Models for behaviour analysis of monolithic wall and precast or monolithic floor slab connections

Engineering Structures 40 (2012) 466–478

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Engineering Structures

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Models for behaviour analysis of monolithic wall and precast or monolithicfloor slab connections

Damir Zenunovic a, Radomir Folic b,⇑a Faculty of Mining, Geology and Civil Engineering, Tuzla, Bosnia and Herzegovinab Faculty of Technical Sciences, Trg D. Obradovica 6, 21000 Novi Sad, Serbia

a r t i c l e i n f o a b s t r a c t

Article history:Received 15 July 2011Revised 14 February 2012Accepted 1 March 2012Available online 11 April 2012

Keywords:Semi-rigidConnectionsPrecast and monolithic slabMonolithic wallComparative analysisExperimentNumericalFEMStrut and Tie model

0141-0296/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.engstruct.2012.03.007

⇑ Corresponding author. Tel.: +381 21 485 2644; faE-mail address: [email protected] (R. Folic).

This paper presents the comparative analyses of experimental and numerical research results of RC con-nections between the precast slab and monolithic wall (Type 1), and both monolithic slab and wall ele-ments (Type 2). Type 1 was applied in prefabricated building system developed in Tuzla, Bosnia andHerzegovina, as the most sensitive part of the structure. In order to provide a favourable response ofthe bearing structure connections under seismic load (Type 1) it is dislocated in the span. In order to com-pare the behaviour of listed structure connection types, three specimens made of precast slabs and mono-lithic wall as well as three specimens made of monolithic slabs and walls were tested under quasi-staticload. Thus, the mathematical models are proposed in order to analyse both types of connection, based onthe exact method of displacement and FEM. Furthermore, the stiffness matrix is modified by introducingthe stiffness parameter (semi-rigid) connection. The approximate Strut and Tie model is proposed accord-ing to the stress field analysis obtained by FEM.

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1. Introduction

In modern civil engineering, prefabricated constructions make asignificant share of building structures. Generally, prefabricatedstructure is composed from two sets: elements and connections.Existing prefabricated building systems are improved in a way toachieve the quality with high reliability. Joints/connections arethe most sensitive parts of the precast structures. Selection of thesystem of connections influences not only the structure safety,but durability, production and erection procedure. Behaviour ofprecast concrete structures depends substantially on their quality.Precisely assessing the behaviour of the precast connection is afundamental problem in the analysis of prefabricated structuresas well as creating a numerical model of real behaviour. Also, thereare various prefabricated building systems, and it is the problem ofuniformity of numerical models.

By calculation procedure connections are treated ideally rigid orof the hinges (pinned). In reality, all connections are semi-rigid. To-tal precast frame analysis may therefore be carried out by substitut-ing rigid joint connections with ones of finite strength androtational stiffness. The behaviour may be described in terms ofthe well known moment-rotation data, but in the case of precast

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slab – monolith wall connections, the semi rigidity is due to materialand deformations. Relatively different behaviour of connections ofelements is introduced through the semi-rigid connections, i.e. con-nections yielding. It is defined through determination of relativedeformation parameters (strain and rotation) of elements in jointarea. In the paper [1] rotational spring stiffness-connection ratiorelation is explained and revealed. The finite element analysis onfour types of precast connections which are pinned, rigid and semirigid is presented in the paper [2]. The stiffness of the connectionsis obtained from the slope of the total load versus deflection curvein the elastic domain. Model of precast joint using 3D solid elementsand surface-to-surface contact elements between the beam/columnfaces and interface grout in the vicinity of the connection was pre-sented in the paper [3]. Results from the finite element analysis cor-relate fairly well with experimental results.

Experimental researches are necessary for defining the charac-teristics of connection yielding. A lot of researches have been per-formed and it represents a considerable data base of characteristicsof precast connections [4–13]. Some of them are presented and de-scribed in the paper [14]. According to the findings modificationsof monolithic structure calculation are made. However, problemof unification and application of uniform analytical approach fordifferent precast connections still exist. In order to attain a goalit is necessary to increase the data base for connection characteris-tics, with further experimental and analytical researches.

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D. Zenunovic, R. Folic / Engineering Structures 40 (2012) 466–478 467

Experiences of the above-mentioned experimental researchesare used in the preparation of the experimental program for the re-search presented in the paper. Prefabricated building system called‘‘MMS’’, which is scheduled for construction of residential build-ings, was developed in Tuzla, Bosnia and Herzegovina, in the80th of the last century. Bearing structure is designed with mono-lithic reinforced concrete walls (MRCW) and prefabricated rein-forced concrete slab (PRCS) (Fig. 1). Connection of MRCW andPRCS is dislocated in the span in order to prevent damage ofwall–slab joint during the earthquake (Fig. 1). According to the pa-per [15] precast connections fail to emulate properly monolithicconstruction. In case when emulation of monolithic constructionis desired, beam rotations inside the joint should be minimised.The paper recommends forcing the concentration of beam rota-tions far from the column faces, i.e. relocating the beam plastichinges.

