Demountable construction: Analysis of the behaviour of a 1:5 scale floor bay. Part 2: Series D-cyclic loading experiments

1:5 scale floor bay: Part 2

Series D: cyclic loading experiments

Ir. J . Stroband/ing. J . J . Kolpa

t U Delft

Faculty of Civil Engineering

_ _ _ _ Division of Mechanics and Structures

2 1 Section of Concrete Structures Stevin Laboratory Delft University of Technology

May 1987

DEMOUNTABLE CONSTRUCTION

Analysis of the befiaviour of a 1:5 scale floor bay; Part 2

Series D - cyclic loading experiments

ir. J. Stroband ing. J.J. Kolpa

Mailing address:

Delft University of Technology Concrete Structur-es Group Stevinlaborntory II

Stevinweg 1 2628 CN Delft The Netherlands

Technische Universiteit Delft Bibliotheek Faculteit der Civiele Techniek

(Bezoekadres Stevinweg 1) STl_ Postbus 5048

ACKNOWLEDGEMENTS

The research which forms the basis of this report was carried out in the Stevin Laboratory of the Delft University of Technology, in close collaboration with CUR Committee D7. The authors wish to thank the members of that Committee for their stimulating contributions to the discussions. They also wish to record their indebtedness to Mr.A.van Rhijn and Mr.G.F. Liqui Lung, both on the Laboratory's staff, for their conscientious execution of the experimental investigations and to the CUR (Civiltechnical Centre Execution and Regulation) for financial support provided.

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Reproduction of this report as a whole or in part is allowed only with the authors' permission in writing.

CONTENTS page

Notations

Summary

1. INTRODUCTION 11

2. FORCES AND DEFORMATION? 13

2.1 Description of the tests 13 2.2 Description of the anticipated behaviour of the floor 17

2.2.1 Shear behaviour at transverse joints 20

2.2.1.1 Monotonically increasing load 24

2.2.1.2 Cyclic loading 27 2.3 Lateral deflection of the longitudinal edge 31

2.4 Longitudinal forces 35

2.5 Failure load 36

3. TEST RESULTS OF SERIES D AND COMPARISON WITH THE RESULTS OF SERIES C AND

WITH A STRUCTURAL ANALYSIS 39

3.1 Longitudinal forces at the transverse joints at grid lines 2, 5, 7

and 11 41 3.1.1 Longitudinal forces at grid line 7 42

3.1.2 Longitudinal forces at grid line 5 42 3.1.3 Longitudinal forces at grid line 11 42 3.1.4 Longitudinal forces at grid line 2 42 3.1.5 Distribution of the longitudinal forces at grid lines A and D 43

3.2 Shear deformations in the transverse joints at grid lines 2 and 11 43

3.2.1 Shear deformation at grid line 2 45 3.2.2 Shear deformation at grid line 11 45 3.2.3 Shear deformation in the longitudinal joints at grid line B near

grid line 11 45 3.2.4 Shear deformation in the longitudinal joints between grid lines

1 and 2 46 3.3 Crack width in the transverse joints at grid lines 2, 5, 7 and 11 47

3.3.1 Crack width at grid line 7 48 3.3.2 Crack width at grid line 5 49 3.3.3 Crack width at grid line 11 49 3.3.4 Crack width at grid line 2 50 3.4 Lateral deflection of the longitudinal edge 50

3.5 Deformation of the ends of the floor bay 52

3.6 Failure load 52

ANALYSIS OF TEST RESULTS OF SERIES D ' 55

4.1 Forces and deformations at grid line 7 55 4.2 Forces and deformations at grid line 5 62 4.3 Forces and deformations at grid line 11 65

4.3.1 Shear friction effect in the transverse joint at grid line 11 66 4.3.2 Deformations in longitudinal eind transverse joints at point Bll 73

4.4 Forces and deformations at grid line 2 90 4.5 Lateral deflection of the longitudinal edge 93 4.6 Deformation of the ends of the floor bay 97

4.7 Failure of the floor bay 97 4.7.1 Failure behaviour 97 4.7.2 Failure load 99

CONCLUSIONS 101

Ac cross-sectional area of concrete As cross-sectional area of steel

Ec modulus of elasticity of concrete in compression Es modulus of elasticity of steel in tension

EI stiffness F point load

H horizontal force M bending moment

Nc direct compressive force in concrete Ns direct tensile force in steel

Q total horizontal load R reaction force

V shear force

Vu ultimate shear force

fc design value of compressive strength of concrete fs design value of tensile strength of steel

n number of cycles s shear deformation w crack width

X depth of compressive zone 6 deflection

£c strain of concrete

Ecy strain of concrete at fc

Ecu maximum compressive strain of concrete Es strain of steel

Esy strain of steel at fs Esu maximum strain of steel (p rotation

X curvature

G"c normal stress occurring in the concrete (7u n o r m n l 3tr(i3s occtirr'ing in I.IKJ s t e e l r Hh(!ar- s t r e s s

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SUNMARY

With a view to the rational use of energy and raw materials, on the one hand, and having regard to the environmentally detrimental effects of demolition (nuisance caused by demolition work, problems with the disposal of rubble), on the other, it will have to established by research to what extent demountable construction methods offer a meaningful alternative to the methods now in current use.

By "demountable construction" is understood a building method characterized by the fact that the structural connections are so designed that the component parts of the structure can be dismantled (demounted) intact or with only minimal damage and are, if possible, suitable for re-use.

Within the context of such research the behaviour of a demountable floor bay, constructed to a 1:5 scale, was investigated in the Stevin Laboratory of the Delft University of Technology. This floor bay was composed of reinforced concrete hollow core slabs supported on beams which in turn rested on columns.

The results of the research, which was carried out in collaboration with CUR-VB Committee D7 "Demountable construction", are given in Stevin Report No. 5-84-4

[1]. Stevin Report No. 5-85-14 [2] gives an analysis of the results of the measurements relating to part of that research, namely, comprising tests performed with a monotonically increasing load and with repeated loading applied within the plane of the floor bay.

The present report gives an analysis of the results of measurements performed in the tests of series D with alternating cyclic loading. After a general introductory chapter. Chapter 2 contains a description of the tests in question, followed by some considerations on the anticipated behaviour of a floor bay (composite panel) assembled from precast concrete units. Next, in Chapter 3, the test results of series D are compared with those of series C for monotonically increasing load and with the results of a calculation based on a simplified mathematical model. On the basis of the data presented in Chapter 3 an analysis of the behaviour of the floor bay is attempted in Chapter 4.

5.

The analysis shows that there is a significant difference in behaviour between a transverse joint subjected predominantly to bending moment and a transverse joint subjected predominantly to shear force. In the former case the measured values are in reasonably good agreement with the calculated values. In the latter a very different pattern of behaviour emerges. This behaviour is, however, so complex that it cannot as yet be reliably analysed. That would require complementary detailed research.

