- Email: [email protected]
Finite element modelling of single lap screw connections in steel sheeting under static shear
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Thin-Walled Structures Vol. 27, No. 2, pp. 165-185, 1997 ~ 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0263-8231/97 $17.00 + .00 I (96)00037-7
Finite Element Modelling of Single Lap Screw Connections in Steel Sheeting under Static Shear
L. Fan, J. R o n d a l and S. Cescotto Department M.S.M., Universit6 de Li6ge, 6 Quai Banning, B-4000, Liege, Belgium
A BSTRA CT The behaviour of sheeting connections is important in thin walled structures, especially when diaphragm skin action is considered for these structures. Thousands of tests have been carried out in many countries but the stud), results are still not satisfactory. We felt that it would be very helpful to do more studies on this subject not only by experimental analysis but numerically, as well. Therefore, a finite element model is proposed, in this study, to simulate single lap screw connections in steel sheeting of different thicknesses, under static shear. This model can be used to predict the ultimate resistance, deJbrmation, screw rotation and stress distribution of the connections. The results show a good correlation with the test results. This model can therefore be used in further relevant parametric studies and with little change adapted to model the connections with other types or fasteners. © 1997 Elsevier Science Ltd. All rights reserved.
1 INTRODUCTION With the relatively recent increases in the power and capacity of computers and the rapid development in computer software, numerical simulation has become more and more popular in almost all research fields. In mechanical engineering, numerical analysis has become a very powerful tool. Together with the experimental analysis, it can provide a insight into such factors like stresses and strains, which is impossible or very expensive to obtain from tests. It can also supplement the lack of testing. If some numerical simulations have been calibrated against experimental tests, a large number of similar simulations can be done with different parameters instead of doing the corresponding tests. This is often more economical. 165
166
L. Fan, J. Rondal, S. Cescotto
The behaviour of connections in thin walled structures is an important problem. Especially, when diaphragm skin action is considered for the design, it becomes crucial. Thousands of tests have been carried out in Europe, North America, and many other countries. However, the criteria determining the design resistance in Europe or other countries are still not satisfactory. Generally, they give excessively conservative values but in some cases they have a tendency to give unsafe values. We felt that it would be very helpful to do additional studies on the mechanical properties of the connections, by means of not only experimental analysis but theoretical as well. Therefore, in this study, a finite element model, using the L A G A M I N E finite element computer program, ~ is proposed to simulate single lap screw connections in steel sheeting of different thicknesses, subjected to a static shear load. This model can be used for the prediction of the resistance, deformation, screw rotation and stress distribution of a connection up to the ultimate resistance. The results show that this method is in good agreement with the test results. This model can therefore be used in further relevant parametric studies and utilised to analyse connections with other types of fasteners. This simulation was directed at modelling the behaviour of a connection in which failure is caused by the yielding of the connected steel sheets, but not by the failure of the screw itself. The objective of the simulation was not only to assess the connection deflection (global deformation and screw rotation) but also to compute the stresses in the screw vicinity of connected steel sheets. Computer modelling of such a connection is rather complicated. Attention must be paid to: large non-linearities in almost all the involved materials, the contact problems between materials, and the possible local fracture of the connected steel sheets in the screw vicinity at loading levels approaching the ultimate resistance of the connection. Many people have done simulations of bolted connections under tension with relatively thicker connected materials (normally the thickness is larger than 10mm). For screw connections in sheeting with thicknesses ranging from 0-4 to 3.0ram under static shear, the simulation experiments have not been reported so far.
2 EXPERIMENTAL WORK The standard form for shear testing recommended by the E.C.C.S Committee 2 was adopted. To give an idea of the structure to be analysed, Fig. 1 shows a tested specimen of a 1.50/0.63 (mm) connection with one screw. Hundreds of tests with steel sheet thicknesses of 0.63 8.0 mm and a
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Fig. 1. A tested specimen of a 1.50/0.63 screw connection showing the structure to be analysed,
screw diameter of 6-3mm were carried out. They were conducted on a servo-regulated hydraulic machine with displacement control. The instant test data (load, deformation and conducting time) were recorded during the test with a data acquisition program. The loading speed is about 0.8 mm/min. The representation of such a test result is in the form of a load-displacement (i.e. load-global deformation) curve. Some of the test results are shown in Fig. 8.
3 FINITE E L E M E N T M O D E L L I N G 3.1 Software and hardware After a preliminary investigation it was decided that the L A G A M I N E finite element computer program ~ would be used to do the numerical simulation. The program L A G A M I N E has been developed by the M.S.M. Department at the University of Li+ge, Belgium, as a research program. The program is a non linear computer program with strong features in the non linear computations with large inelastic strains and contact problems. The program possesses a versatile element library and rich constitutive
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L. Fan, J. Rondal, S. Cescotto
laws. Its input files can be easily generated with pre-processor. As the output file of L A G A M I N E , the results of the final stage or any stages in the process of a simulation can be saved. With its post-processor the nodal coordinates, displacements, stresses, state variables, etc. can be saved in a screen readable and printable file, or in a graphic file. The computer used in the simulation is an Alpha AXP Workstation: DEC 3000 Model 400. The main relevant properties are as follows: (a) operation system: VMS V6.1; (b) C P U speed: clock speed--133MHz, SPECint92--74.7, SPECfp92--112-5; (c) RAM: 64 Mbytes.
