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Frontiers of Structural and Civil Engineering

Front. Struct. Civ. Eng.    2014, Vol. 8 Issue (4) : 337-353     https://doi.org/10.1007/s11709-014-0081-0
RESEARCH ARTICLE |
Shear-flexural strength mechanical model for the design and assessment of reinforced concrete beams subjected to point or distributed loads
Antonio MARÍ1,*(),Antoni CLADERA2,Jesús BAIRÁN1,Eva OLLER1,Carlos RIBAS2
1. Department of Construction Engineering, Universitat Politècnica de Catalunya, Barcelona 08034, Spain
2. Department of Physics, University of Balearic Islands, Palma de Mallorca, Spain
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Abstract

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 318-08 provisions and they fit very well the available experimental results from the recently published ACI-DAfStb 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.

Keywords shear strength      mechanical model      reinforced concrete      design      assessment      shear tests     
Corresponding Authors: Antonio MARí   
Online First Date: 11 December 2014    Issue Date: 12 January 2015
 Cite this article:   
Antonio MARí,Antoni CLADERA,Jesús BAIRáN, et al. Shear-flexural 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): 337-353.
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http://journal.hep.com.cn/fsce/EN/10.1007/s11709-014-0081-0
http://journal.hep.com.cn/fsce/EN/Y2014/V8/I4/337
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Antonio MARí
Antoni CLADERA
Jesús BAIRáN
Eva OLLER
Carlos RIBAS
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 υ w = 167 f c t E c ( 1 + 2 E c G f f c t 2 d ) , (4)
longitudinal reinforcement υ s > 0 v l = 0.25 x / d - 0.05 , (5a)
υ s = 0 v l = 0 , (5b)
transversal reinforcement υ s = 0.85 ρ w f y w f c t , (6)
compression chord υ c = ζ [ ( 0.88 + 0.70 υ s ) x / d + 0.02 ] , (7)
ζ = 1.2 - 0.2 ? a 0.65 ( a i n m e t e r s ) , (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 un-cracked 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
fcm/MPa 13 139 16 125
r/% 0.14 6.64 0.50 4.73
a/d 2.40 8.10 2.45 5.00
Asw·fyw /MPa - - 0.32 3.07
Vtest /kN 7 1308 87 1172
Tab.2  Range of variables in the employed databases
Vtest/Vpred 720 beams without stirrups 85 beams with stirrups
EC-2 ACI318-08 MC10Lev II proposal EC-2 ACI318-08 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
(Vtest/Vpred)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
(Vtest/Vpred)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
dmax maximum aggregate size
fck characteristic value of the cylinder concrete compressive strength
fcm mean value of the cylinder concrete compressive strength
fct uniaxial concrete tensile strength
fctm mean value of the concrete tensile strength
fyw yield strength of the transverse reinforcement
h overall depth of concrete section
s longitudinal coordinate from the support
scr location of the section where the critical shear crack starts
smx average crack spacing of inclined cracks along the beam axis
sm? average crack spacing of inclined cracks
su location of the shear critical section
vc dimensionless contribution to the shear strength of the un-cracked concrete chord
vl dimensionless contribution to the shear strength of the longitudinal reinforcement.
vs dimensionless contribution to the shear strength of the transverse reinforcement
vu dimensionless ultimate shear force
vu,0 dimensionless ultimate shear force of beams and one-way slabs without transverse reinforcement
vw dimensionless shear force resisted along the crack
x neutral axis depth
xw vertical projection of length along the crack where the tensile stresses are extended
z lever arm
As longitudinal reinforcement area
Asw area per unit length of the transverse reinforcement
C compression force in the un-cracked concrete chord
Ec modulus of elasticity of concrete
Es modulus of elasticity of steel
Gf concrete fracture energy
K? constant
M bending moment
Mcr cracking moment
Rt ratio between the principal tensile stress and the tensile strength
T tensile force in the longitudinal reinforcement
V shear force
Vc contribution to the shear strength of the un-cracked concrete chord
Vl contribution to the shear strength of the longitudinal reinforcement
Vpred predicted value of the ultimate shear force
Vs contribution to the shear strength of the transverse reinforcement
Vsd design shear force
Vtest experimental value of the ultimate shear force
Vu ultimate shear force
Vu,0 ultimate shear force of beams and one-way slabs without transverse reinforcement
Vw shear force resisted along the crack
?e modular ratio (Es/Ec)
?ct,cr concrete strain at the beginning of macro-cracking
?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)
Tab.4  Notations
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