75
Journal of Sustainable Architecture and Civil Engineering 2017/3/20

Introduction

*Corresponding author: volhasannikava@list.ru

Estimation of the Restrained 
Strains and Self-stresses 
under Two-way Restraint 
Conditions Received  

2017/09/18

Accepted after  
revision 
2017/11/21

Journal of Sustainable 
Architecture and Civil Engineering
Vol. 3 / No. 20 / 2017
pp. 75-86
DOI 10.5755/j01.sace.20.3.19082
© Kaunas University of Technology

Estimation of the 
Restrained Strains 
and Self-stresses 
under Two-way 
Restraint Conditions

JSACE 3/20

http://dx.doi.org/10.5755/j01.sace.20.3.19082

Volha Sannikava*, Viktar Tur
Brest State Technical University, Faculty of Civil Engineering 
Moskovskaya 267, 224017, Brest, Republic of Belarus

Models for the restrained expansion strains under two-way restraint conditions both with their 
advantages and disadvantages are presented. An implementation of the modified strains development 
model (MSDM) for the assessment of the two-way restrained expansion strains is proposed. The 
reliability of the proposed model for the two-way restraint conditions is confirmed by the experimental 
results based on the tests of the expansive concrete plane specimens with the mesh reinforcement in 
the mid-depth of the cross section.

Keywords: design model, expansive concrete, restrained expansion strains, two-way restraint conditions.

As was shown in Semianiuk et al. 2017, the development of concrete technologies in the last de-
cades has opened the door to extended use of the High performance Concrete (HPC). Shrinkage 
strains perform the most considerable problem of practical application of the High strength and 
High performance concretes because of inducing the cracks formation at early age that makes 
worse the serviceability performance of concrete structures. Shrinkage processes are typical of 
any concrete system and depend on many factors such as type of cement, type of mineral ad-
mixtures, water-cement ratio and volume concentration of aggregate (Tazawa et al. 2000). Many 
studies have proved the effectiveness of the expansive concretes in the compensating of shrink-
age (Tanimura et al. 2007) as well as in realization of the chemical prestressing of the structures 
(Mikhailov and Litver 1974). 1974). In general, as was shown in Semianiuk et al. 2017, models for 
the estimation of the restrained strains and self-stresses in expansive concrete at early age are 
based on one of two basic concepts: conservation law of chemical energy or initial strain calcu-
lation.

To design self-stressed concrete structures it’s necessary to take into account the initial stress-
strain state caused by the concrete expansion under restraint conditions. The dependence of re-
strained strains from the reinforcement arrangement (uniaxial, biaxial, triaxial), stiffness charac-
teristics of restraint, curing conditions is confirmed by numerous investigations of self-stressed 
elements (Litver and Petukhov 1984, Litver and Sabajeva 1984, Marchuk 2002, Mikhailov and 
Gershvald 1984, Mikhailov and Litver 1974,). It should be noticed that because of the features of 
the expansive concrete process under two-way restraint conditions the self-stressed concrete ob-



Journal of Sustainable Architecture and Civil Engineering 2017/3/20
76

tain more solid structure than that of under one-way restraint conditions. Therefore the shrinkage 
strains become less apparent for the self-stressed elements under two-way restraint conditions. 
Moreover these elements made with high expansion energy capacity concrete perform the ex-
treme values of self-stressing in comparison with the element under uniaxial restraint (Litver and 
Sabajeva 1984). 

Consequently, design model for the numerical determination of the restrained strains and 
self-stresses at the expansive stage of the concrete should be done in favour of that which could 
adequately reproduce chemical, physical and mechanical processes during the self-stressing 
concrete expansion under various design and technological factors.

Models of estimation the restrained expansion strains and self-stresses at the 
expansion stage of the concrete under two-way restraint conditions
The following models for the evaluation of the restrained expansion strains in the reinforcement 
for elements under two-way restraint conditions have been suggested.

