MODELLING THE FLEXURAL TENSILE STRENGTH OF MASONRY

15th International Brick and Block Masonry Conference Florianópolis – Brazil – 2012

MODELLING THE FLEXURAL TENSILE STRENGTH OF MASONRY Schmidt, Ulf1; Hannawald, Joachim2; Koster, Matthias3; Graubohm, Markus4; Brameshuber, Wolfgang5 1

4

5

Dipl.-Ing., Materialprüfungs- und Versuchsanstalt Neuwied GmbH, Germany, [email protected] 2

PhD, Institute of Building Materials Research (ibac), RWTH Aachen University, Germany, [email protected]

3

PhD, Institute of Building Materials Research (ibac), RWTH Aachen University, Germany, [email protected]

Dipl.-Ing., Institute of Building Materials Research (ibac), RWTH Aachen University, Germany, [email protected]

PhD, Professor, Institute of Building Materials Research (ibac), RWTH Aachen University, Germany, [email protected]

The load bearing capacity of masonry with respect to out-of-plane horizontal loads, like for example earth pressure or wind loads, is decisively affected by its flexural tensile strength. The flexural load bearing behaviour of masonry, in turn, is determined by a large number of influences, e. g. the material properties of its components masonry unit and mortar, the bond behaviour between the masonry unit and the mortar, and, in the case of a horizontally spanning wall, the dimensions of the units, the length of the overlap, the masonry thickness, whether the cross joints are filled with mortar or not and so on. In order to characterize the resistance of masonry to out-of-plane horizontal loads and to investigate the different influencing factors, a numerical model on the basis of Finite Element Methods (FEM) was developed where different material laws were assigned to the masonry units and the joints. The material laws of the masonry units and the joints used in the FE calculations, in particular the post-failure behaviour, were previously determined on the basis of small-sized test specimens. The model was validated by comparing the numerical results with experimental data obtained by flexural tensile tests on small masonry walls. For this purpose extensive deformation measurements were carried out on the tested masonry walls and the influence of an additional vertical load applied (normal stress) on the flexural strength was analysed. In an extended sensitivity analysis different material properties of the masonry components and varying geometric boundary conditions, e. g. filled/unfilled head joints, different overlaps, and masonry thicknesses, were investigated. The calculated stress distributions in the masonry walls immediately reveal the inherent failure mechanisms which are hardly accessible by means of experimental investigations alone. On the basis of the numerical results calculation equations to evaluate the flexural tensile strength of masonry were derived. Keywords: masonry, flexural tensile strength, Finite Element Model


INTRODUCTION The load bearing capacity of masonry under horizontal out-of-plane loads (i.e. loads normal to the wall’s surface), e.g. wind or earth pressure, chiefly depends on its flexural tensile strength. Depending on the arrangement of the supports of the masonry wall the resulting stresses are parallel to the bed joints or perpendicular to them. In both cases two failure mechanisms can be observed: Failure of the masonry unit and/or failure of the bond between unit and mortar in the bed joint (Figure 1).

unit failure joint failure

Figure 1: Bending load perpendicular (left) and parallel (right) to the bed joints In this article only the flexural tensile strength of masonry parallel to the bed joints is considered (Figure 1 right) and the different influencing factors on it are investigated. For this purpose bending tests on small masonry walls were simulated with Finite Element Methods using the FE software DIANA 9. Within the framework of the Finite Element Model both the masonry units and the bed and head joints were modelled. The material laws of the masonry units and the bed joints entering the calculations where previously determined on the basis of small-sized test specimens. The model was validated by comparing the numerical results to experimental data obtained by flexural tensile tests. Extensive parameter studies were performed by varying the masonry thickness and the overlap of the masonry units. MATERIALS For the investigations two different calcium silicate units, two clay units, two autoclaved aerated concrete units, and two lightweight aggregate concrete units were used (Figure 2). At first only the solid unit materials without core holes were considered.

