30. Masonry Properties

List of symbols:

\(\begin{array}{ll}f_{m} & = \text { compressive strength of masonry } \\ f_{m, l} & = \text { compressive strength of masonry in longitudinal direction } \\ f_{b} & = \text { compressive strength of unit } \\ f_{\text {mo }} & = \text { compressive strength of mortar } \\ f_{b t, l} & = \text { tensile strength of units in longitudinal direction of the units } \\ f_{b t, s} & = \text { splitting tensile strength of units } \\ f_{t} & = \text { tensile strength of masonry } \\ f_{v} & = \text { cohesion } \\ a & = \text { coefficient } \\ b & = \text { coefficient } \\ c & = \text { coefficient } \\ c_{1} & = \text { ratio } E_{m} / f_{m} \\ c_{2} & = \text { ratio } f_{b t, l} / f_{b} \\ c_{3} & = \text { ratio } f_{b t, s} / f_{b} \\ Y_{1} & = \text { a log-normal variable taking into account uncertainties in the calculation of the compressive strength of masonry } \\ Y_{2} & = \text { a log-normal variable taking into account uncertainties in the calculation of the modulus of elasticity } \\ Y_{3} & = \text { a log-normal variable taking into account uncertainties in the calculation of the tensile strength of masonry } \\ Y_{4} & = \text { a log-normal variable taking into account uncertainties in the calculation of the tensile strength of unit } \\ Y_{5} & = \text { a log-normal variable taking into account uncertainties in the calculation of the cohesion } \\ Y_{6} & = \text { a log-normal variable taking into account uncertainties in the calculation of the friction coefficient } \\ \varepsilon & = \text { strain } \\ \mathcal{E}_{u} & = \text { ultimate strain } \\ E_{0} & = \text { modulus of elasticity } \\ \mu & = \text { friction coefficient } \\ \sigma & = \text { stress } \\ h_{k} & = \text { buckling length of wall } \\ t & = \text { thickness of wall }\end{array}\)

Abbreviations:

\(\begin{array}{ll}\text { LC } & \quad \text { Lightweight concrete } \\ \text { NC } & \quad \text { Normal density concrete } \\ \text { C } & \quad \text { Concrete } \\ \text { CS } & \quad \text { Calcium silicate } \\ \text { AAC } & \quad \text { Autoclave aerated concrete } \\ \text { CB } & \quad \text { Clay brick } \\ \text { GPM } & \quad \text { General purpose mortar } \\ \text { TLM } & \quad \text { Thin layer mortar }\end{array}\)

30.1. Scope

Masonry is made of units and mortar which show strong regional differences in the material properties and dimensions. Since stochastic treatment of masonry is relatively new, all the values given in the following represent prior values and may be subject to updating techniques. Also, the values (except the values for the compressive strength of masonry) are values derived from tests on single elements. The values might be different for full walls due to spatial variation.

This section especially deals with unreinforced masonry made of artificial units that meet the requirements of the national manufacturing codes. Masonry is widely used as an unreinforced material but in certain countries reinforced solutions are common (e.g. in earthquake regions). For reinforced masonry, additional information is required for the stochastic modelling of the reinforcing steel (s. chapter 26.).

30.2. Basic Properties

The main property of masonry is the compressive strength \(f_{m}\) which is determined on standard test specimens (so called RILEM specimens which are 5 units tall, 2.5 units wide and 1 unit thick) tested according to standard conditions (RILEM TC 76-LUM). Other properties, such as the modulus of elasticity are related to the compressive strength. However, some properties - like the friction coefficient - directly depend on the used type of units and mortar.

Masonry only provides very small tensile strength perpendicular to the bed joints which therefore is often neglected. However, the flexural tensile strength of masonry may have a large effect on the load-carrying capacity of slender walls and should be considered in a probabilistic analysis. The flexural tensile strength parallel to the bed joints often has to be taken into account if walls are subjected to lateral loads (wind, earth pressure).

The shear capacity of masonry walls depends on the tensile strength of the units (shear failure) and on the cohesion between units and mortar (sliding failure). Additionally, the friction coefficient is influencing the tensile strength parallel to the bed joints and the sliding shear capacity.

30.2.1. Compressive strength of masonry

The compressive strength of masonry depends on the compressive strength of units and mortar. The following equation (30.1) is recommended for the determination of the compressive strength.

