An approximate method for the ultimate shear strength of horizontally curved composite plate girders.
Shanmugam, Nandivaram E. ; Basher, Mohammed A. ; Rashid, Khalim A. 等
Introduction
Horizontally curved plate girders are frequently employed in the
construction of modern highway bridges (Hall 1996) in view of their
aesthetic appearance and ease of construction where the road alignment
requires such curvature. Ultimate strength of these girders under
external loads is notably decreased in view of the initial curvature. As
the curvature increases excessive deflections may occur. Vertical
displacements and rotations are not independent but coupled in the
horizontally curved beams. In practice, rotation is often restrained by
providing lateral supports such as cross frames, diaphragms and bracings
connecting two or more parallel girders. Comprehensive studies on
horizontally curved composite plate girders appear to be very limited in
the published literature. With the growth in rapid transport systems
that rely on close deflection tolerances it is imperative to study the
behavior of such curved girders closely, in order to achieve an
efficient and effective design approach. An experimental investigation
on full-scale horizontally curved steel plate girders has been carried
out by Zureick et al. (2002). They studied in detail the overall
behavior and ultimate shear strength. Horizontally curved steel plate
girders of medium size with and without web openings were tested to
failure (Shanmugam et al. 2003; Lian, Shanmugam 2003) in order to
examine the ultimate load behavior and load carrying capacity. Finite
element modelling of the girders was also proposed. Jung and White
(2006) have reported results obtained from the finite element analyses
of full-scale curved girders tested by Zureick et al. (2002). Both the
elastic shear bucking and full nonlinear maximum shear strength
responses have been considered.
Composite action between plate girders and concrete slab has been
studied in the early eighties in respect of straight girders. An
experimental investigation on steel-concrete composite plate girders
under combined shear and negative bending was carried out by Allison et
al. (1982). Based on the experimental results and Cardiff analytical
model (Porter et al. 1975; Evans et al. 1978), Allison et al. (1982)
proposed equations to determine the collapse load of composite plate
girders. In a recent study on horizontally curved composite plate
girders Shanmugam et al. (2009) and Basher et al. (2009, 2011) have
researched into the behavior and ultimate shear strength by using the
finite element modelling and proposed simple methods to determine the
shear strength of these girders. The study included the effects of web
openings and trapezoidally corrugated webs. This paper is concerned with
an approximate method to predict the ultimate shear strength of
horizontally curved composite plate girders. Effects of openings in the
web panels and trapezoidally corrugated webs are also accounted for in
the method. The accuracy of the method is established by comparing the
results with the corresponding values obtained by using the finite
element method.
1. Ultimate shear strength
1.1. Shear strength of straight girders with solid webs
An equilibrium solution based on tension filed theory was proposed
by Porter et al. (1975) and Evans et al. (1978) to calculate the
ultimate strength of straight steel plate girders in which it was
assumed that shear in a plate girder is resisted by the web plate up to
the elastic load (Fig. 1(a)). Any further increase in the load results
in formation of buckles parallel to the tensile direction (Fig. 1(b)).
Small band of web plate along the tensile diagonal begins to behave in a
mode similar to tension member of an N-truss. Membrane tension in the
web is developed which enables the web to sustain loads well in excess
of the elastic critical load. A consequence of this membrane tension is
the inward pulling of the flanges, under increasing loads. Eventually
plastic hinges are formed in the flanges leading to collapse of the
girder (Fig. 1(c)). In view of this type of behavior observed, collapse
load of a plate girder is assumed to consist of three components viz.