Model 1. In accordance with the TCP Codes (2010), the strain in the center of gravity of reinforce-
ment at the section by the time of the expansive concrete stabilization is expressed as:

Methods for 
estimation 

of the 
restrained 

strains and 
stresses 

in the 
expansive 

concrete at 
early age

. . 0
1

s CE CE d s e w
l s

f k k k k k
E ρ

ε = ⋅ ⋅ ⋅ ⋅ ⋅
ρ ⋅

(1)

1, 57
0, 0057

l

l

kρ
⋅ρ

=
+ρ

(2)

where .CE df  – design self-stress of the con-
crete that is taken from the table 6, 7 TCP 
Codes (2010) depending on the self-stress-
ing grade of the expansive concrete; lρ – to-

tal cross sectional reinforcement ratio; sE  – Young’s modulus of the reinforcement; sk – factor that 
depends on the reinforcement arrangement in plane and equals 1,2 for the biaxial reinforcement 
arrangement in the concrete element; kρ – factor that depends on the total cross sectional rein-
forcement ratio for the given direction ( lρ ) and is defined as follows:

ek  – factor that represents the influence of 
the reinforcement eccentricity; wk  – factor 
that counts the influence of the expansive 
concrete initial compressive strength at the 

beginning of moisture curing; 0k – factor that counts self-stressed element curing conditions at the 
expansion stage of the concrete.

From this model it could be seen that two-way axial restraint conditions are considered only by 
the factor sk from the Eq. (1).
Model 2. According to the experimental data obtained by Marchuk (2002) the empirical expres-
sions which consider the interference of the restraint in the principal axes directions on the re-
strained expansion strains are received:

. . .0

. . .0

(0, 98 16 12 ) ,
(0, 98 16 12 ) ,

CE x ly lx CE x lx ly

CE x lx ly CE x lx ly

if
if

ε = + ⋅ρ − ⋅ρ ⋅ε ρ ≤ρ

ε = + ⋅ρ − ⋅ρ ⋅ε ρ >ρ

(3)

. . .0

. . .0

(0, 98 16 12 ) ,
(0, 98 16 12 ) ,

CE y lx ly CE y ly lx

CE y ly lx CE y ly lx

if
if

ε = + ⋅ρ − ⋅ρ ⋅ε ρ ≤ρ

ε = + ⋅ρ − ⋅ρ ⋅ε ρ >ρ

(4)

 _ in the direction x

 _ in the direction y

where . .0CE xε  and . .0CE yε  – restrained expansion strains of the expansive concrete in the direction 
x without considering the restraint in the direction y and otherwise in the direction y without con-
sidering the restraint in the direction x respectively:



77
Journal of Sustainable Architecture and Civil Engineering 2017/3/20

1/ 2 , 1/ 2 , , , 1/ 2, 1/ 2
1

( ) [( ) ( , )] ( )
i

x i x c j i j x T as ds i
j

t J t t t t+ + ∆ +
=

ε = ∆σ ⋅ +ε∑ (8)

..
. .0 . .0;

CE d yCE d x
CE x CE y

lx s ly s

f kf k
E E

ρρ ⋅⋅ε = ε =
ρ ⋅ ρ ⋅

(5)

where lxρ  and lyρ  – reinforcement ratio in the 
directions x and y respectively; .CE df , sE , kρ – 
as in the Eq. (1).

As seen from the above models 1 and 2, the components (0, 98 16 12 )lx ly+ ⋅ρ − ⋅ρ  and 
(0, 98 16 12 )ly lx+ ⋅ρ − ⋅ρ  from the Eqs. (3) and (4) are, in fact, the refined values of the factor sk from 
the Eq. (1).

Model 3. In the work Litver and Petukhov (1984) for the symmetrical reinforced plane self-stressed 
elements with the reinforcement ratio lρ  the self-stress is defined in accordance with:

. .20,702 0,0278
.2 . .2

CE df
CE CE d lf

− ⋅σ = ⋅ρ (6)
where . .2CE df  – the value of the self-stress 
which is defined in the dynamometrical 
rings with stiffness equals 1%lρ = .