3.2 Structural modelling Global structure The form of the structural assembly used in the experimental analysis was chosen in this numerical simulation, but with a slight reduction in the dimension of the connected sheet plane because, according to the experimental results, this reduction will not change anything in the behaviour of a connection. Since both the loading and geometry of the structure, except the screw thread, satisfy symmetric conditions, a cross-cut along a central line, Fig. 3(b), was actually chosen for the simulation in order to save computation time. As shown in Fig. 3, a 3-D model was chosen for the analysis. Of course, a 3-D model takes much more computation time than a 2-D model but to simulate a 3-D problem with a 2-D model, sufficient knowledge of the modelling with both a 3- and 2-D model must be obtained and compared in order to convert the 3- into a 2-D model. This was not attempted for this research program. Screw thread The main function of the screw thread is to prevent the screw from being moved along its axial direction, but not much from being rotated. To model this function and to limit the number of elements, only one thread was applied, Fig. 2. In dealing with the asymmetrical condition of the screw thread, very good care has been taken in modifying the thread form in order to obtain results closest to the experimental ones. Choosing continuous bodies The entire connection was defined as three continuous bodies: the first is the thinner sheet, the second is the thicker sheet and the third is composed
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of the screw shaft, the screw head, the aluminium alloy and the neoprene washer. In reality, the screw and washers do not form a continuous body. However, we consider that linking these bodies does not influence the behaviour o f a connection or very little, but can appreciably reduce computation time. Contact interfaces The contact interfaces include the ones between thinner sheet and thicker sheet, between thinner sheet and neoprene washer, between thinner sheet and screw shaft, between thicker sheet and screw shaft, and between thicker sheet and screw thread. To model the interaction o f these contacts, contact elements were used and consequently, appropriate friction, cohesion coefficients and so on, were chosen for these interfaces. Assumption on screw shaft and head A n o t h e r assumption is made: since in screw connections, as long as failure does not take place in the screw itself, the screw rigidity is always m u c h greater than those o f other materials. Our simulation is not aimed at modelling the failure m o d e of the screw itself. In the
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simulation, the screw shaft and head were regarded as a linear elastic material with, more or less, the same Young's modulus as that of the connected steel sheet.
Pre-stress established by tightening the screw After the assembly of a connection, by tightening the screw, pre-stress has been created between the interfaces of the thinner sheet and the thicker sheet, the thinner sheet to the neoprene washer, the thinner and thicker sheets to the screw thread. This step is finalised with the completion of the assembly being ended. In the simulation, the prestress was created as follows: (a) let the neoprene washer penetrate into the thinner sheet to a certain predetermined depth (the a m o u n t was based how much pre-stress we wanted to apply); (b) during the first step of computation, the penetrating part of the neoprene washer was pushed back because of the presence of contact elements and the neoprene washer was compressed. So the pre-stress was created during the end of the first step.
Boundary conditions In this model, Fig. 3(b) and (e), the ends of the two connected sheets were fixed in three directions, X, Y and Z, and, along the symmetry line. All the materials in the assembly were fixed in the Y direction in order to meet the conditions of symmetry. These boundary conditions implied that not only the global deformation along the X direction (loading direction), but also the screw rotation should be examined.
Loading In loading, the uniform displacement was applied, with a maximum increment for each step of approximately 0.004mm, to the eight nodes at the end of the thin sheet along the positive X direction, see Fig. 3(b) and
(e). The nature of the problem The problem at hand was a 3-D static one. From experimental investigations, we expected it to be with large material and contact non-linearities. Probably, the local cracks of the connected steel sheet in the screw vicinity would occur at a high level loading stage to reach the ultimate resistance of the connection.
Steel sheeting under static shear
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3.3 Choice of finite elements and layout of mesh
Choice of finite elements Elements for strained bodies The shell 3-D element has been widely used in sheeting modelling, because the element is flat and subjected to simultaneous bending stresses and membrane stresses. Prior to our numerical simulation, from experimental studies, it had become clear that the inelastic strain in the connected sheets in the screw vicinity could be high. The shell 3-D element in L A G A M I N E is inadequate for the computation of large membrane inelastic strain. Besides, in the area below the screw head, the sheet is submitted to transverse compression, which is not compatible with a shell hypothesis where the transverse stress is assumed to be zero. It was decided, therefore, that a three dimensional mixed solid element called JET3D, 3 with eight-nodes and one integration point and with three global degrees of freedom per node, would be used for the simulation. The use of the JET3D element rather than a standard element, was based on the consideration that some bending would occur in the elements, even when using a very small element size. Even very small element distortion can make a standard eight-nodes element lock, especially when the aspect ratio of the element is relatively large. The formulation of the JET3D element prevents volumetric and shear locking. This element was used for all materials (two connected steel sheets, screw, aluminium alloy and neoprene washer) in a connection.
Contact element In addition, to account for the interaction between materials, as mentioned above: sheet to sheet, sheet to screw shaft and so on, three dimensional contact elements with nine integration points were employed. These contact elements were developed by Charlier and Habraken, 4 to model contact with friction in three dimensions while taking into account large displacements and rotations between a strained body and a rigid one or between two strained bodies.
Layout of mesh After testing a few models, it becames clear that the size of the global stiffness matrix must be as small as possible, because the computation time for solving the equations is too long to be acceptable for subsequent parametric studies. The experience obtained from experimental analysis determines to a certain degree whether the mesh should be refined or not. In the meantime, the purpose of this simulation must be kept in mind.