It also appears from the analysis that the horizontal (i.e., lateral) deflection of the floor bay under alternating cyclic loading is considerably greater and that the loadbearing capacity (failure load) is considerably lower than under a once-only load.

1. INTRODUCTION

In 1977 the CUR-VB initiated the development of structures that are simple to dismantle or demount and therefore referred to as "demountable structures". In the then following years a large number of tests were performed on a floor bay consisting of precast concrete units. This floor structure is characterized by the fact that the forces are transmitted solely via the joints and with the aid of simple connections between the units, not through in-situ concrete or an in-situ concrete topping.

The "diaphragm action" of this floor - constructed to a 1:5 scale - under horizontal loading in its own plane was investigated. The set-up and execution of the research are described in Stevin Report No. 5-84-4 [1], which also contains an overview of the results obtained in the tests.

The object of the research was to gain more insight into the pattern and action of the forces in a floor bay consisting entirely of precast concrete units. The calculation of the internal forces and the deformations of such a structure is highly complex. An accurate analysis is impracticable without the aid of a computer.

All the same, for estimating the dimensions of the precast units, the forces acting in the connections and the loadbearing capacity of the floor bay at • the design stage, and for analysing the stability of the building, the designer requires a simple calculation method that is sufficiently reliable for preliminary use at this stage.

For employing a computer program for the final calculations the designer should have at his disposal sufficient input data to enable the behaviour of the structure to be determined. Also, at that stage of the work he will require a simple method of calculation for verifying the validity of the results yielded by the computer.

A proposal for a simplified calculation of this kind has been worked out for a monotonically increasing load in Stevin Report 5-84-14 [2]. In the present report it will be investigated to what extent the fundamental points relating

to monotonie loading are also valid for alternating cyclic loading.

The aim of the present report is to contribute to finding the answers to the questions arising from these considerations. At the same time it must be realized that, having regard to the nature of the research, this report can do neither more nor less than add to the understanding of the behaviour of a rigid floor bay. In order to proceed from here to generally-valid design rules will require, on the one hand, detailed research into the transmission of force in joints between precast floor slabs and, on the other, the development of a theoretical model with which such transmission of force can be described.

It is perhaps needless to add that, although the tests were performed against the background of the design of demountable structures, the results are valid also in the context of precast concrete constructions in general, where rationalization and efficiencies are of primary importance.

2. FORCES AND DEFORMATIONS

In this chapter the anticipated behaviour of a floor bay (composite panel) subjected to cyclic horizontal loading is examined in its general aspects. This analysis relates to the tests comprised in series D as described in Stevin Report 5-84-4: "Demountable construction. Description and results of tests on a 1:5 scale demountable floor bay" [1].

First, the structure of the bay under investigation, and the manner in which the tests were performed, are outlined in Sec. 2.1. Next, in Sec. 2.2 to 2.5, the anticipated behaviour of such a bay, particularly as regards the action of the forces and as regards the deformations, is described.

2.1 Description of the tests

The constructional features of the floor bay are shown in Fig. 2.1. It is composed of beams extending longitudinally and of hollow core slabs extending transversely. The slabs are supported on the beams through strips of hair felt serving as bearings. The beams in turn are supported through hair felt strips on the column heads. The spaces between adjacent hollow slabs and between the slabs and the beams are filled with low-strength mortar. In the longitudinal direction the beams are joined end-to-end by a threaded bar (MS) at each connection. The ends of the beams are provided with threaded sockets for the purpose. The column heads are supported through roller bearings on the steel testing rig, so that virtually frictionless bearing conditions are obtained.

The shear walls along the short sides of the floor bay, at grid lines 1 and 14, are simulated by steel strips whose stiffness corresponds to that of a solid wall in an actual building. At three points along each short side the bay is connected to the steel strip, each connection being located at mid-length of a slab. These features are shown in Fig. 2.2.

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means of pneumatic jacks. For applying this loading, distributing beams, interconnected by tie-rods, were provided on grid lines A and D.

Two ranges of alternating cyclic loading were applied:

(a) with maximum and minimum total load of + 6 kN and - 6 kN; (b) with maximum and minimum total load of +12 kN and -12 kN.

Range (a) corresponds to the characteristic value of the wind loading.

The relation between the loading and the measurements performed is shown in Fig. 2.3.

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TOTAL NUMBER OF CYCLES

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TOTAL NUMBER OF CYCLES TOTAL NUMBER OF CYCLES

Each load cycle represented was followed by a series of 200 cycles in which no measurements were performed. The total numbers of cycles are also indicated in the diagrams.

More detailed information on the constructional features of the floor bay and on the execution of the tests is given in Stevin Report 5-84-4 [1] already referred to.

2.2 Description of the anticipated behaviour of the floor

The horizontal loads produce bending moments and shear forces within the plane of the floor bay. For a bay provide with shear walls at its short sides the bending moment and shear force diagrams are as shown in Fig. 2.4.

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Fig. 2.4 Bending moment and shear diagrams,

In consequence of the bending moments there develops at every cross—section a strain distribution which depends, on the one hand, on the nature of the materials of which tliat section is composed and, on the other, on whether or not cracking occurs there.

At the grid lines 2 to 13, where the connections between the longitudinal beams are located, the section consists entirely of jointing mortar. If cracks are formed when the tensile stresses that can be resisted are exceeded, the strain diagram will have the form represented in Fig. 2.5.b. With the aid of the C-E diagrams of the materials used it is possible to calculate the distribution of stresses and forces as shown in Fig. 2.5.c. The longitudinal forces in the longitudinal beam connections and the compressive forces in the compressive

zone can be calculated by applying the two equilibrium equations Y H = 0 emd Y M = 0 .

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Fig. 2.5 Strain and stress distribution at a cracked section.

Between the grid lines there is a continuous beam, so that the cross-section of the floor bay will remain uncracked for a long time. Because of this, the strain distribution and the associated stress distribution will have entirely different shapes from those at the grid lines. The strain and stress distributions that can be expected at an uncracked section between the grid lines are shown schematically in Fig. 2.6.b and c.

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Fig. 2.6 Strain and stress distribution at an uncracked section.

The schematization of the structure as described above was adopted in Stevin Report 5-85-14 [2] for the purpose of memual calculation, although the two schematized systems will in reality influence each other. The approach adopted for the purpose; is an approximation, partly because the shear deformation in the longitudinal joints is neglected in this approach.

Since there is no question of a homogeneous section, it is very likely that discontinuities will occur in the strain distribution at the longitudinal joints, more particularly under cyclic loading. In the analysis of the tests with monotonically increasing load it was, in Stevin Report 5-85-14 [2], already noted that the shear stresses in the longitudinal joints can attain high values under the characteristic wind loading.

