Shear and torsion interaction of hollow core...

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Shear and torsion interaction

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Technical Report 3, Rev. 1

Analyses of hollow core floors

December 2004

The content of the present publication is the sole responsibility of its publisher(s) and in no way represents the view of the Commission or its services.

Analyses of hollow core floors KARIN LUNDGREN, MARIO PLOS Department of Structural Engineering and Mechanics Report 04:7, rev. 1 Concrete Structures CHALMERS UNIVERSITY OF TECHNOLOGY Göteborg, Sweden 2004

REPORT 04:7, rev. 1

Analyses of hollow core floors

KARIN LUNDGREN, MARIO PLOS

Department of Structural Engineering and Mechanics Concrete Structures

CHALMERS UNIVERSITY OF TECHNOLOGY

Göteborg, Sweden 2004

ANALYSES OF HOLLOW CORE FLOORS KARIN LUNDGREN, MARIO PLOS

© KARIN LUNDGREN, MARIO PLOS, 2004

ISSN 1651-9035 Report 04:7, rev. 1 Archive no. 82 Department of Structural Engineering and Mechanics Concrete Structures Chalmers University of Technology SE-412 96 Göteborg Sweden Telephone: + 46 (0)31-772 1000 Cover: Forces on a hollow core unit in a floor system. Department of Structural Engineering and Mechanics Göteborg, Sweden 2004

ANALYSES OF HOLLOW CORE FLOORS KARIN LUNDGREN, MARIO PLOS Department of Structural Engineering and Mechanics Concrete Structures Chalmers University of Technology

ABSTRACT

To obtain background information for code prescriptions for the shear and torsion interaction in hollow core floors, finite element analyses of some basic cases were made. A simplified global model for complete floors developed in Lundgren et al. (2004) was used, where the height of the hollow core units is included. Analyses where the longitudinal joints between the hollow core units were assumed to act as hinges were also carried out.

Several set-ups were modelled, in total 30 floors. 12 floors with an even number of hollow core units (2, 4, and 6) were loaded with a point load on the joint in the centre of the floor. Furthermore, one example of each load case described in the Annex C in prEN 1168, was analysed. These examples include both line and point loads, and with and without a third line support. In all the analyses, the chosen span was 12 m, and the hollow core units were 400 mm high and 1200 mm wide.

The analyses show that the maximum torsional moment was reduced when the height of the hollow core units was included in the analyses, compared to when hinges were assumed between the hollow core units. Also the bending moment was reduced, in most of the studied set-ups. This reduction was rather small. On the other hand, the shear force increased when the height of the hollow core units was included, especially the shear force at the supports.

In four of the studied set-ups, a direct comparison to the present code, CEN/TC229 (2004), regarding the load distribution could be done. In CEN/TC229 (2004), it is not clear if the given distributions are valid for the bending moments at mid span or the shear forces at the supports. It is important to note that the distribution of these differ. No clear answer to which of these results that were considered in the code could be found.

Key words: Hollow core, shear and torsion interaction, transfer in joints

I

Contents

1 INTRODUCTION 1

1.1 Modelled cases 1

1.2 Modelling method 1

1.3 Chosen input 4

2 LOAD IN THE MIDDLE OF A FLOOR, ON THE LONGITUDINAL JOINT 7

2.1 Modelled set-ups 7

2.2 Results 8

3 LOAD CASES IN THE PRELIMINARY EUROPEAN CODE 14

3.1 Set-up C1 with line loads 14 3.1.1 Edge line load 15 3.1.2 Centre line load 18

3.2 Set-up C2 with point load in centre 20

3.3 Set-up C3 with point load at edge 23

3.4 Set-up C5 with three supported edges and line load 25

3.5 Set-up C6 with three supported edges and point load at mid span 28

4 COMPARISON OF RESULTS WITH DIFFERENT MODELLING TECHNIQUES 32

5 CONCLUSIONS 34

6 REFERENCES 36

APPENDIX A. Effect of varying shear stiffness (D22) in interface simulating the longitudinal joint A1

APPENDIX B. Results from analyses with point load in the middle of the floor, on the longitudinal joint B1

APPENDIX C. Results from analyses of set-up C1 in prEN, with line loads C1

APPENDIX D. Results from analyses of set-up C2 in prEN, with pont load in centre D1

APPENDIX E. Results from analyses of set-up C1 in prEN, with pont load at edge E1

CHALMERS, Structural Engineering and Mechanics, Report 04:7, rev. 1

II

APPENDIX F. Results from analyses of set-up C1 in prEN, with three supported edges and line load F1

APPENDIX G. Results from analyses of set-up C1 in prEN, with three supported edges and point load G1


III

Preface The original version of this report was published in November 2004. This report (rev.1) is revised from its original version in the following:

The analyses of C2-a were in the original version by mistake carried out with the shear stiffness in the joints D22 3·1010 N/m. In this version, this is corrected, so that the shear stiffness D22 was 1·109 N/m also in these analyses.

•

•

•

The quote Mmid/Mmid(analysis a) in Tables 5 and 6, in rows C3-b0 and C3-b05 was in the original version by mistake calculated for hollow core unit No. 3. In this version, it is instead calculated for hollow core unit No. 1, where the maximum bending moment is.

The figure G2c in Appendix G is corrected.

Göteborg, December 2004.


IV

Notations Roman upper case letters D22 Shear stiffness in interface E Young’s modulus F Force Fn Force per length in normal direction in longitudinal joint Ft Force per length in vertical shear direction in longitudinal joint Fl Force per length in longitudinal shear direction in longitudinal joint G Shear modulus I Moment of inertia It Torsional moment of inertia L Span M Bending moment Mmid Bending moment at mid span Mmid0 Bending moment at mid span if all load was applied on one unit P Point load Q Line load Ry Reaction force per length in vertical direction T Torsional moment TA Torsional moment at support A Tmax Maximum torsional moment V Shear force VA Shear force at support A VA0 Shear force at support A if all load was applied on one unit Vmax Maximum shear force

Roman lower case letters x Coordinate along main support line y Coordinate in vertical direction z Coordinate along hollow core

Greek letters δn Displacement in normal direction in longitudinal joint δt Displacement in shear direction in longitudinal joint σ Stress τ Shear stress


V

1 Introduction To obtain background information for code prescriptions for the shear and torsion interaction in hollow core floors, finite element analyses of some basic cases were made. A simplified global model for complete floors developed in Lundgren et al. (2004) was used.

1.1 Modelled cases Several set-ups were modelled. Floors with an even number of hollow core units (2, 4, and 6) loaded with a point load on the joint in the centre of the floor were analysed. Furthermore, one example of each load case described in the Annex C in prEN 1168, see CEN/TC229 (2004), was analysed. Each of the analysed set-ups is described in chapters 2-3, together with results. In all analyses, the chosen span was 12 m, and the hollow core units were 400 mm high and 1200 mm wide.

1.2 Modelling method Beam elements were used to model the hollow core units; each beam element representing the whole cross-section of one hollow core unit. The beam elements used were three-noded and three-dimensional, using a two-point Gauss integration scheme; see TNO (2002).

The hollow core units are in practice connected to each other along longitudinal joints that are grouted in situ. Two alternatives were tested for the description of these joints: see Figures 1 and 2. In both alternatives, point interface elements were used between slave nodes that were tied in all three directions to the nodes describing the hollow core units. One difference between the two alternatives was the position of the slave nodes.

In alternative a, the slave nodes are positioned in the middle of the edges of the hollow core units. The point interface was assumed to carry both tensile and compressive forces in both the normal and transverse directions. High values of the stiffnesses in both the normal direction and the transverse direction was chosen; thus this modelling is approximately equal to a hinge.

In alternative b, the slave nodes are positioned in the corners of the hollow core units. The point interfaces were assumed to carry shear forces; both in the direction along the hollow core units, and in the vertical direction. In the normal direction, compressive forces were allowed but a very small stiffness was used for the tensile side, thus reducing the tensile stresses, in order to represent cracked concrete in the joint. This modelling technique was earlier used to model several floor tests described in literature, and was shown to give a good agreement with measurements, see Lundgren et al. (2004).


1

t

n

δ n

σ n

δ t

τ

t

n

δ n

σ n

δ t

τ

Slave nodes, positioned in the middle of the edges of the hollow core units Point interface elements between the slave nodes

Main nodes, defining the beam elements

(a) (b)

Figure 1 Principle of the model used, alternative a for the joints: (a) Adjacent hollow core beam elements interact through point interface elements between slave nodes, rigidly connected to the beam nodes, and (b) Principal normal and shear response of the interface elements.

t

n

δ n

σ n

δ t

τ

Slave nodes, positioned in the corners of the hollow core units Point interface elements between the slave nodes

t

n

δ n

σ n

δ t

τ

Main nodes, defining the beam elements

(a) (b)

Figure 2 Principle of the model used, alternative b for the joints: (a) Adjacent hollow core beam elements interact through point interface elements between slave nodes, rigidly connected to the beam nodes, and (b) Principal normal and shear response of the interface elements.

It can be noted that with modelling technique b, the hollow core units will block each other for torsion, due to that normal forces will be applied eccentrically on the hollow core units that rotate, see Figure 3a. Thus, the torsional moments are expected to be reduced when compared to modelling technique a. Another effect described in alternative b is that the longitudinal shear forces transferred in the joints will contribute to the bending moment, see Figure 3b. This leads to that also the bending moment can be expected to be slightly reduced compared to modelling technique a. In reality, the normal forces are most likely transferred rather close to the corners, as different rotations of two adjacent hollow core units will cause contact mainly at the corner with a small extension. For the longitudinal shear force, it can be argued that it mainly is transferred where compression is achieved. Therefore, it was considered to be a reasonable assumption to connect the adjacent hollow core units at the corners.


2

(a) (b)

Figure 3 (a) The hollow core units will block each other for torsion in modelling technique b. (b) The longitudinal shear forces transferred in the joints will result in a contribution to the bending moment in modelling technique b.

An example of a model used is shown in Figure 4. As can be seen, on one of the sides, the nodes were supported in all directions, and also for rotations in two directions; i.e. for torsion and bending in horizontal direction. On the other side, again the same rotations were zero. Also the displacements in two of the directions were prevented, i.e. in vertical direction and in the transverse direction. In the direction along the slabs, the displacements of the hollow core units were tied to each other. In the load cases where a third line support was assumed, stiff links were used between the closest beam element and the support line. This way of modelling corresponds to the use of slave nodes; the reason why stiff links were used here was that it is not possible to support slave nodes in the same direction as they are tied. The nodes at the third support line were supported only in the vertical direction.

Beam elements to model single hollow core units

Interface elements to model joints

Tie beams modelled by supporting the beam elements in this direction

Deformations tied to each other in the direction along the slabs

Figure 4 Example of a model used for the investigated floors. Double arrows mean support for rotation, single arrow supports for displacement.


3

1.3 Chosen input The input needed for the analyses are the material properties, the geometry of the cross-sections, number of nodes along each hollow core unit, and properties for the point interface modelling the longitudinal joints. The analyses were carried out assuming linear behaviour of the concrete, and included only the effect of loading. Thus, the effect of prestress and dead weight were not included. As these effects are active already when the longitudinal joints are grouted, these effects do not result in any contact forces between the hollow core units. Young’s modulus for the concrete was assumed to be 30 GPa, and Poisson’s ratio was assumed to be 0.15.

The beam elements were assigned the cross-sectional properties of the 400 mm high and 1200 mm wide hollow core unit by defining the cross-section with 17 zones, see Figure 5. To obtain correct torsional stiffness of the beam elements, two factors which in TNO (2002) are called “shear factors” were adjusted. The torsional stiffness was evaluated from pure torsion tests carried out on single hollow core units; see Pajari (2003). Details about the input and the corresponding stiffness are given in Table 1.

y

x

Figure 5 The cross-section of the beam elements defined with zones.

Table 1 Cross-sectional properties.

Height [mm] 400

Shear factors 4.06, 4.06

EIx [MNm2] 133

EIy [MNm2] 841

GIt [MNm2] 105

Ix [m4] 4.44·10-3

Iy [m4] 2.80·10-2

It [m4] 8.02·10-3


4

The number of nodes along each hollow core unit was chosen large enough to describe the contact forces acting in the longitudinal joint. Based on experience from earlier work, see Lundgren and Plos (2004), the number of nodes along each hollow core unit was chosen to 61.

