Equivalent post-buckling models for the exural behaviour of. steel connection.

Transcription

1 Computers and Structures 77 (000) 65±64 Equivalent post-buckling models for the exural behaviour of steel connections L. Sim~oes da Silva a, *, Ana Gir~ao Coelho b, Eliseu Lucena Neto c a Departamento de Engenharia Civil, Fac. de Ciencias e Tecnologia, Universidade de Coimbra, Polo II ± Pinhal de Marrocos, 3030 Coimbra Codex, Portugal b Instituto Superior de Engenharia de Coimbra, 3030, Coimbra, Portugal c ITA, , S. Jose dos Campos, SP, Brazil Received 9 May 999; accepted December 999 Abstract Analytical solutions for the evaluation of the behaviour of steel connections are presented which are able to reproduce their full non-linear behaviour. Because usual models for the analysis of steel connections consist of translational springs and rigid links whereby the springs exhibit a non-linear force±deformation response, usually taken as a bi-linear approximation, they require an incremental non-linear analysis. Using a substitute elastic post-buckling model where each bi-linear spring is replaced by two equivalent elastic springs in the context of a post-buckling stability analysis using an energy formulation, closed-form solutions are obtained for a connection loaded in bending. Application to a beam-to-column welded connection using the component (spring) characterisation of code regulations yields the same results in terms of moment resistance and initial sti ness, being additionally able to trace the full unsti ening response. Ó 000 Elsevier Science Ltd. All rights reserved. Keywords Steep; Joints; Bending; Component method; Nonlinear behaviour. Introduction Currently, the evaluation of the behaviour of connections relies on the independent evaluation of strength, sti ness and ductility properties in the context of the socalled component method []. Although the rst two, for a calibrated range of steel and composite connections, are covered by independent procedures that yield a moment resistance and an initial sti ness [], the latter still remains quite unexplored, despite some recent attempts at providing a more quantitative guidance [3]. In spite of these advances in connection behaviour over the traditional approach of pinned or fully rigid response, no analytical procedure able to predict the full * Corresponding author. Tel ; fax address luis_silva@gipac.pt (L. Sim~oes da Silva). non-linear moment±rotation curve of a joint, based on the load±deformation response of the contributing components is currently available. This paper presents analytical expressions for the evaluation of the response of steel connections based on equivalent post-buckling models.. Equivalent models.. Introduction The component method consists of idealising a connection as a mechanical model composed of translational springs and rigid links, whereby the springs (components) represent a speci c part of a joint that, depending on the type of loading, make an identi ed contribution to one or more of its structural properties, as illustrated in Fig. for a typical welded beamto-column steel connection /00/$ - see front matter Ó 000 Elsevier Science Ltd. All rights reserved. PII S (00)0005-

2 66 L. Sim~oes da Silva et al. / Computers and Structures 77 (000) 65±64 Fig.. Welded beam-to-column steel connection (a) connection geometry and (b) mechanical model. In the model of Fig. (b), spring k represents the behaviour of the column web in compression [4], which is characterized by a non-linear curve with an initial elastic response and a subsequent sti ness reduction with limited ductility, as shown in Fig. (a). Similarly, springs k and k 3 represent, respectively, the column web panel in shear [5] and in tension [6], which typically present the non-linear behaviour of Fig. (b), practically with unlimited ductility after an initial linear elastic response. It is thus clear that a bi-linear force±displacement approximation adequately represents the component behaviour, involving the identi cation of four properties for each component, namely initial elastic sti ness (k e ), limit load (F C ), post-limit sti ness (k p ) and limit displacement (D f ), schematically shown in Fig. 3, the yield displacement, D y, being de ned as (D y ˆ F C =k e ). The springs and rigid links model of Fig. (b), with the assumed bi-linear behaviour of each spring (component) requires an incremental non-linear analysis when loaded in bending. Here, analytical moment±rotation curves are obtained using equivalent elastic models, replacing each bi-linear spring with an equivalent elastic system which yields the same response, in the context of a post-buckling elastic analysis. This approach was successfully employed in the context of the well-known Shanley model for the analysis of compressed columns in the elasto-plastic range [7]... Non-linear model for equivalent elasto-plastic springs... Elasto-plastic spring in compression The basic building block of an equivalent model corresponds precisely to replacing each elasto-plastic (bi-linear) spring with an equivalent elastic system. Here, the two degree-of-freedom system of Fig. 4 is proposed, which consists of one elastic spring with sti ness k e (linear elastic sti ness of component) and a second elastic spring with sti ness k p (post-limit sti ness of component) and resistance F C ˆ P B = applied as a pre-compression, the degrees of freedom being de ned as follows Q ± total displacement, Q ± rotation of rigid links. Clearly, this model exhibits distinct behaviour in tension and compression, the latter being of most interest because of the inherent bifurcational behaviour. Assuming a positive compressive load, and using an energy approach, the following total potential energy function is obtained Fig.. Individual component behaviour (a) column web in compression and (b) column web panel in shear or in tension. Fig. 3. Typical force±displacement diagram for generic component.

