Shear and torsion interaction of hollow core slabs

Transcription

1 Competitive nd Sustinble Growth Contrct Nº G6RD-CT--6 Sher nd torsion interction of hollow core slbs HOLCOTORS Technicl Report, Rev. Anlyses of hollow core floors December The content of the present publiction is the sole responsibility of its publisher(s) nd in no wy represents the view of the Commission or its services.

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3 Anlyses of hollow core floors KARIN LUNDGREN, MARIO PLOS Deprtment of Structurl Engineering nd Mechnics Report :7, rev. Concrete Structures CHALMERS UNIVERSITY OF TECHNOLOGY Göteborg, Sweden

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5 REPORT :7, rev. Anlyses of hollow core floors KARIN LUNDGREN, MARIO PLOS Deprtment of Structurl Engineering nd Mechnics Concrete Structures CHALMERS UNIVERSITY OF TECHNOLOGY Göteborg, Sweden

6 ANALYSES OF HOLLOW CORE FLOORS KARIN LUNDGREN, MARIO PLOS KARIN LUNDGREN, MARIO PLOS, ISSN 6-9 Report :7, rev. Archive no. 8 Deprtment of Structurl Engineering nd Mechnics Concrete Structures Chlmers University of Technology SE- 96 Göteborg Sweden Telephone: + 6 ()-77 Cover: Forces on hollow core unit in floor system. Deprtment of Structurl Engineering nd Mechnics Göteborg, Sweden

7 ANALYSES OF HOLLOW CORE FLOORS KARIN LUNDGREN, MARIO PLOS Deprtment of Structurl Engineering nd Mechnics Concrete Structures Chlmers University of Technology ABSTRACT To obtin bckground informtion for code prescriptions for the sher nd torsion interction in hollow core floors, finite element nlyses of some bsic cses were mde. A simplified globl model for complete floors developed in Lundgren et l. () ws used, where the height of the hollow core units is included. Anlyses where the longitudinl joints between the hollow core units were ssumed to ct s hinges were lso crried out. Severl set-ups were modelled, in totl floors. floors with n even number of hollow core units (,, nd 6) were loded with point lod on the joint in the centre of the floor. Furthermore, one exmple of ech lod cse described in the Annex C in pren 68, ws nlysed. These exmples include both line nd point lods, nd with nd without third line support. In ll the nlyses, the chosen spn ws m, nd the hollow core units were mm high nd mm wide. The nlyses show tht the mximum torsionl moment ws reduced when the height of the hollow core units ws included in the nlyses, compred to when hinges were ssumed between the hollow core units. Also the bending moment ws reduced, in most of the studied set-ups. This reduction ws rther smll. On the other hnd, the sher force incresed when the height of the hollow core units ws included, especilly the sher force t the supports. In four of the studied set-ups, direct comprison to the present code, CEN/TC9 (), regrding the lod distribution could be done. In CEN/TC9 (), it is not cler if the given distributions re vlid for the bending moments t mid spn or the sher forces t the supports. It is importnt to note tht the distribution of these differ. No cler nswer to which of these results tht were considered in the code could be found. Key words: Hollow core, sher nd torsion interction, trnsfer in joints I

8 Contents INTRODUCTION. Modelled cses. Modelling method. Chosen input LOAD IN THE MIDDLE OF A FLOOR, ON THE LONGITUDINAL JOINT 7. Modelled set-ups 7. Results 8 LOAD CASES IN THE PRELIMINARY EUROPEAN CODE. Set-up C with line lods.. Edge line lod.. Centre line lod 8. Set-up C with point lod in centre. Set-up C with point lod t edge. Set-up C with three supported edges nd line lod. Set-up C6 with three supported edges nd point lod t mid spn 8 COMPARISON OF RESULTS WITH DIFFERENT MODELLING TECHNIQUES CONCLUSIONS 6 REFERENCES 6 APPENDIX A. Effect of vrying sher stiffness (D ) in interfce simulting the longitudinl joint A APPENDIX B. Results from nlyses with point lod in the middle of the floor, on the longitudinl joint B APPENDIX C. Results from nlyses of set-up C in pren, with line lods C APPENDIX D. Results from nlyses of set-up C in pren, with pont lod in centre D APPENDIX E. Results from nlyses of set-up C in pren, with pont lod t edge E II CHALMERS, Structurl Engineering nd Mechnics, Report :7, rev.

9 APPENDIX F. Results from nlyses of set-up C in pren, with three supported edges nd line lod F APPENDIX G. Results from nlyses of set-up C in pren, with three supported edges nd point lod G CHALMERS, Structurl Engineering nd Mechnics, Report :7, rev. III

10 Prefce The originl version of this report ws published in November. This report (rev.) is revised from its originl version in the following: The nlyses of C- were in the originl version by mistke crried out with the sher stiffness in the joints D N/m. In this version, this is corrected, so tht the sher stiffness D ws 9 N/m lso in these nlyses. The quote M mid /M mid (nlysis ) in Tbles nd 6, in rows C-b nd C-b ws in the originl version by mistke clculted for hollow core unit No.. In this version, it is insted clculted for hollow core unit No., where the mximum bending moment is. The figure Gc in Appendix G is corrected. Göteborg, December. IV CHALMERS, Structurl Engineering nd Mechnics, Report :7, rev.