Therefore, the analysis of dislocated connection of the walls andslabs is based on the experiences of previous empirical and theoret-ical basis. The program of experimental and numerical research ofthese connections was made in the period from 2004 to 2007. Con-nections of two slab types: 1-precast, 2-monolithic and monolithicwall are studied. The research objective was to define precast con-nection rigidity and investigate its behaviour in relation to mono-lithic connection. Preliminary the results of the experimentalstudies were published in the paper [14], and the research wasdescribed in details. In this paper, the details of the experimental re-search have been highlighted which is essential for the presentationof comparative experimental and numerical research. The moreprecise mathematical model for connection analysis is proposedwith appliance of finite element method (FEM). The stiffness matrixis modified by introduction of semi-rigid connections which givessatisfactory results. Additionally, the approximate Strut and Tiemodel is formulated for precast slab and monolithic wall connectionanalysis.

2. Experimental program

The research program consists of comparative and numerical re-search, using the experience and recommendations from the re-search presented in [4–8,16–18]. Experimental investigations

Fig. 1. Specimen with precast slab and monolithic wall.

include testing of three specimens with precast connection (Fig. 1)and three comparative monolithic specimens. The aim of the testsis the comparison of precast and monolithic connection behaviour,as well as definition of connection yielding. Thickness of PRCS is16 cm; while monolithic part of structure (MRCW) has the thicknessof 15 cm. Monolithic specimens are performed with the same geo-metrical parameters. Recommendations given in [16,27] are usedin selection of geometrical relation of specimens (Fig. 2). Geometryof the specimen on the Fig. 1 is accepted according to the geometricrelations presented on the Fig. 2, and relation between the lever armof the load of cantilever (d) and the thickness of the precast slab (h).Findings of the experiment show that the effect of short bracket islost in the relation d/h = 5.0. In this case the relation is d/h = 90/16 = 5.625. In accordance with the proposal of the authors of treatedprecast system, connection of MRCW and PRCS is performed with‘‘tooth’’ and dislocated in the span. The bottom of connection is dis-located 5.0 cm from wall internal surface, while the connection topis dislocated 12.0 cm, due to technological reasons. Adopted widthof the specimen is 50 cm.

Monolithic parts of the structure (MRCW and joint) are per-formed from concrete class C 25/30, while PRCS is made of concreteC 30/37. Reinforcement (rebar) S 400/500 (fyk = 400 MPa) is used inthe construction. Longitudinal rebars (loops) are £12/15 cm, anddistributing rebars £10/30 cm. Rebars for loops stiffening are£12. Walls are reinforced with mesh reinforcement fyk = 500 MPa(Figs. 3 and 4).

During the manufacture of the specimens, samples of in-builtmaterials (concrete and steel) were tested in order to determinetheir mechanical properties (Fig. 5). The results of the tests are pre-sented in the paper [14] and Table 1.

The specimens are placed into a claw press of maximum capac-ity of 6500 kN, which is used to restrain the specimen. The walls ofspecimen are pressed by loading that causes normal stresses of0.4fck. The degree of constraint is controlled by deflectometer atthe measurement points U6 and U7 (Fig. 6). The samples are loadedby a hand pumped hydraulic press of 100 N measurement accu-racy. Loading of specimens was applied in 4 stages up to the ser-vice load Pserv. The service load of specimen (Pserv = 17.18 kN) isdefined as the loading at the wall/slab joint caused by internalforces of the same intensity as in a real structure with the slab

Fig. 2. Geometric relationship of specimens.

Fig. 3. Slab reinforcement.

Fig. 4. Wall reinforcement.

Fig. 5. Concrete and steel samples.

Table 1Average testing values of concrete and reinforcement samples.

Tests of characteristics Averagevalues

Concrete compressive strength (MPa) (evaluatedon cubes 15 � 15 � 15 cm)

Precastplate

47.37

Monolithicwall

37.55

Monolithicmodel

49.63

Modulus of elasticity (GPa) (evaluated on cylinders15 � 30 cm)

Precastplate

34.85

Monolithicwall

33.05

Reinforcement strength (MPa) (evaluated onsamples 150 mm in length)

Yieldstrength

568

Tensionstrength

698

468 D. Zenunovic, R. Folic / Engineering Structures 40 (2012) 466–478

range of 6.15 m. Namely, the distance of the bearing walls of MMSsystems is 6.15 m.

After the stages P/2 and P specimens were unloaded. After that,they were loaded up to failure. The following measurements wereperformed for every load stage:

� Strain of reinforcement loops. Steel bolts welded on barswere used (Figs. 3 and 4). Measurements were executedwith mechanical extensometer type Demac with measure-ment base of 250, 150 and 50 mm and measurement accu-racy of 0.001 mm. Point of measurements, marked as Da, ispresented in Figs. 7 and 8.