(30.1)\[f_{m}=K \cdot f_{b}^{\alpha} \cdot f_{m o}^{\beta} \qquad\left(f_{b} \text { and } f_{m} \text { represent mean values }\right)\]

In case of thin layer mortar, the mortar quality does not influence the compressive strength of masonry and so the relationship becomes

(30.2)\[f_{m}=K \cdot f_{b}^{\alpha} \qquad\left(f_{b} \text { represents the mean value }\right)\]

with

\(f_{m} =\) mean of the compressive strength of masonry

\(f_{b} =\) mean of the compressive strength of units in vertical direction

\(f_{\text {mo }}=\) mean of the compressive strength of mortar

\(K=\) coefficient, see Table 30.1

\(\alpha=\) coefficient, see Table 30.1

\(\beta =\) coefficient, see Table 30.1

The slenderness \(\lambda=h_{k} / t\) plays an important role in the determination of the masonry compressive strength. To transform the obtained compressive strength to the reference slenderness of \(\lambda=5\), the following equation can be used.

(30.3)\[k_{\lambda=5}=\left(0.966+0.00136 \cdot \lambda_{\text {specimen }}^{2}\right)\]

The parameters \(K, \alpha\), and \(\beta\) are prior values and should be updated by use of test data. The compressive strength of masonry \(f_{m, l}\) in longitudinal direction (only for masonry with grouted head joints) is thought to be 50% of the masonry compressive strength \(f_{m}\) according to equation (30.1) and (30.2).

Table 30.1 Parameters K, α, β for the determination of the compressive strength of masonry according to [1] for a slenderness of the specimen of λ = 10

	Masonry		\(n\)	\(K\)	\(\alpha\)	\(\beta\)
	Units
Material	Unit type	Mortar
LC	full and hollow blocks	TLM LWM GPM	35 80 167	0.85 0.85 0.85	0.84 0.58 0.73	0 0.15 0.07
LC	full blocks	LWM	21	0.70	0.66	0.16
LC	hollow blocks	LWM	59	0.86	0.57	0.14
LC	full blocks	GPM	61	0.85	0.72	0.09
LC	hollow blocks	GPM	106	0.89	0.69	0.05
LC	full blocks	TLM	20	0.63	1.00	0
AAC	regular unit	NM	140	0.98	0.68	0.02
AAC	regular unit	NM	140	0.99	0.69	0
AAC	regular unit	LM	17	0.80	0.64	0.09
AAC	regular unit	LM	17	0.99	0.64	0
AAC	plane element	DM	162	0.63	1.00	0
AAC	plane element	DM	162	0.83	0.86	0
NC	hollow block	GPM	15	0.03	1.82	0.23
CS	full	GPM	276	0.70	0.74	0.21
CS	block	GPM	24	0.44	0.92	0.17
CS	perforated	GPM	108	0.85	0.57	0.20
CS	hollow	GPM	70	0.99	0.64	0.05
CS	plane elements	TLM	66	0.53	1.00	0
CB	full	GPM	55	0.73	0.73	0.16
CB	perforated	GPM	342	0.55	0.56	0.46
CB	lightweight perforated	TLM	9	0.75	0.72	0
CB	lightweight perforated	LWM 21	17	0.67	0.50	0.05
CB	lightweight perforated	LWM 21	17	0.18	1.00	0
CB	lightweight perforated	LWM 36	13	0.47	0.82	0
CB	lightweight perforated	LWM 36	13	0.28	1.00	0
CB	lightweight perforated	GPM	28	0.26	0.82	0.42

30.2.2. Modulus of elasticity

The modulus of elasticity is commonly related to the compressive strength of masonry. For the stochastic modelling, the following equation (30.4) is recommended.

(30.4)\[E_{m}=c_{1} \cdot f_{m}\]

with

\(E_{m}=\) mean of modulus of elasticity of masonry

\(c_{1}=\) ratio \(E_{m} / f_{m}\) according to Table 30.2

The parameter \(c_{1}\) is a prior value and should be updated by use of test data.

Table 30.2 Values for \(c_{1}\) according to [1]

Type of unit	Mortar	\(\boldsymbol{c}_{\boldsymbol{1}}\) [-]
CS	GPM, TLM	500
AAC	GPM	520
AAC	TLM	560
LC	GPM	1040
LC	TLM	930
Perforated clay bricks	GPM	1170
Perforated clay bricks	TLM	1190
Perforated clay bricks	lightweight	1480

30.2.3. Flexural tensile strength of masonry

The flexural tensile strength of masonry can be divided into flexural tensile strength perpendicular and parallel to the bed joints. It is a strongly scattering property and depends mainly on the mortar. The execution of the head joints (unfilled (uf) or filled (f)) was also found to have small influence. Since only thin layer mortar is providing sufficiently reliable cohesion, tests of the flexural tensile strength were more or less only conducted on masonry with TLM. Values of the flexural tensile strength for masonry with TLM can be found in Table 30.3 and Table 30.4.

