A unified approach for determining the ultimate strength of RC members subjected to combined axial force, bending, shear and torsion
A unified approach for determining the ultimate strength of RC members subjected to combined axial force, bending, shear and torsion
Pu Wang 0 1
Zhen Huang 0 1
0 School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiaotong University , Shanghai , P.R. China
1 Editor: Jun Xu, Beihang University , CHINA
This paper uses experimental investigation and theoretical derivation to study the unified failure mechanism and ultimate capacity model of reinforced concrete (RC) members under combined axial, bending, shear and torsion loading. Fifteen RC members are tested under different combinations of compressive axial force, bending, shear and torsion using experimental equipment designed by the authors. The failure mechanism and ultimate strength data for the four groups of tested RC members under different combined loading conditions are investigated and discussed in detail. The experimental research seeks to determine how the ultimate strength of RC members changes with changing combined loads. According to the experimental research, a unified theoretical model is established by determining the shape of the warped failure surface, assuming an appropriate stress distribution on the failure surface, and considering the equilibrium conditions. This unified failure model can be reasonably and systematically changed into well-known failure theories of concrete members under single or combined loading. The unified calculation model could be easily used in design applications with some assumptions and simplifications. Finally, the accuracy of this theoretical unified model is verified by comparisons with experimental results.
Data Availability Statement: All relevant data are
within the paper.
Funding: The authors acknowledge the National
Natural Science Foundation of China for its
financial support of this research project (No.
Competing interests: The authors have declared
that no competing interests exist.
Failure of reinforced concrete (RC) members in extreme loading events is typically caused by
different combinations of axial force, bending, shear and torsions, and the failure mechanisms
can be highly complex. Compared to steel structural design and calculation theory, the design
and calculation theory of RC members under the combined actions of tensile/compressive
axial force, bending, shear and torsion is relatively imperfect and a unified failure theory that
can be used worldwide has yet to be developed. The majority of design codes use experimental
formulas to calculate the bearing capacity of RC members under combined loading actions;
however, experimental data and theoretical formulation for these cases are lacking.
Currently, the theories for RC members subjected to axial forces and bending moments
used in various countries are mostly identical, and this theory is widely accepted. However,
there are many different theories and design methods for RC members subjected to shear and
torsion loading. The existing research on the ultimate strength of RC members subjected to a
combination of the four loads is far from conclusive, and experimental research is lacking,
which makes theoretical research even more difficult.
Many theories have been established for RC members subjected to combined loads. These
theories mainly include statistical analysis methods, truss models and limit equilibrium theory.
Because the failure mechanisms of RC members are highly complex under combined loading,
statistical analysis methods are widely used in construction applications. Statistical analysis
methods have been used to establish semi-experimental equations based on regression analyses
of experimental results. Statistical analysis methods typically aim to prevent the members from
experiencing brittle shear failure but do not attempt to accurately predict the ultimate strength.
Statistical analysis methods create concise equations and are convenient for many applications.
However, statistical analysis methods lack mechanical models and require vast experimental
Truss models have led to numerous advancements in research on the shear and torsion
loading of concrete over the past several decades. The most representative theories are
compression field theory (CFT) and softened truss model theory. In 1973, Collins first proposed
the deformation compatibility condition for RC members under shear, which is the Mohr
deformation compatibility condition . Mitchell and Collins  subsequently established
CFT for RC members under shear and torsion in 1974 using the Mohr deformation
compatibility condition, equilibrium condition and uniaxial stress-strain relationship of concrete.
In 1981, Vecchio and Collins  quantitatively analysed the softening effect in the
stressstrain relationship of concrete in a multi-axial stress state and introduced the softening
stressstrain relationship into CFT, which was a significant breakthrough in the research on RC
members under shear. CFT assumes that the concrete tension stress is zero after cracking even
though the residual tensile stress in the concrete between the inclined cracks is not actually
zero. To consider the influence of the residual tensile stress, Vecchio and Collins  proposed
the modified compression field theory (MCFT) based on CFT. The key improvement of
MCFT is that it considers the tension stiffening of RC elements between the inclined cracks
and restricts the concrete tensile stress of the concrete by checking the local equilibrium at the
cracks. MCFT provides a more accurate evaluation of the ultimate strength than CFT. Vecchio
experimentally determined that the directions of principle strains in concrete differed from
those of the principle stresses when the deformation of concrete was extremely large.