(i) load corresponding to the elastic buckling stage (ii) load resisted
by tension field action and (iii) load contributed by the flanges. As
per the equilibrium solution (Porter et al. 1975; Evans et al. 1978),
the ultimate shear resistance ([V.sub.s]) of a straight steel plate
girder of uniform cross-section with slender webs is given by:
[V.sub.s] = [[tau].sub.cr]dt + [[sigma].sup.y.sub.t](d cot [theta]
- b + c)[sin.sup.2][theta] + [4[M.sub.pf]/c]. (1)
In the above equation, [[tau].sub.cr] refers to the elastic
critical shear stress in the web, [[sigma].sup.y.sub.t] the membrane
stress in the web in post-critical stage, b, web panel width, c,
distance between the hinges formed in the flange, d, web depth, t, web
thickness, [M.sub.pf], plastic moment of the flange and [theta], the
angle of inclination of the tensile membrane stress
[[sigma].sup.y.sub.t]. It has been proposed (Porter et al. 1975; Evans
et al. 1978), based on a number of observations, that 0 can be
calculated as:
[theta] = [2/3][tan.sup.-1](d/b). (2)
The value of c is determined from the equation:
c = [2/sin [theta]] [M.sub.pf]/[square root of
([[sigma].sup.y.sub.t]t)]. (3)
Membrane stress, [[sigma].sup.y.sub.t] is obtained from:
[[sigma].sup.y.sub.t] = [-3/2][[tau].sub.cr]sin 2[theta] + [square
root of ([[sigma].sup.2.sub.yw] + [[tau].sup.2.sub.cr]{[([3/2]sin
2[theta]).sup.2] - 3})], (4)
in which [[sigma].sub.yw] is yield stress for the web material.
[FIGURE 1 OMITTED]
Shanmugam and Baskar (2006) extended this method to determine the
ultimate shear capacity of straight composite plate girders. It was
assumed that the ultimate shear capacity of straight composite plate
girders ([V.sub.ult]) may be obtained as a sum of the shear resistance
of bare steel plate girder ([V.sub.s]), as given in Eqn (1) and the
contribution by concrete slab to the shear capacity ([V.sub.c]), i.e.:
[V.sub.ult] = [V.sub.s] + [V.sub.c]. (5)
Test results (Baskar, Shanmugam 2003; Shanmugam, Baskar 2003) have
shown that width of the tension field increases due to composite action
between steel girder and concrete slab, and the final sway mechanism in
the steel part of the composite girders is similar to the mechanism
proposed for steel plate girders subjected to shear loading. Three
different failure mechanisms in the concrete slab at different stages of
loading were observed during the tests. At the initial stages of loading
and up to the elastic buckling of web, the composite girders acted like
composite beams with hot rolled sections and, the slab was subjected to
flexural hairline cracking near the load point and over the two end
supports. After the initial hairline cracks in the slab, the girders
continued to resist loading without any noticeable change in the
stiffness of the girder. The second mode of cracks that were observed in
the girders under shear loading was a cone shaped pullout failure cracks
at the end of the girders. These cracks affected the stiffness of the
composite girder to a certain extent but allowed the girder to continue
to resist further load. At the ultimate condition the concrete slab was
subjected to sudden failure in the form of split tensile failure, as
shown in Figure 2(a). Based on the experimental observations, a failure
mechanism is assumed as shown in Figure 2(b) to calculate the ultimate
shear carrying capacity.
[FIGURE 2a OMITTED]
[FIGURE 2b OMITTED]
It is assumed herein that the tension field theory for steel plate
girders (Porter et al. 1975; Evans et al. 1978) is applicable to
determine the shear capacity of steel part (Vs) of the composite plate
girders. The diagonal tension field in the web of steel-concrete
composite plate girder is partly anchored to the concrete slab through
composite action and therefore, shear carrying capacity is enhanced. The
extent of tension field anchored by the slab depends on the plastic
hinge location, angle of inclination of the tension field, tensile
strength of concrete, and the shear strength of concrete slab. Composite
action increases the load carrying capacity of the girder due to
additional anchorage to the tension field. The contribution by the slab
to shear carrying capacity of steel-concrete composite plate girder is
given by:
[V.sub.c] = [b.sub.c] x [T.sub.l] x [f.sub.ta], (6)
where: [b.sub.c] is effective width of the slab; [T.sub.l] anchor
length and [f.sub.ta] allowable split tensile stress of concrete. The
value of [b.sub.c] may be taken as the effective width of concrete slab
determined based on the code provisions. For an isolated composite
girder such as those considered in the present study the whole width of
the slab may be assumed effective especially under shear loading.
