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FACULTY OF CIVIL ENGINEERING
Eng. Attila PUSKÁS
DEFORMATIONS OF WIDE BEAMS
PhD THESIS
- ABSTRACT -
Scientific Coordinator:
Prof.Univ.PhD.Eng. Zoltán KISS
Prof.Univ.DHC.Eng. Mircea MIHAILESCU
CHAIRMAN: - Prof.Univ.PhD.Eng. Mihai Iliescu - Dean, Faculty of Civil Engineering,
Technical University of Cluj-Napoca
MEMBERS: - Prof.Univ.PhD.Eng. Zoltan KISS - Scientific Coordinator, Technical
University of Cluj-Napoca
- Prof.Univ.PhD.Eng. Radu PASCU – Reviewer, Techincal University of
Civil Engineering Bucharest
- Prof.Univ.PhD.Eng. Valeriu STOIAN – Reviewer, “Politehnica” University
of Timișoara
- Prof.Univ.PhD.Eng. Nicolae TĂRANU - Reviewer, “Gheorghe Asachi”
Technical University of Iaşi
- Prof.Univ.PhD.Eng. Adrian M. IOANI – Reviewer, Technical University of
Cluj-Napoca
- 2012 -
Deformations of wide beams
Page 1
Contents
Abstract ....................................................................................................................................... 2
Chapter 1 – Introduction............................................................................................................... 2
Chapter 2 – Considerations regarding calculus of wide beams ....................................................... 2
Chapter 3 – Numerical analysis of the wide beam ......................................................................... 3
Chapter 4 – Experimental program ............................................................................................... 5
Chapter 5 – Interpretation of results. Conclusions. ........................................................................ 7
Selective References ................................................................................................................... 10
Appendixes ................................................................................................................................ 12
I. Full papers of the author from the domain of the thesis ......................................................12
II. Design of the wide beam using FETT-Abacus software .........................................................12
III. Test of beam PA-G1 ...............................................................................................................12
IV. Test of beam PA-G2 ...............................................................................................................12
V. Test of beam PA-G1 ...............................................................................................................12
The thesis is structured in 5 chapters and 5 appendixes, with a total of 227and 63 pages respectively,
having 361 figures, 41 tables and 105 references.
The abstract retains the structure and numbering of chapters, figures, tables, relations and references.
For a better understanding it is recommended to study the full thesis.
Deformations of wide beams
Page 2
Abstract
Wide beams are horizontal structural elements presenting multiple advantages when using for
multistory building having generally larger openings.
A special case is the use of wide precast prestressed concrete beams, where excessive initial
deformations of the beam before the continuization of the structure might adversely affect
subsequent behavior of the structure.
The thesis is studying behavior of wide precast prestressed concrete beams, by mean of
analytical, numerical and experimental studies.
Chapter 1 – Introduction
As introduction the chapter presents short history of reinforced concrete frame structures,
marked by the achievements of several personalities, followed by the evolution of the
prestressed concrete and sequences in prefabrication of reinforced concrete structures.
Despite of the mechanical disadvantages of wide beams with respect to the regular ones
multi-story building having wide beams are used when complex optimization parameters are
imposed. The second part of the chapter shows the rationality of wide beams, examples of
structures using wide reinforced concrete beams, and studies on wide concrete beams,
performed by researchers as Siah et al [89], Stehle et al [91], Benavent-Climet [30] and
Sherwood [88].
Existing standard limitations of wide beams regarding their dimensions are described
afterward, and the aim of the thesis, formulated as study of the effect of width, loading pattern
and transversal reinforcement ratio of wide beams on their deformations, and evaluation of
the prestressed beam design method based on Eurocod 2 applied for designing wide beams.
Chapter 2 – Considerations regarding calculus of wide beams
Based on the influence of the cross-section dimensions presented in fig. 2.2, the author
establishes the limit of the wide beams at b/h≥1.
Fig. 2.2: Influence of the cross-section on the deformation of the beam
0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0
De
flec
tio
n
h/b ratio
A - const
h - const
b - const
wide beams regular beams
Deformations of wide beams
Page 3
Work stages of reinforced concrete and prestressed concrete beams are detailed afterward.
