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FINITE ELEMENT ANALYSIS OF TAPERED-HAUNCHED CONNECTIONS E. S. B. Machaly1, S. S. Safar2 and A.E. Ettaf3 ABSTRACT: Corner connections are crucial parts for the safety and adequate performance of steel frames. Extensive studies were made on the performance of square connections (without haunches) or those equipped with short triangular haunches. However, little was mentioned in literature or codes of practice about the behavior of tapered- haunched connections. This was in spite of their extensive use in medium-to-large span steel frames to develop more economic design and aesthetically pleasing views. In this paper, the behavior of tapered-haunched connections was investigated using the finite element method. The general-purpose finite element program, ANSYS was utilized in the analysis incorporating material nonlinearity and geometric imperfections to capture the interaction of yielding and buckling on the different failure modes of such connections. The magnitude of geometric imperfections was based on the allowable limit stipulated by the Egyptian Code of practice. Flow of stresses, yielding zones at failure, distribution of forces on stiffeners and carrying load capacity were determined analytically for a connection proportioned by the conventional design method. It was noticed that the location of the plastic zone as well as the connection capacity were mainly affected by web slenderness and stiffeners configuration. Moreover, it was concluded that the true forces in the stiffeners were overestimated by the conventional method. KEYWORDS: Tapered Connections, Haunched Connections, Beam-to-Column Corner Connections, Geometric Imperfections, Material Non-linearity, Finite Element Analysis. _____________________________________________________________________ 1. Professor of Steel Structures and Bridges, Faculty of Engineering, Cairo University. 2. Associate Professor of Steel Structures and Bridges, Faculty of Engineering, Cairo University. 3. Teaching Assistant, Structural Engineering Department, Faculty of Engineering, Cairo University. 1. INTRODUCTION: Tapered haunches are widely used in medium-to-large span frames in order to minimize the cost of steel in framed structures and to give a more pleasant view. Figure 1 shows the main dimensions and the geometric configuration of a typical tapered-haunched connection. + Fig. 1. Geometric configuration of a typical tapered haunched connection The analysis of tapered knees was carried out previously by several researchers using both elastic and plastic theories ([1], [2], [3] and [4]). One of the remarkable attempts in this field was by Fisher [1], who proposed a simplified analytical model for the tapered connection based on simple plastic theory. The location of the neutral axis was found to be slightly shifted towards the upper tension flange in case of identical haunch flanges thicknesses. This was attributed to the effect of the haunch slope. To account for the impact of axial force on the plastic moment of the connection, interaction curves were proposed in which the reduction in the plastic moment capacity was only required if the axial force value was greater than 15% of the force that causes section yielding. Moreover the effect of shear-moment interaction on the plastic capacity was reflected in adopting a reduced yield stress for longitudinal stresses based on Von-Mises yielding criterion [7]. The location of the critical section was found to be dependent on the slope of the haunch. If this slope was greater than a certain value the plastic hinge will be formed at the junction of beam and haunch and if it was smaller than that value, the plastic hinge will be formed at the corner. Such certain value was designated as the haunch critical angle at which the yielding will occur throughout the haunch length, and was estimated to be 12° for the range of haunch dimensions considered in the study [1]. The critical unsupported length for the haunch flange was found to be five times the haunch flange width unless special precautions were made to control strains. Strain control and enforcing strain hardening at certain points to limit spreading of yielding throughout the haunch length can be achieved by either increasing the haunch slope or increasing the haunch flange thickness. A stiffener was assumed at each point of abrupt change in flange direction. Stiffeners located at flange tips were used to resist the unbalanced vertical component in the sloping flange of the haunch assuming no contribution for the web. On the other hand, the diagonal stiffener was designed to resist the greater of the unbalanced flange force or the shear force in excess of the web panel zone capacity. The web contribution was also