Identification of Bending Moment in Semi-Fixed Beam

Abstract

Paper presents procedure of identification of supporting moment in the statically indeterminable, partly fixed reinforced concrete beam. Examined beam is a part of reinforced concrete frame, where rate of fixing is subjected variation under influence of continually increasing loading. Bending moment at the fixing end remains the unknown internal force in the beam. This internal force is determined on the basis of identification of theoretical and experimentally measured deflections in the selected sections of element. Theoretical deflections are computed numerically integrated curvature equation of the beam axis under assumption that effects of physical nonlinearity of reinforced concrete reveal in the process of loading. These effects describes quantitatively the Muraszew's theory of stiffness changes. Boundary problem of partly fixed beam is solved applying iterative procedure characteristic the initial problem applying integration schema ahead with increased accuracy.

Full text

Procedia Engineering 57 ( 2013 ) 746 – 753
1877-7058 © 2013 The Authors. Published by Elsevier Ltd.
Selection and peer-review under responsibility of the Vilnius Gediminas Technical University
doi: 10.1016/j.proeng.2013.04.094
11th International Conference on Modern Building Materials, Structures and Techniques,
MBMST 2013
Identification of Bending Moment in Semi-Fixed Beam
Jaroslaw Malesza
*
Department of Structural Mechanics, Faculty of Civil and Environmental Engineering, Bialystok University of Technology, Wiejska 45 E, 15-351
Bialystok, Poland
Abstract
Paper presents procedure of identification of supporting moment in the statically indeterminable, partly fixed reinforced concrete beam.
Examined beam is a part of reinforced concrete frame, where rate of fixing is subjected variation under influence of continually increasing
loading. Bending moment at the fixing end remains the unknown internal force in the beam. This internal force is determined on the basis
of identification of theoretical and experimentally measured deflections in the selected sections of element. Theoretical deflections are
computed numerically integrated curvature equation of the beam axis under assumption that effects of physical nonlinearity of reinforced
concrete reveal in the process of loading. These effects describes quantitatively the Muraszew’s theory of stiffness changes. Boundary
problem of partly fixed beam is solved applying iterative procedure characteristic the initial problem applying integration schema ahead
with increased accuracy.
© 2013 The Authors. Published by Elsevier Ltd.
Selection and peer-review under responsibility of the Vilnius Gediminas Technical University.
Keywords: bending stiffness of RC section; generalized bending law of RC beam; parametric identification.
1. Introduction
Assignation of most unfavorable loadings combination and compliance of the load bearing capacity and serviceability
limit states remain the basic requirements and expectations in process of designing the reinforced concrete structures.
Application of the elastic analysis methods as a rule are used in determination of the internal forces in the structure.
Statically undetermined systems require determination of cross-sectional stiffness distribution. Initial sectional stiffness is
determined by their geometry, materials modulus of deformations of concrete and reinforcing steel, and displacement and
the cross-sectional area of the main reinforcement. During the stressing increase of reinforced concrete structure the initial
stiffness distribution undergoes changing, which are influencing the internal forces including supernumerary values.
Physical nonlinearity of reinforced concrete is expressed by relation of the cross-sections stiffness to their stressing. Non-
elastic properties of tensile as well as compressed concrete with its limited deformational abilities remain the factors of that
nonlinearity. Cracks formation generates the change of cross-section geometry. Spatial 3D blocks are forming between the
cracks where strain and stress state does not correspond the beam reaction. Hence cracking contributes non-uniform
deformation of reinforced concrete elements. Changing of cracking the cross-section geometry causes degradation of its
stiffness in the process of loading. Degradation intensity along the element length is varying. Reinforced concrete element
converts in result into element with continually developing distribution of spatial stiffness. Negligence in static analysis of
reinforced concrete physical non-linearity shall be considered as an assumption of strongly simplifying problem. Internal
forces determined applying the elastic analysis methods are directly proportional to the external loading. Evaluation of this
loading on the design level does not allow to determine the internal forced distribution in the stages preceding exhaust of the
load bearing capacity of structure. The bending moments determined in the critical sections in the span and in the adjoining
nodal sections retain constant proportions independently to the intensity of stressing of the structure. This is influencing the
correctness of designing decisions concerning distribution of reinforcement in sections and along the critical parts of
* Corresponding author.
E-mail address: j.malesza@pb.edu.pl
Available online at www.sciencedirect.com
© 2013 The Authors. Published by Elsevier Ltd.
Selection and peer-review under responsibility of the Vilnius Gediminas Technical University