In work stage I calculus of structural elements subjected to bending is based on simplifying
hypotheses. In case of simple loadings, based on the similarity of the wide beams to the one
way slabs, calculus of the wide beams according to the theory of elasticity can be performed
accepting the limitations for thin slabs having small displacements, even if in some situations
wide beams are beyond the mentioned limitations. For the loads corresponding to the erection
phase of the beams from the structure presented in chapter 1, longitudinal and transversal
deformations of the beam are calculated in several phases.
Fig. 2.33: Longitudinal deformation of the beam for the superposition of loads
Fig. 2.34: Transversal deformation of the beam for the superposition of loads
Influence of width on deformations is studied based on the calculation according to the theory
of elasticity by the presented method.
In work stage II supplementary hypotheses are necessary for the calculus of the wide beam.
When calculating prestressed beams according Eurocod 2, effect of the width is neglected,
and the actual cross section is transformed into the idealized one. The initial stage of the beam
for the experimental studies is determined based on the calculation method of the prestressed
beam. Limit of the IInd
work stage is determined by calculating the value of the ultimate
moment by the Morsch method.
Chapter 3 – Numerical analysis of the wide beam
The analysis of the wide beam is done by modeling in Abaqus the prestressed beam by the
finite element method, taking into consideration the nonlinear behavior of the used materials
(Lubliner et al [62], Lee and Fenves [62], [63]). The stress-strain relationship for the concrete
is determined based on the model proposed by Mander [65], once the compression resistance
of the concrete being determined.
Analysis of the beam is performed for reinforcement ratios ρwc=0.168% and ρwc=0.084%, for
concentrated loads applied in the middle of the opening, on the sides of the beam, and for
distributed loads, using two steps of applying the loads; the first step is used for applying the
prestressing, while the second step is used for applying the load incrementally, monotone
increasingly.
0
5
0 1.9 3.8 5.7 7.6
d [
mm
]
Beam length [m]
Latură grindă
Axul grinzii
3.957 3.888 3.957
3.00
4.00
-600 0 600
d [
mm
] Lățime grindă [mm]
I+II+III
Deformations of wide beams
Page 4
Fig. 3.6: Stress-strain relationship used for concrete modeling [65]
Comparison of the deformations for the different type of loads is done by realizing the
bending moment-deformation diagrams.
Fig. 5.10: Bending moment-deformation diagrams for the studied beams
Deformations of beams are depending on the transversal reinforcement ratio and type of
applied loading.
Fig. 5.9: Transversal deformation of beam after prestress transfer
-10.0
0.0
10.0
20.0
30.0
40.0
50.0
60.0
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
σ [N
/mm
2]
ε [‰]
0
100
200
300
400
500
600
700
800
900
1000
-30 -15 0 15 30 45 60 75 90 105 120 135 150 165 180 195 210
Ben
din
g m
om
ent
[kN
m]
Deflection [mm]
q, ρwc=0.164%
F, ρwc=0.168%
F, ρwc=0.084%
q, ρwc=0.084%
F, ρwc=0.084%, majorat
-15.916
-15.871
-15.916
-15.926 -15.896 -15.926
-15.898
-15.830
-15.898
-15.950 -15.876
-15.950 -16.00
-15.75
-600 0 600
d [
mm
]
Lățime grindă [mm]
F,δwc=0.168% F, δwc=0.084% q, δwc=0.168% q, δwc=0.084%
Deformations of wide beams
Page 5
Fig. 5.11: Longitudinal deformations of beams for F=88.42 kN and q=15.789 kN/m2 (on edges)
Fig. 5.12: Transversal deformations of beams for F=88.42 kN and q=15.789 kN/m2
Chapter 4 – Experimental program
In order to study the behavior of wide prestressed precast beams, three elements, namely PA-
G1, PA-G2 and PA-G3 were tested, loaded by concentrated loads applied on the middle of the
opening, on the sides of the beams. The differences between the beams PA-G1 and PA-G2,
and PA-G3 are in the qualities of the materials used (due to concreting in different time),
transversal reinforcement ratio (ρwf=0.084% and ρwc=0.164%, respectively), and number of
load cycles applied.
In order to compare the calculated stress loss and stress distribution in the middle section of
the beam, at the transfer phase the strain variation in the active reinforcement (4) and upper
(2) and lover (2) passive reinforcements was measured by mean of strain gages displaced on
the studied reinforcements.
Fig. 4.18: Strain gages disposal for active
reinforcement
Fig.4.22: Strain gages disposal for passive
reinforcement
Using the Hooke’s law, stress variation in reinforcements can be calculated.