neglected in resisting the localized unbalanced component of flange forces. The main drawbacks of the conventional approach were identified in neglecting web contribution in resisting the unbalanced flange force components, prohibiting the use of un-stiffened or semi-stiffened connections and imposing the use of a flange brace at every point of abrupt change in direction which may be difficult in some circumstances. Moreover, there was no explicit design procedure or a certain recommended connection configuration provided by such conventional methods. In this work, a finite element model was established for a typical tapered-haunched connection proportioned by the conventional method [1]. Both material nonlinearity and geometric imperfections were included in the analysis to incorporate possible interaction of yielding and buckling at failure. The purpose of the analysis established herein was to assess numerically the general behavior of tapered knees and investigate the strength of un-stiffened and semi-stiffened connections. 2. FINITE ELEMENT MODELING: The complexities involved in the real behavior of knee connections impose the use of a sophisticated finite element model. Both material and geometric nonlinearities must be incorporated in the model to initiate the interaction between buckling and yielding. For this reason, the general purpose finite element program ANSYS [5] was used for its powerful capabilities in solving such complicated problems that include special features such as plasticity and large deformation effects. 2.1 Geometric Configuration: The tapered knee geometric configuration was selected as per Fisher [1] and Machaly [2]. Dimensioning of the tapered knee components was performed in accordance with the Egyptian Code of Practice for Steel Construction and Bridges [6] to support the straining actions induced on the beam-to-column connection of a portal frame with 30 m span and with 8 m height when subjected to gravity and wind loads. The beam section was selected as an I- shaped section with web plate 386x7 mm and flanges 280x14 mm, i.e. W(386x7/280x14) while the column section was designed as W(485x7/280x15). The haunch flange dimensions were taken similar to those of the beam flange, whereas the haunch web thickness was increased to 8 mm. The slope of the taper was assumed to be equal to 9°. 2.2 Modeling of Connection Plate Elements: Figure 2 illustrates the finite element model selected for the connection in which flange and web plates were modeled by an iso-parametric finite strain shell element in ANSYS element library designated as Shell 181[5]. Such an element was selected for its higher stability and better modeling for large deformation problems. Shell 181 is a four nodded shell element with six degrees of freedom at each node. It has both bending and membrane capabilities. It is suitable for analyzing thin to moderately thick shell structures. It is well suited for linear, large rotation and/or large strain non linear applications. The elements special features include: Plasticity, stress stiffening, large deflection and initial stress import. Fig. 2. Finite element model for tapered haunched connection 2.3 Boundary Conditions and Loading: The finite element model was extended beyond the haunch tips to a distance equals to twice the beam depth horizontally and equals to twice the column depth vertically in order to minimize end conditions effect on results. The far end of the column was restrained in the three spatial directions x, y and z. The connection upper flange was restrained in the out-of-plane direction at points corresponding to purlins locations. At the haunch tips, an out-of-plane restraint was provided whereas the corner re-entrant point was left un-braced in the out- of-plane direction. The model was loaded by reporting the bending moment, normal force and shear force at the location of the beam end from the portal frame analysis and converting such forces into equivalent nodal forces [9]. The computed nodal forces were scaled to produce the value of the beam theoretical plastic moment [1] at the haunch-to-beam junction. Due to the small values of normal and shear forces, the reduction of the beam plastic moment due to moment-shear-normal interaction was not considered during load application. 2.4 Material Model: The idealized stress-strain curve for mild steel based on elastic- perfectly plastic behavior was employed [8]. The value of the yield stress was taken as 2.4t/cm2 and the Young's modulus was chosen to be 2100 t/cm2. Isotropic hardening and Von Mises yield criterion were employed throughout the non- linear analysis. 