Construction and Building Materials 25 (2011) 4294–4298
Contents lists available at ScienceDirect
Construction and Building Materials
journal homepage: www.elsevier.com/locate/conbuildmat
The relationship between porosity and strength for porous concrete
C. Lian a,⇑, Y. Zhuge b, S. Beecham a
a
b
School of Natural and Built Environments, University of South Australia, Adelaide, South Australia, Australia
Faculty of Engineering and Surveying, University of Southern Queensland, Brisbane, Queensland, Australia
a r t i c l e
i n f o
Article history:
Received 19 November 2010
Received in revised form 5 April 2011
Accepted 23 May 2011
Available online 14 June 2011
Keywords:
Porous concrete
Compressive strength
Porosity
a b s t r a c t
As for many porous media, the strength of porous concrete is significantly affected by the porosity of its
internal structure. This paper describes the development of a mathematical model to characterize the
relationship between compressive strength and porosity for porous concrete by analyzing empirical
results and theoretical derivations. The suitability of existing equations for porous concrete is assessed
and a new model is then proposed. The new model, which was derived from Griffith’s theory, presents
a better agreement with the experimental data for porous concrete. It is demonstrated that the proposed
model could provide a better prediction of porous concrete compressive strength based on the material
porosity.
Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction
It is well known that the mechanical behavior of a building
material is predominately dependent on its composited structure.
The presence of pores can adversely affect the material’s mechanical properties such as failure strength, elasticity and creep strains
[1]. Porous concrete, which differs from conventional concrete, has
a large volume of air voids. Currently it is mainly utilized in permeable pavements and infiltration beds [2]. For maximizing the benefit of its permeability, several studies have been conducted to
reveal the relationship between pore features and the hydraulic
or acoustic conductivity of porous concrete [3,4]. But as a construction material, porous concrete also needs to be able to withstand
traffic loads. It is also important to determine how its mechanical
performance is affected by the presence of pores.
In a previous experimental investigation [5], the compressive
strength of porous concrete has been tested. This testing could
be used as an index to characterize the mechanical capacity of porous concrete in this study. On the other hand, the pore structure of
a porous material can be characterized by a number of parameters
including pore size, pore connectivity, pore surface roughness and
pore volume fraction (porosity). Of these, the porosity is regarded
as the primary parameter of porous material microstructures [6].
Normally the strength of a porous material is influenced by porosity, the other parameters listed above having less influence. Thus,
in this study the porosity is chosen as an independent variable to
relate to the material strength. The objective of this study is to
establish a quantitative relationship between porosity and compressive strength of porous concrete.
⇑ Corresponding author. Tel.: +61 8 83029941; fax: +61 8 83025721.
E-mail address: chunqi.lian@unisa.edu.au (C. Lian).
0950-0618/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.conbuildmat.2011.05.005
The influence of porosity on the strength of cement paste has already been investigated [7–9]. In these studies, hydrated cement
paste was considered the main source of pores within conventional
concrete. However, porous concrete contains a higher fraction of
large macroscopic pores which are required to achieve sufficient
hydraulic conductivity. Therefore, the suitability of the existing
relationships between porosity and strength developed for normal
concrete need to be examined and extended for porous concrete.
This is the purpose of the work described in this paper.
2. Experimental study
2.1. Mix compositions
The compositions used to prepare porous concrete in this study consisted of
coarse aggregates, ordinary Portland cement and water. However, admixtures such
as quarry sand, silica fume and superplasticizer were also used in some of the mixes
to produce a strength variation. Two groups of samples were developed. The mixture proportions for each were summarized in Table 1. The first group was produced with only coarse aggregate, cement and water. Quartzite, limestone and
dolomite were used as coarse aggregates and three gradings were selected (G1:
13.2–4.75 mm; G2: 9.5–6.7 mm; G3: 9.5–4.75 mm). The second group was made
with additives including 7% silica fume and 0.8% superplasticizer by weight of cement and some quarry sands as fine aggregates. For this second group, dolomite
was used as the coarse aggregate. In this second group, the water to cement ratio
was incrementally changed from 0.30 to 0.38. In this way, samples of different
strength and porosity were obtained. The preparation and mixing procedures are
discussed in [5].
2.2. Compressive strength
The compressive strength of porous concrete was determined through sample
tests according to AS1012.9-1999. The samples were cylinders of 100 mm diameter
and 200 mm height. After 24 h the samples were removed from the steel moulds
and moist cured until the day of testing. The curing condition complied with
AS1012.8.1-2000. Prior to testing, the samples were weighed to determine the den-
C. Lian et al. / Construction and Building Materials 25 (2011) 4294–4298
Table 1
28-Day mix proportioning and compressive strength of porous concrete.