Therefore, Vecchio [5±7] introduced shear slip into the deformation compatibility condition to
consider the difference between the stress and strain directions and also established the disturbed
stress field model (DSFM).
In 1988, Hsu  established the rotating angle-softened truss model (RA-STM) based on
the equilibrium condition, deformation compatibility condition and softened stress-strain
relationship. The RA-STM is similar to MCFT; however, the RA-STM uses the experimental
stress-strain relationship of the reinforcement embedded in the concrete instead of the
relationship of the bare reinforcement and does not require checking the local equilibrium at the
cracks. Hsu found that the RA-STM was only valid when the degrees of the inclined cracks
were between 33Ê and 57Ê and that it could not consider the shear capacity of concrete. Pang
and Hsu  established the fixed angle-softened truss model (FA-STM) in 1996. The FA-STM
has a larger scope of application and can consider the shear capacity of concrete. The FA-STM
is considerably more complex than the RA-STM because the equilibrium condition and
deformation compatibility conditions are more complex; in addition, FA-STM adopts an additional
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shear constitutive relationship for cracking concrete. Hsu and his team  made significant
developments in truss model theory and referred to their work as a unified theory of
The ultimate strength of RC members can also be investigated via limit equilibrium theory.
In 1968, Gvozdev  established the limit equilibrium theory for warped failure surfaces
considering the equilibrium and constitutive relations. Gvozdev established equations for the
ultimate strength of RC members based on limit equilibrium theory by making assumptions
about the shape of the failure surface. Huang [12, 13] established equations for the ultimate
strength of RC members with box sections and rectangular sections by modifying the limit
equilibrium theory for a warped failure surface. However, in traditional limit equilibrium
theory, it is often difficult to determine the shapes of the failure surfaces, and the shear capacity of
concrete is neglected; these simplifications lead to errors. Nielson  proposed a stress yield
criterion for RC slabs that can consider the shear capacity of concrete and is valid for thin slabs
with uniform reinforcement. Based on the yield criterion, Huang Z and Liu XL 
established a unified model for RC box section members with uniform reinforcement.
In recent years, many studies have focused on the ultimate strength of RC members under
combined loading. Huang L and Lu Y  studied the overall interactions between different
types of loads and established semi-empirical equations for the ultimate strength of
symmetrical RC members under combined loading. Rossi and Recupero [17, 18] established analytical
formulations for the truss action and arch action, respectively, and calculated the ultimate
shear strength of RC members under combined axial forces, bending moments, and shear
forces. Panjehpour, Chai, and Voo  improved the strut-and-tie model (STM) for deep RC
beams using experiments and the finite element method. Different STMs may be established
for RC members, and establishing a good STM relies on the experience of calculators. The
ultimate strength of RC beams with ratios of shear span to effective depth that are less than 3 can
be accurately evaluated using proper STMs.
In this paper, a total of 15 RC beams were fabricated to study the ultimate strength of RC
members. These beams had identical geometric parameters and reinforcement. The beams
were subjected to different load combinations of axial force, bending moment, shear force and
torsion. The experimental results are presented and discussed in detail. This experimental
research intends to determine how the ultimate strength of RC members changes with changes
in load combinations. Then, a theoretical model was established by determining the shape of
the warped failure surface, assuming a proper stress distribution on the failure surface, and
considering the equilibrium conditions. The model attempts to develop a concise expression
and has a certain level of accuracy for calculating the ultimate strength of RC members
subjected to different load combinations. Finally, the accuracy of this model was verified by
comparisons with experimental results.
RC member design
The length of the 15 experimental beams was 2,200 mm, including a middle experimental
segment and two clamped end segments. The experimental segment was 1,200 mm long, and
each clamped end was 500 mm long. The cross section of the beams was 240 mm×240 mm.
The ratio of the effective length to width l0/b was 5, where l0 is the distance between the lateral
supports and b is the width of the beam. The concrete cover for the outermost reinforcement
was 20 mm.
All of the beams were made using the same concrete and reinforcement. Nine concrete test
cubes and three samples for each steel species were made to test the material strength. The
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Fig 1. Layout and cross sections of the test beams.
proportions of concrete mix were 368 kg/m3 P.O. 42.5 ordinary Portland cement, 185 kg/m3
water, 637 kg/m3 medium sand, and 1,184 kg/m3 stone. The measured compressive strength
of the concrete cubes fcu was 45.4 N/mm2. In the middle experimental segment, the
longitudinal steel bars were 4;16 mm in the corners. The yield strength fyl and ultimate tensile strength
ful of the ;16 mm rebar were 498 and 648 N/mm2, respectively. The transverse reinforcements
were ;8@60 mm. The yield strength fyt and ultimate tensile strength fut of the ;8 mm rebar
were 450 and 670 N/mm2, respectively. The clamped ends were reinforced with more steel
rebar to ensure that the experimental beams failed in the middle experimental segment. The
beam dimensions and reinforcement are shown in Fig 1.