747
Jaroslaw Malesza / Procedia Engineering 57 ( 2013 ) 746 – 753
elements. Effects of nonlinear behavior of the reinforced concrete beams can be considered throughout the static analysis
where evolutional stiffness changes developed by loading are taken into account. An acceptable way in formulation of these
changes is the assumption that determination of deflection for reinforced concrete element is based on curvature of its
deflected axis,
2
2
() ()
()
()
dyx Mx
kx
Bx
dx
=− =
(1)
Initial heterogeneity of the stiffness distribution in the beam as well as change of the cross-sectional stiffness due to
bending moment is considered in the formula (1), hence B(x) ≡ B(x,M).
Many proposals being considered as a trial of the theoretical seizure of the stiffness changes phenomena in concrete
structures are found in literature. Most of these methods assume that relation between the curvature k and bending moment
M can be approximated applying selected function M(k). The function may have the curvilinear character [1]. The other
function has the sectional-linear dependence as it is presented in [2]. Such dependence reflects strong change of stiffness in
result of cracking without description of progressed defect of this stiffness. Proposal of the other non-linear law M-k
considered as the basis determining deflections are presented in [3].
An analytical model for fly ash ordinary grade concrete was developed for obtaining the complete moment curvature
diagram in [4]. The ultimate moments obtained from the proposed analytical procedure are found to be in good agreement
with the experimental values. Presented in [5] approach considers RC/LRC as a homogenous material whose stress-strain
characteristics are derived based on the moment-curvature relationship of the structural component. Numerical studies on
Laced Reinforced Concrete beams where structural element consists of equal reinforcement in tension and compression
faces along with lacings, are carried out and the results are compared with those of the experimental values to validate the
proposed approach. The application of the proposed approach for analysis of uniformly loaded LRC slabs is demonstrated.
The proposed approach can be extended to ordinary RC by modifying the ultimate stress and strain values.
Moment-curvature plots and moment-axial force interaction diagrams are essential tools for understanding the load-
deformation behavior of structural elements. To make the use of these tools simpler, numerical program was created [6] that
can generate moment-curvature plots and moment-axial force interaction diagrams for reinforced concrete sections. These
plots can include the effects of confinement provided by transverse reinforcement on the stress-strain relationship of
concrete.
Paper [7] presents several modifications to existing analytical methods for predicting the behavior of reinforced concrete
beams under severe concentrated loads. The modifications include failure criteria for the cross section at the collapse
condition, formation and development of plastic hinges, and influence of shear on flexural performance of the beam. This
approach allows to study the response of reinforced concrete beams under the combined influence of flexure, shear, and
axial thrust from impose of loading to collapse. The procedure was used to derive moment-curvature and load-deflection
relationships and obtained results compared well with experimental data.
A trilinear moment-curvature curve is constructed in [8] by describing as linear segments each of uncracked, cracked,
and yielded behavior of reinforced concrete beams. Moment-curvature and load-deflection curves derived by Branson's
formula and a recent nonlinear model by Bazant and Oh have been compared with those produced by the trilinear model for
several example beams. The main advantage of the trilinear model is in computer applications related to analysis and design
of reinforced concrete beams and frames.
The stress-strain curves for concrete in compression obtained in [9] from the flexural tests are remarkably similar to
those generated from uni-axially loaded specimens. The analysis based on the usual flexural theory, but using the stress-
strain curves for uni-axially loaded specimens, gives close predictions of the experiment data on moment-curvature
relationship and ultimate moment capacity of the beams.
Nonlinear characteristics of constitutive materials were mathematically modeled according to Eurocode 2, to estimate the
moment-curvature response [10]. The closed form solutions were verified with those from numerical procedures.
Calculations of M-c relationship were based on their nonlinear characteristic for different ratios of reinforcement in tension
and compression. Failure criteria in stress space is characterized by parabolic stress-strain relations. Elastic limit stress and
strain at cracking were limited to 0.2% and 0.35% (as prescribed by the Code). Tensile stress in concrete were ignored.
Basing on model of laminated material, in which the mechanical properties of separated layers may have discreet
different values depending on the volume of external load. A laminated medium model witch different geometrical and
mechanical properties of layers are used for determination of deflection. Values of normal stresses in axial direction of
beam are constant in each i-layer [11-12].
The theory of W.I. Murashev adopted for practical application in Polish Standards, former [13] and later [14] can be
considered as an example of conception of stiffness changes considering section (weakening) attenuation in result of
cracking. The curvature of the element section between cracks according to this theory is computed on the base of the