Sample
number
Mix proportions
Coarse
aggregate
Group 1 (no additives)
1–1
Q-G2
1–2
Q-G3
1–3
L-G2
1–4
L-G3
1–5
D-G1
1–6
D-G2
1–7
D-G3
W/C
ratio
Percentage of
sand
cement hydration process: 0.25 is taken as the ratio of hydration water to cement by
weight, so the non-evaporable water mass is 0.25 times the anhydrous cement mass
Pc; and the volume of this water reduces to approximately 0.75 of the original volume after chemically hydrating the cement [9]. Thus, the total porosity can be calculated as:
Compressive
strength (MPa)
p¼1
0.36
0.36
0.36
0.36
0.36
0.36
0.36
0
0
0
0
0
0
0
4295
11.8
15.5
15.5
14.0
15.5
15.8
19.0
qb
;
qt
ð2Þ
where p is the total porosity and qb is the bulk density of the sample.
2.3.3. Relationship between effective porosity and total porosity
In this study, silica fume was employed in Group 2 to improve the strength of
porous concrete and consequently the gravity of qc was slightly adjusted. For calculating qc, the relative gravity for cement and silica fume was taken as 3.15 and 2.2
respectively. Hence, qc = (1 + 0.07)/(1/3.15 + 0.07/2.2) = 3.063 for Group 2. The measured effective porosity and estimated total porosity for each mix are shown in
Fig. 1. The relationship is approximately linear.
The best fitted regression line for the data is given by:
Group 2 (7% silica fume and 0.8% plasticizer by weight of cement)
2–1
D-G3
0.38
18
23.3
2–2
D-G3
0.36
18
33.2
2–3
D-G3
0.34
18
46.2
2–4
D-G3
0.32
18
40.5
2–5
D-G3
0.30
18
40.3
2–6
D-G3
0.30
15
43.0
2–7
D-G3
0.28
18
24.3
p ¼ 1:28pe 18:11:
ð3Þ
2
Note: (1) Q – quartzite; L – limestone; D – dolomite; (2) the percentage of sand was
based on the weight of coarse aggregate.
sity. The samples were then sulphur capped and the unconfined compressive
strengths were tested at 7 and 28 days. The 28-day results were taken as representative values of compressive strengths of porous concrete and the tested average
values for each batch are listed in Table 1.
2.3. Porosity
Ghafoori and Dutta [10] stated that the majority of pores in porous concrete are
formed by the spaces left between coarse aggregates and they distinguished between porosity and air void content. In their research, the fraction of measureable
voids migrated by fluids in their experiments was termed porosity and the sum of
measureable voids between aggregates plus entrained or entrapped air in the cement paste was termed air content. In other words, the porosity of porous concrete
could be defined differently. In this study, for clarity, the measureable voids are defined as the effective porosity since this relates to permeability and the overall air
content is accordingly defined as total porosity.
2.3.1. Effective porosity
The effective porosity was determined by testing the volume of water displaced
by samples. The sample was firstly oven dried at 110 °C and then immersed in
water for up to 24 h. By measuring the difference in the water level before and after
immersing the sample, the volume of water repelled by the sample (Vd) can be
readily determined. Subtracting Vd from the sample bulk volume (Vb) yields the volume of open pores. This volume was then expressed as a percentage as an effective
porosity percentage: pe = (Vb Vd)/Vb 100%.
The R value of 0.947 shows a good correlation between measured effective
porosity and estimated total porosity. A similar relationship was also determined
by Ghafoori and Dutta [10] but with different coefficient values. This could be
attributed to the different mix proportions and compaction energy applied in making the samples.
3. Existing models for cementitious materials
Before creating a quantitative model to characterize the relationship between porosity and compressive strength for porous
concrete, it is worth noting that the influence of porosity on
strength has been investigated for several engineering materials,
such as ceramics, metals, plaster and rocks [1]. The research presented in this paper focuses developing a mathematical model between total porosity and compressive strength of porous concrete.
Historically, four general types of model have been developed
[19,20] for cement-based materials, as summarized in Table 2.
In Eqs. (a)–(c), the porosity (p) and the corresponding strength
(f) of a porous material are related through a parameter, r0, which
is the material strength when porosity is zero. In Eq. (d), p0 is the
porosity when the material has zero strength. Chindaprasirt et al.