Table 30.3 Flexural tensile strength \(f_{t, m, p a}\) of masonry (parallel to bed joints) according to [1]

Unit		Head joints\(^a\)	Number of tests \(n\)	\(f_{t, m, p a}\) in \(\mathbf{N} / \mathbf{m m}^{2}\)	Range of values in N/mm²
Type	Compressive strength of brick \(f_{b}\) in \(\mathrm{N} / \mathrm{mm}^{2}\)
Lightweight perforated CB	>8, >12	uf	2	0.21	0.20 ; 0.22
CS	>12, >20	f	6	0.51	0.36 \(\div\) 0.69
CS	>28	f	2	1.05	0.96 ; 1.14
CS	>8\(\ldots\)>28	uf	11	0.71	0.38 \(\div\) 0.97
Perforated CS	>12	f	4	0.48	0.45 \(\div\) 0.51
Perforated CS	>12	uf	4	0.25	0.29 \(\div\) 0.35
AAC	>	\(\mathrm{f}\)	3	0.23	0.22 \(\div\) 0.23
AAC	>4 \(\ldots\) >8	\(\mathrm{f}\)	7	0.45	0.28 \(\div\) 0.64
AAC	>2, >4	uf	6	0.20	0.16 \(\div\) 0.24

\(^a\) uf= unfilled, f= filled

Table 30.4 Flexural tensile strength of masonry \(f_{t, m, p e r}\) (perpendicular to bed joints) according to [1]

Unit		Head joints\(^a\)	Number of tests \(n\)	\(f_{t, m, p a}\) in \(\mathbf{N} / \mathbf{m m}^{2}\)	Range of values in \(\mathrm{N} / \mathrm{mm}^{2}\)
Unit		Head joints\(^a\)	\(n\)	\(f_{t, m, p a}\) in \(\mathbf{N} / \mathbf{m m}^{2}\)	Range of values in N/mm²
:—:	:—:	:—:	:—:	:—:	:—:
Type	Compressive strength of brick \(f_{b}\) in N/mm
Lightweight perforated CB	>8, >12	uf	3	0.28	0.26 \(\div\) 0.30
\(\mathrm{CS}\)	-	uf/f	8	0.56	0.35 \(\div\) 0.73
Perforated CS	-	f	4	0.34	0.23 \(\div\) 0.48
AAC	>2 \(\ldots\)>8	uf/f	23	0.40	0.25 \(\div\) 0.81
Concrete blocks	>2 \(\ldots\)>6	f	5	0.33	0.22 \(\div\) 0.44

\(^a\) uf= unfilled, f= filled

30.2.4. Tensile strength of the units

For the tensile strength in longitudinal direction of the units:

(30.5)\[f_{b t, l}=c_{2} \cdot f_{b}\]

For the splitting tensile strength of the units:

(30.6)\[f_{b t, s}=c_{3} \cdot f_{b}\]

with

\(f_{b t, l}=\) mean of tensile strength of the unit in longitudinal direction

\(f_{b t, s}=\) mean of splitting tensile strength of the units

\(c_{2}=\operatorname{ratio} f_{b t, l} / f_{b}\)

\(c_{3} = \operatorname{ratio} f_{b t, s} / f_{b}\)

For the large variety of masonry products which all differ in perforation and material, the prior values given in Table 30.5 can be used.

Table 30.5 Values \(c_{2}\) and \(c_{3}\) according to [1]

Type of unit	Number of tests \(\left(\boldsymbol{c}_{\mathbf{2}}\right)\)	\(\boldsymbol{c}_{\mathbf{2}}[-]\)	Number of tests \(\left(\boldsymbol{c}_{\mathbf{3}}\right)\)	\(\boldsymbol{c}_{\mathbf{3}}[-]\)
CS	18	0.063	40	0.070
Perforated CS	19	0.035	31	0.060
CB	9	0.040	9	0.070
Perforated CB	20	0.030	29	0.040
Lightweight perforated CB	54	0.010	-	-
Lightweight concrete hollow block	8	0.080	10	0.090
Full lightweight concrete block	23	0.080	13	0.110
AAC, plane element	24	0.110	24	0.090
AAC, plane element 2	7	0.180	9	0.150
AAC, plane element 4,6,8	8	0.110	9	0.120
Hollow Concrete block	2	0.080	3	0.040

30.2.5. Cohesion

The values \(f_{v, m}\) for the cohesion can be taken from Table 30.6. Please note that these values have been derived from the characteristic values provided in [30.01] by application of the stochastic models presented in Table 30.13 and assuming the characteristic values to represent 5%-quantiles. The values for the cohesion are prior values and require updating by use of test data.