748 Jaroslaw Malesza / Procedia Engineering 57 ( 2013 ) 746 – 753
average strains of tensile steel and outlining fiber of compressed concrete. These strains depend on adequate values
appearing in the cracked section. The cross-sections stiffness along the parts of beam under uniform sign of bending
moment fixed as a constant in intervals, determined for the extremely stressed section.
Relations between stiffness and curvature of deflected elements decidedly influence in determination of elements
deflections. Deflections of reinforced concrete beams were proposed to be determined for stiffness of the most stressed
critical section of the simply supported beam. The possibilities of application of such an approach in the statically
undetermined continuous beams were analyzed in the work [2].
The aim of the work is determination of the supporting bending moment evolution in result of continual stiffness defect
in the statically undetermined reinforced concrete beam. The beam remains a part of frame semi-fixed on the supports and it
is subjected of increasing action of loading. Procedure of semi-fixing supports has non-elastic character. It remains a semi-
stiff fixing with the indeterminate stiffness modulus and depends on the level of loading. Fixing joint exhibits effects of
cracks which are influencing on distribution of bending moments in the beam under continually changing loading. The
physical law of bending sections according to W. I. Murashov being a base to determine deflections according to [13] are
taken into consideration and it is compared with the solutions actually proposed in the Standard [14]. The problem is solved
assisting the experimental tests results where are obtained deformations of beam in the form of angle of rotation in the partly
fixed section or displacements of selected points located along the span of beam.
2. Experimental data
The initial beam is divided into sectional parts as in figure 1 with similar non-differing more than for 5% stiffness. This
division is resulted from arrangement of reinforcing bars in the beam.
Fig.1. Static scheme of the frame beam, loadings and initial stiffness distribution
The bending moments identification has been conducted basing on results of beam displacements and on the base of the
angle of rotation of frame joint obtained from experimental data of P. Bodzak and A. Czkwianianc presented in the work
[15]. Monolithic reinforced concrete frame in the form of capital letter H has been investigated. The beam of the frame was
loaded with two concentrated forces. Loading was rising from zero value to 180 kN with constant step of 10 kN. The beam
of the frame with the cross-section 20×30 cm and span of 2.76 m was reinforced in the way adequate to the distribution of
the different signs bending moments: negative and positive. Three segments along the length of the beam with varying
stiffness located symmetrically from the centre of beam were distinguished. The stiffness of the particular segments were
marked B
01
, B
02
, B
03
. Supporting parts 0.30 m are doubly reinforced top and bottom symmetrically with
A
s1
=
A
s2
= 6.16 cm
2
. Indirect segments length of 0.58 m are bottom reinforced A
s1
= 8.17 cm
2
and top reinforcement
A
s2
= 3.13 cm
2
. These segments envelopes parts of beam where bending moments are changing signs. Span segment length
of 1.00 m reinforced with bottom A
s1
= 8.17 cm
2
and top A
s2
= 2.26 cm
2
envelope middle part of beam. Stirrups of diameter
ø8mm makes the transverse reinforcement. The statical scheme of the beam is presented in figure 1. Beam of the frame was
concrete made with the compressive strength marked in the cube test f
c,cube
= 41.8 MPa. The other parameters of concrete
were obtained from the test in values: design compressive strength f
c
= 27.5 MPa, tensile strength f
ct
= 3.14 MPa, and
average longitudinal modulus of elasticity E
cm
= 23 300 MPa.
In the experiment displacements and strains of the selected points of structure were recorded. Displacements of the beam
were measured twice, directly after load rising and before the sequent step of loading. Displacements measured in the
sections: 0, 1, 2, 3 indicated on Figure 1 were considered in the analysis due to symmetrical arrangement of reinforcement.
Diagrams of displacements in figure 2 displays characteristic singularities of deformational reaction for reinforced
concrete. Three-segmental run of changes evidently appears in the line illustrating deflection of the middle point of beam
under rising loading. The stage of elastic strains of the phase before cracking occurred until loading reaches value of 35 kN.
The process of identification was not influenced by the value of cracking moment obtained in the experiment. The value of
this moment were determined on the base of Standard recommendations [13].
Analysis were conducted in the range of exploitation loadings to the value of 0,8·P
max
= 145 kN. Deflections diagrams
within this stage are curvilinear in general without abrupt increment of displacements.
Effects of reinforcement yielding appears in the phase of the load bearing capacity exhaust under loading (P > 145 kN).
They do not lead to the element failure in result of plastic strengthening of reinforcement while concrete does not undergo