The magnitude of the discontinuities in the strain diagram depends on the T-s relationship of a cracked joint. Strain diagrams for a cracked and an uncracked section are shown in Fig. 2.7.

Fig. 2.7 Strain diagram for a cracked and an uncracked section.

In analysing the action of forces and the deformations expected to occur the following aspects will successively be considered:

1) Shear behaviour at transverse joints (Sec. 2.2.1) 2) Lateral deflection of the longitudinal edge (Sec. 2.2.2) 3) Longitudinal forces (Sec. 2.2.3) 4) Failure load (Sec. 2.2.4)

This analysis relates only to a floor bay subjected to cyclic loading in its own plane.

2.2.1 Shear behaviour at transverse joints

As already explained above, the horizontal loading produces bending moments and shear forces in the floor bay. In Fig. 2.8 it is shown what form the stress distribution due to the moments and to the shear forces may have.

At the joints located at grid lines 2 to 13, tensile forces develop in the longitudinal beam connections and compressive forces develop in the compressive

zone of such a joint. In this schematization it is furthermore assumed that the cracked part of the joint does not contribute to the transfer of shear forces, these being resisted entirely in the compressive zone.

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Fig. 2.8 Stress distribution at a transverse joint. •

The magnitude of the shear force that can be resisted depends on the f-tf relationship of the uncracked joint. If this relationship is known from tests, then the magnitude of this shear force associated with this stress distribution can be calculated.

If shear displacement occurs in consequence of the shear strength being exceeded, the crack width will increase because of the frictional effect at the rough crack faces (shear friction). As a result, additional tensile forces will develop in the longitudinal connections and will in turn give rise to additional compressive forces in the joint, so that the shear strength is increased.

The stress distribution at a transverse joint under such conditions is not known with certainty, but could conceivably be as shown schematically in Fig. 2.9.

It can reasonably be supposed that under cyclic loading the shear friction effect will occur sooner than under monotonically increasing load, since a continuous crack is formed already after one load cycle. In general, the shear strength at a cracked joint will be lower than at an uncracked joint, so that shear displacement will occur sooner and the shear friction effect thus be able to develop.

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Fig. 2.9 Stress distribution at a transverse joint after shear displacement has ocurred.

Tests have shown that, when shear displacement occurs, there exists a relationship between the shear stress (tc), the normal stress (Cc), the shear strain (s) and the crack width (w). In the literature this relationship is usually expressed in two diagrams, namely, a r-G"-s diagram and a w-s diagram, as shown schematically in Fig. 2.10.

Tc = shear stress Oe= normal stress s = shear deformation w = crack width

Fig. 2.10 Schematic representation of a Tc-Cc-s-w relationship at a cracked joint.

These diagrams will be briefly considered in the following sections of this chapter. A comprehensive analysis of the action of the forces at a crack has been given in a doctoral thesis by Walraven [5] and in a CEB Bulletin d'Information by Tassios [3]. In Stevin Report 5-78-12 [4] Walraven gave an overview of the literature relating to the transfer of force in cracks in concrete. From the results reported by various investigators it emerges that there is a notable difference in shear behaviour of cracks between a monotonically increasing load, on the one hand, and a cyclically acting load, on the other. The results obtained with monotonie load increase will be considered in Sec. 2.2.1.1, and cyclic loading will then be considered in Sec. 2.2.1.2.

2.2.1.1 Monotonically increasing load

The r-s relationship experimentally determined by Loeber and Paulay [6] is given in Fig. 2.11. Their tests were performed with different constant values of the crack width in conjunction with monotonie load increase.

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Fig. 2.11 Shear stress-displacement ratio for constant crack width, according to [6].

Fig. 2.12 gives the relationship between the shear stress and the shear displacement obtained from tests with a constant ratio between shear stress and crack width, i.e., r/w-constant. Those tests, too, were performed by Loeber and Paulay [6],

w = 0.51mm

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Shear displacement (mm )

Fig. 2.12 Mean experimental curve for shear stress-displacement relation-ship with constant shear stress to crack width ratio,

according to [6].

Figs. 2.11 and 2.12 are not directly comparable with each other because the boundary conditions (w=const. and r/w=const. respectively) are different. In order nevertheless to obtain some idea of the difference in behaviour for different boundary conditions, the average values of the results presented in Fig. 2.11 have been drawn in Fig. 2.12.

Diagrams already published in Stevin Report 5-85-14 [2] are reproduced in Fig. 2.13. They represent the shear behaviour of the transverse joint at grid line 2

in test C 206 for monotonie load increase.

In Fig. 2.13a the average shear stress at grid line 2 has been plotted against the average shear deformation along this joint. The points corresponding to crack widths of O.IO mm, 0.15 mm and 0.20 mm, respectively, are also indicated in this diagram. These points have been deduced from Fig. 2.13c.

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Fig. 2.13 Relationship between shear stress, normal stress, shear defonnntion, crack width and coefficient of friction for test C 206 with

In Fig. 2.13b the average normal sti-ess at grid line 2 has been plotted against the average shear deformation. In Fig. 2.13c the average crack width at grid line 2 has been plotted against the average shear deformation. In Fig. 2.13d the average normal stress at grid line 2 has been plotted against the average value of the quotient of shear stress and normal stress. In Fig. 2.13e the average normal stress at grid line 2 has been plotted against the average crack width. The average normal stress was calculated from the sum of the tensile forces acting in the longitudinal connections, divided by the overall area of the transverse joint.

From Fig. 2.13a it appears that the shear stiffness (k=r/s) is less according as the crack width is greater. This phenomenon affects the stress distribution at a section. Thus, in the case of a flexural crack the smallest crack width develops in the compressive zone, so that the greatest resistance to shearing can occur there. Because of this, the shearing action and therefore the shear stress will not be uniformly distributed over the section, but will be concentrated in the compressive zone. The stress distribution will then possibly have a shape as shown in Fig.2.9.

To calculate this stress distribution it is necessary to know the material properties of the mortar joint, more particularly the modulus of elasticity, the shear modulus and the failure envelope for biaxial loading. These properties were investigated in a detailed investigation of joints between floor slabs. The results of the measurements are given in a Stevin Report.

2.2.1.2 Cyclic loading

Laible, White and Gergely [7] performed tests on elements with a shear area Ac=194000 mm2. Their behaviour in shear was investigated for am initial crack width of 0.25 ram, 0.51 mm and 0.76 mm respectively. Fig. 2.14 shows the r-s

relationship and the r-w relationship obtained from tests with cyclic loading and 0.76 mm initial crack width. The ratio between normal stress and crack width was kept constant in these tests, i.e. (r/w=const ant.

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Fig. 2.14 Typical test results: specimen with initial crack width 0,76 mm and restraint stiffness = 600 kN/mm, according to [7].