The experimental device is shown in Fig 2. The beams were supported by two hinged supports
at the ends of the middle experimental segment, and the distance between the two supports
was 1,200 mm. When torsion was needed, torsion restraints were applied at the left hinged
support, and the other hinged support was replaced with a blade bearing to provide vertical
support while allowing free rotation. The external forces were applied with 4 hydraulic jacks. A
100 t hydraulic jack was used to apply the axial compressive force. Two 10 t hydraulic jacks
and a frame were used to apply torsion. The distance between the two 10 t hydraulic jacks was
1,300 mm. Finally, a 50 t hydraulic jack was used to apply vertical force at the middle segment
of the test beams.
The 15 experimental beams were divided into four groups based on the different load
combinations of axial force (N), bending (M), shear (V) and torsion (T). The first two groups had
load combinations without axial force (N), and the last two groups had load combinations with
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Fig 2. Experimental device.
axial force (N). The first group consisted of 2 beams under bending and shear (MV×2) and 2
beams under pure torsion (T×2). The second group consisted of 4 beams under bending, shear
and torsion (MVT×4). The third group consisted of 1 beam under axial force, bending and
shear (NMV×1) and 1 beam under axial force and torsion (NT×1). The last group consisted of
5 beams under axial force, bending, shear and torsion (N0.15MVT×2 and N0.3MVT×3). The
shear span ratio was 3 for all of the load combinations. All of the load combinations and
supports are shown in Fig 3.
The loads were slowly applied incrementally. The loading sequence was compressive axial
force, torsion and vertical force. The load increment of each load step was 60 kN for
compressive axial force and 1.3 kN·m for torsion. When applying vertical force to the beam, the load
increment of each load step was 5 kN before the concrete cracked, 10 kN after cracking
occurred and 5 kN when the loads approached 80% of the estimated ultimate strength of the
RC beam. Each load step was maintained for 5 min, and the concrete strain, middle
deformation and crack developments were observed and recorded at each load step.
The experimental results are shown in Figs 4±9 and Table 1. Fig 4 shows the T-θ curves of the
experimental beams. Fig 5 shows the P-Δ curves at the mid-spans. Figs 6±9 show the
experimental crack graphs. Table 1 lists the cracking loads and the ultimate loads of the experimental
Group 1 (MV and T)
Group 1 consists of four beams, which are referred to as MV1, MV2, T1 and T2. Beam MV1
was subjected to bending moment and shear force and was supported by two hinged supports.
A 50 t hydraulic jack was used to apply a vertical force at the middle of the beam. In the initial
loading procedure, the strain gages indicated that the concrete strain increased linearly in
the beam's middle segment. The first crack appeared on the bottom when the shear force V
reached 21.5 kN. The concrete on the bottom stopped carrying load because of the cracks.
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Fig 3. Combinations of loads and supports.
Along with the loading, new inclined cracks gradually appeared between the supports and the
loading point due to the influence of shear stress. No additional cracks were observed when
the internal force was rearranged. The existing cracks grew longer and wider until one crack
became the critical crack. The longitudinal reinforcement across the critical crack yielded, and
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Fig 4. T-θ curves of the experimental beams.
MV1 failed when the shear force reached 87 kN. The failure procedure of beam MV2 was
similar to that of MV1.
Beams T1 and T2 were loaded under pure torsion. The left end of the beam was fixed, and
the other end was supported by a blade bearing to supply the torsional load. The distance
between the two supports was maintained at 1,200 mm. Two 10 t hydraulic jacks and a frame
were used to apply torsion. The value of the force arm was 1,300 mm for the two jacks. In the
initial loading procedure, the strain gages and the load-rotation curve indicated that the
concrete worked in the elastic stage. Then, the first crack was observed when the torsion reached
10.4 kN·m for T1 and 5.2 kN·m for T2. Along with the torsional increase, additional cracks
with inclination angles of 45Ê gradually appeared and developed to form spiral cracks. The
cracked concrete carried less loads, and the reinforcement across the cracks gradually began to
yield. After the reinforcement yielded, the beam rotation increased more rapidly. Finally, the
front concrete of T1 and the bottom concrete of T2 were crushed.