749
Jaroslaw Malesza / Procedia Engineering 57 ( 2013 ) 746 – 753
crushing. Rise of loading were noticed to the value of loading 180 kN. Tests were ended when displacements in the middle
point of span exceeded 50 mm and the last reading comparing o the preceding characterized 15% of deflection increase.
Description of exhaust of the load bearing capacity does not remain the aim of this work.
Fig.2. Displacements of the beam axis measured at the experimental tests
3. Numerical algorithm of determination of deflection
The theoretical deflections in the mid-span are searched in analyzed case of both sides semi-fixed beam loaded with two
concentrated forces like in figure 1. Determination of deflections resolves itself into twice integration of equation (1) under
assumption of adequate boundary conditions. In case of classic fixing the boundary conditions are to be assumed where
y(0) = y(l) = 0 and y’(0) = y’(l) = 0. Results of experimental tests displayed that in the frame joints occur deformations
being related to rotation and coming from displacements. Fixing of the beam end becomes partial, maintaining the internal
forces of both sides fixed beam. The boundary conditions are therefore modified taking y(0) = f
3
and y’(0) = φ
eksp
, y(l) = f
0
-
f
3
where f
0
and f
3
are the measured beam deflections in the sections 0 and 3 from figure 1, while ϕ
eksp
is the angle of rotation
of fixing section obtained in the experiment and f
0
is the measured displacement in the mid-span of beam 0.The values of
bending moments in the diagram remains unknown due to the statically indeterminate system.
Fig.3. Bending moment diagram alongthe length of the beam.
The form of the bending moment diagram is bi-linear like in figure 3 and the bending moments equations (2) are
determined with accuracy to the value to unknown supporting fixing moment M
U
.
11
22
()
(0,88)
U
PU
MM Px
Mx
MM M Px
=−⋅
⎧
=
⎨
==−⋅−
⎩
(2)
The unknown value of MU are determined utilizing way of numerical integration of equation (1) in the iteration
algorithm. The axis of the beam is divided into nodes with steps Δx. The integration step Δx was assigned as a fraction of
the beam length, as an example Δx = L/N where N = 1000. Searched line of deflected beam is described applying Taylor’s
series. Integral formula is obtained reducing to four initial terms of expansion (3)
( ) () () ()
()
()
()
23
' '' '''
2! 3!
xx
yxxyxyxxyx yx
ΔΔ
+Δ = + ⋅Δ + ⋅ + ⋅
. (3)
The second and the third derivatives of function of deflection is determined on the base of internal forces. Function
y"(x) = M(x)/B
i
is defined, where M(x) represents the bending moment in the current section and B
i
corresponding stiffness

750 Jaroslaw Malesza / Procedia Engineering 57 ( 2013 ) 746 – 753
on the segment 1, 2 or 3. Influence of the shear forces with known distribution along the length of the beam is considered,
taking y"'(x) = T(x)/B
i
.
Integration according to formula (3) is conducted with simultaneous adoption of value M
U
adequate for given level of
loading. Compatibility criterion of theoretical deflection value y(l/2) and measured value f
0
with assumed accuracy decides
on correctness of M
U
selection. In the integrating procedure the cross-section stiffness in the nodes of beam were fixed as a
dependent from bending moment M
i
= M(x
i
). W.I. Murashov’s formula described in [13] was used for this purpose in the
form:
00
0
,
()
0,9
cm cr
f
II cr
a
aa bbc
BEJ whereMM
zh
Bx
BwhereMM
EF EF
=⋅ ≤
⎧
⎪
⋅
⎪
=
=>
⎨
ψ
⎪
+
⎪
⋅ν⋅⋅
⎩
(4)
where:
()
2
12
0,292 1,5 0,1
cr S S ctk
n
M
AAbhf
bh
⎡⎤
=+⋅⋅+⋅⋅⋅⋅
⎢⎥
⋅
⎣⎦
(5)
and formula given in the actual Standard for RC structures [14], assuming the stiffness under short - term acting loading
2
12
0
11
cm II
II
sr
I
I
s
EJ
B
J
J
⋅
=
⎛⎞⎛ ⎞
σ
−β ⋅β ⋅ ⋅ −
⎜⎟⎜ ⎟
σ
⎝⎠⎝ ⎠
. (6)
In the Eq. (6) the fraction of σ
sr
/ σ
s
is substituted by the ratio of the cracking bending moment and searched value of
bending moment in the most stressed section adjacent to frame joint M
cr
/ M
U
and in the span cross-section M
cr
/ M
P
.
Coefficients β
1
and β
2
were taken as equal 1 considering the ribbed bars of the main tensile reinforcement and immediate
loading acting on structure.
4. Results of numerical analysis
Dependence loading P – deflection f in the mid-span; experimental f
0
or theoretically (l/2) are quite in compliance as it is
presented in figure 4. This testifies on correctness of theoretic W. I. Murashov’s theory describing stiffness of reinforced
concrete cross-section under bending. The bending moment M
U
in the support adjacent cross-section remains the main
parameter used in governing due to obtain deflections conformability. This value determines the span bending moment M
P
.
The diagram of variation of both bending moments is presented in Fig. 5.
Fig. 4. Theoretical and experimental deflections of the frame beam