3 L. Sim~oes da Silva et al. / Computers and Structures 77 (000) 65±64 67 Fig. 4. Equivalent elastic system for elasto-plastic spring in compression. V ˆ k e Q L cosq Š k P B p k p L cosq FQ Di erentiation with respect to the various degrees of freedom yields the equilibrium equations Fig. 5. Force±deformation solutions for an equivalent elastic system. < oq ˆ 0 < k e Q Lk e cosq F ˆ 0; () oq ˆ 0 senq 4L k e k p cosq LQ P B ˆ 0 giving Q ˆ Lk e cosq FŠ; 3 k e which can be replaced in Eq. () to give V Q ˆL k p cosq P B L cosq LF cosq P B F k p k e 4 Solution of Eq. () yields a trivial fundamental solution Q ˆ 0; 5 F ˆ k e Q ; and a post-buckling solution >< F ˆ Lk p cosq P B ; > F ˆ ke k p Q P B k e k p 6 Di erentiating Eq. (4) twice with respect to Q, evaluating along the fundamental path and equating to zero yields the critical load d V Q dq or E ˆ d V dq Q ˆ0 ˆ 4L k p cos0 cos 0 Š LP B F cos0 ˆ LP B F 7 F C ˆ P B The resulting force±deformation curves are shown in Fig. 5, clearly reproducing the bi-linear behaviour of the original elasto-plastic spring (component). Under tensile loading, similar reasoning yields simply the linear elastic solution, F ˆ k e Q ; the bifurcational response being absent Elasto-plastic spring in tension As seen in the previous section, the equivalent model of Fig. 4 is not able to reproduce a bi-linear response in tension. Such a situation requires the elastic system of Fig. 6, which, using a similar derivation and assuming F to be positive in tension, yields the same equilibrium solutions of Eqs. (5), (6) and (), and exhibits the required bi-linear response of the component in tension and a linear elastic behaviour when loaded in compression. Fig. 6. Equivalent elastic system for an elasto-plastic spring in tension.

4 6 L. Sim~oes da Silva et al. / Computers and Structures 77 (000) 65±64 V ˆ z k ec k et sen q L c cosq k pc k zl ck ec k et senq P B C L c cosq P C B Mq 3 k pc Di erentiation with respect to q and q gives the equilibrium equations of the system Fig. 7. Equivalent elastic model for a non-linear compression zone..3. Non-linear model for the shear and compression zones Having established a substitute model for one individual bi-linear spring, dealing with a steel connection requires introducing it in the model of Fig. (b). Starting, for simplicity, with an equivalent model where only one component in compression is assumed to reach the unsti ening load, the three degree-of-freedom model of Fig. 7 applies. Besides the total rotation of the joint (q ) and the rotation of the rigid links (q ) already explained, a third degree-of-freedom (q 4 ± axial displacement of the connection) is required, because of the shift in neutral axis caused by asymmetries in spring sti nesses between the tension and compression zones. Combining, for simplicity, the linear elastic sti ness of components and into one single equivalent elastic spring ˆ ; 0 k ec k e k e the total potential energy function for this system can be obtained V ˆ k et q 4 z senq h k ec q 4 z senq i L c cosq k PC B pc k pc oq ˆ z k ec k et senq cosq oq zl ck ec k et cosq cosq M ˆ 0; k ec k et ˆ 4L k pc k ec k et c senq cosq zl ck ec k et senq P B C k ec k L c senq ˆ 0; et which correspond to a fundamental solution M ˆ z k < k ec k et sen q ; q ˆ 0 and a coupled solution M ˆ zkecket k ec k et zsenq zkecket sen q P B C >< kec ket k pc k ec k et k Š cosq ˆ zkecket sen q P B C > kec ket 4L c k pc k ec k et k Š.4. Non-linear model for the tension zone cosq Likewise, should the tensile component present the lowest critical value for the limit load, the equivalent L c cosq Mq Eliminating q 4 as a passive coordinate through di erentiation of Eq. () with respect to q 4 and equating to zero and substituting back into Eq. () yields, successively, ˆ 0 () q 4 oq 4 h ˆ z k ec k et senq i ; L c k ec cosq Fig.. Equivalent elastic model for a non-linear tensile zone.