11 Nottions Romn upper cse letters D E F F n F t F l G I I t L M M mid M mid P Q R y T T A T mx V V A V A V mx Sher stiffness in interfce Young s modulus Force Force per length in norml direction in longitudinl joint Force per length in verticl sher direction in longitudinl joint Force per length in longitudinl sher direction in longitudinl joint Sher modulus Moment of inerti Torsionl moment of inerti Spn Bending moment Bending moment t mid spn Bending moment t mid spn if ll lod ws pplied on one unit Point lod Line lod Rection force per length in verticl direction Torsionl moment Torsionl moment t support A Mximum torsionl moment Sher force Sher force t support A Sher force t support A if ll lod ws pplied on one unit Mximum sher force Romn lower cse letters x Coordinte long min support line y Coordinte in verticl direction z Coordinte long hollow core Greek letters δ n δ t σ τ Displcement in norml direction in longitudinl joint Displcement in sher direction in longitudinl joint Stress Sher stress CHALMERS, Structurl Engineering nd Mechnics, Report :7, rev. V

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13 Introduction To obtin bckground informtion for code prescriptions for the sher nd torsion interction in hollow core floors, finite element nlyses of some bsic cses were mde. A simplified globl model for complete floors developed in Lundgren et l. () ws used.. Modelled cses Severl set-ups were modelled. Floors with n even number of hollow core units (,, nd 6) loded with point lod on the joint in the centre of the floor were nlysed. Furthermore, one exmple of ech lod cse described in the Annex C in pren 68, see CEN/TC9 (), ws nlysed. Ech of the nlysed set-ups is described in chpters -, together with results. In ll nlyses, the chosen spn ws m, nd the hollow core units were mm high nd mm wide.. Modelling method Bem elements were used to model the hollow core units; ech bem element representing the whole cross-section of one hollow core unit. The bem elements used were three-noded nd three-dimensionl, using two-point Guss integrtion scheme; see TNO (). The hollow core units re in prctice connected to ech other long longitudinl joints tht re grouted in situ. Two lterntives were tested for the description of these joints: see Figures nd. In both lterntives, point interfce elements were used between slve nodes tht were tied in ll three directions to the nodes describing the hollow core units. One difference between the two lterntives ws the position of the slve nodes. In lterntive, the slve nodes re positioned in the middle of the edges of the hollow core units. The point interfce ws ssumed to crry both tensile nd compressive forces in both the norml nd trnsverse directions. High vlues of the stiffnesses in both the norml direction nd the trnsverse direction ws chosen; thus this modelling is pproximtely equl to hinge. In lterntive b, the slve nodes re positioned in the corners of the hollow core units. The point interfces were ssumed to crry sher forces; both in the direction long the hollow core units, nd in the verticl direction. In the norml direction, compressive forces were llowed but very smll stiffness ws used for the tensile side, thus reducing the tensile stresses, in order to represent crcked concrete in the joint. This modelling technique ws erlier used to model severl floor tests described in literture, nd ws shown to give good greement with mesurements, see Lundgren et l. (). CHALMERS, Structurl Engineering nd Mechnics, Report :7, rev.

14 t σ n τ δ n δ t Min nodes, defining the bem elements Slve nodes, positioned in the middle of the edges of the hollow core units Point interfce elements between the slve nodes n Figure Principle of the model used, lterntive for the joints: Adjcent hollow core bem elements interct through point interfce elements between slve nodes, rigidly connected to the bem nodes, nd Principl norml nd sher response of the interfce elements. t σ n τ δ n δ t Min nodes, defining the bem elements Slve nodes, positioned in the corners of the hollow core units Point interfce elements between the slve nodes n Figure Principle of the model used, lterntive b for the joints: Adjcent hollow core bem elements interct through point interfce elements between slve nodes, rigidly connected to the bem nodes, nd Principl norml nd sher response of the interfce elements. It cn be noted tht with modelling technique b, the hollow core units will block ech other for torsion, due to tht norml forces will be pplied eccentriclly on the hollow core units tht rotte, see Figure. Thus, the torsionl moments re expected to be reduced when compred to modelling technique. Another effect described in lterntive b is tht the longitudinl sher forces trnsferred in the joints will contribute to the bending moment, see Figure b. This leds to tht lso the bending moment cn be expected to be slightly reduced compred to modelling technique. In relity, the norml forces re most likely trnsferred rther close to the corners, s different rottions of two djcent hollow core units will cuse contct minly t the corner with smll extension. For the longitudinl sher force, it cn be rgued tht it minly is trnsferred where compression is chieved. Therefore, it ws considered to be resonble ssumption to connect the djcent hollow core units t the corners. CHALMERS, Structurl Engineering nd Mechnics, Report :7, rev.

15 Figure The hollow core units will block ech other for torsion in modelling technique b. The longitudinl sher forces trnsferred in the joints will result in contribution to the bending moment in modelling technique b. An exmple of model used is shown in Figure. As cn be seen, on one of the sides, the nodes were supported in ll directions, nd lso for rottions in two directions; i.e. for torsion nd bending in horizontl direction. On the other side, gin the sme rottions were zero. Also the displcements in two of the directions were prevented, i.e. in verticl direction nd in the trnsverse direction. In the direction long the slbs, the displcements of the hollow core units were tied to ech other. In the lod cses where third line support ws ssumed, stiff links were used between the closest bem element nd the support line. This wy of modelling corresponds to the use of slve nodes; the reson why stiff links were used here ws tht it is not possible to support slve nodes in the sme direction s they re tied. The nodes t the third support line were supported only in the verticl direction. Deformtions tied to ech other in the direction long the slbs Bem elements to model single hollow core units Interfce elements to model joints Tie bems modelled by supporting the bem elements in this direction Figure Exmple of model used for the investigted floors. Double rrows men support for rottion, single rrow supports for displcement. CHALMERS, Structurl Engineering nd Mechnics, Report :7, rev.

16 . Chosen input The input needed for the nlyses re the mteril properties, the geometry of the cross-sections, number of nodes long ech hollow core unit, nd properties for the point interfce modelling the longitudinl joints. The nlyses were crried out ssuming liner behviour of the concrete, nd included only the effect of loding. Thus, the effect of prestress nd ded weight were not included. As these effects re ctive lredy when the longitudinl joints re grouted, these effects do not result in ny contct forces between the hollow core units. Young s modulus for the concrete ws ssumed to be GP, nd Poisson s rtio ws ssumed to be.. The bem elements were ssigned the cross-sectionl properties of the mm high nd mm wide hollow core unit by defining the cross-section with 7 zones, see Figure. To obtin correct torsionl stiffness of the bem elements, two fctors which in TNO () re clled sher fctors were djusted. The torsionl stiffness ws evluted from pure torsion tests crried out on single hollow core units; see Pjri (). Detils bout the input nd the corresponding stiffness re given in Tble. y x Figure Tble The cross-section of the bem elements defined with zones. Cross-sectionl properties. Height [mm] Sher fctors.6,.6 EI x [MNm ] EI y [MNm ] 8 GI t [MNm ] I x [m ]. - I y [m ].8 - I t [m ] 8. - CHALMERS, Structurl Engineering nd Mechnics, Report :7, rev.