Group 2 (MVT)
In Group 2, four beams, MVT1, MVT2, MVT3 and MVT4, were loaded under a combination
of bending moment, shear force and torsion. The supports were the same as those of beams
T1 and T2. First, the torsion was first applied by two 10 t hydraulic jacks. Then, the bending
moment and shear force were applied by one 50 t hydraulic jack applying a vertical force P at
the middle of the beams. The torsion applied to MVT1, MVT2, MVT3 and MVT4 was 0.35Tu,
0.55Tu, 0.75Tu and 0.75Tu, respectively. The value of Tu was set to the mean value of the
ultimate torsion of beams T1 and T2 found from the Group 1 results.
The experiment results indicated that the ultimate strength of bending and shearing decreased
with increasing applied torsion force.
MVT1 did not crack under torsion of 0.35Tu. The ultimate strength of MVT1 was very
similar to those of MV1 and MV2, and the failure modes were similar. Therefore, the torsion of
0.35Tu was sufficiently small that the applied torsion did not significantly influence the
ultimate strength of the beam.
MVT2 cracked when the torsion reached 6.5 kN m. With the the torsion and vertical
loading, inclined cracks were observed between the loading point and supports. These cracks
mainly resulted from the shear stress in the concrete, and the inclination angles were
approximately 45Ê. Cracks were observed in the lower part of the beam near the middle segment.
These cracks mainly resulted from the bending moment, and the inclination angles were
approximately 60±70Ê. From the load-deformation curve of the mid-span, as shown in Fig 5,
the deformation started to increase rapidly when the shear force reached 67.5 kN, indicating
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Fig 5. P-Δ curves at the mid-spans.
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Fig 6. Crack graphs of Group 1.
the yielding of the longitudinal reinforcement. When the shear force reached 80 kN, the
concrete in compression was crushed, and MVT2 reached its ultimate strength.
MVT3 and MVT4 cracked when the torsion reached 9.1 and 7.8 kN, respectively. When the
torsion increased to 0.75Tu, cracks gradually developed and appeared as spiral cracks. With
further increases in the vertical force, the concrete was crushed in compression.
Group 3 (NMV and NT)
In Group 3, Beam NMV1 was tested under combined axial force, bending moment and shear
force. The supports of NMV1 were the same as those for MV1 and MV2. A 50 t hydraulic jack
was used to apply the vertical force, just as in the experiments for MV1 and MV2. A 100 t
hydraulic jack was used to apply the axial compressive force on the NMV1 beam. In the first
loading procedure, the axial force was gradually increased to 588 kN, which was equal to 30%
of the total axial bearing capacity (0.3fcA). Next, the vertical force was loaded to apply the
bending moment M and shear force V to NMV1. The first crack was observed when the shear
force V reached 42.5 kN. The cracking load of NMV1 was greater than those of MV1 and
MV2. As the vertical force increased, more cracks appeared between the middle loading point
and the supports until an inclined crack finally became the critical crack. The inclination angle
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Fig 7. Crack graphs of Group 2.
of the critical crack was approximately 28Ê on the front. The concrete in the compressive area
was crushed when the shear force reached 215 kN.
Beam NT1 was loaded under combined axial force and torsion. The supports were the
same as those of T1 and T2. One 100 t hydraulic jack was added to apply an axial compressive
force. Two 10 t hydraulic jacks were used to apply torsion to the beam in the same manner as
in the experiments for T1 and T2. First, an axial compressive force of 588 kN was applied.
Then, a torsion load was applied and increased.
The first crack of beam NT1 was observed considerably later than those of T1 and T2. After
the concrete cracked, the internal stress was transferred to the reinforcement. The
torsionrotation curve illustrates that the rotation of NT1 began increasing rapidly when the torsion
reached 26 kN·m, indicating the yielding of the reinforcement. The critical inclined crack of
NT1 appeared on the bottom, and the inclination angle of the crack was approximately 35Ê.
The ultimate torsion of NT was 28.6 kN·m.
Group 4 (NMVT)
Group 4 consisted of five beams, N0.15MVT1, N0.15MVT2, N0.3MVT1, N0.3MVT2 and
N0.3MVT3, which were loaded under axial forces, bending moments, shear forces and torsion.
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Fig 8. Crack graphs of Group 3.
The supports were the same as those for beams MVT. The axial force was applied first,
followed by torsion and then vertical force.