Table 30.6 Mean of the cohesion \(f_{v, m}\)

\(f_{v, m}\) in \(MN/m^2\)
GPM	GPM	TLM	LWM
20 and higher	0.42	0.65\(^{\mathrm{a}}\)	0.26
10 \(\div\) 19	0.37	0.65\(^{\mathrm{a}}\)	0.26
5 \(\div\) 9	0.28	0.65\(^{\mathrm{a}}\)	0.26
2.5 \(\div\) 4.5	0.14	0.65\(^{\mathrm{a}}\)	0.26
1 \(\div\) 2	0.04	0.65\(^{\mathrm{a}}\)	0.26

\(^a\) divide by 2 if the perforation is more than 15% of the cross-section

30.2.6. Friction coefficient

The friction coefficient \(\mu\) is normally estimated with a mean value of 0.8 . In case of in-plane shear a reduction may become necessary to account for possible overturning of the units.

30.2.7. Stress-strain relationship

In unified form, the stress-strain relationship of masonry under compression is determined by:

(30.7)\[\frac{\sigma}{f_{m}}=\frac{k_{0} \cdot \eta-\eta^{2}}{1+\left(k_{0}-2\right) \cdot \eta} \]

(30.8)\[\eta=\frac{\varepsilon}{\varepsilon_{f}} \]

(30.9)\[k_{0}=\frac{E_{0} \cdot \varepsilon_{f}}{f_{m}}\]

The value \(k_{0}\) represents the slope of the unified stress-strain curve at origin. Values for the different materials can be taken from Table 30.7.

Table 30.7 Parameters for the stress-strain relationship of different kinds of masonry under compression

Lightweight con- crete	Hollow clay bricks and AAC	CS hollow block	CS full block
\(\mathrm{k}_{0}=1.0\)	\(\mathrm{k}_{0}=1.0\)	\(\mathrm{k}_{0}=2.0\)	\(\mathrm{k}_{0}=2.0\)
\(\varepsilon_{\mathrm{f}}=1.2~‰\)	\(\varepsilon_{\mathrm{f}}=2.0~‰\)	\(\varepsilon_{\mathrm{f}}=2.0~‰\)	\(\varepsilon_{\mathrm{f}}=2.0~‰\)
\(\varepsilon_{\mathrm{u}}=1.2~‰\)	\(\varepsilon_{\mathrm{u}}=2.0~‰\)	\(\varepsilon_{\mathrm{u}}=2.5~‰\)	\(\varepsilon_{\mathrm{u}}=3.5~‰\)

30.3. The probabilistic model

30.3.1. Compressive strength

The compressive strength of masonry depends on the compressive strength of units and mortar. The following equation (30.10) is recommended for the determination of the compressive strength.

(30.10)\[f_{m, j}=Y_{1} \cdot f_{m}\]

with

\(f_{m} =\) compressive strength of masonry according to equation (30.1) or (30.2)

\(Y_{1}=\) a log-normal variable accounting for uncertainties in the calculation of the compressive strength, see Table 30.8

Table 30.8 Stochastic Parameters for \(Y_{1}\)

Unit	Mortar	Distr.	Mean	CoV
CS	TLM	LN	1.0	20%
AAC	TLM	LN	1.0	20%
Large sized units CS	TLM	LN	1.0	16%
Large sized units AAC	TLM	LN	1.0	14%
Clay bricks	GPM	LN	1.0	17%

30.3.2. Modulus of elasticity

The modulus of elasticity is commonly related to the compressive strength of masonry. For the stochastic modelling, the following equation (30.11) is recommended. A first estimate for the stochastic parameters of the log-normal variable \(Y_{2}\) can be found in Table 30.7.

(30.11)\[E_{m, j}=Y_{2} \cdot E_{m}\]

with

\(E_{m}=\) modulus of elasticity according to equation (30.4)

\(Y_{2}=\) a log-normal variable

Table 30.9 Stochastic Parameters for \(Y_{2}\)

Unit	Mortar	Distr.	Mean	CoV
all	all types	LN	1.0	25%

30.3.3. Flexural tensile strength of masonry

There is significant lack of data for the tensile strength of masonry. Nevertheless, the values in Table 30.3 may be taken as first estimates for the means. The flexural tensile strength of masonry may be modelled as follows:

(30.12)\[f_{t, j}=Y_{3} \cdot f_{t, m, i}\]

with

\(f_{t, m} =\) mean of flexural tensile strength according to Table 30.3 and Table 30.4

\(Y_{3} =\) a log-normal variable

The stochastic parameters for the log-normal variable \(Y_{3}\) can be found in Table 30.10 depending on the direction of the flexural tensile strength (parallel or perpendicular to the bed joint). Please note, these values only represent prior parameters and require updating and validation by use of test data.

