The analysis of reinforced concrete box girder viaduct defects and their estimation/Gelzbetoniniu dezinio skerspjuvio viaduku pazaidu analize ir ju vertinimas/Defektu analize un tas izvertejums viaduktam ar dzelzsbetona kastveida siju/Armeeritud betoonist kasttalaga viaduktide defektide analuus ja prognoos.
Augonis, Mindaugas ; Zadlauskas, Saulius ; Rudzionis, Zymantas 等
1. Introduction
Prestressed reinforced concrete became firmly used in bridge
construction in the 1950s, after the Second World War. In 1956, the
first prestressed concrete box girder bridge was constructed in the USA
(Hewson 2003) which is still one of the longest bridges in the world
today. In 1969, the first prestressed concrete box girder bridge was
constructed in Lithuania.
At present, there are around 4000 bridges and viaducts in
Lithuania, with the overall length of 93 km. The majority of bridges are
constructed in motorways and only around 14%--in railways. The highest
number of defects and damages are found in reinforced concrete bridges
(Kamaitis 1995). 95% of Lithuanian bridges are reinforced concrete
bridges. The greatest concern has been caused by the prestressed
concrete frame--box girder viaducts constructed over the main Lithuanian
highways during the period of 1968-1983. There are 17 viaducts of this
type reinforced by stressed wire bunches made from wires of high
strength: sixteen viaducts have a span of (16 + 48 + 16) m and one
viaduct has a span of (18 + 36.1 + 18) m. All these viaducts were
erected by the cantilever method. Prefabricated segments are joint
together using epoxy glue. The above mentioned viaducts became the
object of concern from 1995 when, carrying out static experiments, it
was found that some viaducts were in pre-emergency condition. Since then
all viaducts are inspected regularly (viaduct decks are graded twice a
year) and the cracks opening in box girders are observed.
The paper pays more attention to the defects and damages of the
Pareizgupis viaduct (Fig. 1) conducting its thorough theoretical and
analytical research, the results of which are described in the following
chapters.
The static design scheme of the Pareizgupis viaduct is a three-span
six times statically indeterminate frame with hinges at supports. A
mobile hinge is erected in the first support and a fixed hinge in the
other one. The bottom height of a frame beam varies according to the
quadratic parabola and is equal to 170 cm at the support and 120 cm in
the middle of the span. In viaduct cross-section, there are two box
girders joined together by a monolithic reinforced concrete slab and
transverse beams (diaphragms) at the supports and in the middle of the
span. Frame beams are assembled from the segments of different length
(16-19 t in weight).
[FIGURE 1 OMITTED]
[FIGURE 2 OMITTED]
The viaduct piers are the columns of oval cross-section with 38 cm
in thickness and 150 cm in width. Thin supports were designed in order
to get relatively low bending moments in them resulting from the
temperature and plastic deformations of a span construction. The
prestressed wire bunch consists of 24 high-strength wires. There are 24
prestressed high-strength wire bunches at the bottom of the span in one
box girder and 40 at the top of the support. The Pareizgupis viaduct
plan and segment numeration are presented in Fig. 2.
2. Defects and damages of bridges and viaducts
The defects occurring during service life of bridges have a
significant effect on the durability of reinforced concrete structures.
The major and most important defects occurring in prestressed concrete
bridges are their deck cracking (Gribniak et al. 2007; Kaklauskas et al.
2008; Muttoni, Ruiz 2007, 2010) and the opened shear and flexural
cracks. The majority of reinforced concrete bridges built in Europe and
the USA have defects. According to the American scientist Polodny (1985)
cracking in prestressed reinforced concrete bridges can be produced: a)
during design, b) by excess permanent loads, c) by secondary stresses
and overloads, d) associated with bridge operation. Dynamic loads and
overloads cause secondary stresses and large shear stresses later
producing cracking in oblique sections (Wang 2005). In segmental
prestressed concrete bridges with the glued joints of girder segments,
cracks generally occur near segment joints and the places where wire
bunches are corroded (Darmawan 2009; Darmawan, Stewart 2007; Kamaitis
2008; Liang, Wu 2001; Moon et al. 2005; Polodny 1985).
The most frequent defects of prestressed concrete bridges in
Lithuania are thoroughly examined and presented in the books of
Jokubaitis and Kamaitis (2005). The causes of shear cracks forming are
analyzed by Kamaitis (1996, 2000, 2002).