17 The number of nodes long ech hollow core unit ws chosen lrge enough to describe the contct forces cting in the longitudinl joint. Bsed on experience from erlier work, see Lundgren nd Plos (), the number of nodes long ech hollow core unit ws chosen to 6. For modelling of joints s in lterntive, the point interfces describing the behviour of the longitudinl joint were ssigned stiffness in the norml direction of N/m, estimted from the stiffness in compression of the units. The stiffness in the sher direction ws t first lso chosen to N/m to obtin smll deformtions in the joints. However, tht resulted in oscillting results of the trnsferred sher force; therefore the stiffness in the sher direction ws chosen to 9 N/m. In Figure 6, results with vrying stiffness in the sher direction re shown; s cn be seen, the stiffness in the sher direction does not hve ny mjor influence on the results; except tht the trnsferred sher force oscilltes when the stiffness is too lrge chosen. These results re from nlyses of floor with five hollow core units loded with n edge line lod, see Figure. More results from these nlyses cn be found in Appendix A. For modelling of joints s in lterntive b, the point interfces describing the behviour of the longitudinl joint were ssumed to crry sher forces; both in the direction long the hollow core units, nd in the verticl direction, with stiffness of 9 N/m. This stiffness is chosen from comprisons of deformtions over joints in nlyses of floor tests, see Lundgren et l. (). The stiffness in the norml direction for compression ws N/m, while the stiffness in tension ws only N/m, in order to represent crcked concrete in the joint. In the models where gp ws ssumed, compression ws ssumed to be built up first when this gp ws closed, see Figure 7. The stiffness in the norml direction for compression ws chosen from estimtions of compression of the cross-section of the hollow core units. CHALMERS, Structurl Engineering nd Mechnics, Report :7, rev.

18 F n [kn/m] E E E9 F t [kn/m] E E E9 M [knm] E E E9 V [kn] E E E9 - - (c) F [kn] T [knm] E E E9 - - Q [kn/m] - -6 (d) E E E9 Figure 6 (e) Comprison of results from nlyses of Ce-, Q = 7 kn/m, vrying D (sher stiffness in point interfce). Overlpping curves cuse tht some curves re not visible. Contct forces in the joint -, norml forces, nd verticl sher forces. Distribution of (c) bending moment, (d) sher force, nd (e) torsionl moment in hollow core unit. (f) The force from one tie bem to hollow core unit versus the pplied lod. The modelled set-up is shown in Figure. (f) δ. n [mm] - gp. mm no gp σ[mp] - - Figure 7 6 Input for the stresses in the norml direction in the point interfces modelling the longitudinl joint. CHALMERS, Structurl Engineering nd Mechnics, Report :7, rev.

19 Lod in the middle of floor, on the longitudinl joint. Modelled set-ups In erlier work, Lundgren nd Plos (), two connected hollow core units loded with point lod P t the longitudinl joint ws modelled. Here, similr set-ups were investigted, with more hollow core units on ech side of the lod. The point lod ws in ll these nlyses pplied in the mid spn, see Figure 8. The results from the corresponding nlyses of two connected units, nd corresponding set-up with only one unit (Figure 8d) re lso included here, to ese the comprison. For ll these setups, three nlyses were done: ssuming hinges between the hollow core units, including the height without gp, nd including the height nd ssuming gp of. mm. For n overview nd nottions of these nlyses, see Tble. h P h P L L/ z h P h P P L L/ z (c) (d) (e) Figure 8 Modelled set-ups with lod in the middle of floor, on the longitudinl joint. 6 hollow core units, with supports in horizontl direction, 6 hollow core units, (c) hollow core units, nd (d) hollow core units. (e) Corresponding sitution with only one hollow core unit. CHALMERS, Structurl Engineering nd Mechnics, Report :7, rev. 7

20 Tble Anlyses with lod in the middle of the floor, on the longitudinl joint. Nottion Number of elements Modelling lterntive Gp [mm] 6s- 6+supports 6s-b 6+supports b 6s-b 6+supports b b 6 b 6-b 6 b. - -b b -b b. - -b b -b b. ) The sme nlysis is in Lundgren nd Plos () denoted s---inf-. ) The sme nlysis is in Lundgren nd Plos () denoted s---inf-.. Results As there is symmetry in the studied set-ups (Figures 8-d), no sher force will be trnsferred in the longitudinl joint in the centre of the floor. Compressive contct forces will pper t the upper edge of the centre longitudinl joint, while the lower edge of the centre longitudinl joint will open without ny force trnsfer. In the nlyses with gp between the hollow core units, the results re strongly non-liner, see Figure 9, which shows how the force from one tie bem to ech hollow core unit depends on the pplied lod in one of the studied cses. In the erlier studied set-up with only two hollow core units, Lundgren nd Plos (), the results were bi-liner, with breking point t certin pplied lod. Thus, comprisons between nlyses with different prmeters could be done by compring single vlues. Here, s the results re so strongly non-liner, this is not possible. To enble comprison between the different set-ups, the results re therefore compred t chosen lod of P = kn. 8 CHALMERS, Structurl Engineering nd Mechnics, Report :7, rev.

