ACI STRUCTURAL JOURNAL
TECHNICAL PAPER
Title no. 93-S37
Design Procedure for Reinforced Concrete Beams with
Large Web Openings
by Kiang-Hwee Tan and Mohammad A. Mansur
Based on test evidence, guidelines for the placement of large web openings
in reinforced concrete beams are given, following which a simple design
procedure is suggested. Generally, openings should be positioned so that
chords have sufficient concrete area to develop the ultimate compression
block in flexure and adequate depth to provide effective shear reinforcement. They should not be deeper than one-half the beam depth and should
be located not closer than one-half the beam depth from supports or concentrated loads.
For analyses for elastic bending moments and shear forces by conventional methods, the recommended procedure uses an equivalent shear stiffness incorporating an effective length for the opening and considers the
applied shear to be carried in proportion to the flexural stiffness of the
chords. The design of chords for strength follows ACI code provisions.
Cracking at the opening is controlled by proper detailing, while deflections
are calculated using the same analysis procedure but considering cracked
moment of inertia and checking against code requirements.
Keywords: beams (supports); cracking (fracturing); deflection; openings;
reinforced concrete; serviceability; shear tests; stiffness; strength; structural design.
In tall building construction, there is a trend towards
providing openings through the webs of reinforced concrete
beams to accommodate service ducts. This results in a
reduction in the amount of dead space above the false ceiling
and hence a more compact and economical design. In the
past, several methods1-5 have been proposed for strength
design of beams with large web openings under various
loading combinations. However, these methods require the
magnitudes of forces and moments at the center of the
opening to be predetermined, and can therefore be readily
applied to statically determinate beams only. In continuous
beams that generally occur in practice, reduction in stiffness
due to the provision of openings through webs causes a
redistribution of internal forces and moments, the amount of
which needs to be evaluated before a design can proceed.
To facilitate the design process, the authors6 have earlier
proposed the use of an equivalent stiffness for the opening
segment of a beam whereby analysis of a continuous beam
with large web openings can be reduced to a routine procedure.
A comparison of the method with available test results
shows good agreement with respect to both the strength and
deflection of the beams.6,7 With this tool in hand, a procedure suitable for the complete analysis and design of a beam
with large web openings can be formulated. The procedure
is described in this paper and illustrated with an example.
404
RESEARCH SIGNIFICANCE
This paper provides a useful procedure for the complete
analysis and design of reinforced concrete beams with large
web openings. General guidelines for the placement of openings are given. Supported by test results, the procedure
checks for ultimate strength and deflections in both the short
and long term. It also insures that cracking at the opening is
controlled by proper reinforcement detailing.
GENERAL GUIDELINES
A review of the literature1-7 on the behavior and strength
of beams with web openings indicates that the following
guidelines can be used to facilitate the selection of the size
and location of web openings (refer to Fig. 1):
1. For T-beams, openings should preferably be positioned
flush with the flange for ease in construction. In the case of
rectangular beams, openings are commonly placed at middepth of the section, but they may also be placed eccentrically
with respect to depth. Care must be exercised to provide
sufficient concrete cover to the reinforcement for the chord
members above and below the opening. The compression
chord should also have sufficient concrete area to develop
the ultimate compression block in flexure and have adequate
depth to provide effective shear reinforcement.
2. Openings should not be located closer than one-half the
beam depth D to the supports to avoid the critical region for
shear failure and reinforcement congestion. Similarly, positioning of an opening closer than 0.5D to any concentrated
load should be avoided.
3. Depth of openings should be limited to 50 percent of
overall beam depth.
4. The factors that limit the length of an opening are the
stability of the chord members, in particular the compression
chord, and the serviceability requirement of deflection.
When the opening gets bigger, it is preferable to use multiple
openings providing the same passageway instead of using a
single opening.
5. When multiple openings are used, the post separating
two adjacent openings should not be less than 0.5D to insure
that each opening behaves independently.
ACI Structural Journal, V. 93, No. 4, July-August 1996.