The main defects observed in prestressed concrete viaducts are the
following: leakage in expansion joints over mobile and fixed hinges, a
rough, rolling and deteriorated pavement of a carriageway, leakage in
expansion joints of footpaths, the lack of or inadequate viaduct
drainage system, inadequate (leaky) waterproofing on the top of deck
beams, inadequate erection of segment joints resulting in leakage (Fig.
3), the bottom wires of girders affected by corrosion and broken in
several viaducts, the webs of girder segments with shear cracks on the
outside and from the inside (Fig. 4), the segments of a central span
with vertical cracks on the outside (red cracks) and shear cracks in the
inside (green cracks) (Fig. 5). The forming of cracks in the Pareizgupis
viaduct "B" girder is presented in Fig. 5.
The Pareizgupis viaduct has been constructed over the [A.sub.1]
highway carrying the heaviest traffic flow. Since the viaduct has been
constructed, the traffic flows increased significantly. Also, the
weights of heavy vehicles have changed, environmental factors such as
C[O.sub.2], Cl and S[O.sub.2] affect the concrete intensively (Rombach,
Specker 2000) and cause the corrosion of wires. When carrying out the
detailed inspection of the Pareizgupis viaduct, it was found that
dangerous flexural cracks opened in the 5th segment of girder
"B" of the mid-span (Fig. 5, B-5 block). Having measured the
crack width in different places, it was determined that the crack varied
from 0.25 mm to 0.50 mm in height. The max crack width (0.50 mm) was
recorded at the bottom of the fifth segment flange. This crack occurs
along the entire bottom of the box. Having inspected the fifth segment
wires of girder "B" of the mid-span, 48 fully broken wires (2
wire bunches) were found, another two wire bunches were greatly subject
to corrosion. The cross-section of wires decreased from 5 mm to 3 mm in
the corrosive parts (Fig. 6). Additionally, around 20 wires were
released but were not broken. After the visual evaluation and summing up
of all the deteriorated wires, it was determined that from 60 to 80
wires were detrimental to the performance of the viaduct, which
constituted about 3-4 wire bunches.
[FIGURE 3 OMITTED]
[FIGURE 4 OMITTED]
[FIGURE 5 OMITTED]
As a result of corrosion, the cross section area of prestressed
wires and the strength of concrete decrease and the losses of prestress
increase.
3. Design data of the viaduct
The Pareizgupis viaduct design was produced in 1977, according to
the Russian design code CH 200-62 "Techniceskie usloviya
projektirovaniya zeleznodopoznych avtodopoznych igorodskich mostov i
trub" (1962). The design loads of H-30 and HK-80 were chosen. The
H-30 design load consisted of two automobile queues (equivalent load of
one automobile queue [lambda] = 1.76 t/m) located in two traffic lanes
and the load of a crowd of people on the viaduct footpaths ([lambda], =
0.4 t/m). Reliability factor of the load [[gamma].sub.f] = 1.4.
Dynamic factor is calculated by formula (CH 200-62):
[[mu].sub.din] = 1 + (45 - [lambda])/135, (1)
where [lambda]--span length, m.
Dynamic factor for the H-30 load in the mid-span is [[mu].sub.din]
= 1.0. The HK-80 load is a wheel four-axle vehicular load. The weight of
each axle is 20 t. Reliability factor of the wheel load--[[gamma].sub.f]
= 1.1 and dynamic factor--[[mu].sub.din] = 1.0.
The viaduct deck segments have been designed from M400 mark
concrete which, according to LST EN 206-1:2002 "Concrete--Part 1:
Specification, Performance, Production and Conformity", conformed
to C30/37 concrete class. According to the Russian design code CH 365-67
"Ukazaniya po projektirovaniju zeleznobetonych i betonych
konstrukciji mostov i trub" (1967) theoretical concrete compressive
strength is 14 MPa and concrete tensile strength is 2.0 MPa.
[FIGURE 6 OMITTED]
For the calculation of the viaduct deck effects, the computer
program "Midas Civil" was used to simulate the viaduct girder
model. Having evaluated the self weight, the permanent load and design
loads, bending moments were calculated. Characteristic bending moment in
the span middle resulting from the HK-80 load together with the self
weight and permanent load is 8.48 MNm and characteristic bending moment
from the H-30 load together with the self weight and permanent load is
8.59 MNm.
In the central part of the viaduct, there are 576 wires stressed by
the initial stresses of 1100 MPa. Having evaluated prestress losses, the
cracking moment of 8.69 MNm calculated according to the requirements of
the Design Code CH 365-67 was found. Cracking moment is higher than the
moment resulting from design loads, thus there have to be no cracks at
normal and oblique sections of the mid-span.