21 F [kn] P [kn] Figure 9 The force from one tie bem to ech hollow core unit versus the pplied lod, results from nlysis 6s-b. The nottions refer to the hollow core unit, see Figure 8. An exmple of the resulting contct forces, cross-sectionl forces nd moments from one of the nlyses is shown in Figure. In the nlyses where the height of the units is included, the longitudinl joint loded with the point lod typiclly crried compressive contct forces in the upper edge, nd lso compressive forces t the lower edge in the next longitudinl joint, both cting to decrese the torsion. The results from ech of the nlyses re presented in Appendix B. F n [kn/m] upper - lower - upper - lower F t [kn/m] totl - totl M [knm] V [kn] (c) T [knm] (e) (d) CHALMERS, Structurl Engineering nd Mechnics, Report :7, rev. 9

22 Figure Results from nlysis -b, P = kn. Contct forces in the joints, norml forces, nd verticl sher forces. Distribution of (c) bending moment, (d) sher force, nd (e) torsionl moment. The nottions refer to the hollow core unit, see Figure 8. The distribution of the contct forces long the upper edge of the centre longitudinl joint from ll of the nlyses where the height of the units is included re compred in Figure. As cn be seen, very high contct forces were obtined when there ws no gp nd horizontl supports. When there ws n initil gp between the elements, the contct forces were pproximtely the sme whether there were horizontl supports or not. The results re lso pproximtely the sme for nd 6 hollow core units, with or without gp, while the contct forces re slightly lower in the nlyses with two units. The torsionl moment in hollow core unit No. (the loded one) is compred in Figure ; s cn be seen similr results re obtined in ll nlyses except when no gp nd horizontl supports were ssumed. F n [kn/m] b -b 6-b 6s-b -6 F n [kn/m] b -b 6-b 6s-b - -6 CHALMERS, Structurl Engineering nd Mechnics, Report :7, rev.

23 Figure The distribution of the norml contct forces long the upper edge of the centre longitudinl joint, P = kn. Results from nlyses lterntive b: without gp, with gp. mm. T [knm] b -b 6-b 6s-b T [knm] b -b 6-b 6s-b T [knm] s- 6- Figure (c) The distribution of the torsionl moment in one of the loded hollow core units, P = kn. Results from nlyses lterntive b, without gp, lterntive b, with gp. mm, nd (c) lterntive. In Figure, the bending moment t mid spn, nd sher nd torsionl moment t the support in one of the loded hollow core units (P = kn) in the vrious nlyses re compred. As cn be seen, the bending moment t mid spn nd sher force t the CHALMERS, Structurl Engineering nd Mechnics, Report :7, rev.

24 support both decresed for incresing number of hollow core units in the studied setups. This is due to tht the lod is spred to more units. The vrious modelling techniques hd lmost no effect on these results. Furthermore, the presence of gp hd no or very little influence on these results. This is s cn be expected, s the sher force ws ssumed to be trnsferred directly, independent of the gp. M mid [knm] b b 6 Number of units V A [kn] b b 6 Number of units supports supports T A [knm] 8 6 Figure b b 6 Number of units (c) supports Cross-sectionl forces nd moments in one of the loded hollow core units versus number of hollow core units in the studied set-up, P = kn. The results from the nlyses with horizontl supports re lso shown. The bending moment t mid spn, sher force t support, nd (c) torsionl moment t the support. CHALMERS, Structurl Engineering nd Mechnics, Report :7, rev.

25 The torsionl moment t the support (Figure c) ws smller in the nlyses where the height of the units ws tken into ccount. The torsionl moment ws lowest in the nlyses without gp, s the blocking effect strted cting lredy for low lods then. It is lso worth to note tht when the height of the slb ws tken into ccount, the torsionl moment ws rther close to wht ws obtined in single element. When hinges were ssumed between the hollow core units (lterntive ) nd or 6 units were nlysed, the torsionl moment incresed compred to the single element cse due to the trnsfer of sher force to the next neighbouring hollow core unit. When the height of the units ws included, the torsionl moment ws blnce between two countercting effects: it ws incresed compred to the single unit cse due to the trnsfer of sher to the djcent unit, but on the other hnd decresed due to the blocking effect. This effect is illustrted in Figure. Thus, the torsionl moment ws bout the sme mgnitude s in the corresponding single unit cse. Horizontl supports did not influence the torsionl moment when modelling lterntive ws used, while it hd lrge influence in the nlyses using modelling technique b. Especilly in the nlysis of the set-up with supports in horizontl direction nd no gp, the torsionl moment t the support becme lmost zero. This is due to the lrge norml contct forces tht were obtined, see Figure. However, the bsic ssumptions in the model cn be discussed in this cse: for these high compressive forces, the ssumption bout negligible sher deformtions in the crosssection (i.e. stiff rottion) of the hollow core units is questionble. It is lso worth noting tht rigid supports re to be voided for other resons, s they will cuse restrining forces due to for exmple shrinkge or temperture lod. Figure Forces cting on one of the loded hollow core units in the studied setups with or more hollow core units. CHALMERS, Structurl Engineering nd Mechnics, Report :7, rev.

26 Lod cses in the preliminry Europen code In the preliminry Europen code for hollow core units (pren 68), digrms showing lod distributions re given for number of set-ups in Appendix C of CEN/TC9 (). At lest one set-up from ech of the digrms ws nlysed here, both ssuming the longitudinl joints to ct s hinges, nd tking the height into ccount (lterntive nd b, see Figures nd ). The nlyses re denoted ccording to the figures in Appendix C in pren 68. Ech lod cse is presented nd discussed in the following subsections. Results from ll the nlyses re shown in Appendix C-G. Similr s in the previous section, the results were strongly non-liner when gp ws ssumed. To enble comprisons, the results re compred t certin pplied lods. For point lods, the chosen lod ws kn, while line lods were chosen to 7 kn/m.. Set-up C with line lods The set-up C consists of five hollow core units, loded with line lod either t the edge, or in the centre, see Figure. Six nlyses were crried out, denoted: Ce-: Ce-b: Edge lod, no gp, nd ssuming the longitudinl joints to ct s hinges (lterntive ). Edge lod, no gp, nd the longitudinl joints s lterntive b (tking the height of the slb into ccount) Ce-b: Edge lod, gp. mm, nd the longitudinl joints s lterntive b (tking the height of the slb into ccount) Cc-: Cc-b: Centre lod, no gp, nd ssuming the longitudinl joints to ct s hinges (lterntive ). Centre lod, no gp, nd the longitudinl joints s lterntive b (tking the height of the slb into ccount) Cc-b: Centre lod, gp. mm, nd the longitudinl joints s lterntive b (tking the height of the slb into ccount) CHALMERS, Structurl Engineering nd Mechnics, Report :7, rev.