Received July 22, 1994, and reviewed under Institute publication policies. Copyright © 1996, American Concrete Institute. All rights reserved, including the making of
copies unless permission is obtained from the copyright proprietors. Pertinent discussion will be published in the May-June 1997 ACI Structural Journal if received
by January 1, 1997.
ACI Structural Journal/July-August 1996
ACI member Kiang-Hwee Tan is a senior lecturer in civil engineering at the National
University of Singapore. He has been involved in an extensive study on the behavior of
reinforced concrete beams with large web openings. His research interests include
external prestressing in structures, high-performance construction materials, durability design, ferrocement, and fiber reinforced concrete.
Mohammad A. Mansur, FACI, is an associate professor of civil engineering at the
National University of Singapore. In 1988 and 1992 he was a corecipient of ACI's
Maurice P. Van Buren Structural Engineering Award. He is a member of ACI Committee 549, Ferrocement and Other Thin Reinforced Products, and is president of ACI's
Singapore chapter. His research interests include reinforced concrete, fiber concrete,
and ferrocement.
RECOMMENDED DESIGN PROCEDURE
In the design procedure proposed herein, the ACI Code8
has been followed throughout unless otherwise stated. In
general, the design of reinforced concrete structures involves:
1. Structural analysis, whereby the structure is analyzed to
determine the distribution of shear forces and moments due
to ultimate loads. All possible loading combinations are
considered and bending moment and shear force envelopes
are determined accordingly.
2. Strength design, wherein the critical sections are
designed for ultimate strength in bending and in shear.
Strength requirements are fulfilled throughout the whole
structure.
3. Serviceability design, wherein it is insured that the
structure performs its intended functions satisfactorily under
working loads.
Structural analysis
In the case of a statically determinate beam, shear force
and bending moment envelopes can be obtained from statics.
For continuous beams, the method given in Reference 6 can
be followed; that is, the member containing an opening is
considered as a nonprismatic beam with different crosssectional properties: those of a solid section and of the
equivalent section for opening segments. Provided that the
guidelines described earlier are complied with, the problem
can be reduced to an ordinary continuous beam analysis by
introducing two extra joints for each web opening. For this
purpose, it is necessary to estimate the stiffnesses of the
various segments comprising the member.
Stiffnesses of beam segments—Stiffness calculations may
be based on the gross concrete section as permitted by the
ACI Code.8 The modulus of elasticity of concrete according
to Section 8.5.1 of the Code can be taken as
E c = 4730 f c′
(1)
where fc′ is the cylinder compressive strength of concrete (in
MPa) and shear modulus can be taken as
Ec
G = ------------------2(1 + ν)
(2)
where Poisson’s ratio ν for concrete is taken as 0.2 as recommended by BS 8110.9 In calculating the equivalent flexural
stiffness of an opening segment (EI)eq , the moment of inertia
based on the concrete section minus the void due to the
opening should be used.
The equivalent shear stiffness for an opening segment can
be obtained as6
12E c ( I gt + I gb )
( GA ) eq = -----------------------------------2
le
(3)
where Igt and Igb are the gross moment of inertia for top and
bottom chord members, respectively, and le is the effective
length of an opening that has been empirically established as7
lo
l e = ------------------------d 1.5
1 – ⎛ ----o-⎞
⎝ D⎠
(4)
where lo, do, and D are opening length, opening depth, and
beam depth, respectively. The shearing deformation of the
solid segments is generally ignored.
Bending moment and shear force envelopes—The beam
can be analyzed for all possible load combinations by any
elastic method to obtain shear force and bending moment
envelopes. Here, the direct stiffness method10 is recommended because it can be easily modified to include shear
deformations and easily translated into a computer algorithm.
Design for strength
Knowing the bending moment and shear force envelopes,
the solid segments of the beam can be designed in the usual
manner. The recommended design process for the opening
segment is based on the observed Vierendeel behavior of
chord members at an opening. That is, consistent with test
results,1-7 contraflexure points are assumed at midspan of
chord members for which the axial load is obtained by
dividing the beam moment at the center of the opening by the
distance between the plastic centroids of the chord members.
