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TOTAL STRAIN FE MODEL FOR REINFORCED CONCRETEFLOORS ON PILES

H. HOFMEYER1* AND A. A. VAN DEN BOS2

1 ABT consulting engineers, Arnhem, The Netherlands; and Technische Universiteit Eindhoven, Eindhoven, The Netherlands2 ABT consulting engineers, Arnhem, The Netherlands

SUMMARY

A finite element (FE) model using a total strain material model has been developed to predict the behavior ofwarehouse reinforced concrete floors on piles. The material model (not the FE model itself) was calibrated tomaterial tests. The FE model for the floor structure was checked with full-scale experiments. For a warehouse,punching load optimization and surface crack control are important design factors. It is concluded that if cali-brated material models are used, total strain-based FE models are able to indicate surface crack width and punch-ing strength for several types of reinforcement. Furthermore, it is concluded that it is possible to develop a totalstrain material model from material tests. Copyright © 2007 John Wiley & Sons, Ltd.

1. INTRODUCTION

Existing finite element (FE) models for punching shear in concrete plates can be studied from dif-ferent points of view, which are the dimensionality, the aim of the model, and the material model used.

Regarding dimensionality, it should be noted that the application of three-dimensional FE modelsis quite recent (Dyngeland et al., 1994; Ozbolt et al., 1999; Staller, 2000). From 1980 to 1995, FEmodels were often two-dimensional, modeling the plate cross-section only, to save computing time(Loseth et al., 1982; Andrä, 1982; Foeken, 1983; Borst and Nauta, 1985; González-Vidosa et al., 1988;Menétrey, 1994; Hallgren and Bjerke, 2002). Another efficient technique to save computing time wasa two-dimensional model of the plate surface, having multiple layers of degenerated shell elementsover the thickness, thus creating a semi-three-dimensional approach (Marzouk and Chen, 1993;Polaka, 1998).

Concerning the aim of FE models, most are developed for finding the relationship between theamount of reinforcement and the corresponding punching load (e.g., Xiao and O’Flaherty, 2000). Still,other relationships are investigated. Abbasi and others researched the influence of the main tensilereinforcement on the failure mechanism type using a nonlinear FE model (Abbasi et al., 1992). Forthe relationship between membrane stresses in the plate and the punching load, Kuang and Morley presented a model that used a material model that incorporates plasticity (Kuang and Morley, 1993).Hallgren and others presented a FE model for several shear-span-to-depth ratios and studied the cor-responding failure mechanism type (Hallgren, 1996).

The critical factor of FE models for concrete punching is the material model used, which is the rela-tionship between strains and stresses for elastic behavior, crushing and cracking (Selby and Vecchio,1993). Most FE models for realistic structures work with a macro material model, which models the

THE STRUCTURAL DESIGN OF TALL AND SPECIAL BUILDINGSStruct. Design Tall Spec. Build. 17, 809–822 (2008)Published online 11 September 2007 in Wiley Interscience (www.interscience.wiley.com). DOI: 10.1002/tal.387

Copyright © 2007 John Wiley & Sons, Ltd.

*Correspondence to: H. Hofmeyer, Faculty of Architecture, Building, and Planning, Unit Structural Design and ConstructionTechnology (SDCT), PO Box 513/VRT 9.32, 5600 MB Eindhoven, The Netherlands. E-mail: [email protected]

concrete as a continuum and does not incorporate the behavior of cement gel, particles and aggregates(Hofstetter and Mang, 1996; Zimmerman, 2004). Within this macro model, concrete cracking can bemodeled through either a smeared or discrete crack approach (Fib, 2000). The discrete crack approachis based on an a priori determined crack location, with can be accomplished with interface elements(Hillerborg et al., 1976). It cannot be used for the research presented here, because crack locationsare not known. The smeared crack approach reduces the stiffness in an FE at the moment the tensilestrength is reached. Using the smeared crack approach, the strain in the concrete is decomposed intoelastic and crack strains to make an FE calculation possible (Litton, 1974). Because the decomposi-tion gives rise to numerical problems in some instances, another approach was developed: the totalstrain crack model that describes tensile and compressive behavior of concrete with a singlestress–strain relationship (Vecchio and Collins, 1986). This makes the material model less vulnerablefor numerical problems and it can easily be developed using material tests.