Steel sheeting under static shear
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Connected sheets F r o m experimental results, we knew that the deformation of a tested specimen mainly originates from the local area o f connected steel sheets behind the screw and that this area is buckled at a high level of loading. The above two points were proved to be correct by the trial simulations: it was found that the strain gradient in the screw vicinity of the sheets is so large that the element sizes in the sheet plane in that area had to be, more or less, of the same magnitude as the thinner sheet thickness. It was also observed by visual inspection that the sheet was slightly sliding and bending up against the screw in the case of large screw rotation at ultimate loading level. Therefore, this area was intensively refined not only because of the large strain gradient but also the possible sheet buckling, in order to keep the individual elements not too warped to fit the sheet curvature, i.e. not to let the elements be subjected to dominant bending. As result, the same layout with one layer in the transverse direction was chosen for both thinner and thicker sheets. Figure 3(a) shows the layout of the thinner sheet. Viewing this mesh, we can see that some elements far away from the screw possess a large aspect ratio or a corner angle much larger than 90 °. These shape distortions do not affect the simulation results because the strain gradients in those areas are very small and even though these elements produce slight errors, they can not spread but die out locally. 5 Screw shaft and head Even though we were not interested in computing the stresses in screws, in order to obtain the screw displacement during loading so that the screw rotation could be modelled, the screws also had to be discretized to be involved in the model. As few elements as possible were arranged in these bodies with relatively good element shape, in order to reduce the size of the model. Although a larger element size produces a larger error (for the JET3D element, it made the screw either softer or harder than what it actually was), yet this did not affect the results for the area of interest and for the prediction of screw rotation. Because the screw was regarded as linear elastic material with, more or less, the same Young's modulus as the ones of the connected steel sheets, the deformation of the screw would be much less than those of other materials in the connection. Aluminium alloy and neoprene washers Our interest in washers was to see how the mechanical property of a connection is modified by the change of washer size, stiffness and initial pressure, but not to analyse the stress of the washers themselves. Aluminium alloy and neoprene washers were also discretized with large size elements, Fig. 3(c) and (d). O f course, this diminished the accuracy of the prediction of the effect of washers on a connection. But according to our
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L. Fan, J. Rondal, S. Cescotto
experience, in screw connections with neoprene washers, the washer has little effect on the mechanical properties. After adjusting the values used in their constitutive laws to make the deformation of the washer and the rotation of the screw correspond to that o f the tests, we believe our goal of investigating the influence o f the washer on connection behaviour can still be achieved.
Contact elements Contact elements were placed all over the area where the possible contact between materials could occur during loading. In order to more accurately account for the contact force, the finer element mesh in a contact of two surfaces was chosen as the contact element mesh. The number of contact interfaces in a connection were already cited above. Check of the model The results o f several trial meshes were compared prior to choosing the final mesh. They include the following comparisons: (1) a few models with one layer in connected sheets but different coarseness in the sheet plane were compared to choose the appropriate mesh, guided by the principle that the force-displacement curve should approach the one obtained from tests. The mesh is appropriate when further refinement barely influences the results and the stress contours or gradients in the sheets are almost smooth; (2) between a mesh with two layers in the screw vicinity and one layer elsewhere in the sheet and a mesh with one layer for the entire sheet. Of course, from the point of view of accuracy, two or three layers are better than one, but the economic aspect should also be considered. After comparison o f the above two models, the discrepancy o f the results was not important, in particular for obtaining the force-displacement curve of a connection. With the one layer scheme, the accuracy of the stresses was a bit less, especially of the stress distribution across the sheet thickness, but this did not interest us. So, finally, a one layer scheme was chosen. Even then, one computation still took about 6 h. The final mesh is shown in Fig. 3.
3.4 Material modelling In this simulation, the Von Mises yield criterion was chosen to control the plasticity of the materials, the mechanical properties o f which were described by the following constitutive laws.
Steel sheeting under static shear
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Constitutive laws for the materials The mechanical properties o f two steel sheets, aluminium alloy washer and neoprene washer were separately modelled based on four piecewise defined elasto-plastic constitutive laws and the one of screw shaft and head based on a linear elastic constitutive law. The mechanical properties were determined as follows.
Connected steel sheets The material properties obtained from tension coupon tests were used applying these constitutive laws. As an example (Fig. 4) line A B C D represents the true stress-strain curve of a 0.63 m m sheet. But from trial simulations, we found that for some connections with a large ratio of thick to thin sheet, traits, the Von Mises stress in the screw vicinity of the thinner sheet exceeded the ultimate tensile strength of the material. This means that the material mainly under tension stress is already cracked but that the one mainly under compression stress, for normal steel, continues to function. To treat this local cracks, the damage theory can be used. But, unfortunately, a damage law was not available for the J E T 3 D element at the time. So instead, a simple criterion was chosen and coded into the program: (a) the true stress-strain curve of steel sheet material was defined by two parts beyond the ultimate tensile strength (Fig. 4) one for the material under compression and the other for the one under tension; (b) in each iteration of the computation, the mean stress was examined and if it was larger than or equal to zero, the tension curve was used, otherwise, the compression curve was used. O f course, this is a rather crude way to model fracture. The obvious drawback is that whenever a negative mean stress changes its sign at an equivalent strain, the absolute value of which is greater than the one corresponding to the ultimate tensile strength of the sheet material, it becomes inaccurate, because the element stiffness can not be changed radically. Fortunately, after examining the simulation results, we found that this 'sign switch' rarely occurs, and even for the 'sign switching' elements, the Von Mises stress of the elements was just slightly greater than the ultimate tensile strength of the material at the switching point. This means that the influence of those 'sign switch' elements was minor. It was also observed that the sign was always from ' - ' to ~+' in our computation, so this could cause the connections with a higher ratio of traits, to possess a slightly higher resistance than they actually did.