27 Edge line lod Q or Centre line lod Q Q L = m Figure Anlysed set-up C: floor consisting of five hollow core units loded with line lod, either on the edge or in the centre... Edge line lod In Appendix C, the distribution of contct forces, bending moment, sher nd torsionl moment long the hollow core units from the different nlyses with edge line lods re shown. The contct forces in the longitudinl joint closest to the pplied lod re shown in Figure 6-c. While nlysis method resulted in slightly higher verticl sher forces, the norml forces nd the longitudinl sher forces were lot lrger in nlyses where modelling method b ws used. The bending moment, sher nd torsionl moment in the loded hollow core unit, re compred in Figure 6d-f. As cn be seen, the differences between the nlyses re rther smll for this set-up; the bending moment is pproximtely the sme; the sher forces becme slightly smller in nlysis while the torsionl moment is slightly smller in nlysis b. The distribution of the bending moment t mid spn cn be compred with the digrms in CEN/TC9 (), see Figure 7. As cn be seen, the lod is less distributed in the nlyses thn wht is given directly in the pren. It should be noted tht for design in the ultimte limit stte, the bending moment in the loded unit shll ccording to the code be incresed with fctor., nd decresed for the other units. When the distribution fctors given in the code re djusted in this wy (denoted pren ult. in Figure 7), the greement to the nlyses is rther good. The bending moment is more distributed thn the sher force t the supports, compre Figures 7 nd b. In CEN/TC9 (), it is not cler if the given distributions re vlid for the bending moments t mid spn or the sher forces t the supports. Therefore, the distributions given in the code re compred to both types of results. For this set-up, however, the greement ppers to be better to the bending moment. In Figure 8, the torsionl moments t the support re compred to the ones obtined in nlysis. As cn be seen, the torsionl moments re typiclly decresed in the hollow core units with high torsionl moments when the height is tken into ccount. However, in the hollow core units with low torsionl moments, they cn become lrger when the height is tken into ccount. For hollow core unit No. in nlysis Ce-b, the increse expressed s percentge is very lrge (Figure 8b), however, in rel vlues (Figure 8), the difference is smll. CHALMERS, Structurl Engineering nd Mechnics, Report :7, rev.

28 F n [kn/m] b lower b lower F t [kn/m] b b F l [kn/m] b lower b lower M [knm] b b V [kn] (c) b b T [knm] (d) b b Figure 6 (e) Contct forces in the joints, norml forces, verticl sher forces, nd (c) longitudinl sher forces. The distribution of (d) bending moment, (e) sher, nd (f) torsionl moment in the loded hollow core unit. Results from nlyses of set-up C, edge line lod, t n pplied lod of Q = 7 kn/m. (f) 6 CHALMERS, Structurl Engineering nd Mechnics, Report :7, rev.

29 M mid /M mid [-] pren ult. pren b b V A /V A [-] pren ult. pren b b Hollow core unit No. [-] Hollow core unit No. [-] Figure 7 The distribution of the bending moment t mid spn nd the sher t the supports, both evluted t lod of Q = 7 kn/m. Setup C, edge line lod. The numbering of the hollow core units is shown in Figure. T A [knm] Hollow core unit No. [-] T A / T A (nlysis ) [-] b b..9.8 Hollow core unit No. [-] b b Figure 8 The torsionl moment t the supports t n pplied lod of Q = 7 kn/m. The torsionl moment t the supports in reltion to the sme one in nlysis. Set-up C, edge line lod. The numbering of the hollow core units is shown in Figure. CHALMERS, Structurl Engineering nd Mechnics, Report :7, rev. 7

30 .. Centre line lod All results from the nlyses with centre line lod cn be found in Appendix C. The bending moment nd sher in the loded hollow core unit, nd the torsionl moment in the djcent unit re compred in Figure 9. As cn be seen, the torsionl moment is rther much reduced when the height of the units is tken into ccount; else, the differences between the nlyses re smll. The distribution of the bending moment t mid spn corresponds well to wht is directly given in pren 68, see Figure. Agin, the bending moment is more distributed thn the sher t the supports. Modelling lterntive b decresed the mximum torsionl moment to only 7% of the similr in nlysis when no gp ws ssumed, while 87% were obtined when gp of. mm ws ssumed, see Figure b. M [knm] b b V [kn] b b T [knm] b b Figure 9 (c) The distribution of bending moment in the loded hollow core unit, sher in the loded hollow core unit, nd (c) the torsionl moment in the djcent unit. Results from nlyses of set-up C, centre line lod, t n pplied lod of Q = 7 kn/m. 8 CHALMERS, Structurl Engineering nd Mechnics, Report :7, rev.

31 M mid /M mid [-]... pren ult. pren b b V A /V A [-]... pren ult. pren b b.. Hollow core unit No. [-] Hollow core unit No. [-] Figure The distribution of the bending moment t mid spn nd the sher t the supports, both evluted t lod of Q = 7 kn/m. Set-up C, centre line lod. The numbering of the hollow core units is shown in Figure. T A [knm] Hollow core unit No. [-] T A / T A (nlysis ) [-] b b.8.6. Hollow core unit No. [-] b b Figure The torsionl moment t the supports t n pplied lod of Q = 7 kn/m. The torsionl moment t the supports in reltion to the sme one in nlysis. Set-up C, centre line lod. The numbering of the hollow core units is shown in Figure. CHALMERS, Structurl Engineering nd Mechnics, Report :7, rev. 9