� Strain of concrete is marked as Dn (n = 1, 2, 3,. . .) (Figs. 7and 8).

� Deformation (deflection and rotation) of walls, joints andslabs with deflectometers of 50 mm range with accuracyof 0.01 mm and 10 mm range with accuracy of 0.001 mmis presented in Fig. 6.

3. Experimental results and discussion

The movability of the supporting part of the deflectometer, theoperation of hands on the deflectometer, and the clamping ofspecimen were controlled prior to the process of measurement.Specimen was test-loaded with the load intensity of up to P/4.After the calibration of instruments, the measurements startedfor the individual loading stages. The measured values at themeasuring points from U1 to U5 were processed in relation to themeasuring points U6 and U7. Deflectometer measurement of placesU6 and U7 were used as control of model constraints and for deter-mination of the values of slab rotation to the wall.

For example, the data U�1 is,

U�1 ¼ U1 �U6 � ð38:5þ8Þ cm

ð63:9þ7:5Þ cm

h iþ U7 � ð38:5þ8Þ cm

ð59:5þ7:5Þ cm

h i2

The processed measurement results are presented in Table 2.Based on the results, shown in the table, and measuring of con-

crete and reinforcement strain, curves P–D and M–U are devel-oped. The rotation U is defined as the average of the valuesmeasured by deflectometers and extensometers (Fig. 8). Similarprocedure was presented in the paper [19].

The results of the measured deflection are presented in Figs.9–11. Force–displacement curve determined at measurementplace U5, located over applied load is presented in Fig. 9. On Figs.10 and 11 M–U curves for cross-section rotation 3.0 cm far frominternal wall surface and cross-section on distance 20.5 cm farfrom wall are presented.

During the experiment, the crack opening mechanism and thestate just before failure were observed. Micrometer was used formeasuring the width of the crack. First crack on the monolithicand precast specimens opened at the slab – wall joint with the sizeof the force of 18–24 kN, i.e. the size of the bending moment rangesfrom 15 to 19.8 kNm. The angle of the crack was approximately45�. After reaching a length greater than 3 cm the direction ofthe crack was shifted towards the joint, i.e. branch of the crackmade the angle with the vertical axis less than 45�. When theintensity of the load P � 34.0 kN, i.e. the size of the bending mo-ment M � 28 kNm crack in slab–wall joint, by precast specimens,

Fig. 6. Arrangement of deflectometers.

Fig. 7. Arrangement of measurement points.


has the width of 0.4 mm (Fig. 12A). This width is for monolithicspecimens, at the intensity of the load P � 38.0 kN (Fig. 12B.) Atprecast specimen the crack opened at the precast connection ofthe MRCW and PRCS (Fig. 12A), while on the monolithic specimenthe second crack opened at a distance of 15–16 cm from the wall –slab joint (Fig. 12B).

With the increase of load, the length and width of the crack atthe wall–slab joint is also increasing, changing also the crack pat-tern inclination towards the vertical axis (Fig. 12C and D). Afterthe opening of a second crack, the crack opening mechanism inthe precast specimens has the property that the second crack inthe precast connection opens more intensely than the crack atthe wall–slab joint, so that for the load intensity of P = 40 kN, i.e.for the moment of M = 33.0 kNm, the width of each crack is equal,i.e. 0.55 mm (Fig. 12C). In monolithic specimen, with the furtherincrease of load, the next crack opens at a distance of 15 cm fromthe second crack (Fig. 12D).

At the load of P = 48 kN, i.e. the size of the bending momentM = 40.0 kNm crack at the junction of the wall–slab for the precast

specimen reaches a maximum value of width w = 0.8 mm andlength l = 12 cm. At the end, its path is nearly vertical. At the sametime cracks in the dislocated precast connection has a width ofw = 1.0 mm. The next crack open on the regular grid of 12–13 cm. Further cracks open following the pattern of the regular gridfrom 12 to 13 cm. At the above load intensity, reinforcement yield-ing occurs. Cracks at the joint of the wall–slab are still not devel-oped and gradually close with further development of the crackin the precast connection (Fig. 13A). Maximum implemented loadon the precast specimen is Pmax = 52 kN, while on the monolithicspecimen is Pmax = 58 kN. Thus the experimentally determined fail-ure load for the monolithic connection is for 11.5% higher than forthe prefabricated connection. At these load intensities, reinforce-ment yielding, structural deterioration and decline of force athydraulic press occur.

The damage of the model shows the characteristics of ductilefracture. In fact, registered at the fracture site the opening of thecracks have the width wmax = 20 mm and strain of reinforcement60‰. This ductility is the result of double reinforced section.

Fig. 8. Arrangement of extensometers.

Fig. 9. P–D average curves over applied load.