The axial compressive force on N0.15MVT1 and N0.15MVT2 was 294 kN, which was equal
to 0.15fcA. The torsion on N0.15MVT1 was 0.55Tu, and the torsion on N0.15MVT2 was 0.75Tu.
The failure procedures of N0.15MVT1 and N0.15MVT2 were similar, but the ultimate strength
of N0.15MVT1 was higher. No cracks were observed when only axial force and torsion were
applied. The compressive force caused N0.15MVT1 and N0.15MVT2 to crack later than MVT2,
MVT3 and MVT4. The cracks were flexural and shear cracks and developed more severely in
the additive shear stress zone with τv+τT.
The inclination angles of the cracks were smaller than those of MVT2, MVT3 and MVT4
due to the existence of normal axial compressive stress. The concrete covers started to peel off
due to these cracks in the additive shear stress zone with τv+τT. The load-deformation curves
of the middle segment indicate that N0.15MVT1 and N0.15MVT2 still showed notable ductility
behaviour during loading.
The axial compressive forces on N0.3MVT1, N0.3MVT2 and N0.3MVT3 were 588 kN, which
was equal to 0.3fcA. The torsion on N0.3MVT1 was 0.55Tu. The torsion on N0.3MVT2 and
N0.3MVT3 was 0.75Tu. The cracking modes of N0.3MVT1, N0.3MVT2 and N0.3MVT3 were
similar to those of N0.15MVT1 and N0.15MVT2, but the higher compressive force made the
inclination angles of cracks even lower in N0.3MVT1, N0.3MVT2 and N0.3MVT3. Furthermore,
the higher compressive force caused N0.3MVT1, N0.3MVT2 and N0.3MVT3 to exhibit brittle
behaviour during the loading and failure.
Discussion of the results
The ultimate strength of RC members is affected by many factors, such as the material
properties, reinforcement ratio, member sizes, loading methods, shear span ratio and loading
combination. This paper focuses on the influence of different loading combinations on the ultimate
strength. The result is appropriate for members with ordinary material, normal reinforcement,
and normal span ratios.
The following conclusions were drawn from the research and experimental results:
1. Torsion forces could reduce the ultimate capacity of beams under combined loading due to
three main factors. First, torsion can increase the tension stress of the longitudinal
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Fig 9. Crack graphs of Group 4.
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reinforcement and cause a compression zone on the opposite side. Second, shear stress
caused by torsion can decrease the concrete compressive strength. Finally, torsion may peel
off the concrete covers under high shear stress.
2. The compressive axial force could affect the ultimate strength of the beams. An appropriate
compressive force can increase the ultimate strength of the beams by decreasing the tensile
stress of the longitudinal reinforcement and could work as additional longitudinal
reinforcement in the tension zone. Furthermore, an appropriate compressive force could
improve the shear strength of the concrete. However, a high axial compressive force may
decrease the ultimate strength of the beams by increasing the compressive stress in the
concrete compression zone and may crush the compressive concrete before failure of the tensile
3. Cracking angles change with changes in the load combinations. The critical cracks
constitute the twist failure surface of the beams, and the twist failure surface affects the ultimate
strength of the beams. When the bending moment dominates the combined load, the
inclination angles of the critical cracks are approximately 90Ê, such as in beams MV1 and MV2.
When pure torsion or shear force dominates the combined load, the inclination angles of
the critical cracks are approximately 45Ê. When an axial compressive force is applied, the
axial force changes the direction of the principal stress in the concrete and thus the cracking
Theoretical approach to the failure model
Based on the above experimental research on 15 RC beams under combined axial force,
bending, shear and torsion loading, a simplified failure model is established as the theoretical
approach. This failure model corresponds well with the above experimental research results
and is fairly accurate. Additionally, the model attempts to provide a concise expression for
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The ultimate strength of RC members under combined loading is highly complex. The
following basic assumptions are adopted to simplify the calculation of the failure model.
1. The effect of the load path was neglected. The model is suitable for members whose ultimate
strengths are not highly sensitive to the load path.
2. The dowel action of the rebar was neglected.
3. The stresses of the longitudinal reinforcement were calculated based on the Bernoulli truss
model. There are two classic models for the shear strength of RC beams, namely, the STM
and the Bernoulli truss model. The STM is more appropriate for expressing the force
transfer mechanism when the shear span ratio is λ<3. The stress distribution is even throughout
the majority of the beam when the shear span ratio is λ 3; thus, the Bernoulli truss model
is more appropriate. In most engineering applications, the shear span ratios of the majority
of beams and columns are no less than 3. Thus, the Bernoulli truss model was adopted in
4. The stresses of the transverse web reinforcement across the failure surface are assumed to
reach the yield strength. When the bending moment dominated the combined load, the
transverse web reinforcement may not yield. In this situation, a reduction factor ; is
introduced to approximately consider the real stress in the web reinforcement.