The most notable aspect of the behaviour represented in Fig. 2.14 is the great difference between the first cycle and subsequent cycles in so far as the r-s relationship is concerned. In general, the rising branch of this relationship was substantially linear during the first load cycle. In subsequent cycles, however, the r-s relationship was markedly non-linear, as is clearly apparent from Fig. 2.14.

Since hardly any shear stress is needed to return the specimens to their neutral position, this phenomenon is evidently caused by local crushing of the material.

With the data presented in Figs. 2.14a and 2.14b a fresh diagram can be drawn, in which the crack width is plotted against the shear deformation, as has been done in Fig. 2.15.

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cle 1 ' - I S - 1 0 - 0 5 0 0.5 10 1.5 shear dispLacement ( m m )

Fig. 2.15 Relationship between shear deformation and crack width under cyclic (alternating) loading; according to [7].

It appears from Fig. 2.14 that, after the first load cycle, a fairly large amount of slip can occur without any appreciable load having to be applied. This means that the crack width will also undergo little change in the cyclic loading range concerned, as is clearly seen at the 15th cycle in Fig. 2.15. Subsequently, the actual shear friction effect develops, which means that the crack width then further increases.

After each cycle a certain increase in the shear deformation can be expected to occur - due to crushing of the material - in conjunction with come increase in crack width.

The influence of the initial crack width upon the shear deformation for a shear stress of 1.26 N/mm^ and a restraint stiffness of 600 kN/mm is represented in Fig.2.16a. The influence of the restraint stiffness upon the shear deformation for an initial crack width of 0.76 mm is represented in Fig. 2.16b.

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Fig. 2.16 Relationship between shear deformation and initial crack width and between shear deformation and restraint stiffness; according to [7].

The results recorded in these diagrams can be regarded as typical of the behaviour of cracks in concrete under cyclic loading.

Fig. 2.17 shows the relationship between the shear deformation and the crack width for the above-mentioned tests. As already stated above,' these tests were performed with cyclic loading, a restraint stiffness of 600 kN/mm and an initial crack width of 0.25 mm, 0.51 mm and 0.76 mm respectively. It appears from this diagram that specimens with 0.25 mm initial crack width behaved very differently from those with 0.51 ram or 0.76 mm initial crack width.

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0.25 Q50 0.75 _ crack width ( m m ]

Fig. 2.17 Shear deformation as a function of crack width and cycling, restraint 600 kN/mm, according to [7].

Cracked joints can be expected to display the same sort of behaviour as described above for cracked concrete. The relationship between the shear deformation and the crack width will in that case be an important property, the principal parameters then being:

- the quality of the jointing material; - the roughness of the crack faces; - the initial crack width;

- the stiffness of the connection perpendicularly to the joint; - the cyclic loading range applied;

- the number of load cycles

2-3 Lateral deflection of the longitudinal edge

The lateral deflection of the longitudinal edge under the action of a horizontal load will be produced in two ways:

a) by deformation due to bending moments (see Fig. 2.18a);

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Fig. 2.18 Lateral deflection of floor bay due to bending moments (a) and to shearing in transverse joints (b).

In the tests performed in series D on the floor bay the maximum and the minimiom value chosen for the cyclic loading was approximately 0.4 times the failure load under monotonically increasing load.

For these values the tensile and the compressive stresses are still within the linear-elastic range of behaviour. Hence it is to be expected that with increasing number of load cycles there will be hardly any increase in lateral deflection due to bending moments.

The situation is different with regard to lateral deflection due to shear deformation in the transverse joints, as can be deduced from Figs. 2.19 and 2.20. The Q-6 diagram of tests C 206 is shown in Fig. 2.19, which relates to a test with a preformed 0.3 mm wide crack, a tensile force F = 3.04 kN acting at the ends of the floor bay and monotonically increasing load. From this diagram it appears that the increase in lateral deflection is greater than could be expected on the basis of structural analysis. Fig. 2.20 shows the lateral deflection lines for the same test for total values of 12 kN and 30 kN, respectively, for the horizontal load.

Q (kN) 32 28 24 20 16 12 8 k 1 2 3 4 5 d e f l e c t i o n at D7 ( m m )

Fig. 2.19 Q-6 diagram for test C 206; according to [2].

Both the measured and the calculated deflection curve is represented in the above-mentioned diagrams. The deflection curve as it would have been if no shearing had occurred is also shown.

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Fig. 2.20 Relationship between measured and calculated deflection curves for test C 206; according to [2].

It appears from these diagrams that for a total horizontal load of 12 kN the increase in the lateral deflection due to shear deformation at grid line 2 is still quite small. On the other hand, for a total horizontal load of 30 kN this increase is already very considerable.

For monotonically increasing load it is therefore apparent that under the given conditions the effect of shear deformation begins to manifest itself only after the total load has reached a value of 12 kN.

Having regard to the difference in shearing behaviour of a tremsverse joint due to monotonically increasing and to cyclic loading respectively, it can be presumed that the lateral deflection will increase with increasing number of load cycles.

2.4 Longitudinal forces

Like the lateral deflection of the longitudinal edge, the magnitude of the forces in the longitudinal connections (the bars connecting the beams end-to-end) is likewise affected by the following factors:

a) elastic deformation due to the bending moment;

b) shearing behaviour in the transverse joints due to shear force (i.e., the transverse force acting in the floor bay) .

The configuration of forces for pure bending and the calculation of the longitudinal forces have already been described in Sec. 2.2. This method of calculation is valid in so far as no shear displacement occurs in the tranverse joints.

If the shear strength is exceeded, a shear displacement will occur, so that, in consequence of the wedge action of the rough crack faces, the crack width will increase. As a result of this, additional forces will develop in the longitudinal connections. This effect has already been dealt with in Sec. 2.2.1.

The influence of both the bending moment and the shear force on the total tensile force in the longitudinal connections is determined by the ratio of shear force to bending moment. The higher this ratio is, the greater is the influence that the shearing behaviour has on the magnitude of the longitudinal force.

Thus, near the ends of the floor bay, where the ratio of shear to bending moment is high, the magnitude of the longitudinal forces is determined chiefly by the shearing behaviour of the joint. On the other hand, in the central region, where the shear/moment ratio is low, the magnitude of the longitudinal forces will be determined almost entirely by the bending moment.

An analysis for numerically calculating the longitudinal forces encounters difficulties because in the present stage of research not enough information is as yet available on the deformation behaviour of a cracked joint. Such information may be provided by the investigations of CUR-VB Committee C 43 "Joints in precast concrete floors" [8].

2-5 Failure load

Particular attention will be paid to the influence of cyclic loading upon the failure load in this report.

In principle, a floor bay may fail in one of four ways: a) by yielding of the longitudinal connections;

b) by failure of the compressive zone; c) by shear failure of a transverse joint; d) by shear failure of a longitudinal joint.