KSd and KSe Calcium Silicate Units, PPb and PPc Autoclaved Aerated Concrete Units, VMzb and VMzc Clay Units, LBa and LBd Lightweight Concrete Units

Figure 2: Tested masonry units


As mortars a thin layer mortar, a general purpose mortar and an epoxy resin – as a material with very high bond strength – were used. MATERIAL LAWS OF THE MASONRY UNITS UNDER TENSILE LOAD In case of unit failure the flexural tensile strength of masonry parallel to the bed joints is mainly influenced by the material behaviour of the masonry units under tensile load. The tensile strength and Young’s modulus of the masonry units were determined by tensile tests on cylinders, which were gained by coring in the direction of the units’ length. Unfortunately, due to the brittleness of the material the determination of the post-failure behaviour was not possible. Hence, the post-failure behaviour was determined inversely on the basis of deformation-controlled three-point bending tests on notched prisms extracted from the units in the direction of their length. To this end, the bending tests were simulated with DIANA. The flexural tensile prisms were discretised with two-dimensional continuum elements obeying a linear elastic material law. In the area above the notch, where fracturing occurs during the bending test, interface elements were arranged. A multi-linear stress-crack opening diagram (SRD) was assigned to the interface elements and the SRD was varied until the calculated load-deflection curves best matched the experimental results, see Hannawald (2006). The resulting SRDs and Young’s modulus of the different masonry units are shown in Figure 3. σZ in N/mm²

4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 w1 0.0 0.00

0.7

KSd

σZ in N/mm² PPb

0.6

E = 9150 N/mm², w1 = 18.4 µm

KSe E = 12050 N/mm², w1 = 8.3 µm

E = 1800 N/mm², w1 = 12.1 µm

0.5

PPc

0.4

E = 1000 N/mm², w1 = 13.1 µm

0.3 0.2 0.1 0.01

0.02

0.03

0.04

0.05

0.0 0.00

0.01

0.02

0.03

w in mm 2.5

σZ in N/mm²

2.0 1.5

Lba E = 13300 N/mm², w1 = 103.7 µm

LBd

1.0 0.5

0.05

0.05

w in mm σZ in N/mm²

E = 3000 N/mm², w1 = 23.9 µm

0.0 0.00

0.04

0.10

0.15

0.20

10.0 9.0 8.0 7.0 6.0 5.0 4.0 3.0 2.0 1.0 0.0 0.00

VMzb E = 16000 N/mm², w1 = 7.0 µm

VMzc E = 16500 N/mm², w1 = 11.0 µm

0.01

0.02

0.03

w in mm

Figure 3: Material laws of the masonry unit materials Calcium Silicate Units (KSd and KSe) Autoclaved Aerated Concrete Units (PPb and PPc) Lightweight Concrete Units (LBa and LBd) Clay Units (VMzb and VMzc)

0.04

0.05

w in mm


MATERIAL LAWS OF THE MORTAR JOINTS UNDER TENSILE AND SHEAR LOAD In the FE simulation of the bending tests of the wall the mortar joints are characterised by a homogenized material law which encompasses the mortar as well as the interface between masonry unit and mortar. The material law for the bond joints submitted to tensile stresses is described by a strain-stress curve up to the maximum load and a stress-crack opening relation after a crack has occurred. Again, tensile and bending tests on specimens consisting of masonry unit and mortar in conjunction with numerical simulations were carried out in order to identify the corresponding material law in much the same manner as in the case of the masonry units. The shear strength of the bed joints as a function of the stresses normal to the joint significantly affects the flexural tensile strength of the masonry. In case of thin layer mortars the material law for the bed joints under shear stress was determined with torsion tests on hollow cylinders with three different superimposed load levels perpendicular to the joint, see Schmidt (2008a). The hollow cylinders were obtained by concentric coring of the masonry units. For each test specimen two cylinders were glued together with thin layer mortar. As an example, the results of the torsion tests on hollow cylinders made of calcium silicate units combined with thin layer mortar for three different superimposed normal stresses are displayed in Figure 4 (left). τM denotes the torsional stress and vM the twisted arc length. The black curves are exponential fits and ΦR denotes the residual friction coefficient. 1.4

