Microsoft Word - 31-3390_sETASR_V10_N2_pp5512-5519
Engineering, Technology & Applied Science Research Vol. 10, No. 2, 2020, 5512-5519 5512
www.etasr.com Alkhafaji & Izzet: Prestress Losses in Concrete Rafters with Openings
Prestress Losses in Concrete Rafters with Openings
Falah Jarass Aied Alkhafaji
Department of Civil Engineering
College of Engineering
University of Baghdad, Iraq
falahgaras@gmail.com
Amer Farouk Izzet
Department of Civil Engineering
College of Engineering
University of Baghdad, Iraq
amer.f@coeng.uobaghdad.edu.iq
Abstract—In this paper, experimental work was conducted to
evaluate the losses in prestressing force of 13 (12 perforated and 1
solid) simply supported prestressed concrete rafters. All beams
had the same dimensions and reinforcements. The tested beams
were divided into four main groups and three additional
subgroups were driven. These groups were classified according to
size, number, and configuration of the openings, and the
orientation of the posts (vertical or inclined). Regarding the
prestress losses that have been affected by the cross-section
properties, the provision of the codes is applicable only to
prismatic solid beams, so non-prismatic or moreover perforated
beams also need to be studied. This paper aims to propose a
method based on the same code provisions but taking into
consideration the cross-section variation along the beam length.
The proposed method divides the overall length of the rafter into
a number of assumed prismatic segments with heights measured
at centers. Then, the overall prestress loss is found as the sum of
these segments weighed by the ratios of the length of each beam
segment to the overall length. The experimental results of the
proposed method ranged from 84.749% to 95.607% denoting its
validity.
Keywords-prestress losses; rafter beams; openings
I. INTRODUCTION
The stresses in the tendons of prestressed concrete members
are decreasing with time, but at a decreasing rate, and
asymptotically level off after a long time. The total stress
reduction during the lifespan of the member is called total
prestress loss [1-3]. The reduction in the prestressing force can
be grouped into two categories:
• Immediate elastic loss during the fabrication or construction
process, including elastic shortening of the concrete,
anchorage losses, and frictional losses.
• Time-dependent losses such as creep, shrinkage, and those
due to temperature effects and steel relaxation, all of which
are determinable at the service-load limit state of stress in
the prestressed concrete element.
An exact determination of the magnitude of these losses,
particularly the time-dependent ones, is not feasible, since they
depend on many interrelated factors. Empirical methods of
estimating losses differ with the different codes of practice or
recommendations. The degree of rigor of these methods
depends on the approach chosen and the accepted practice of
record [4-5].
The presence of openings in gable reinforced concrete
beams has many advantages such as flexibility, easier handling,
and, most importantly, reduced overall weight. Furthermore,
concrete has very low to no maintenance cost and high fire
resistance, therefore reinforced concrete gable beams can be
used as a good alternative option to design roofs for
warehouses, industrial buildings, and airplane hangars instead
of steel sections [6-10]. The link structural elements (posts)
between the upper and the lower chords of the rafter beam have
many advantages such as avoiding Vierendeel truss to prevent
shear failure and allowing a beam to enhance its bending
capacity and ductility [11]. The fact that concrete is not
efficient in resisting tensile stress makes very difficult reaching
a long span beam in design, so the addition of prestressing
reinforcements has become necessary to reach span lengths
which cannot be reached by using ordinary reinforcements [12-
13]. To estimate losses such as elastic shortening, creep,
shrinkage, and relaxation in beams, the empirical methods
recommended by the codes are only applicable to prismatic
beams. No special recommendations have been included
regarding rafter beams with or without openings. This study
has proposed a method to estimate prestress losses in rafter
beams, which are divided into a number of segments that are
assumed to be prismatic along their lengths with heights
measured at centers.
II. EXPERIMENTAL INVESTIGATION
The experimental program consisted of casting and testing
of 13 rafter beams, including 12 beams with openings
(perforated) and one reference solid beam without openings.
All the tested beams had the same rafter geometry, i.e. a
rectangular cross-section of 100mm width and height of
400mm at center tapered to 250mm at the two ends, while the
overall length was 3000mm with a clear span of 2800mm.
Figure 1 shows the geometrical details of the tested beams.