Failure due to yielding of the longitudinal connections at the raid-section of the floor bay is hardly likely to be affected by the number of load cycles unless this number is so high as to cause steel fatigue. Under service load, at which the cycles occur, this is certainly not the case. Similar considerations apply to failure of the compressive zone. Failure due to shear will, on the other hand, depend on the number of load cycles and on the magnitude of the

From test series C it had already emerged that increasing the number of cycles in repeated loading has an unfavourable effect on the failure load, as appears from Fig. 2.21. In both tests indicated there, failure was due to shearing in a transverse joint.

Alternating cyclic loading can be presumed to have an even more unfavourable effect because in that case crushing of the jointing material will occur in two directions, whereas this occurs in only one direction when repeated loading is applied. Vu (kN) ^C206 " ~ ' ~ -""-—--___ ^ —-.,__ ~"'~~^ C 306-/ 1 6 200 10 100 1000 10000 n = number of cycles

Fig. 2.21 Relationship between shear force at failure and number of cycles of repeated loading; according to [2].

Shear stresses occur not only in the transverse but also in the longitudinal joints. From a computer calculation with the ZEFE program [9] as well as from a manual calculation given in Stevin Report 5-85-14 [2] it appears that with monotonically increasing load the shear stresses in the longitudinal joints can be very considerable. Fig. 2.22 gives some idea of the deformation behaviour as calculated with ZEFE.

Fig. 2.22 Shear deformation at the longitudinal joints; according to ZEFE [9].

Although no information on this is as yet available, it can also be presumed that the number of cycles must affect the shearing behaviour in the longitudinal joints and thus possibly affects the failure behaviour of a floor bay.

3. TEST RESULTS OF SERIES D AND COMPARISON WITH THE RESULTS OF SERIES C AND WITH A STRUCTURAL ANALYSIS.

In Chapter 5 of Stevin Report 5-85-14 [2] the calculated values for a number of behaviour patterns are compared with the measured values of series C for monotonically increasing load. The comparative investigation described in that

report relates to: - longitudinal forces;

- lateral (horizontal) deflection of the longitudinal edge; - deformation of the ends of the bay at grid lines 1 and 14; - deformation of the transverse joints;

- failure load.

In the present report the measured values of some tests in series D with cyclic (alternating) loading are compared as far as possible with the measured values of test C 401 in series C.

Also included in the relevant diagrams representing the results are the calculated values obtained with the aid of a simplified mathematical model. These calculations are given in, or are derivable from, the appendices to Stevin Report 5-85-14 [2]. These results are indicated by "calculated" in the diagrams concerned.

Where possible, the values obtained from a calculation with the ZEFE computer program are likewise included in these diagrams. The main results from that calculation have been described in Stevin Report 5-83-19 [9]. In the diagrams

the results in question are indicated by "ZEFE".

The measured results for the following behaviour patterns have been compared with the calculated and/or measured values for monotonically increasing load:

Sec. 3.1: i.ongi Lud inul forces ul. I.he trnnsverse joints at grid lines 2, 5, 7 and 11.

Sec. 3.3: Crack width in the transverse joints at grid lines 2, 5, 7 and 11. Sec. 3.4: Lateral deflection of the longitudinal edge;

Sec. 3.5: Deformation of the ends of the floor bay; Sec. 3.6: Failure load.

The comparisons relate only to the tests D 5, D 105 and D 205 in series D, with a loading range from +12 kN to -12 kN.

For a correct interpretation of the results it is necessary to bear in mind that series D was in fact a continuation of series C, when the ends of the floor bay had been repaired after failure. This means that even before the tests in series D were begun there were already some longitudinal cracks present which may have affected the results.

Another aspect which may have had an unfavourable influence on the results was that a number of tests had to be performed in order to try out the behaviour of the structure under cyclic loading. For this reason a total of 6200 load cycles were applied in the range from +6 kN to -6 kN before the tests D 5, D 105 amd D 205 were begun.

In these last-mentioned tests the floor bay was subjected only to cyclic loading, not combined with other variables. The boundary conditions for the ends of the bay were also kept constant. At the ends the bay was connected only via the hollow core slabs to the steel reaction strips, as has already been described in Sec. 2.1.

In test D 5 failure occurred due to shearing at grid line 12 after 400 load cycles. After the joint had been repaired, testing continued (as test D 105) up to 1000 cycles. No failure occurred in this case. Next, test D 205 was performed, resulting in failure due to shearing at grid line 13 after 1000 cycles. Actually, tests D 105 and D 205 can together be regarded as constituting one test.

explain. It is probably bound up with the fact that these joints were repaired after test series C and therefore had no "previous history". Another explanation could be that no bending moment occurs at those grid lines, so that no flexural cracks liable to have an unfavourable effect on the stress distribution in the transverse joint could develop there.

3.1 Longitudinal forces at the transverse joints at grid lines 2, 5, 7 and 11

The measured values of the tensile forces in the longitudinal connections at grid lines 2, 5, 7 and 11 are plotted in three kinds of diagram in Figs. Al to

» A20. These represent the relationships between:

a) the bending moment and the tensile forces in the longitudinal connections; b) the magnitude of the tensile forces at a section, compared with one

another, after a certain number of cycles;

c) the tensile forces in the longitudinal connections and the number of cycles.

Fig. 3.1 gives an overview of the diagrams contained in Figs. Al to A20, which show the calculated values for once-only loading as well as the measured values of test C 401 for a monotonically increasing load. The tensile forces mentioned in (b) and (c) were measured at the maximum and the minimum loads of +12 kN and -12 kN respectively. However, for reasons associated with the control system adopted, some deviation from these load values occurred after a series of cycles. M+ M--D(A) •C(B) -B(C) F i g . 3 . 1 O v e r v i e w o f F i g s . Al I o A20.

3.1.1 Longitudinal forces at grid line 7

In Figs. Al to A3, for tests D 5, D 105 and D 205, the tensile forces in the longitudinal connections at points A7, B7, C7 and D7 have been plotted against the bending moment at grid line 7. In Fig. A4, for tests D 5, D 105 and D 205, the relationship between the tensile forces at grid line 7 for a maximum load Q= +12 kN and a minimum load Q= -12 kN is represented in two diagrams in each case. For test D 5 these forces after 1, 200 and 400 cycles, and for tests D 105 and D 205 after 1, 200, 400, 600, 800 and 1000 cycles, have been plotted.

In Fig. A5, for tests D 5, D 105 and D 205, the tensile forces in the longitudinal connections at grid line 7 have been plotted against the number of cycles. The left-hand diagrams relate to a maximum load Q= +12 kN, the right-hand diagrams to a minimum load Q= -12 kN.

3-1-2 Longitudinal forces at grid line 5

The longitudinal forces at grid line 5 have, in the same way as described in Sec. 3.1.1, been plotted in three kinds of diagram in Figs. A6 to AlO. A comparison with test C 401 with a monotonically increasing load was not possible because in this test the joint at grid line 5 had not cracked.