-1.5
-1.0
-0.5
0.0
0.5
0 1.9 3.8 5.7 7.6d[m
m]
Beam length [m] q, ρwc=0.168% q, ρwc=0.084% F, ρwc=0.168% F, ρwc=0.084%
0.088 0.064 0.088
0.080 0.084 0.080
0.756 0.695 0.756
0.769 0.678 0.769
0.00
0.50
1.00
-600 0 600
d [
mm
]
Lățime grindă [mm] q, ρwc=0.168%
q, ρwc=0.084%
F, ρwc=0.168%
F, ρwc=0.084%
1 2 3 4 1 2
4 3
Deformations of wide beams
Page 6
Fig. 4.20: Stress variation in tendons, at ½ section of the beam
During experiments, the beams were subjected to 3, 5, and 9 load cycles, respectively,
displacements in several points and strain variation being measured.
Fig. 4.28: Displacement transducers disposal
During experiments, the beams were subjected to 3, 5, and 9 load cycles, respectively,
displacements in several points and strain variation being measured. Differences in behavior
of the beams were remarked during the tests, due to differences in the used materials,
transversal reinforcement ratio, initial camber and load cycles applied.
Fig. 4.117: Longitudinal deformation of beams after consuming the camber
1,310
1,315
1,320
1,325
1,330
1,335
1,340
1,345
1,350
0 30 60 90 120 150 180 210
Stre
ss v
aria
tio
n[N
/mm
2 ]
Transfer time [sec]
ToronMijloc 1
ToronMijloc 2
ToronMijloc 3
ToronMijloc 4
-5
0
5
0 1.9 3.8 5.7 7.6
d [
mm
]
Beam length[m]
PA-G1, F=126.00 kN
PA-G2, F=110.00 kN
PA-G3, F=90.00 kN
Deformations of wide beams
Page 7
Fig. 4.118: Transversal deformation of beams after consuming the camber
Fig. 4.108: Strain variation on the side of the beam at the middle of the opening
Fig. 4.110: Strain variation on the upper face of the beam at the middle of the opening
Chapter 5 – Interpretation of results. Conclusions.
Influence of the width on the deflections of the beams using the principles from theory of
elasticity were established for uniform distributed loads and concentrated loads, displaced on
a transversal strip of 20 cm on the middle of the opening.
0.0
1.0
-600 0 600
De
form
atio
n [
mm
]
Beam width [mm]
PA-G1,F=126.00 kN
PA-G2,F=110.00 kN
PA-G3,F=90.00 kN
0
5
10
15
20
25
-2,000 -1,500 -1,000 -500 0 500 1,000 1,500 2,000
Be
am h
eig
ht
[cm
]
Strain[μm/m]
Ciclul 1
Ciclul 2
Ciclul 3
Ciclul 4
Ciclul 5
Cclul 6
Ciclul 7
Ciclul 8
Ciclul 9
0
200
400
600
800
1,000
1,200
1,400
1,600
-600 -400 -200 0 200 400 600
Stra
in [μ
m/m
]
Beam width [mm]
Ciclul 1
Ciclul 2
Ciclul 3
Ciclul 4
Ciclul 5
Ciclul 6
Ciclul 7
Ciclul 8
Ciclul 9
Deformations of wide beams
Page 8
Fig. 5.1: Transversal deformation of the beam with variable width, supposed to q=constant
Fig. 5.3: Relative transversal deformation of the beam with variable width, supposed to
P=constant
Using the assimilation of the wide beam with one way thin slab having small deformations for
width/height ratio smaller than 6 the relative transversal deformation of the beam is less than
1% for both layout of the service loads.
Fig. 5.15: Force-deflection diagrams for beams modeled and tested
Differences in force-deflection diagrams of the modeled and tested beams are due to time
dependent characteristics of concrete, differences of stress-strain curves for materials used in
-19.2
-19.1
-19.0
-18.9
-100 -80 -60 -40 -20 0 20 40 60 80 100d
[m
m]
Lățime grindă [mm]
b=200 cm
b=180 cm
b=150 cm
b=120 cm
b=90 cm
b=60 cm
b=40 cm
b=25 cm
-0.2
-0.1
0.0
-100 -80 -60 -40 -20 0 20 40 60 80 100
d [
mm
]
Lățime grindă [mm]
b=200 cm
b=180 cm
b=150 cm
b=120 cm
b=90 cm
b=60 cm
b=40 cm
b=25 cm
0
50
100
150
200
250
300
350
400
450
-45 -30 -15 0 15 30 45 60 75 90 105 120 135 150
Forc
e [
kN]
Deflection [mm]
ρwc=0.168%
ρwc=0.084%
PA-G1
PA-G2
PA-G3
Deformations of wide beams
Page 9
modeling and tests, and the effect of the cyclic loads, producing residual deformations even in
the elastic domain of behavior.