[T.sub.l] is determined from the assumed failure mechanism shown in
Figure 2. It is assumed that the anchor plane lies along the line CE and
inclined at an angle of [[phi].sub.2] with respect to the connecting
steel flange (assumed horizontal). The magnified view of the portion
CDEF is shown in Figure 3. The node E in the concrete slab lies on a
vertical line through node D. The anchor plane is assumed to lie in the
plane connecting the nodes C and E. Even though the anchor plane is
between the nodes C and E, the effective anchor length, [T.sub.l], is
taken as the length between the nodes C and F and determined from the
geometry in Figure 3.
[FIGURE 3 OMITTED]
The allowable split tensile stress, [f.sub.ta], along the anchor
plane is a parameter dependant on the angle of inclination of the anchor
plane, angle of inclination of tension field and the maximum allowable
shear and tensile strength of concrete. At the ultimate load condition,
it is assumed that the shear strength of concrete will reach to its
maximum value. The allowable split tensile stress, [f.sub.ta], along the
anchor plane is taken as [f.sub.ta] = [f.sub.tu] - [v.sub.cuinlined] in
which [f.sub.tu] is the measured ultimate split tensile stress of
concrete. Codes and researchers have recommended different values of
[v.sub.cu] for an un-reinforced and minimum reinforced concrete
sections. Narayanan et al. (1989), related this value [v.sub.cu]
approximately to the concrete cube strength, [f.sub.cu], as [v.sub.cu] =
0.3[([f.sub.cu]).sup.0.5]. This value of [v.sub.cu] is considered in the
present study to determine the contribution of concrete slab on shear
carrying capacity of composite plate girders.
2. Horizontally curved composite plate girders with solid webs
The method described above was applied to the horizontally curved
composite girders with a modification factor ([K.sub.c]) to account for
curvature of the girder. The modification factor, [K.sub.c] for shear
force was obtained by considering the bending of horizontally curved
beam (Pytel, Singer 1987) shown in Figure 4 as:
[K.sub.c] = [3/2[alpha]](sin [alpha] - sin [[alpha]/3]), (7)
where [alpha] is the included angle at the centre of curvature. The
curvature factor [K.sub.c] can be used to obtain shear force at
specified points along the curved girder based on the values
corresponding to straight girders. It should be noted that the above
equation for [K.sub.c] is independent of number of transverse stiffeners
and, the included angle '[alpha]' is for a single curved span
only. The ultimate shear strength of a horizontally curved composite
plate girder can thus be calculated as:
[V.sub.ult(curved)] = [K.sub.c]([V.sub.s] + [V.sub.c]). (8)
[FIGURE 4 OMITTED]
It is necessary to assess the accuracy of any new method proposed.
Therefore, horizontally curved composite plate girders with solid webs
C1, C2, C1-C and C2-C reported earlier by the authors (Shanmugam et al.
2009) were considered in this study. The straight girder values of
[V.sub.s], [V.sub.c] and [V.sub.ult] for the eight girders were
determined first and the ultimate strength for the horizontally curved
girders was obtained as [V.sub.ult(curved)] equal to the product of
[V.sub.ult] and [K.sub.c]. Shear strength values for the girders
obtained from finite element analyses using LUSAS, [V.sub.ultFEA] are
listed along with [V.sub.ult(curved)] in Table 1 for comparison. It can
be seen from the table that there is a close agreement between the
finite element and predicted values, the maximum deviation being 4% thus
confirming the accuracy of the proposed method. Despite the
approximations, the proposed method is found to be capable of predicting
the shear capacity of the horizontally curved composite plate girders to
an acceptable accuracy.