The shear force acting at the center of the opening is distrib-
Fig. 1—Guidelines for locating web openings.
ACI Structural Journal/July-August 1996
405
uted between the chord members according to their relative
flexural stiffnesses. Such an assumption has been found to
give a realistic distribution of applied shear7 and simplifies
calculation. The moments at the ends of the chord member
are then calculated from statics. The steps involved are
summarized as follows.
Forces and moments in chord members—Determine the
ultimate design bending moment Mm and shear force Vm at
the middle of the opening segment from bending moment
and shear force envelopes, and calculate axial forces Nt and
Nb (positive for compression) acting, respectively, in the top
and bottom chords as
M
N t = -------mz
(5)
Nb = –Nt
(6)
where z is the distance between the plastic centroids of the
top and bottom chords.
Distribute the applied shear between the top and bottom
chords as
I gt ⎞
V t = V m ⎛ -----------------⎝I + I ⎠
gt
gb
(7)
I gb ⎞
V b = V m ⎛ -----------------⎝I + I ⎠
gt
gb
(8)
2
(9)
2
ωl Vt l o
M 2 = – --------o + --------8
2
(10)
Vb lo
M 3 = – ---------2
(11)
Vb lo
M 4 = ---------2
(12)
Fig. 2—Free-body diagram of opening segment.
406
Kl u
M 1b
-------- < 34 – 12 --------r
M 2b
(13)
in which effective length factor K is taken as 1, lu is the
unsupported length of the compression chord, and r is the
radius of gyration. The values of M1b and M2b can be taken
as M3 and M1, respectively, with the signs as dictated by
Section 10.11.4.1 of the ACI Code.8 According to Section
10.11.3 of the Code,8 r can be taken as
r = 0.3dc
where Igt and Igb are the gross moment of inertia of the top
and bottom chords, respectively.
Calculate moments at the ends of chord members from
statics (refer to Fig. 2)
ωl Vt l o
M 1 = – --------o – --------8
2
where ω is the uniformly distributed load acting directly on
the top chord and M is the moment. Subscripts 1, 2, 3, and 4
designate the opening corners, as shown in Fig. 2.
Stability of compression chord—When the section being
analyzed is a T-beam, the effective width of flange in determining the properties and capacities of the compressive strut
should not exceed the limits set by the ACI Code8 in Section 8.7.
Where the opening segment is subjected to positive bending
(for example, in the midspan region of a continuous beam),
the compression (top) chord will be restrained by the continuity of the slab and thus may be considered as a member
braced against sidesway for which, according to Section
10.11.4.1 of the Code,8 the effects of slenderness may be
neglected when
(14)
where dc can be taken as the depth of the compression chord.
However, when an opening segment is subjected to negative
bending (for example, in between the inflection points and
the support of a continuous beam), the compression (bottom)
chord should be considered as a member unbraced against
sidesway for which, according to Section 10.11.4 of the
Code,8 the effects of slenderness may be neglected when
Kl u
-------- < 22
r
(15)
If Eq. (13) or (15) are not satisfied, the moment magnification method as described in Section 10.11.5 may be used
to design the compression chord. However, it is suggested
that the dimensions of the chord be revised so as to eliminate
the effects of slenderness.
Design of longitudinal reinforcement for chord members—
The longitudinal reinforcement in the top and bottom of the
solid section adjacent to the opening should be continued
throughout the opening segments. Additional reinforcement
required to resist the combined moment and axial force in
each chord member is designed, and, as a trial, it could be
such that each chord is symmetrically reinforced. With the
reinforcement for the chord members so decided, the corresponding idealized column interaction diagrams can be
constructed by the method of strain compatibility. The critical
combinations of bending moment and axial load for the chord
members as determined earlier are then plotted in the interaction
diagrams. If all the combinations fall within the appropriate
interaction diagrams, the reinforcement provided will be sufficient. Otherwise, a revision of reinforcement is necessary. Also,
the flexural capacity of the top chord should be sufficient to
support any external loading directly applied to it.