Model 4. In the case of nonsymmetrical restraint in two directions in the paper Mikhailov and Ger-
shvald (1984) the system of equations is used:

1/1/

.

. .

1/ 1/
.

. .

(1 ) 0

(1 ) 0

nn

CE x

CE x CE y x

n n
CE y

CE y CE x y

A A
B

C C k

A A
B

C C k

    σ −λ − −λ − =    σ + σ +   

  σ 

−λ − −λ − =    σ + σ +  

(7)

where .CE xσ  and .CE yσ  – self-stresses in the concrete in the direction x and y respectively; A, B, C – 
empirical coefficients; 210x lx sk E

−= ρ ⋅ ⋅ , 210y ly sk E
−= ρ ⋅ ⋅  – stiffness characteristics of restraint in 

the direction x and y respectively; λ – Poisson’s ratio of the expansive concrete at early age that is 
accepted to be equal to 0,46..0,48 in accordance with Mikhailov and Gershvald (1984).

It should be mentioned that all proposed design models of defining the stress-strain parameters 
of self-stressed elements under two-way restraint conditions have some significant disadvantag-
es. First, all formulas contain either the empirical factors or expressions which are restricted by 
research conditions. Second, no one formula counts the early age expansive concrete behavior as 
well as the contribution of elastic and plastic (creep) concrete strains on the restrained strains. As 
in the researches Tsuji (1984) these models are based on the energy conservation law and allow 
to calculate the self-stresses or restrained expansion strains only by the end of the expansive con-
crete stabilization. It’s significant that at practical application of listed models some contradictions 
have been evolved (TCP Code 2010) as they are being caused by different correlations of the free 
expansion strains and the standard self-stressing grade of the expansive concrete .CE df .
Model 5. A more advanced finite element method along with a practical calculation method based 
on the beam theory considers the mentioned disadvantages (Ito et al. 2004). In accordance with 
this model for the self-stressed reinforced concrete elements in the case of the uniaxial reinforce-
ment arrangement the strain in the x-direction at the i-th time interval is calculated by:

where ,( )x c j∆σ  – incremental self-stress in the x-direction at the j-th time interval; , , , 1/ 2, 1/ 2( )x T as ds it t∆ +ε  – 
difference of the free strain due to the temperature change and shrinkage or expansion of the 



Journal of Sustainable Architecture and Civil Engineering 2017/3/20
78

where s  – empirical factor that takes into 
account cement type; a  – empirical factor 
that takes into account initial setting time 
effect; ,28et  – temperature adjusted con-
crete age at 28 days; t  – temperature ad-
justed concrete age at t   days.

0,5
,28

,28( ) exp 1
e

c c

t a
E t E s

t a

  − 
  = ⋅ −   −   

(10)

The temperature adjusted concrete age at t  days is determined by the equation (Ito et al. 2004):

where it∆  – number of days where a tem-
perature ( )T C°  prevails; ( )iT t∆  – tem-
perature in [ ]C°  during the time period it∆ ; 

0 1T C= ° .
1 0

4000
exp 13, 65

273 ( ) /

n

i
i i

t t
T t T=

 
= ∆ ⋅ − + ∆ 
∑ (11)

The creep coefficient 0( , )t tϕ  for the expansive concrete at early age is defined in accordance with 
fib Model Code (2010) by the equation:

0 0 0( , ) ( , )ct t t tϕ = ϕ ⋅β (12)

( )20 0 ,285, 31 ( ) 1 1,11c cE t Eϕ = ⋅ − + (13)

0,3

0
0

0

( , )
( )c í

t t
t t

t t
 −

β =  β + − 
(14)

where 0ϕ  – notional creep coefficient that 
is estimated with regard to the relative 
Young’s modulus 0 ,28( ) /c cE t E  (the ratio of the 

Young’s modulus of the expansive concrete at 0t  to that of the expansive concrete at 28 days in the 
temperature adjusted concrete age) by the expression suggested in the paper Ito et al. (2004):

0( , )c t tβ  – coefficient to describe the devel-
opment of creep with time after loading 
and it is calculated in accordance with fib 

