INTRODUCTION
Understanding the response of reinforced concrete beam under pure torsion is crucial due to the fact that the elastic theory is not applicable for reinforced concrete composite material. The nonlinear finite element analysis may be one of the best solutions for this type of problem. There is an immense work in the field of nonlinear finite element analysis of reinforced concrete structures (William and Tanabe, 2001). The work concerning the study of the torional behavior of reinforced concrete beams using nonlinear finite element method is very limited; on the contrary; a lot of experimental and theoretical works based on either elastic or plastic theory have been done in this area (Hsu, 1968; Fang and Shiau, 2004). The effects of longitudinal reinforcement on the torsional capacity of rectangular beams under combined torsion and moment are found to be marginal through the experimental work done by Aryal (2005) and Rahal (2000). Recently, HaoJan et al. (2006) experimentally investigated the effects of aspect ratio of the cross section and the variation of volume ratio of transverse to longitudinal reinforcement on the cracking and ultimate strength of reinforced rectangular concrete beams under pure torsion. The effect of the span on torsional behavior of reinforced concrete beam has not been yet investigated. So the present work is an attempt to predict the nonlinear response of cantilever reinforced concrete rectangular beam under pure torsion using the finite element package ANSYSV10 (2005) aiming to predict the effect of the span of cantilever beam on its torsional response and the safety margin in the (AC131805, 2005) code provisions, highlight the effectiveness of the torsional reinforcement in the pre and post cracking stages of rectangular reinforced concrete beams under pure torsion, predict variation of the stresses in the transverse and longitudinal torsional reinforcements at different stages of loading. Six cantilever beams with different length varies from 0.5 to 3 m with 0.5 m increments are analyzed under torsional moment applied at the free end of the beam. The cross section, longitudinal and transverse reinforcement are kept constant for all the beams and this is the main limitations of the present study.
DETAILS OF THE BEAMS
The six cantilever beams has been assigned the notations, B1, B2, B3, B4, B5
and B6 having span of 0.5, 1, 1.5, 2, 2.5 and 3 m length, respectively. Each
beam is reinforced with 10 mm bar diameter (area = 79 mm^{2}) vertical
closed stirrups spaced at 150 mm center to center and 6Φ12 (area =113 mm^{2}
each) longitudinal reinforcement bars. These are the required torsional reinforcement
for the beam as per the (ACI31805) design code to carry a factored torque (Tu)
equal to 23 kN.m using concrete cylinder strength (f'c) equal to 30 MPa with
steel yield strength (fy) equal to 400 MPa.

Fig. 1: 
Details of the analyzed Beam 
Since the cantilever beam is statically
determinant structure, the torque will be constant along the span and the required
torsional reinforcements are constant along the span and identical for the six
beams. The details of the adopted cross section and arrangement of loading are
shown in Fig. 1.
FINITE ELEMENT MODELLING
Three different types of element are used to model each of the reinforced
concrete beams; the first one is the Solid65 concrete brick element which is
used for 3D modeling of concrete with or without reinforcing bars (rebar). This
element has eight nodes with three degrees of freedom per nodetranslations
in the global x, y and z directions. The element is capable of handling plastic
deformation, cracking in three orthogonal directions and crushing. It has also
the ability to model the reinforcement as equivalent smeared within the element
with the proper orientation. The adopted element size is 50x50x50 mm such that
the number of Solid65 concrete elements used for B1 to B6 beams are 600, 1200,
1800, 2400, 3000 and 3600 elements, respectively. The second element type is
the Solid45 brick element used for the loading steel plate, which is used to
avoid local failure of concrete at the load locations. Each plate is 150 mm
depth, 100 mm width and 30 mm thick modeled by 6 Solid45 elements. These plates
located at the two staggered opposite sides of the beam free end as shown in
Fig. 1. Equal load with opposite direction is uniformly applied over these
two plates to develop the required torsion. The third element type is the Link8
element which was used to model the steel reinforcement. This element is a 3D
spar element and it has two nodes with three translational degrees of freedom
per node. This element is also capable of handling plastic deformation and is
used to represent the steel bars. The length of each Link8 element is 50 mm
perfectly connected to the nodes of the concrete element. Mesh for beam B6 is
shown in Fig. 2, in which some of the concrete elements are
removed to show the steel elements.