An FE model based on a total strain approach, which indicates punching strength and surface crack width for specific types of reinforcement, does not exist. However, Van den Bos and Hofmeyershowed that such a model is needed for the design of warehouse floors where surface crack controland punching load optimization are the two most important design criteria (Van den Bos and Hofmeyer, 2005). This article presents a total strain FE model that indicates punching strength andsurface crack width for several types of reinforcement. The material model (not the FE model itself)is calibrated to material tests. The FE model of the floor structure is verified against full-scale experiments that were carried out at Technische Universiteit Eindhoven (Van den Bos and Hofmeyer,2005).

2. MATERIAL MODEL, MATERIAL TESTS

Two types of concrete were used in the investigation: non-reinforced concrete and fiber-reinforcedconcrete. Both types are low shrinkage and made with the following constituents: water–cement ratio0·50; cement content 330kg/m3; 75% ENV 197-1:1992 class III/B Portland blast furnace cement; 25%type III rapid-hardening Portland cement; coarse aggregate: 75% parts smaller than 32mm, 25% partssmaller than 16mm and 160mm slump. The fiber-reinforced concrete was reinforced with 35kg/m3

Bekaert Dramix RC 65/60 BN (Figure 1).The concrete was tested in compression, pure tension, and shear. By the use of FE models

that simulated the material tests, it was possible to obtain input variables for the FE material model.

2.1 Compression tests

Standard cube compression tests were carried out as shown in Figure 2. The loading speed was 2mm/min. The average compression strength and standard deviation of all 30 days aged cubes were 40·0N/mm2 and 1·8N/mm2, respectively, for eight fiber-reinforced cubes and 41·0N/mm2

810 H. HOFMEYER AND A. A. VAN DEN BOS

Copyright © 2007 John Wiley & Sons, Ltd. Struct. Design Tall Spec. Build. 17, 809–822 (2008)DOI: 10.1002/tal

60 mm

0.9 mm

fy >1000 N/mm2

Bekaert Dramix RC 65/60 BN

Figure 1. Bekaert reinforcement fiber

and 1·7N/mm2 for 11 plain cubes. Curve A in Figure 2 displays a typical load–deformation relationfor plain concrete. Average stress–strain values from all experiments (curve C) were used as input foran FE model of the compression test (curve B). Figure 2 shows that the results from the FE analysiscompare very closely to the experimental data. For non-reinforced and fiber-reinforced concrete, thesame FE input (curve C) can be used as the test results for the two materials are almost identical.

2.2 Pure tension tests

For investigating the behavior of concrete in tension, a test set-up was used that was originally devel-oped for studying the post-peak behavior of unreinforced masonry in tension (Van der Pluijm, 1999).The tension plates of the set-up shown in Figure 3 remain perfectly parallel during load application,

TOTAL STRAIN FE MODEL FOR RC FLOORS 811


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Concrete cube,150*150*150 mm

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Figure 2. Concrete compression test, plain and fiber-reinforced concrete

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Concrete cube,150*150*150 mm,saw cuts result in circularcross-section

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Figure 3. Concrete pure tension test, plain concrete



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Figure 4. Concrete pure tension test, fiber-reinforced concrete

thereby allowing non-uniform crack distributions to occur. This yields a typical S-shape segment inthe descending branch of the experimental load–strain curve A. Saw cuts in the concrete cube createa circular cross-section with a diameter of 123mm. The average tension strength and standard devia-tion for four fiber-reinforced experiments are 2·41N/mm2 and 0·18N/mm2, respectively. For the twotests on non-reinforced concrete these values are 1·56N/mm2 and 0·06N/mm2. There is a differencein tensile strength for plain and fiber concrete that is only found for pure tension tests. Still, puretension tests are used in favor of bending tests because pure tension tests results are closest to actualtensile strength and therefore they yield the best results if a FE model is used with a total strain mate-rial model.

A typical load–deformation relation for non-reinforced concrete is shown in Figure 3 by curve A.Curve C represents average stress–strain values for the two non-reinforced concrete tests. They wereused as input for an FE model of a pure tension test (curve B). Figure 3 shows that the results froman FE analysis of a non-reinforced concrete specimen compare very closely to the experimental data.Figure 4 shows the same procedure for fiber-reinforced concrete.

2.3 Shear tests

For the investigation of concrete subject to shear only, vertical shear failure surfaces in the concretespecimens are created by applying horizontal offset saw cuts as shown in Figure 5. Subjecting thecubes to compression allows the concrete between the saw cuts to fail in shear. Curve A in Figure 5shows a typical load–deformation relationship. In the process of calibrating the FE model for shear,it was found that the shear modulus G of the concrete should be multiplied by 0·2 after cracking,which is indicated by FEM input curve C. Using this input, the FE model shows a load–deformationbehavior shown by curve B and this curve is similar to experimental curve A.