Model Code (2010) by the equation:

where t  – temperature adjusted concrete 
age; 0t  – temperature adjusted concrete 
age of loading age; 

íβ – coefficient repre-
senting the effect of loading age on the rate 
of creep development and it is defined with 

regard to the relative Young’s modulus ,28( ) /c cE t E  (the ratio of the Young’s modulus of the expan-
sive concrete at t  to that of the expansive concrete at 28 days in the temperature adjusted concrete 
age)  by the expression suggested in the paper Ito et al. (2004):

1/ 2
1/ 2

,28

( , )1
( , )

( )
i j

i j
c j c

t t
J t t

E t E
+

+

ϕ
= + (9)

where ( )c jE t  – Young’s modulus of the ex-
pansive concrete at jt  days in temperature 
adjusted concrete age; ,28cE  – Young’s mod-
ulus of the expansive concrete at 28 days in 

temperature adjusted concrete age; 1/ 2( , )i jt t+ϕ  – creep coefficient of the expansive concrete at 1/ 2it +  
days under constant self-stress applied at jt  days.
The Young’s modulus of the expansive concrete at early age is calculated in accordance with the Euro-
code 2 model (2004):

expansive concrete in the x-direction from the time interval 1/ 2t  to the 1/ 2it + ; 1/ 2( , )i jJ t t+  – creep 
compliance function which is defined by: 

,28

,28
,28

0, 000001, 0 ( ) / 0, 346

( )
40, 5 0, 346 0, 485, 0, 346 ( ) / 1, 0

í c c

c
í c c

c

if E t E

E t
if E t E

E

β = ≤ <


 
β = ⋅ − + ≤ <   

 

(15)



79
Journal of Sustainable Architecture and Civil Engineering 2017/3/20

The difference of strain at i-th time interval and (i-1)-th time interval is obtained:

1

, 1/ 2 ( 1) 1/ 2 , 1/ 2 , , , ,
1 ,28

( , )
( ) ( ) ( ) ( ) ( , ) ( ) ( )

i
i j

x c i x i x i x c i i i x c j x T as ds i
j c

t t
t t J t t

E

−

+ − + + ∆
=

 ∆ϕ
∆ε = ε −ε = ∆σ ⋅ + ∆σ ⋅ + ∆ε 

  
∑

1

, 1/ 2 ( 1) 1/ 2 , 1/ 2 , , , ,
1 ,28

( , )
( ) ( ) ( ) ( ) ( , ) ( ) ( )

i
i j

x c i x i x i x c i i i x c j x T as ds i
j c

t t
t t J t t

E

−

+ − + + ∆
=

 ∆ϕ
∆ε = ε −ε = ∆σ ⋅ + ∆σ ⋅ + ∆ε 

  
∑

(16)

1/ 2 ( 1) 1/ 2( , ) ( , ) ( , )i j i j i jt t t t t t+ − +∆ϕ = ϕ −ϕ (17)

with:

Thus the incremental stress in the x-direction at the i-th time interval is calculated as follow:

1
,

, , , , ,
1 ,28

1/ 2
,28

( )( )
( ) ( ) ( , ) ( )

( )
1 ( , )

i
x c jc i

x c i x c i i j x T as ds i
c i j c

i i
c

E t
t t

E t Et t
E

−

∆
=

+

  ∆σ
∆σ = ∆ε − ⋅∆ϕ − ∆ε      + ⋅ϕ

∑ (18)

where 1/ 2( , )i it t+ϕ – creep coefficient of the expansive concrete at 1/ 2it +  days under constant self-
stress applied at it  days; , , ,( )x T as ds i∆∆ε  – incremental free strain due to the temperature change 
and shrinkage or expansion of the expansive concrete in the x-direction at the i-th time interval; 

( , )i jt t∆ϕ  – incremental creep coefficient of the expansive concrete at the i-th time interval under 
constant self-stress applied at the j-th time interval.