Fig. 2: 
Finite element mesh for beam B6 
MATERIALS MODELS
Concrete: ANSYS software has the ability to model different types of
material properties. Cracking of concrete and stressstrain relation in tension
is modelled by a linear elastic tension stiffening relationship. The cracking
stress of concrete is taken equal to the modulus of rupture calculated according
to the (ACI31805) code and it is equal to 3.5 MPa. In the present study the
ability of concrete to transfer shear forces across the crack interface is accounted
for by using two different shear retention factor (β) for cracked shear
modulus, it was assumed equal to 0.3 for opened crack and 0.7 for closed crack.
The concrete constitutive relationships under multiaxial state of stresses,
is based upon the William and Warnke (1975) model which are considered as an
appropriate model to describe the concrete failure. In this model the yield
condition is approximated by five parameters and it is used to distinguish linear
from nonlinear and elastic from inelastic deformations using the failure envelope
defined by a scalar function of stress f(σ) = 0 through a flow rule, using
incremental stressstrain relations. The author present the equivalent uniaxial
stressstrain relationship for the concrete under compression by using the equation
suggested by Desayi and Krishnan (1964) to compute the multilinear isotropic
stressstrain curve for the concrete (Mac Gregor, 1992) in the form:
Where, E_{c} is the elastic modulus of concrete calculated as per the
(ACI31805) code and it is equal to 26017MPa for fc = 30 MPa, ε strain
corresponding to stress σ and ε_{0} is the strain at peak
stress equal to (2f_{c}/E_{c}) and fc is the concrete compressive
strength.

Fig. 3: 
Multilinear stressstrain isotropic hardening curve for concrete 
Ten points are used to represent the stress strain curve using Eq.
1 starting from the elastic stress limit (0.3fc) up to fc = 30 MPa and last
point is corresponding to ultimate strain (εu) equal to 0.003 at peak stress
fc = 30 MPa. The resulting multilinear isotropic hardening stressstrain curve
for concrete is shown in Fig. 3.
Steel: In the present study a Bilinear Kinematics Hardening (BKIN) is adopted for steel bars and loading plates. The yield stress is equal to 400 MPa and the hardening modulus is taken equal to 200 MPa; while the elastic modulus is equal to 200000 MPa and Poisson's ratio 0.3.
RESULTS AND DISCUSSION
Each of the six beams is analyzed under increasing incremental torque equal
to 0.5 kN.m up to failure. The torque is induced by a uniform load applied over
the staggered opposite plate at the free end of the beam. The ultimate load
is assumed to be reached when the solution is not converged. The predicted variations
of twist angle with the applied torque for all the beams are shown in Fig.
4, which shows that ductility of the beams in the post cracking stages is
significantly increased by increasing the length of the beam.
The predicted cracking torque (Tcr), elastic torque (Te), ultimate torque (Tu)
and the percent safety margin (reserved strength) over the design torque (23
kN.m) are shown Table 1, where (Te) is the torque that limits
the linear part of the torquetwist curve. The Table 1 also
contains comparison between the predicted elastic torsional stiffness (Ke) and
that based on beam theory (GJ/L), where (Ke) represent the initial slope (linear
part) of the torquetwist curve, G is shear modulus, J is the torsional rigidity
which for these beams cross section is equal to (0.002795 m^{3}) and
L is the span of the beam.

Fig. 4: 
Torquetwist angle for the analyzed beams 
Table 1: 
Predicted cracking, elastic and ultimate torque and safety
margin 

The ultimate torque shown in Table 1 is the torque at which
divergence occurred. It is clear that the predicted torsional stiffness (Ke)
for B1 is about 50% greater than that calculated by the elastic theory and the
difference get reduced for beams with longer span and it becomes 7% for the
longer beam B6. The Table 1 also shows that the cracking and
ultimate torques for beam B1 (having span/depth ratio equal to 1) is about 17%
more than that of B4, B5 and B6 which have same failure torque (37.5 kN.m).
It can be seen that the three values of the torque (Tu, Te and Tcr) are reduced
by increasing the span from 0.5 to 2 m and these values become constant for
the beams having span equal to or more than 2 m (Fig. 5).
Thus it can be stated that, when the span to depth ratio is equal to or more
than 4, the span has no effect on the torsional strength of cantilever beam
under pure torsion.
Figure 6 and 7 show a sample of the crack
pattern of beam B2 at torque level Te = 31.5 kN.m and at 32 kN.m, respectively.
These two figures show the drastic change in the rotation of the beam and the
wide spread of the cracks along the span when the torque exceeds the (Te) value
by only 0.5 kN.m increment. Cracks are first started at the vicinity of the
loaded plate as shown in Fig. 6, then these cracks are widely
spread along the span of the beam at torque equal to 32 kN.m as shown in
Fig. 7.

Fig. 5: 
Variations of Tu, Te and Tcr with the span of cantilever 

Fig. 6: 
Crack pattern for B4 at torque equal to 31.5 kN.m 

Fig. 7: 
Crack pattern for B4 at torque equal to 32 kN.m 
This explains the sudden change in the twist angle when the torque
exceeds (Te) for all the beams as shown in Fig. 4.
To view the effectiveness of steel reinforcement, only sample of the predicted
results is presented here. The variation of axial stress for longitudinal bar
at mid depth along the span of the beam B4 at different stages of loading is
shown in Fig. 8, where each element length is 50 mm. The chosen
torques are 31.5, 32 and 37 kN.m which is only 0.5 kN.m less than failure torque
(37.5 kN.m). Figure 9 shows the same but for stirrups (spaced
at 150 mm, except the last 3 stirrups spaced at 50 mm) at mid depth of the beam.