3. FE MODEL FOR FULL-SCALE EXPERIMENTS

At the Technische Universiteit Eindhoven, full-scale experiments have been carried out on warehouseslabs to investigate surface crack behavior and punching strength for specific reinforcement types (Vanden Bos and Hofmeyer, 2005) (Figure 6). A steel boundary frame was used to simulate the boundary



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Figure 6. Full-scale experiments



Table 1. Test specimens for full-scale experiments

Number Concrete Reinforcement

1 Fibera

2 Fibera (B) Square pile matb

3 Fibera (C) Circular pile matc

4 Normal5 Normal (B) Square pile matb

6 Normal (C) Circular pile matc

7 Normal (A) Normal matd

(B) Square pile matb

8 Normal (A) Normal matd

(B) Square pile matb

(D) Bent-up barse

a Concrete fiber reinforcement equals 35kg/m3 Bekaert Dramix RC 65/60 BN.b The square pile mat consists of 10 bars (diameter 8mm, length 1000mm) in eachdirection, cross-sectional bar distance 100mm. Upper surface of the mat distance 30mm. Only positioned above the pile.c Circular pile mat consists of one bar (diameter 8mm, length 17000mm) circular bentusing an increasing radius of 75mm for each revolution. On this mat, 7 u-shaped bars(diameter 8mm, length 1000mm) are welded.d Normal mat consists of bars (diameter 8mm) in each direction, cross-sectional bardistance 100mm. Positioned above the whole test specimen.e Bent-up bars are two trapezoidal 45 degrees bent bars (diameter 8mm) in each direc-tion, cross-sectional bar distance 200mm.

conditions in practice (Van den Bos and Hofmeyer, 2005). Data for the test specimens are given inTable 1. Figure 7 shows the four types of reinforcement.

3.1 ACI code predictions

American Standard ACI 318-02 is used, without any safety factor, for predicting the punching loadof the experiments (Table 2). The ACI code performs well. It does not permit use of the normal mat(A), pile mat (B), and circular pile mat (C) as shear reinforcement. Even the bent-up bars are notallowed as shear reinforcement because the test specimen depth is too low. If still a prediction is made,the value permitted is lower than for the non-shear-reinforced sections because, for the concrete shearpart, a reduced value should be used.

3.2 Model geometry

An FE model of the floor structure is made using the general FE program Diana (Diana, 2004). Becausethe full-scale experiments are symmetric, only one quarter of the structure needs to be modeled. Thisquarter will be completely built with volume elements. The geometry of the FE model is shown inFigure 8. The concrete floor slab comprises two volume layers while the boundary frame structureconsists of three volume layers.

3.3 Boundary conditions

Symmetry conditions are applied along two sides as shown in Figure 8. The support is fixed in the z-direction, but is free to rotate. This also applies to the experiments.



3100 mm

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1000 mm1050 m m 1050 m m

30 mm

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1000 mm

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(B) Square pile mat (D) Bent-up bars

(B) Square pile mat

(C) Circularpile mat

160 mm

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φ

Bent-up bars 8 mm,distance 200 mm

φ

Bars 8 mm,circular and U-shaped

φ

Figure 7. Reinforcement types A–D

Table 2. Experiments versus FE model

Ultimate load (kN) Number of cracks Crack width (mm)

Exp.ACI 318-02 observed

Ultimate FE (in LVDTExp. Exp. load (kN) model range) FE model Exp. FE model

1 361 384 9 (9) Distributed 1·44 1·452 458 400 20 (15) Distributed 1·03 0·753 440 438 27 (13) Distributed 0·74 0·434 349 343 337 6 (5) Distributed 3·51 5·525 353 343 355 9 (8) Distributed 1·36 2·086 351 343 382 23 (12) Distributed 0·55 0·547 422 343 393 24 (11) Distributed 0·66 0·798 496 291 412 24 (10) Distributed 0·81 0·75

3.4 Elements

The FEs used in the analysis are CTP45 (Diana, 2004). These are isoparametric wedge elements with15 nodes. The reinforcement shown in Figure 7 is modeled by specific Diana ‘embedded reinforcingbars’ (Diana, 2004) for each individual steel rod.