Being the most rational today, the deformation model 5 can be used for the self-stressed rein-
forced concrete elements under arbitrary restrained conditions and allows to evaluate the stress-
strain relationship of the self-stressed elements from the early age to the concrete expansion 
stabilization. If the free strain function and the incremental self-stress at (i-1)-th time interval are 
known, the incremental self-stresses and correspondingly the incremental restrained expansion 
strains at i-th time interval in the x-direction are calculated. 

However, the presented above model has got one important contradiction, what has been con-
firmed by own experimental data. As noticed in the researches Tur and Semianiuk (2016) and 
Semianiuk et al. (2017) the expansion of the expansive concrete element at the elementary i-th 
time interval occurs in the presence of the cumulative force induced by the restraint at the end of 
the (i-1)-th time interval (Fig. 1).

Fig. 1 
Scheme of the 
addition cumulative 
force action

ΔPx.mt – cumulative force 
induced by the restraint in the 
x-direction;
Δlx.s – increment of restraint 
length in the x-direction caused 
by the concrete expansion;

Δlx.mt – increment of restraint 
length in the x-direction caused 
by the cumulative force;
Ac, Ec – parameters of concrete 
section;
As, Es – parameters of restraint.

Thus, the main theses of the deformation model (model 5) modified by taking into account the 
additional cumulative force induced by the restraint have allowed to formulate the design model 
for self-stressed elements under two-way restraint conditions both for symmetrical and nonsym-
metrical restraint in two directions.



Journal of Sustainable Architecture and Civil Engineering 2017/3/20
80

Implementation of the MSDM-model for the assessment  
of the two-way restrained strains
The general theoretical background of the modified early age strains development model (MSDM) 
for concrete with high expansion energy capacity in the case of the uniaxial reinforcement ar-
rangement has been presented in the works Tur and Semianiuk (2016) and Semianiuk et al. (2017). 

It should be noticed that because of the bond conditions the expansive concrete and the reinforce-
ment deform compatibly:

, , ,( ) ( )x c i x s i s x lE∆σ = ∆ε ⋅ ⋅ρ (19)

1/ 2 ,
,28

( )
( )

1 ( , )

c i

c i
i i s x l

c

E t
D

E t
t t E

E +

=
 
+ ⋅ϕ ⋅ ⋅ρ  

 

(21)

In the present investigations, considering 
the additional cumulative force, the Eq. (9) 
and the law of signs (expansive strains are 

taken with “plus”, shrinkage strains are taken with “minus”), the incremental restrained expansion 
strains in the x-direction at the i-th time interval are calculated as follows:

1 1

, , , ,
1 1,28 ( 1) 1/ 2

( , )
( ) ( ) ( ) ( )

1 ( )

i i
i j s

x s i cf i x c j x s j x l
j jc c i

t t ED
D E E t

− −

= = − +

  ∆ϕ
∆ε = ∆ε − ∆σ ⋅ − ∆ε ⋅ρ ⋅   +    

∑ ∑ (20)

where ( )cf i∆ε  – incremental free expansion 

strain at the i-th time interval; 
1

,
1
( )

i

x s j
j

−

=

∆ε∑ – 
resultant restrained expansion strains from 
the cumulative force at the end of (i-1)-th 
time interval of the concrete expansion; 

( 1) 1/ 2( )c iE t − +  – Young’s modulus of the expansive concrete at the end of (i-1)-th time interval.
The details of the Eq. (10) are presented in the work Tur et al. (2016).