3. Horizontally curved composite plate girders with web openings
For plate girder with web openings additional load in the post
critical stage is carried by the membrane stresses,
[[sigma].sup.y.sub.t] which form two tension bands in the web, one above
and the other below the openings, as shown in Figure 5 (Narayanan,
Rockey 1981). The distance c between hinges and the angle of inclination
of the panel diagonal [[theta].sub.d] are shown in the figure. A
suitable value for the angle of inclination of the tensile membrane
stress [[theta].sub.d] is chosen appropriate to the perforated web.
Narayanan and Avanessian (1983a, b) have suggested that for a perforated
web the angle may be taken as:
[[theta].sub.d] = [2/3][theta](1 - [[d.sub.o]/d]) or
[[theta].sub.d] = [2/3][theta](1 - [[b.sub.o]/b]), (9)
whichever is smaller.
[d.sub.o] and [b.sub.o] are, respectively the depth and width of
the rectangular openings, d and b being, respectively depth and width of
the web panel.
[FIGURE 5 OMITTED]
The ultimate strength, [V.sub.s] of a straight plate girder
containing centrally located circular or square opening can be obtained
as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (10)
where [([[tau].sub.cr]).sub.red] is the reduced value of elastic
critical shear stress of the web containing a hole given as:
[([[tau].sub.cr]).sub.red] = k[[[pi].sup.2]E/12(1 -
[v.sup.2])][(t/d).sup.2]. (11)
Narayanan and Der Avanessian (1983a, b) have demonstrated that in
calculating [([[tau].sub.cr]).sub.red], the value of k appropriate to a
web fixed at its edges should be used. This is justified as, when there
is a hole, the stiffness of the flange is much higher than that of the
web and the behaviour of the web plate is closer to the one having
encastre' supports. The value of k is obtained as:
k = [k.sub.o](1 - [[d.sub.o]/d]), (12)
where:
[k.sub.o] = 8.98 + 5.6[(d/b).sup.2] for [b/d] > 1.0. (13)
Membrane stress can be evaluated as:
[[sigma].sup.y.sub.t] =
[-3/2][([[tau].sub.cr]).sub.red]sin2[[theta].sub.d] +
[square root of ([[sigma].sup.2.sub.yw] +
[[tau].sup.2.sub.cr]{[([3/2]sin 2[theta]).sup.2] - 3})]. (14)
The contribution by concrete slab to shear capacity of the girder
is determined based on the assumed failure mechanism shown in Figure 6.
Since there is no observed buckled pattern of webs for composite plate
girders with web openings, the buckled pattern corresponding to girder
with solid webs is assumed approximately for this case.
[FIGURE 6 OMITTED]
Ultimate shear strength, [V.sub.s of the straight steel girder with
web openings, is obtained by Eqn (10) and the corresponding values for
composite girder, straight or curved, are determined using Eqns (5) and
(8), respectively. The accuracy of the method is assessed by considering
the girder C1 referred in the previous section. Circular or square
openings, of size ranging from 0.1D to 0.5D, D being the girder depth,
were introduced in each of the web panels and the girders with web
openings analyzed using the proposed method. The straight girder values
of [V.sub.s], [V.sub.c] and [V.sub.ult] and the curved girder values for
the ten girders, [V.sub.ult(curved)] are summarized in Table 2. The
corresponding shear strength values, [V.sub.ultFEA] for the girders
obtained from the finite element analyses using LUSAS, are also listed
in Table 2 for comparison. It can be seen from the table that there is a
close agreement between the finite element and predicted values, the
maximum deviation being 6% thus confirming the accuracy of the proposed
method.
4. Horizontally curved composite plate girders with trapezoidally
corrugated webs
Corrugated profile in webs similar to those shown in Figure 7
provides a kind of uniformly distributed stiffening in the transverse
direction of a girder and also increases the out-of-plane stiffness and
buckling strength. Compared to plate girders with stiffened flat webs a
girder with a corrugated web enables the use of thinner webs, thus a
higher load-carrying capacity is achieved for a lower cost. Girders with
trapezoidally corrugated webs are considered in this paper.