5 L. Sim~oes da Silva et al. / Computers and Structures 77 (000) 65±64 69 ˆ z k ec k et senq cosq zl ck ec k et cosq cosq oq model of Fig. applies, the same degrees of freedom being valid..5. General non-linear model of connection Finally, a more general model can be proposed which caters for the bi-linear component behaviour both in the tensile and compressive zones. From the model of Fig. 9, using the same procedure as above yields in succession V ˆ h k ec q 4 h k et q 4 k pc k pt z i senq L c cosq z i senq L t cosq 3 PC B k pc PT B k pt ˆ 0 () q 4 oq 4 h ˆ V ˆ z Fig. 9. General equivalent elastic model. L c cosq L t cosq 3 M ; 7 q z k ec k et senq L c k ec cosq L t k et cosq 3 k ec k et sen q L c cosq L t cosq 3 cosq k pc k pt k k zl ck ec k et senq zl tk ec k et senq P B C L c P B T L t i ; k ec k et cosq 3 4L c L t cosq cosq 3 P C B P T B M q ; 9 k pt k pc zl tk ec k et cosq cosq 3 M ˆ 0; ˆ 4L c k pc oq ˆ 0; oq 3 ˆ 4L t k senq cosq zl ck ec k et senq P B C L c senq k ec k et 4L c L t senq cosq 3 k pt k senq 3 cosq 3 zl tk ec k et senq P B T k ec k L t senq 3 et ˆ 0; k ec k et 4L c L t cosq senq 3 and the following solutions (i) Fundamental solution M ˆ z k sen q k ec k et ; >< q ˆ 0; > q 3 ˆ 0 (ii) Non-linear solution in q M ˆ zkecket k ec k et zsenq zkecket sen q P B C kec ket k pc k ec k et k Š >< cosq ˆ zkecket sen q P B C kec ket ; 4L c k pc k ec k et k Š > q 3 ˆ 0 (iii) Non-linear solution in q 3 M ˆ zkecket k ec k et zsenq zkecket sen q P B T kec ket k pt k ec k et k Š >< q ˆ 0; > cosq 3 ˆ zkecket sen q P B T kec ket 4L t k pt k ec k et k Š cosq ; cosq ; 0 3