Figures 2 and 3 exhibit the details of the solid prestressed rafter
concrete beam and the beams with openings respectively. Mild
steel reinforcement of 4, 6, and 12mm bar diameters were used
while seven-wire low-relaxation strands, Grade 270 with
Ø12.7mm diameter were used as prestressing steel. The tested
beams were divided into four main groups (A, B, C, and D).
These groups were classified according to the studied variables
which are: size, openings number, posts inclination, and
configuration of the openings. Table I exhibits the grouping
according to these variables as follows:
Corresponding author: Falah Jarass Aied Alkhafaji
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• Main groups
To study the effect of openings’ width versus the number of
openings along the beam length, it is worth to highlight that
Group A and B had vertical posts whereas Group C had
inclined posts. Group A and C had the same upper and lower
chords depth of 100mm whereas it was 75mm for Group B.
Group D has been prepared in order to study the effect of the
increasing of the opening area in the case of using the same
number of circular openings.
• Subgroups
Group E has been driven to investigate the effect of
increasing opening height or in other words decreasing depth of
the upper and lower chords of the beam (beams have been
taken from Groups A and B). Group F was driven to find the
effect of post configurations linked between the upper and
lower beam chords, i.e. the geometric shape of the quadratic
openings (beams have been adopted from Groups A and C).
Group G consists of beams with the same number of openings
but with different openings configurations in order to compare
circular and quadratic openings (beams have been chosen from
Groups A, B, C and D).
A. Measurements of Prestress Losses
Prestress losses were determined at different stages (at
transfer of prestressing force and just before loading test). A
special high accuracy electrical resistance measuring device
was used for this purpose. Electrical resistance strain gauges
(FLA-6-11, length=6mm) with gauge factor of 2.09±1%, and
resistance of 120.4Ω were fixed on the strand and bridged to
the data logger. An initial reading was recorded on applying the
prestressing force, and another reading was conducted before
the load test. The electrical resistance is converted to strain
through the following equation:
0
0
iR R
kR
ε
−
= (1)
where ��=initial electrical resistance at the moment of
prestress transfer, ��=electrical resistance immediately before
testing, and �=strain gauge factor (for this type of strain gauge,
�=2.09). Through (1) the electrical resistance is converted to
strain. This strain is compared with the strain calculated from
the elongation which was measured immediately after applying
the prestressing [14].
B. Prestressing Process
For post-tensioning concrete gable beams, after attaining
the age of 57 days for concrete, the prestressing process has
been done according to the following sequence:
• After fixing the strain gauges on the strand (7-wire strand,
12.7mm diameter), it was inserted through the PVC duct
which has been embedded in the mold before concrete
casting.
• Bridging the strain gauges wires to strain indicator (data
logger)
• Attaching the predesigned end bearing steel plates with
adequate grips at the beam ends.
• Applying the prestressing force (110kN) from one end
according to the ACI-318M-14 [15] limitations.
• Finally, releasing the jack and measuring the strand
elongation and the strain which has been occurred in the
strand. The corresponding strain was monitored and
compared with the reading of the pressure gauge of the
hydraulic jack.
TABLE I. DETAILS OF TESTED BEAMS
Group
Beam
mark
Shape of openings
Number of
openings
Total area of
openings (mm
2
)
Width of
openings (mm)
Height of upper
chord (mm)
Height of lower
chord (mm)
A
PGB ------ ------ 0 ------ ------ ------
PGT6 Trapezoidal 6 180000 200 100 100
PGT8 Trapezoidal 8 174000 150 100 100
PGT10 Trapezoidal 10 144000 100 100 100
B
PGB ------ ------ 0 ------ ------ ------
PGTH6 Trapezoidal 6 240000 200 75 75
PGTH8 Trapezoidal 8 234000 150 75 75
PGTH10 Trapezoidal 10 195000 100 75 75
C
PGB ------ ------ 0 ------ ------ ------
PGP6
Trapezoidal with
inclined posts
6 154000 200 100 100
PGP8
Trapezoidal with
inclined posts
8 151000 150 100 100
PGP10
Trapezoidal with
inclined posts
10 138000 100 100 100
D
PGB ------ ------ 0 ------ ------ ------
PGC1 Circular 8 184200 D 75 75
PGC2 Circular 8 128000 0.83D 100 100
PGC3 Circular 8 82000 0.67D 120 120
Engineering, Technology & Applied Science Research Vol. 10, No. 2, 2020, 5512-5519 5514
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Fig. 1. Geometrical details of the tested beams
III. EXPERIMENTAL RESULTS AND DISCUSSION
The prestressing losses were monitored. These losses result
from end anchorage slip, strand friction with the duct, strand
relaxation and shrinkage, and creep of concrete. Table II shows
the prestress losses and the residual prestress (effective
prestress). It was observed that the losses ranged from 17.465%
to 20.309% of the initial prestress depending on size, number
of openings, posts inclination, and openings configuration.