4. Analysis of the main concrete tensile and compressive stresses
As can be seen in Fig. 5, shear cracks are also opened in the
central part of the viaduct (blocks 6), therefore the aim was set to
determine the main concrete stresses in the midspan. The same problem
was also analyzed in the article of Plos and Gylltoft (2006). Moreover,
in order to evaluate the potential effect of cross-sectional reduction
on the results, the main concrete stresses of the viaduct mid-section
deck (resulting from design loads) were calculated according to two
methods: the requirements of the Design Code CH 365-67 and the FEM.
Using the FEM, the Pareizgupis viaduct was simulated with regard to
its real dimensions. Also, the top and bottom wire bunches were located
precisely as was indicated in the viaduct design. The viaduct box girder
cross-sectional model is presented in Fig. 7.
[FIGURE 7 OMITTED]
[FIGURE 8 OMITTED]
[FIGURE 9 OMITTED]
[FIGURE 10 OMITTED]
[FIGURE 11 OMITTED]
According to the Design Code CH 365-67, the main concrete stresses
may be calculated as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2)
where [[sigma].sub.x]- stresses in the dire ction of x axis;
[[sigma].sub.y]- stresses in the direction of y axis; [tau]- tangential
stresses.
Having calculated, according to the above mentioned code, the main
stresses of reduced cross-section in the middle of the bottom flange,
the middle of the top flange and the web middle of the central part, it
was determined that the main compression stresses did not exceed 14 MPa
and the main tensile stresses did not exceed 2 MPa. The main concrete
tensile and compressive stresses resulting from the design H-30 load and
the total permanent loads, calculated according to the FEM, are
presented in Fig. 8, and the stresses resulting from the design HK-80
applied on the middle part of cross section and the total permanent
loads are presented in Fig. 9.
In the case of asymmetric loading, the main highest tensile and
compressive stresses are formed in the girder on which the design HK-80
load is applied. The distribution of the main stresses in this type of
loading is presented in Fig. 10.
Under asymmetric HK-80 load, compressive stresses are formed in the
bottom flange, the top flange and the web of girder "A", when
load is applied on girder "B". Tensile stresses are formed in
the bottom flange and the junction of the bottom flange and the web of
girder "B"; compressive stresses are formed in the top flange
and the junction of the top flange and the web. The main tensile
stresses of the bottom flange of girder "B" are equal to 1.9
MPa in the flange middle and the tensile stresses of 2.1 MPa and 3.1 MPa
are formed near the edges of the flange. Having performed the analysis
of the main stresses under symmetrical and asymmetric loads, the
conclusion can be made that under asymmetric loads higher tensile
stresses are formed, the occurrence of which may cause the cracking of
concrete, especially if the wire corrosion is estimated.
In order to evaluate differences of the main stresses of the real
box section and the reduced I-section and considering that it is not
possible to calculate the main stresses of asymmetrically loaded
elements according to the codes, the reduced I-section (Fig. 11) and the
box section (Fig. 12) of the Pareizgupis viaduct were simulated.
[FIGURE 12 OMITTED]
Both cross-sections were calculated from the load of the same
combination: the self girder weight, decking weight, design H-30 load
and precompresion load.
The distribution curves of both simulated cross-sections main
stresses according to distance from viaduct middle were completed. The
main stresses of these models in the junction of the web and the top
flange are presented in Fig. 13, in the middle of web--in Fig. 14, in
the junction of the web and the bottom flange--in Fig. 15, and in the
bottom of the bottom flange--in Fig. 16.
It can be seen in the presented diagrams that the distribution of
the main stresses is different at the same point in the box section and
I-section under the same load. The largest difference in compressive
stresses is noticeable in the top flange. The major disagreement of the
main stresses occurs in the junction of the web and bottom flange and
the bottom of the bottom flange. The main compressive stresses dominate
across all the lenght in I-section of the web middle and in the box
section the main stresses change from compressive stresses to tensile
stresses.