6 60 L. Sim~oes da Silva et al. / Computers and Structures 77 (000) 65±64 (iv) Non-linear solution in q and q 3 M ˆ zkecket k ec k et zsenq L c cosq L t cosq 3 Šcosq ; >< cosq ˆ zkecket sen q P B C kec ket 4Ltkecket cos q 3 ; 4L c k pc k ec k et k Š cosq 3 ˆ zkecketkpc sen q k P B T P B C P > B T kec ket kpc 4L t k k pc k pt k ec k et k pck pt Š 4 3. Application to beam-to-column welded connections 3.. Component characterisation In order to illustrate the application of the equivalent elastic models, one connection con guration was chosen from the database SERICON II (Klein 05.00) [], corresponding to a welded beam-to-column steel connection, described in Fig., which was tested by Klein at the University of Innsbruck in 95. As described above, the rst step consists in establishing the components properties, initial sti ness k i, resistance Fi C, post-limit sti ness k pi and maximum displacement D f i. From Fig., three contributing components are identi ed, namely () column web in shear, () column web in compression, (3) column web in tension. Using the speci cations of the revised Annex J of Eurocode 3 [], the results shown in Table were obtained for the initial sti ness of each component, where k, k and k 3 denote the initial sti ness of components, and 3, respectively, non-dimensionalised with respect to YoungÕs modulus. It is noted that the remaining quantities are described in Ref. []. From Eq. (0), k ec ˆ E ˆ kn=m and k k k et ˆ k 3 E ˆ kn=m For the resistance (limit load) of each component, again following the revised Annex J [], corresponding results are shown in Table. It is noted that no ductility limits were imposed on each component because of lack of data. Also, the postlimit sti ness is currently not covered by code regulations. 3.. Numerical model To con rm the results from the analytical model, a numerical model was implemented using the non-linear nite element system LUSAS [9], shown in Fig. 0 and initially analysed using linear elastic properties for the springs (corresponding to initial sti ness) and subsequently analysed with bi-linear properties for the springs. The de nition of the nite element model is summarised in Table Results Having established in Table that the critical component was the column web in shear, an initial comparison was performed using only a bi-linear approximation for this component, the remaining being kept linear elastic. Table 4 and Fig. show the corresponding results, values for the post-limit sti ness k pc being chosen as zero to match the EC3 prediction. Next, a second comparison was made with all components as bi-linear springs, shown in Fig.. It is clear that the numerical and analytical results are similar, and in agreement with the initial sti ness predictions of EC3. It is interesting to note in Fig., the various equilibrium paths corresponding to the various components reaching their resistance. It should be noted Table Initial sti ness of each component Column web in shear (J.4. ()) Column web in compression (J.4. ()) Column web in tension (J.4. (3)) k ˆ 03A vc =bz k ˆ 07b eff;c;wc t wc =d c k 3 ˆ 07b eff;t;wc t wc =d c A vc ˆ 30 mm b eff;c;wc ˆ 49.0 mm b eff;t;wc ˆ 49.0 mm z ˆ h b t f;b ˆ 0 mm d c ˆ 9 mm d c ˆ 9 mm b b Table J4! b k ˆ 03 30= 0 ˆ 35 mm k ˆ =9 ˆ 7936 mm k 3 ˆ =9 ˆ 7936 mm

7 L. Sim~oes da Silva et al. / Computers and Structures 77 (000) 65±64 6 Table Resistance of each component Column web in shear Column web in compression Column web in tension J35 V wd;rd p ˆ 09f y;wc A vc = J.3.5. J cm0 f y;wc ˆ 75 MPa F c;wc;rd ˆ xb eff;c;wc t wc f y;wc =c M0 F t;wc;rd ˆ xb eff;t;wc t wc f y;wc =c M0 A vc ˆ 30 mm but F c;wc;rd 6 xqb p eff;c;wc t wc f y;wc =c M b eff;c;wc ˆ 490 mm c M0 ˆ 3 k p ˆ 093 b eff;c;wc d wc f y;wc =Etwc f y;wc ˆ 75 MPa V wd;rd ˆ 70 kn b eff;c;wc ˆ 490 mm b b Table J4! b d c ˆ 9 mm Table J5! b ˆ ) x ˆ x ˆ q ˆ b eff;c;wc twc=avc f y;wc ˆ 75 MPa c M0 ˆ c M ˆ E ˆ 0 GPa p k p ˆ = ˆ 056 F t;wc;rd ˆ 93 kn k p ˆ 056 < 0673 ) q ˆ 0 4 b b Table J4! b Table J5! b ˆ ) x ˆ x ˆ q ˆ 074 c M0 ˆ c M ˆ 3 b eff;c;wc twc=avc F c;wc;rd ˆ 93 kn Without safety coecients V wd ˆ 7 kn F c;wc ˆ kn F t;wc ˆ kn Fig. 0. Finite element model.