Aluminium alloy washer The mechanical property of the aluminium alloy washer was unknown. It was first adopted f r o m 6 and later modified according to the conformity of the deflection of the aluminium alloy washer with the corresponding
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Neoprene washer A compression test was performed on the neoprene washer together with the aluminium alloy washer. For simplicity, the deformation of the aluminium alloy washer was ignored due to the large difference in stiffness between the two materials. Figure 6 shows the nominal stress--strain curve obtained from the test. For the constitutive law of the neoprene, the nominal stress-strain curve was used (instead of true stress-strain curve which is normally used for highly deformable material). The material of the neoprene washer which is compressed out of the cover of the aluminium alloy washer is considered as no longer functional (Fig. 7). Screw body The object of the exercise was to let the screw transfer the internal force and to predict screw rotation. With this purpose in mind, the screw shaft and head were regarded as an elastic material with the same Young's modulus as the one of the connected steel sheet so that no failure and large deformation would occur in the screw body. Constitutive laws for the contacts between materials A three dimensional contact law 4 was applied. The contact condition o f the law is enforced via a penalty method. To avoid penetration between contacting materials, the penalty coefficient for the contact pressure and for the shear frictional stress were chosen as high as possible, provided that the c o m p u t a t i o n can converge. For example, the penalty coefficients for the contact pressure and for the shear frictional stress were both chosen as 5 x l 0 4 N / m m for the contact of thinner sheet to thicker sheet, and as 3 x l 0 4 N / m m for the contact of thinner sheet to screw shaft. In using this law, we should also choose an appropriate contact friction and contact cohesion coefficient for each contact, which was guided by the following principles: (l) closest to the reality; or (2) if the concerned part of the structure was simplified in the simulation, the choice should be made in a way that the simplified structure with the chosen coefficient could better simulate the real structure than with the coefficient which was closer to reality. For example, in the real structure, there is no cohesion in any of the contacts. But according to the above point two, we have compared
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the results of the simulations with or without cohesion for the contact between screw shaft and thinner steel sheet and finally, an appropriate cohesion coefficient was used. Here the cohesion was used to model a part o f the screw thread function because we used only one thread. Five of these contact laws were used for the interfaces: • thin sheet to thick sheet with friction and cohesion coefficient equal to 0.13 and 0, respectively; • neoprene washer to thin sheet with friction and cohesion coefficient equal to 0.3 and 0, respectively; • screw shaft to thin sheet with friction coefficient equal to 0.25 (normally 0-25 is slightly higher for such a contact of steel on steel). Since we did not put a screw thread between this contact, taking 0.25 was to model the screw thread effect to prevent the thin sheet from prematurely sliding up along the screw shaft with the rotation of the screw) and cohesion coefficient equal to 5 N/mm2; • screw shaft to thick sheet. In reality, this contact is the same as the one between screw shaft and thin sheet. But because of the thread, the friction and cohesion coefficient were taken as 0-18 and 0, respectively; • screw thread to thick sheet with friction and cohesion coefficient equal to 0.18 and 0, respectively. In the above, the choice of the friction coefficients was referred to Ref. [7].
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4 COMPARISON OF THE RESULTS BETWEEN THE S I M U L A T I O N A N D T H E TESTS As examples, two results of simulation with sheet thicknesses of 0.63/3.00, 0-63/1.50, and 0-63/0.63 are separately presented here. Figure 8 shows the curves of force~lisplacement and screw rotation~lisplacement while Fig. 9 represents the non-smoothed distributions of Von Mises stress in the deformed 0.63/1.50 screw connection. F r o m Fig. 8, it is clear that the simulation can well reach the onset point o f the ultimate resistance of a connection. The results of force, screw rotation and displacement o f the simulation agreed well with the experimental results. F o r 0-63/0.63 screw connection, the stiffness of the simu-
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Steel sheeting under static shear
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lated connection more approaches to the greatest one of the tested connections and is slightly larger than the mean stiffness. This is normal, because under a static shear load the screw connections with two very thin connected sheets possess more rigid movement between the connected materials than other screw connections, but, unfortunately, the rigid movement could not be modelled in this numerical simulation method. Figure 9(b) shows that the stress distribution is rather smooth. This means that the selected mesh is fine enough to compute the stresses.
5 CONCLUSION The results o f the simulation show that the above proposed finite element model, using the non linear finite element computer program L A G A M I N E , provides a simulation which agrees well with the tests. It also gives good results in computing the stresses and strains in the connected sheets. Therefore, this model can be used for the prediction of the ultimate resistance, deformation, screw rotation and stress distribution of connections. It can further be applied to relevant parametric studies and with minor modification, to connections with other types of fasteners.
REFERENCES 1. Cescotto, S., Habraken, A.-M., Radu J.-P. and Charlier R., Some recent developments in computer simulations of metal forming processes. Proceedings of 9th International Conference on Computer Methods in Mechanics, Vol. 4. Krakow-Rytro, Poland, 1989, pp. 19-52. 2. E.C.C.S Committee TC7, Working Group TWG 7.2. The design and testing of connections in steel sheeting and sections. European Recommendations for Steel Construction, Publication No. 21, May 1983. 3. Jetteur, P. and Cescotto, S., A mixed finite element for the analysis of large inelastic strains. International Journal of Numerical Methods in Engineering, 1991, 31,229-239. 4. Charlier, R. and Habraken, A. M., Numerical modellisation of contact with friction phenomena by the finite element method. Journal of Computers and Geotechnics, 1990, 9, 59-72. 5. Cook, R. D., Malkus, D. S. and Plesha, M. E., Concepts and Applications of Finite Element Analysis. John Wiley and Sons, New York, 1989. 6. E.C.C.S. Committee T2, Aluminium Alloy Structures. European Recommendation for aluminium alloy structures. European Convention for Constructional Steelwork, 1978. 7, Miner, D. F. and Seastone, J. B., Handbook of Engineering Materials. John Wiley and Sons, New York, 1955.