For modelling of joints as in alternative a, the point interfaces describing the behaviour of the longitudinal joint were assigned a stiffness in the normal direction of 3·1010 N/m3, estimated from the stiffness in compression of the units. The stiffness in the shear direction was at first also chosen to 3·1010 N/m3 to obtain small deformations in the joints. However, that resulted in oscillating results of the transferred shear force; therefore the stiffness in the shear direction was chosen to 1·109 N/m3. In Figure 6, results with varying stiffness in the shear direction are shown; as can be seen, the stiffness in the shear direction does not have any major influence on the results; except that the transferred shear force oscillates when the stiffness is too large chosen. These results are from analyses of a floor with five hollow core units loaded with an edge line load, see Figure 15. More results from these analyses can be found in Appendix A.

For modelling of joints as in alternative b, the point interfaces describing the behaviour of the longitudinal joint were assumed to carry shear forces; both in the direction along the hollow core units, and in the vertical direction, with a stiffness of 1·109 N/m3. This stiffness is chosen from comparisons of deformations over joints in analyses of floor tests, see Lundgren et al. (2004). The stiffness in the normal direction for compression was 3·1010 N/m3, while the stiffness in tension was only 1·104 N/m3, in order to represent cracked concrete in the joint. In the models where a gap was assumed, compression was assumed to be built up first when this gap was closed, see Figure 7. The stiffness in the normal direction for compression was chosen from estimations of compression of the cross-section of the hollow core units.


5

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

3E101E101E9

F n [kN/m]

z/L [-]

0

20

40

60

80

0 0.2 0.4 0.6 0.8 1

3E101E101E9

F t [kN/m]

z/L [-]

(a) (b)

-500

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00 0.2 0.4 0.6 0.8 1

3E101E101E9

M [kNm] z/L [-]

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0

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0 0.2 0.4 0.6 0.8 1

3E101E101E9

V [kN]

z/L [-]

(c) (d)

-400

-200

0

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0 0.2 0.4 0.6 0.8 1

3E101E101E9

T [kNm]

z/L [-]

-6-5-4-3-2-10

0 20 40 60 80 1003E101E101E9

Q [kN/m]

F [kN]

(e) (f)

Figure 6 Comparison of results from analyses of C1e-a, Q = 70 kN/m, varying D22 (shear stiffness in point interface). Overlapping curves cause that some curves are not visible. Contact forces in the joint 1-2, (a) normal forces, and (b) vertical shear forces. Distribution of (c) bending moment, (d) shear force, and (e) torsional moment in hollow core unit 1. (f) The force from one tie beam to hollow core unit 1 versus the applied load. The modelled set-up is shown in Figure 15.

-15

-10

-5

0-1.5 -1 -0.5 0 0.5δn[mm]

σ[MPa]

gap 0.5 mm no gap

Figure 7 Input for the stresses in the normal direction in the point interfaces modelling the longitudinal joint.


6

2 Load in the middle of a floor, on the longitudinal joint

2.1 Modelled set-ups In earlier work, Lundgren and Plos (2004), two connected hollow core units loaded with a point load 2P at the longitudinal joint was modelled. Here, similar set-ups were investigated, with more hollow core units on each side of the load. The point load was in all these analyses applied in the mid span, see Figure 8. The results from the corresponding analyses of two connected units, and a corresponding set-up with only one unit (Figure 8d) are also included here, to ease the comparison. For all these set-ups, three analyses were done: assuming hinges between the hollow core units, including the height without a gap, and including the height and assuming a gap of 0.5 mm. For an overview and notations of these analyses, see Table 2.

(a)

2P h

1 1 2 3 3 2

L

L/2

z

(b)

2P h

1 1 2 3 3 2

L

L/2

z

(c)

2P h

1 1 2 2

(e)

P

(d)

2Ph

1 1

Figure 8 Modelled set-ups with load in the middle of a floor, on the longitudinal joint. (a) 6 hollow core units, with supports in horizontal direction, (b) 6 hollow core units, (c) 4 hollow core units, and (d) 2 hollow core units. (e) Corresponding situation with only one hollow core unit.


7

Table 2 Analyses with load in the middle of the floor, on the longitudinal joint.

Notation Number of elements

Modelling alternative

Gap [mm]

6s-a 6+supports a 0

6s-b0 6+supports b 0

6s-b05 6+supports b 0.5

6-a 6 a 0

6-b0 6 b 0

6-b05 6 b 0.5

4-a 4 a 0

4-b0 4 b 0

4-b05 4 b 0.5

2-a 2 a 0

2-b0 1 2 b 0

2-b05 2 2 b 0.5

1) The same analysis is in Lundgren and Plos (2004) denoted s4-12-0-inf-05. 2) The same analysis is in Lundgren and Plos (2004) denoted s4-12-05-inf-05.

2.2 Results As there is symmetry in the studied set-ups (Figures 8a-d), no shear force will be transferred in the longitudinal joint in the centre of the floor. Compressive contact forces will appear at the upper edge of the centre longitudinal joint, while the lower edge of the centre longitudinal joint will open without any force transfer. In the analyses with a gap between the hollow core units, the results are strongly non-linear, see Figure 9, which shows how the force from one tie beam to each hollow core unit depends on the applied load in one of the studied cases. In the earlier studied set-up with only two hollow core units, Lundgren and Plos (2004), the results were bi-linear, with a breaking point at a certain applied load. Thus, comparisons between analyses with different parameters could be done by comparing single values. Here, as the results are so strongly non-linear, this is not possible. To enable a comparison between the different set-ups, the results are therefore compared at a chosen load of P = 200 kN.


8

01020304050

0 100 200 300 400 500

123

P [kN]

F [kN]

Figure 9 The force from one tie beam to each hollow core unit versus the applied load, results from analysis 6s-b05. The notations refer to the hollow core unit, see Figure 8.

An example of the resulting contact forces, cross-sectional forces and moments from one of the analyses is shown in Figure 10. In the analyses where the height of the units is included, the longitudinal joint loaded with the point load typically carried compressive contact forces in the upper edge, and also compressive forces at the lower edge in the next longitudinal joint, both acting to decrease the torsion. The results from each of the analyses are presented in Appendix B.

-250-200-150-100-50

00 0.2 0.4 0.6 0.8 1

1-2 upper1-2 lower1-1 upper1-1 lower

F n [kN/m] z/L [-]

-30

-20

-10

00 0.2 0.4 0.6 0.8 1

1-2 total1-1 total

F t [kN/m] z/L [-]

(a) (b)

-400

-300

-200

-100

0

100

0 0.2 0.4 0.6 0.8 1 12

M [kNm]z/L [-]

-100

-50

0

50

100

0 0.2 0.4 0.6 0.8 1

12

V [kN]

z/L [-]

(c) (d)

-80-60-40-20

020406080

0 0.2 0.4 0.6 0.8 1

12

T [kNm]

z/L [-]

(e)


9

Figure 10 Results from analysis 4-b0, P = 200 kN. Contact forces in the joints, (a) normal forces, and (b) vertical shear forces. Distribution of (c) bending moment, (d) shear force, and (e) torsional moment. The notations refer to the hollow core unit, see Figure 8.

The distribution of the contact forces along the upper edge of the centre longitudinal joint from all of the analyses where the height of the units is included are compared in Figure 11. As can be seen, very high contact forces were obtained when there was no gap and horizontal supports. When there was an initial gap between the elements, the contact forces were approximately the same whether there were horizontal supports or not. The results are also approximately the same for 4 and 6 hollow core units, with or without gap, while the contact forces are slightly lower in the analyses with two units. The torsional moment in hollow core unit No. 1 (the loaded one) is compared in Figure 12; as can be seen similar results are obtained in all analyses except when no gap and horizontal supports were assumed.

-600

-500

-400

-300

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-100

00 0.2 0.4 0.6 0.8 1

2-b04-b06-b06s-b0

z/L [-]F n [kN/m]

(a)

-600

-500

-400

-300

-200

-100

00 0.2 0.4 0.6 0.8 1

2-b054-b056-b056s-b05

z/L [-]F n [kN/m]

(b)


10

Figure 11 The distribution of the normal contact forces along the upper edge of the centre longitudinal joint, P = 200 kN. Results from analyses alternative b: (a) without gap, (b) with gap 0.5 mm.

-100

-50

0

50

100

0 0.2 0.4 0.6 0.8 1

2-b04-b06-b06s-b0

T [kNm]

z/L [-]

(a)

-100

-50

0

50

100

0 0.2 0.4 0.6 0.8 1

2-b054-b056-b056s-b05

T [kNm]

z/L [-]

(b)

-150-100-50

050

100150

0 0.2 0.4 0.6 0.8 1

2-a4-a6s-a6-a

T [kNm]

z/L [-]

(c)

Figure 12 The distribution of the torsional moment in one of the loaded hollow core units, P = 200 kN. Results from analyses (a) alternative b, without gap, (b) alternative b, with gap 0.5 mm, and (c) alternative a.

In Figure 13, the bending moment at mid span, and shear and torsional moment at the support in one of the loaded hollow core units (P = 200 kN) in the various analyses are compared. As can be seen, the bending moment at mid span and shear force at the


11

support both decreased for increasing number of hollow core units in the studied set-ups. This is due to that the load is spread to more units. The various modelling techniques had almost no effect on these results. Furthermore, the presence of a gap had no or very little influence on these results. This is as can be expected, as the shear force was assumed to be transferred directly, independent of the gap.

0100200300400500600

1 2 3 4 5 6

ab05b0

M mid [kNm]

Number of units

supports

(a)

020406080

100

1 2 3 4 5 6

ab05b0

V A [kN]

Number of units

supports

(b)

020406080

100120

1 2 3 4 5 6

ab05b0

T A [kNm]

Number of units

supports

(c)

Figure 13 Cross-sectional forces and moments in one of the loaded hollow core units versus number of hollow core units in the studied set-up, P = 200 kN. The results from the analyses with horizontal supports are also shown. (a) The bending moment at mid span, (b) shear force at support, and (c) torsional moment at the support.


12

The torsional moment at the support (Figure 13c) was smaller in the analyses where the height of the units was taken into account. The torsional moment was lowest in the analyses without gap, as the blocking effect started acting already for low loads then.

It is also worth to note that when the height of the slab was taken into account, the torsional moment was rather close to what was obtained in a single element. When hinges were assumed between the hollow core units (alternative a) and 4 or 6 units were analysed, the torsional moment increased compared to the single element case due to the transfer of shear force to the next neighbouring hollow core unit. When the height of the units was included, the torsional moment was a balance between two counteracting effects: it was increased compared to the single unit case due to the transfer of shear to the adjacent unit, but on the other hand decreased due to the blocking effect. This effect is illustrated in Figure 14. Thus, the torsional moment was about the same magnitude as in the corresponding single unit case.

Horizontal supports did not influence the torsional moment when modelling alternative a was used, while it had a large influence in the analyses using modelling technique b. Especially in the analysis of the set-up with supports in horizontal direction and no gap, the torsional moment at the support became almost zero. This is due to the large normal contact forces that were obtained, see Figure 11a. However, the basic assumptions in the model can be discussed in this case: for these high compressive forces, the assumption about negligible shear deformations in the cross-section (i.e. stiff rotation) of the hollow core units is questionable. It is also worth noting that rigid supports are to be avoided for other reasons, as they will cause restraining forces due to for example shrinkage or temperature load.

Figure 14 Forces acting on one of the loaded hollow core units in the studied set-ups with 4 or more hollow core units.


13

3 Load cases in the preliminary European code In the preliminary European code for hollow core units (prEN 1168), diagrams showing load distributions are given for a number of set-ups in Appendix C of CEN/TC229 (2004). At least one set-up from each of the diagrams was analysed here, both assuming the longitudinal joints to act as hinges, and taking the height into account (alternative a and b, see Figures 1 and 2). The analyses are denoted according to the figures in Appendix C in prEN 1168. Each load case is presented and discussed in the following subsections. Results from all the analyses are shown in Appendix C-G. Similar as in the previous section, the results were strongly non-linear when a gap was assumed. To enable comparisons, the results are compared at certain applied loads. For point loads, the chosen load was 400 kN, while line loads were chosen to 70 kN/m.