Since some prestressed reinforcement was found broken in the
inspected viaduct, the main concrete tensile stresses were calculated
according to the FEM having accepted the assumption that two wire
bunches were broken:
--Under the symmetrical H-30 and HK-80 load, tensile stresses of
2.0 MPa are formed in the bottom flange of girder "A" and
tensile stresses of 2.4 MPa are formed in the bottom flange of girder
"B";
--Under the asymmetrical HK-80 load, tensile stresses of 1.5 MPa
are formed in the bottom flange of girder "A" and tensile
stresses of 2.9 MPa are formed in the bottom flange of girder
"B".
5. Calculation of reinforcement stress according to crack width
In order to find out the reinforcement stress increment because of
broken wires in the Pareizgupis viaduct (where flexural cracks are the
largest and reach 0.35 mm in the middle of span), it was tried to use a
flexural crack width. The crack width was calculated according to the
Lithuanian Construction Technical Regulation STR 2.05.05:2005
"Design of Concrete and Reinforced Concrete Structures" valid
in Lithuania and the requirements of ENV 1992-2:1996 Eurocode 2: Design
of Concrete Structures--Part 2: Concrete Bridges. According to the code
valid in Lithuania, the crack width is calculated as follows:
[w.sub.k] = [delta][[phi].sub.l][eta] [[sigma].sub.s]/[E.sub.s]
20[(3.5 - 100[[rho].sub.1]).sup.3] [square root of [[??].sub.s], (3)
where [delta], [[phi].sub.l], [eta]--coefficients;
[[sigma].sub.s]--stresses in wire bunches, MPa; [E.sub.s]--the modulus
of elasticity of wires, MPa; [[rho].sub.1]--reinforcement ratio of
element cross-section; [[??].sub.s]--wire bunch diameter, m.
According to the requirements of ENV 1992-2:1996 the crack width is
calculated as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (4)
where [[phi].sub.s]--wire bunch diameter, m;
[[rho].sub.p,eff]--reinforcement ratio; [f.sub.ctm]--the average
concrete tensile strength, MPa.
[FIGURE 17 OMITTED]
[FIGURE 18 OMITTED]
[FIGURE 19 OMITTED]
The stresses in wire bunches were calculated by STR 2.05.05:2005
requirements because ENV 1992-2:1996 does not present such calculations.
Firstly, the crack width, cracking moment and stresses in wire
bunches were calculated for all wires, then up to four wire bunches were
broken and it was determined how the calculated parameters changed. The
dependence of parameters on the cross-sectional area of stressed wires
is presented in Figs 17 and 18.
It can be seen in Fig. 17 that flexural cracks are likely to open
when nearly 8 wires were broken, whereas during the visual inspection it
was determined that 48 wires (2 wire bunches) were already fully broken
and others were damaged by corrosion or released (Fig. 6).
The variation in crack width, calculated according to the methods
of the STR 2.05.05:2005 and ENV 19922:1996 with regard to the increment
of stresses in wire bunches is presented in Fig. 19.
According to STR 2.05.05:2005 the real crack (0.35 mm) will open
when reinforcement stress increment will be ~280 MPa. From the stress
increment equation (STR 2.05.05:2005), such value (external bending
moment is equal to 8.59 MNm, chapter 3) is obtained if cross sectional
area of prestressed reinforcement decreases by ~5.2 wire bunches.
According to ENV 19922:1996, at the real crack width the reinforcement
stress increment should be ~195 MPa. From the same equation (STR
2.05.05:2005), the obtained decrease of prestressed reinforcement is
~3.2 wire bunches.
Having compared the widths of flexural cracks calculated by the
above mentioned methods, it was noticed that [w.sub.k] (STR
2.05.05:2005) and [w.sub.k] (ENV 1992-2:1996) calculating values differ
up to 1.4 times, whereas the calculated stresses [[sigma].sub.s] (STR
2.05.05:2005) and [[sigma].sub.s] (ENV 1992-2:1996) are not different.
Jokubaitis and Juknevicius (2009), when analyzing the calculation of
flexural crack widths of prestressed reinforced concrete beams according
to the methods of STR 2.05.05:2005 and ENV 1992-2:1996, obtained similar
results as in our calculations.
6. Forecasting of bridge deflections
Due to of various defects, irregular cracks are formed and
developed in bridges and the deflection of bridge increases. Based on
these parameters (which depend on defects), it is convenient to evaluate
the present viaduct status and at the same time the reliability of a
viaduct. It is not easy to accurately determine the widths of cracks
opened in a bridge and their development because the crack width varies
across its height.