3. FINITE ELEMENT RESULTS: In this section the finite element results are presented. Three connection configurations were studied based on stiffener configuration. At first, the case of un-stiffened connection was solved with no edge or diagonal stiffeners. Second, a semi-stiffened connection with diagonal stiffener only was considered and finally the case of a fully stiffened connection with both diagonal and edge stiffeners was solved. The analysis of each configuration was conducted to determine the elastic buckling load, plastic limit load neglecting geometric imperfections and the inelastic limit load considering material nonlinearity and geometric imperfections. 3.1 Case of Un-Stiffened Connection: 3.1.1 Elastic Buckling Analysis: Figure 3 illustrates the contour plot of out-of-plane displacements corresponding to the fundamental buckling mode of the un-stiffened connection. It was noticed that the connection lost its stability primarily due to web buckling at corner re-entrant under the effect of the concentrated unbalanced flange forces at that point. The buckling load did not exceeded 0.33 times the plastic moment of the adjacent beam section, Mp. Fig.3. Contour plot of out-of-plane displacements at the fundamental buckling mode of un-stiffened Connection, cm 3.1.2 Inelastic Analysis Neglecting Geometric Imperfections: In this section, the plastic limit load of the connection was computed by conducting a nonlinear static analysis for a geometrically perfect connection configuration. The Von Mises yield criterion with isotropic hardening was employed. The load was applied on the model incrementally and solution obtained at each load step iteratively using the full Newton Raphson technique. The analysis was terminated when the limit load was reached. The contour plot of the equivalent stresses at the plastic limit load is illustrated in Fig.4. Figure 4 shows that the yielding zone was concentrated near the corner re- entrant zone due to the localized effect of the concentrated unbalanced flange forces. The connection could not develop the beam plastic moment and the plastic limit load did not exceed 0.87 Mp at which excessive deflection of the beam took place and the connection lost its stiffness. Fig.4. Equivalent stress distribution at plastic limit load neglecting geometric imperfections, t/cm2 3.1.3 Inelastic Buckling Analysis Considering Geometric Imperfections: To initiate the interaction between yielding and buckling, geometric imperfection were introduced in the model. This was done by scaling the amplitude of the first buckling mode (Fig. 3) of the connection to the allowable limit in the Egyptian Code [6] such that the bowing in web would not exceed hw/150 and the lateral deformation in the flange would not exceed hw/75. The scaled buckled shape was used to modify the nodal coordinates of the perfect configuration. The analysis was conducted using the full Newton-Raphson technique as illustrated in Sec 3.1.2. The analysis was terminated when the limit load at which the connection loses its stability was reached. Figure 5 illustrates the equivalent stresses distribution computed at the limit load. It was noticeable that the yielding zone was still concentrated at the web panel zone. Nevertheless, it was extended towards the tension flange. The imperfections had also initiated some yielding in the column flange. However, it was the deterioration of the web panel zone capacity which limited the connection capacity. The failure was mainly attributed to inelastic buckling of the web at the web panel zone. The limit load did not exceed 0.28 Mp. Fig. 5. Equivalent stresses distribution at limit load considering geometric imperfections, t/cm2 Based on the above results, only the elastic buckling and the inelastic analysis will be considered for the remaining connection configurations to be studied herein since it was believed that the plastic limit load ignoring geometric imperfections is a bit theoretical and can hardly be achieved in practice. 3.2 Case of Semi-Stiffened Connection: The effect of adding diagonal stiffeners in the web panel zone was explored. An overall increase in strength was expected since diagonal stiffeners will support part of the unbalanced flange force component at the corner re-entrant and will strengthen the panel zone against shear buckling. 