3.1 Set-up C1 with line loads The set-up C1 consists of five hollow core units, loaded with a line load either at the edge, or in the centre, see Figure 15. Six analyses were carried out, denoted:

C1e-a: Edge load, no gap, and assuming the longitudinal joints to act as hinges (alternative a).

C1e-b0: Edge load, no gap, and the longitudinal joints as alternative b (taking the height of the slab into account)

C1e-b05: Edge load, gap 0.5 mm, and the longitudinal joints as alternative b (taking the height of the slab into account)

C1c-a: Centre load, no gap, and assuming the longitudinal joints to act as hinges (alternative a).

C1c-b0: Centre load, no gap, and the longitudinal joints as alternative b (taking the height of the slab into account)

C1c-b05: Centre load, gap 0.5 mm, and the longitudinal joints as alternative b (taking the height of the slab into account)


14

1 2 3 4 5

Centre line load Q

L = 12 m

Edge line load Q or

Q

Figure 15 Analysed set-up C1: a floor consisting of five hollow core units loaded with a line load, either on the edge or in the centre.

3.1.1 Edge line load

In Appendix C, the distribution of contact forces, bending moment, shear and torsional moment along the hollow core units from the different analyses with edge line loads are shown. The contact forces in the longitudinal joint closest to the applied load are shown in Figure 16a-c. While analysis method a resulted in slightly higher vertical shear forces, the normal forces and the longitudinal shear forces were a lot larger in analyses where modelling method b was used. The bending moment, shear and torsional moment in the loaded hollow core unit, are compared in Figure 16d-f. As can be seen, the differences between the analyses are rather small for this set-up; the bending moment is approximately the same; the shear forces became slightly smaller in analysis a while the torsional moment is slightly smaller in analysis b.

The distribution of the bending moment at mid span can be compared with the diagrams in CEN/TC229 (2004), see Figure 17a. As can be seen, the load is less distributed in the analyses than what is given directly in the prEN. It should be noted that for design in the ultimate limit state, the bending moment in the loaded unit shall according to the code be increased with a factor 1.25, and decreased for the other units. When the distribution factors given in the code are adjusted in this way (denoted prEN ult. in Figure 17a), the agreement to the analyses is rather good. The bending moment is more distributed than the shear force at the supports, compare Figures 17a and b. In CEN/TC229 (2004), it is not clear if the given distributions are valid for the bending moments at mid span or the shear forces at the supports. Therefore, the distributions given in the code are compared to both types of results. For this set-up, however, the agreement appears to be better to the bending moment.

In Figure 18, the torsional moments at the support are compared to the ones obtained in analysis a. As can be seen, the torsional moments are typically decreased in the hollow core units with high torsional moments when the height is taken into account. However, in the hollow core units with low torsional moments, they can become larger when the height is taken into account. For hollow core unit No. 5 in analysis C1e-b0, the increase expressed as a percentage is very large (Figure 18b), however, in real values (Figure 18a), the difference is small.


15

-80

-60

-40

-20

00 0.2 0.4 0.6 0.8 1 b0 lower

b05 lowera

F n [kN/m]

z/L [-]

0

20

40

60

0 0.2 0.4 0.6 0.8 1

b0b05a

F t [kN/m]

z/L [-]

(a) (b)

-150

-50

50

150

0 0.2 0.4 0.6 0.8 1

b0 lowerb05 lowera

F l [kN/m]

z/L [-]

-500

-400

-300

-200

-100

00 0.2 0.4 0.6 0.8 1

b0b05a

M [kNm]

z/L [-]

(c) (d)

-300-200-100

0100200300

0 0.2 0.4 0.6 0.8 1

b0b05a

V [kN]

z/L [-]

-600-400-200

0200400600

0 0.2 0.4 0.6 0.8 1

b0b05a

T [kNm]

z/L [-]

(e) (f)

Figure 16 Contact forces in the joints, (a) normal forces, (b) vertical shear forces, and (c) longitudinal shear forces. The distribution of (d) bending moment, (e) shear, and (f) torsional moment in the loaded hollow core unit. Results from analyses of set-up C1, edge line load, at an applied load of Q = 70 kN/m.


16

00.10.20.30.40.50.6

1 2 3 4 5

prEN ult.prENab0b05

M mid /M mid0 [-]

Hollow core unit No. [-]

00.10.20.30.40.50.6

1 2 3 4 5

prEN ult.prENab0b05


V A /V A0 [-]

(a) (b)

Figure 17 The distribution of (a) the bending moment at mid span and (b) the shear at the supports, both evaluated at a load of Q = 70 kN/m. Set-up C1, edge line load. The numbering of the hollow core units is shown in Figure 15.

-400

-300

-200

-100

01 2 3 4 5

ab0b05

T A [kNm] Hollow core unit No. [-]

0.8

0.9

1

1.1

1.2

1 2 3 4 5

ab0b05

T A / T A (analysis a) [-]


(a) (b)

Figure 18 (a) The torsional moment at the supports at an applied load of Q = 70 kN/m. (b) The torsional moment at the supports in relation to the same one in analysis a. Set-up C1, edge line load. The numbering of the hollow core units is shown in Figure 15.


17

3.1.2 Centre line load

All results from the analyses with a centre line load can be found in Appendix C. The bending moment and shear in the loaded hollow core unit, and the torsional moment in the adjacent unit are compared in Figure 19. As can be seen, the torsional moment is rather much reduced when the height of the units is taken into account; else, the differences between the analyses are small. The distribution of the bending moment at mid span corresponds well to what is directly given in prEN 1168, see Figure 20a. Again, the bending moment is more distributed than the shear at the supports. Modelling alternative b decreased the maximum torsional moment to only 57% of the similar in analysis a when no gap was assumed, while 87% were obtained when a gap of 0.5 mm was assumed, see Figure 21b.

-300

-200

-100

00 0.2 0.4 0.6 0.8 1

ab0b05

M [kNm]

z/L [-]

-150-100-50

050

100150

0 0.2 0.4 0.6 0.8 1

ab0b05

V [kN]

z/L [-]

(a) (b)

-150-100-50

050

100150

0 0.2 0.4 0.6 0.8 1

ab0b05

T [kNm]

z/L [-]

(c)

Figure 19 The distribution of (a) bending moment in the loaded hollow core unit, (b) shear in the loaded hollow core unit, and (c) the torsional moment in the adjacent unit. Results from analyses of set-up C1, centre line load, at an applied load of Q = 70 kN/m.


18

0

0.1

0.2

0.3

0.4

1 2 3 4 5

prEN ult.prENab0b05

M mid /M mid0 [-]


0

0.1

0.2

0.3

0.4

1 2 3 4 5

prEN ult.prENab0b05


V A /V A0 [-]

(a) (b)

Figure 20 The distribution of (a) the bending moment at mid span and (b) the shear at the supports, both evaluated at a load of Q = 70 kN/m. Set-up C1, centre line load. The numbering of the hollow core units is shown in Figure 15.

-150-100

-500

50

100150

1 2 3 4 5ab0b05


0.4

0.60.8

11.2

1.4

1 2 3 4 5

ab0b05



(a) (b)

Figure 21 (a) The torsional moment at the supports at an applied load of Q = 70 kN/m. (b) The torsional moment at the supports in relation to the same one in analysis a. Set-up C1, centre line load. The numbering of the hollow core units is shown in Figure 15.


19

3.2 Set-up C2 with point load in centre The set-up C2 consists of five hollow core units, loaded with a point load in the centre of the span, see Figure 22. This set-up was modelled not only for the span 12 m, but also for spans of 4, 7 and 14 m, taking the height of the slab into account (alternative b). For the span 12 m, three analyses were carried out, denoted:

C2-a: No gap, and assuming the longitudinal joints to act as hinges.

C2-b0: No gap, and the longitudinal joints as alternative b (taking the height of the slab into account)

C2-b05: Gap 0.5 mm, and the longitudinal joints as alternative b (taking the height of the slab into account)

Point load P

1 2 3 4 5

P

L/2 = 6 m L/2 = 6 m

Figure 22 Analysed set-up C2: a floor consisting of five hollow core units loaded with a point load in the centre of the span.

In Figure 23, some results from the analyses where the joint was modelled taking the height of the slab into account (alternative b) with varying spans are shown. In Figure 23a, the transverse distribution of the bending moment at mid-span in the analyses is compared with curves given in CEN/TC229 (2004). As can be seen, the analyses show less distribution of the bending moment at mid span than is given directly by the prEN. It is important to note that the transverse distribution of the shear at the supports was not the same as the transverse distribution of the bending moment at mid-span; compare Figures 23a and 23b. The shear at the supports was more distributed among the hollow core units than the bending moment. Note, however, that the maximum shear force in the loaded hollow core unit was not at the support: as part of the shear force is transferred to the neighbouring hollow core units, the maximum shear force in the loaded hollow core unit was at the position of the point load; see Figure 24b.


20

0

10

20

30

40

50

0 5 10 15

Distribution of bending moment at mid span [%]

Span [m]

α1

α2

α3

0

10

20

30

40

50

0 5 10 15

Distribution of shear at the supports [%]

Span [m]

α1

α2α3

Figure 23 Results from analyses (unfilled markers) of set-up C2 are compared with curves given in CEN/TC229 (2004) (filled markers). (a) Transverse distribution of bending moment at mid-span; (b) Transverse distribution of shear at the supports. The subscripts indicate the number of the hollow core unit; see Figure 22.

For the span 12 m, some more results are shown in Figures 24-26. As can be seen, the assumption of the behaviour of the longitudinal joint affected the torsional moment, which became less when the height of the slab was taken into account.

-400

-300

-200

-100

00 0.2 0.4 0.6 0.8 1

ab0b05

M [kNm]

z/L [-]

-200

-100

0

100

200

0 0.2 0.4 0.6 0.8 1

ab0b05

V [kN]

z/L [-]

(a) (b)

-100

-50

0

50

100

0 0.2 0.4 0.6 0.8 1

ab0b05

T [kNm]

z/L [-]

(c)

Figure 24 The distribution of (a) bending moment in the loaded hollow core unit, (b) shear in the loaded hollow core unit, and (c) the torsional moment in the adjacent unit. Results from analyses of set-up C2, point load in mid span, at an applied load of P = 400 kN.


21

00.050.1

0.150.2

0.250.3

1 2 3 4 5

prEN ult.prENab0b05

M mid /M mid0 [-]


00.050.1

0.150.2

0.250.3

1 2 3 4 5

ab0b05prEN ult.prEN


V A /V A0 [-]

(a) (b)

Figure 25 The distribution of (a) the bending moment at mid span and (b) the shear at the supports, both evaluated at a load of P = 400 kN. Set-up C2, point load in mid span. The numbering of the hollow core units is shown in Figure 22.

-100

-50

0

50

100

1 2 3 4 5ab0b05


0.4

0.60.8

11.2

1.4

1 2 3 4 5

ab0b05



(a) (b)

Figure 26 (a) The torsional moment at the supports at an applied load of P = 400 kN. (b) The torsional moment at the supports in relation to the same one in analysis a. Set-up C2, point load in mid span. The numbering of the hollow core units is shown in Figure 22.


22

3.3 Set-up C3 with point load at edge The set-up C3 consists of five hollow core units, loaded with a point load at the edge, see Figure 27. Three analyses were carried out for this set-up, denoted:




Point load P

1 2 3 4 5P

L/2 = 6 m L/2 = 6 m

Figure 27 Analysed set-up C3: a floor consisting of five hollow core units loaded with a point load at the edge.

The bending moment, shear and torsional moment in the loaded hollow core unit are compared in Figure 28. While the bending moment became almost the same in the different analyses, there were small differences in the shear and the torsional moment; the shear was slightly smaller in analysis a while the torsional moment was slightly larger. In Figure 29a, the distribution of the bending moment is compared to what is given in prEN; as can be seen, the bending moment was less distributed in the analyses than directly given in prEN. The shear at the supports was approximately equally distributed as the bending moment at mid span in the analyses using alternative b; compare Figures 29a and b. When comparing the different modelling techniques, the torsional moment was reduced in alternative b in the hollow core units that have the highest torsional moments, and increased in the one that had only low torsional moment (Figure 30).