According to [20], Fig. 10.22, p. 350, the increase of the percentageof reinforcement in the compressed zone improves the ductility.Displacement ductility is Uu/Uy > 16.0 for the relationship of rein-forcement in compressed and tension zone q0/q = 0.75.

On the basis of the experiments, in [11] the beam-column jointfailure was classified as:

– Beam failure.– Joint failure.

Experiments on the specimens designated as RK1 and RK4,where the joint reinforcement is consistent with that in the speci-men presented in this paper that the beam failure occurs at thegeometric relation of hbeam/hcolumn 6 1.25, while the joint failure oc-curs at hbeam/hcolumn P 1.50. Crack paths immediately before thefailure are presented in [11, p. 657].

The crack-opening mechanism, as identified by experiments onprecast and monolithic models, indicate a slab failure instead of ajoint failure, which corresponds to the relations given in [11].Namely, the geometric relation is hslab/hwall = 16/15 = 1.07. This

Table 2The measurement results of deflectometer for different phases of load.

Load phases Precast specimensAverage values (mm)

U�1 U�2 U�3 U�4

0 0 0 0 0I phase – P/4 �0.23 +0.18 �0.51 +0.33II phase – P/2 �0.51 +0.40 �0.94 +0.71Unloaded �0.04 +004 �0.06 +0.05III phase – 3P/4 �0.78 +0.61 �1.38 +1.31IV phase – P �0.97 +0.77 �1.95 +1.95Unloaded �0.09 +0.07 �0.14 +0.20

1.33P �1.17 +0.93 �2.45 +2.531.89 P �1.46 +1.35 �3.20 +3.322.44 P �2.15 +2.10 �4.42 +4.562.78 P �3.13 +3.15 �5.90 +6.50

conclusion may be confirmed also by comparing the experimen-tally identified crack paths (Figs. 11–13) with Fig. 4a in [11].

A similar type of failure can be seen in [5, p. 132], type ‘‘O’’. Onpage 133 of this paper there is a conclusion that the cracks openingmechanism depends on the relation mE = Eb,p/Eb,j, where Eb,p is theelasticity modulus of the concrete of prefabricated elements andEb,j is the elasticity modulus of concrete in the joint. With the iden-tified value of relation mE of 0.9 6mE 6 1.12 the crack opens eitherin the concrete of slab or in the concrete of joint, instead of thelocation of failure, i.e. at the joint of the prefabricated and mono-lithic part of the structure. If 0.9 > mE > 1.12, the crack opens atthe joint. On the tested precast specimen, the relation mE is 1.05.Also in [5, p. 132], it was noted that in the vast majority of exper-iments, the first cracks open at the joint.

With the introduction of the relation b = er/fe, where er – strainat the failure, ef – strain at the opening of the first crack, afterextensive experimental research in [5] the following possibleforms of failure were identified:

– 5 < b < 8 an elastic crack opens with concrete crush,– 8 < b < 12 simultaneous failure,– b > 12 a plastic type of crack opens, failure of the reinforcement

or failure of the reinforcement anchorage.

On the tested precast and monolithic models, the relation be-tween failure strain and the strain at the first crack opening isb � 28.

It is important to emphasise that the testing was done unilater-ally by quasi-static load, and therefore should take into account the

Monolithic specimensAverage values (mm)

U�5 U�1 U�2 U�3 U�4 U�5

0 0 0 0 0 00.67 �0.11 +0.09 �0.21 +0.21 0.201.39 �0.28 +0.27 �0.53 +0.53 0.670 0 0 �0.04 +0.04 02.10 �0.52 +0.50 �0.94 +0.95 1.512.71 �0.73 +0.73 �1.40 +1.44 2.510.31 �0.08 +0.08 �0.14 +0.14 0.20

4.23 �1.03 +1.04 �1.80 +1.82 4.026.96 �1.24 +1.44 �2.32 +2.42 6.03

10.07 �1.52 +1.90 �3.01 +3.14 8.2013.53 �1.78 +2.37 �3.99 +3.80 10.50

Fig. 10. M–U average curves of slab cross-section close to the wall.

Fig. 11. M–U average curves on distance 20.5 cm from the wall.


effects of dynamic loads, which will be the subject of furtherresearch.