Formulations of the proposed model
The simplified failure model is based on the ultimate equilibrium method. The first step for
establishing the model is to determine the warped failure surface. As shown in Fig 10, 4
parameters were adopted to describe the surface: the three critical crack angles θl, θr, and θb and the
depth of the compressive concrete zone x. In reality, the depth of the compressive zone is
different on the left and right sides, but a mean depth of x is used to simplify the expression.
The depth of the compression zone x can be calculated based on the Bernoulli truss model
where As is the area of longitudinal reinforcement in the tension area, As' is the area of
longitudinal reinforcement in the compression area, b is the width of the beam section, x is the depth
of the compression zone, σs is the stress in the tensile reinforcement, fy’ is the yield strength of
Fig 10. Warped failure surface.
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the compressive reinforcement, fc is the compressive strength of concrete, Es is the modulus of
elasticity of the reinforcement, εcu is the ultimate compressive strain of concrete, and h0 is the
effective height of the section, which is the distance from centre of tensile reinforcement to the
The stress distribution in the section under bending is assumed to be as shown in Fig 11.
The stress on the section under torsion is assumed to be distributed along the external
box section. The width of the box section t is calculated based on the results of Rahal and
Collins' model . The stress under different single external forces can be expressed as follows:
1. Axial force
2. Bending moment
3. Shear force
Fig 11. Stress distributions on the section under bending.
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where t is calculated based on Rahal and Collins's model.
The second step to establish the model is to determine the stresses of the longitudinal and
transverse reinforcement across the failure surface. The stresses of the longitudinal
reinforcement can be obtained from Eq 2. If shear or torsion dominates the combined load, then the
transverse reinforcement is assumed to yield when the member reaches its ultimate strength.
However, the transverse reinforcement may not yield if the reinforcement is excessive or if the
shear force and torsion are small. Thus, a factor ; is used to reduce the stress in the model. To
simplify the model, the factor ; is calculated using Eq 9. This equation is based on the ratio of
the external forces to the ultimate forces of the transverse reinforcement across the failure
surface in the left flange. Because the stress of the transverse reinforcement in the left flange is no
less than that in the right flange, this simplification tends to be conservative.
T2AT0c hcor V 2Vc
where nl is the number of reinforcements across the crack in the left flange, Tc is the resistant
torsion of concrete and Vc is the resistant shear force of concrete. The calculations of nl, Tc and
Vc is given in equations later in this chapter.
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The third step is to establish the formulation through the ultimate equilibrium. Eq 10 can
be obtained from the equilibrium of the failure surface in the left zone.
In Eq 10, the shear strength of concrete τc is calculated based on the Tasuji-Slate-Nilson.
failure criterion. The factor γ is added to consider the strength reduction after cracking.
According to experimental research, γ is set to 0.5.
1 fc=ftfcs fc=fts2
hcor V2 nl Asv1fyv Fl
Fl b2h0 tc hcorttc
ni hcorcotyi ; i l; r; b
V nr Asv1fyv Fr
bcor nb Asv1fyv Fb
b2h0 tc hcorttc
Eq 13 can be established from the bending moment equilibrium at the centre of the
M N h2
hcorcot yl bcorcot yb
hcorcot yr bcorcot yb
By substituting Eqs 10±12 into Eqs 13 and 14 can be expressed as a unified expression:
All load combinations
Statistical analysis method
All load combinations
All load combinations
Axial force, bending
moment, and shear force
Bending moment and
All load combinations
All load combinations
Solve simultaneous equations of equilibrium condition,
deformation compatibility condition and constitutive relationship
Solve simultaneous equations of equilibrium condition,
deformation compatibility condition and constitutive relationship
Establish analytical formulations for the truss action and arch
action and then calculate the ultimate shear strength of RC
Establish an STM for members based on load-transferring
Establish yield equations based on the stress yield criterion  for
Determine the shape of the warped failure surface, assume the
stress distribution on the failure surface, and then establish
equations based on the equilibrium condition
4 Asv1fyv ssAs h0
4 Asv1fyvAcor ssAs h0
where as’ is the distance from the centre of the compressive reinforcement to the nearby edge.
