7/28/2019 Design Charts for Corbels http://slidepdf.com/reader/full/design-charts-for-corbels 1/4 Paper: A l-Shawi Paper Design charts for corbels F. A. N. Al-Shawi, BEng, PhD Sheffield Hallam University Synopsis The designof corbels using the strut-and-tie model entails a trial-and-error procedure in the determination of tension reinforcement. This is a laborious method and mightalso lead to unacceptable depth ofthecompression block or the verti cal shear capacity being exceeded. This meansthat the whole procedure needs to be repeated until all design criteria are satisfied. In this paper; design charts based on BS 8110and EC2 recommendations are presented. These charts take into consideration all constraints. Therefore, they can be used as a design aid. Notation is theareaof tension reinforcement is theareaof concrete cross-section are the distances from thelineof action of the load to theroot of the corbel in BS 8 l0 and EC2, respectively is the width of corbel is the compressive force developed by the concrete (strut force) is the effective depth at the root of thecorbel is the total tensile force to bedeveloped by the tension reinforcement is the vertical force acting on thecorbel (EC2) is thecharacteristic compressive strength of concretecylinders is the characteristic compressive strength of concrete cubes (BS 81 10) arethe characteristic strengths of steel reinforcement in BS 8 IO and EC2 respectively are the depths at the root of corbel in BS 81 10 and EC2, respectively arethe horizontal forces acting on the corbel in BS 81 10 andEC2, respectively is the non-dimensional factor, =V/bdfc,,(BS 81 10) is thedepth factor for shear resistance is thedesign axial force (compression positive) is theratio of the depth of the compression zoneto the effective depth of the corbel =x/d is the non-dimensional factor, =A,f,/bdf,.,, (BS 81 1 0) is the. tensile force in the strut-and-ti e model is thevertical force acting on the corbel (BS 8 I O) is the design shear resistance without shear reinforcement is the maximumdesign shear force that can becarried without crushing is the design shear stress at a cross-section (BS 8110) is thedesign concrete shear stress (BS 81 10) is the depth of the compression zone at the root of corbel =a,/d (B S 8 10) =a,/d (EC2) is the shear enhancement factor is thepartial safety factor for concrete is the partial safety factor for steel reinforcement is the angle between the direction of the strut compressive force and the horizontal is thecoefficient of friction between the contact surfaces atthe support is the efficiency factor used in the assessment of shear strength is the average stress in concrete due to axial force is thebasic design shear strength of members without shear reinforcement =F,/bdf,k (EC2) = A,fJ bdf,.k (EC2) Charts based on BS 8110’ The recommendations, as laid out in clause 5.2.7, are as follows: (a) The distance a,. between the lineof the reaction to the supported load V and the root of the corbel is lessthan d (the effectivedepth of the root of the corbel); and (b) the depth at the outer edge of the contact areaof the supported load is not lessthan one-half of the depth at the root of the corbel. Thedesign simplifyingassumptionsarethat the concrete andreinforcement may be assumed to act as elements of a simple strut-and-tie system. Note that other systems are also available?. Referring to Fig l(a) V= C sin 8 T= CCOS~ where (seeclause 3.4.4.4 for the stressblock) and 1 -0.45n : tan 8= - hence, (1 -0.45n) sin = JW .... (1) .... ( 2) .... ( 3) .... ( 4) .... ( 5) and, =J ....( 6) substitution of eqns. (3), (S) and (6) in eqn. (1) (noting that ‘yc = l .SO, as given in clause 2.4.4) gives v=-) 1, .9x b a (1 -0.45n) 1 .... ( 7) a2+ 1-0.45n)’I dividing by bdf,.,, (noting that x = nd) and introducing the parameter K =V&&.,,, eqn. (7) becomes (0.202K+0.182a)n2-(0.9K+0.405a)n+K(l +a2)=0 .... 8) solving the above quadratic equation gives (0.9K + 0.40%~) ,/(0.9K + 0.405~~)~ 4(0.202K + 0.182a)K(1+ a’) n= 2(0.202K + 0. 182~~) ....( 9) Having found n, the tensile force T can be evauated from (2) as T= (0.y) 0.9 x b a2 a2+ (1- 0.45n)’ JlO) If thehorizontal fri ctional force H ( = pV) s addedto the above force, then the total tensil e force is given by: 222 The Structural EngineerNolume 74/No 13/2 July 1996
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Design charts for corbelsF.A. N. Al-Shawi,BEng, PhDSheffield Hallam University
SynopsisThe designof corbels using the strut-and-tie model entails atrial-and-error procedure in the determination of tensionreinforcement. This is a laborious method and mightalso lead tounacceptable depth ofthe compression block or the vertical shearcapacity being exceeded. This means that the whole procedureneeds tobe repeated until all design criteria are satisfied.
In thispaper; design chartsbased on BS 8110and EC2recommendations are presented. These charts take intoconsideration all constraints. Therefore, they can be used as adesign aid.
Notationis the area of tension reinforcement
is the area of concrete cross-section
are the distances from the line of action of the load to the root of
the corbel in BS 8 l0and EC2, respectively
is the width of corbel
is the compressive force developed by the concrete (strut force)
is the effective depth at the root of the corbel
is the total tensile force to be developed by the tension
reinforcement
is the vertical force acting on the corbel (EC2)
is the characteristic compressive strength of concrete cylinders
is the characteristic compressive strength of concrete cubes(BS8110)
are the characteristic strengths of steel reinforcement inBS8 I Oand EC2 respectively
are the depths at the root of corbel in BS 8110and EC2,
respectively
are the horizontal forces acting on the corbel in BS 8110and EC2,
respectively
is the non-dimensional factor,
=V/bdfc,,(BS 8110)
is the depth factor for shear resistance
is the design axial force (compression positive)
is the ratio of the depth of the compression zone to the effective
depth of the corbel=x/d
is the non-dimensional factor,
=A,f,/bdf,.,,(BS8110)
is the. tensile force in the strut-and-tie model
is the vertical force acting on the corbel (BS 8 I O)
is the design shear resistance without shear reinforcement
is the maximum design shear force that can be carried without