τM in N/mm²

MT in Nm 1600

σN = 1.0 N/mm² / ΦR = 0.60

1.2

1000

σN = 0.2 N/mm² / ΦR = 0.62

0.8

σN = 0.6 N/mm²

1200

σN = 0.56 N/mm² / ΦR = 0.63

1.0

σN = 1.0 N/mm²

1400

σN = 0.2 N/mm²

800

0.6

σN = 0.0 N/mm²

600

0.4

400

0.2

200

0.0

0

0

1

2

3

4

5

vM in mm

0

20

40

60

-3

80

ϕ in 10 rad

Figure 4: left:

Shear stress – displacement curves of a thin layer mortar determined on hollow cylinders (calcium silicate) right: Comparison of the experimentally and numerically determined torsional moment – rotation angle - curves on solid cylinders (calcium silicate) with thin layer mortar

Due to the shrinking of the mortar in the joint, hollow cylinders for testing could not be produced with general purpose mortar. Therefore, the corresponding material laws were determined on the basis of torsion tests on solid cylinders. At first the thin layer mortar was used for the torsion tests on solid cylinders and the tests were simulated with the FE software DIANA. The material of the masonry units was discretised with three-dimensional continuum elements with linear elastic behaviour. The mortar was represented by interface elements obeying the material law “combined crackingshearing-crushing” (see Lourenco and Rots (1996) and van Zijl (1999)) implemented in DIANA. For the mortar the material law shown in Figure 4 left was applied. On the right


hand side of Figure 4 the measured (grey) and the calculated (black) torsional moment – rotation angle curves are shown. The experimental and numerical results correspond well demonstrating that the used material law for the bed joints is correct. On the basis of the analysis of the numerical determined stress distribution in the solid cylinder - mortar joint a calculation approach has been developed. With this calculation approach it was also possible to determine the essential parameters of the material law of the general purpose mortar with torsion tests on solid cylinders. EXPERIMENTAL INVESTIGATIONS OF THE FLEXURAL STRENGTH OF MASONRY WALLS The experimental determination of the flexural strength of masonry was carried out according to the German standard DIN EN 1052-2:1999-10. The small wall specimens used were 4 stone layers high and 4.5 units long (Figure 5). For the top and the bottom layer of the walls halved units were used for reasons that will become clear later. A description of the testing device can be found in Schmidt (2003).

Figure 5: 4-point bending test on masonry walls The tests were conducted in a deformation-controlled way in order to be able to determine the post-failure behaviour, if necessary. During testing, the deflection in the centre of the wall was measured relative to the displacement of the supports. Furthermore, comprehensive deformation measurements were carried out to obtain data of the resulting strains for comparing with the results of the FE calculations. The arrangement of the measuring points is displayed in Figure 5. The configuration of the measuring points was adjusted to the expected mode of failure as well as to the length of the overlap of the masonry units. The images 1 to 3 in Figure 5 give an impression of the variation and number of the measuring points arranged at the surface of the walls. FINITE ELEMENT MODEL OF THE MASONRY WALLS Just as in the previous cases the simulation of the flexural tension tests of the walls was carried out with the FE software DIANA 9. The units were discretised with three-dimensional continuum elements with 20 nodes and quadratic displacement approximation. The masonry


mortar as well as the interface between masonry unit and mortar was modelled with interface elements in a homogenized way. Due to reasons of symmetry only one half of the walls has to be discretised. Half units for the top and bottom layer were used (cf. Figure 5) because in this case the half wall is still symmetric. On the symmetry plane, all nodes perpendicular to the symmetry plane were fixed. The bending load was introduced by imposing a defined displacement in both load introduction stripes at the back side of the wall. The discretised wall model is displayed in Figure 6. head joint load introduction bed joint support

axis of symmetry

unit

pseudo joint (=unit material)

