ACI STRUCTURAL JOURNAL TECHNICAL PAPER

Title no. 86-S3

Torsion in Concrete Deep Beams with an Opening

by Ali A. Akhtaruzzaman and Abul Hasnat

Twenty-six concrete deep beams with and without a transverse sional strength equations of shallow beams with and
opening were tested to failure under torsion. The variables were span- without openings, as given by Hsu 2 and Mansur and
depth ratio, concrete strength, and size and location of the opening.
The presence of an opening significantly reduced torsional strength.
Hasnat, 3 respectively.
For span-depth ratios greater than 3.0, the torsional strength of
beams remained practically constant. It increased significantly, how- RESEARCH SIGNIFICANCE
ever, as the span-depth ratio decreased. The crack inclinations on the The experimental results described herein show the
beam surfaces followed a similar trend and were also influenced by
concrete strength and opening size. A modification has been sug- effect of span-depth ratio on torsional strength of con-
gested to the strength equations of Mansur and Hasnat for shallow crete deep beams with and without an opening. Cur-
beams under torsion to include the effect of span-depth ratio for deep rent codes do not have any design provisions for deep
beams. beams under torsion. The torsional strength increases
significantly at span-depth ratios less than 3.0. The
Keywords: beams (supports); concretes; cracking (fracturing); deep beams;
deep beam effect results in steeper tension cracks at
openings; span-depth ratio; strength; torsion.
lower span-depth ratios, and consequently vertical as
Beams whose span-depth ratio is about 5 or less and well as longitudinal reinforcements are required, as with
that are loaded at the top or compression faces are used deep beams under shear.
occasionally as structural members. Such members are
categorized by the ACI Building Code 1 as deep beams. EXPERIMENTAL PROGRAM
Deep beams, like shallow members, may be subjected Twenty-six beams having a constant cross section of
to torsion, bending, and shear. Although a consider- 100 x 400 mm (4 x 16 in.) and lengths ranging from 0.7
able amount of research has been carried out on plain to 2.3 m (28 to 92 in.) were tested. The center-to-center
and reinforced concrete shallow beams, little has been span-depth ratio flh was varied from 1.0 to 5.0, corre-
reported on deep beams to predict their behavior under sponding to clear span-depth ratio f"!h of 0.75 to 4.75,
such loading combinations. The current ACI Building to study the relative behavior of deep beams, moder-
Code 1 has provisions for the design of deep beams sub- ately deep beams, and nearly shallow beams. The test
jected to flexure and shear only. No recommendations specimens consisted of solid beams as well as beams
have been made for the effect of torsion on deep with a transverse circular opening of 110 or 200 mm
beams, either acting alone or in combination with other (4.4 or 8.0 in.) diameter. In general, the opening was
types of loading. located symmetrically at midspan. In two beams, how-
This investigation studied the effect of torsion on ever, they were located eccentrically at midspan. Two
plain concrete deep beams with and without a trans- different grades of concrete having nominal compres-
verse opening in the web. Analysis of deep beams un- sive strengths of 20.7 and 34.5 MPa (3000 and 5000 psi)
der torsion is quite complex. Introduction of a trans- were used. All the beams were of plain concrete, except
verse opening in the beam web for the passage of ser- two in which reinforcements were used around the
vice ducts further complicates the analysis. Because of opening. Details of the beams are shown in Fig. 1.
the complexities involved, no theoretical analysis for The beams were grouped in seven different series,
predicting the torsional strength of deep beams has each designated by two letters and a two-digit number.
been attempted. Instead, an experimental approach has The A and B stand for nominal concrete strengths of
been adopted. The main parameters were span-depth
Received Mar. 16, 1987, and reviewed under Institute publication policies.
ratio, concrete strength, and opening size and location. Copyright© 1989, American Concrete Institute. All rights reserved, including
For predicting the torsional strength of deep beams, the the making of copies unless permission is obtained from the copyright propri-
etors. Pertinent discussion will be published in the November-December 1989
experimental data have been used to modify the tor- ACI Structural Journal if received by July I, 1989.