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ELSEVIER
Pll:
S0263-823
Thin-Walled Structures Vol. 27, No. 2, pp. 165-185, 1997 ~ 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0263-8231/97 $17.00 + .00 I (96)00037-7
Finite Element Modelling of Single Lap Screw Connections in Steel Sheeting under Static Shear
L. Fan, J. R o n d a l and S. Cescotto Department M.S.M., Universit6 de Li6ge, 6 Quai Banning, B-4000, Liege, Belgium
A BSTRA CT The behaviour of sheeting connections is important in thin walled structures, especially when diaphragm skin action is considered for these structures. Thousands of tests have been carried out in many countries but the stud), results are still not satisfactory. We felt that it would be very helpful to do more studies on this subject not only by experimental analysis but numerically, as well. Therefore, a finite element model is proposed, in this study, to simulate single lap screw connections in steel sheeting of different thicknesses, under static shear. This model can be used to predict the ultimate resistance, deJbrmation, screw rotation and stress distribution of the connections. The results show a good correlation with the test results. This model can therefore be used in further relevant parametric studies and with little change adapted to model the connections with other types or fasteners. © 1997 Elsevier Science Ltd. All rights reserved.
1 INTRODUCTION With the relatively recent increases in the power and capacity of computers and the rapid development in computer software, numerical simulation has become more and more popular in almost all research fields. In mechanical engineering, numerical analysis has become a very powerful tool. Together with the experimental analysis, it can provide a insight into such factors like stresses and strains, which is impossible or very expensive to obtain from tests. It can also supplement the lack of testing. If some numerical simulations have been calibrated against experimental tests, a large number of similar simulations can be done with different parameters instead of doing the corresponding tests. This is often more economical. 165
166
L. Fan, J. Rondal, S. Cescotto
The behaviour of connections in thin walled structures is an important problem. Especially, when diaphragm skin action is considered for the design, it becomes crucial. Thousands of tests have been carried out in Europe, North America, and many other countries. However, the criteria determining the design resistance in Europe or other countries are still not satisfactory. Generally, they give excessively conservative values but in some cases they have a tendency to give unsafe values. We felt that it would be very helpful to do additional studies on the mechanical properties of the connections, by means of not only experimental analysis but theoretical as well. Therefore, in this study, a finite element model, using the L A G A M I N E finite element computer program, ~ is proposed to simulate single lap screw connections in steel sheeting of different thicknesses, subjected to a static shear load. This model can be used for the prediction of the resistance, deformation, screw rotation and stress distribution of a connection up to the ultimate resistance. The results show that this method is in good agreement with the test results. This model can therefore be used in further relevant parametric studies and utilised to analyse connections with other types of fasteners. This simulation was directed at modelling the behaviour of a connection in which failure is caused by the yielding of the connected steel sheets, but not by the failure of the screw itself. The objective of the simulation was not only to assess the connection deflection (global deformation and screw rotation) but also to compute the stresses in the screw vicinity of connected steel sheets. Computer modelling of such a connection is rather complicated. Attention must be paid to: large non-linearities in almost all the involved materials, the contact problems between materials, and the possible local fracture of the connected steel sheets in the screw vicinity at loading levels approaching the ultimate resistance of the connection. Many people have done simulations of bolted connections under tension with relatively thicker connected materials (normally the thickness is larger than 10mm). For screw connections in sheeting with thicknesses ranging from 0-4 to 3.0ram under static shear, the simulation experiments have not been reported so far.
2 EXPERIMENTAL WORK The standard form for shear testing recommended by the E.C.C.S Committee 2 was adopted. To give an idea of the structure to be analysed, Fig. 1 shows a tested specimen of a 1.50/0.63 (mm) connection with one screw. Hundreds of tests with steel sheet thicknesses of 0.63 8.0 mm and a
Steel sheeting under static shear
167
Fig. 1. A tested specimen of a 1.50/0.63 screw connection showing the structure to be analysed,
screw diameter of 6-3mm were carried out. They were conducted on a servo-regulated hydraulic machine with displacement control. The instant test data (load, deformation and conducting time) were recorded during the test with a data acquisition program. The loading speed is about 0.8 mm/min. The representation of such a test result is in the form of a load-displacement (i.e. load-global deformation) curve. Some of the test results are shown in Fig. 8.
3 FINITE E L E M E N T M O D E L L I N G 3.1 Software and hardware After a preliminary investigation it was decided that the L A G A M I N E finite element computer program ~ would be used to do the numerical simulation. The program L A G A M I N E has been developed by the M.S.M. Department at the University of Li+ge, Belgium, as a research program. The program is a non linear computer program with strong features in the non linear computations with large inelastic strains and contact problems. The program possesses a versatile element library and rich constitutive
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laws. Its input files can be easily generated with pre-processor. As the output file of L A G A M I N E , the results of the final stage or any stages in the process of a simulation can be saved. With its post-processor the nodal coordinates, displacements, stresses, state variables, etc. can be saved in a screen readable and printable file, or in a graphic file. The computer used in the simulation is an Alpha AXP Workstation: DEC 3000 Model 400. The main relevant properties are as follows: (a) operation system: VMS V6.1; (b) C P U speed: clock speed--133MHz, SPECint92--74.7, SPECfp92--112-5; (c) RAM: 64 Mbytes.