23

-600

-400

-200

00 0.2 0.4 0.6 0.8 1

b0b05a

M [kNm]

z/L [-]

-200

-100

0

100

200

0 0.2 0.4 0.6 0.8 1

b0b05a

V [kN]

z/L [-]

(a) (b)

-250

-150

-50

50

150

250

0 0.2 0.4 0.6 0.8 1

b0b05a

T [kNm]

z/L [-]

(c)

Figure 28 The distribution of (a) bending moment, (b) shear, and (c) torsional moment in the loaded hollow core unit. Results from analyses of set-up C3, point load at edge, at an applied load of P = 400 kN.

00.10.2

0.30.40.5

1 2 3 4 5

prEN ult.prENab0b05

M mid /M mid0 [-]


00.1

0.20.3

0.40.5

1 2 3 4 5

prEN ult.prENab0b05


V A /V A0 [-]

(a) (b)

Figure 29 The distribution of (a) the bending moment at mid span and (b) the shear at the supports, both evaluated at a load of P = 400 kN. Set-up C3, point load at edge. The numbering of the hollow core units is shown in Figure 27.


24

-250

-200

-150

-100

-50

01 2 3 4 5

ab0b05


0.8

0.9

1

1.1

1.2

1 2 3 4 5

ab0b05



(a) (b)

Figure 30 (a) The torsional moment at the supports at an applied load of P = 400 kN. (b) The torsional moment at the supports in relation to the same one in analysis a. Set-up C3, point load at edge. The numbering of the hollow core units is shown in Figure 27.

3.4 Set-up C5 with three supported edges and line load The set-up C5 consists of five hollow core units, loaded with a line load. In CEN/TC229 (2004), the reaction force from several placements of the load are shown. Here, one placement of the load was chosen, at the centre of the floor, see Figure 31. Three analyses were carried out for this set-up, denoted:




Line load Q

Q

1 2 3 4 5

L = 12 m

Figure 31 Analysed set-up C5: a floor consisting of five hollow core units with three supported edges loaded with a line load in the centre.


25

The bending moment and shear in the loaded hollow core unit and torsional moment in the adjacent unit, and also the reaction force along the support lines are compared in Figure 32. As can be seen, there were some differences between the different modelling techniques; of approximately equal size for both the bending moment, shear force and torsional moment. In Table 3, the reaction force at the third line support as a part of the total applied load is tabulated, and compared to what is given in prEN. As can be seen, the reaction force is larger in prEN than was obtained in these analyses.

-200

-150

-100

-50

00 0.2 0.4 0.6 0.8 1

ab0b05

M [kNm]

z/L [-]

-125

-75

-25

25

75

125

0 0.2 0.4 0.6 0.8 1

ab0b05

V [kN]

z/L [-]

(a) (b)

-300-200-100

0100200300

0 0.2 0.4 0.6 0.8 1

ab0b05

T [kNm]

z/L [-]

01020304050

0 0.2 0.4 0.6 0.8 1

ab0b05

R y [kN/m]

z/L [-] (c) (d)

-1000

-500

0

500

1000

0 1 2 3 4 5 6

ab0b05

R y [kN/m]

x [m]

(e)

Figure 32 The distribution of (a) bending moment in the loaded hollow core unit, (b) shear in the loaded hollow core unit, and (c) the torsional moment in the adjacent unit. The distribution of the reaction force (d) along the third support line, and (e) along one of the main support lines (assuming linear distribution at each hollow core unit). Results from analyses of set-up C5, three supported edges and line load, at a load of Q = 70 kN/m.


26

Table 3 Amount of applied load carried by the third line support for set-up C5 with a line load.

CEN/TC229 (2004)

Analysis C5-b0

Analysis C5-b05 1

Analysis C5-a

Amount of loading [%]

64 43 42 39

1) Evaluated at a load of Q = 70 kN/m

The distribution of bending moment at mid span and shear at the supports is shown in Figure 33. Note that the sum is not equal to 100%; as some part of the load is carried by the third line support. The total bending moment was therefore reduced to around 50% of what would be obtained if the third support line was not present. Modelling technique b gave results with negative reaction force in the hollow core unit closest to the third support line; i.e. lifting of the slab. This was not the case in the analyses where modelling technique a was used; however, when the effect of the torsional moment was included, the total effect was lifting of the corners, and the difference between the different modelling techniques was small, see Figure 32e. The torsional moments at the supports are shown in Figure 34. As can be seen, large torsional moments were obtained, compare Figures 34a and 21a, which are for similar load set-ups except for the third line support.

-0.1

0

0.1

0.2

0.3

1 2 3 4 5

ab0b05

M mid /M mid0 [-]

Hollow core unit No. [-] -0.1

0

0.1

0.2

0.3

1 2 3 4 5

ab0b05


V A /V A0 [-]

(a) (b)

Figure 33 The distribution of (a) the bending moment at mid span and (b) the shear at the supports, both evaluated at a load of Q = 70 kN/m. Set-up C5, three supported edges and line load. The numbering of the hollow core units is shown in Figure 31. Note that the sum is not equal to 100 %; as some load is carried by the third support line, see Table 3.


27

-100

0

100

200

300

1 2 3 4 5

ab0b05

T A [kNm]


00.20.40.60.8

11.21.4

1 2 3 4 5

ab0b05



(a) (b)

Figure 34 (a) The torsional moment at the supports at an applied load of Q = 70 kN/m. (b) The torsional moment at the supports in relation to the same one in analysis a. Set-up C5, three supported edges and line load. The numbering of the hollow core units is shown in Figure 31.

3.5 Set-up C6 with three supported edges and point load at mid span

The set-up C6 consists of five hollow core units, loaded with a point load. In CEN/TC229 (2004), the reaction force from several placements of the load are shown. Here, one placement of the load was chosen, at the centre of the floor, see Figure 35. Three analyses were carried out for this set-up, denoted:




Point load P

1 2 3 4 5

P

L/2 = 6 m L/2 = 6 m

Figure 35 Analysed set-up C6: a floor consisting of five hollow core units with three supported edges loaded with a point load in the centre.


28

The bending moment and shear in the loaded hollow core unit and torsional moment in the adjacent unit, and also the reaction force along the support lines are compared in Figure 36. Small differences were obtained in the different analyses. The reaction force carried by the third line support is shown in Table 4, and compared to what is given in prEN. There is most likely something wrong with the values that can be read from a diagram in prEN. The scale of the diagram (Figure C.6 in CEN/TC229 (2004)) appears to be wrong, as more than 100% of the applied load is carried by the third line support for several load cases. It was therefore assumed that the whole load is carried by the third support line when the load is applied directly on the support, and the scale was adjusted to fit this. The second value shown in Table 4 was then obtained, which corresponds rather well with the values obtained in the analyses here.

-300

-200

-100

00 0.2 0.4 0.6 0.8 1

ab0b05

M [kNm]

z/L [-]

-200

-100

0

100

200

0 0.2 0.4 0.6 0.8 1

ab0b05

V [kN]

z/L [-]

(a) (b)

-200

-100

0

100

200

0 0.2 0.4 0.6 0.8 1

ab0b05

T [kNm]

z/L [-]

01020304050

0 0.2 0.4 0.6 0.8 1

ab0b05

R y [kN/m]

z/L [-] (c) (d)

-600-400-200

0

200400600

0 1 2 3 4 5 6

ab0b05

R y [kN/m]

x [m]

(e)

Figure 36 The distribution of (a) bending moment in the loaded hollow core unit, (b) shear in the loaded hollow core unit, and (c) the torsional moment in the adjacent unit. The distribution of the reaction force (d) along the third support line, and (e) along one of the main support lines (assuming linear distribution at each hollow core unit). Results from analyses of set-up C6, three supported edges and point load, at a load of P = 400 kN.


29

Table 4 Amount of applied load carried by the third line support for set-up C6 with a point load.

CEN/TC229 (2004)

Analysis C6-b0

Analysis C6-b05 2

Analysis C6-a

Amount of loading [%]

97 / 66 1 63 61 57

1) The scale in this figure (Figure C.6 in CEN/TC229 (2004)) appears to be wrong. If it is assumed that the whole load is carried by the third support line when the load is applied directly on the support, the second value is obtained.

2) Evaluated at a load of P = 400 kN

The distribution of bending moment and shear at the supports are shown in Figure 37. Again, note that the sum is not equal to 100%; as some part of the load is carried by the third line support. The total bending moment was therefore reduced to around 60% of what would be obtained if the third support line was not present. Modelling technique b gave results with negative reaction force in the hollow core unit closest to the third support line; i.e. lifting of the slab. This was not the case in the analyses where modelling technique a was used; however, when the effect of the torsional moment was included, the total effect was lifting of the corners, and the difference between the different modelling techniques was small, see Figure 36e. The torsional moments at the supports are shown in Figure 38. As can be seen, large torsional moments were obtained, compare Figures 38a and 26a, which are for similar load set-ups except for the third line support.

-0.1

0

0.1

0.2

0.3

1 2 3 4 5

ab0b05

M mid /M mid0 [-]

Hollow core unit No. [-] -0.1

0

0.1

0.2

0.3

1 2 3 4 5

ab0b05


V A /V A0 [-]

(a) (b)

Figure 37 The distribution of (a) the bending moment at mid span and (b) the shear at the supports, both evaluated at a load of P = 400 kN. Set-up C6, three supported edges and point load. The numbering of the hollow core units is shown in Figure 35. Note that the sum is not equal to 100 %; as some load is carried by the third support line, see Table 4.


30

-100-50

050

100

150200

1 2 3 4 5

ab0b05

T A [kNm]

Hollow core unit No. [-]0

0.20.40.60.8

11.2

1 2 3 4 5

ab0b05



(a) (b)

Figure 38 (a) The torsional moment at the supports at an applied load of P = 400 kN. (b) The torsional moment at the supports in relation to the same one in analysis a. Set-up C6, three supported edges and point load. The numbering of the hollow core units is shown in Figure 35.


31

4 Comparison of results with different modelling techniques

In Tables 5 and 6, cross-sectional moments and shear forces are compared to the corresponding ones in analysis alternative a. This evaluation was done for the same applied load as in earlier presented results. For the shear force and the torsional moment, both the ones at the supports and the maximum obtained are compared – for some set-ups they were the same, and for others not. The maximum bending moment was in all the studied set-ups in mid span. As can be seen, when modelling technique b was used, the torsional moment decreased, especially when no gap was assumed. The reduced torsional moment varied from 49 to 93% when there was no gap, and 76 to 95% when the assumed gap was 0.5 mm. One exception was the analysis with horizontal supports and no gap; there the torsional moment became as small as 3 or 26% of the corresponding one in analysis a, depending on if it was the torsional moment at the support or the maximum torsional moment which was studied.

Table 5 Results from analyses with alternative b, no gap, compared to results from analyses with alternative a.

Analysis TA/TA (analysis a) Tmax/Tmax(analysis a) VA/VA (analysis a) Vmax/Vmax (analysis a) Mmid/Mmid(analysis a)

6s-b0 0.03 0.26 1.29 0.94 0.96

6-b0 0.49 0.56 1.32 0.98 0.96

4-b0 0.56 0.69 1.16 1.00 0.98

2-b0 0.80 0.93 1.00 1.00 1.00

C1e-b0 0.87 0.87 1.19 1.19 0.95

C1c-b0 0.57 0.57 1.03 1.03 0.97

C2-b0 0.56 0.65 1.10 1.00 0.96

C3-b0 0.86 0.89 1.45 1.00 0.95

C5-b0 0.81 0.81 1.09 1.09 0.92

C6-b0 0.89 0.89 1.04 1.01 0.91

Average1 0.71 0.76 1.15 1.03 0.95

Stand. dev. 2

0.16 0.14 0.14 0.06 0.03

1) Average except analysis 6s-b0.

2) Standard deviation except analysis 6s-b0.


32

Also the bending moment was reduced, in most of the studied set-ups, when modelling technique b was used compared to modelling technique a. This reduction was rather small. On the other hand, the shear force increased when modelling technique b was used compared to modelling technique a, especially the shear force at the supports.

Table 6 Results from analyses with alternative b, 0.5 mm gap, compared to results from analyses with alternative a.