The proposed model is different from that given in existing studies. Table 2 shows the
differences between the proposed model and models from the literature.
Comparisons between the proposed theoretical model and experimental results
retical model. The bending moment at the intersection point of the first inclined crack and
the longitudinal reinforcement were used in the calculation. This intersection point was
approximately h0 from the middle span of the beam. The concrete covers may peel off when
the members are under pure torsion or 0.75Tu of pure torsion. Thus, the value of Acor should
be decreased; 0.6Acor was used in the calculation.
Fig 12 compares the comparison between the proposed theoretical model results and the
experimental results. This figure illustrates that the model coincides well with the experimental
results for the majority of members. Only NMV1 and MVT3 deviate noticeably from the
experimental results. For NMV1, the model result is 12% lower than the experimental result.
This difference may be caused by the hardening of the longitudinal reinforcement.
Considering the discreteness of the concrete experimental results, this error is acceptable. MVT3 and
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Fig 12. Comparison of the proposed model results and experimental results.
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MVT4 experience the same conditions. For MVT4, the calculated result coincides well with
the experiment result. For MVT3, the calculated result is 22% lower than the experimental
result. Fig 7 illustrates that the external concrete of MVT4 was damaged more seriously than
that of MVT3, as the external concrete of MVT3 was still solid. If Acor was used in the
calculation instead of 0.6Acor, then calculated result for MVT3 is 78.34 kN, which is closer to the
experimental result of 75 kN. This result indicates that the external concrete may occasionally
peel off when approximately 75% of the pure torsion strength is applied and that the
experimental results show discreteness. The lower calculation value should be used for safety.
This paper conducts experimental research on the ultimate strength of RC members. Fifteen
beams with identical rectangular sections and reinforcement were tested. These beams were
divided into 4 groups and subjected to different combinations of axial force, bending moment,
shear force and torsion. The ultimate strength, load-deformation behaviour and cracking
graphs were recorded, and the experimental results were discussed. Then, a unified theoretical
model for estimating the strength of these beams based on the limit equilibrium analysis was
The following conclusions can be drawn from this study:
1. The experimental research demonstrates how the ultimate strength of RC members changes
due to changes in combined loads. The experimental results conform to common sense and
can serve as a benchmark for theoretical analysis.
2. Torsion can reduce the ultimate capacity of beams under combined loading, and the axial
compressive force could affect the ultimate strength of beams. An appropriate compressive
force can increase the ultimate strength of the beam and the shear strength of the concrete.
However, a high compressive axial force may decrease the ultimate strength of the beam.
3. The crack inclination angles change with changes in combined load. The critical cracks
constitute the twist failure surface of the beams, which affects the ultimate strength of the
beams. When a compressive axial force is applied, the axial force changes the direction of
the principal stress in the concrete and thus changes the crack inclination angles.
4. According to the experimental research, a unified theoretical model was established by
determining the shape of the warped failure surface, assuming an appropriate stress
distribution on the failure surface, and considering the equilibrium conditions. The model
focuses on the influence of axial force on the spatial angle of the ultimate failure surface of
members, on the brittle or ductile failure model, and on the ultimate capacity of the
members. Based on the limit equilibrium analysis, the geometric shape and detailed dimensions
of the failure surface of an RC member can be determined for varying combined loads.
5. The unified theoretical formulas can be simplified into commonly accepted formulas for
RC members subjected to only one force of the four loads: a single axial force, bending,
shear or torsion. The formulas can also be used to calculate the combined ultimate strength
under combined loading, which illustrates that the unified theoretical model is a more
general model and confirms the validity of the model.
6. The accuracy of the proposed unified theoretical model is proven with experimental
research. The agreement between the experimental results and model illustrates that the
proposed theoretical model can accurately estimating the ultimate strength of rectangular
RC members under combined axial force, bending, shear and torsion loading.
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The authors acknowledge the National Natural Science Foundation of China for its financial
support of this research project (No. 51178265).
Conceptualization: PW ZH.
Data curation: PW.
Formal analysis: PW.
Funding acquisition: ZH.
Investigation: PW ZH.
Project administration: ZH.
Writing ± original draft: PW.
Writing ± review & editing: PW ZH.
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Collins MP . Torque-twist characteristics of reinforced concrete beams . 1973 : 211 ±231 Mitchell D. Diagonal compression field theory-A rational model for structural concrete in pure torsion .