751
Jaroslaw Malesza / Procedia Engineering 57 ( 2013 ) 746 – 753
Proportional and linear increase of bending moments is observed till loading reaches value of 42 kN. In the experimental
investigations this loading generates the first cracks along the mid-segment of the beam. The value of cracking loading
corresponds to cracking bending moment close to 23 kNm being determined on the basis of Eq. (5) of the Standard [13].
The span bending moment reaches value of the cracking moment earlier than the bending moment in the section adjacent to
supports, which the cross-section is strongly reinforced. Cracking appearing generates loss of stiffness and disturbance of
proportional relations of analyzed bending moments.
Fig. 5. Diagram of variation of the span bending moment M and supporting bending moment M
U
, under assumption of the cross-section stiffness changes
according to PN-B-03264:1984 and 2002
Selected characteristic points 1, 2, … 10 presented in figure 5 were marked due to analyze changes of stiffness on the
path of bending moments variations in the process of loading. The ratios of the supporting bending moment to the span
bending illustrates Figure 6. They are were determined in those points. This ratio remains the stiffness measure of semi-
fixing beam in the frame columns. Its characteristic values are set in the open polygon where the evolution of semi-fixed
support of beam can be isolated. The ratio is constant in the range up to the loading forceending linear-elastic stressing of
span cross-section (P = 32 kN = 0,75 P
cr
).This indicates that distribution of the global moment for beam M
g
= P · a on the
supporting moment and the span bending moment is realized proportionally. Effects of cracking in the span and adjacent to
the supports appear in the range of loading from P = 32 kN to P = 83 kN. Inter relation of bending moments in those
sections is non-stabilized. Averaging prognosis of this variation correctly corresponding with the non-cracked phase of
beam is obtained when line is drawn between value of ratio in point 2 and adequate value in point 6. The effectiveness of
supporting section in transmission of bending moments is increasing continually in the range of loading from 83 kN to
146 kN. The terminal value of loading P = 146 kN corresponds to value of bending moment M
U
, where in the supporting
section the internal lever arm is equal z = d – a
2
. The force in the top tensile reinforcement is equal to plastifying force on
the level of characteristic loading Z
k
= A
s1
· f
yk
= 250 kN. The above allows for conclusion that, subsequent process of
loading leads to decrease of ratio M
U
/ M
P
, hence the span cross-section becomes more effective in transmission of the
global bending moment. This prognosis confirm segment 8-10 of broken line in Fig. 6.
The further subsequent process of loading causes the phase of exhaust of the load bearing capacity and generally are
appearing phenomena assisting yielding of tensile reinforcement.
Varying stiffness of partly-fixed frame beam in the columns can be determined in the formula K = M
U
/ φ
eksp
, what is
presented in Fig. 6. General conformity of variation character of curve K and ratio M
U
/ M
P
is observed.
Results of numerical analysis in the aspect of changes of stiffness decrements in the process of loading are presented in
Fig. 7. As it comes from the spatial distribution of these losses they are concentrated mainly along the segments in the span
and at joint with columns. In the experiment cracking of both segments were very intensive. They were divided by non-
cracked intermediate segment.

752 Jaroslaw Malesza / Procedia Engineering 57 ( 2013 ) 746 – 753
Fig. 6. Ratio of fixing moment value to the span bending moment and stiffness of partial fixing K
Fig.7a. Evolution of stiffness distribution along the length of the frame beam in different stages of loading, according to PN-B-03264:1984
Fig.7b. Evolution of stiffness distribution along the length of the frame beam in different stages of loading, according to PN-B-03264:2002.