Analysis TA/TA (analysis a) Tmax/Tmax(analysis a) VA/VA (analysis a) Vmax/Vmax (analysis a) Mmid/Mmid(analysis a)

6s-b05 0.72 0.72 1.30 1.00 0.86

6-b05 0.76 0.76 1.18 1.01 0.95

4-b05 0.79 0.79 1.12 1.00 0.98

2-b05 0.95 0.95 1.00 1.00 1.00

C1e-b05 0.94 0.94 1.23 1.23 0.95

C1c-b05 0.87 0.87 1.14 1.14 0.97

C2-b05 0.86 0.86 1.25 1.00 0.97

C3-b05 0.92 0.92 1.51 1.01 0.96

C5-b05 0.88 0.88 1.22 1.22 0.92

C6-b05 0.86 0.86 1.28 1.01 0.93

Average1 0.87 0.87 1.21 1.07 0.96

Stand. dev. 2

0.06 0.06 0.13 0.09 0.02

1) Average except analysis 6s-b05.

2) Standard deviation except analysis 6s-b05.


33

5 Conclusions Some conclusions can be been drawn from the work:

The maximum torsional moment was reduced when the height of the hollow core units was included in the analyses, compared to when hinges were assumed between the hollow core units. In the investigated set-ups (without horizontal restraint), the torsional moment varied from 49 to 93% when there was no gap, and 76 to 95% when the assumed gap was 0.5 mm.

•

•

•

•

•

•

The largest reduction of the torsional moment was obtained in a set-up with horizontal restraints at the edges of the floor; in that case the torsional moment became almost zero. However, the basic assumptions in the model can be discussed in this case: for these high compressive forces, the assumption about negligible shear deformations in the cross-section (i.e. stiff rotation) of the hollow core units is questionable. It is also worth noting that rigid supports are to be avoided for other reasons, as they will cause restraining forces due to for example shrinkage or temperature load.

Also the bending moment was reduced, in most of the studied set-ups, when modelling technique b was used compared to modelling technique a. This reduction was rather small. On the other hand, the shear force increased when modelling technique b was used compared to modelling technique a, especially the shear force at the supports.

In the analyses of floors with load on the longitudinal joint in the centre of the floor, the torsional moment was rather close to what was obtained in a single element, when the height of the units was included. The torsional moment was a balance between two counteracting effects: it was increased compared to the single unit case due to the transfer of shear to the adjacent unit, but on the other hand decreased due to the blocking effect.

In four of the studied set-ups, a direct comparison to the present code, CEN/TC229 (2004), regarding the load distribution could be done. In CEN/TC229 (2004), it is not clear if the given distributions are valid for the bending moments at mid span or the shear forces at the supports. Therefore, the distributions given in the code were compared to both types of results. No clear answer to which of these results that were considered in the code could be found. In three of the four set-ups, the bending moment at mid span was less distributed in the analyses than what is given directly by the current code. The fourth set-up corresponded well. It should be noted that for design in the ultimate limit state, the bending moment in the loaded unit shall according to the code be increased with a factor 1.25, and decreased for the other units. When the distribution factors given in the code were adjusted in this way, the agreement was rather good for the three set-ups that else showed a difference.

In two of the studied set-ups, a direct comparison to the present code, CEN/TC229 (2004), regarding the amount of reaction force carried by the third line support could be done. In both of them, the reaction force in the code was larger than resulting from the analyses. However, for one of these set-ups,


34

the scale in the diagram in the code is questionable. When a reasonable assumption about that was done, the reaction force at the third line support corresponded well with the results for that set-up.

In all of the studied set-ups, line loads resulted in that the bending moment at mid span was more distributed than the shear at the supports. Point loads, on the other hand, resulted in that the shear at the supports was more distributed than the bending moment at mid span.

•


35

6 References CEN/TC229 (2004): Precast concrete products - Hollow core slabs. Final draft,

European Standard prEN 1168, 2004.

Lundgren K., Broo H. and Engström B. (2004): Analyses of hollow core floors subjected to shear and torsion. Accepted for publication in Structural Concrete in August 2004.

Lundgren K. and Plos M. (2004): Analyses of two connected hollow core units. 04:04, Department of Structural Engineering and Mechanics, Concrete Structures, Chalmers University of Technology, Göteborg, Sweden.

Pajari M. (2003): Pure torsion tests on single slab units. RTE50-IR-25/2002, Technical Research Centre of Finland, VTT Building and Transport.

TNO (2002): DIANA Finite Element Analysis, User's Manual release 8.1, TNO Building and Construction Research, 2002.


36

Appendix A. Effect of varying shear stiffness (D22) in interface simulating the longitudinal joint

0

0.5

1

1.5

20 0.2 0.4 0.6 0.8 1

1-22-33-44-5

F n [kN/m] z/L [-]

0

20

40

60

800 0.2 0.4 0.6 0.8 1

1-22-33-44-5

F t z/L [-]

(a) (b)

-500-400-300-200-100

0100

0 0.2 0.4 0.6 0.8 1

12345

M [kNm]z/L [-]

-200

-100

0

100

200

0 0.2 0.4 0.6 0.8 1

12345

V [kN]

z/L [-]

(c) (d)

-400

-200

0

200

400

0 0.2 0.4 0.6 0.8 1

12345

T [kNm]

z/L [-] -6

-4

-2

0

2

4

0 20 40 60 80 100

12345

Q [kN/m]

F [kN]

(e) (f)

Figure A1 Results from analysis C1e-a, Q = 70 kN/m, D22 = 3·1010. Contact forces in the joints, (a) normal forces, and (b) vertical shear forces. Distribution of (c) bending moment, (d) shear force, and (e) torsional moment. (f) The force from one tie beam to each hollow core unit versus the applied load. The notations refer to the number of the hollow core unit; see Figure 15.


A1

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

1-22-33-44-5

F n [kN/m]

z/L [-]

0

20

40

60

80

0 0.2 0.4 0.6 0.8 1

1-22-33-44-5

F t [kN/m]

z/L [-]

(a) (b)

-500-400-300-200-100

0100

0 0.2 0.4 0.6 0.8 1

12345

M [kNm]z/L [-]

-200

-100

0

100

200

0 0.2 0.4 0.6 0.8 1

12345

V [kN]

z/L [-]

(c) (d)

-400

-200

0

200

400

0 0.2 0.4 0.6 0.8 1

12345

T [kNm]

z/L [-] -6

-4

-2

0

2

4

0 20 40 60 80 100

12345

Q [kN/m]

F [kN]

(e) (f)



A2

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

1-22-33-44-5

F n [kN/m]

z/L [-]

0102030405060

0 0.2 0.4 0.6 0.8 1

1-22-33-44-5

F t [kN/m]

z/L [-]

(a) (b)

-500

-400

-300

-200

-100

00 0.2 0.4 0.6 0.8 1 1

2345

M [kNm]z/L [-]

-200

-100

0

100

200

0 0.2 0.4 0.6 0.8 1

12345

V [kN]

z/L [-]

(c) (d)

-400

-200

0

200

400

0 0.2 0.4 0.6 0.8 1

12345

T [kNm]

z/L [-] -6

-4

-2

0

2

4

0 20 40 60 80 100

12345

Q [kN/m]

F [kN]

(e) (f)



A3

Appendix B. Results from analyses with point load in the middle of the floor, on the longitudinal joint

-600-500-400-300-200-100

00 0.2 0.4 0.6 0.8 1 2-3 upper

2-3 lower1-2 upper1-2 lower1-1 upper1-1 lower

F n [kN/m] z/L [-]

-50-40-30-20-10

00 0.2 0.4 0.6 0.8 1

2-3 total1-2 total1-1 total

F t [kN/m] z/L [-]

(a) (b)

-300

-200

-100

0

100

0 0.2 0.4 0.6 0.8 1123

M [kNm]

z/L [-]

-100

-50

0

50

100

0 0.2 0.4 0.6 0.8 1

123

V [kN]

z/L [-]

(c) (d)

-40

-20

0

20

40

0 0.2 0.4 0.6 0.8 1

123

T [kNm]

z/L [-]

(e)

Figure B1 Results from analysis 6s-b0, P = 200 kN. Contact forces in the joints, (a) normal forces, and (b) vertical shear forces. Distribution of (c) bending moment, (d) shear force, and (e) torsional moment. The notations refer to the hollow core unit, see Figure 8.


B1

-250-200-150-100-50

00 0.2 0.4 0.6 0.8 1 2-3 upper


F n [kN/m] z/L [-]

-40

-30

-20

-10

00 0.2 0.4 0.6 0.8 1


F t [kN/m] z/L [-]

(a) (b)

-300

-200

-100

0

100

0 0.2 0.4 0.6 0.8 1123

M [kNm]

z/L [-]

-100

-50

0

50

100

0 0.2 0.4 0.6 0.8 1

123

V [kN]

z/L [-]

(c) (d)

-100

-50

0

50

100

0 0.2 0.4 0.6 0.8 1

123

T [kNm]

z/L [-]

(e)

Figure B2 Results from analysis 6s-b05, P = 200 kN. Contact forces in the joints, (a) normal forces, and (b) vertical shear forces. Distribution of (c) bending moment, (d) shear force, and (e) torsional moment. The notations refer to the hollow core unit, see Figure 8.


B2

-3

-2

-1

0

1

0 0.2 0.4 0.6 0.8 12-31-21-1

F n [kN/m]

z/L [-]

-40

-30

-20

-10

00 0.2 0.4 0.6 0.8 1

2-31-21-1

F t [kN/m] z/L [-]

(a) (b)

-300

-200

-100

0

100

0 0.2 0.4 0.6 0.8 1123

M [kNm]

z/L [-]

-100

-50

0

50

100

0 0.2 0.4 0.6 0.8 1

123

V [kN]

z/L [-]

(c) (d)

-100

-50

0

50

100

0 0.2 0.4 0.6 0.8 1

123

T [kNm]

z/L [-]

(e)

Figure B3 Results from analysis 6s-a, P = 200 kN. Contact forces in the joints, (a) normal forces, and (b) vertical shear forces. Distribution of (c) bending moment, (d) shear force, and (e) torsional moment. The notations refer to the hollow core unit, see Figure 8.


B3

-300

-200

-100

00 0.2 0.4 0.6 0.8 1 2-3 upper


F n [kN/m] z/L [-]

-40

-30-20

-10

00 0.2 0.4 0.6 0.8 1


F t [kN/m] z/L [-]

(a) (b)

-300

-200

-100

0

100

0 0.2 0.4 0.6 0.8 1123

M [kNm]

z/L [-]

-100

-50

0

50

100

0 0.2 0.4 0.6 0.8 1

123

V [kN]

z/L [-]

(c) (d)

-60-40-20

0204060

0 0.2 0.4 0.6 0.8 1

123

T [kNm]

z/L [-]

(e)

Figure B4 Results from analysis 6-b0, P = 200 kN. Contact forces in the joints, (a) normal forces, and (b) vertical shear forces. Distribution of (c) bending moment, (d) shear force, and (e) torsional moment. The notations refer to the hollow core unit, see Figure 8.


B4

-250-200-150-100-50

00 0.2 0.4 0.6 0.8 1 2-3 upper


F n [kN/m] z/L [-]

-40

-30

-20

-10

00 0.2 0.4 0.6 0.8 1


F t [kN/m] z/L [-]

(a) (b)

-300

-200

-100

0

100

0 0.2 0.4 0.6 0.8 1123

M [kNm]

z/L [-]

-100

-50

0

50

100

0 0.2 0.4 0.6 0.8 1

123

V [kN]

z/L [-]

(c) (d)

-100

-50

0

50

100

0 0.2 0.4 0.6 0.8 1

123

T [kNm]

z/L [-]

(e)



B5

-4

-3

-2

-1

0

1

0 0.2 0.4 0.6 0.8 12-31-21-1

F n [kN/m]

z/L [-]

-40

-30

-20

-10

00 0.2 0.4 0.6 0.8 1

2-31-21-1

F t [kN/m] z/L [-]

(a) (b)

-300

-200

-100

0

100

0 0.2 0.4 0.6 0.8 1123

M [kNm]

z/L [-]

-100

-50

0

50

100

0 0.2 0.4 0.6 0.8 1

123

V [kN]

z/L [-]

(c) (d)

-100

-50

0

50

100

0 0.2 0.4 0.6 0.8 1

123

T [kNm]

z/L [-]

(e)

Figure B6 Results from analysis 6-a, P = 200 kN. Contact forces in the joints, (a) normal forces, and (b) vertical shear forces. Distribution of (c) bending moment, (d) shear force, and (e) torsional moment. The notations refer to the hollow core unit, see Figure 8.