[21,22] have demonstrated that the exponential relationship (type
c in Table 2) proposed by Ryshkevitch [17] was valid for describing
porous concrete. Fig. 2 shows how an exponential regression equation can be fitted to the data from the present study.
From Fig. 2, the fitted exponential curve yields the equation:
r = 231 exp (0.09p), with an R2 value of 0.90, which is lower than
the value of 0.96 obtained by Chindaprasirt et al. [22]. This may be
2.3.2. Total porosity
The strength of concrete is affected by the volume of its overall voids [9]. In the
complex microstructure of concrete, the pores can be present from the nano-scale
to the macro-scale. The difficulty of accurately testing the total porosity of porous
concrete arises from its unique microstructure. Compared with the pores within cement paste, the interconnected voids between coarse aggregate are larger by several millimetres. Although it is well known that the method of mercury intrusion
porosimetry (MIP) is effective for observing the pore configuration in normal concrete, the large amount of connected voids within porous concrete will cause dripping and leakage of mercury if pressure is applied. Thus, the method of MIP is not
feasible for porous concrete. Vacuum sealing apparatus is more appropriate to test a
relatively accurate porosity for porous concrete in laboratory research [11]. However, in practice, setting up such a delicate apparatus is challenging for concrete
manufacturers and a simpler method is preferred. In the literature, Kearsley and
Wainwright [12] have successfully used the Hoff equation [13] to estimate the total
porosity of foam concrete. Similarly, Zheng [14] has presented an equation to estimate the total porosity of porous concrete, which was analogous to the Hoff equation, but incorporated the aggregate proportions for porous concrete. This is shown
in the following equation:
qt ¼ 100
q
100 þ P c þ 0:25P c
qw ;
þ qPc þ ð0:25Pc 0:75Þ
ð1Þ
c
where qt is the theoretical density, Pc is the cement to aggregate ratio by weight, qc
is the specific gravity of cement, qw is the unit weight of water and q is the aggregate
apparent density. It can be seen that this equation was derived by understanding the
Fig. 1. Relationship between measured effective porosity and estimated total
porosity.
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C. Lian et al. / Construction and Building Materials 25 (2011) 4294–4298
Table 2
Empirical models relating porosity and strength of cement-based materials.
(a)
(b)
(c)
(d)
Equations
Mathematic
law
Constant
Derivation
r = r0
Linear
b
Power
n
Exponential
c
Logarithmic
k
Derived by Hasselmann [15]
originally for glass
Derived by Balshin [16] for
powder metals
Proposed by Ryshkevitch [17]
for ceramics and rocks
Proposed by Schiller [18] for
non-metallic brittle materials
(1 bp)
r = r0
(1 p)n
r = r0
exp (cp)
r = k ln
(p0/p)
because they tested the strength of hardened cement paste used in
the porous concrete and also the fineness modulus of aggregates.
These values were used to calibrate the equation constants. This
paper tries to theoretically calibrate the model for situations when
the strength of the hardened cement paste is not available.
4. Proposed model
4.1. Theoretical derivation
Griffith’s model of fracture [23] is usually taken as a classic theory to explain how the mechanical performance is related to porosity. Griffith found that the critical stress incurs crack propagation
within a brittle material and can be expressed by:
rffiffiffiffiffiffiffiffi
2Ec
;
r¼
pa
ð4Þ
where r is the stress at the fracture (Pa), E is the elasticity modulus
(Pa), c is the fracture surface energy (J/m2) and a is the half length of
an internal crack (m).
When considering this criterion for a porous material, the effective value of E and r need to be determined. This is because the
presence of pores affects both elasticity and fracture energy. The
elasticity and fracture energy are both reduced compared to the
pore-free solid material. After mixing well and compacting the porous concrete, the cement paste wraps around the aggregate and
behaves as one unit with air voids. Therefore all the pores, including both the large interconnected voids between aggregates and
the small ones in the paste, are taken as defects that can lead to
fracture of the porous concrete. The failure stress of the paste matrix can be determined by Eq. (4).
Various equations have been developed to describe the influence of pore content on Young’s Modulus and surface energy for
Fig. 3. The proposed model for compressive strength versus porosity.
different materials [24,25]. In the present study, two different
methods were considered when choosing the appropriate empirical equations for E and c to use in Eq. (4). Firstly, Rice [26] observed
the reduction of Young’s modulus as E = E0 exp (tp), where E0 is
the elastic modulus of the material at zero porosity with t as a constant. He also determined the fracture energy of pores as: c = c0
exp (qp), where c0 is the fracture energy at zero porosity and q
is a constant. This showed the variation of the fracture energy
would be the same as that of the Young’s modulus in terms of
porosity. If these empirical relationships are assumed for porous
concrete, the following relationship can be derived from Eq. (4):
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffi
2E0 eqp c0 etp
2E0 c0 mp
mp
¼
e
r¼
¼ ke ;
pa
pa
ð5Þ
where k and m are constants.