753
Jaroslaw Malesza / Procedia Engineering 57 ( 2013 ) 746 – 753
5. Final conclusions
Experimental investigation exhibits that beam connection in the frame joint is rigid with stiffness undergoing changing as
a result of rotation of frame joint as well as its deformation. Composite state of stress is generated in joint. Results of
analysis indicate that assumption of partly – fixing support is correct to determination of bending moment distribution on
the basis of identification procedure. Assumed conformity criterion of theoretical and experimental deflections became the
basis in determination of unknown internal force; the bending moment in the fixing joint in the analyzed statically
indeterminate scheme. That bending moment decides on distribution of the banding moments along the length of all
element. The segment of linear decrements of bending moments testifying the proportional response of structure on
introduced loading makes 20% of all range of loading. All process of loading can be divided into phases of differing
increased of both internal forces. Results of analysis indicates that process of redistribution of internal forces undergoes
between the span section and the column adjacent sections of the frame beam.
Evolution of spatial stiffness variations along the length of element confirm strong cracking at the adjacent to joint cross
sections as well as along the middle-segment of the frame beam.
Achieved good conformity of experimentally obtained and theoretical deflections results confirms of W. I. Murashov’s
bending theory of reinforced concrete as well as it confirms the correctness of proposed identification to determination of
internal forces approach in the static indeterminate beam system.
Acknowledgements
The research has been financed by projects No. S/WBiIS/3/08 to be realized in period of 2008-2012.
References
[1] Kuczynski, W., 1971. Concrete structures. Continual theory of bending. PWN. Warsaw.
[2] Knauf, M., 1979. On computation of deflections of reinforced concrete beams applying bi-line method. Editions of Warsaw University of Technology,
Science Fascicle. Building. Z. 65. Warsaw.
[3] Malesza, J. 2008. Nonlinear M-k law as the base in determination of reinforced concrete beams deflection, Bialystok University of Technology, Science
Fascicle. Building. Z. 32, pp. 103-114.
[4] Dakshina Murthy, Sudheer Reddy. 2010. Moment-Curvature Characteristics of Ordinary Grade Fly Ash Concrete Beams, International Journal of Civil
and Structural Engineering 1(3), pp. 497-508.
[5] Anandavalli, A., Lakshmann, N., Knight, G.M.S., 2012. Simplified Approach for Finite Element Analysis of Laced Reinforced Concrete Beams, ACI
Structural Journal 109(January), pp. 91-100.
[6] Alaoui, S., Klingner, R.E., 2007. Reinforced Concrete Section under Moment and Axial Load, Concrete International 29( October), pp. 63-67.
[7] Krauthammer, T., Rauthammer-Shashiriar, S., Shanaa, H.M., 1987. Analysis of Reinforced Concrete Beams Subjected to Severe Concentrated Load,
ACI Structural Journal 84(November), pp. 473-480.
[8] Alwis, W.A.M., 1990. Trilinear Moment-Curvature Relationship for Reinforced Concrete Beams, ACI Structural Journal 87(May), pp. 276-283.
[9] Mansur, M.A., Chin, M.S., Wee, T.H., 1997. Flexural Behavior of High Strength Concrete Beams, ACI Structural Journal 94(November), pp. 663-674.
[10] Chandrasekaran, S., Nunziante, L., Serino, G., Carannante, F. 2011. Curvature Ductility of RC Section Based on Eurocode: Analytical Procedure,
KSCE Journal of Civil Engineering 15(1), pp. 131-144.
[11] Bulavs, F., Radinsh, I., Tirans, N., 2003. “Method of Prediction of the Deflection of Reinforced Concrete Beams Considering Cracking”, Scientific
Proceedings of Riga Technical University. Architecture and construction science. RTU. Riga.
[12] Bulavs, F., Radinsh, I., Tiraus, N., 2004. “Forecasting of Deflection of Reinforced Concrete Beams Strengthened with Carbon Plastic Sheets”, The 8th
International Conference „Modern Building Materials, Structures and Techniques”. May 19-24. Vilnius, Lithuania.
[13] PN-B-03264:1984. Polish Standard. Concrete, reinforced concrete and prestressed structures. Static analysis and design.
[14] PN-B-03264:2002. Polish Standard. Concrete, reinforced concrete and prestressed structures. Static analysis and design.
[15] Bodzak, P., Czkwianianc, A., 2001. Experimental investigations of reinforced concrete monolithic frames – zone of edge node, Experimental tests of
concrete elements and structures. Science Fascicle No. 11. Lodz.