Eqn (1) is modified to account for corrugated webs assuming tension
zone developing across the diagonal of a web panel. Global (overall)
buckling of a corrugated web panel is characterized by diagonal buckling
over several corrugation panels. The critical shear stress for this mode
is estimated by considering the corrugated web as an orthotropic plate.
The critical shear stress of this mode, [[tau].sub.crg] is obtained as
(Galambos 1998):
[[tau].sub.crg] =
[k.sub.g][[([D.sub.y][D.sup.3.sub.x]).sup.0.25]/[d.sup.2][t.sub.w]],
(15)
where: [k.sub.g] is the global shear buckling coefficient which
depends solely on the web top and bottom constraints. [k.sub.g] is 36
for steel girders (Luo, Edlund 1994; Sayed-Ahmed 2001, 2005). The
factors [D.sub.x] and [D.sub.y] are the flexural stiffness per unit
corrugation about the longitudinal and transverse axes, respectively
(Fig. 7). [D.sub.x] and [D.sub.y] are given as (Elgaaly et al. 1996;
Johnson, Cafolla 1997):
[D.sub.x] = [E/[[b.sub.h] +
[d.sub.tz]]]([[b.sub.h][t.sub.w][h.sup.2]/4] + [[t.sub.w][h.sup.3]/12
sin[beta]]); (16)
[D.sub.y] = ([[b.sub.h] + [d.sub.tz]]/[[b.sub.h] +
a])(E[t.sup.3.sub.w]/12). (17)
[FIGURE 7 OMITTED]
It is assumed in this case that buckling occurs diagonally over a
number of corrugations within a web panel as shown in Figure 8 and,
therefore, global buckling governs the collapse of the girder. The
failure of the girder occurs when the buckled portion of the corrugated
web panel yields and a mechanism is developed with hinges formed in the
flanges. The ultimate shear strength of the steel part of the girder is
assumed to be given by:
[V.sub.s] = [[tau].sub.crg]d[t.sub.w] +
[[sigma].sup.y.sub.t][t.sub.w] [sin.sup.2] [theta](d cot [theta] - b) +
2c[t.sub.w][[sigma].sup.y.sub.t] [sin.sup.2] [theta]. (18)
The value of [theta] and c can be evaluated by Eqns (2) and (3),
respectively. [[sigma].sup.y.sub.t] is given as:
[[sigma].sup.y.sub.t] = [-3/2][[tau].sub.crg]sin2[theta] + [square
root of ([[sigma].sup.2.sub.yw] + [[tau].sub.crg.sup.2]{[([3/2]sin
2[theta]).sup.2] - 3})]. (19)
[FIGURE 8 OMITTED]
The contribution by concrete slab to shear capacity of the girder
is determined based on the assumed failure mechanism shown in Figures 2
and 3 and, it is given by Eqn (6). The ultimate strength obtained as
above for a straight girder is multiplied by a constant [K.sub.c] given
in Eqn (7) to account for the curvature in a curved girder.
Girders with different values of corrugation width, corrugation
height and corrugation inclinations as shown in Table 3 were analyzed
for ultimate strength using the proposed method in order to verify the
accuracy of the method. For all the fifteen composite girders, CT1-CT15,
the contributions by steel part of the straight girders, [V.sub.s] were
determined first and then the increase in shear resistance due to
concrete slab, [V.sub.c] obtained. The combined values [V.sub.ult] give
the ultimate shear strength for straight girders which when multiplied
by the curvature factor [K.sub.c] yields the corresponding values for
curved girders, [V.sub.ul(curved)]. These values of ultimate strength
and the corresponding values determined by finite element analyses are
summarized along with the comparison between the two values in Table 3.
It can be seen from the table that there is a close agreement between
the finite element and predicted values, the maximum deviation being 8%
thus confirming the accuracy of the proposed method. Despite the
approximations the proposed method is found to be capable of predicting
the shear capacity of the horizontally curved composite plate girders
with corrugated webs to an acceptable accuracy.