Aci J 1974 ; 71 .https://doi.org/10.14359/7103 Vecchio F, Collins MP . Stress-strain characteristics of reinforced concrete in pure shear: Delft , 1981 .
Vecchio FJ , Collins MP . The Modified Compression-Field Theory for Reinforced Concrete Elements Subjected to Shear . ACI Structural Journal 1986 ; 83 ( 2 ): 219 ± 231 . https://doi.org/10.14359/10416 Vecchio FJ. Disturbed Stress Field Model for Reinforced Concrete: Formulation . Journal of Structural Engineering 2000 ; 126 ( 9 ): 1070 ± 1077 . https://doi.org/10.1061/(asce) 0733 - 9445 ( 2000 ) 126 : 9 ( 1070 ) Vecchio FJ . Disturbed stress field model for reinforced concrete: Implementation . Journal of Structural Engineering 2001 ; 127 ( 1 ): 12 ± 20 . https://doi.org/10.1061/(asce) 0733 - 9445 ( 2001 ) 127 : 1 ( 12 ) Vecchio FJ , Lai D , Shim W , Ng J . Disturbed Stress Field Model for Reinforced Concrete: Validation.
Journal of Structural Engineering 2001 ; 127 ( 4 ): 12 ± 20 . https://doi.org/10.1061/(asce) 0733 - 9445 ( 2001 ) 127 : 4 ( 350 ) Hsu TTC . Softened Truss Model Theory for Shear and Torsion . ACI Structural Journal 1988 ; 85 ( 6 ): 624 ± 635 .https://doi.org/10.14359/2740 Pang XB, Hsu TTC . Fixed Angle Softened Truss Model for Reinforced Concrete . ACI Structural Journal 1996 ; 93 ( 2 ): 197 ± 207 .https://doi.org/10.14359/1452 Hsu TTC, Mo YL . Unified Theory of Concrete Structures . Wiley, 2010 .
Gvozdev AA. Research on Reinforced Concrete Beams under Combined Bending and Torsion in the Soviet Union . 1968 : Huang Z , Liu XL . Modified Skew Bending Model for Segmental Bridge with Unbonded Tendons . Journal of Bridge Engineering 2006 ; 11 ( 1 ): 59 ± 63 . https://doi.org/10.1061/(asce) 1084 - 0702 ( 2006 ) 11 : 1 ( 59 ) Wang P , Huang Z , Sun L . A Unified Model of Ultimate Capacity of RC Members with a Rectangular Section under Combined Actions . International Conference on Sustainable Development of Critical Infrastructure. Proceedings of the 2014 International Conference on Sustainable Development of Critical Infrastructure; 2014 ; Shanghai: ASCE; 2014 . https://doi.org/10.1061/9780784413470.020 14.
Nielsen MP . Limit analysis and concrete plasticity . Rnford Onr 1984 : Huang Z , Liu XL . Unified Approach for Analysis of Box-Section Members under Combined Actions .
Journal of Bridge Engineering 2007 ; 12 ( 4 ): 494 ± 499 .https://doi.org/10.1061/(asce) 1084 - 0702 ( 2007 ) 12 : 4 ( 494 ) Huang L , Lu Y. Unified Calculation Method for Symmetrically Reinforced Concrete Section Subjected to Combined Loading . ACI Structural Journal 2013 ; 110 ( 1 ): 127 ± 136 .https://doi.org/10.14359/51684336 Rossi PP, Recupero A. Ultimate Strength of Reinforced Concrete Circular Members Subjected to Axial Force, Bending Moment, and Shear Force . Journal of Structural Engineering 2013 ; 139 ( 6 ): 915 ± 928 .
https://doi.org/10.1061/(asce)st. 1943 - 541x .0000724 Rossi PP . Evaluation of the ultimate strength of R.C. rectangular columns subjected to axial force, bending moment and shear force . Engineering Structures 2013 ; 57 ( 4 ): 339 ± 355 .https://doi.org/10.
1016/j.engstruct. 2013 . 09 .006 Panjehpour M , Chai HK , Voo YL . Refinement of Strut-and-Tie Model for Reinforced Concrete Deep Beams . PLoS One 2015 ; 10 ( e01307346 ). https://doi.org/10.1371/journal.pone.0130734 Rahal KN , Collins MP . Simple model for predicting torsional strength of reinforced and prestressed concrete sections . ACI Structural Journal 1996 ; 93 ( 6 ): 658 ± 666 . https://doi.org/10.14359/512