B6

-250-200-150-100-50

00 0.2 0.4 0.6 0.8 1


F n [kN/m] z/L [-]

-30

-20

-10

00 0.2 0.4 0.6 0.8 1

1-2 total1-1 total

F t [kN/m] z/L [-]

(a) (b)

-400

-300

-200

-100

0

100

0 0.2 0.4 0.6 0.8 1 12

M [kNm]z/L [-]

-100

-50

0

50

100

0 0.2 0.4 0.6 0.8 1

12

V [kN]

z/L [-]

(c) (d)

-80-60-40-20

020406080

0 0.2 0.4 0.6 0.8 1

12

T [kNm]

z/L [-]

(e)



B7

-250-200-150-100-50

00 0.2 0.4 0.6 0.8 1


F n [kN/m] z/L [-]

-30

-20

-10

00 0.2 0.4 0.6 0.8 1

1-2 total1-1 total

F t [kN/m] z/L [-]

(a) (b)

-400

-300

-200

-100

0

100

0 0.2 0.4 0.6 0.8 1 12

M [kNm]z/L [-]

-100

-50

0

50

100

0 0.2 0.4 0.6 0.8 1

12

V [kN]

z/L [-]

(c) (d)

-100

-50

0

50

100

0 0.2 0.4 0.6 0.8 1

12

T [kNm]

z/L [-]

(e)



B8

-4

-3

-2

-1

0

1

0 0.2 0.4 0.6 0.8 1 1-21-1

F n [kN/m]

z/L [-]

-40

-30

-20

-10

00 0.2 0.4 0.6 0.8 1

1-21-1

F t [kN/m] z/L [-]

(a) (b)

-400

-300

-200

-100

0

100

0 0.2 0.4 0.6 0.8 1 12

M [kNm]

z/L [-]

-100

-50

0

50

100

0 0.2 0.4 0.6 0.8 1

12

V [kN]

z/L [-]

(c) (d)

-100

-50

0

50

100

0 0.2 0.4 0.6 0.8 1

12

T [kNm]

z/L [-]

(e)

Figure B9 Results from analysis 4-a, P = 200 kN. Contact forces in the joints, (a) normal forces, and (b) vertical shear forces. Distribution of (c) bending moment, (d) shear force, and (e) torsional moment. The notations refer to the hollow core unit, see Figure 8.


B9

-150

-100

-50

00 0.2 0.4 0.6 0.8 1

1-1 upper1-1 lower

F n [kN/m] z/L [-]

-600

-400

-200

00 0.2 0.4 0.6 0.8 1

1

M [kNm]

z/L [-]

(a) (b)

-120

-20

80

0 0.2 0.4 0.6 0.8 11

V [kN]

z/L [-]

-60-40-20

0204060

0 0.2 0.4 0.6 0.8 11

T [kNm]

z/L [-]

(c) (d)

Figure B10 Results from analysis 2-b0, P = 200 kN. (a) Normal contact forces in the joints. Distribution of (b) bending moment, (c) shear force, and (d) torsional moment. The notations refer to the hollow core unit, see Figure 8.

-100-80-60-40-20

00 0.2 0.4 0.6 0.8 1

1-1 upper1-1 lower

F n [kN/m] z/L [-]

-600

-400

-200

00 0.2 0.4 0.6 0.8 1

1

M [kNm]

z/L [-]

(a) (b)

-120

-20

80

0 0.2 0.4 0.6 0.8 11

V [kN]

z/L [-] -60-40-20

0204060

0 0.2 0.4 0.6 0.8 11

T [kNm]

z/L [-]

(c) (d)

Figure B11 Results from analysis 2-b05, P = 200 kN. (a) Normal contact forces in the joints. Distribution of (b) bending moment, (c) shear force, and (d) torsional moment. The notations refer to the hollow core unit, see Figure 8.


B10

-5

-3

-1

1

0 0.2 0.4 0.6 0.8 11-1

F n [kN/m]

z/L [-]

-600

-400

-200

00 0.2 0.4 0.6 0.8 1

1

M [kNm]

z/L [-]

(a) (b)

-125-100

-75-50-25

0255075

100125

0 0.2 0.4 0.6 0.8 11

V [kN]

z/L [-]-80-60-40-20

020406080

0 0.2 0.4 0.6 0.8 11

T [kNm]

z/L [-]

(c) (d)

Figure B12 Results from analysis 2-a, P = 200 kN. (a) Normal contact forces in the joints. Distribution of (b) bending moment, (c) shear force, and (d) torsional moment. The notations refer to the hollow core unit, see Figure 8.


B11

Appendix C. Results from analyses of set-up C1 in prEN, with line loads

1 2 3 4 5

Centre line load Q

L = 12 m

Edge line load Q or

Q

Figure C1 Analysed set-up C1: a floor consisting of five hollow core units loaded with a line load, either on the edge or in the centre.

Table C1 The bending moment at mid span, in kNm. Set-up C1, edge line load Q = 70 kN/m. The notations refer to the number of the hollow core unit; see Figure C1.


Analysis C1e-b0

Analysis C1e-b05

Analysis C1e-a

1 372 373 393

2 289 289 295

3 232 230 226

4 194 193 183

5 174 175 163

Table C2 The reaction force at the supports, in kN. Set-up C1, edge line load Q = 70 kN/m. The notations refer to the number of the hollow core unit; see Figure C1.


Analysis C1e-b0

Analysis C1e-b05

Analysis C1e-a

1 206 212 173

2 77 71 89

3 56 53 63

4 45 43 49

5 33 38 43


C1

Table C3 The torsional moment at support, in kNm. Set-up C1, edge line load Q = 70 kN/m. The notations refer to the number of the hollow core unit; see Figure C1.


Analysis C1e-b0

Analysis C1e-b05

Analysis C1e-a

1 -345 -371 -396

2 -200 -204 -239

3 -132 -130 -148

4 -81 -71 -81

5 -42 -23 -26

Table C4 The bending moment at mid span, in kNm. Set-up C1, centre line load Q = 70 kN/m. The notations refer to the number of the hollow core unit; see Figure C1.


Analysis C1e-b0

Analysis C1e-b05

Analysis C1e-a

1 238 237 233

2 257 258 259

3 270 271 278

4 257 258 259

5 238 237 233

Table C5 The reaction force at the supports, in kN. Set-up C1, centre line load Q = 70 kN/m. The notations refer to the number of the hollow core unit; see Figure C1.


Analysis C1e-b0

Analysis C1e-b05

Analysis C1e-a

1 60 55 66

2 87 86 84

3 122 135 118

4 87 86 84

5 60 55 66


C2

Table C6 The torsional moment at support, in kNm. Set-up C1, centre line load Q = 70 kN/m. The notations refer to the number of the hollow core unit; see Figure C1.


Analysis C1e-b0

Analysis C1e-b05

Analysis C1e-a

1 54 35 39

2 73 112 129

3 0 0 0

4 -73 -112 -129

5 -54 -35 -39


C3

-80

-60

-40

-20

00 0.2 0.4 0.6 0.8 1

1-2 upper1-2 lower2-3 upper2-3 lower3-4 upper3-4 lower4-5 upper4-5 lower

F n [kN/m] z/L [-]

-100

1020

304050

0 0.2 0.4 0.6 0.8 1

1-2 total2-3 total3-4 total4-5 total

F t [kN/m]

z/L [-] (a) (b)

-150-100-50

050

100150

0 0.2 0.4 0.6 0.8 1


F l [kN/m]

z/L [-]-400

-300

-200

-100

0

100

0 0.2 0.4 0.6 0.8 1

12345

M [kNm]

z/L [-]

(c) (d)

-300-200-100

0100200300

0 0.2 0.4 0.6 0.8 1

12345

V [kN]

z/L [-]-400

-200

0

200

400

0 0.2 0.4 0.6 0.8 1

12345

T [kNm]

z/L [-]

(e) (f)

-200

-100

0

100

200

300

0 20 40 60 80 100

12345

Q [kN/m]

F [kN]

(g)

Figure C2 Results from analysis C1e-b0, Q = 70 kN/m. Contact forces in the joints, (a) normal forces, (b) vertical shear forces, and (c) longitudinal shear forces. Distribution of (d) bending moment, (e) shear force, and (f) torsional moment. (g) The force from one tie beam to each hollow core unit versus the applied load. The notations refer to the number of the hollow core unit; see Figure C1.


C4

-12-10-8-6-4-20

0 0.2 0.4 0.6 0.8 11-2 upper1-2 lower2-3 upper2-3 lower3-4 upper3-4 lower4-5 upper4-5 lower

F n [kN/m] z/L [-]

0

10

20

30

40

50

0 0.2 0.4 0.6 0.8 1


F t [kN/m]

z/L [-]

(a) (b)

-150-100-50

050

100150

0 0.2 0.4 0.6 0.8 1


F l [kN/m]

z/L [-]

-400

-300

-200

-100

0

100

0 0.2 0.4 0.6 0.8 1

12345

M [kNm]

z/L [-]

(c) (d)

-300-200-100

0100200300

0 0.2 0.4 0.6 0.8 1

12345

V [kN]

z/L [-]-400

-200

0

200

400

0 0.2 0.4 0.6 0.8 1

12345

T [kNm]

z/L [-]

(e) (f)

-20-10

010203040

0 20 40 60 80 100

12345

Q [kN/m]

F [kN]

(g)

Figure C3 Results from analysis C1e-b05, Q = 70 kN/m. Contact forces in the joints, (a) normal forces, (b) vertical shear forces, and (c) longitudinal shear forces. Distribution of (d) bending moment, (e) shear force, and (f) torsional moment. (g) The force from one tie beam to each hollow core unit versus the applied load. The notations refer to the number of the hollow core unit; see Figure C1.


C5

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

1-22-33-44-5

F n [kN/m]

z/L [-]

0102030405060

0 0.2 0.4 0.6 0.8 1

1-22-33-44-5

F t [kN/m]

z/L [-]

(a) (b)

-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1

1-22-33-44-5

F l [kN/m]

z/L [-]-500

-400

-300

-200

-100

00 0.2 0.4 0.6 0.8 1 1

2345

M [kNm]z/L [-]

(c) (d)

-200

-100

0

100

200

0 0.2 0.4 0.6 0.8 1

12345

V [kN]

z/L [-]

-400

-200

0

200

400

0 0.2 0.4 0.6 0.8 1

12345

T [kNm]

z/L [-]

(e) (f)

-6

-4

-2

0

2

4

0 20 40 60 80 100

12345

Q [kN/m]

F [kN]

(g)

Figure C4 Results from analysis C1e-a, Q = 70 kN/m. Contact forces in the joints, (a) normal forces, (b) vertical shear forces, and (c) longitudinal shear forces. Distribution of (d) bending moment, (e) shear force, and (f) torsional moment. (g) The force from one tie beam to each hollow core unit versus the applied load. The notations refer to the number of the hollow core unit; see Figure C1.


C6

-50

-40

-30

-20

-10

00 0.2 0.4 0.6 0.8 1


F n [kN/m] z/L [-]

-30-20-10

0

102030

0 0.2 0.4 0.6 0.8 1


F t [kN/m]

z/L [-] (a) (b)

-50

-30

-10

10

30

50

0 0.2 0.4 0.6 0.8 1


F l [kN/m]

z/L [-]

-300

-200

-100

0

100

0 0.2 0.4 0.6 0.8 1

12345

M [kNm]

z/L [-]

(c) (d)

-200

-100

0

100

200

0 0.2 0.4 0.6 0.8 1

12345

V [kN]

z/L [-]-100

-50

0

50

100

0 0.2 0.4 0.6 0.8 1

12345

T [kNm]

z/L [-]

(e) (f)

-150-100-50

050

100150

0 20 40 60 80 100

12345

Q [kN/m]

F [kN]

(g)

Figure C5 Results from analysis C1c-b0, Q = 70 kN. Contact forces in the joints, (a) normal forces, (b) vertical shear forces, and (c) longitudinal shear forces. Distribution of (d) bending moment, (e) shear force, and (f) torsional moment. (g) The force from one tie beam to each hollow core unit versus the applied load. The notations refer to the number of the hollow core unit; see Figure C1.