3.2 Structural modelling Global structure The form of the structural assembly used in the experimental analysis was chosen in this numerical simulation, but with a slight reduction in the dimension of the connected sheet plane because, according to the experimental results, this reduction will not change anything in the behaviour of a connection. Since both the loading and geometry of the structure, except the screw thread, satisfy symmetric conditions, a cross-cut along a central line, Fig. 3(b), was actually chosen for the simulation in order to save computation time. As shown in Fig. 3, a 3-D model was chosen for the analysis. Of course, a 3-D model takes much more computation time than a 2-D model but to simulate a 3-D problem with a 2-D model, sufficient knowledge of the modelling with both a 3- and 2-D model must be obtained and compared in order to convert the 3- into a 2-D model. This was not attempted for this research program. Screw thread The main function of the screw thread is to prevent the screw from being moved along its axial direction, but not much from being rotated. To model this function and to limit the number of elements, only one thread was applied, Fig. 2. In dealing with the asymmetrical condition of the screw thread, very good care has been taken in modifying the thread form in order to obtain results closest to the experimental ones. Choosing continuous bodies The entire connection was defined as three continuous bodies: the first is the thinner sheet, the second is the thicker sheet and the third is composed
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of the screw shaft, the screw head, the aluminium alloy and the neoprene washer. In reality, the screw and washers do not form a continuous body. However, we consider that linking these bodies does not influence the behaviour o f a connection or very little, but can appreciably reduce computation time. Contact interfaces The contact interfaces include the ones between thinner sheet and thicker sheet, between thinner sheet and neoprene washer, between thinner sheet and screw shaft, between thicker sheet and screw shaft, and between thicker sheet and screw thread. To model the interaction o f these contacts, contact elements were used and consequently, appropriate friction, cohesion coefficients and so on, were chosen for these interfaces. Assumption on screw shaft and head A n o t h e r assumption is made: since in screw connections, as long as failure does not take place in the screw itself, the screw rigidity is always m u c h greater than those o f other materials. Our simulation is not aimed at modelling the failure m o d e of the screw itself. In the
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simulation, the screw shaft and head were regarded as a linear elastic material with, more or less, the same Young's modulus as that of the connected steel sheet.
Pre-stress established by tightening the screw After the assembly of a connection, by tightening the screw, pre-stress has been created between the interfaces of the thinner sheet and the thicker sheet, the thinner sheet to the neoprene washer, the thinner and thicker sheets to the screw thread. This step is finalised with the completion of the assembly being ended. In the simulation, the prestress was created as follows: (a) let the neoprene washer penetrate into the thinner sheet to a certain predetermined depth (the a m o u n t was based how much pre-stress we wanted to apply); (b) during the first step of computation, the penetrating part of the neoprene washer was pushed back because of the presence of contact elements and the neoprene washer was compressed. So the pre-stress was created during the end of the first step.
Boundary conditions In this model, Fig. 3(b) and (e), the ends of the two connected sheets were fixed in three directions, X, Y and Z, and, along the symmetry line. All the materials in the assembly were fixed in the Y direction in order to meet the conditions of symmetry. These boundary conditions implied that not only the global deformation along the X direction (loading direction), but also the screw rotation should be examined.
Loading In loading, the uniform displacement was applied, with a maximum increment for each step of approximately 0.004mm, to the eight nodes at the end of the thin sheet along the positive X direction, see Fig. 3(b) and
(e). The nature of the problem The problem at hand was a 3-D static one. From experimental investigations, we expected it to be with large material and contact non-linearities. Probably, the local cracks of the connected steel sheet in the screw vicinity would occur at a high level loading stage to reach the ultimate resistance of the connection.
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3.3 Choice of finite elements and layout of mesh
Choice of finite elements Elements for strained bodies The shell 3-D element has been widely used in sheeting modelling, because the element is flat and subjected to simultaneous bending stresses and membrane stresses. Prior to our numerical simulation, from experimental studies, it had become clear that the inelastic strain in the connected sheets in the screw vicinity could be high. The shell 3-D element in L A G A M I N E is inadequate for the computation of large membrane inelastic strain. Besides, in the area below the screw head, the sheet is submitted to transverse compression, which is not compatible with a shell hypothesis where the transverse stress is assumed to be zero. It was decided, therefore, that a three dimensional mixed solid element called JET3D, 3 with eight-nodes and one integration point and with three global degrees of freedom per node, would be used for the simulation. The use of the JET3D element rather than a standard element, was based on the consideration that some bending would occur in the elements, even when using a very small element size. Even very small element distortion can make a standard eight-nodes element lock, especially when the aspect ratio of the element is relatively large. The formulation of the JET3D element prevents volumetric and shear locking. This element was used for all materials (two connected steel sheets, screw, aluminium alloy and neoprene washer) in a connection.
Contact element In addition, to account for the interaction between materials, as mentioned above: sheet to sheet, sheet to screw shaft and so on, three dimensional contact elements with nine integration points were employed. These contact elements were developed by Charlier and Habraken, 4 to model contact with friction in three dimensions while taking into account large displacements and rotations between a strained body and a rigid one or between two strained bodies.
Layout of mesh After testing a few models, it becames clear that the size of the global stiffness matrix must be as small as possible, because the computation time for solving the equations is too long to be acceptable for subsequent parametric studies. The experience obtained from experimental analysis determines to a certain degree whether the mesh should be refined or not. In the meantime, the purpose of this simulation must be kept in mind.
