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 wereoverestimated by the conventional method.

KEYWORDS:

Tapered Connections, Haunched Connections, Beam-to-Column Corner Connections, Geometric Imperfections, Material Non-linearity, Finite Element Analysis.

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  1. Professor of Steel Structures and Bridges, Faculty of Engineering, CairoUniversity.
  2. Associate Professor of Steel Structures and Bridges, Faculty of Engineering, CairoUniversity.
  3. Teaching Assistant, Structural Engineering Department, Faculty of Engineering, CairoUniversity.

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.

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Fig. 1. Geometric configuration of a typical tapered haunched connection

The analysis of tapered knees was carried out previously by several researchersusing 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 greaterof theunbalanced 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 taperwas assumedto 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.

3.1Case of Un-StiffenedConnection:

3.1.1 Elastic Buckling Analysis:

Figure 3 illustrates the contour plot of out-of-plane displacements corresponding to the fundamental buckling mode ofthe 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 usingthe 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.

3.2Case 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-planedeformation 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 Mpto 0.72 Mp.

3.3Case 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 bucklingmode 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.