The effect of the considered parameters on the prestress
losses are: For Group A, B, and C, when the number of
openings increases, the prestress losses decrease. This might be
due to the minimization of total area of openings and increasing
number of posts which have a positive effect on prestress losses
and the behavior of the beam in general. The following
comparison demonstrates the decreasing ratio in prestress
losses with increasing number of openings:
• Group A: 2.558 and 4.33% for PGT8 and PGT10 beams
respectively in relation to beam PGT6.
• Group B: 2.175 and 3.403% for PGTH8 and PGTH10
beams respectively in relation to beam PGTH6.
• Group C: 2.888 and 4.764% for PGP8 and PGP10 beams
respectively in relation to beam PGP6.
• Group D (circular openings) it can be noticed that, the
prestress losses decrease with reducing size of circular
openings. The decreasing ratio was 4.25% and 6.441% for
GC2 and GC3 beams respectively in relation to GC1.
• Group E compares the beams of Groups A and B. The
decreasing depth of both upper and lower chords (Group B)
by 25% (elative to that of Group A) led to reduced prestress
losses by 3.824% for beam PGT6 related to beam PGTH6
(Group EI), 4.201% for beam PGT8 related to beam
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PGTH8 (Group EII), and 4.749% for beam PGT10 related
to beam PGTH10 (Group EIII) respectively.
• Group F consists of a comparison between the beams of
Groups A and C. The beams of the two groups have the
same number of openings and post dimensions but they
differ in their inclinations. The prestress losses were
0.438% for beam PGP6 related to beam PGT6 (Group FI),
0.775% for beam PGP8 related to beam PGT8 (Group FII),
and 6.567% for beam PGP10 related to beam PGT10
(Group FIII).
(a)
(b)
Fig. 2. (a) Reinforcement details for solid rafter PGB, (b) section A-A (all
dimensions are in mm)
(a)
(b)
(c)
Fig. 3. (a) Details of reinforcement for rafter with openings PGT6, (b)
section B-B, (c) section C-C
• Group G consists of beams having eight openings but
different configurations (circular and quadratic openings),
with both opening shapes are restricted by the same chord
depth. The results indicate a decrease in the prestress losses
on the beams with circular openings in comparison with the
beams with quadratic openings. The decrease ratios are:
2.488% for beam PGC2 related to beam PGT8 (Group GI),
2.437% for beam PGC1 related to beam PGTH8 (Group
GII), and 1.726% for beam PGC2 related to beam PGP8
(Group GIII).
IV. PRESTRESS LOSSES WITH THE PROPOSED METHOD
The variation in cross-section properties should be
considered in order to calculate the prestress losses in solid or
perforated rafter beams. The method suggests dividing a rafter
beam into a set of segments that are considered as prismatic
parts with heights measured at centers (Figure 4). For the
perforated beams, the number of segments should be chosen in
a manner such that the segment is either solid or with openings.
The overall prestress loss is found as the summation of the
contributions of the beam subdivisions weighed by the ratio of
the length of the beam segment to the overall length. The PGT6
beam has been taken as an example to show the steps of the
estimation process in details. The same procedure was used for
the other tested beams.
1) Instantaneous Losses
a) Anchorage Seating Loss
Usually long tendons will be less affected by seating loss.
For short tendons, the seating loss should be detected and
subtracted from the applied prestressing force [16]. Assuming
∆A=1.5 mm and L=3000 mm and by (2):
����
�� �� (2)
substituting we get ����=98.75MPa.
b) Elastic Shortening
No elastic shortening occurred because only one strand was
used in each beam: ���� 0
c) Friction Losses
The strand is straight therefore there is no curvature, α=0.
Assuming that wobble coefficient k=0.002, by (3) we get:
���� ����μ � � � �� (3)
substituting we get: ����=6.25MPa. The stress remaining in
the restressing strand after all instantaneous stresses is:
��� ��� � ����� � ���� � ����� (4)
substituting we get: ��� =116-(105.8+6.25)=1003.95Mpa. The
net prestressing force is calculated by:
�� ��� �� (5)
substituting we get: �� ��� �� =99391N.