Table 30.10 Stochastic Parameters for \(Y_{3}\)

Unit	Mortar	Direction	Distr.	mean	CoV
CS	TLM	parallel	LN	1.0	30%
CS	TLM	perpen.	LN	1.0	35%
AAC	TLM	parallel	LN	1.0	26%
AAC	TLM	perpen.	LN	1.0	30%
Lightweight perforated CB	TLM	parallel	LN	1.0	30%
Lightweight perforated CB	TLM	perpen.	LN	1.0	35%
Concrete blocks	TLM	perpen.	LN	1.0	30%

30.3.4. Tensile strength of the units

The shear capacity of masonry walls is strongly influenced by the tensile strength of units. As for many materials, the tensile strength can be described as function of the compressive strength. It was found that the tensile strength in longitudinal direction describes the shear capacity of CS and AAC units well while in case of CB, the splitting tensile strength is preferable.

For the tensile strength in longitudinal direction of the units of CB and AAC units:

(30.13)\[f_{b t, l, j}=Y_{4} \cdot f_{b t, l}\]

For the splitting tensile strength of CB units:

(30.14)\[f_{b t, s, j}=Y_{4} \cdot f_{b t, s}\]

with

\(f_{b t, l}=\) the tensile strength of the units in the longitudinal direction according to equation (30.5)

\(f_{b t, s}=\) the splitting tensile strength of the units according to equation (30.6)

\(Y_{4}=\) a log-normal variable accounting for the large scatter of the tensile strength of the units. Values can be found in Table 30.11.

Table 30.11 Stochastic Parameters for \(Y_{4}\)

Unit	Distr.	Mean	CoV
CS	LN	1.0	26%
AAC	LN	1.0	16%
Clay bricks	LN	1.0	24%

30.3.5. Cohesion

The cohesion may be modelled according to equation (30.15). A first estimate for the stochastic parameters for the random variable \(Y_{5}\) may be taken from Table 30.12 for different combinations of units and mortar.

(30.15)\[f_{v, j}=Y_{5} \cdot f_{v, m}\]

with

\(f_{v, m}=\) mean of the cohesion according to Table 30.6

\(Y_{5}=\) a log-normal variable according to Table 30.12

Table 30.12 Stochastic Parameters for \(Y_{5}\)

Unit	Mortar	Dist.	Mean	CoV
AAC	TLM	LN	1.0	35%
CS	TLM	LN	1.0	35%
CB	IIa	LN	1.0	40%

30.3.6. Friction coefficient

The friction coefficient may be modelled according to equation (30.16) by use of the stochastic parameters found in Table 30.11.

(30.16)\[\mu_{j}=Y_{6} \cdot \mu_{m}\]

with

\(\mu_{m} =\) 0.8

\(Y_{6}=\) a log-normal variable according to Table 30.13

Table 30.13 Stochastic Parameters for \(Y_{6}\)

Unit	Distr.	Mean	CoV
CS	LN	1.0	19%
AAC	LN	1.0	19%

References

[30.01]

European Committee for Standardization. Design of masonry structures. Standard EN 1996-1-1 (Eurocode 6), Beuth Verlag, Berlin, 2005. AC: 2009.

Additional References

1. Schubert, P. (2010): Eigenschaftswerte von Mauerwerk, Mauersteinen und Mauermörtel, Mauerwerk-Kalender 2010, Ernst & Sohn, Berlin

2. Glowienka, S. (2007): Zuverlässigkeit von Mauerwerkswänden aus großformatigen Steinen, doctoral thesis, Technische Universität Darmstadt, Darmstadt, in German

3. Schueremans, L. (2001): Probabilistic Evaluation of structural masonry, doctoral thesis, University of Leuven, Belgium, 2001

4. Glock, C. (2004): Traglast unbewehrter Beton- und Mauerwerkswände, doctoral thesis, Technische Universität Darmstadt, Darmstadt, in German

5. Brehm, E. (2011): Reliability of Unreinforced Masonry Bracing Walls, doctoral thesis, ISBN 978-3-942886-02-4, Technische Universität Darmstadt