20 ACI Structural Journal I January-February 1989

Ali A. Akhtaruzzaman is an assistant professor of civil engineering at the King
cally loaded arm placed on the top of the beam at the
Abdulaziz University, Jeddah, Saudi Arabia. He obtained his PhD in 1973 from support, as shown in Fig. 2. For the plain concrete
Imperial College of Science and Technology, University of London, England. specimens, the inclination of the failure crack on all
His research interests are primarily in plain concrete, lightweight aggregate
concrete, and reinforced concrete structures.
four faces was recorded. For beams with reinforce-
ments around the opening, strain measurements of the
Abu/ Hasnat is a professor of civil engineering at the King Abdulaziz Univer- reinforcing bars were taken at different stages of load-
sity, Jeddah, Saudi Arabia. He obtained his PhD in 1965 from Texas A & M
University, College Station, Texas. His engineering experience includes design
ing.
and construction of highway bridges and multistoried structures, and his re-
search interests are primarily in lightweight aggregate concrete and reinforced
and prestressed concrete structures.
TEST RESULTS
All beams either of plain concrete or with reinforce-
ments around the opening failed suddenly, with a
20.7 and 34.5 MPa (3000 and 5000 psi), respectively. compression hinge forming at the lateral side of the
The P stands for plain concrete beams and the R stands beam. For plain concrete beams with an opening, the
for beams with reinforcements around the opening. The failure surface invariably passed through the opening,
two-digit numbers 00, II, and 20 indicate the diameter often along the diameter. However, for beams with re-
of the opening in em. Each individual specimen of a inforcements around the opening, the failure occurred
series is designated by a hyphenated digit after the se- at a solid unreinforced section, indicating adequate
ries number, indicating its span-depth ratio f!h. For strengthening of the opening section by the reinforce-
example, AP 20-3 represents a plain concrete beam ment scheme used.
having a nominal concrete strength of 20.7 MPa (3000
psi), with a transverse circular opening of 20 em (8 in.) Crack pattern
diameter and tested with flh of 3.0. Those beams with The failure-crack inclinations with the beam axis
an eccentric opening were marked with an asterisk * were different in the four faces of the beams, as can be
before the series number. seen from Table l. The crack inclination depended on
All beams were tested under pure torsion. Torque span-depth ratio, concrete strength, and diameter of the
was applied in small increments through an eccentri- opening. Fig. 3 through 6 show crack inclination plot-
ted against span-depth ratio for the four different faces
Series AP 00, BPOO of the beams of the different series.

l DI ho400mm Tension crack on vertical face - The angle of crack

inclination varied approximately from 42 to 65 deg for

Series AP11
span t (variable) J ,f--t b= 100mm
the different series tested, as shown in Fig_ 3. For any

S 180x 22.6 Tors1on arm

l
Series AP10, BP 20
0
J
§tnomm

~I 100mm
l
Series •Ap 20
0
J Spec1al
Torsional
bearing

(All plates are 20x 100 mm)

Side View

l 0 grmm Front Vie>w

d.= 200mm

J Details of Loading Arrangement

Top surtace
crack

J All bars 6 mm diameter

Clear cover 10 mm
t lh = 1,2,3,4 and 5

Fig. 1-Details of beams Fig. 2- Test setup

ACI Structural Journal I January-February 1989 21
Table 1 - Failure crack inclination with 70
beam axis Ql APOO D
~ AP11 Ll
Angle between the beam axis and: ,c:n
Ql AP20 0

Beam Tension Compression Top surface Bottom ~

u
60 •AP20
BPOO
•X
series and crack, crack, crack, surface crack,
number deg deg deg deg
~
u
c
BP20 •
AP00-1 54.2 50.4 77.5 78.6 0