32 . Set-up C with point lod in centre The set-up C consists of five hollow core units, loded with point lod in the centre of the spn, see Figure. This set-up ws modelled not only for the spn m, but lso for spns of, 7 nd m, tking the height of the slb into ccount (lterntive b). For the spn m, three nlyses were crried out, denoted: C-: C-b: C-b: No gp, nd ssuming the longitudinl joints to ct s hinges. No gp, nd the longitudinl joints s lterntive b (tking the height of the slb into ccount) Gp. mm, nd the longitudinl joints s lterntive b (tking the height of the slb into ccount) Point lod P P L/ = 6 m L/ = 6 m Figure Anlysed set-up C: floor consisting of five hollow core units loded with point lod in the centre of the spn. In Figure, some results from the nlyses where the joint ws modelled tking the height of the slb into ccount (lterntive b) with vrying spns re shown. In Figure, the trnsverse distribution of the bending moment t mid-spn in the nlyses is compred with curves given in CEN/TC9 (). As cn be seen, the nlyses show less distribution of the bending moment t mid spn thn is given directly by the pren. It is importnt to note tht the trnsverse distribution of the sher t the supports ws not the sme s the trnsverse distribution of the bending moment t mid-spn; compre Figures nd b. The sher t the supports ws more distributed mong the hollow core units thn the bending moment. Note, however, tht the mximum sher force in the loded hollow core unit ws not t the support: s prt of the sher force is trnsferred to the neighbouring hollow core units, the mximum sher force in the loded hollow core unit ws t the position of the point lod; see Figure b. CHALMERS, Structurl Engineering nd Mechnics, Report :7, rev.

33 Distribution of bending moment t mid spn [%] α α α Spn [m] Distribution of sher t the supports [%] α α α Spn [m] Figure Results from nlyses (unfilled mrkers) of set-up C re compred with curves given in CEN/TC9 () (filled mrkers). Trnsverse distribution of bending moment t mid-spn; Trnsverse distribution of sher t the supports. The subscripts indicte the number of the hollow core unit; see Figure. For the spn m, some more results re shown in Figures -6. As cn be seen, the ssumption of the behviour of the longitudinl joint ffected the torsionl moment, which becme less when the height of the slb ws tken into ccount. M [knm] b b V [kn] b b - - T [knm] b b Figure (c) The distribution of bending moment in the loded hollow core unit, sher in the loded hollow core unit, nd (c) the torsionl moment in the djcent unit. Results from nlyses of set-up C, point lod in mid spn, t n pplied lod of P = kn. CHALMERS, Structurl Engineering nd Mechnics, Report :7, rev.

34 M mid /M mid [-] Hollow core unit No. [-] pren ult. pren b b V A /V A [-] Hollow core unit No. [-] b b pren ult. pren Figure The distribution of the bending moment t mid spn nd the sher t the supports, both evluted t lod of P = kn. Set-up C, point lod in mid spn. The numbering of the hollow core units is shown in Figure. T A [knm] Hollow core unit No. [-] T A / T A (nlysis ) [-]. - - b b Hollow core unit No. [-] b b Figure 6 The torsionl moment t the supports t n pplied lod of P = kn. The torsionl moment t the supports in reltion to the sme one in nlysis. Set-up C, point lod in mid spn. The numbering of the hollow core units is shown in Figure. CHALMERS, Structurl Engineering nd Mechnics, Report :7, rev.

35 . Set-up C with point lod t edge The set-up C consists of five hollow core units, loded with point lod t the edge, see Figure 7. Three nlyses were crried out for this set-up, denoted: C-: C-b: C-b: No gp, nd ssuming the longitudinl joints to ct s hinges. No gp, nd the longitudinl joints s lterntive b (tking the height of the slb into ccount) Gp. mm, nd the longitudinl joints s lterntive b (tking the height of the slb into ccount) Point lod P P L/ = 6 m L/ = 6 m Figure 7 Anlysed set-up C: floor consisting of five hollow core units loded with point lod t the edge. The bending moment, sher nd torsionl moment in the loded hollow core unit re compred in Figure 8. While the bending moment becme lmost the sme in the different nlyses, there were smll differences in the sher nd the torsionl moment; the sher ws slightly smller in nlysis while the torsionl moment ws slightly lrger. In Figure 9, the distribution of the bending moment is compred to wht is given in pren; s cn be seen, the bending moment ws less distributed in the nlyses thn directly given in pren. The sher t the supports ws pproximtely eqully distributed s the bending moment t mid spn in the nlyses using lterntive b; compre Figures 9 nd b. When compring the different modelling techniques, the torsionl moment ws reduced in lterntive b in the hollow core units tht hve the highest torsionl moments, nd incresed in the one tht hd only low torsionl moment (Figure ). CHALMERS, Structurl Engineering nd Mechnics, Report :7, rev.

36 M [knm] b b V [kn] b b -6 - T [knm] b b - Figure 8 (c) The distribution of bending moment, sher, nd (c) torsionl moment in the loded hollow core unit. Results from nlyses of set-up C, point lod t edge, t n pplied lod of P = kn. M mid /M mid [-].... pren ult. pren b b V A /V A [-].... pren ult. pren b b.. Hollow core unit No. [-] Hollow core unit No. [-] Figure 9 The distribution of the bending moment t mid spn nd the sher t the supports, both evluted t lod of P = kn. Set-up C, point lod t edge. The numbering of the hollow core units is shown in Figure 7. CHALMERS, Structurl Engineering nd Mechnics, Report :7, rev.

37 T A [knm] Hollow core unit No. [-] T A / T A (nlysis ) [-] b b.9.8 Hollow core unit No. [-] b b Figure The torsionl moment t the supports t n pplied lod of P = kn. The torsionl moment t the supports in reltion to the sme one in nlysis. Set-up C, point lod t edge. The numbering of the hollow core units is shown in Figure 7.. Set-up C with three supported edges nd line lod The set-up C consists of five hollow core units, loded with line lod. In CEN/TC9 (), the rection force from severl plcements of the lod re shown. Here, one plcement of the lod ws chosen, t the centre of the floor, see Figure. Three nlyses were crried out for this set-up, denoted: C-: C-b: C-b: No gp, nd ssuming the longitudinl joints to ct s hinges. No gp, nd the longitudinl joints s lterntive b (tking the height of the slb into ccount) Gp. mm, nd the longitudinl joints s lterntive b (tking the height of the slb into ccount) Line lod Q Q L = m Figure Anlysed set-up C: floor consisting of five hollow core units with three supported edges loded with line lod in the centre. CHALMERS, Structurl Engineering nd Mechnics, Report :7, rev.