In the case of two-way restraint conditions the stresses in the principal axes directions are con-
nected by means of Poisson’s ratio ( µ ). There are two points of view concerning the Poisson’s 
ratio of the early age concrete. Some researchers (Oluokun et al. 1991, Truman et al. 1991) re-
gard that one remains constant during expansion stage whereas others (Mikhailov and Gershvald 
1984, Byfors 1980, De Schutter and Taerwe 1996) have proved experimentally the evolution of 
the Poisson’s ratio during the hydration. In the present study the Poisson’s ratio of the early age 
concrete is taken 0, 47µ =  according to Mikhailov and Gershvald (1984).
Thus, for the two-way restraint conditions the incremental restrained expansion strains at the i-th 
time interval in the principal axes directions are calculated by:

1 1
,

, , , ,
1 1,28 ( 1) 1/ 2

1 1
,

, , , ,
1 1,28

( , )
( ) ( ) ( ) ( )

1 ( )

( , )
( ) ( ) ( ) ( )

1

i i
i j x s

x s i cf i x c j x s j x l
j jc c i

i i
i j y s

y s i cf i y c j y s j y l
j jc

t t ED
D E E t

t t ED
D E

− −

= = − +

− −

= =

  ∆ϕ
∆ε = ∆ε − ∆σ ⋅ − ∆ε ⋅ρ ⋅   +    

  ∆ϕ
∆ε = ∆ε − ∆σ ⋅ − ∆ε ⋅ρ ⋅   +    

∑ ∑

∑ ∑
( 1) 1/ 2( )c iE t − +









(22)

The incremental self-stresses at the i-th time interval in the principal axes directions are obtained 
as follows:

( ) ( ) ( )( )
( ) ( ) ( )( )

, , , , , ,

, , , , , ,

x x s x s x l y s y s y li i i

y y s y s y l x s x s x lii i

E E

E E

 ∆σ = ∆ε ⋅ ⋅ρ −µ⋅ ∆ε ⋅ ⋅ρ

∆σ = ∆ε ⋅ ⋅ρ −µ⋅ ∆ε ⋅ ⋅ρ

(23)



81
Journal of Sustainable Architecture and Civil Engineering 2017/3/20

The total self-stresses and restrained expansion strains in the principal axes directions at given 
time interval are defined by:

( ) ( ) ( )
( ) ( ) ( )

( ) ( ) ( )
( ) ( ) ( )

, , ,1 1

, , ,1 1

;
x x x x s x s x si i i i i i

y y y y s y s y si i i i i i

− −

− −

 σ = σ + ∆σ ε = ε + ∆ε 
 
σ = σ + ∆σ ε = ε + ∆ε  

(24)

The iterative calculation continues until the end of the expansive concrete stabilization period.

Experimental investigations
For the verification of the proposed model for the restrained expansion strains estimation at the 
expansion stage of the concrete under two-way restraint conditions the experimental investiga-
tions were conducted.

The present investigations were carried out based on the three series of the expansive concrete 
plane specimens with the mesh reinforcement in the mid-depth of the cross section (Fig. 2). The 
restraint rate in the principal axes directions was taken as the variable parameter of samples.

series a b

I 3Ø12mm 3Ø5mm 

II 3Ø12mm 3Ø12mm 

III 3Ø5mm 3Ø5mm 
1 – expansive concrete, 
2 – mesh reinforcement

Fig. 2 
Details of the 
experimental 
specimens

Geometry and reinforcement characteristics of the expansive concrete specimens are listed in the 
Table 1. The view of the mold with mesh reinforcement and the specimen after casting is present-
ed in the Fig. 3.

Table 1 
Geometry and 
reinforcement 
characteristics of 
specimens

series
specimen 
marking

specimen 
geometry, 
mm, bxlxh

x-direction y-direction

Asx, mm
2

reinforcement 
ratio ρlx, %

Asy, mm
2

reinforcement 
ratio ρly, %

I
P1-1
P1-2

500x500x70

339 0,97 58,9 0,16

II
P2-1
P2-2

339 0,97 339 0,97

III P3-1 58,9 0,16 58,9 0,16

Asx– reinforcement area in the x-direction; Asy– reinforcement area in the y-direction.



Journal of Sustainable Architecture and Civil Engineering 2017/3/20
82

The expansive cement was prepared in laboratory environment and consisted of the following 
components by weight: Portland cement CEM I 42,5R – 74%; high-alumina cement (HAC) – 14%; 
gypsum – 12%.