2) Time Dependent Losses
a) Stage (I): Losses After 24hr of the Force Transfer
• Relaxation loss:
For t1=1hr, t2=24hr:
���! ��� "#$%&' (#$%&)*+ , -
./0
./1
� 0.554 (6)
substituting we get: ���!=6.89 MPa.
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TABLE II. PRESTRESS LOSSES AND STRESSES IN PRESTRESSED CONCRETE BEAMS UP TO THE MOMENT OF TESTING
Group
Beam's
labeling
Prestress
stress
(MPa)
Age of
beam at
testing
(days)
Age of beam
transferred to
testing (days)
Instantaneous
losses after
transfer (MPa)
Time dependent
losses at testing
(MPa)
Total
prestress
losses,
(MPa) ∆PFT
Residual
prestress in
strands, (effective
prestress) (MPa)
Presstress
loss/initial
prestress (%)
Increasing
ratio of
losses% (1)
Main groups
A
PGB 1116 105 48 100.313 94.596 194.909 921.090 17.465 0
PGT6 1116 115 58 107.088 110.894 217.982 898.0176 19.533 11.837
PGT8 1116 120 63 105.494 106.912 212.406 903.594 19.033 8.9766
PGT10 1116 122 65 104.751 103.787 208.538 907.462 18.686 6.9922
B
PGB 1116 105 48 100.313 94.596 194.909 921.091 17.465 0
PGTH6 1116 123 66 105.122 121.528 226.650 889.350 20.309 16.284
PGTH8 1116 125 68 104.244 117.476 221.720 894.280 19.867 13.755
PGTH10 1116 127 70 105.208 113.729 218.936 897.063 19.618 12.327
C
PGB 1116 105 48 100.313 94.596 194.909 921.090 17.465 0
PGP6 1116 128 71 101.489 115.539 217.028 898.972 19.447 11.348
PGP8 1116 129 72 97.996 112.764 210.760 905.240 18.885 8.1322
PGP10 1116 130 73 105.789 100.899 206.688 909.312 18.520 6.0427
D
PGB 1116 105 48 100.313 94.596 194.909 921.091 17.465 0
PGC1 1116 131 74 98.013 118.303 216.316 899.684 19.383 10.982
PGC2 1116 132 75 97.727 109.396 207.123 908.878 18.559 6.2657
PGC3 1116 133 76 103.649 98.733 202.382 913.618 18.135 3.8338
Secondary groups
EI
PGB 1116 105 48 100.313 94.596 194.909 921.091 17.465 0
PGT6 1116 115 58 107.088 110.894 217.982 898.018 19.533 11.837
PGTH6 1116 123 66 105.122 121.528 226.650 889.350 20.309 16.284
EII
PGB 1116 105 48 100.313 94.596 194.909 921.091 17.465 0
PGT8 1116 120 63 105.494 106.912 212.406 903.593 19.033 8.9766
PGTH8 1116 125 68 104.244 117.476 221.720 894.240 19.867 13.755
EIII
PGB 1116 105 48 100.313 94.596 194.909 921.091 17.465 0
PGT10 1116 122 65 104.751 103.787 208.538 907.462 18.686 6.992
PGTH10 1116 127 70 105.207 113.729 218.937 897.063 19.618 12.327
FI
PGB 1116 105 48 100.313 94.596 194.909 921.091 17.465 0
PGT6 1116 115 58 107.088 110.894 217.982 898.018 19.533 11.837
PGP6 1116 128 71 101.489 115.539 217.027 898.973 19.447 11.348
FII
PGB 1116 105 48 100.313 94.596 194.909 921.091 17.465 0
PGT8 1116 120 63 105.494 106.912 212.406 903.594 19.033 8.977
PGP8 1116 129 72 97.997 112.764 210.761 905.240 18.885 8.132
FIII
PGB 1116 105 48 100.313 94.596 194.909 921.091 17.465 0
PGT10 1116 122 65 104.751 116.464 221.215 894.785 19.822 6.640
PGP10 1116 130 73 105.789 100.899 206.688 909.312 18.520 6.043
GI
PGB 1116 105 48 100.3134 94.596 194.9094 921.0906 17.465 0
PGT8 1116 120 63 105.4942 106.912 212.4062 903.5938 19.0328 8.9766
PGC2 1116 132 75 97.72652 109.396 207.1225 908.8775 18.5594 6.2657
GII
PGB 1116 105 48 100.3134 94.596 194.9094 921.0906 17.465 0
PGTH8 1116 125 68 104.244 117.476 221.72 894.28 19.8674 13.755
PGC1 1116 131 74 98.01274 118.303 216.3157 899.6843 19.3831 10.982
GIII
PGB 1116 105 48 100.3134 94.596 194.9094 921.0906 17.465 0
PGP8 1116 129 72 97.99647 112.764 210.7605 905.2395 18.8853 8.1322
PGC2 1116 132 75 97.72652 109.396 207.1225 908.8775 18.5594 6.2657
(1)
5678 �9:;<�( 5678 ��=>�
5678 ��=>� ∗ 100
• Creep loss: ��A! 0
• Shrinkage loss: ���B 0
• Tendon stress at the end of Stage I:
�� ��� � ���! (7)
substituting we get: �� 997.06MPa.