Therefore, in order to evaluate bridge reliability, it is more
convenient to apply the parameter of bridge deflection which is easier
to determine and the determined values are more reliable (when new
cracks are produced, the development of the first cracks is getting
slow, whereas bridge deflection is getting higher in this case).
According to the viaduct deflection data of 13 inspection years,
the Pareizgupis viaduct has deflected most intensively from all the
unreinforced viaducts. The deflection variation curves of the four
viaducts during the inspection period and the flows of heavy traffic on
each viaduct per day were presented for the analysis (Fig. 20). It can
be seen in Fig. 20 that viaduct deflections depend on its transport
flow.
Therefore, it is important to evaluate this fact in the analysis of
bridge behaviour. In order to determine the reliability of reinforced
concrete box girder bridges, three viaducts were chosen in which the
flow of moving heavy transport varied from 285 to 332 vpd (Fig. 20). For
the evaluation of reliability, structural sustainability equations were
chosen (Kudzys et al. 1992). The probability that ultimate deflection
will be reached is calculated according to the following expression:
Q = 1 - F[EY - Ey/[[sigma].sup.2]Y + [[sigma].sup.2]y], (5)
where EY, Ey, [[sigma].sup.2]Y, [[sigma].sup.2]y,--the average
values of parameters Y and y and their dispersion.
Probability index of structural sustainability:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (6)
where k--the number of element types (k = 1); n--the number of
analyzed deflections (n = 3).
Having evaluated the average values of results (Fig. 20), it was
obtained that the increment of deflection [v.sub.f,m]=1.015 mm/year, its
dispersion [[sigma].sup.2][v.sub.f] = 0.204 [mm.sup.2]/[(year).sup.2].
The average critical deflection assumed [f.sub.crit] = 25 mm and
distribution dispersion [[sigma].sup.2][f.sub.crit] = 1.4 [mm.sup.2].
Having calculated the dispersion of deflection, it was obtained
[[sigma].sup.2] f = 1.259 [mm.sup.2].
Taking the initial deflection equal to its average value at the end
of inspection (2008) f = 11.67 mm, the probability and reliability
values, calculated according to expressions (5) and (6), are presented
in Table 1.
Taking that viaduct sustainability has to be not lower than 95%, it
was obtained that the critical viaduct deflection will be reached in
less than 9 years. In this case the critical deflection is chosen
conditionally because it is difficult to determine deflection before the
inspection (observations started about 20 years after the beginning of
operation) and the initial bridge camber. The aim of this calculation
was to show that it is not difficult to forecast the residual bridge
operating time using the probability method. However, using this method
it is more difficult to evaluate concrete creep which has a direct
effect on bridge deflection and is the most evident during the first
several years.
In order to evaluate the potential effect of creep, which is damped
in the course of time, on the deflections of analyzed viaducts, creep
coefficient and modulus of deformation of M400 mark concrete (design
based concrete of the viaduct deck) with regard to its composition were
calculated according to the method proposed by Bazant and Baweja (1995).
The concrete composition is the following: cement--425 kg/[m.sup.3],
sand--820 kg/[m.sup.3], chrushed granit--975 kg/[m.sup.3], water and
cement ratio--0.40. The obtained variation curves of creep coefficient
and deformation modulus for the period of 30 years are presented in Figs
21 and 22.
[FIGURE 20 OMITTED]
[FIGURE 21 OMITTED]
[FIGURE 22 OMITTED]
Having calculated, according to the method proposed by Bazant and
Baweja (1995), deformation modulus for the M400 mark concrete
composition, it was obtained that deformation modulus decreased by 30%
in 21 years due to concrete creep, when design based elasticity modulus
was equal to 32 000 MPa. The most intensive variations in deformation
modulus occurred in the first three years, and then variation stabilized
in 14 years and changed only marginally. Since the analyzed viaducts
were started to be inspected after 20 years, the effect of creep on
deflection was minimal.
According to the FEM, the deflection of the Pareizgupis viaduct
from the design (HK-80, total permanent and precompression) loads was
theoretically calculated with the software "Midas Civil",
evaluating the concrete deformation modulus but not taking into account
the cracking. The design based viaduct deflection from the short time
load was equal to 14.98 mm and, after the evaluation of concrete creep,
deflection increased up to 21.84 mm. 34 years have passed since the
beginning of the Pareizgupis viaduct construction. The viaduct
deflection has been inspected since 1995. During 13 years of observation
the viaduct deflected by 24 mm. Having evaluated, according to the FEM,
the concrete creep at this period and the factor that about 4 wire
bunches out of 24 of girder "B" were detrimental to the
performance, the average viaduct deflection value of 25.03 mm was
obtained.