Design of shear reinforcement for chord members—The
shear forces carried by the top and bottom chords are given
by Eq. (7) and (8), respectively. Knowing these forces, the
required amount of reinforcement can be designed in a
ACI Structural Journal/July-August 1996
manner similar to reinforced concrete beams and slabs.
However, according to Section 11.3 of the Code,8 the effects
of axial forces in the chord members must be accounted for
in design. For a T-beam where the opening is placed flush
with the flange, the top chord can be considered as a slab.
Although the flange may be too shallow for effective placement of shear reinforcement, the shear stresses are usually
low and, consequently, shear reinforcement would not be
necessary in the top chord.
Design of post between openings—The post should be
designed as a solid segment to carry the total applied shear.
The contribution of the corner reinforcement at the two adjacent openings should be ignored.
Design for serviceability
The two important serviceability requirements to be met
are cracking and deflection.
Cracking—Assuming that the crack control requirements
of the solid segments are met either by proper reinforcement
detailing or by physical calculation, the following crack
control provisions are recommended for the critical sections
at corners of the opening. At each vertical edge of the
opening, a combination of vertical stirrups and diagonal bars
would be used with a shear concentration factor η of 2 such
that at least 75 percent of the shear resistance is provided by
the diagonal bars.4 Thus, for each side of the opening, the
required area of vertical stirrups Av is given by
( ηV )A v = 0.25
---------------------φf yv
(16)
in which V, φ, and fyv are the design shear, capacity reduction
factor, and yield stress of stirrups, respectively. The vertical
stirrups should be placed as close to the edge of the opening
as permitted by the required concrete cover. The required
area of diagonal reinforcement Ad is given as
( ηV )A d = 0.75
---------------------φf yd sin ϕ
(17)
where fyd is the yield stress and ϕ is the angle of inclination of
the diagonal bars to the beam axis. To avoid confusion during
construction and to account for any possible load reversal, the
same amount of diagonal reinforcement should be provided
both at the top and bottom corners of the opening.
Deflections—The indirect way of satisfying the serviceability requirement of deflection by limiting the span-effective
depth ratio is not valid for a beam with openings. Therefore,
an estimate of the actual service load deflection is necessary.
For this purpose, the method used for the analysis of the
beam at ultimate load may be used. Since the reinforcement
details are fully known, a conservative estimate of service
load deflection may be obtained and checked against code
requirements by using the cracked moment of inertia of
various segments. The equivalent shear stiffness of the
opening segment [Eq. (3)] can also be calculated using the
cracked moment of inertia of the chord members.
NUMERICAL DESIGN EXAMPLE
A three-span, continuously reinforced concrete T-beam
containing a web opening in the interior span is shown in
Fig. 3. This represents a typical feature of an office floor in
which the air-conditioning ducts run along the corridor
between two rows of office rooms. The beam carries a dead
load Gk, including self-weight of 14 kN/m and a uniformly
imposed load Qk of 11.3 kN/m. The material properties are:
fc′ = 30 MPa; fy (longitudinal steel) = 460 MPa; and fyv (transverse steel) = 250 MPa. Following the proposed procedure,
a suitable design for the beam with particular emphasis on
the opening segment is given as follows.
Step 1: Structural analysis
a) Calculate stiffnesses of beam segment
From Eq. (1) and (2), Ec = 26,000 MPa and G = 10,800
MPa. Since the effective slab width as T-beam flange =
1500 mm (ACI Code,8 Section 8.10.2), the gross moment of
inertia of the solid section, opening section, top chord, and
bottom chord are, respectively, Igs = 1.03 × 1010 mm4, Igo =
9.81 × 109 mm4, Igt = 4.22 × 108 mm4, and Igb = 3.91 × 108
mm4. Also, effective length of opening lo = 1114.5 mm
[refer to Eq. (4)] and cross-sectional area of solid segment
Fig. 3—Three-span continuous beam with web opening.