3.2.1 Elastic Buckling Analysis: Figure 6 shows the contour plot of the out-of-plane displacements of the first buckling mode. It was noticeable that although the corner re-entrant lacks an out of plane support (Sec. 2.3), the diagonal stiffener greatly reduced the out- of-plane deformation at that point. The first buckling mode was mainly composed of lateral displacement of the flange resembling the out-of-plane flexural buckling mode of a pinned-pinned strut. The buckling load was increased to 1.67 Mp. Fig. 6. Contour plot of out-of-plane displacements at the first buckling mode of a diagonally stiffened tapered knee, cm 3.2.2 Inelastic Buckling Analysis: The nonlinear static analysis of an imperfect diagonally stiffened tapered knee showed that at limit load the yielding zone was transferred from the web panel zone to the haunch tips (see Fig. 7). This was attributed to the additional strength provided by the diagonal stiffener at the web panel zone and also due to the effect of the concentrated unbalanced flange forces at the haunch tips. It is to be noted that the yield zone formed at the haunch tip with the beam was not symmetrically generated at the haunch tip with the column since the column cross sectional dimensions provide larger load carrying capacity (Sec. 2.1). The limit load of the connection was increased due to the addition of the diagonal stiffener from 0.28 Mp to 0.72 Mp. 3.3 Case of Fully Stiffened Connection: In this case additional stiffeners are introduced at the haunch tips to resist the unbalanced flange force components at such points (see Fig. 1). 3.3.1 Elastic Buckling Analysis: Figure 8 shows the contour plot of the out-plane displacement at the first buckling mode of the connection. The buckling mode was mainly composed of web local buckling without buckling in the flanges. The buckling load of the connection was increased to 3Mp. This result reflects the major influence of stiffeners in controlling the buckling behavior of the connection. 3.3.2 Inelastic Buckling Analysis: The analysis of the fully stiffened connection revealed that failure was primarily due to the flange yielding at haunch tips rather than web yielding that was prohibited by the effect of diagonal and edge stiffeners (see Fig 9). The limit load of the connection was increased to reach 0.98 Mp indicating that the connection adequate capacity was maintained by the addition of diagonal and edge stiffeners. Since failure was mainly attributed to yielding, the effect of geometric imperfections was insignificant and the plastic moment of the beam section was almost reached. Fig.7. Equivalent stresses distribution at limit load for a semi-stiffened knee, t/cm2 Fig. 8. Contour plot of out-of-plane displacements at the fundamental buckling mode of a fully stiffened connection A summary of the finite element results was illustrated in Table 1. For each connection configuration studied, the connection capacity together with the type and location of failure was listed for the purpose of comparison. It was concluded that the addition of stiffeners not only pronounced significantly the connection elastic buckling load, but also altered the buckled shape. The elastic buckling load of the fully stiffened connection was almost ten times as much as the un-stiffened connection. On the other hand, the addition of stiffeners had a less significant effect on the plastic limit load since failure took place by yielding. The addition of diagonal stiffeners increased the plastic limit load by 5 % whereas the addition of edges stiffeners increased the plastic limit load by 10% approximately. Fig.9. Contour plot of equivalent stresses at limit load for fully stiffened connection, t/cm2 The limit load of un-stiffened knee revealed that failure was mainly attributed to buckling of the web at corner re-entrant since it was slightly less than the elastic buckling load. On the other hand, the plastic limit load greatly exceeded the elastic buckling load. Therefore, it was noticeable that geometric imperfections had minor effect on such connections unlike semi-stiffened knees that were greatly affected by geometric imperfections (the limit load was reduced by 26% by the introduction of geometric imperfections) since failure at limit load was mainly attributed to inelastic buckling. On the other hand, fully stiffened connections were insignificantly affected by geometric imperfections since failure was caused by flange yielding at haunch tips. Table 1 Summary of Finite Element Results Un-stiffened Semi-stiffened Fully-Stiffened Connection Connection Connection Analysis Moment Type & Moment Type & Moment Type & Case Capacity location Capacity location Capacity location of Failure of Failure of Failure Web Buckling Buckling buckling Elastic 0.33 Mp of web at 1.67 Mp of haunch 3.0 Mp near Buckling corner flange diagonal stiffener Yielding Web Yielding of web at yielding of flanges Plastic 0.87 Mp corner 0.91 Mp at haunch 1.0 Mp at haunch tips tips Buckling Web Yielding of web at yielding of flanges Inelastic 0.28 Mp panel 0.67 Mp at haunch 0.98 Mp at haunch Buckling zone tips tips 4. COMPUTATION OF FORCES IN STIFFENERS: The simple analysis adopted by the conventional method [1, 2] for computing forces in stiffeners by considering equilibrium of flange forces is inaccurate as it ignores the interaction between the