In order to determine connection yielding relative rotation ofPRCS and MRCW has been analysed. Analysis of the experimental

Fig. 12. Crack opening mechanism: (A) Precast specimens-opening of 2nd crack. (B) Mcracks. (D) Monolithic specimens-development of cracks.

results is very difficult, due to the fact that sometimes it is impos-sible to separate relative rotation inside connection from flexibleslab rotation. In order to overcome mentioned problem relativerotation is determined, by using the pattern given in papers [21–23], as the sum of:

� Slab end rotation deformation within the connection zone (dueto the slab and curvature along a plastic hinge length lP)

U ¼ M � lp

Ec � IIðIIÞð1Þ

� Slab–wall interface rotation due to the joint openings

U ¼ rs � le

Es � dð2Þ

� Rotation by relevant cross-section deformability

U ¼ MAs � Es � zII � ðd� xÞ ð3Þ

where lP – length of the plastic hinge, Ec – Young’s modulus of concrete,II – second moment of non-cracked cross-section, III – second momentof cracked cross-section, s – reinforcement stress, Es – Young’s modulus

onolithic specimens- opening of 2nd crack. (C) Precast specimens-development of

Fig. 13. Failure mechanism of precast specimens: (A) Crack pattern. (B) Precast specimens by yielding of reinforcement.

Fig. 14. Comparative curves of moment – relative rotation.

Table 3Average degree of connection yielding.

After Monolithic specimens Precast specimens

[16] a/a + 2 = 0.91 a/a + 2 = 0.89[21] c = 0.89 c = 0.87


of steel, le – anchorage length or length over which a stress distributionalong the rebar is uniform, d – effective depth, As – reinforcement area,zII – lever arm of internal forces.

Measurements are performed out of plastic hinge, at distance of20.5 cm from the specimen wall. Moment – relative rotation exper-imental curves and calculated curves, with real geometrical char-acteristics and characteristics of materials used in experiments,are given in Fig. 14.

When analysing presented curves it may be concluded thatmonolithic connection can be considered semi-rigid. Performedexperiments show that monolithic and precast specimens havesimilar behaviour up to load of 1.33Pservice. This means that in thementioned range of load intensity tested precast connection canbe considered as monolithic.

Factor of connection yielding exposed to service load is obtainedaccording to the recommendations in [8,16,21,22]. According to thepaper [21, p. 2], the joints can be classified as low-strength joints,mid-strength joints and high-strength joints. The classification isdone according to the degree of restraints c, which is defined bythe formula:

c ¼ 1þ 3EISL

� ��1

ð4Þ

According to [21], the joints with a degree of restraining ofc > 0.67 are considered as high-strength compounds, i.e. the sup-port moments are larger than the mid-span moments.

The joint stiffness of precast models is

Sprecast ¼13:26

0:00378¼ 3507:94 kNm=rad

The joint stiffness of monolithic models is

Smonolit ¼13:33

0:00355¼ 3754:93 kNm=rad

Recommendations of defining the degree of joints yielding(semi-rigidity) and the calculated bending moment at the supportis given in [16, pp. 54–60], with the following equations:

a ¼ Ksup � lf

Bfð5Þ

Msup;y ¼a

2þ a�Msup;r ð6Þ

where Ksup – bending stiffness of the joint, lf, Bf – slab span andstiffness, Msup,y – moment at support with semi-rigid connections,Msup,r – moment at support with rigid connections.

The results are presented in Table 3.Yielding of precast connection is in increase mode relatively to

monolithic connection for the range over 1.33P. Rotation of precastconnection relatively to monolithic is presented in Fig. 15.

Based on the experimental results, the following conclusionsmay be drawn:

� Researches performed on monolithic specimens show thateven monolithic structure can also be considered semi-rigid.

� Degradation of investigated precast connection start at loadintensity greater then service load, and it is insignificant upto ultimate load. Bearing capacity of precast connection isproved to be at the same value as monolithic.

� Yielding of connection rigidity increase in the range over1.33Pservice.

4. Numerical model based on matrix formulated displacementmethod

Connection bearing capacity and deformability depend on thetype of connection structure, ways of execution, and load distribu-tion from horizontal to vertical structure elements. Experimentaldetermination of support moments is necessary for creating a rel-evant mathematical model. The calculation of RC frame with semi-rigid connections can be done by using the Force Method and the

Fig. 16. Geometrical-statically meaning of first column of yielding matrix.


Displacement Method, as for the RC frame with rigid connection.At statically indeterminate structures with relatively small numberof statically unknown values the Force method is appropriate, butwith relatively large number of statically unknown values the Dis-placement Method is more suitable [24]. Therefore, the Displace-ment Method is more convenient for the generalisation of theproblem and its solving.

During the analysis of the experimental results, the definition ofconnection stiffness presented in paper [25], was used, startingwith the base formula,

Kjr ¼ M=/ ð7Þ

According to [8], the connection stiffness or the average moduleof deformability was identified based on the experimental curvespresented in Fig. 14. With the identified parameters of connectiondeformability, it is possible to formulate conditions for the compat-ibility of Force Method or the conditions for the balance of the Dis-placement method. The application of the Force Method wasdescribed in details in paper [24]. This paper presents the applica-tion of the Displacement Method.