C7

-25

-20

-15

-10

-5

00 0.2 0.4 0.6 0.8 1


F n [kN/m] z/L [-]

-40

-20

0

20

40

0 0.2 0.4 0.6 0.8 1


F t [kN/m]

z/L [-] (a) (b)

-50

-30

-10

10

30

50

0 0.2 0.4 0.6 0.8 1


F l [kN/m]

z/L [-]

-300

-200

-100

0

100

0 0.2 0.4 0.6 0.8 1

12345

M [kNm]

z/L [-]

(c) (d)

-200

-100

0

100

200

0 0.2 0.4 0.6 0.8 1

12345

V [kN]

z/L [-]

-150-100-50

050

100150

0 0.2 0.4 0.6 0.8 1

12345

T [kNm]

z/L [-]

(e) (f)

-40

-20

0

20

40

0 20 40 60 80 100

12345

Q [kN/m]

F [kN]

(g)

Figure C6 Results from analysis C1c-b05, Q = 70 kN. Contact forces in the joints, (a) normal forces, (b) vertical shear forces, and (c) longitudinal shear forces. Distribution of (d) bending moment, (e) shear force, and (f) torsional moment. (g) The force from one tie beam to each hollow core unit versus the applied load. The notations refer to the number of the hollow core unit; see Figure C1.


C8

-1.5

-1

-0.5

0

0.5

0 0.2 0.4 0.6 0.8 1

1-22-33-44-5

F n [kN/m]

z/L [-]

-40

-20

0

20

40

0 0.2 0.4 0.6 0.8 1

1-22-33-44-5

F t [kN/m]

z/L [-] (a) (b)

-0.4

-0.2

0

0.2

0.4

0 0.2 0.4 0.6 0.8 1

1-22-33-44-5

F l [kN/m]

z/L [-] -300

-200

-100

00 0.2 0.4 0.6 0.8 1 1

2345

M [kNm]z/L [-]

(c) (d)

-150

-50

50

150

0 0.2 0.4 0.6 0.8 1

12345

V [kN]

z/L [-]

-150

-50

50

150

0 0.2 0.4 0.6 0.8 1

12345

T [kNm]

z/L [-]

(e) (f)

-4

-2

0

2

4

0 20 40 60 80 100

12345

Q [kN/m]

F [kN]

(g)

Figure C7 Results from analysis C1c-a, Q = 70 kN. Contact forces in the joints, (a) normal forces, (b) vertical shear forces, and (c) longitudinal shear forces. Distribution of (d) bending moment, (e) shear force, and (f) torsional moment. (g) The force from one tie beam to each hollow core unit versus the applied load. The notations refer to the number of the hollow core unit; see Figure C1.


C9

Appendix D. Results from analyses of set-up C2 in prEN, with point load in centre

Point load P

1 2 3 4 5

P

L/2 = 6 m L/2 = 6 m

Figure D1 Analysed set-up C2: a floor consisting of five hollow core units loaded with a point load in the centre of the span.

Table D1 The bending moment at mid span, in kNm. Set-up C2, point load in mid span, P = 400 kN. The notations refer to the number of the hollow core unit; see Figure D1.


Analysis C2-b0

Analysis C2-b05

Analysis C2-a

1 196 193 188

2 239 239 240

3 329 335 344

4 239 239 240

5 196 193 188


D1

Table D2 The reaction force at the supports, in kN. Set-up C2, point load in mid span, P = 400 kN. The notations refer to the number of the hollow core unit; see Figure D1.


Analysis C2-b0

Analysis C2-b05

Analysis C2-a

1 35 33 40

2 43 42 40

3 44 50 40

4 43 42 40

5 35 33 40

Table D3 The torsional moment at support, in kNm. Set-up C2, point load in mid span, P = 400 kN. The notations refer to the number of the hollow core unit; see Figure D1.


Analysis C2-b0

Analysis C2-b05

Analysis C2-a

1 31 20 24

2 40 62 72

3 0 0 0

4 -40 -62 -72

5 -31 -20 -24


D2

-70-60-50-40-30-20-10

00 0.2 0.4 0.6 0.8 1


F n [kN/m] z/L [-]

-150-100

-500

50100

150

0 0.2 0.4 0.6 0.8 1


F t [kN/m]

z/L [-]

(a) (b)

-30-20-10

0102030

0 0.2 0.4 0.6 0.8 1


F l [kN/m]

z/L [-]-400

-300

-200

-100

00 0.2 0.4 0.6 0.8 1 1

2345

M [kNm]

z/L [-]

(c) (d)

-200

-100

0

100

200

0 0.2 0.4 0.6 0.8 1

12345

V [kN]

z/L [-]

-50

-30

-10

10

30

50

0 0.2 0.4 0.6 0.8 1

12345

T [kNm]

z/L [-]

(e) (f)

-60-40-20

0204060

0 100 200 300 400

12345

P [kN]

F [kN]

(g)

Figure D2 Results from analysis C2-b0, P = 400 kN. Contact forces in the joints, (a) normal forces, (b) vertical shear forces, and (c) longitudinal shear forces. Distribution of (d) bending moment, (e) shear force, and (f) torsional moment. (g) The force from one tie beam to each hollow core unit versus the applied load. The notations refer to the number of the hollow core unit; see Figure D1.


D3

-40

-30

-20

-10

00 0.2 0.4 0.6 0.8 1


F n [kN/m] z/L [-]

-150-100-50

050

100

150

0 0.2 0.4 0.6 0.8 1


F t [kN/m]

z/L [-] (a) (b)

-30-20-10

0102030

0 0.2 0.4 0.6 0.8 1


F l [kN/m]

z/L [-]

-400

-300

-200

-100

00 0.2 0.4 0.6 0.8 1 1

2345

M [kNm]

z/L [-]

(c) (d)

-200

-100

0

100

200

0 0.2 0.4 0.6 0.8 1

12345

V [kN]

z/L [-]

-100

-50

0

50

100

0 0.2 0.4 0.6 0.8 1

12345

T [kNm]

z/L [-]

(e) (f)

-10

-5

0

5

10

0 100 200 300 400

12345

P [kN]

F [kN]

(g)

Figure D3 Results from analysis C2-b05, P = 400 kN. Contact forces in the joints, (a) normal forces, (b) vertical shear forces, and (c) longitudinal shear forces. Distribution of (d) bending moment, (e) shear force, and (f) torsional moment. (g) The force from one tie beam to each hollow core unit versus the applied load. The notations refer to the number of the hollow core unit; see Figure D1.


D4

-1

-0.5

0

0.5

0 0.2 0.4 0.6 0.8 1

1-22-33-44-5

F n [kN/m]

z/L [-]

-150-100

-500

50100150

0 0.2 0.4 0.6 0.8 1

1-22-33-44-5

F t [kN/m]

z/L [-] (a) (b)

-0.3-0.2-0.1

00.10.20.3

0 0.2 0.4 0.6 0.8 1

1-22-33-44-5

F l [kN/m]

z/L [-]-400

-300

-200

-100

0

100

0 0.2 0.4 0.6 0.8 1

12345

M [kNm]z/L [-]

(c) (d)

-200

-100

0

100

200

0 0.2 0.4 0.6 0.8 1

12345

V [kN]

z/L [-]

-100

-50

0

50

100

0 0.2 0.4 0.6 0.8 1

12345

T [kNm]

z/L [-]

(e) (f)

-1.5-1

-0.50

0.51

1.5

0 100 200 300 400

12345

P [kN]

F [kN]

(g)

Figure D4 Results from analysis C2-a, P = 400 kN. Contact forces in the joints, (a) normal forces, (b) vertical shear forces, and (c) longitudinal shear forces. Distribution of (d) bending moment, (e) shear force, and (f) torsional moment. (g) The force from one tie beam to each hollow core unit versus the applied load. The notations refer to the number of the hollow core unit; see Figure D1.


D5

Appendix E. Results from analyses of set-up C3 in prEN, with point load at edge

Point load P

1 2 3 4 5P

L/2 = 6 m L/2 = 6 m

Figure E1 Analysed set-up C3: a floor consisting of five hollow core units loaded with a point load at the edge.

Table E1 The bending moment at mid span, in kNm. Set-up C3, point load at the edge, P = 400 kN. The notations refer to the number of the hollow core unit; see Figure E1.


Analysis C3-b0

Analysis C3-b05

Analysis C3-a

1 478 481 503

2 254 252 254

3 187 185 180

4 149 149 140

5 131 133 123


E1

Table E2 The reaction force at the supports, in kN. Set-up C3, point load at the edge, P = 400 kN. The notations refer to the number of the hollow core unit; see Figure E1.


Analysis C3-b0

Analysis C3-b05

Analysis C3-a

1 71 74 49

2 39 37 46

3 34 33 40

4 31 29 34

5 25 27 31

Table E3 The torsional moment at support, in kNm. Set-up C3, point load at the edge, P = 400 kN. The notations refer to the number of the hollow core unit; see Figure E1.


Analysis C3-b0

Analysis C3-b05

Analysis C3-a

1 -181 -194 -210

2 -126 -130 -154

3 -89 -88 -102

4 -56 -50 -58

5 -28 -16 -19


E2

-300-250-200-150-100

-500

0 0.2 0.4 0.6 0.8 11-2 upper1-2 lower2-3 upper2-3 lower3-4 upper3-4 lower4-5 upper4-5 lower

F n [kN/m] z/L [-]

-20

0

20

40

60

80

0 0.2 0.4 0.6 0.8 1


F t [kN/m]

z/L [-] (a) (b)

-100

-50

0

50

100

0 0.2 0.4 0.6 0.8 1


F l [kN/m]

z/L [-]

-500

-400

-300

-200

-100

00 0.2 0.4 0.6 0.8 1 1

2345

M [kNm]

z/L [-]

(c) (d)

-200

-100

0

100

200

0 0.2 0.4 0.6 0.8 1

12345

V [kN]

z/L [-]

-200

-100

0

100

200

0 0.2 0.4 0.6 0.8 1

12345

T [kNm]

z/L [-]

(e) (f)

-100

-50

0

50

100

0 100 200 300 400

12345

P [kN]

F [kN]

(g)

Figure E2 Results from analysis C3-b0, P = 400 kN. Contact forces in the joints, (a) normal forces, (b) vertical shear forces, and (c) longitudinal shear forces. Distribution of (d) bending moment, (e) shear force, and (f) torsional moment. (g) The force from one tie beam to each hollow core unit versus the applied load. The notations refer to the number of the hollow core unit; see Figure E1.


E3

-100

-80

-60

-40

-20

00 0.2 0.4 0.6 0.8 1


F n [kN/m] z/L [-]

0

20

40

60

80

0 0.2 0.4 0.6 0.8 1


F t [kN/m]

z/L [-] (a) (b)

-100

-50

0

50

100

0 0.2 0.4 0.6 0.8 1


F l [kN/m]

z/L [-]

-500

-400

-300

-200

-100

00 0.2 0.4 0.6 0.8 1 1

2345

M [kNm]

z/L [-]

(c) (d)

-200

-100

0

100

200

0 0.2 0.4 0.6 0.8 1

12345

V [kN]

z/L [-]

-200

-100

0

100

200

0 0.2 0.4 0.6 0.8 1

12345

T [kNm]

z/L [-]

(e) (f)

-10

-5

0

5

10

0 100 200 300 400

12345

P [kN]

F [kN]

(g)

Figure E3 Results from analysis C3-b05, P = 400 kN. Contact forces in the joints, (a) normal forces, (b) vertical shear forces, and (c) longitudinal shear forces. Distribution of (d) bending moment, (e) shear force, and (f) torsional moment. (g) The force from one tie beam to each hollow core unit versus the applied load. The notations refer to the number of the hollow core unit; see Figure E1.