b) Stage (II): Losses after 58 Days:
• Creep loss: �� ��� �� = 98.709kN
rD EF�F (8)
ẛH � � �0�F × -1 �
:0'
J0'
4 � KL0 :0EF (9)
For part No.1: i=1, l1=0.6m (length of segment), X1=0.3m,
Y1=
0.15
0.3
1.45
× =0.031m, H1=0.25+Y1=0.281m (height of
sectional center of segment), B1=0.1m (beam width),
A1=B1×H1=0.0281m
2
(cross section area of segment), so:
I1=
>)×B)
*D =0.00018497m
2
, 1
1
2 2
0.00658mC
C
I
i A
r = = and
M* = "B)D � NOP, =0.0905m. Taking as h=0.25m we have:
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Q* R* × S × ϪH =0.6kN/m and w2=0.0155kN/m. R is:
� UVW(VX/YZ0Z[\×Ϫ]D =0.963kN, and:
MD1 � × ^* � �Q* × ^* × _* � QD × `'D × _D� 0.2617
ẛH � �
�)
�])
I "1 � :)'J)', �
KL) :)
EH� 7.75769MPa.
With ẛH � I `0�=3.103MPa we have:
ẛH ∑ ẛH � I `0�b�c* (10)
�H 4700 I f�’h (11)
i �/j�] (12)
substituting in (10)-(12) we get: ẛH =7.718MPa, EC=29725, and
n=6.65. For a posttensioned beam: KCR=1.6, so:
���A! i �A! ẛH (13)
substituting we have ���A! =82.12MPa. The same steps are
repeated for the other segments (portions) of the beam, and
then the total prestress loss due to the creep is found after
multiplying fcsi by Li/L for all segments as in (10). The
calculations are shown in Table III.
• Shrinkage loss:
���B� 2.8 I 10(m �no� Eqr��1 � 0.06 V0�0 ��100 � �o�� (14)
���B ∑ ��no� I `��b�c* (15)
For i=1, l1=0.6m, X1=0.3m, Y1=31.034mm we get:
H1=281.03mm, B1=100mm, Vi=28103mm
2
, Si=(B1+H1)× 2
=762.069mm
2
(surface area of segment), KSH=0.77, RH=70%,
Epsi=197500,and:
��no* 2.8 I 10 � 6 �no� Eqrt "1 � 0.06V0�0, �100 � �o�� =>
��no�=34.0994, while ��no* I `*� =13.6397845MPa and
��no ∑ ��no� I `��b�c* =34.48MPa.
The same steps are repeated for the other segments of the
beam, and then the total prestress loss due to shrinkage is the
cumulative summarization of the ��no� multiplied by Li/L for
all parts as in (15). The calculations are shown in Table IV.
c) Relaxation Loss after 43 Days
fps=997.06Mpa, t1=24hr, t2=1032hr, and
Δf6w ��� "#$%x' (#$%x)*+ , -
./0
./1
� 0.554=8.02MPa. The total
losses are: ��yz ���N� � ��no � ����=124.622MPa
��: �� � ��yz=872.438MPa
The same calculations are repeated for all beams. Table V.
shows the prestress losses by the proposed method and stresses
in prestressed openings. For simplification reasons, the
equivalent trapezoidal shape has the same area as the original
opening and uniform posts are considered. The same
simplification is used for Group D, where equivalent
rectangulars are positioned.