AP00-2 48.4 45.2 76.1 74.1 ·m 50

AP00-3 42.2 37.6 76.1 54.4 a.
AP00-4 42.3 37.3 60.4 65.4 E
AP00-5 43.1 37.6 60.4 60.4 8-
0
API I-I 52.3 48.4 81.9 81.9
APII-2 49.1 46.4 75.3 76.8 .~ 40
-;o
APII-3 47.2 42.0 76.1 72.3 c
APII-4 47.1 41.0 72.3 72.3 :a
APII-5 49.0 45.6 74.1 70.6 .s
AP20-I 65.4 54.7 76.1 65.4 30~----~----~----~----~----~----~
AP20-2 54.3 49.3 74.3 0 3 4 5 6
64.3
AP20-3 52.2 46.8 70.6 69.9
AP20-4 Span- depth ratio, Vh
50.0 46.0 72.3 72.3
AP20-5 52.5 47.8 74.1 67.8
*AP20-3 56.4 49.0 72.3 56.4 Fig. 4-Effect of span-depth ratio on compression
*AP20-4 48.5 45.0 78.6 78.6 crack inclination
BP00-1 54.7 47.2 70.6 67.8
BP00-3 46.1 39.4 59.6 61.3
0
BP20-I 58.3 54.7 62.3 65.4 ll
BP20-2 50.2 47.2 64.3 63.2
0
BP20-3 44.6 41.4 72.3 69.2
BP20-4 46.1 43.9 65.4 65.4 .>:
60 •
BP20-5 48.4 42.5 66.0 58.0 u
~
u
•
Ql
u
70 ~:J 70
APOO D Ul
Ql
AP11 ll
Ql
0, AP20 0
2
,Ql
60 *AP20 • 0
BPOO X .§ 60
.>:
v
~
• 8P20 • -:;;
.~
u
u
c .s
.Q 50
Ill
c 50~----L-----~----~----~----~--~
! 0 2 3 4 5 6
0 Span-depth ratio, 1/h
c
0
~ 40
.C:::
u Fig. 5-Effect of span-depth ratio on top surface crack
.s inclination
30~----~--~~--~~--~----~-----J ways lower than the corresponding tension crack. It
0 2 3 4 5 6
varied approximately from 37 to 55 deg, depending
Span-depth ratio, 1/h
upon span-depth ratio, concrete strength, and opening
size, as shown in Fig. 4. In general, the characteristics
Fig. 3-Effect of span-depth ratio on tension crack in- of the compression cracks were similar to those of ten-
clination sion cracks.
Tension cracks on top and bottom surfaces - The
series, the angle of inclination remained practically angle of inclination of top-surface cracks varied ap-
constant for flh from 5.0 to 3.0. Thereafter it in- proximately from 59 to 82 deg and that of bottom-sur-
creased rapidly with the decrease of span-depth ratio. face cracks from 54 to 82 deg, as shown in Fig. 5 and
For a given concrete strength, it was observed that at 6, respectively. Although for flh greater than 3.0 the
any span-depth ratio, the smaller the opening size, the angle of inclination was smaller than that for f I h less
smaller the inclination of the tension crack - the angle than 3.0, no systematic variation was observed.
of inclination being lowest for the solid section. Also, for
identical opening sizes, the higher the concrete strength, Torsional strength
the smaller the inclination of the tension crack. Typical Table 2 shows the experimental results of the beams
tension crack patterns are shown in Fig. 7. of different series and the corresponding theoretical
Compression crack on vertical face - The angle of torsional strengths computed on the basis of the equa-
inclination of a compression crack was found to be al- tion proposed by Mansur and Hasnat 3 for plain con-
22 ACI Structural Journal I January-February 1989
 "' 90
"'0, APOO 0
4
AP11
"'
-o
AP20 0

""'
~u
BO
•
•AP20
BPOO
•
X
BP20 •
~"'
:,
"' 70
E
0

.8
0
c 60
0

~
u
.£
50~--~~--~~--~----~----~----~
0 4 5 6

Span -depth ratio, 1/h

Fig. 6-Effect of span-depth ratio on bottom surface

crack inclination

crete shallow beams with a circular transverse opening.

The equation is given by

(l)