This equation is similar to the empirical one which was observed by Chindaprasirt et al. [22]. Kendall et al. [27] proposed
an alternative method. They applied a different function of Young’s
modulus: E = E0 (1p)3 with fracture energy c = c0 exp (tp) into a
fracture criterion formula and the predicted failure stress was
compared with the test results from their study of concrete made
Fig. 2. Relationship between compressive strength and total porosity.
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C. Lian et al. / Construction and Building Materials 25 (2011) 4294–4298
Table 3
Analytical and experimental data for modeling.
Sample
number
Cement to aggregate
ratio
Specific gravity of
binder
Aggregate density (103
kg/m3)
Sample
density
Total porosity
(%)
Compressive strength
(MPa)
Y
x1
x2
1–1
0.22
0.22
0.22
3.15
3.15
3.15
2.65
2.65
2.65
1831
1734
1763
30.90
34.57
33.48
12.00
12.00
11.50
4.97
4.97
4.89
0.37
0.42
0.41
0.31
0.35
0.33
1–2
0.22
0.22
0.22
3.15
3.15
3.15
2.65
2.65
2.65
1880
1800
1840
29.06
32.08
30.57
17.50
14.50
14.50
5.72
5.35
5.35
0.34
0.39
0.36
0.29
0.32
0.31
1–3
0.22
0.22
0.22
3.15
3.15
3.15
2.74
2.74
2.74
1985
2105
1947
27.55
23.17
28.94
15.50
19.50
11.50
5.49
5.94
4.88
0.32
0.26
0.34
0.28
0.23
0.29
1–4
0.22
0.22
0.22
3.15
3.15
3.15
2.74
2.74
2.74
1960
1920
1900
28.46
29.92
30.65
15.50
13.00
13.50
5.48
5.13
5.21
0.34
0.36
0.37
0.28
0.30
0.31
1–5
0.22
0.22
0.22
3.15
3.15
3.15
2.70
2.70
2.70
1940
1940
1900
28.15
28.15
29.63
17.00
16.50
13.00
5.67
5.61
5.13
0.33
0.33
0.35
0.28
0.28
0.30
1–6
0.22
0.22
0.22
3.15
3.15
3.15
2.70
2.70
2.70
1863
1895
1880
31.00
29.81
30.37
15.00
17.00
15.50
5.42
5.67
5.48
0.37
0.35
0.36
0.31
0.30
0.30
1–7
0.22
0.22
0.22
3.15
3.15
3.15
2.70
2.70
2.70
1920
1980
1920
28.89
26.67
28.89
17.00
22.50
17.50
5.67
6.23
5.72
0.34
0.31
0.34
0.29
0.27
0.29
2–1
0.25
0.25
0.25
3.06
3.06
3.06
2.70
2.70
2.70
2080
2060
2040
22.96
23.70
24.44
30.50
31.50
28.00
6.84
6.90
6.66
0.26
0.27
0.28
0.23
0.24
0.24
2–2
0.25
0.25
0.25
3.06
3.06
3.06
2.70
2.70
2.70
2120
2120
2140
21.47
21.47
20.73
34.50
32.00
33.00
7.08
6.93
6.99
0.24
0.24
0.23
0.21
0.21
0.21
2–3
0.25
0.25
0.25
3.06
3.06
3.06
2.70
2.70
2.70
2240
2240
2240
17.02
17.02
17.02
49.00
46.50
43.00
7.78
7.68
7.52
0.19
0.19
0.19
0.17
0.17
0.17
2–4
0.25
0.25
0.25
3.06
3.06
3.06
2.70
2.70
2.70
2160
2200
2140
19.98
18.50
20.72
39.50
42.00
40.00
7.35
7.48
7.38
0.22
0.20
0.23
0.20
0.19
0.21
2–5
0.25
0.25
0.25
3.06
3.06
3.06
2.70
2.70
2.70
2180
2180
2140
19.24
19.24
20.72
41.00
41.00
39.00
7.43
7.43
7.33
0.21
0.21
0.23
0.19
0.19
0.21
2–6
0.25
0.25
0.25
3.06
3.06
3.06
2.70
2.70
2.70
2140
2140
2200
20.72
20.72
18.50
42.00
44.00
43.00
7.48
7.57
7.52
0.23
0.23
0.20
0.21
0.21
0.19
2–7
0.25
0.25
0.25
3.06
3.06
3.06
2.70
2.70
2.70
1960
1960
1920
27.39
27.39
28.87
23.00
26.50
23.50
6.27
6.55
6.31
0.32
0.32
0.34
0.27
0.27
0.29
with polymers. In light of this, a combined functional model based
on this approach is proposed for porous concrete as shown in the
following equation:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2E0 ð1 pÞm c0 enp
r¼
;
pa
ð6Þ
Squaring both sides and taking the natural logarithm of each
side:
2 ln r ¼ m lnð1 pÞ np þ ln A;
ð7Þ
which now can be regarded as a linear equation of the form:
Y ¼ mx1 þ nx2 þ c;
ð8Þ
with Y ¼ 2 ln r; x1 ¼ lnð1 pÞ; x2 ¼ p and c ¼ ln A:
ð9Þ
where m and n are new material constants for porous concrete.