3.5 Material model and properties

A total strain approach is used with fixed smeared cracking; i.e., the crack direction is fixed after crack initiation. For this approach, a compression and a tension stress–strain curve are used. Compression curve C in Figure 2 is employed for both non-reinforced and fiber-reinforced concrete. For tension, curve C in Figure 3 is used for non-reinforced concrete and Figure 4 is used for fiber-reinforced concrete. Curve C in Figure 5 is employed for shear in both types of concrete.

The steel reinforcement behaves elastically (E = 210 000N/mm2) up to the Von Mises yield stress of 440N/mm2, from which perfect plastic behavior has been assumed. The members of the steelboundary frame, shown in Figure 6, consist of grout-filled square hollow steel sections of 200 ×200 × 5mm. The calculated elastic modulus of equivalent uniform beams measuring 200 × 200mmis 90 000N/mm2. They are assumed to be subject to elastic behavior only.

3.6 Load

The displacements in the z-direction of all element nodes of the steel loading plate are coupled. Oneof these nodes is given a prescribed displacement in the z-direction. The reaction force of this pre-scribed displacement equals the load on the plate. In every element node, forces are applied to modelthe dead load of the concrete plate.



Volumes forsupport 300*300 mm

Surface forsteel loading plate

Volumes forsteel frame

Lines for squarepile reinforcement mat

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uy=0, rotx=0, rotz=0

ux=0,roty=0,rotz=0

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Figure 8. 1/4 FE model geometry volumes

3.7 Solving strategy

A common Newton–Rhapson solving strategy is used to sequentially solve the separate linear systemsof equations. To increase the solving speed, a line-search algorithm is used. Because the load is a rep-resented by a prescribed displacement, no problems occur in finding the ultimate load and in contin-uation after the ultimate load.

4. FE MODEL RESULTS

The following FE model results will be compared to the full-scale experiments: the ultimate load andload–displacement curve (1), the crack pattern, crack widths, and failure geometry (2), and the straingauge readings (3). The results from experiment 2 have been selected for further discussion as its rein-forcement layout is mostly used in practice. Results of other full-scale experiments are summarized.

4.1 Ultimate load and load–displacement curve (1)

Table 2 gives the ultimate punching load for the full-scale experiments and the FE model. Indepen-dent of the specific reinforcement type, the FE model predicts the ultimate load well. This is clearlyshown in Figure 9. Figure 10 shows the load–displacement curves for Experiment 2. The stiffness ofthe experiment (curve A) seems to be significantly smaller than the stiffness of the model (curve B).At this stage the flexibility of the test-rig has not yet been taken into account in the FE model. Thedeflection in the test-rig was measured to be 24mm at the ultimate load. Including this flexibility inthe FE analysis of the floor structure leads to a load–displacement relationship represented by curveC in Figure 10.

4.2 Crack pattern, crack opening widths, and failure geometry (2)

The number of cracks and their width are also given in Table 2. The experimental crack width is obtained from the measured deformation of the surface over a specific distance and dividing by the number of cracks over this length. The deformation is measured by a linear variable differential transformer (LVDT) over 300mm, as shown in Figure 6. The FE model assumes distributed cracks only, as shown in Figure 11. To obtain the FE crack width, the maximum FE



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Model: ORTHOLC1: Load case 1Step: 38 LOAD: 7.2Gauss EL.EKNN1 EKNNMax = .125E-1Min = 0

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Figure 11. Fine and equally distributed crack pattern for the FE model (upper surface above load application)(Diana 8.1.2. TNO Diana BV, Delft, The Netherlands)

crack strain is multiplied by the LVDT measurement length and then divided by the number of cracks found in the experiments. Figure 12 shows the crack widths from the experiments and the FE model. The FE model slightly underestimates the crack width for fiber-reinforced concrete asshown by Experiments 1, 2 and 3, but is qualitatively correct for the different types of reinforcement:applying a square pile mat (Experiment 2) or a circular pile mat (Experiment 3) decreases the

crack width. Omitting reinforcement (experiment 4) yields large cracks, both in the experiment andthe FE model. The FE model predicts cracks width well for traditionally reinforced experiments (experiments 6–8).