in which the value of() is obtained from the expression

sec 30 - 2 secfJ + ~h = 0 (2) Fig. 7- Typical tension crack patterns

Table 2 - Test results

-------
Average concrete Theoretical torque,
Beam Opening Experimental strength T, T,, T,,
-·-
series Span diameter torque using using T, T,
f f', f, using
and f. - d,, T", f, f: .. f: .. using
number m h mm kN-m MPa MPa MPa kN-m kN-m f f~
AP00-1 0.4 I - 3.600 26.9 3.73 2.52 4.227 2.856 0.85 1.26
AP00-2 0.8 2 - 3.371 26.9 3.73 2.52 4.227 2.856 0.80 1.18
AP00-3 1.2 3 - 2.945 22.3 3.45 2.31 3.883 2.600 0.76 1.13
AP00-4 1.6 4 - 2.945 22.3 3.45 2.31 3.883 2.600 0.76 1.13
AP00-5 2.0 5 - 2.905 20.9 3.31 2.22 3.753 2.517 0.77 1.15
API I-I 0.4 I 110 2.649 20.9 3.31 2.22 3.001 2.013 0.88 1.32
APII-2 0.8 2 110 2.476 20.9 3.31 2.22 3.001 2.013 0.83 1.23
APII-3 1.2 3 110 2.395 24.2 3.59 2.32 2.472 2.101 0.74 1.14
APII-4 1.6 4 110 2.353 24.2 3.59 2.32 2.472 2.101 0.72 1.12
APII-5 2.0 5 110 2.350 22.5 3.40 2.28 2.429 2.065 0.76 1.14
AP20-I 0.4 I 200 2.292 27.2 3.80 2.42 2.009 1.708 0.86 1.34
AP20-2 0.8 2 200 2.115 27.2 3.80 2.42 2.009 1.708 0.79 1.24
AP20-3 1.2 3 200 1.930 25.0 3.62 2.39 1.989 1.691 0.75 1.14
AP20-4 1.6 4 200 1.945 25.0 3.62 2.39 1.989 1.691 0.76 1.15
AP20-5 2.0 5 200 1.988 27.4 3.73 2.46 2.049 1.742 0.75 1.14
*AP20-3 1.2 3 200 2.014 23.3 3.52 2.39 1.993 1.694 0.81 1.19
*AP20-4 1.6 4 200 1.985 23.3 3.52 2.39 1.993 1.694 0.80 1.17
AR20-3 1.2 3 200 3.106 24.7 3.56 2.38 3.151 2.678 0.78 1.16
AR20-5 2.0 5 200 2.962 24.7 3.56 2.38 3.151 2.678 0,74 I. II
BP00-1 0.4 I - 4.022 38.9 4.35 3.16 3.696 3.142 0.93 1.28
BP00-3 1.2 3 - 3.645 38.9 4.35 3.16 3.696
--
3.142 0.84 1.16
---- .
BP20-I 0.4 I 200 2.932 40.7 4.38 3.14 2.613 2.221 0.95 1.32
BP20-2 0.8 2 200 2.710 40.7 4.38 3.14 2.613 2.221 0.88 1.22
BP20-3 1.2 3 200 2.576 40.7 4.38 3.14 2.613 2.221 0.83 1.16
BP20-4 1.6 4 200 2:545 39.1 4.32 3.27 2.722 2.314 0.83 1.10
BP20-5 2.0 5 200 2.592 39.1 4.32 3.27 2.722 2.314 0.85 1.12
Mean (excluding Series AR and all beams with f!h < 3) 0.78 1.13
1 mm ; 0.394 m.; I MPa ; 145.03 ps1; I kN-m = 8.85 m.-k1p.

ACI Structural Journal I January-February 1989 23

 of deep beams is approximately constant for f!h ~ 3.0.
APOO
AP11
c
6.
For e/ h < 3.0, the torsional strength gradually in-
AP20 o creases; the smaller the span-depth ratio, the greater the
•AP20 e
BPOO X torsional strength.
BP20 •
This increase in torsional strength may be explained
as follows: The reaction of the torsional load at the top
of the beam over the support is dispersed over a large
area, producing compressive stresses in the confine-
ment zone. Fig. 9 shows that when span-depth ratio is
large, the confinement zone is localized near the sup-
ports and the elements around the opening are not sub-
jected to these compressive stresses. The torsional
strength of beams in such cases can be predicted by Eq.
Span- d{>pth ratio, Uh
( l) and (3). However, when span-depth ratio is small,
Fig. 8-Effect of span-depth ratio on experimental tor- the confining zones may overlap (Fig. 9) and the ele-
sional moment capacity ments around the opening are subjected to compressive
stresses in addition to torsional shearing stresses. This
r Torsion arm
load
stress combination results in an increase in the tor-
sional strength of the beams, as can be observed from

T the Mohr's stress circles in Fig. 9.