3.1.3 Longitudinal forces at grid line 11

The longitudinal forces at grid line 11 have, in the same way as described in Sec. 3.1.1, been plotted in three kinds of diagram in Figs. All to A15. A comparison with test C 401 with a monotonically increasing load was not possible because in this test no measurements were performed at grid line 11.

3.1.4 Longitudinal forces at grid line 2

The longitudinal forces at grid line 2 have, in the same way as described in Sec. 3.1.1, been plotted in three kinds of diagram in Figs. A16 to A20.

A comparison with test C 4Ö1 was not possible because in this test no cracking had occurred at grid line 2.

3.1.5 Distribution of the longitudinal forces at grid lines A and D

The tensile forces in the longitudinal connections at grid lines A and D for a horizontal loading Q= +12 kN and -12 kN are shown Fig. 3.2. In this diagram the calculated values are compared with the measured values of test C 401 for a once-only load and with the measured values of test D 205 for cyclic loading

(after 1000 cycles).

? „ 1 2 3 4 5 6 7

grid lines

9 10 11 12 13 14

Fig. 3.2 Tensile forces in the longitudinal connections at grid lines A and D.

3-2 Shear deformations in the transverse joints at grid lines 2 and 11

In Figs. A21 to A35 the measured values of the shear deformations at grid lines 2 and 11 have been plotted in three kinds of diagram. These represent the relationships between:

c) the shear deformation and the number of cycles.

V+

f

V-w

Fig. 3.3 Overview of Figs. A21 to A35.

Fig. 3.3 gives an overview of the diagrams contained in Figs. A21 to A35. The coding of the measured points which has been adopted in these diagrams is illustrated by the following example:

2-BA = the measured point on grid line 2, at grid line B on the side nearest grid line A.

The location of the measured points on a transverse joints is indicated in Fig. 3.4.

(D

©

2-A8 2-BA -•- '— :ü 1- . 2-C8y ^ -•-' - — ' — D ^ = 1-measuring points - 2-DC

plan of floor _{Strom diagram}

Comparing the measured values with a numerical analysis is not possible at this stage of the research because characteristic properties of a cracked joint are not known. Nor is it possible to compare the measured values from tests under cyclic loading and under monotonically increasing load because in test C 401, which could potentially have made this possible, the joint at grid line 2 did not crack and no measurements were performed at grid line 11.

3.2-1 Shear deformation at grid line 2

For tests D 5, D 105 and D 205 the shear deformations at grid line 2 have been plotted against the shear force at that grid line in Figs. A21 to A23. For each test these relationships are represented in six diagrams corresponding to the number of measured points on the grid line.

For tests D 5, D 105 and D 205 the shear deformations at six points on grid line 2 have been plotted against the crack width at these points in Figs. A24 to A26. This relationship after 1, 200 and 400 cycles is shown for test D 5,

and after 1, 200, 400, 600, 800 and 1000 cycles for tests D 105 and D 205. \

For tests D 5, D 105 and D 205 the shear deformations at grid line 2 have been plotted against the number of cycles in Fig. A27. The left-hand diagrams relate to a load Q= + 12 kN (approx.), the right-hand diagrams to a load Q= -12 kN

(approx.). . • ;

3.2.2 Shear deformation at grid line 11

In Fig. A28 to A34 the shear deformation at grid line 11 has been plotted in three kinds diagram in the some way as has been described in Sec. 3.2.1.

3.2.3 Shear deformation in the longitudinal joints at grid line B near grid line 11

deformation were performed on both sides of grid line 11. The locations of these measurements are indicated in Fig. 3.5, which shows the coding adopted for the measured points represented in the diagrams.

®

(A)

Ki^AB

i A ^ > - _ ^ i ^ 1 0 / 1 2 B C - i o n r 11-OB^

©

®:

' ^ ^ f ^ T BC-11/12 11-BC CD 31 11-DC

Fig. 3.5 Location of measurements performed at grid line 11.

In Fig. A35, for test D 205, the shear deformation has been plotted against the number of load cycles. The left-hand diagram relates to a load Q= +12 kN

(approx.), the right-hand diagram to a load Q= -12 kN (approx.).

In Fig. A37, for test D 205, the shear deformation has been plotted against the crack width. This relationship after 1, 200, 400, 600, 800 and 1000 cycles is shown.

3.2.4 Shear deformation in the longitudinal joints between grid lines 1 and 2

width and shear deformation were performed between grid lines 1 and 2. The locations of these measurements and the coding adopted for them are indicated in Fig. 3.6.

©

©

©

D c

©

A

•X-®

'T—

'2 ZL ^ BA-1, '2 ^ BC-1)'2 I C8-1''2 CD-I'2 ^ D C - 1 '2

Fig. 3.6 Location of measurements performed between grid lines 1 and 2.

In Fig. A36, for tests D 5, D 105 and D 205, the shear deformations have been plotted against the number of cycles. The left-hand diagrams relate to a load Q= +12 kN (approx.), the right-hand diagrams to a load Q= - 12 kN (approx.).

3.3 Crack width in the transverse joints at grid lines 2, 5, 7 and 11

The measured values of the crack widths at grid lines 2, 5, 7 and 11 have been plotted in four kinds of diagram in Figs. A38 to A61. These represent the relationships between:

a) the crack width and the tensile force in the corresponding longitudinal connections;

c) the crack width and the number of load cycles;

d) the distribution of the crack width along a transverse joint (except at grid line 5 ) .

Fig. 3.7 gives an overview of the diagrams contained in Figs. A38 to A61.

Fig. 3.7 Overview of Figs. A38 to A61.

3.3.1 Crack width at grid line 7

\\ • i

In Figs. A38 to A40, for tests D 5, D 105 and D 205, the crack widths at grid line 7 have been plotted against the corresponding tensile force in the longitudinal connections. These measurements were performed at six different points on this grid line. The calculated values for once-only loading are also indicated in these diagrams.

Comparison with the measured results from test C 401 is possible only at point 7-DC, no measurements having been performed at other points on grid line 7.

The distribution of the crack width along grid line 7 is shown in Fig. A41. For test D 5 the distribution after 1, 200 and 400 cycles, and for tests D 105 and D 205 after 1, 200, 400, 600, 800 and 1000 cycles, is shown here.

In Fig. 42, for tests D 5, D 105 and D 205, the crack widths at grid line 7 have been plotted against the number of load cycles. The left-hand diagrams relate to a maximum load Q= +12 kN (approx.), the right-hand diagrams to a minimum load Q= -12 kN (approx.).

3.3.2 Crack width at grid line 5

In Fig. A43, for tests D 5, D 105 and D 205, the crack widths at grid line 5 for the points 5-AB and 5-DC have been plotted against the corresponding longitudinal forces. The calculated values for once-only loading are also included in these diagrams. Comparison with test C 401 is not possible because in that test no cracking occurred at grid line 5.