4.2. Examination of the proposed model
To assess the validity of the proposed model, a regression analysis was performed on Eq. (6) based on the available experimental
data. The proposed model was shown in Fig. 3. It can be seen from
the figure that Eq. (6) is complicated as a combined format of
power and exponential relations. Thus, in order to utilize a linear
least-squares regression technique, Eq. (6) has to be rearranged:
First of all, 2Ep0ac0 ¼ A is assumed, regardless of the possible different pore sizes formed in different samples. Then:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r Að1 pÞm enp :
The values of Y, x1 and x2 are calculated in Table 3.
4.3. Results and discussion
The multiple linear regression run by least square method generates the best fitted plane and parameters, as shown in Fig. 4. The
regression results are: m = 5.96 and n = 10.01 when c = 10.61 for
Eq. (8), which is: Y = 5.96x1 10.01x2 + 10.61. The coefficient of
determination R2 for this equation is estimated to be 0.99 and
the standard error of estimated Y is 0.306. This indicates that the
model could describe the correlation between compressive
strength and porosity for porous concrete with acceptable
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C. Lian et al. / Construction and Building Materials 25 (2011) 4294–4298
(3) With a large set of data on porosity and tested compressive
strength, a new model using Griffith’s fracture theory has
been proposed. It has been shown that the proposed model
provides a stronger relationship between the compressive
strength and the porosity of porous concrete, with a model
regression statistic R2 of up to 0.99. This represents a significant improvement over the simple exponential equation.
Other statistics also verified that this semi-empirical model
could predict the compressive strength of porous concrete
based on the material porosity.
Acknowledgment
The authors would like to express their special thanks to Mr.
David Carver for his technical assistance during the experimental
work.
References
Fig. 4. Relationship between Y and X1, X2.
accuracy. Moreover, the F statistic is calculated as 4361.7, with an
extremely small probability of 1.857E 42, indicating that the observed relationship did not occur by chance. This means that the
proposed model is reliable for predicting the failure compressive
strength of porous concrete.
In addition, the proposed model predicts a zero strength when
the material is assumed to be fully porous, conquering the limitation of the exponential function which cannot make sense when
the porosity is close to 1. Therefore it offers a wider range for application. On the other hand, it is noticed that while the current proposed model is inclusive of all three aggregate types used in this
study, some factors such as aggregate shape and absorption were
not accounted for separately. Another constant B is suggested to
be employed in the proposed Eq. (6) for future work to account
for these additional factors. In this case, a general format for this
future model could be:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r ¼ B ð1 pÞm enp ;
where B could be determined when these additional factors are
tested and quantified.
5. Conclusions
The dependence of compressive strength on porosity for porous
concrete was analysed empirically and theoretically in this paper.
The following conclusions can be drawn:
(1) The effective porosity of porous concrete has been measured. However, since the non-intrusive pores weaken the
strength of concrete, the total porosity was estimated and
then compared with the effective porosity. It has been demonstrated that the estimated total porosity has a good correlation with the measured effective porosity. This estimation
method could be used when total porosity testing apparatus
is not available.
(2) Existing equations relating compressive strength and porosity for cement-based materials were presented and a potential equation for porous concrete has been appraised by
fitting to the experimental data. It has been shown that
without extra knowledge of paste strength, the exponential
function derived using experimental data resulted in a relatively low correlation coefficient.
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