connections component based on their relative stiffness. On the other hand, the effect of stiffener thickness on connection moment capacity was not explicitly demonstrated. In this section, the effect of stiffener thickness on moment capacity in a fully stiffened connection was demonstrated. The stiffener thickness was varied from 0.5 to 2 times the thickness stipulated by the conventional approach. The variation in elastic buckling load and plastic limit load was recorded in each case and normalized with respect to their respective values when stiffener thickness computed by the conventional method was utilized. Figure 10 illustrates the results obtained. It is to be noted that the limit load was excluded in such comparison since the failure of fully stiffened connections was mainly caused by yielding rather than by inelastic buckling (Table 1). 1.4 1.2 1 Relative strength 0.8 0.6 Elastic Buckling 0.4 Elastic Plastic 0.2 0 0 0.5 1 1.5 2 2.5 Relative thickness Fig. 10 Effect of stiffeners thickness on the connection moment capacity Figure 10 shows that the elastic buckling load is greatly pronounced when the stiffener thickness increases from 0.5 to 1.0 times the conventional thickness since the buckling mode in such region included local buckling in stiffeners. When stiffener thickness exceeded the conventional thickness, the connection capacity was increased by 18% due to pronounced stiffener thickness whereas the stiffener local buckling modes were excluded. The increase in the plastic limit load was nearly negligible for stiffeners thickness greater than or equal to 0.7 the thickness suggested by the conventional method. This indicated that providing a smaller thickness than such limit will result in a fast deterioration of strength. On the other hand, an evaluation of forces developed in stiffeners showed that the conventional approach is generally conservative. Figures 11 and 12 illustrate the growth of vertical force component in edge and diagonal stiffeners, respectively, by loading. Vertical component of stiffener forces computed by the conventional method together with the vertical component of flange forces were also plotted in Figs. 11 & 12 for comparison. By investigating the results presented in Figs. 11 & 12, the following can be concluded: 1- Forces in stiffeners vary linearly with the applied moment. This was because the connection components in the vicinity of the diagonal stiffener or the web zone around the edge stiffeners remain elastic over a major part of loading. 2- Force developed in edge stiffener was nearly 65% of corresponding force deduced from the conventional method. Moreover, force in diagonal stiffener was only 50% of its conventional value. This was mainly attributed to two reasons : a) The conventional method assumes that the bending moment is solely supported by flanges. This overestimates the value of the flange forces. In fact a part of the bending moment (about 16%) is resisted by the web. b) The conventional method totally ignores the web contribution in resisting any unbalanced component of flange concentrated force although the web has some capacity [9]. 3- Vertical component of stiffener force does not coincide with the vertical component of flange forces. Edge stiffener supports a lower load compared to flange vertical component whereas the diagonal stiffener carried a higher load than expected from applying joint equilibrium using the finite element results. This was attributed to the positive contribution of the web in case of edge stiffeners. On the other hand, the web exerted additional force on diagonal stiffeners due to shear deformation of the web panel zone. VERTICAL FORCE IN FLANGE 10 Vertical FORCE IN STIFFENER 8 Conventional force Force(t) 6 4 2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Fraction of load Force growth in the edge stiffener Fig. 11. Growth of force in edge stiffeners by loading VERTICAL FORCE IN FLANGE 40 Vertical FORCE IN STIFFENER 35 Conventional force 30 25 Force 20 15 10 5 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Fraction of loading Force growth in diagonal stiffener Fig.12. Growth of vertical force component in diagonal stiffeners by loading 5. SUMMARY AND CONCLUSIONS: In this paper the finite element analysis of a typical tapered knee designed by the conventional approach was conducted to predict the connection behavior and evaluate the design assumptions adopted by the conventional design method. Based on stiffeners used, three connection configurations were studied designated herein as: un-stiffened, semi-stiffened and fully stiffened connections. The analysis of each connection configuration was conducted three times to compute the elastic buckling load, the plastic limit load and the limit load. Although the