In Fig. A44, for tests D 5, D 105 and D 205, the crack widths at grid line 5 for the points 5-AB and 5-DC have been plotted against the total shear force at grid line 5, The calculated values for once-only loading are also included in these diagrams.

In Fig. A45, for tests D 5, D 105 and D 205, the crack widths for the points 5-AB and 5-DC have been plotted against the number of load cycles.

3.3.3 Crack width at grid line 11

In Figs. A46 to A48 the crack widths at grid line 11 have been plotted against the corresponding longitudinal force in the same way as described in Sec. 3.3.1. Here, too, the measured values are compared with the calculated values for once-only loading. Comparison with test C 401 is not possible because in that test no cracks were measured at grid line 11.

In Figs. A49 to A51, for tests D 5, D 105 and D 205, the crack widths at six different points on grid line 11 have been plotted against the total shear force at grid line 11. The calculated values for once-only loading are also included in these diagrams.

The distribution of the crack width along grid line 11 is shown in Fig. A52.

In Fig. A53 the crack widths have been plotted against the nimiber of cycles in the same way as described in Sec. 3.3.1.

In Figs. A54 to A56 the crack widths at grid line 2 have been plotted against the corresponding longitudinal force in the same way as described in Sec. 3.3.1. Comparison with test C 401 is possible only for point 2-DC. In that test no cracking or tensile force occurred at the other points.

In Figs. A57 to A59, for tests D 5, D 105 and D 205, the crack widths at six different points on grid line 2 have been plotted against the total shear force at grid line 2. The calculated values for once-only loading are also included

in these diagrams. Comparison with test C 401 is possible only for the points 2-CD and 2-DC. In that test no cracking occurred at the other points.

The distribution of the crack width along grid line 2 is shown in Fig. A60.

In Fig. A61 the crack widths have been plotted against the number of cycles in

the same way as described in Sec. 3.3.1. '

I I .

I i 3-4 Lateral deflection of the longitudinal edge > f

In Figs. A62 to A68 the values for the deformation of the longitudinal edge are shown in four kinds of diagram for tests D 5, D 105 and D 205. The measured values for tests C 401 and C 206 in series C are also included in these diagrams, these being tests performed with monotonically increasing load. Test C 401 relates to a floor bay without tensile force acting at the ends and without preformed cracking of a transverse joint. Test C 206 relates to a floor bay with a total tensile force F = 3.04 acting at the ends and a 0.3 mm wide preformed crack at grid line 2.

The values calculated with the aid of a simplified mathematical model (see Appendix D to Stevin Report 5-85-14 [2]) have also been included in these diagrams.

The values calculated with the aid of the ZEFE computer program are likewise represented in Figs. A62, A64 and A66. In these diagrams the lateral deflections of the longitudinal edge at point D7 have been plotted against the total horizontal load for tests D 5, D 105 and D 205 respectively.

The Q-5 diagram after 1, 200 and 400 load cycles for test D 5, after 1 and 1000 cycles for test D 105, and after 1, 400 and 1000 cycles for test D 205, is presented here.

In Figs. A63a, A65a and A67a the lateral deflection of the longitudinal edge at grid line D is shown for a maximum load Q= +12 kN and a minimum load Q= -12 kN, respectively, for tests D 5, D 105 and D 205.

Likewise for Q= +12 kN and Q= -12 kN, the increase in lateral deflection after a certain number of load cycles in relation to the first load cycle is shown in Figs. A63b, A65b and A67b. This increase in deflection is shown in order to reveal more clearly what effect the shear deformation in the transverse joints has on the total lateral deflection.

In Figs. A63a, A65a, A67a and in Figs. A63b, A65b, A67b the effect of the measured shear displacement at grid line 11 on the deflection curves is represented. From the shape of these lines it appears that at grid line 12 in test D 5 and at grid line 13 in tests D 105 and D 205 there must also have occurred a shear deformation. The magnitude of these deformations is not known because no measurements were performed on grid lines 12 and 13. The estimated shear behaviour of these joints is shown dotted in these diagrams.

In Figs. A68a, b and c, for tests D 5, D 105 and D 205, the lateral deflection curves have been drawn, neglecting the deflection due to the shear displacements in the transverse joints.

3.5 Deformation of the ends of the floor bay

In Fig. A69, for tests D 5, D 105 and D 205, the deformation of the ends of the bay at grid line 1 is shown in three diagrams. The deformation has been plotted for a total maximum and minimum horizontal load Q= +12 kN and Q= -12 kN respectively. The deformation is shown after 1, 200 and 400 cycles for test D 5, after 1 and 1000 cycles for test D 105, and after 1, 400 and 1000 cycles

for test D 205.

In Fig. A69 the measured values are compared with the calculated values. Also included are the measured values from test C 401 in series C for monotonically increasing load.

3.6 Failure load

For the tests C 206 and C 306 in series C and D 5, D 105 and D 205 an overview of the failure load, the failure mode, the shear force at failure and the number of load cycles are given in Table 3.1. The tests in series C are, however, not entirely comparable with those in series D because in C 206 and C 306 the ends of the floor bay were loaded by a total horizontal tensile force F=3.04 kN and moreover there was a preformed crack at grid line 2 in test C 206 and at grid line 13 in test C 306. The measured values given in Table 3.1 are represented in a diagram in Fig. 3.8, where the number of cycles has been plotted to a logarithmic scale against the shear force Vu at failure (ultimate shear force).

The number of data on the failure load Vu in relation to the number of load cycles is insufficient to make predictions as to the number of cycles to failure under a different horizontal load.

The dotted lines between tests C 206 and C 306 and those between tests C 206 and D 105 and D 205 indicate a possible relationship between the shear force and the number of cycles to failure.

Table 3.1 Overview of failure loads. monotonically increasing load repeated loading alternating cyclic load tost C206 C306 D5 D105+D205 failure load (kN) +30.3 -+12.5 -11.8 + 12.1 -12.5 number of cycles -6200 400 2000 failure mode shearing at grid line 2 shearing at grid line 13 shearing at grid line 12 shearing at grid line 13 shear force Vu at failure (kN) +13.9 +10.2 +4.7 -4.4 +5.5 -5.7

The fact that in test D 5 failure at grid line 12 occurred at a lower value of the load and at a lower number of cycles than in the case of tests D 105 and D 205 is possibly due to the fact that in test C 306 the joint at grid line 12 had already been subjected to 6200 cycles of repeated loading of different magnitudes. Grid line 13, on the other hand, had been repaired before the start

V„ ( k N )

10

V-^C206 " • • • - • — - _ a x i s 2 \ ï- •' j—repeated loadinc ,— cyclic loading

t

D 5 - V a x i s 12 1 1 1 4 0 0 ~ - - - — é ^ C 3 0 6 - ^ axis 13 \ 0 205 axis 13

1 1

1 1 1 |-2000 6200

10

100

1000 10000

n = n u m b e r of c y c l e s

Fig. 3.8 Relationship between shear force at failure Vu and number of load cycles.