Figure 6: FE model of the flexural tensile test Since the DIANA software does not feature “hanging nodes” the node lines along the head joints had to be continued through the masonry units and pseudo joints inevitably occurred in the units. For the interface elements of the head and bed joints the “combined crackingshearing-crushing model” was chosen. A linear elastic material behaviour was assigned to the pseudo joints with the same stiffness as for the unit material. The tensile fracture behaviour of the masonry units was characterised by the “total strain crack model” with smeared crack formation. For the tensile softening the approach of Hordijk (Hordijk (1991)) was assumed. For the parameters of the material laws the previously obtained results of the tests on the small-sized specimens were used. The measured stress – crack opening diagrams (SRD) had to be converted into stress – strain curves before they could be used in the FE calculations by adopting a crack bandwidth. As crack bandwidth the distance between two adjacent Gauss points in the direction of the longitudinal flexural tensile stresses parallel to the bed joints was chosen. During the calculations locally different widths of the crack growth areas occurred. Thus, deviations from the crack bandwidth assumption were inevitable. With this approach however, good correlations between the numerical simulations and the experimental results could be achieved for the different masonry unit – mortar combinations. As an example, Figure 7 shows a comparison of the measured and calculated load – deflection curves of autoclaved aerated concrete masonry with different unit overlaps l0.


10

F in kN

10

PPc/TLM

8

F in kN PPc/TLM l0=50mm/fi

8

l0=300mm/fi

l0=50mm/fi

6 4 2

6

l0=300mm/uf

4

l0=100mm/uf

2

l0=300mm/uf

l0=300mm/fi

l0=100mm/uf

l0=50mm/uf

l0=50mm/uf 0

0 0.0

0.5

1.0

1.5

2.0

f in mm

0.0

0.5

1.0

1.5

2.0

f in mm

Figure 7: Measured (left) and calculated (right) load-deflection curves of autoclaved aerated concrete masonry (PPc) with thin layer mortar (TLM) in a fourpoint bending test and different unit overlaps l0 fi – filled head joint, uf – unfilled head joint The measured deformations in the region of the units as well as in the joints also correlate well with the numerical results (Schmidt (2008b)). This indicates that the proposed model correctly predicts the flexural tensile strength of masonry. INFLUENCE OF THE MASONRY THICKNESS AND THE OVERLAP OF THE UNITS ON THE FLEXURAL TENSILE STRENGTH In an extensive sensitivity analysis the different influence factors on the flexural tension strength of masonry were investigated. In this article the case of unit failure when the head joints are unmortared is exclusively presented. The mortar was assumed to behave linear elastic. To avoid stress peaks due to different stiffnesses the same Young’s modulus as for the unit material was chosen. The influence of the material of the masonry units, the geometry of the units as well as the length of their overlap on the bending strength was determined. Furthermore, the stress distributions in the masonry were analysed. As an example, in Figure 8 the calculated ratio of the tensile stresses in a calcium silicate unit within the plane formed by the head joints of the neighbour stone layers to its tensile strength is displayed. The stresses vary over the height of the unit and their maximum before cracking occurs beneath the head joints of the neighbour stone layers.


250

unit height in mm

250 d = 175 mm

200

unit height in mm

200

KSe 150

150 l /h 0

100

= 0.2

100

l0 /h = 0.4 l /h 0

50

= 0.75

50

l0 /h = 1.0

0

0 0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0 σZ / f t

σZ / f t

Figure 8: Stress distribution (ratio tensile stress / tensile strength) in the unit for first crack (left) and maximum load (right)

The calculated ratio of the masonry flexural tensile strength to the masonry unit tensile strength as a function of the ratio of the overlap to the height of the units is shown in Figure 9 for different masonry thicknesses and the different unit materials. ffl,mw / ft,u 0.7

→0

0.6 0.5 0.4 0.3 0.2 0.1

10 mm 40 mm 70 mm 115 mm 175 mm 240 mm 365 mm →∞

0.0

0.5

1.0

70 mm

0.4

115 mm

0.3

365 mm

0.2

PPc 0.0

1.5

l0/h

70 mm 115 mm

0.8

0.5

0

ffl,mw / ft,u

1.0

ffl,mw / ft,u

0.1

KSe

0

1.2

0.6

0.4

0.5

1.0

ffl,mw / ft,u 70 mm

0.3

365 mm

365 mm

0.2

0.6 0.4

1.5

l0/h

0.1

0.2

VMzb

LBa 0

0.0 0.0

0.5

1.0

1.5

l0/h

0.0

0.5

1.0

1.5

l0/h

Figure 9: Ratio masonry flexural strength / unit tensile strength depending on the ratio overlap / unit height for different masonry thicknesses