7. Conclusions
Having performed the analysis of the main stresses of the
Pareizgupis viaduct with regard to the effect of symmetrical and
asymmetrical loads, it was found that under asymmetrical loads much
higher tensile stresses (~30%) and compressive stresses (~13%) are
formed at the sides of box girder than under symmetrical loads.
The main stresses of the viaduct box section and reduced I-section
calculated according to the FEM were obtained quite different. The main
compressive stresses under asymmetrical loads in the box section top
flange are ~32% higher and tensile stresses in the bottom flange are
~52% higher than in the reduced I-section.
Having estimated, after the visual inspection, that ~4 wire bunches
are broken (~17% of prestress reinforcement) the increase of 9.6% in
reinforcement stress (including a prestress of 1100 MPa) was calculated
according to the FEM.
Due to relatively small losses of reinforcement area (up to 20% in
the discussed case), the relation of losses with the crack moment
values, the increase of reinforcement stress and the crack width is
close to a line. In such case, it is not difficult to estimate the
possible losses of reinforcement area according to the crack width;
however, the width varies not uniformly in time and for this reason it
does not reflect the increase of defects directly.
It is proposed to estimate the reliability of the examined viaduct
by its deflection with regard to variation of parameters in time. When
evaluating reliability parameters in the discussed case (such as the
velocity of deflection increase, its dispersion, etc.) it is necessary
to take the volume of traffic flows and the possible pavement roughness
into consideration.
doi: 10.3846/bjrbe.2012.02
Received 20 May 2011; accepted 23 November 2011
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Mindaugas Augonis (1) ([mail]), Saulius Zadlauskas (2), Zymantas
Rudzionis (3), Algis Pakalnis (4)
(1,2,3) Kaunas University of Technology, Studentu g. 48, 51367
Kaunas, Lithuania
(4) SE "Transport and Road Research Institute", I. Kanto
g. 25, P.O. Box 2082, 44009 Kaunas, Lithuania
E-mails: (1) mindaugas.augonis@ktu.lt; (2)
saulius.zadlauskas@stud.ktu.lt;
(3) zymantas.rudzionis@ktu.lt; (4) a.pakalnis@tkti.lt
Table 1. Probability and sustainability indexes of the analyzed
viaducts
Operation Probability that
forecasting time bridge deflection Sustainability
from the chosen will reach critical index, P
moment, year value Q
2 0 1
4 6.00E-06 0.99998
6 0.00318 0.99905
8 0.0118 0.96522
9 0.0478 0.86641
10 0.1416 0.6539
12 0.617 0.15708
13 0.9522 0.05746
Fig. 13. Distribution curves of the main stresses of the girder
and box models in the junction of the web and the top flange
The stress distribution The stress distribution
curve of box model curve of girder model
1.0 -6.0 -9.1
2.0 -5.7 -8.1
3.0 -5.8 -8.5
4.0 -5.9 -8.7
5.0 -6.0 -8.8
Note: Table made from line graph.
Fig. 14. Distribution curves of the main stresses of the girder and
box models in the web middle
The stress distribution The stress distribution
curve of box model curve of girder model
1.0 -5.0 -6.2
2.0 -4.2 -5.6
3.0 -4.1 -5.8
4.0 -4.1 -5.9
5.0 -4.0 -6.0
Note: Table made from line graph.
Fig. 15. Distribution curves of the main stresses of the girder and
box models in the junction of the web and the bottom flange
The stress distribution The stress distribution
curve of box model curve of girder model
1.0 -4.5 -5
2.0 -2 -3.8
3.0 -1.4 -3.5
4.0 -0.9 -3.2
5.0 -0.7 -3.1
Note: Table made from line graph.
Fig. 16. Distribution curves of the main stresses of the girder and
box models in the bottom of the bottom flange
The stress distribution The stress distribution
curve of box model curve of girder model
1.0 -4.5
2.0 -1.3 -3.1
3.0 0.8 -2.7
4.0 1.0 -2.4
5.0 1.1 -2.3
Note: Table made from line graph.