ACI Structural Journal/July-August 1996
407
Aso = 3.6 × 105 mm2. Hence, for the solid segment EcIgs =
2.67 × 1014 Nmm2 and GAso = 3.89 × 109 N, while for
opening segment (EI)eq = 2.54 × 1014 Nmm2 and (GA)eq = 2.03
× 108 N [refer to Eq. (3)].
b) Obtain bending moment and shear force
envelopes
The proposed method of analysis was used to analyze the
beam for all possible factored load combinations (refer to
Fig. 4). The equivalent segmented beams for the analysis are
shown in Fig. 5 and the results are presented in Fig. 6 with 0
percent moment redistribution at ultimate.
Table 1—Summary of flexural design for critical
solid sections
Location
Moment, kNm
Main reinforcement
Span AB
Support B
172.8
–184.8
4Y16 (bottom)
12Y10 (top)
5Y12 (top), 12Y10 (bottom)
Span BC
94.8, –22.3
Support C
–194.2
12Y10 (top)
Span CD
170.7
4Y16 (bottom)
Note: Shear reinforcement consisting of R8 at 200-mm spacing to be provided
throughout beam.
Step 2: Design for strength
The critical solid sections were designed for bending and
shear in the usual manner. Table 1 summarizes the final
output. For the opening segment, the axial loads and shear
forces in chord members (refer to Fig. 2) are evaluated from
bending moment Mm and shear force Vm at the center of
opening for each loading pattern using Eq. (5) through (8)
and shown in Table 2, taking z = 400 mm. The secondary
moments at the critical end sections calculated using Eq. (9)
through (12) are shown in Table 3.
The opening segment is subjected to negative bending;
therefore, compression chord is considered as unbraced
against sidesway. Since r = 0.3 × 250 = 75 mm [refer to Eq. (14)]
and lu = 900 mm, Klu/r = 1 × 900 / 75 = 12 < 22 [refer to
Eq. (15)]; hence, the compression chord is satisfactory with
regard to stability.
Longitudinal reinforcement—From Table 1, it is seen that
the solid section adjacent to the opening is provided with 12
Y10 bars (high-strength deformed bars of 10-mm diameter) and four Y16 (high-strength deformed bars of 16-mm
diameter). Hence, for symmetrical arrangement of reinforcement, provide the top chord with an additional 12 Y10 bars
at the bottom and the bottom chord with an additional four
Y16 bars at the top as a first trial.
For the tension chord, only the bending-tension interaction
curves are needed, while for the compression chord bendingcompression interaction curves are required. The interaction
curves are shown in Fig. 7 together with the bending momentaxial force combinations for the end sections of the chord
members. Since all the points fall within the respective interaction diagrams, the amount of reinforcement provided is
satisfactory.
Fig. 4—Loading combination: (a) Loading Case 1; (b)
Loading Case 2; and (c) Loading Case 3.
Fig. 5—Equivalent segmented beam: (a) continuous beam
with opening; (b) equivalent segmented beam for ultimate
load analysis; (c) equivalent segmented beam for service
load analysis.
408
Fig. 6—Shear force and bending moment envelopes
ACI Structural Journal/July-August 1996
The uniformly distributed loading is carried directly by the
top chord over the opening segment and creates a midspan
moment of 38.8 × 0.92/24 = 1.31 kNm that is less than the
pure bending moment capacity of 49.6 kNm for the top chord
[refer to Fig. 7].
Shear reinforcement—The maximum shear of 95.7 kN at
the opening center occurs for Loading Cases 1 and 3 [refer to
Fig. 6]. Therefore, from Table 2, design shear Vt = 49.8 kN
and Vb = 45.9 kN. The top chord is subjected to combined
bending and axial tension; thus, the shear strength of
concrete is [ACI Code,8 Section 11.3.2.3]
( – 174.3 )
φV c = 0.85 × 0.17 1 + 0.29 -------------------------1500 ( 150 )
30 ( 1500 ) ( 95 ) = 112.7 kN < 49.8 kN
The top chord is treated as a slab and no links are required.