TABLE III. CALCULATION STEPS OF CREEP LOSS FOR BEAM PGT6 (ALL DIMENSIONS ARE IN m)
(a)
(b) (c) (d)
Fig. 4. Dividing the beam to a set of segments. (a) Details and number of segments, (b) section 1-1, (c) section 2-2, (d) section 3-3
Part No. i li Xi Yi Hi Bi Aci Ici r
2
i ei pi MDi ẛcsi ẛcsi×l/L
1 1 0.6 0.3 0.031 0.281 0.1 0.0281 0.00018497 0.00658 0.0905 98.71 0.2597 7.75769 3.103
2 2 0.2 0.7 0.072 0.2 0.1 0.02 6.6667E-05 0.00333 0.05 98.71 0.5167 8.24952 1.1
3 3 0.1 0.85 0.088 0.338 0.1 0.0338 0.00032159 0.00952 0.119 98.71 0.585 7.04862 0.47
4 4 0.2 1 0.103 0.2 0.1 0.02 6.6667E-05 0.00333 0.05 98.71 0.6408 8.15642 1.087
5 5 0.1 1.15 0.119 0.369 0.1 0.0369 0.00041858 0.01134 0.1345 98.71 0.6849 6.72019 0.448
6 6 0.2 1.3 0.134 0.2 0.1 0.02 6.6667E-05 0.00333 0.05 98.71 0.7131 8.10221 1.080
7 7 0.05 1.425 0.147 0.397 0.1 0.0397 0.00052306 0.01316 0.1487 98.71 0.7134 6.45419 0.215
8 8 0.05 1.475 - 0.4 0.1 0.04 0.00053333 0.01333 0.15 98.71 0.7309 6.42646 0.214
∑li 1.5
ẛcs 7.718
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TABLE IV. CALCULATION STEPS OF SHRINKAGE LOSS FOR BEAM PGT6 (ALL DIMENSIONS ARE IN m)
Part No. l Xi Yi Hi Vi Si KSH Eps RH ∆fSHi ∆fSHi×Li/Ltotal
1 600 300 31.034 281.03 28103 762.069 0.77 197500 70 34.0994 13.639
2 200 700 - 200 20000 800 0.77 197500 70 35.1658 4.688
3 100 850 87.931 337.93 33793 875.862 0.77 197500 70 33.9463 2.263
4 200 1000 - 200 20000 800 0.77 197500 70 35.1658 4.688
5 100 1150 118.97 368.97 36897 937.931 0.77 197500 70 33.8785 2.2585
6 200 1300 - 200 20000 800 0.77 197500 70 35.1658 4.688
7 50 1425 147.41 397.41 39741 994.828 0.77 197500 70 33.8237 1.127
8 50 1475
400 40000 1000 0.77 197500 70 33.819 1.127
∑L 1500
∆fSH 34.482
TABLE V. PRESTRESS LOSSES BASED ON THE PROPOSED METHOD AND STRESSES IN PRESTRESSED CONCRETE BEAMS UP TO THE MOMENT OF TESTING
Group Beam
Instantaneous losses
(1)
Time dependent losses (2)
∆PFT (II) fps fpe
∆PFT
(1)+(2)
Prestress
loss/initial
prestress (%)
Increasing
ratio of
losses% (1)
At transfer (I) At the age of test (II)
∆fPA ∆fES ∆fPF ∆fPR ∆fCR ∆fSH ∆fPR ∆fCR ∆fSH
A
PGB(ref) 98.75 0 6.25 6.89 0 0 7.65 76.47 33.975 118.095 997.06 878.965 229.985 20.608 0
PGT6 98.75 0 6.25 6.89 0 0 8.02 82.12 34.482 124.622 997.06 872.438 236.512 21.193 2.838
PGT8 98.75 0 6.25 6.89 0 0 8.18 81.54 34.48 124.2 997.06 872.86 236.09 21.155 2.654
PGT10 98.75 0 6.25 6.89 0 0 8.25 80.9 34.4 123.55 997.06 873.51 235.44 21.097 2.371
B
PGB(ref) 98.75 0 6.25 6.89 0 0 7.65 76.47 33.975 118.095 997.06 878.965 229.985 20.608 0
PGTH6 98.75 0 6.25 6.89 0 0 8.28 82.284 34.61 125.174 997.06 871.886 237.064 21.243 3.078
PGTH8 98.75 0 6.25 6.89 0 0 8.33 81.703 34.609 124.642 997.06 872.418 236.532 21.195 2.846
PGTH10 98.75 0 6.25 6.89 0 0 8.39 81.028 34.505 123.923 997.06 873.136 235.813 21.130 2.534
C
PGB(ref) 98.75 0 6.25 6.89 0 0 7.65 76.47 33.975 118.09 997.06 878.965 229.985 20.608 0
PGP6 98.75 0 6.25 6.89 0 0 8.42 81.204 34.43 124.05 997.06 873.006 235.944 21.142 2.591
PGP8 98.75 0 6.25 6.89 0 0 8.45 80.84 34.43 123.72 997.06 873.34 235.61 21.112 2.445
PGP10 98.75 0 6.25 6.89 0 0 8.47 80.429 34.169 123.06 997.06 873.992 234.958 21.054 2.162
D
PGB(ref) 98.75 0 6.25 6.89 0 0 7.65 76.47 33.975 118.095 997.06 878.965 229.985 20.608 0
