000 mm^{2}, modulus of elasticity (E) = 200 kN/mm^{2}, ambient temperature (T_{0}) = 20°C, elevated temperature (ΔT) = 50°C, length of beam (L) = 6000 mm, coefficient of thermal expansion (α) = 1.5 × 105 °C^{1}, linear symmetric transverse load (W) = 10 kN/m, and the perpendicular axial force (tightening force) on the friction bolts (Pb) = 50 kN (total force). The slope of slant endplate connection (θ) and the friction coefficient factor (μs = tan φ) are varied.
In addition, a two dimensional numerical finite element model is created using the commercially available software, ABAQUS, for comparison. In the ABAQUS model, the slant support components are modeled using the continuum tetrahedral brick elements and the beam is constructed using the beam element. The contacts between endplates are simulated by the nodetosurface formulation. For this illustration, we have considered four angles (θ = 15˚, 30˚, 45˚, 60˚) using three different friction factors (μs= 0.2, 0.3, 0.5). The results of one sample study where θ = 30˚ and μs is 0.2 are shown in Figure 16 for 4 cases: Using general bolts and friction bolts for each case of before and after elevated temperature. The force diagram shows that the amount of axial force due to symmetric transverse load before any thermal effect (Pi, W) is equal to 33.61 kN, but when friction bolts are used (Pi, W + Pb), the amount of axial force decreases to 18.89 kN. It is demonstrated in Figure 16 that the friction bolts can decrease the induced axial force in the beam before elevated temperature due to just symmetric transverse load. After temperature change, the minimum requirement of axial force in the beam for initiation of crawling (Pi, W + ΔT) is equal to 86.54 kN, and when the friction bolt is used, the minimum requirement (Pi, W + Pb + ΔT) is 116.46 kN, which is higher. Also, from the results of other cases of after elevated temperature for various angles, it can be found that if the friction bolt is used instead of general bolts, a higher axial force is required to initiate crawling of the beam on the slant plane. The detail is shown next.
Schematics of ABAQUS model with arrows denoting the axial force for all considered cases for θ = 30˚ and μs = 0.2. Unit of forces is kN
In Figures 1722, results from three separate methods are given: ABAQUS software modeling, direct stiffness method and analytical method. Here, for verifying the results of two finite element methods (direct stiffness and ABAQUS software modeling), the analytical solutions for equations obtained by (Zahmatkesh et al., 2014bZahmatkesh F., Osman M.H., Talebi E., Kueh A.B.H. (2014b). Analytical study of slant endplate connection subjected to elevated temperatures. Steel and Composite Structures 17(1): 4767.). are used. It is obvious that the axial forces from analytical method coincide with those of direct stiffness. The results produced from ABAQUS match those analytical and numerical although some disagreements can be observed in Figures 18 and 21. Reduction in axial force, which is dependent on the use of friction bolts, can be seen when making comparison of results shown in Figures 17 and 20 in the case of temperaturefree condition. Conversely, axial force increases in presence of elevated temperature when comparing the beams using general bolts with those using friction bolts. In this case, the existence of elevated temperature increases considerably the axial force when the friction bolts are employed, in consistence with the response displayed in Figure 16. Making use of the axial forces that are resulted from Figures 1722 (after elevated temperature), the minimum elevated temperature required for upward sliding can be obtained.