The expansive concrete mix composition per 1 m3 was as follows: expansive cement – 500kg; fine 
aggregate – 750kg; coarse aggregate – 1150kg; water – 215kg. The water-to-cement ratio (w/c) 
was 0,43.

For every series of specimens the individual expansive concrete mix was prepared. The self-stress-
ing grade of the expansive concrete was controlled in accordance with STB 2101 (2011). The free 
strains development of every series of specimens were measured on the prismatic samples bx-
h=50x50mm, l=200mm. The characteristics of the expansive concrete are listed in the Table 2. 

After demolding at an expansive concrete age of one day, the specimens were cured up to 28 days 
under moisture conditions.  

Fig. 3 
The view of specimen 

before (a) and after (b) 
casting.

a b

Table 2
Characteristics of the 

expansive concrete

series
free strain at the concrete 

expansion stabilization cfε , %
young’s modulus at 28 days 

age ,28cmE , GPa
self-stressing grade at the concrete 

expansion stabilization .ce df , MPa

I 0,117 42,66 1,4

II 0,437 23,1 2,4

III 0,334 33,3 2,1

The restrained expansion strains and corresponding self-stresses experimental data of plane 
specimens in the principal axes directions from one day to the expansive concrete stabilization 
along with those calculated according to the proposed model and the model described in the work 
Ito et al. (2004) are presented in the Fig. 4-6.

As could be seen from the Fig. 4-6, providing the low restraint level equals ρl=0,16%   the re-
strained expansion strains values calculated both according to the proposed model and design 
model Ito et al. (2004) have the similar meaning during the whole period of the expansive concrete 
stabilization (series I, direction y; series III, directions x and y). Simultaneously, providing the high 
restraint level equals ρl=0,97% corresponding data have the similar meaning only within four-
days initial period of stabilization. But at the end of the expansion concrete stabilization period the 
divergence of calculated restrained strains values extends to 40% (series II) and 26% (series I, di-

Verification 
of the 

proposed 
model 
based 
on the 

experimental 
results



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Journal of Sustainable Architecture and Civil Engineering 2017/3/20

Fig. 4
The restrained expansion 
strains and self-stresses 
in the principal axes 
directions for the series I

Fig. 5
The restrained expansion 
strains and self-stresses 
in the principal axes 
directions for the series II

rection x). At the same time in the case of nonsymmetrical restraint conditions in the direction with 
the low restraint level equals ρl=0,16% the self-stresses calculated both according to the proposed 
model and design model Ito et al. (2004) have the divergence up to 40% (series I, direction y). This 
result is explained by the influence of the restraint in the perpendicular direction.

Comparing both experimental and calculated data of the restrained expansion strains and cor-
responding self-stresses it should be pointed out that values defined according to the proposed 
model have more similar meaning to corresponding experimental data over the whole expansive 
concrete stabilization period.

As with the elements under uniaxial restraint conditions (Semianiuk et al. 2017) in the case of two-
way restraint conditions failing the addition cumulative force the wide disagreement has appeared 



Journal of Sustainable Architecture and Civil Engineering 2017/3/20
84

Fig. 6 
The restrained 

expansion strains 
and self-stresses in 

the principal axes 
directions for the 

series III

between the experimental and calculated according to the model Ito et al. (2004) restrained expan-
sion strains and corresponding self-stresses  values. This divergence has especially observed for 
series II providing the high restraint level (ρl=0,97%) and high self-stressing grade of the expansive 
concrete ( .ce df =2,4 MPa).
The comparison of the final restrained expansion strains values at the end of the expansion con-
crete stabilization period defined according to the available design models and corresponding ex-
perimental data for every series of plane specimens is shown in the Table 3.