Fig. 8 also shows that the presence of a transverse

l opening significantly decreases the torsional strength of

beams. For the AP beam series, a 110 mm diameter
opening (d,,!h = 0.275) and a 200 mm diameter open-
~~=\
o-' = confining strf;)SS
t,'t'-.:. torsi01al shearing ing (d"! h = 0. 500) reduced the torsional capacity an
CaY For !th ( 3 stress
reaction average of about 22 and 35 percent, respectively, when
compared with beams of solid section.
A comparison between Series AP20 and BP20 indi-
cates that a 66 percent increase in nominal concrete
strength from 21.7 to 34.5 MPa (3000 to 5000 psi) in-
creased the torsional strength an average of only 30
percent. This indicates that torsional strength is pro-
portional to the square root of concrete strength.
(b) For !th >3 At failure-, cr; =tensile strength A comparison of the test results of Beams * AP20-3
of concrete
and *AP20-4 with AP20-3 and AP20-4 shows that an
eccentricity of the center of the opening by 0.15h re-
j-r'j ":>l'tl hence, T' > T
sulted in a slight increase in the torsional strength. In a
Fig. 9-Load dispersion and Mohr's stress circles rectangular beam, shearing stress due to torsion is
maximum at the middle of its deeper side and zero at
The equation for torsional strength of shallow beams of the corners. Any loss of material from a beam cross
solid section can be obtained by putting d" = 0 in Eq. section decreases its torsional capacity. However, the
(l) and (2), and is given by loss of material from a region of high stress at mid-
depth causes more reduction in torsional strength. This
T1 = 0.85 b2h /,
r (3) accounts for the slightly higher torsional strength of
3
beams with an eccentric opening.
which is the equation proposed by Hsu 2 for torsional The torsional strength of shallow beams as given by
strength of plain concrete beams of solid section. Eq. (1) and (3) is independent of span-depth ratio.
Table 2 shows that when j, is used as the tensile However, the test results show that at span-depth ra-
strength of concrete, Eq. (1) and (3) overestimate tios less than 3.0, the torsional strength of deep beams
greatly the torsional strength of the beams, particularly with or without an opening increases with the decrease
for span-depth ratios greater than 2.0. However, when of span-depth ratio. The equations thus need to be
j;P is taken as the tensile strength of concrete, a closer modified at lower ranges of span-depth ratios. Fig. 10
agreement, slightly on the conservative side, was ob- presents a dimensionless plot of T,/T, versus flh for the
tained between the experimental and the theoretical re- different series. It shows that for all the series, TjT, is
sults. Similar conclusions were also previously reported nearly constant for flh ~ 3.0. For flh < 3.0, TjT, in-
by the authorsY Accordingly, j;P has been used in the creases approximately linearly. As a lower-bound ap-
equations to obtain the theoretical torsional strengths. proach, the torsional strength of deep beams can be ex-
Fig. 8 shows the plot of experimental torsional pressed by the following equations
strength T,, against span/depth ratio flh for the differ-
ent series_ The curves show that the torsional strength T" = 1.1 T, for flh ~ 3.0 (4)
24 ACI Structural Journal I January-February 1989
and
APOO c.
APll 6
Tu = T1(1.34 -0.08 f!h) for flh < 3.0 (5)
1.3
AP20
•AP 20 e
BPOO X
BP20 •
in which T is obtained from Eq. (1 ), (2), and (3) with
1
~

/, replaced by J:P.
,.?

In actual reinforced concrete frames where deep

beams are used, large column loads may produce local-
.
0

cr
1.2

ized high compressive stresses in the beams near the 1.1

joints. For beams with small span-depth ratios, these
high compressive stresses acting along with torsional
shearing stresses will appreciably increase the torsional 1.0 '-0_ __ J L . __ __j__ _____!_ _ __ j __ __J.__ ____J