Relation between axial force in the beam with connection slanting (θ) subjected to symmetric transverse load and elevated temperature (friction factor, μs = tan φ = 0.2, using normal bolts)
Relation between axial force in the beam with connection slanting (θ) subjected to symmetric transverse load and elevated temperature (friction factor, μs = tan φ = 0.3, using normal bolts)
Relation between axial force in the beam with connection slanting (θ) subjected to symmetric transverse load and elevated temperature (friction factor, μs = tan φ = 0.5, using normal bolts)
Relation between axial force in the beam with connection slanting (θ) subjected to symmetric transverse load and elevated temperature (friction factor, μs = tan φ = 0.2, using friction bolts)
Relation between axial force in the beam with connection slanting (θ) subjected to symmetric transverse load and elevated temperature (friction factor, μs = tan φ = 0.3, using friction bolts)
Relation between axial force in the beam with connection slanting (θ) subjected to symmetric transverse load and elevated temperature (friction factor, μs = tan φ = 0.5, using friction bolts)
Variation of movement initiation elevated temperatures, ΔTm, in the beam with slanting connection, θ, due to symmetric transverse load with normal and friction bolts, Pb, are shown in Figures 23 and 24, respectively. Again, three separate methods are used and compared. The curves of direct stiffness method match perfectly those of analytical method whereas the curves from ABAQUS model although diverge are almost parallel with them before a certain minimum inclination angle, after which the agreements are excellent. Such divergence in response may be attributed to the contact behavior between 1D middle beam elements and the sidebeam solid elements. Since it has been exhibited that analytical model, current numerical model and experimental results are closely matching, the divergence by the ABAQUS model can be related to nonconforming modeling setting that captures the structural response only after a certain minimum angle of slanted connection inclination. Between 0° to 60° of connection slanting, for direct stiffness and analytical approaches, if the angle of endplate, ϑ, is equal to the angle of friction factor, φ, ΔTm for the beam goes to infinity. If the slant connection's angle, ϑ, is equal to 90º, tan ϑ, goes to infinity. As a result, the movementelevated temperature, ΔTm, goes to zero. This means that if the endplate approaches the vertical position, the elevated temperature requirement for primary movement will approach infinity and if the endplate approaches the horizontal position, the elevated temperature requirement for primary movement approaches zero. Making comparison between the amounts of minimum elevated temperature for movement, ΔTm, it can be found that by increasing the friction factor from zero to 0.5, the amount of minimum elevated temperature for movement, ΔTm, will be increased. This means the friction force that resists against upward crawling at the end of beam needs higher elevated temperature to initiate movement. Also, the normal force of friction bolts can increase the friction force against upward movement of the beam, causing increase in both axial force and ΔTm.
Relation between movement initiation elevated temperature, ΔTm, and angle of slant connection subjected to symmetric transverse load and elevated temperature (in the case of normal bolts)
Relation between movement initiation elevated temperature, ΔTm, and angle of slant connection subjected to symmetric transverse load and elevated temperature (in the case of friction bolts)
6 CONCLUSION
This paper reports on the development of a linear one dimensional finite element model of the steel beam with a slant endplate connection by means of direct stiffness matrix method and ABAQUS software modeling when it is subjected to a uniform elevated temperature (under 100° C and elastic field of steel material under thermal conditions applicable to an early stage of fire incident) and a symmetric transverse load. It has been shown that the current formulation agrees excellently in terms of the elastic structural response with the conducted test results. From the comparison of results of the direct stiffness model with those of analytical, it is demonstrated that the numerical models are able to estimate practically identical amount of initial axial force and elevated temperature for initiation of movement of a steel beam when a slant endplate is used as connection. The parametric investigation shows that the thermal expansion of the restrained beam may cause higher axial force that can reduce the primary strength of the beam element. It is found that by changing connections' detail using the slant endplate connection replacing that of vertical, the induced axial force is damped by the actions of friction sliding and movement on the slope surface of endplate. Increase in the friction of endplates contributes to a rise in the axial force in the beam. The use of friction bolts reduces the axial force in the beam in contrast to that using general bolts in absence of elevated temperature. It occurs conversely when the beams are exposed to elevated temperature.
From the models, it has been shown that the minimum requirement of elevated temperature for crawling at the end of beam, ΔTm, increases when the amount of friction factor changes from zero to 0.5 (with the increase of the friction resistance), because the friction force resists against upward crawling of the ends of beam. In the case where the friction bolts are used in presence of elevated temperature, the friction bolts increase the friction force against upward movement of the beam, and thus increasing ΔTm. It is notable to see that by varying the symmetric transverse load, the friction factor, the inclination of the endplate, and the normal force of friction bolts using the present model, we can potentially obtain the optimum design that has enough ability to absorb the huge axial force induced due to temperature increase in the beam before any yielding and buckling can occur.
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Publication Dates

Publication in this collection
July 2016
History

Received
24 Apr 2015 
Reviewed
10 July 2015 
Accepted
11 July 2015