The bottom chord is subjected to combined bending and
axial compression; thus, the shear strength of concrete can be
taken as [ACI Code,8 Section 11.3.1.2]
( 58 )
φV c = 0.85 × 0.17 1 + 0.073 ----------------------300 ( 250 )
30 ( 300 ) ( 192 ) = 45.6 kN > 45.9 kN
The bottom chord is treated as a beam; therefore, a
minimum amount of links must be provided. The maximum
spacing limit is d/2 = 192/2 = 96 mm [ACI Code,8 Section
11.5.4.1]. Therefore, provide stirrups of R8 (mild steel bars
of 8-mm diameter) at 90-mm spacing.
The previous design has been based on bending moment
and shear force distributions derived using gross section
properties. With the reinforcement detail so determined, a
more refined analysis may be carried out using the cracked
section properties if necessary.
Step 3: Design for serviceability
a) Cracking
Crack control requirements of the solid segments are met
by proper reinforcement detailing. For the opening segment,
the maximum shear at the left edge occurs for Loading Case 1
and 3. That is, Vt = 95.7 + 38.8 × 0.45 = 113.2 kN. Assuming
η = 2, ϕ = 45 deg, and fyd and fyv = 460 MPa, the required area
of vertical stirrups and diagonal bars [refer to Eq. (16) and
(17)] is
2
0.75 ( 2 ) ( 78.2 )
A v = ---------------------------------------------- = 424.3 mm
0.85 ( 460 ) ( sin 45° )
Therefore, provide one Y10 vertical stirrup (area of 157.1
mm2) at 40 mm away from the edge of the opening and four
Y12 diagonal bars (area of 452.4 mm2 at both top and bottom
corners.
b) Deflections
Short-term deflection—Assuming cracked moment of
inertia for various segments, the beam is analyzed using the
proposed method for a service load of (Gk + Qk) = 25.3 kN/m.
The maximum deflection is found to occur in Span AB at
2.98 m from Support A with values of 10.3, 5.7, and 4.6 mm,
respectively, due to the total load, dead load only, and live
load only. According to the ACI Code,8 Section 9.5.2.6, the
allowable short-term deflection for floors not supporting or
attached to nonstructural elements likely to be damaged by
large deflections is L/360 = 7000/360 = 19.4 mm > 4.6 mm;
hence, the design is satisfactory.
Table 2—Calculated shear and axial forces at
midspan of chord members
Loading
case
Mm, kNm
Vm, kN
Vt, kN
Vb, kN
Nt = –Nb,
kN
1
2
–69.7
–81.8
95.7
48.3
49.8
25.1
45.9
23.2
174.3
204.5
3
–23.2
95.7
49.8
45.9
58.0
Table 3—Moments at critical end sections of chord
members
Loading case
1
Top chord
M2, kNm
M1, kNm
Bottom chord
M3, kNm
M4, kNm
–22.0
14.1
–19.5
2
–8.4
5.6
–7.6
19.5
7.6
3
–23.0
15.1
–20.7
20.7
2
0.25 ( 2 ) ( 113.2 )
A v = ------------------------------------- = 144.7 mm
0.85 ( 460 )
2
0.75 ( 2 ) ( 113.2 )
A v = ---------------------------------------------- = 614.1 mm
0.85 ( 460 ) ( sin 45° )
Therefore, provide one Y10 vertical stirrup (area of 157.1
mm2) at 40 mm away from the edge of the opening and four
Y16 diagonal bars (area of 804 mm2) at both top and
bottom corners.
For the right edge of the opening, Vr = 95.7 – 38.8 × 0.45 =
78.2 kN. Hence, the required area of vertical stirrups and
diagonal bars [refer to Eq. (16) and (17)] are
2
0.25 ( 2 ) ( 78.2 )
A v = ---------------------------------- = 100 mm
0.85 ( 460 )
ACI Structural Journal/July-August 1996
Fig. 7—Linearized interaction diagrams for chord members.
409
Fig. 8—Reinforcement details at opening segment.