Note: previous history of the tests

Test C 30n: 25 tests wi.l:h inoaoLon i ca 1 I y incri^asing load;

TOKI; Ü 5 : (5200 rfp<-al.(Hl loading cy(.-le;;o(' I.eslH C 'Ai)V> al. (lirfert-nl values oC the load and 6200 alternating cycles between Q = +6 kN and -6 kN; Test D 205: inclusive of 1000 alternating cycles of test D 105 between

4. ANALYSIS OF TEST RESULTS OF SERIES D

The analysis of the test results of series D will be performed on the basis of the measured values as assembled in Chapter 3 of this report. In this context the data used for comparison are the results of the measurements obtained in test series C with monotonically increasing load and the calculated values obtained with a simplified mathematical model [2] and, where possible, with the ZEFE computer program [9].

In this analysis the following will be considered: Sec. 4.1: Forces and deformations at grid line 7. Sec. 4.2: Forces and deformations at grid line 5. Sec. 4.3: Forces and deformations at grid line 11. Sec. 4.4: Forces and deformations at grid line 2. Sec. 4.5: Lateral deflection of the longitudinal edge. Sec. 4.6: Deformations of the ends of the floor bay. Sec. 4.7: Failure of the floor bay.

The conclusions drawn from this analysis are valid only for the floor bay under investigation. In drawing generally-valid conclusions from these results it is necessary to proceed with caution because, among other reasons, the geometric features of the floor bay were not varied in this experimental research, only one type of longitudinal connection was applied, and only one grade of jointing mortar was used in all the tests. Thus this analysis provides a qualitative

indication of the behaviour of the bay rather than a quantitative assessment which would enable the behaviour of any type of floor to be predicted.

4.1 Forces and deformations at grid line 7

On the basis of a structural analysis a linear relationship can be expected to exist between the tensile forces in the longitudinal connections and the bending moment at the relevant section.

As appears from Figs. Al to A3, there is reasonably good agreement between the measured and the calculated values of the tensile forces at the points A7 and D7, At the points B7 and C7 there is less good agreement,

It is, on the basis of a structural analysis, also to be expected that at a section the tensile forces acting there vary linearly in relation to one another.

From Fig. A4 it appears, however, that - both for a positive load with tension at D7 and for a negative load with tension at A7 - the agreement between the measured and the calculated values is not good.

For a positive load it is notable that the tensile forces at B7 are larger than at C7, whereas a structural analysis indicates that the reverse would have to be the case. For a negative load the tensile forces do indeed show a linear relationship with one another, yet there is a considerable difference between the calculated and the measured values.

In order to investigate whether these differences in calculated and measured values disturbed the internial equilibrium of the structure, the relationship between the internal and the external bending moment at grid line 7 is shown in

Fig. 4.2. Here, for each test, the external moment has been plotted against the internal moment in two diagrams. Equilibrium between the two moments exists if the plotted line is straight and forms an angle of 45° with the co-ordinate axes.

The external moment was calculated from the sum of the reactions at grid lines 1 and 14. The external moment at grid line 7 is expressed by:

Me = 1.26 (Ri + Ri4)

The internal moment at the transverse joints cemnot be accurately determined with the aid of the measured results because the position of the resultant

obtain some idea of the magnitude of the internal moment, the resultant compressive force will be assumed to coincide with grid line A, an assumption which yields the highest possible value of this moment. The internal moment is then expressed by the following formulae (see Fig. 4.1):

SI " f o r p o s i t i v e l o a d : Mi = Nsi x 2 . 8 8 + Ns2 x 1.68 + Ns3 x 1.20 -Nsi \^ t ^ ^ •^ N s

Y-Mi

for negative load: Mi - Ns4 x 2.88 + Ns3 x 1.68 + Ns2 x 1.20

Fig. 4.1 Schema of forces for the determination of the internal moment.

From Fig. 4.2 there is seen to be good agreement between the internal and the external moment. The discontinuity of the section therefore has no effect on this. It is also apparent that the number of load cycles has no effect on this either.

»»»»»»» » o 5 » »»»»»»»

Fig. 4.2 Relationship between internal and external moment at grid line 7; tests Ü 5, D 105 and D 205.

The above-mentioned differences between the calculated and the measured values may be due to the following possible causes:

a) There occurs a discontinuity in the action of the force in consequence of shear deformation in the longitudinal joints.

b) The linear behaviour assumed for the longitudinal connections in the calculation is not in accordance with the actual behaviour.

In order to ascertain the influence of these two possible causes, the occurrence of cracking will first be brought into the analysis.

In Fig. 4.3, representing the floor bay on plan, the cracking of the bay at the end of test series D is shown. It appears from this diagram that a continuous crack had developed at grid line B, a fairly long crack at grid line A and grid

line C, and some short cracks at grid line D. Besides, it has already been shown in Sec. 2.5 that, on the basis of a structural analysis, the shear stresses in the longitudinal joints are likely to have been very high.

®

©

© © ® 0 ® ® 0 ® 0 ® ® ® ( § ) ®

joint repaired after test D5 M M failure in stiear after 2000 cycles

T

13 « 720 = 9 360 — crack test D 205

Fig. A41, showing the distribution of the crack width along grid line 7, reveals some notable aspects. For one thing, it appears from this diagram that both for a positive and for a negative load the crack width exhibits a reasonably continuous distribution. From this it can be inferred that virtually no shear deformation in the longitudinal joints on both sides of grid line 7 can have occurred. On the basis of this the behaviour of the tensile forces at grid line 7 is also likely to have been continuous. However, according to Fig. A4, this was not the case under positive load. Under negative load, on the other hand, it was indeed so.

Secondly, it appears from Fig. A41 that the crack width distribution under positive load is different from that under negative load. With positive load the crack width is considerably greater than with negative load of equal magnitude. However, the reverse situation occurs in the case of the tensile forces (see Fig. A4), where the tensile force at A7 under negative load is greater than the tensile force at D7 under positive load.

Thirdly, it appears from Fig. A41 that the compression movement across the joint, particularly under negative load, was large in the compressive zone. Presumably this was due mainly to the previous history of the tests in question. In preceding tests in series C the floor bay had always been subjected to positive load. It is conceivable that jointing material which had been crushed during those tests prevented complete closure of the joint, so that a large compression movement could occur under negative load.

From Figs. A38 to A40 it appears that at most of the longitudinal connections there is no linear relationship between the crack width and the tensile forces in these connections. It does appear, however, that after a certain amount of deformation the slope of the calculated and that of the measured values are in good agreement with each other.