elastic buckling load may not govern the connection capacity, the elastic buckling was established to identify critical regions that are susceptible to fail by instability and to determine the fundamental buckling mode that is identified as the worst imperfection shape. The plastic analysis of the geometrically perfect connection was conducted to obtain the plastic limit load that is only governed by yielding and to identify critical zones that fail by yielding. The inelastic buckling analysis was established to determine the limit load that accounts for the interaction between buckling and yielding. Finally, forces that develop in stiffeners were evaluated and compared to conventional design method with emphasis on the effect of stiffener thickness on connection carrying capacity. The analysis conducted herein indicated that the connection failure mechanism depends mainly on stiffener configuration. Un-stiffened connection almost failed by elastic buckling of the web panel zone and achieved 30% of the beam plastic moment capacity approximately. Since failure of un-stiffened connections was governed by elastic buckling, the capacity of such connections was insensitive to geometric imperfections. In case of semi-stiffened connections, failure occurred by inelastic buckling of the web at haunch tips due to unbalanced flange forces. Effect of geometric imperfections was significant in this connection configuration that achieved almost 67% of the beam plastic moment. When stiffeners were used at all points of abrupt change in the flange direction, the connection failed by yielding of flanges at haunch tips provided that stiffeners width-to-thickness ratio was selected to prohibit stiffener local buckling. It was noticed that fully stiffened connections achieved the beam plastic moment and were insensitive to geometric imperfections. Evaluation of stiffener forces revealed that the conventional method generally overestimates the forces that develop in stiffeners by at least 50% due to ignoring the resistance of web to concentrated forces. Provided that local buckling requirements of the stiffeners were satisfied, the stiffener thickness stipulated by the conventional method could be reduced by 30% without reduction in connection capacity. Finally, the analysis conducted herein provided some light on the complex behavior of such connections. Due to lack of experimental research work on such connection type, a comprehensive analytical work is recommended to evaluate the effect of connection geometric configuration on behavior and to help develop the conventional design equations to have a better reflection on the real connection behavior. REFERENCES : 1. Fisher, J.W.,Lee, G.C., Yura, J.A. and Driscoll, J.r., "Plastic Analysis and Tests of Haunched Corner Connections" WRC Bulletin No. 91,NewYork, 1969. 2. Machaly, E.B., "Behavior, Analysis and Design of Steel Work Connections", 4th Edition, 2000. 3. Osgood, W.R., "A theory of flexure for beams with non parallel extreme fibres", Journal of Applied Mechanics, ASME, 1939. 4. Bleich, F., "Design of Rigid Framed Knees", American Institute of Steel Construction, Chicago, IL, 1943. 5. Desalvo, G.J., and Gorman, R.W., "ANSYS User's Manual", Swanson Analysis Systems, Houston, PA, 1989. 6. Egyptian Code of Practice for Steel Construction and Bridges (Allowable Stress Design), Code no. (205), 1st Edition 2001. 7. Bakhoum, M., “Structural Mechanics”, first Edition, 1992. 8. Salmon, C.G. and Johnson, J.E.," Steel Structures: Behavior and Design", Harper and Row Publishers, 4th Edition, 1996. 9. El-Banna, A.A., " Analysis and Design of Tapered and Curved Beam- to-Column Connections", M.Sc Thesis, Cairo University, 2005. جحليل سلوك الوصالث النسلوبت باسجخدام طريقت العناصر النحددة النلخص حوعب اهّصالج اهرنٌيث فٓ اإلطاراج اهيعدٌيث دّرا حيّيا فٓ اهححنى فىٓ درةىث اايىاً ّاادا اهيرضٓ هحوم اهيٌشآج . ّ هلد أةريج اهعديد يً ااةحاخ عوٓ اهّصالج اهرنٌيث اهيسحلييث أّ حوىم اهحٓ حزّد فيِا اهنيراج ةاةرةث يدوديث ، ةيٌيا هى يذنر اهندير عً اهّصالج اهيسوّةث عوٓ اهرغى يً اسحخدايِا اسحخدايا ّاسعا فٓ االطاراج ذاج اهةحّر اهّاسعث ّ اهيحّسطث االحساع ، هيا يٌحج عٌِىا يً ّفر فٓ نيياج اهحديد اهيسحخديث ّ هيا حعطيَ يً شنل ةياهٓ يةِر.ّ هِذا فولد حى اهحرنيز فىٓ ُذا اهةحخ عوٓ دراسث سوّم اهّصالج اهيسوّةث ، ّ ذهم ةاسحخداى ةرٌايج طريلث اهعٌاصر اهيحددت ّ ANSYSةحضييً اهالخطيث فٓ نل يً سوّم يادت اهحديد ّ اهشنل اهٌِدسٓ هوّصوث . ّهلد احفق عوٓ أً حؤخذ أكصٓ كييث الٌحراف أْ عٌصر يً عٌاصر اهّصوث ةاهلييث اهيسيّح ةِا فٓ اهنىّد اهيصرْ 1220.ّ ةاسحخداى ُذٍ اهفوسفث حى اةرا دراسث ييحدت هدراسث اهحّزيعاج اهيححيوث اعصاب اهحلّيث. ّهلد ّةد أً ييناٌينيث االٌِيار يعحيد ةصّرت أساسيث عوٓ نيفيث حّزيع أعصاب اهحلّيث .نيا ّةد أً اهلّْ اهحليليث فٓ ااعصاب حلل نديرا عً حوم اهيلدرت ةاهطرق اهحلويديث.

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