Long-term deflection—Assuming that 20 percent of the
live load is being sustained for a period of 5 years. Following
the ACI Code,8 Section 9.5.2.5, the additional long-term
deflection due to shrinkage and creep may be determined by
multiplying the immediate deflection caused by the
sustained load by a factor λ = 2/(1 + 50ρ′) where ρ′ is the
ratio of the compression steel at the location when maximum
deflection occurs. Since ρ′ = 942/(300 × 542) = 0.0058, the
value of λ = 1.55. For a sustained load of (1.0Gk + 0.2Qk) =
16.3 kN/m, the calculated maximum deflection is 6.6 mm;
hence, the total long-term deflection = 10.3 + 1.55 × 6.6 =
20.6 mm. This is less than the allowable long-term deflection
(ACI Code, Section 9.5.2.6) of L/240 = 7000/240 = 29.2 mm.
Therefore, the design is acceptable.
Step 4: Reinforcement detailing
The reinforcement details follow the normal detailing procedure for continuous beams. Fig. 8 shows the final arrangement
of reinforcement for the opening segment of the beam.
CONCLUSIONS
This paper gives design guidelines for the placement of
openings in the webs of reinforced concrete beams. The
analysis and design procedure recommended here is based
on the state-of-the-art information that had been verified by
experimental studies. It incorporates the provisions of the
ACI Building Code8 and is expected to yield a satisfactory
design for a reinforced concrete beam with large web openings that is subjected to combined bending and shear.
ACKNOWLEDGMENTS
This paper is based on research work that was supported by the National
University of Singapore under Research Grant No. RP 880619.
410
CONVERSION FACTORS
1 in. = 25.4 mm
1 ksi = 6.895 MPa
1 kip = 4.448 kN
REFERENCES
1. Nasser, K. W.; Acavalos, A.; and Daniel, H. R., “Behavior and Design of
Large Openings in Reinforced Concrete Beams,” ACI JOURNAL, Proceedings
V. 64, No. 1, Jan. 1967, pp. 25-34.
2. Barney, G. B.; Corley, W. G.; Hanson, J. M.; and Parmalee, R. A.,
“Behavior and Design of Prestressed Concrete Beams with Large Openings,”
PCI Journal, V. 22, No. 6, Nov.-Dec. 1977, pp. 32-61.
3. Mansur, M. A., “Combined Bending and Torsion in Reinforced Concrete
Beams with Rectangular Openings,” Concrete International: Design &
Construction, V. 5, No. 11, Nov. 1983, pp. 51-58.
4. Mansur, M. A.; Tan, K. H.; and Lee, S. L., “Design Method for
Reinforced Concrete Beams with Large Openings,” ACI JOURNAL,
Proceedings V. 82, No. 4, July-Aug. 1985, pp. 517-524.
5. Mansur, M. A., “Ultimate Strength Design of Beams with Large
Openings,” International Journal of Structures, V. 8, No. 2, July-Dec.
1988, pp. 107-125.
6. Mansur, M. A.; Huang, L. M.; Tan, K. H.; and Lee, S. L., “Deflections of
Reinforced Concrete Beams with Web Openings,” ACI Structural
Journal, V. 89, No. 4, July-Aug. 1992, pp. 391-397.
7. Tan, K. H., and Mansur, M. A., “Reinforced Concrete T-Beams with
Large Web Openings in Positive and Negative Moment Regions,” ACI
Structural Journal, V. 93, No. 3, May-June 1996, pp. 277-289.
8. ACI Committee 318, “Building Code Requirements for Reinforced
Concrete (ACI 318-89) and Commentary (ACI 318R-89),” American
Concrete Institute, Detroit, 1989, 353 pp.
9. British Standard Institution, “Structural Use of Concrete (BS 81101985),” Part 1, London, 1985, 55 pp.
10. Weaver, W., Jr., and Gere, J. M., Matrix Analysis of Framed Structures,
second edition, Van Nostrand Reinhold Co., New York, 1980, 491 pp.
ACI Structural Journal/July-August 1996