For Experiment 2, the crack width was studied as a function of the load, both for the experimentand the FE model. The crack width values were obtained using the same method as presented in theprevious paragraph. Table 3 and Figure 13 present the results and these results show that the FE model



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Table 3. Data FE model for experiment 2

Maximum crack strainCalculation step no. Load (kN) at integration points Crack width (mm)

6 97 0·000375 0·0314 208 0·002070 0·1818 251 0·002990 0·2623 300 0·004150 0·3629 348 0·005490 0·4838 400 0·008570 0·75

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Figure 13. Total crack opening width over several (3–4) cracks above the pile head, typical experiment (number 2) and FE model

gives a good indication of the size and progress of the crack width. This is also valid for all the otherfull-scale experiments and FE models.

The failure geometry of experiment 2 is shown in Figure 14. In this figure, a contour graph is shownof the incremental displacements after the last load increment. Incremental displacements are theincrease of displacements for one load increment. The punching cone has an upper diameter of



XY

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Figure 14. Incremental displacements (top figure) and punching cone geometry (bottom figure)(Diana 8.1.2. TNO Diana BV, Delft, The Netherlands)

440mm and a lower diameter of 220mm. In relation to a floor thickness of 160mm, the spread angleis 55 degrees. Similar measurements were obtained from the experiments.

4.3 Strain gauge readings (3)

Three strain gauges were positioned on the bottom concrete surface (Figure 6). For each gauge a comparison was made between the experiment and the FE model. For experiment 2, Figure 15 shows strains in the x-direction from the gauge. The FE model gives reasonably accurate results for the strains on the surface of the concrete floor slab and thus gives additional verification of the FE model.

5. CONCLUSIONS

An FE model employing a total strain approach was developed for the analysis of warehouse concrete floors with specific reinforcement on pile foundations. The total strain material model was calibrated to experimental tests for compression, tension, and shear. The FE model of the floor structure was verified by full-scale experiments for ultimate load, crack pattern, crack width,deformation and strain measurements. The FE model performed well enough to use it for design purposes.

The research shows that it is possible to develop and calibrate a total strain material model basedon material tests.

ACKNOWLEDGEMENTS

The authors wish to record their appreciation of J. C. D. Hoenderkamp for his contribution to thismanuscript.

REFERENCES

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Andrä H-P. 1982. Zum Tragverhalten des Auflagerbereichs von Flachdecken. PhD thesis, Universität Stuttgart,Germany.



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Figure 15. Reading of the x-direction of strain gauge

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Diana. 2004. Diana version 8.1.2. TNO Diana BV, Delft, The Netherlands.Dyngeland T, Hoiseth K, Opheim E, Hole A. 1994. Nonlinear analyses of reinforced concrete members subjected

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Fib. 2000. Punching of structural concrete slabs. Technical Report Bulletin 12. International Federation for Structural Concrete, Lausanne, Switzerland.

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González-Vidosa F, Kotsovos M, Pavlovic M. 1988. Symmetrical punching of reinforced concrete slabs: an ana-lytical investigation based on nonlinear finite element modeling. ACI Structural Journal 85: 241–250.

Hallgren M. 1996. Puching shear capacity of reinforced high strength concrete slabs. Doctoral thesis, TRITA-BKNBulletin 23. KTH Stockholm, Royal Institute of Technology, Stockholm.

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Kuang JS, Morley CT. 1993. A plasticity model for punching shear of laterally restrained slabs with compressivemembrane action. International Journal of Mechanical Sciences 35(5): 371–385.

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Marzouk H, Chen Z. 1993. Finite element analysis of high strength concrete slabs. ACI Structural Journal 90:505–513.

Menétrey P. 1994. Numerical analysis of punching failure in reinforced concrete structures. Thèse No. 1279.Département de Génie Civil, EPFL, Lausanne.

Ozbolt J, Mayer U, Vocke H, Eligehausen R. 1999. Das FE-Programm MASA in Theorie und Anwendung. Beton-und Stahlbetonbau 94(10): 403–412.

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Selby RG, Vecchio FJ. 1993. Three-dimensional constitutive relations for reinforced concrete. Technical Report93-02. Department of Civil Engineering, University of Toronto, Canada.

Staller M. 2000. Analytische und numerische Untersuchungen des Durchstanztragverhaltens punktgestützterStahlbetonplatten. PhD thesis, TU München.

Van den Bos AA, Hofmeyer H. 2005. Experiments investigating concrete floor punching using specific rein-forcement. Heron 50(2): 93–108 (available online at http://heron.tudelft.nl).

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NOTATION

E Young’s modulus (N/mm2)G Shear modulus (N/mm2)ν Poisson’s constant



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