Table 3 
The restrained expansion 

strains values of the plane 
specimens at the end of 
the expansive concrete 

stabilization

calculated restrained expansion strains calc
sε

design model

series I
ρlx=0,97%, ρly=0,16%

series II
ρlx=0,97%, ρly=0,97%

series III
ρlx=0,16%, ρly=0,16%

,
calc
s xε , ‰ ,

calc
s yε , ‰ ,

calc
s xε , ‰ ,

calc
s yε , ‰ ,

calc
s xε ,‰

,
calc
s yε ,‰

proposed model 0,794 1,054 2,166 2,166 3,050 3,050

model 1 (TCP Codes 2010) 0,860 3,070 1,470 1,470 4,610 4,610

model 2  (Marchuk 2002) 0,792 2,850 1,240 1,240 3,793 3,793

model 5  (Ito et al. 2004) 1,071 1,160 3,662 3,662 3,284 3,284

experimental data  expsε

exp
,s xε , ‰

exp
,s yε , ‰ exp,s xε , ‰

exp
,s yε , ‰

exp
,s xε , ‰

exp
,s yε , ‰

P1-1 P1-2 P1-1 P1-2 P2-1 P2-2 P2-1 P2-2 P3-1 P3-1

0,872 0,978 1,161 1,146 2,058 2,589 2,327 2,381 3,132 2,731

design model exp calcs sε ε

proposed model 1,098 1,230 1,101 1,087 0,950 1,195 1,074 1,099 1,027 0,895

model 1  (TCP Codes 2010) 1,014 1,130 0,378 0,373 1,400 1,761 1,583 1,619 0,679 0,592

model 2  (Marchuk 2002) 1,101 1,230 0,407 0,402 1,660 2,088 1,876 1,920 0,825 0,720

model 5   (Ito et al. 2004) 0,814 0,913 1,000 0,987 0,562 0,707 0,635 0,650 0,953 0,831



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Journal of Sustainable Architecture and Civil Engineering 2017/3/20

As follows from the Table 3 the restrained expansion strains values calculated according to the 
models 1 (TCP Codes 2010) and 2 (Marchuk 2002) differ considerably from the corresponding ex-
perimental data both towards overestimation (series I, III) and underestimation (series II). Where-
as the data of proposed model represent a good agreement with the corresponding experimental 
values (from 2,6% to 18,6 %).

Conclusions1 A design model for the restrained expansion strains defining under two-way restraint con-ditions is presented. This model is based on the modified early age strain development 
model (MSDM) for the elements under uniaxial restraint conditions (Tur and Semianiuk 2016, 
Semianiuk et al. 2017).

2 Having a good agreement with the experimental data, the proposed design model is ad-equate enough for the restraint expansion strains estimation under two-way restraint 
conditions from the early age period to the expansion concrete stabilization. The experimental 
and calculated in accordance with the proposed model restrained expansion strains and cor-
responding self-stresses values have the similar meaning over the whole expansion concrete 
stabilization period (maximum divergence of values equals 22%). At the same time the compar-
ison of the experimental restrained expansion strains values and those calculated according to 
the available design models (model 1 TCP Codes (2010), model 2 Marchuk (2002), model 5 Ito et 
al. (2004), proposed model) demonstrates a good agreement of the proposed model data with 
the corresponding experimental data (maximum divergence of values equals 18,6%).

3 The main advantage of proposed design model is the consideration of the actual curing conditions of the expansive concrete specimens such as temperature, Young’s modulus of 
the expansive concrete at given time interval and real free expansion strains development law 
of the expansive concrete from the early age to the end of stabilization period.

4 According to the proposed model the restrained expansion strains in the case of different reinforcement ratio in the principal axes directions might be estimated.

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VOLHA SANNIKAVA

MSc, engineer

Brest State Technical University, Faculty of Civil 
Engineering, Department “Geotechnics and 
transport communications”

Main research area

Self-stressing concrete structures

Address

267, Moskovskaya str., 224017, Brest, Belarus
Tel. +375 29 7255952
E-mail: volhasannikava@list.ru

VICTAR TUR

Professor, DSc, PhD

Brest State Technical University, Faculty of Civil 
Engineering, Department “Technology of concrete 
and building materials”

Main research area

Self-stressing concrete structures, reliability of 
building structures

Address

267, Moskovskaya str., 224017, Brest, Belarus
Tel. +375 33 6729404
E-mail: profturvic@gmail.com

About the 
authors