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Connected sheets F r o m experimental results, we knew that the deformation of a tested specimen mainly originates from the local area o f connected steel sheets behind the screw and that this area is buckled at a high level of loading. The above two points were proved to be correct by the trial simulations: it was found that the strain gradient in the screw vicinity of the sheets is so large that the element sizes in the sheet plane in that area had to be, more or less, of the same magnitude as the thinner sheet thickness. It was also observed by visual inspection that the sheet was slightly sliding and bending up against the screw in the case of large screw rotation at ultimate loading level. Therefore, this area was intensively refined not only because of the large strain gradient but also the possible sheet buckling, in order to keep the individual elements not too warped to fit the sheet curvature, i.e. not to let the elements be subjected to dominant bending. As result, the same layout with one layer in the transverse direction was chosen for both thinner and thicker sheets. Figure 3(a) shows the layout of the thinner sheet. Viewing this mesh, we can see that some elements far away from the screw possess a large aspect ratio or a corner angle much larger than 90 °. These shape distortions do not affect the simulation results because the strain gradients in those areas are very small and even though these elements produce slight errors, they can not spread but die out locally. 5 Screw shaft and head Even though we were not interested in computing the stresses in screws, in order to obtain the screw displacement during loading so that the screw rotation could be modelled, the screws also had to be discretized to be involved in the model. As few elements as possible were arranged in these bodies with relatively good element shape, in order to reduce the size of the model. Although a larger element size produces a larger error (for the JET3D element, it made the screw either softer or harder than what it actually was), yet this did not affect the results for the area of interest and for the prediction of screw rotation. Because the screw was regarded as linear elastic material with, more or less, the same Young's modulus as the ones of the connected steel sheets, the deformation of the screw would be much less than those of other materials in the connection. Aluminium alloy and neoprene washers Our interest in washers was to see how the mechanical property of a connection is modified by the change of washer size, stiffness and initial pressure, but not to analyse the stress of the washers themselves. Aluminium alloy and neoprene washers were also discretized with large size elements, Fig. 3(c) and (d). O f course, this diminished the accuracy of the prediction of the effect of washers on a connection. But according to our
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experience, in screw connections with neoprene washers, the washer has little effect on the mechanical properties. After adjusting the values used in their constitutive laws to make the deformation of the washer and the rotation of the screw correspond to that o f the tests, we believe our goal of investigating the influence o f the washer on connection behaviour can still be achieved.
Contact elements Contact elements were placed all over the area where the possible contact between materials could occur during loading. In order to more accurately account for the contact force, the finer element mesh in a contact of two surfaces was chosen as the contact element mesh. The number of contact interfaces in a connection were already cited above. Check of the model The results o f several trial meshes were compared prior to choosing the final mesh. They include the following comparisons: (1) a few models with one layer in connected sheets but different coarseness in the sheet plane were compared to choose the appropriate mesh, guided by the principle that the force-displacement curve should approach the one obtained from tests. The mesh is appropriate when further refinement barely influences the results and the stress contours or gradients in the sheets are almost smooth; (2) between a mesh with two layers in the screw vicinity and one layer elsewhere in the sheet and a mesh with one layer for the entire sheet. Of course, from the point of view of accuracy, two or three layers are better than one, but the economic aspect should also be considered. After comparison o f the above two models, the discrepancy o f the results was not important, in particular for obtaining the force-displacement curve of a connection. With the one layer scheme, the accuracy of the stresses was a bit less, especially of the stress distribution across the sheet thickness, but this did not interest us. So, finally, a one layer scheme was chosen. Even then, one computation still took about 6 h. The final mesh is shown in Fig. 3.
3.4 Material modelling In this simulation, the Von Mises yield criterion was chosen to control the plasticity of the materials, the mechanical properties o f which were described by the following constitutive laws.
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Constitutive laws for the materials The mechanical properties o f two steel sheets, aluminium alloy washer and neoprene washer were separately modelled based on four piecewise defined elasto-plastic constitutive laws and the one of screw shaft and head based on a linear elastic constitutive law. The mechanical properties were determined as follows.
Connected steel sheets The material properties obtained from tension coupon tests were used applying these constitutive laws. As an example (Fig. 4) line A B C D represents the true stress-strain curve of a 0.63 m m sheet. But from trial simulations, we found that for some connections with a large ratio of thick to thin sheet, traits, the Von Mises stress in the screw vicinity of the thinner sheet exceeded the ultimate tensile strength of the material. This means that the material mainly under tension stress is already cracked but that the one mainly under compression stress, for normal steel, continues to function. To treat this local cracks, the damage theory can be used. But, unfortunately, a damage law was not available for the J E T 3 D element at the time. So instead, a simple criterion was chosen and coded into the program: (a) the true stress-strain curve of steel sheet material was defined by two parts beyond the ultimate tensile strength (Fig. 4) one for the material under compression and the other for the one under tension; (b) in each iteration of the computation, the mean stress was examined and if it was larger than or equal to zero, the tension curve was used, otherwise, the compression curve was used. O f course, this is a rather crude way to model fracture. The obvious drawback is that whenever a negative mean stress changes its sign at an equivalent strain, the absolute value of which is greater than the one corresponding to the ultimate tensile strength of the sheet material, it becomes inaccurate, because the element stiffness can not be changed radically. Fortunately, after examining the simulation results, we found that this 'sign switch' rarely occurs, and even for the 'sign switching' elements, the Von Mises stress of the elements was just slightly greater than the ultimate tensile strength of the material at the switching point. This means that the influence of those 'sign switch' elements was minor. It was also observed that the sign was always from ' - ' to ~+' in our computation, so this could cause the connections with a higher ratio of traits, to possess a slightly higher resistance than they actually did.