Conclusions
An approximate method to predict the ultimate shear strength of
horizontally curved composite plate girders has been presented. Girders
with solid webs, perforated webs and trapezoidally corrugated webs have
been considered in the study. The proposed method is based on the
Cardiff model for the collapse behaviour of straight plate girders which
is governed by tension field action in the web panels. Effects of
parameters such as curvature in plan, composite action between the steel
part of the girder and the reinforced concrete slabs, openings in the
web and corrugated webs have been examined in the study. A number of
girders covering the parameters mentioned above have been analyzed using
the proposed method and the ultimate shear strength has been predicted.
These results are compared with the corresponding values determined by
the nonlinear finite element analyses in order to assess the accuracy of
the proposed method. It has been observed from the comparison that the
proposed method is capable of predicting the ultimate shear strength
with an acceptable accuracy and hence the method could be used for
design office applications. It should, however, be noted that the
proposed method and the accompanying verifications are purely based on
theoretical studies and, the accuracy of web capacity predicted needs
confirmation through experimental studies. The authors hope to report
such experimental studies in due course.
doi: 10.3846/13923730.2013.801913
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Final report to Professional Services Industries, Inc. and Federal
Highway Administration.
Nandivaram E. SHANMUGAM (a), Mohammed A. BASHER (b), Khalim A.
RASHID (a)
(a) Department of Civil and Structural Engineering, Universiti
Kebangsaan Malaysia, 43600 UKMBangi, Selangor, Malaysia
(b) Department of Building and Structural Engineering Technology,
Technical College in Mosul, 101 Festival str., Iraq
Received 23 Feb 2012; accepted 08 Jun 2012
Corresponding author: Nandivaram E. Shanmugam
E-mail:
[email protected]
Nandivaram E. SHANMUGAM. Professor at the Department of Civil and
Structural Engineering, Universiti Kebangsaan Malaysia (UKM). He
obtained his PhD degree from the University of Wales (Cardiff) in 1978.
He has taught at undergraduate and graduate levels for more than 45
years. He has published more than 200 scientific papers in international
journals and conference proceedings. He is a member of the editorial
board of Journal of Constructional Steel Research, Thin-Walled
Structures, Journal of Structural Stability and Dynamics, International
Journal of Steel Composite Structures, International Journal of Steel
Structures and IES Journal Part A: Civil and Structural Engineering. He
is a Chartered Engineer (CEng), Fellow of the Institution of Structural
Engineers, London, (FIStructE), Fellow of the Royal Institution of Naval
Architects (FRINA), Fellow of the American Society of Civil Engineers
(FASCE), Fellow of the Institution of Engineers, Singapore (FIES) and
Fellow of the Institution of Engineers, India (FIEI). Research interests
include steel plated structures, steel-concrete composite construction,
long-span structures and connections, cold-formed steel structures,
elastic and ultimate load behaviour of steel structures, etc.
Mohammed A. BASHER. Senior Lecturer at the Department of Building
and Structural Engineering Technology, Technical College-Mosul, Iraq. He
obtained his PhD degree from the National University of Malaysia (UKM)
in 2010. He has taught at undergraduate and graduate levels for more
than a year. He has published 11 scientific papers in international
journals and conference proceedings. He is a Chartered Engineer (CEng),
Fellow of the National Institution of Technical Education, and Fellow of
the Institution of Engineers, Iraq. Research interests include steel
plated structures, steel-concrete composite construction.
Khalim A. RASHID. Associate Professor at the Department of Civil
and Structural Engineering, Universiti Kebangsaan Malaysia (UKM). He
obtained
his PhD degree from the University of Birmingham in 2000. He taught
at undergraduate and graduate levels for more than 20 years. He has
published more than 40 scientific papers in national and international
journals and conference proceedings. He is a corporate member of the
Board of Engineers Malaysia (BEM) and Institution of Engineers Malaysia
(IEM). His research interests include reinforced concrete structures,
structural steelwork, bridge engineering, and industrialized building
systems.