Transversal deformation of beams compared for tests and models for loads close to the
reference load shows differences smaller than 1 mm, for both transversal reinforcement ratio.
Fig. 5.16: Relative transversal deformation of beams at tests and in modeling for load close
to the reference load
Values of ultimate loads obtained during the experiments are 7÷13% higher than the value of
the calculated ultimate load according to Eurocod 2. Even if neither the longitudinal
compression stress nor the transversal tension stress achieves the ultimate limit, failure of
concrete appears in the axis on the upper face of beam in the middle of the opening, due to the
effect of the biaxial effort (longitudinal compression and transversal tension).
Fig. 5.20: Beam failure at biaxial effort (longitudinal compression & transversal tension)
Taking into account the results obtained in modeling and tests, appropriate behavior of the
wide precast prestressed beam dimensioned according Eurocod 2 can be remarked even if
concentrated load is applied.
The obtained results permits considering the method presented in Eurocod 2 for prestressed
beam appropriate and sufficient for dimensioning wide prestressed beams due to transversal
deformations smaller than 1% for b/h ratios less than 6. For this situation the relative
displacements of the edges with respect to the axis of the beams are sufficiently small, even in
0.704
0.000
0.510
0.692
0.000
0.552 0.360
0.000
0.680
0.122
0.000
0.122
0.139 0.000 0.139
0.00
0.25
0.50
0.75
-600 0 600
d [
mm
]
Beam width [mm] PA-G1, F=126.00kN
PA-G2, F=110.00kN
PA-G3, F=90.00kN
Model ρwc=0.168%
Model ρwc=0.084%
Deformations of wide beams
Page 10
case of loads close to the ones causing failure, so that they fit in permissible deviations set out
in NE 012/2 – 2010 [9], permitting neglecting the width influence for prestressed beams
having b/h<6.
Transversal deformation of the beam is produced even in case of uniform distributed load, not
only in case of concentrated loads applied on the sides. It was remarked that longitudinal
concave deformation is producing transversal convex deformation, and vice-versa.
Initial camber of the prestressed beams has major influence on the deformation of the beams
and on the crack appearance. By prestressing the beam transversal convex deformation is
obtained, thus limiting the relative deflection of the edges to the axis for loads at work stage I,
obtaining relative transversal deformations of neglecting size.
The experiments and models show influence of the load pattern on the actual ultimate loads
and on the failure modes, but also proper behavior of the beams even in case of concentrated
loads applied in the middle of the opening.
The most important conclusion of the thesis is that designing wide precast prestressed beams
neglecting the influence of the width can be applied for beams having b/h<6, due to relative
transversal deformations smaller than 1%, the relative displacement of the edges to the axis
being smaller, even in the middle of the opening, than the permissible deviations established
in NE 012/2 – 2010 [9].
Selective References
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978-2-88394-083-3, February 2008;
[5] *** CSA A23.3-04/2005, Design of Concrete Structures, Canadian Standards Association
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și beton precomprimat. Partea 2: Executarea lucrărilor din beton
[10] *** NE 013-2002: Cod de practică pentru execuția elementelor prefabricate din beton, beton armat și
beton precomprimat
[12] *** NZS3101: The New Zealand standard for the design of concrete structures. Wellington, New Zealand:
Standards New Zealand; 1995
[13] *** P100-1/2006: Cod de proiectare seismică — Partea I — Prevederi de proiectare pentru clădiri
[17] *** SR EN 1992-1-1. Eurocod 2: Proiectarea structurilor de beton, 2004
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Page 11
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Appendixes
I. Full papers of the author from the domain of the thesis
II. Design of the wide beam using FETT-Abacus software
III. Test of beam PA-G1
IV. Test of beam PA-G2
V. Test of beam PA-G1