8 6 L. Sim~oes da Silva et al. / Computers and Structures 77 (000) 65±64 Table 3 Numerical model Rigid links Joint elements (Component) Pins Column web in shear Column web in compression Column web in tension Finite element BM3 JPH3 Geometric properties Thin beam A ˆ 00 m ; I x ˆ I y ˆ m 4 Eccentricity e ˆ 0 Material Elastic, isotropic Joint (sti ness, 3DOF) E ˆ kpa k ex ˆ k ex ˆ k ex ˆ k ex ˆ 0 0 t ˆ 0.3 k ey ˆ k ey ˆ k ey ˆ k ey ˆ 0 0 q ˆ 7.50 ton/m 3 k eh ˆ k eh ˆ k eh ˆ k eh ˆ k px ˆ 00 k px ˆ 00 k px ˆ 00 k px ˆ 0 0 k py ˆ k py ˆ k py ˆ k py ˆ 0 0 k eh ˆ k eh ˆ k eh ˆ k eh ˆ F ˆ 70 kn F ˆ 93 kn F ˆ 93 kn F ˆ 0 5 kn Table 4 Comparison between the analytical and numerical models Analytical model Numerical model Spring elongation D lt ˆ m D lt ˆ m D lc ˆ m D lc ˆ m Bending moment M ˆ kn m M ˆ kn m Rotation q ˆ / ˆ rad / 36 ˆ rad Spring force F t ˆ 69.9 kn F t ˆ 70.0 kn F c ˆ )70.0 kn F c ˆ )70.0 kn Rotation centre x ˆ m x ˆ m Fig.. Comparative graph. that Eq. (4) does not apply in this case because the post-limit sti ness was chosen as zero. Because experimental test results were available for this particular connection, it was possible to use them to calibrate the post-limit sti ness of the components, as well as adjusting the actual value of the moment resistance of the connection ± Table 5 summarises the data. Using the same procedure as before, an initial calibration was performed with only the critical component with non-linear properties, illustrated in Fig. 3 with a

9 L. Sim~oes da Silva et al. / Computers and Structures 77 (000) 65±64 63 Fig.. Comparative graph. Table 5 Data used in the general non-linear model of the connection k ec ˆ kn=m k et ˆ kn=m k pc ˆ kn=m k pt ˆ kn=m PC B ˆ 650 kn P T B ˆ 795 kn F ˆ P B C ˆ 35 kn F ˆ P T B ˆ 3975 kn Fig. 3. Comparative graph. subsequent full non-linear implementation, shown in Fig Concluding remarks The four degree-of-freedom elastic model presented in this paper was able to reproduce the full non-linear moment±rotation response of a steel connection. In particular, because of the post-buckling nature of the analysis, it was possible to pinpoint directly the bifurcation points which correspond to the ``yield'' points of each component. An accurate prediction of the post-limit response of the connection requires the knowledge of the post-limit sti ness (k p ) of each component. The analytical results presented here can very easily be used to calibrate the post-limit sti ness against experimental results, as was shown above. Current work on the same model loaded in compression seems very promising, opening the way to the prediction of the behaviour of steel connections under combined loading (bending moment and axial force).

10 64 L. Sim~oes da Silva et al. / Computers and Structures 77 (000) 65±64 Fig. 4. Comparative graph. Acknowledgements The nantial support from the ``Ministerio da Ci^encia e Tecnologia'' ± PRAXIS XXI research project PRAXIS/P/ECM/353/99 and PRODEP II (Subprograma ) is acknowledged. References [] Weynand K, Jaspart JP, Steenhuis M. The sti ness model of revised Annex J of Eurocode 3. Proceedings of the Third International Workshop on Connections, 995 May ±3; Trento, Italy. [] Eurocode 3, ENV-993--, Revised Annex J, Design of Steel Structures, CEN, European Committee for Standardization, Document CEN/TC 50/SC 3 - N 49 E. Brussels, 997. [3] Sim~oes da Silva L, Santiago A, Vila Real P. Ductility of steel connections. Canadian J Civil Engng, submitted for publication. [4] Kuhlmann U. In uence of axial forces on the component web under compression. COST-C ± Working Group Meeting, Thessaloniki, C/WD/99-0, May, 99. [5] Jaspart JP. Etude de la semi-rigidite des noeuds poutrecolonne et son in uence sur la resistance des ossatures en acier. PhD Thesis, Department MSM, University of Liege, 99. [6] Shi YJ, Chan SL, Wong YL. Modelling for moment rotations characteristics for end-plate connections. J Struct Engng 996;300±6. [7] Hunt GW, Burgan B. Hidden asymmetries in the Shanley model. J Mech Phys Solids 95;33()3±94. [] Cruz PJS, Silva LAPS, Rodrigues DS, Sim~oes RAD. Database for the semi-rigid behaviour of beam-to-column connections in seismic regions. J Construct Steel Res 99;46(0)±3. [9] LUSAS User Manual. FEA Ltd, 996.