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Neoprene washer A compression test was performed on the neoprene washer together with the aluminium alloy washer. For simplicity, the deformation of the aluminium alloy washer was ignored due to the large difference in stiffness between the two materials. Figure 6 shows the nominal stress--strain curve obtained from the test. For the constitutive law of the neoprene, the nominal stress-strain curve was used (instead of true stress-strain curve which is normally used for highly deformable material). The material of the neoprene washer which is compressed out of the cover of the aluminium alloy washer is considered as no longer functional (Fig. 7). Screw body The object of the exercise was to let the screw transfer the internal force and to predict screw rotation. With this purpose in mind, the screw shaft and head were regarded as an elastic material with the same Young's modulus as the one of the connected steel sheet so that no failure and large deformation would occur in the screw body. Constitutive laws for the contacts between materials A three dimensional contact law 4 was applied. The contact condition o f the law is enforced via a penalty method. To avoid penetration between contacting materials, the penalty coefficient for the contact pressure and for the shear frictional stress were chosen as high as possible, provided that the c o m p u t a t i o n can converge. For example, the penalty coefficients for the contact pressure and for the shear frictional stress were both chosen as 5 x l 0 4 N / m m for the contact of thinner sheet to thicker sheet, and as 3 x l 0 4 N / m m for the contact of thinner sheet to screw shaft. In using this law, we should also choose an appropriate contact friction and contact cohesion coefficient for each contact, which was guided by the following principles: (l) closest to the reality; or (2) if the concerned part of the structure was simplified in the simulation, the choice should be made in a way that the simplified structure with the chosen coefficient could better simulate the real structure than with the coefficient which was closer to reality. For example, in the real structure, there is no cohesion in any of the contacts. But according to the above point two, we have compared
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the results of the simulations with or without cohesion for the contact between screw shaft and thinner steel sheet and finally, an appropriate cohesion coefficient was used. Here the cohesion was used to model a part o f the screw thread function because we used only one thread. Five of these contact laws were used for the interfaces: • thin sheet to thick sheet with friction and cohesion coefficient equal to 0.13 and 0, respectively; • neoprene washer to thin sheet with friction and cohesion coefficient equal to 0.3 and 0, respectively; • screw shaft to thin sheet with friction coefficient equal to 0.25 (normally 0-25 is slightly higher for such a contact of steel on steel). Since we did not put a screw thread between this contact, taking 0.25 was to model the screw thread effect to prevent the thin sheet from prematurely sliding up along the screw shaft with the rotation of the screw) and cohesion coefficient equal to 5 N/mm2; • screw shaft to thick sheet. In reality, this contact is the same as the one between screw shaft and thin sheet. But because of the thread, the friction and cohesion coefficient were taken as 0-18 and 0, respectively; • screw thread to thick sheet with friction and cohesion coefficient equal to 0.18 and 0, respectively. In the above, the choice of the friction coefficients was referred to Ref. [7].
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4 COMPARISON OF THE RESULTS BETWEEN THE S I M U L A T I O N A N D T H E TESTS As examples, two results of simulation with sheet thicknesses of 0.63/3.00, 0-63/1.50, and 0-63/0.63 are separately presented here. Figure 8 shows the curves of force~lisplacement and screw rotation~lisplacement while Fig. 9 represents the non-smoothed distributions of Von Mises stress in the deformed 0.63/1.50 screw connection. F r o m Fig. 8, it is clear that the simulation can well reach the onset point o f the ultimate resistance of a connection. The results of force, screw rotation and displacement o f the simulation agreed well with the experimental results. F o r 0-63/0.63 screw connection, the stiffness of the simu-
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lated connection more approaches to the greatest one of the tested connections and is slightly larger than the mean stiffness. This is normal, because under a static shear load the screw connections with two very thin connected sheets possess more rigid movement between the connected materials than other screw connections, but, unfortunately, the rigid movement could not be modelled in this numerical simulation method. Figure 9(b) shows that the stress distribution is rather smooth. This means that the selected mesh is fine enough to compute the stresses.
5 CONCLUSION The results o f the simulation show that the above proposed finite element model, using the non linear finite element computer program L A G A M I N E , provides a simulation which agrees well with the tests. It also gives good results in computing the stresses and strains in the connected sheets. Therefore, this model can be used for the prediction of the ultimate resistance, deformation, screw rotation and stress distribution of connections. It can further be applied to relevant parametric studies and with minor modification, to connections with other types of fasteners.
REFERENCES 1. Cescotto, S., Habraken, A.-M., Radu J.-P. and Charlier R., Some recent developments in computer simulations of metal forming processes. Proceedings of 9th International Conference on Computer Methods in Mechanics, Vol. 4. Krakow-Rytro, Poland, 1989, pp. 19-52. 2. E.C.C.S Committee TC7, Working Group TWG 7.2. The design and testing of connections in steel sheeting and sections. European Recommendations for Steel Construction, Publication No. 21, May 1983. 3. Jetteur, P. and Cescotto, S., A mixed finite element for the analysis of large inelastic strains. International Journal of Numerical Methods in Engineering, 1991, 31,229-239. 4. Charlier, R. and Habraken, A. M., Numerical modellisation of contact with friction phenomena by the finite element method. Journal of Computers and Geotechnics, 1990, 9, 59-72. 5. Cook, R. D., Malkus, D. S. and Plesha, M. E., Concepts and Applications of Finite Element Analysis. John Wiley and Sons, New York, 1989. 6. E.C.C.S. Committee T2, Aluminium Alloy Structures. European Recommendation for aluminium alloy structures. European Convention for Constructional Steelwork, 1978. 7, Miner, D. F. and Seastone, J. B., Handbook of Engineering Materials. John Wiley and Sons, New York, 1955.