There is a difference between the classical and matrix formu-lated Displacement Method. The system consists of system ele-ments (bar, or other finite element) connected at discrete points.The analysis of the discrete model consists of the element analysisand the system analysis. The element analysis starts from basicequations of the theory of beam that is the theory of slab, andestablishing the relation between generalised forces and general-ised displacements in joints at the end of the elements. In orderto analyse the behaviour of the structure it is necessary to formu-late a mathematical model. Starting equations for the formulationof the problem are differential equations, which describe the elas-tic line of the elements. The analytical solution of the equations is acomplex mathematical problem, which can be solved only in spe-cial cases. Therefore, the discretisation method has to be used,which is the transformation of differential equations in difference,so that the problem can be analysed numerically. That can be at-tained with the discretisation of the structure with finite elements.The system of equations for the problem solution is appropriate forthe matrix equation and the use of computer.

fPg ¼ ½K�fdg ð8Þ

With the previous formulation, the influence in elements withrigid or pinned connections can be defined with sufficient accu-racy. In order to analyse semi-rigid connections the modificationof stiffness matrix is necessary,

Fig. 15. Rotation of precast specimens relatively to monolithic specimens.

½K�� ¼ ½K� � ½C� ð9Þ

throughout the implementation of yield system with semi-rigidjoints matrix [C] for example (Fig. 16) in the equation

½C� ¼

1 0 0 0� k�21

S/1� k�22

S/� k�23

S/� k�24

S/

0 0 1 0� k�41

S/� k�42

S/� k�43

S/1� k�44

S/

266664

377775 ð10Þ

with S/ ¼M/

— experimental connection stiffness ð11Þ

Through matrix transformation Eq. (9) can be transformed inequation:

½K�� ¼ ½K� � ½I� � ½S/�½K��

ð12Þ

With the solution of matrix Eq. (12) the developed form of mod-ified stiffness matrix is obtained:

Fig. 17. Frame with slab–wall yielding connection.

Fig. 18. Terms of 1st column of modified stiffness matrix.


K� ¼ EI

12þaþb

L3ð1þxÞ6þb

L2ð1þxÞ � 12þaþb

L3ð1þxÞ6þa

L2ð1þxÞ

6þb

L2ð1þxÞ4þb

Lð1þxÞ � 6þb

L2ð1þxÞ2

Lð1þxÞ

� 12þaþb

L3ð1þxÞ �6þb

L2ð1þxÞ12þaþb

L3ð1þxÞ � 6þaL2ð1þxÞ

6þaL2ð1þxÞ

2Lð1þxÞ � 6þa

L2ð1þxÞ4þa

Lð1þxÞ

2666666664

3777777775

ð13Þ

with a ¼ 12EIS/L

; b ¼ 12EIS/L

; x ¼ 112ð4 � ðaþ bÞ þ a � bÞ:

[I] – unit matrix,S/ – connection stiffness,E – Young’s modulus,I – second moment of area of section.

Modified matrix of interpolation functions is,

f/�gT ¼ f/gT � ½C� ð14Þ

so that vector of equivalent loads is,

fP�eg ¼Z L

0f/�gT � pðxÞdx ð15Þ

To test the support (end) moment calculation by the use of theDisplacement Method and modified stiffness matrix, finite elementmodels were programmed by software SAP2000 N application. Onthat occasion models with BEAM and SHELL elements were made.Rectangular finite elements were used during modelling process.The size of finite elements was tested and it was shown that finiteelements with ratio of elements sides and elements depth x:y:z =1:1:1 give the best results.

5. Calculation of support moments for tested models

With the application of procedure presented in Section 4 thecalculation of moments at slab–wall connections for specimenswhich are tested during the experimental research described in[14], was carried out. Modified stiffness matrix [K] and modifiedvector of equivalent loads were defined for the frame in Fig. 17.Terms of 1st column of modified stiffness matrix were describedin Fig. 18.

Modified stiffness matrix has the form:

For the specific case with slab–wall yielding connection loadedwith uniform distributed load, the vector of equivalent loads hasthe form:

K ¼

ð4þ bÞ � EsIsLs �ð1þxÞ þ 2 � 4EwIw

hw0 6þb

Ls� EsIs

Ls �ð1þxÞ

0 EsAsLsþ 2 � 12EwIw

h3w

0

6þbLs� EsIs

Ls �ð1þxÞ 0 2 � EwAwhwþ 12þaþb

L2s� EsIs

Ls �ð1þxÞ

2EsIsLs �ð1þxÞ 0 6þb

Ls� EsIs

Ls �ð1þxÞ

0 � EsAsLs

0

� 6þbLs� EsIs

Ls �ð1þxÞ 0 � 12þaþb

L2s� EsIs

Ls �ð1þxÞ

266666666666666664

P0 ¼

m�q�L2s

12

0q�Ls

2

�m�q�L2s

12

0q�Ls

2

8>>>>>>>>>><>>>>>>>>>>:

9>>>>>>>>>>=>>>>>>>>>>;

m ¼

6EsIs

Ls � Suþ 1

1þ 8EsIs

Ls � Suþ 12

S2u

EsIs

Ls

� �2 ð17Þ

withLs, As, Is, Es – geometrical data and Young’s modulus of prefabri-cated slab,hw, Aw, Iw, Ew – geometrical data and Young’s modulus ofmonolithic wall,q – loads on slab.