38 The bending moment nd sher in the loded hollow core unit nd torsionl moment in the djcent unit, nd lso the rection force long the support lines re compred in Figure. As cn be seen, there were some differences between the different modelling techniques; of pproximtely equl size for both the bending moment, sher force nd torsionl moment. In Tble, the rection force t the third line support s prt of the totl pplied lod is tbulted, nd compred to wht is given in pren. As cn be seen, the rection force is lrger in pren thn ws obtined in these nlyses. M [knm] b b V [kn] b b - - T [knm] b b R y [kn/m] (c) b b R y [kn/m] (d) b b - 6 x [m] - Figure (e) The distribution of bending moment in the loded hollow core unit, sher in the loded hollow core unit, nd (c) the torsionl moment in the djcent unit. The distribution of the rection force (d) long the third support line, nd (e) long one of the min support lines (ssuming liner distribution t ech hollow core unit). Results from nlyses of set-up C, three supported edges nd line lod, t lod of Q = 7 kn/m. 6 CHALMERS, Structurl Engineering nd Mechnics, Report :7, rev.

39 Tble Amount of pplied lod crried by the third line support for set-up C with line lod. CEN/TC9 () C-b C-b C- Amount of loding [%] 6 9 ) Evluted t lod of Q = 7 kn/m The distribution of bending moment t mid spn nd sher t the supports is shown in Figure. Note tht the sum is not equl to %; s some prt of the lod is crried by the third line support. The totl bending moment ws therefore reduced to round % of wht would be obtined if the third support line ws not present. Modelling technique b gve results with negtive rection force in the hollow core unit closest to the third support line; i.e. lifting of the slb. This ws not the cse in the nlyses where modelling technique ws used; however, when the effect of the torsionl moment ws included, the totl effect ws lifting of the corners, nd the difference between the different modelling techniques ws smll, see Figure e. The torsionl moments t the supports re shown in Figure. As cn be seen, lrge torsionl moments were obtined, compre Figures nd, which re for similr lod setups except for the third line support. V A /V A [-] M mid /M mid [-]..... b b. -. Hollow core unit No. [-] -. Hollow core unit No. [-] b b Figure The distribution of the bending moment t mid spn nd the sher t the supports, both evluted t lod of Q = 7 kn/m. Set-up C, three supported edges nd line lod. The numbering of the hollow core units is shown in Figure. Note tht the sum is not equl to %; s some lod is crried by the third support line, see Tble. CHALMERS, Structurl Engineering nd Mechnics, Report :7, rev. 7

40 T A [knm] - Hollow core unit No. [-] b b T A / T A (nlysis ) [-] Hollow core unit No. [-] b b Figure The torsionl moment t the supports t n pplied lod of Q = 7 kn/m. The torsionl moment t the supports in reltion to the sme one in nlysis. Set-up C, three supported edges nd line lod. The numbering of the hollow core units is shown in Figure.. Set-up C6 with three supported edges nd point lod t mid spn The set-up C6 consists of five hollow core units, loded with point lod. In CEN/TC9 (), the rection force from severl plcements of the lod re shown. Here, one plcement of the lod ws chosen, t the centre of the floor, see Figure. Three nlyses were crried out for this set-up, denoted: C6-: C6-b: C6-b: No gp, nd ssuming the longitudinl joints to ct s hinges. No gp, nd the longitudinl joints s lterntive b (tking the height of the slb into ccount) Gp. mm, nd the longitudinl joints s lterntive b (tking the height of the slb into ccount) Point lod P P L/ = 6 m L/ = 6 m Figure 8 Anlysed set-up C6: floor consisting of five hollow core units with three supported edges loded with point lod in the centre. CHALMERS, Structurl Engineering nd Mechnics, Report :7, rev.

41 The bending moment nd sher in the loded hollow core unit nd torsionl moment in the djcent unit, nd lso the rection force long the support lines re compred in Figure 6. Smll differences were obtined in the different nlyses. The rection force crried by the third line support is shown in Tble, nd compred to wht is given in pren. There is most likely something wrong with the vlues tht cn be red from digrm in pren. The scle of the digrm (Figure C.6 in CEN/TC9 ()) ppers to be wrong, s more thn % of the pplied lod is crried by the third line support for severl lod cses. It ws therefore ssumed tht the whole lod is crried by the third support line when the lod is pplied directly on the support, nd the scle ws djusted to fit this. The second vlue shown in Tble ws then obtined, which corresponds rther well with the vlues obtined in the nlyses here. M [knm] b b V [kn] b b T [knm] R y [kn/m] b b R y [kn/m] 6-6 (c) x [m] b b (d) b b Figure 6 (e) The distribution of bending moment in the loded hollow core unit, sher in the loded hollow core unit, nd (c) the torsionl moment in the djcent unit. The distribution of the rection force (d) long the third support line, nd (e) long one of the min support lines (ssuming liner distribution t ech hollow core unit). Results from nlyses of set-up C6, three supported edges nd point lod, t lod of P = kn. CHALMERS, Structurl Engineering nd Mechnics, Report :7, rev. 9

42 Tble Amount of pplied lod crried by the third line support for set-up C6 with point lod. CEN/TC9 () C6-b C6-b C6- Amount of loding [%] 97 / ) The scle in this figure (Figure C.6 in CEN/TC9 ()) ppers to be wrong. If it is ssumed tht the whole lod is crried by the third support line when the lod is pplied directly on the support, the second vlue is obtined. ) Evluted t lod of P = kn The distribution of bending moment nd sher t the supports re shown in Figure 7. Agin, note tht the sum is not equl to %; s some prt of the lod is crried by the third line support. The totl bending moment ws therefore reduced to round 6% of wht would be obtined if the third support line ws not present. Modelling technique b gve results with negtive rection force in the hollow core unit closest to the third support line; i.e. lifting of the slb. This ws not the cse in the nlyses where modelling technique ws used; however, when the effect of the torsionl moment ws included, the totl effect ws lifting of the corners, nd the difference between the different modelling techniques ws smll, see Figure 6e. The torsionl moments t the supports re shown in Figure 8. As cn be seen, lrge torsionl moments were obtined, compre Figures 8 nd 6, which re for similr lod setups except for the third line support. M mid /M mid [-]. V A /V A [-] b b Hollow core unit No. [-] -. Hollow core unit No. [-] b b Figure 7 The distribution of the bending moment t mid spn nd the sher t the supports, both evluted t lod of P = kn. Set-up C6, three supported edges nd point lod. The numbering of the hollow core units is shown in Figure. Note tht the sum is not equl to %; s some lod is crried by the third support line, see Tble. CHALMERS, Structurl Engineering nd Mechnics, Report :7, rev.