PGC1 98.75 0 6.25 6.89 0 0 8.5 84.953 34.365 127.818 997.06 869.242 239.708 21.479 4.227
PGC2 98.75 0 6.25 6.89 0 0 8.53 80.915 34.23 123.675 997.06 873.385 235.565 21.108 2.426
PGC3 98.75 0 6.25 6.89 0 0 8.55 79.048 34.06 121.658 997.06 875.402 233.548 20.927 1.549
V. COMPARISON BETWEEN THE EXPERIMENTAL AND THE
PROPOSED METHOD’S RESULTS
As demonstrated in Table VI, the experimental results
converge to the ones of the proposed method. The ratio of
experimental to the proposed method’s results ranged from
84.749% to 95.607% denoting the validity of the proposed
estimation.
TABLE VI. RESULT COMPARISON
Group Beam
Experimental
loss
Proposed
loss
Experimental loss /
Proposed loss %
A
PGB 194.909 229.985 84.749
PGT6 217.982 236.512 92.165
PGT8 212.406 236.09 89.968
PGT10 208.538 235.44 88.574
B
PGB 194.909 229.985 84.749
PGTH6 226.650 237.064 95.607
PGTH8 221.720 236.532 93.738
PGTH10 218.937 235.81377 92.843
C
PGB 194.909 229.985 84.749
PGP6 217.028 235.944 91.983
PGP8 210.760 235.61 89.453
PGP10 206.688 234.958 87.968
D
PGB 194.909 229.985 84.749
PGC1 216.316 239.708 90.241
PGC2 207.123 235.565 87.926
PGTC3 202.382 233.548 86.656
VI. CONCLUSION
• Increasing the number of quadratic openings along the
beam length from 6 to 8 and then to 10 decreased the
prestress losses by an average of 2.54% and 4.166%,
respectively.
• The prestress losses decrease with reducing size of the
circular openings by 17% and 33% from 4.25% to 6.441%.
• Decreasing the depth of both upper and lower chords for
perforated beams by 25%, i.e. by increasing openings’
height, led to increase prestress losses by an average of
4.258%.
• The average decrease in the prestress losses in beams
having inclined posts in comparison with those having
vertical ones was 2.593%.
• The average losses decrease in prestressing force for beams
with circular openings in comparison with that of quadratic
ones was 2.217%.
• The result of the proposed estimated method converges
with the experimental results. This accordance ranged from
84.749% to 95.607%.
Engineering, Technology & Applied Science Research Vol. 10, No. 2, 2020, 5512-5519 5519
www.etasr.com Alkhafaji & Izzet: Prestress Losses in Concrete Rafters with Openings
APPENDIX
�A Area of net concrete section
�� Area of prestressed steel in tension zone
�h Modulus of elasticity of concrete
�{O Modulus of elasticity of prestressed steel
��� Initial prestress stress in prestressed steel
��� Stress in prestressed steel at jacking stage
f’c Cylinder concrete compressive strength at 28 days
�� Stress in prestressed steel at nominal flexural strength
RH Relative humidity
|A Second moment of area of net concrete section about an axis through its centroid
� Wobble coefficient
MD Dead load moment
�� Initial prestress force
μ Curvature friction coefficient
� Total angular change of prestressing tendon profile in radians from
tendon jacking end to any point x
∆A Slip in tendon from anchorage
���� Prestress losses due to anchorage seating
���� Prestress losses due to elastic shortening
���� Prestress losses due to friction
Δf6w Prestress losses due to relaxation
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