strength of the beams. The increase will possibly be

Span-depth rat1o, !th
larger than predicted by Eq. (5), which is based upon
the test results of beams in which the induced confining
compressive stresses were relatively small. When deep Fig. 10- Variation of TufT, with f!h
beams are used as transfer girders to carry columns, the
entire span may be under high induced compressive
Although the experimental work provided insight
stresses. It is known that flexural shear strengths of
into the behavior and crack patterns of deep beams un-
deep beams are two to three times higher than those of
der torsion, it was limited in scope and objective. Fur-
shallow beams because of induced compressive stresses.
ther investigations are necessary, especially for a ra-
Similar increases in torsional strength of deep beams
tional design of reinforcement for deep beams, with or
may also be expected. However, further research is
without openings, for span-depth ratios less than 3.0.
necessary in this direction.
Torsion reinforcement consists of a combination of
horizontal bars and vertical stirrups. The sharp crack ACKNOWLEDGMENT
The experimental work was carried out at the Structural Engineer-
inclination for deep beams with flh < 3.0, however,
ing Laboratory of King Abdulaziz University, Jeddah, Saudi Arabia.
indicates that, as with flexural shear, the addition of The financial support by the University under Research Grant No. 04-
horizontal bars will be more effective than vertical stir- 301 is gratefully acknowledged.
rups in increasing the torsional strength.

NOTATION
CONCLUSIONS b =beam width, mm
The following conclusions can be drawn from this d, = diameter of opening, mm
investigation: .!: = concrete cylinder compressive strength, MPa
1. Torsional strength of deep beams with or without .f = modulus of rupture of concrete, MPa
f~ = splitting tensile strength of concrete, MPa
a transverse opening remains practically constant for
h = overall depth of beam, mm
U h ~ 3.0 and can be adequately predicted by equa-
f = center-to-center span, m
tions proposed by Mansur and Has nat' with/, replaced i, = clear span, m
by J;p. For f! h < 3.0, the torsional strength varies in- T, = theoretical torsional strength, kN-m
versely with f!h ratio. A modification to Mansur and T,, = experimental torsional strength, kN-m
Hasnat's equations 3 has been proposed by the authors fJ = inclination of failure surface to normal cross section of
beam, deg
for predicting the torsional strength at lower f I h ratios.
2. Torsional strength of deep beams is influenced by
concrete strength and opening size. An increase of con- REFERENCES
crete strength from 20.7 to 34.5 MPa (3000 to 5000 psi) I. ACI Committee 318, "Building Code Requirements for Rein-
increases the torsional strength by about 30 percent. forced Concrete (ACI 318-83)," American Concrete Institute, De-
troit, 1983, Ill pp.
This indicates that torsional strength is proportional to
2. Hsu, Thomas, T. C., "Torsion of Structural Concrete-Plain
the square root of concrete strength. Concrete Rectangular Sections," Torsion of Structural Concrete, SP-
Introduction of an opening significantly reduced the 18, American Concrete Institute, Detroit, 1968, pp. 203-238.
torsional strength. An increase of d,,!h from 0.275 to 3. Mansur, M. A., and Hasnat, A., "Concrete Beams With Small
0.50 decreases the torsional strength from about 78 to Opening Under Torsion," Proceedings, ASCE, V. 105, STII, Nov.
1979, pp. 2433-2447.
65 percent of the torsional strength of solid section.
4. Hasnat, A., and Akhtaruzzaman, A. A., "An Experimental in-
3. The angle of inclination of tension or compression vestigation to Determine the Ultimate Strength of Reinforced Con-
crack on the vertical faces depends on span-depth ra- crete Beams Containing an Opening Under Bending and Torsion,"
tio, concrete strength, and opening diameter. It is ap- Final Report, Research Project No. 01-21, Scientific Research Ad-
proximately constant for f/ h ~ 3.0 but increases ministration, King Abdulaziz University, Jeddah, Nov. 1983, 74 pp.
5. Hasnat, A., and Akhtaruzzaman, A. A., "Reinforced Concrete
sharply for f!h < 3.0. This indicates that for deep
Beams Containing an Opening Under Combined Torsion, Bending
beams with very low span-depth ratios, the addition of and Shear," Final Report, Research Project No. 03-206, Scientific
horizontal bars will be more effective than vertical stir- Research Administration, King Abdulaziz University, Jeddah, Feb.
rups in resisting torsional moment. 1985, 76 pp.

ACI Structural Journal I January-February 1989 25