The numerical analysis of real behaviour of PRCS and MRCWconnection was done on the models where slab and wall weremodelled with BEAM and SHELL elements, whereas the connectionbehaviour was modelled with PLASTIC LINK elements.

Testing of proposed models was performed during modelling,and it was proved that finite elements model with side ratio1:1:1 gave the best results. The model with finite elements without

2EsIsLs �ð1þxÞ 0 � 6þb

Ls� EsIs

Ls �ð1þxÞ

0 � EsAsLs

0

6þbLs� EsIs

Ls �ð1þxÞ 0 � 12þaþbL2

s� EsIs

Ls �ð1þxÞ

ð4þ bÞ � EsIsLs �ð1þxÞ þ 2 � 4EwIw

hw0 � 6þb

Ls� � EsIs

Ls �ð1þxÞ

0 EsAsLsþ 2 � 12EwIw

h3w

0

� 6þbLs� EsIs

Ls �ð1þxÞ 0 2 � EwAwhwþ 12þaþb

L2s� EsIs

Ls �ð1þxÞ

377777777777777775ð16Þ


modelling of connection work mechanism which is generally ap-plied in design practice was used as a starting comparative model.

Afterwards, models with slab–wall connections modelled withone plastic link element (monolithic yielding model) were tested(Fig. 19a). Plastic link element was modelled with the applicationof experimentally obtained M–U curve for monolithic model. Theprecast model was made in the same way. The behaviour of precastmodels in the failure area was modelled with two plastic connec-tion elements being serially joined (Fig. 19b). First linked elementwas modelled by monolithic yielding models, whereas work mech-anism of the second link element was modelled by applying themoment – relative rotation curve of the precast model.

The results of the performed analysis are presented in Table 4.The values of support (end) moments for load stage equivalent toload applied during the experimental research were analysed.Comparative support moment – load curves for proposed numeri-cal models are given in Fig. 20.

Material properties and geometrical parameters of the calcula-tion model are adopted on the basis of the results of the experi-mental research.

According to the presented experimental and calculated curvesand data in Table 4, it can be seen that monolithic connections alsohave a specific yielding level in relation to absolutely rigid joints.Values of the yielding ratio for precast connection in relation tomonolithic connection can be specified from the diagram or Table4. Here are presented two values:

� Relative yielding for service load – Dserv. = 1.28 kNm.� Relative yielding for service load multiplied with global safety

factor 1.75 – D1.75 = 2.34 kNm.

Therefore, calculated yielding levels for precast connection inrelation to monolithic connection are:

Fig. 19. Modelling of slab–wall connection yielding: (

� ccalc,serv. = 0.92 (Eq. (4)).� ccalc,1.75 = 0.90.

In paper [14] are presented experimentally obtained values ofyielding level for service load,

c¼exp;serv: ¼ 0:907� 0:913

and for 1.75 service load,

cexp;1:75 ¼ ð0:907� 0:913Þ � 0:989

cexp;1:75 ¼ 0:897� 0:903

The previously performed numerical analysis and comparisonwith the analysis of experimental results show good agreementof results and sufficient accuracy of presented mathematical modelfor the calculation of researched connection. The deviation of cal-culation values in relation to the experimental, for service load, is0.77–1.43%, and for 1.75 � service load is 0.33%.

The increase of connection yielding in the studied precast jointsrelative to monolithic models is 17.95–20%. Thus, it can be con-cluded that the procedures used for the monolithic concrete struc-ture cannot be applied on precast concrete structures as well.Yielding of researched dislocated precast connection in relationto monolithic connection is maximum 10%. Thus, it can be con-cluded that the proposed dislocation of precast connection isefficient.

6. Proposed Strut and Tie model

During the experimental research of monolithic and precastspecimens, mechanism of crack opening was registered. The mech-

a) Monolithic yielding model. (b) Precast model.

Table 4Values of support moments obtained through numerical analysis of monolithic and precast models (kNm).

Load (kN/m2) BEAM elements model SHELL elements model

Absolutely rigid Monolithic yielding Precast Absolutely rigid Monolithic yielding Precast

0.75 �11.5 �9.78 �8.23 �11.5 �8.5 �7.51.50 �13.0 �11.26 �9.43 �13.0 �9.7 �8.82.25 �15.0 �12.77 �10.61 �15.0 �11.0 �9.93.00 �16.6 �14.37 �11.81 �16.6 �12.4 �11.24.50 �20.0 �17.58 �14.51 �20.0 �15.1 �13.66.00 �24.0 �20.67 �17.09 �24.0 �17.7 �15.97.50 �27.5 �23.76 �19.58 �27.5 �20.4 �18.29.00 �31.5 �26.85 �22.01 �31.5 �23.0 �20.2

12.00 �38.0 �32.80 �26.05 �38.0 �28.2 �23.615.00 �46.0 �33.33 �2.62 �46.0 �1.0 -0.918.00 �53.0 �2.05 – �53.0 – –

Fig. 20. Calculation curves support moment – load (M–p) for analysed numericalmodels.

