Shearflexural strength mechanical model for the design and assessment of reinforced concrete beams subjected to point or distributed loads
Antonio MARÍ^{1,}^{*}(),Antoni CLADERA^{2},Jesús BAIRÁN^{1},Eva OLLER^{1},Carlos RIBAS^{2}
1. Department of Construction Engineering, Universitat Politècnica de Catalunya, Barcelona 08034, Spain 2. Department of Physics, University of Balearic Islands, Palma de Mallorca, Spain
A mechanical model recently developed for the shear strength of slender reinforced concrete beams with and without shear reinforcement is presented and extended to elements with uniformly distributed loads, specially focusing on practical design and assessment in this paper. The shear strength is considered to be the sum of the shear transferred by the concrete compression chord, along the crack, due to residual tensile and frictional stresses, by the stirrups and, if they exist, by the longitudinal reinforcement. Based on the principles of structural mechanics simple expressions have been derived separately for each shear transfer action and for their interaction at ultimate limit state. The predictions of the model have been compared to those obtained by using the EC2, MC2010 and ACI 31808 provisions and they fit very well the available experimental results from the recently published ACIDAfStb databases of shear tests on slender reinforced concrete beams with and without stirrups. Finally, a detailed application example has been presented, obtaining each contributing component to the shear strength and the assumed shape and position of the critical crack.
Online First Date: 11 December 2014Issue Date: 12 January 2015
Cite this article:
Antonio MARí,Antoni CLADERA,Jesús BAIRáN, et al. Shearflexural strength mechanical model for the design and assessment of reinforced concrete beams subjected to point or distributed loads[J]. Front. Struct. Civ. Eng.,
2014, 8(4): 337353.
Fig.1 Sequence of cracking evolution in a shear failing element. Adapted from Ref. [1]
Fig.2 Distribution of shear stresses in the imminent failure situation and qualitative distribution of the different contributing actions
Fig.3 Position of the shear critical section in the beam
contributing component
final simplified dimensionless
expressions
cracked concrete web
υw=167fctEc(1+2EcGffct2d),
(4)
longitudinal reinforcement
υs>0→vl=0.25x/d0.05,
(5a)
υs=0→vl=0,
(5b)
transversal reinforcement
υs=0.85ρwfywfct,
(6)
compression chord
υc=ζ[(0.88+0.70υs)x/d+0.02],
(7)
ζ=1.20.2?a≥0.65(ainmeters),
(8)
Tab.1 Summary of simplified expressions of dimensionless shear contributing components
Fig.4 Contribution of cracked concrete to shear resistance
Fig.5 Shear transfer mechanisms considered
Fig.6 Contribution of uncracked concrete chord to shear resistance
Fig.7 Beam subject to uniformly distributed loads. Shear forces and cracking pattern. Adapted from Ref. [27] (unit: m)
720 beams without stirrups
85 beams with stirrups
min
max
min
max
b/mm
50
3005
125
457
d/mm
65
3000
198
1200
f_{cm}/MPa
13
139
16
125
r/%
0.14
6.64
0.50
4.73
a/d
2.40
8.10
2.45
5.00
A_{sw}·f_{yw} /MPa


0.32
3.07
V_{test }/kN
7
1308
87
1172
Tab.2 Range of variables in the employed databases
V_{test}/V_{pred}
720 beams without stirrups
85 beams with stirrups
EC2
ACI31808
MC10Lev II
proposal
EC2
ACI31808
MC10Lev III
proposal
average
1.07
1.22
1.31
1.05
1.52
1.26
1.21
1.06
median
1.03
1.22
1.27
1.02
1.53
1.25
1.22
1.06
standard deviation
0.249
0.349
0.272
0.192
0.377
0.240
0.209
0.165
COV/%
23.34
28.67
20.81
18.28
24.86
19.10
17.28
15.54
minimum
0.40
0.26
0.51
0.54
0.53
0.68
0.75
0.65
(V_{test}/V_{pred})_{5%}
0.75
0.64
0.97
0.81
0.91
0.90
0.92
0.83
maximum
2.65
2.99
3.09
2.26
2.47
2.02
1.86
1.59
(V_{test}/V_{pred})_{95%}
1.55
1.81
1.78
1.37
2.16
1.65
1.58
1.31
Tab.3 Verification of the different shear design procedures
Fig.8 Correlation between the predictions and the experimental results
Fig.9 Correlation between the predictions and the experimental results of RC beams without shear reinforcement
Fig.10 Correlation between the predictions and the experimental results of RC beams with shear reinforcement
Fig.11 Beam geometry for the application example
a
shear span
b
width of concrete section
d
effective depth to main tension reinforcement
d_{max}
maximum aggregate size
f_{ck}
characteristic value of the cylinder concrete compressive strength
f_{cm}
mean value of the cylinder concrete compressive strength
f_{ct}
uniaxial concrete tensile strength
f_{ctm}
mean value of the concrete tensile strength
f_{yw}
yield strength of the transverse reinforcement
h
overall depth of concrete section
s
longitudinal coordinate from the support
s_{cr}
location of the section where the critical shear crack starts
s_{mx}
average crack spacing of inclined cracks along the beam axis
s_{m?}
average crack spacing of inclined cracks
s_{u}
location of the shear critical section
v_{c}
dimensionless contribution to the shear strength of the uncracked concrete chord
v_{l}
dimensionless contribution to the shear strength of the longitudinal reinforcement.
v_{s}
dimensionless contribution to the shear strength of the transverse reinforcement
v_{u}
dimensionless ultimate shear force
v_{u,0}
dimensionless ultimate shear force of beams and oneway slabs without transverse reinforcement
v_{w}
dimensionless shear force resisted along the crack
x
neutral axis depth
x_{w}
vertical projection of length along the crack where the tensile stresses are extended
z
lever arm
A_{s}
longitudinal reinforcement area
A_{sw}
area per unit length of the transverse reinforcement
C
compression force in the uncracked concrete chord
E_{c}
modulus of elasticity of concrete
E_{s}
modulus of elasticity of steel
G_{f}
concrete fracture energy
K_{?}
constant
M
bending moment
M_{cr}
cracking moment
R_{t}
ratio between the principal tensile stress and the tensile strength
T
tensile force in the longitudinal reinforcement
V
shear force
V_{c}
contribution to the shear strength of the uncracked concrete chord
V_{l}
contribution to the shear strength of the longitudinal reinforcement
V_{pred}
predicted value of the ultimate shear force
V_{s}
contribution to the shear strength of the transverse reinforcement
V_{sd}
design shear force
V_{t}_{est}
experimental value of the ultimate shear force
V_{u}
ultimate shear force
V_{u,0}
ultimate shear force of beams and oneway slabs without transverse reinforcement
V_{w}
shear force resisted along the crack
?_{e}
modular ratio (E_{s}/E_{c})
?_{ct,cr}
concrete strain at the beginning of macrocracking
?_{ct,u}
ultimate tensile strain
?_{s}
strain at the longitudinal reinforcement
ξ
size effect factor
?
distance from the neutral axis
?
inclination angle of the strut
?
longitudinal tension reinforcement ratio
?_{w}
transverse reinforcement ratio
?_{1}, ?_{2}
principal stresses
?_{x}
normal stress in the longitudinal direction
?_{y}
normal stress in the transverse direction
?_{w}
normal stress in a horizontal fiber in the cracked web(DOI: 10.1080/15732479.2014.964735)
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