From the results of Figure 9 the following calculation equation can be derived by a regression analysis: ffl,ma ft,u

=

(l0 / h)2.45 (l0 / h)2.45 d1.165 + 3485.8 + 4.4885 d1.165 0.314 + 0.303 ⋅ 2−0.0114 d

with ffl,ma ft,u h l0 d x’

⋅ x'

(1)

masonry flexural tensile strength [N/mm²] unit tensile strength unit height overlap masonry thickness material correction factor to account for the gradient of the softening curve = 1.0 (Calcium Silicate Units KSe) = 0.97 (Autoclaved Aerated Concrete Unit PPc) = 2.35 (Lightweight Concrete Unit LBa) = 0.72 (Clay Unit VMzb).

As is shown in Figure 9, the curves resulting from Eq. (1) match well the results of the FE calculations. Currently, Eq. (1) is further verified and work is underway to simplify this equation for practical application. The following conclusions can be drawn so far: • The masonry flexural tensile strength decreases with decreasing overlap. • The ratio of the flexural tensile strength of the masonry to the tensile strength of the unit decreases with increasing unit width (masonry thickness). • The ratio of masonry to unit tensile strength tends to decrease with increasing brittleness of the material. • At the same geometry, the scaling of the unit dimensions (height and length) are of inferior importance for the flexural tensile strength of masonry.

SUMMARY AND OUTLOOK A numerical model to characterise the flexural load bearing behaviour of masonry parallel to the bed joints in case of unit failure was developed and calibrated on wall tests. In an extensive sensitivity analysis the influence of the masonry thickness and the overlap of the units were investigated. On the basis of the numerical results a calculation equation to describe the flexural strength of the masonry in case of unit failure was derived. In further investigations this equation will be simplified for practical application and the case of bond failure also will be taken into account.

ACKNOWLEDGEMENTS The financial support by the German Research Foundation (DFG) is gratefully acknowledged. .


REFERENCES Hannawald, J.: Determining the Tensile Softening Diagram of Concrete-Like Materials Using Hybrid Optimisation. Dordrecht : Springer, 2006. - In: Measuring, Monitoring and Modeling Concrete Properties. An International Symposium Dedicated to Prof. Surendra P. Shah, Northwestern University, USA, (Konsta-Gdoutos, M.S. (Ed.)), p. 179-187 Hordijk, D.A.: Local Approach to Fatigue of Concrete. Delft University of Technology, PhD thesis, 1991 Lourenco, P.B. ; Rots, J.G.: A Multisurface Anisotropic Model for Quasi-Brittle Materials. Chichester : Wiley, 1996. - In: Proceedings of the Third ECCOMAS Computational Fluids Dynamics Conference, Paris, 9 - 13 September 1996 (Desideri, J.-A. ; Hirsch, C. (Eds.)), 7 pages Schmidt, U. ; Schubert, P.: Flexural Strength of Masonry. Madison : Omnipress, 2003. - In: Proceedings of the Ninth North American Masonry Conference, Clemson University of Clemson, South Carolina, June 1-4, 2003, p. 674-685 Schmidt, U. ; Hannawald, J. ; Brameshuber, W.: Theoretical and Practical Research on the Flexural Strength of Masonry. Callaghan : University of Newcastle, 2008. - In: Proceedings of the 14th International Brick and Block Masonry Conference, Sydney, 17 - 20 February 2008a, (Masio, M. ; Totoev, Y. ; Page, A. ; Sugo, H. (Eds.)) ISBN 9 7819 2070 1-92-5 Schmidt, U.: Flexural Strength of Masonry. Stuttgart : Universität Stuttgart, 2008. - In: 7th International PhD Symposium in Civil Engineering, Proceedings, Stuttgart, September 11th 13th, 2008b, (Eligehausen, R. ; Gehlen, C. (Eds.)) Zijl van, G.: Computational Modelling of Masonry Creep and Shrinkage. PhD thesis, Delft University of Technology, 1999