Fig. 22. Preliminary STM for monolithic specimen.


anism of crack opening, registered during the testing of precastspecimens, was modelled with finite elements (Fig. 21).

FEM model was used as a basis for making Strut and Tie model.The analysis started from the preliminary Strut and Tie model(STM) for the monolithic specimen (Fig. 22).

Naturally, in this model the effects of local stress and behaviourmechanism of the dislocated precast connection were not covered.So, the preliminary Strut and Tie model served as a comparativemodel. In order to formulate the Strut and Tie model, with suffi-

Fig. 21. FEM model: (a) Stress distribution. (b) Stress c

cient accuracy for design practice, a detailed analysis of stress in-side the precast connection was performed through FEM model(Fig. 23). The proposal of Strut and Tie model for researched pre-cast specimen is given in Fig. 24.

In the connection area, three different areas of stress trajectorypath can be separated regarding the deviation in comparison to themonolithic model, marked in Fig. 24 as areas C, T and C(T). Area T isthe area of crack opening in dislocated precast connection. In thatarea there is a stress concentration in reinforcement, which is dis-tributed in the surrounding non-cracked concrete. In the vicinity ofupper turning point of precast connection tension area passes inthe area where the connection is partially stressed with tensionand partially with compression (area C(T)). Tensioned bars werepresented with tie 3–50. Shear in connection, that is the transfer

oncentration in reinforcement at location of crack.

Fig. 23. FEM model, principal stresses: (a) r1, (b) r2.

Fig. 24. Proposal of STM for Precast specimens.

Fig. 25. Axial forces in compressed bars and ties of proposed STM.


of shear load, was modelled with tie 3–9. Transferred area C(T) wasmodelled with struts 8–9, 40–8 and tie 7–8. The concentration ofcompression stress near the lower turning point of precast connec-tion, area C, was modelled with strut 6–7. The proposed Strut andTie model represents the approximation of complex stress state inthe vicinity of precast connection with resultant forces of stressfields. The accuracy of the model depends on the number of strutsand ties, and the level of model detailing. The proposed model de-scribes basic properties of the designed precast connection workmechanism. Axial forces in struts and ties of the proposed modelare presented in Fig. 25. The effect of strength connection yieldingwas obtained through the introduction of the effects of crack open-ing in dislocated precast connection, with respect to the stress tra-jectory obtained by applying FEM model. Therefore, the force intensioned bars (tie 2–3) is smaller by 7% in relation to the mono-lithic model. Also, compression in joint 1 is smaller by 16%. Thementioned values are approximate and on safe side, and can be ap-plied for preliminary numerical analysis of work mechanismparameter of the researched precast connection.

7. Conclusions

Application of Displacement Method, with modification of stiffnessmatrix through the introduction of yielding of joints, can be defined bya mathematical model that describes the mechanism of structures(assembly) made of monolithic walls and precast slab. The paper pre-sents a modified stiffness matrix obtained in the experiment by intro-ducing stiffness of MRCW-PRCS joint monolithic [14,26]. By the

application of the proposed mathematical model, the degree of yield-ing of the investigated monolithic precast connection was 17.95–20%.The analysis of monolithic joints with the same geometric characteris-tics was performed through the comparative experimental andnumerical study. It was found that monolithic joints are also semi-ri-gid. Relative yielding of the precast joint compared to the monolithic isup to 10%.

Based on the experimental tests on 3 precast and 3 monolithicspecimens it was also proved that the designed precast connectionenable dislocation of joints opening outside of the wall and dissipa-tion of seismic energy with similar bearing capacity as monolithic.In accordance with fib Bulletin 27 [27] prefabricated system withinvestigated prefabricated connection can be used as equivalentmonolithic system. Investigations are performed with static load.For final conclusions it is necessary to perform investigations withcyclic load.

For the design and detailing purpose, the proposed Strut and TieModel, which analysed the yielding of the joint, can be used inpractice [28]. The accuracy of calculation depends on the complex-ity of the model and is achieved through refinements of Strut andTie model.

Acknowledgements

The work reported in this paper is a part of the investigationwithin the research project TR 36043 supported by the Ministryfor Science and Education, Republic of Serbia. This support is grate-fully acknowledged (R. Folic).

Appendix A. Supplementary data

Supplementary data associated with this article can be found, in theonline version, at http://dx.doi.org/10.1016/j.engstruct.2012.03.007.

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