Fig. 8: 
Variation of longitudinal steel stress at mid depth along
span of beam B4 

Fig. 9: 
Variation of stress at mid depth of stirrups along the span
of beam B4 
By scrutinizing these two figures, it can be noticed that up to the torque
(Te = 31.5 kN.m), which is more than the cracking torque (22 kN.m), the stresses
in both longitudinal and transverse reinforcements is almost negligible, after
that and due to the wide spread of the cracks along the span at T = 32 kN.m
(Fig. 7), the steel stress suddenly increased and it has the
same trend of distribution near the failure torque (37 kN.m). The variation
of stresses for both types of reinforcement show that the peak stresses taken
place at 450, 1200 and 1600 mm from fixed end and these may indicate the locations
of formation of major cracks. None of the two types of reinforcement reached
yield stress (400 Mpa).

Fig. 10: 
Steel stress variation at different stages of loading up to
failure 
To trace the variation of steel stresses in all the beams during different
stages of loading up to failure, the variation of stresses in the steel element
that experienced maximum stress for all the beams are shown in Figure
10, which shows drastic increase of stress when the load exceeds the (Te)
value listed in Table 1. Figure 10 also
shows that the steel stress in all the beams have never reached the yield stress
except for beam B1 which sustained the largest ultimate torque (43.5 kN.m) and
this is more than the design torque (23 kN.m) by 87%. The figure also shows
that the stresses in steel in the beams are almost negligible until the torque
exceeds the (Te) value which is also more than cracking torque by 3243% and
more than the ACI31805 design torque (23 kN.m) as well.
It is worth to mention that from the experimental work of HaoJan et al.
(2006) for beams with transverse to longitudinal steel ratio ρ_{t}/ρ_{l}
= 1.0 and total steel ratio ρ_{total} = 1.0%, that the ratios of
Tu/Tcr is about 1.321.59. In the present study the values of ρ_{t}
=0.435, ρ_{l} = 0.452, ρ_{total }= 0.887% and ρ_{t}/ρ_{l}=0.96,
the predicted Tu/Tcr ratios vary from 1.89 for beam B1 to 1.7 for beam B6 (Table
1) and considering the difference in the aspect ratio of the cross section
of the beams and material properties used by HaoJan et al. (2006) and
the present study, the predicted results can be considered fair enough compared
with the experimental results by HaoJan et al. (2006), particularly
for the beams with longer span.
CONCLUSIONS AND RECOMMENDATIONS
From the predicted results of the nonlinear analysis of reinforced concrete
rectangular cantilever beams of different length, having same section and steel
reinforcement subjected to pure torsion the following can be stated:
• 
The beams designed as per the ACI31805 code to carry a factored
torsion Tu = 23 kN.m, have shown a safety margin over the designed torsion
equal to 89, 78, 67, 63, 63 and 63% for beams having span equal to 0.5,
1, 1.5, 2, 2.5 and 3 m, respectively. This shows the importance of the beam
length, which are normally not taken into consideration in the design procedure. 
• 
The maximum cracking torque, elastic limit torque and ultimate torque
are predicted for the beam (B1) which has the shorter span with span to
depth ratio equal to (1), these values get reduced with increasing the span
and become constant when the span to depth ratio is equal to or more than
(4) for this particular beam. This may be attributed to the effect of St.
Venant principle. 
• 
The stresses in the longitudinal and transverse reinforcements remain
negligible even after the initiation of cracks and these stresses drastically
increased after the wide spread of the cracks along the beam length. The
steel attained yielding only in beam B1 (with shorter span) at a very high
torque (43 kN.m), which is almost double the design torque. 
• 
The predicted torsional stiffness for beam B1 with span to depth ratio
equal to unity is 50 percent more than that of the elastic beam theory;
while that of beam B6 with span to depth ratio equal to 6 the difference
is only 7 percent, so the author suggests including the torsional stiffness
of the reinforced concrete beam in the procedure for design of reinforced
concrete beam for torsion. 
• 
There is a need to carry out more rigorous study to separately investigate
the effectiveness of each type of the torsional reinforcement of reinforced
concrete beams subjected to pure torsion by varying the ratio of each type
and to predict which type is more effective. 
• 
The limitations of the present study are that the beam cross section and
torsional reinforcement are kept constant. So more work are needed to study
the effect of varying these parameters. 