Table 1. Comparison of results for girders with solid webs
Girders [b.sub.f] [V.sub.c] [V.sub.s] [V.sub.
mm kN kN ultpred]
kN
C1 203.3 565 1588 2153
546.6 834 1801 2635
C1-C 203.3 798 1696 2494
546.4 1007 1946 2953
C2 203.3 482 1503 1985
557.3 681 1742 2423
C2-C 203.3 879 1677 2556
556.3 952 1837 2789
Girders [V.sub. [V.sub. [V.sub.
ult(curved)] ultFEA] ult(curved)]/
kN kN [V.sub.
ultFEA]
C1 1433 1419 1.01
1754 1703 1.03
C1-C 1660 1643 1.01
1966 1927 1.02
C2 1317 1304 1.01
1608 1546 1.04
C2-C 1696 1713 0.99
1851 1833 1.01
Table 2. Comparison of results for girders with perforated webs
Girder [theta] c [T.sub.l] [V.sub.c] [V.sub.s]
mm mm kN kN
C1-Cr 0.1D 17 713 73 653 1900
C1-Cr 0.2D 15 769 71 587 1686
C1-Cr 0.3D 14 795 69 525 1527
C1-Cr 0.4D 12 894 67 438 1225
C1-Cr 0.5D 10 1041 65 419 1112
C1-Sq 0.1D 15 794 71 604 1950
C1-Sq 0.2D 13 873 70 578 1662
C1-Sq 0.3D 11 991 67 498 1587
C1-Sq 0.4D 10 1171 65 438 1117
C1-Sq 0.5D 9 1289 64 404 1040
Girder [V.sub.ult] [V.sub. [V.sub. [V.sub.
kN ult(curved)] ultFEA] ult(curved)]/
kN kN [V.sub.
ultFEA]
C1-Cr 0.1D 2553 1699 1668 1.02
C1-Cr 0.2D 2273 1513 1480 1.02
C1-Cr 0.3D 2052 1366 1320 1.03
C1-Cr 0.4D 1663 1107 1130 0.98
C1-Cr 0.5D 1531 1019 1040 0.98
C1-Sq 0.1D 2554 1700 1644 1.03
C1-Sq 0.2D 2249 1497 1457 1.03
C1-Sq 0.3D 2085 1388 1302 1.06
C1-Sq 0.4D 1555 1035 1093 0.95
C1-Sq 0.5D 1444 953 1005 0.95
Table 3. Comparison of shear strength for girders with solid webs
Girder [beta] h [b.sub.h] a [V.sub.c] [V.sub.s]
mm mm mm kN kN
CT1 37 300 20 501 823 2578
CT2 37 300 100 501 858 2850
CT3 37 300 300 501 814 2504
CT4 37 300 500 501 796 1999
CT5 37 300 750 501 774 1855
CT6 45 100 500 141 721 1864
CT7 45 200 500 283 756 1983
CT8 45 300 500 424 822 2148
CT9 45 400 500 566 862 2256
CT10 45 500 500 700 924 2253
CT11 90 100 500 100 718 2282
CT12 90 200 500 200 792 2487
CT13 90 300 500 300 846 2568
CT14 90 400 500 400 904 2769
CT15 90 500 500 500 948 3014
Girder [V.sub.ult] [V.sub. [V.sub. [V.sub.
kN ult(curved)] ultFEA] ult(curved)]/
kN kN [V.sub.
ultFEA]
CT1 3401 2245 2388 0.94
CT2 3708 2447 2523 0.97
CT3 3318 2190 2274 0.96
CT4 2795 1845 1922 0.96
CT5 2629 1735 1827 0.95
CT6 2585 1706 1796 0.95
CT7 2739 1808 1903 0.95
CT8 2970 1960 2042 0.96
CT9 3118 2058 2144 0.96
CT10 3177 2097 2231 0.94
CT11 3000 1980 1833 1.08
CT12 3279 2164 2004 1.08
CT13 3414 2253 2187 1.03
CT14 3673 2424 2309 1.05
CT15 3962 2615 2467 1.06