E4

-2

0

2

4

6

8

0 0.2 0.4 0.6 0.8 1

1-22-33-44-5

F n [kN/m]

z/L [-]

020406080

100

0 0.2 0.4 0.6 0.8 1

1-22-33-44-5

F t [kN/m]

z/L [-]

(a) (b)

-0.6-0.4-0.2

00.20.40.6

0 0.2 0.4 0.6 0.8 1

1-22-33-44-5

F l [kN/m]

z/L [-]

-500

-400

-300

-200

-100

00 0.2 0.4 0.6 0.8 1 1

2345

M [kNm]

z/L [-]

(c) (d)

-200

-100

0

100

200

0 0.2 0.4 0.6 0.8 1

12345

V [kN]

z/L [-]

-300-200-100

0100200300

0 0.2 0.4 0.6 0.8 1

12345

T [kNm]

z/L [-]

(e) (f)

-2

-1

0

1

2

0 100 200 300 400

12345

P [kN]

F [kN]

(g)

Figure E4 Results from analysis C3-a, P = 400 kN. Contact forces in the joints, (a) normal forces, (b) vertical shear forces, and (c) longitudinal shear forces. Distribution of (d) bending moment, (e) shear force, and (f) torsional moment. (g) The force from one tie beam to each hollow core unit versus the applied load. The notations refer to the number of the hollow core unit; see Figure E1.


E5

Appendix F. Results from analyses of set-up C5 in prEN, with three supported edges and line load

Line load Q

Q

1 2 3 4 5

L = 12 m

Figure F1 Analysed set-up C5: a floor consisting of five hollow core units with three supported edges loaded with a line load in the centre.

Table F1 The bending moment at mid span, in kNm. Set-up C5, three supported edges and line load, Q = 70 kN/m. The notations refer to the number of the hollow core unit; see Figure F1.


Analysis C5-b0

Analysis C5-b05

Analysis C5-a

1 37 39 38

2 104 107 116

3 151 153 170

4 158 159 172

5 149 149 156


F1

Table F2 The reaction force at the supports, in kN. Set-up C5, three supported edges and line load, Q = 70 kN/m. The notations refer to the number of the hollow core unit; see Figure F1.


Analysis C5-b0

Analysis C5-b05

Analysis C5-a

1 -17 -19 12

2 50 53 46

3 97 109 89

4 65 65 61

5 42 36 46

Table F3 The torsional moment at support, in kNm. Set-up C5, three supported edges and line load, Q = 70 kN/m. The notations refer to the number of the hollow core unit; see Figure F1.


Analysis C5-b0

Analysis C5-b05

Analysis C5-a

1 192 197 203

2 190 210 238

3 65 63 69

4 -35 -75 -91

5 -36 -23 -27


F2

-80

-60

-40

-20

00 0.2 0.4 0.6 0.8 1


F n [kN/m] z/L [-]

-50

-30

-10

10

30

0 0.2 0.4 0.6 0.8 1


F t [kN/m]

z/L [-] (a) (b)

-125-100-75-50-25

0255075

100

0 0.2 0.4 0.6 0.8 1


F l [kN/m]

z/L [-]

-200

-150

-100

-50

0

50

0 0.2 0.4 0.6 0.8 1

12345

M [kNm]

z/L [-]

(c) (d)

-100

-50

0

50

100

0 0.2 0.4 0.6 0.8 1

12345

V [kN]

z/L [-]

-200

-100

0

100

200

0 0.2 0.4 0.6 0.8 1

12345

T [kNm]

z/L [-]

(e) (f)

-200

-100

0

100

200

0 20 40 60 80 100

12345

Q [kN/m]

F [kN]

01020304050

0 0.2 0.4 0.6 0.8 1

R y [kN/m]

z/L [-] (g) (h)

Figure F2 Results from analysis C5-b0, Q = 70 kN/m. Contact forces in the joints, (a) normal forces, (b) vertical shear forces, and (c) longitudinal shear forces. Distribution of (d) bending moment, (e) shear force, and (f) torsional moment. (g) The force from one tie beam to each hollow core unit versus the applied load. (h) The distribution of the reaction force along the third support line. The notations refer to the number of the hollow core unit; see Figure F1.


F3

-25

-20

-15

-10

-5

00 0.2 0.4 0.6 0.8 1


F n [kN/m] z/L [-]

-50

-30

-10

10

30

0 0.2 0.4 0.6 0.8 1


F t [kN/m]

z/L [-] (a) (b)

-100-75-50-25

0255075

100

0 0.2 0.4 0.6 0.8 1


F l [kN/m]

z/L [-]

-200

-150

-100

-50

0

50

0 0.2 0.4 0.6 0.8 1

12345

M [kNm]

z/L [-]

(c) (d)

-125-100-75-50-25

0255075

100125

0 0.2 0.4 0.6 0.8 1

12345

V [kN]

z/L [-]

-250

-150

-50

50

150

250

0 0.2 0.4 0.6 0.8 1

12345

T [kNm]

z/L [-]

(e) (f)

-60-40-20

0204060

0 20 40 60 80 100

12345

Q [kN/m]

F [kN]

01020304050

0 0.2 0.4 0.6 0.8 1

R y [kN/m]

z/L [-] (g) (h)

Figure F3 Results from analysis C5-b05, Q = 70 kN/m. Contact forces in the joints, (a) normal forces, (b) vertical shear forces, and (c) longitudinal shear forces. Distribution of (d) bending moment, (e) shear force, and (f) torsional moment. (g) The force from one tie beam to each hollow core unit versus the applied load. (h) The distribution of the reaction force along the third support line. The notations refer to the number of the hollow core unit; see Figure F1.


F4

-2

-1.5

-1

-0.5

00 0.2 0.4 0.6 0.8 1 1-2

2-33-44-5

F n [kN/m]

z/L [-]

-50

-30

-10

10

30

0 0.2 0.4 0.6 0.8 1

1-22-33-44-5

F t [kN/m]

z/L [-]

(a) (b)

-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1

1-22-33-44-5

F l [kN/m]

z/L [-]

-200

-150

-100

-50

00 0.2 0.4 0.6 0.8 1 1

2345

M [kNm]

z/L [-]

(c) (d)

-100

-50

0

50

100

0 0.2 0.4 0.6 0.8 1

12345

V [kN]

z/L [-]

-300

-100

100

300

0 0.2 0.4 0.6 0.8 1

12345

T [kNm]

z/L [-]

(e) (f)

-6-4-20246

0 20 40 60 80 100

12345

Q [kN/m]

F [kN]

01020304050

0 0.2 0.4 0.6 0.8 1

R y [kN/m]

z/L [-] (g) (h)

Figure F4 Results from analysis C5-a, Q = 70 kN/m. Contact forces in the joints, (a) normal forces, (b) vertical shear forces, and (c) longitudinal shear forces. Distribution of (d) bending moment, (e) shear force, and (f) torsional moment. (g) The force from one tie beam to each hollow core unit versus the applied load. (h) The distribution of the reaction force along the third support line. The notations refer to the number of the hollow core unit; see Figure F1.


F5

Appendix G. Results from analyses of set-up C6 in prEN, with three supported edges and point load

Point load P

1 2 3 4 5

P

L/2 = 6 m L/2 = 6 m

Figure G1 Analysed set-up C6: a floor consisting of five hollow core units with three supported edges loaded with a point load in the centre.

Table G1 The bending moment at mid span, in kNm. Set-up C6, three supported edges and point load, P = 400 kN. The notations refer to the number of the hollow core unit; see Figure G1.


Analysis C6-b0

Analysis C6-b05

Analysis C6-a

1 38 37 35

2 122 124 131

3 239 245 263

4 165 165 175

5 129 127 131


G1

Table G2 The reaction force at the supports, in kN. Set-up C6, three supported edges and point load, P = 400 kN. The notations refer to the number of the hollow core unit; see Figure G1.


Analysis C6-b0

Analysis C6-b05

Analysis C6-a

1 -17 -17 5

2 16 19 13

3 26 32 19

4 26 26 23

5 22 19 25

Table G3 The torsional moment at support, in kNm. Set-up C6, three supported edges and point load, P = 400 kN. The notations refer to the number of the hollow core unit; see Figure G1.


Analysis C6-b0

Analysis C6-b05

Analysis C6-a

1 130 135 141

2 123 134 151

3 48 47 51

4 -13 -35 -44

5 -15 -12 -15


G2

-80

-60

-40

-20

00 0.2 0.4 0.6 0.8 1


F n [kN/m] z/L [-]

-125-100

-75-50-25

0255075

100

0 0.2 0.4 0.6 0.8 1


F t [kN/m]

z/L [-]

(a) (b)

-75-50-25

0255075

0 0.2 0.4 0.6 0.8 1


F l [kN/m]

z/L [-]

-250

-150

-50 0 0.2 0.4 0.6 0.8 1 12345

M [kNm]

z/L [-]

(c) (d)

-200

-100

0

100

200

0 0.2 0.4 0.6 0.8 1

12345

V [kN]

z/L [-]

-150

-50

50

150

0 0.2 0.4 0.6 0.8 1

12345

T [kNm]

z/L [-]

(e) (f)

-100

-50

0

50

100

0 100 200 300 400

12345

P [kN]

F [kN]

01020304050

0 0.2 0.4 0.6 0.8 1

R y [kN/m]

z/L [-] (g) (h)

Figure G2 Results from analysis C6-b0, P = 400 kN. Contact forces in the joints, (a) normal forces, (b) vertical shear forces, and (c) longitudinal shear forces. Distribution of (d) bending moment, (e) shear force, and (f) torsional moment. (g) The force from one tie beam to each hollow core unit versus the applied load. (h) The distribution of the reaction force along the third support line. The notations refer to the number of the hollow core unit; see Figure G1.


G3

-50

-40

-30

-20

-10

00 0.2 0.4 0.6 0.8 1


F n [kN/m] z/L [-]

-125-100-75-50-25

0255075

100

0 0.2 0.4 0.6 0.8 1


F t [kN/m]

z/L [-]

(a) (b)

-75-50-25

0255075

0 0.2 0.4 0.6 0.8 1


F l [kN/m]

z/L [-]

-250

-150

-50 0 0.2 0.4 0.6 0.8 1 12345

M [kNm]

z/L [-]

(c) (d)

-200

-100

0

100

200

0 0.2 0.4 0.6 0.8 1

12345

V [kN]

z/L [-]

-150

-50

50

150

0 0.2 0.4 0.6 0.8 1

12345

T [kNm]

z/L [-]

(e) (f)

-20

-10

0

10

20

0 100 200 300 400

12345

P [kN]

F [kN]

01020304050

0 0.2 0.4 0.6 0.8 1

R y [kN/m]

z/L [-] (g) (h)

Figure G3 Results from analysis C6-b05, P = 400 kN. Contact forces in the joints, (a) normal forces, (b) vertical shear forces, and (c) longitudinal shear forces. Distribution of (d) bending moment, (e) shear force, and (f) torsional moment. (g) The force from one tie beam to each hollow core unit versus the applied load. (h) The distribution of the reaction force along the third support line. The notations refer to the number of the hollow core unit; see Figure G1.


G4

-1.5

-1

-0.5

00 0.2 0.4 0.6 0.8 1 1-2

2-33-44-5

F n [kN/m]

z/L [-]

-125-100

-75-50-25

0255075

100

0 0.2 0.4 0.6 0.8 1

1-22-33-44-5

F t [kN/m]

z/L [-]

(a) (b)

-0.6-0.4-0.2

00.20.40.6

0 0.2 0.4 0.6 0.8 1

1-22-33-44-5

F l [kN/m]

z/L [-]

-300

-200

-100

00 0.2 0.4 0.6 0.8 1 1

2345

M [kNm]

z/L [-]

(c) (d)

-200

-100

0

100

200

0 0.2 0.4 0.6 0.8 1

12345

V [kN]

z/L [-]

-150

-50

50

150

0 0.2 0.4 0.6 0.8 1

12345

T [kNm]

z/L [-]

(e) (f)

-2

-1

0

1

2

0 100 200 300 400

12345

P [kN]

F [kN]

0

10

20

30

40

0 0.2 0.4 0.6 0.8 1

R y [kN/m]

z/L [-] (g) (h)

Figure G4 Results from analysis C6-a, P = 400 kN. Contact forces in the joints, (a) normal forces, (b) vertical shear forces, and (c) longitudinal shear forces. Distribution of (d) bending moment, (e) shear force, and (f) torsional moment. (g) The force from one tie beam to each hollow core unit versus the applied load. (h) The distribution of the reaction force along the third support line. The notations refer to the number of the hollow core unit; see Figure G1.


G5

Shear and torsion interaction of hollow core...

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