43 T A [knm] - Hollow core unit No. [-] - b b T A / T A (nlysis ) [-] Hollow core unit No. [-] b b Figure 8 The torsionl moment t the supports t n pplied lod of P = kn. The torsionl moment t the supports in reltion to the sme one in nlysis. Set-up C6, three supported edges nd point lod. The numbering of the hollow core units is shown in Figure. CHALMERS, Structurl Engineering nd Mechnics, Report :7, rev.

44 Comprison of results with different modelling techniques In Tbles nd 6, cross-sectionl moments nd sher forces re compred to the corresponding ones in nlysis lterntive. This evlution ws done for the sme pplied lod s in erlier presented results. For the sher force nd the torsionl moment, both the ones t the supports nd the mximum obtined re compred for some set-ups they were the sme, nd for others not. The mximum bending moment ws in ll the studied set-ups in mid spn. As cn be seen, when modelling technique b ws used, the torsionl moment decresed, especilly when no gp ws ssumed. The reduced torsionl moment vried from 9 to 9% when there ws no gp, nd 76 to 9% when the ssumed gp ws. mm. One exception ws the nlysis with horizontl supports nd no gp; there the torsionl moment becme s smll s or 6% of the corresponding one in nlysis, depending on if it ws the torsionl moment t the support or the mximum torsionl moment which ws studied. Tble Results from nlyses with lterntive b, no gp, compred to results from nlyses with lterntive. T A /T A (nlysis ) T mx /T mx (nlysis ) V A /V A (nlysis ) V mx /V mx (nlysis ) M mid /M mid (nlysis ) 6s-b b b b Ce-b Cc-b C-b C-b C-b C6-b Averge Stnd dev. ) Averge except nlysis 6s-b. ) Stndrd devition except nlysis 6s-b. CHALMERS, Structurl Engineering nd Mechnics, Report :7, rev.

45 Also the bending moment ws reduced, in most of the studied set-ups, when modelling technique b ws used compred to modelling technique. This reduction ws rther smll. On the other hnd, the sher force incresed when modelling technique b ws used compred to modelling technique, especilly the sher force t the supports. Tble 6 Results from nlyses with lterntive b,. mm gp, compred to results from nlyses with lterntive. T A /T A (nlysis ) T mx /T mx (nlysis ) V A /V A (nlysis ) V mx /V mx (nlysis ) M mid /M mid (nlysis ) 6s-b b b b Ce-b Cc-b C-b C-b C-b C6-b Averge Stnd dev. ) Averge except nlysis 6s-b. ) Stndrd devition except nlysis 6s-b. CHALMERS, Structurl Engineering nd Mechnics, Report :7, rev.

46 Conclusions Some conclusions cn be been drwn from the work: The mximum torsionl moment ws reduced when the height of the hollow core units ws included in the nlyses, compred to when hinges were ssumed between the hollow core units. In the investigted set-ups (without horizontl restrint), the torsionl moment vried from 9 to 9% when there ws no gp, nd 76 to 9% when the ssumed gp ws. mm. The lrgest reduction of the torsionl moment ws obtined in set-up with horizontl restrints t the edges of the floor; in tht cse the torsionl moment becme lmost zero. However, the bsic ssumptions in the model cn be discussed in this cse: for these high compressive forces, the ssumption bout negligible sher deformtions in the cross-section (i.e. stiff rottion) of the hollow core units is questionble. It is lso worth noting tht rigid supports re to be voided for other resons, s they will cuse restrining forces due to for exmple shrinkge or temperture lod. Also the bending moment ws reduced, in most of the studied set-ups, when modelling technique b ws used compred to modelling technique. This reduction ws rther smll. On the other hnd, the sher force incresed when modelling technique b ws used compred to modelling technique, especilly the sher force t the supports. In the nlyses of floors with lod on the longitudinl joint in the centre of the floor, the torsionl moment ws rther close to wht ws obtined in single element, when the height of the units ws included. The torsionl moment ws blnce between two countercting effects: it ws incresed compred to the single unit cse due to the trnsfer of sher to the djcent unit, but on the other hnd decresed due to the blocking effect. In four of the studied set-ups, direct comprison to the present code, CEN/TC9 (), regrding the lod distribution could be done. In CEN/TC9 (), it is not cler if the given distributions re vlid for the bending moments t mid spn or the sher forces t the supports. Therefore, the distributions given in the code were compred to both types of results. No cler nswer to which of these results tht were considered in the code could be found. In three of the four set-ups, the bending moment t mid spn ws less distributed in the nlyses thn wht is given directly by the current code. The fourth set-up corresponded well. It should be noted tht for design in the ultimte limit stte, the bending moment in the loded unit shll ccording to the code be incresed with fctor., nd decresed for the other units. When the distribution fctors given in the code were djusted in this wy, the greement ws rther good for the three set-ups tht else showed difference. In two of the studied set-ups, direct comprison to the present code, CEN/TC9 (), regrding the mount of rection force crried by the third line support could be done. In both of them, the rection force in the code ws lrger thn resulting from the nlyses. However, for one of these set-ups, CHALMERS, Structurl Engineering nd Mechnics, Report :7, rev.