Corbel Design Paper

ASIAN JOURNAL OF CIVIL ENGINEERING (BUILDING AND HOUSING) VOL. 6, NOS. 1-2 (2005) PAGES 21-33

DESIGN OF A DOUBLE CORBEL USING THE STRUT-AND-TIE

METHOD

Bhupinder Singh∗1, Yaghoub Mohammadi2 and S.K. Kaushik3 1Department of Civil Engineering, National Institute of Technology,

Jalandhar-144 011, India, 2Department of Civil Engineering, University of Mohaghegh Ardebili, Ardebil, Iran,

3Department of Civil Engineering, IIT Roorkee, Roorkee-247 667, India

ABSTRACT

The strut-and-tie method can be used for the design of regions of structures where the basic assumptions of flexure theory, namely plane sections remaining plane before and after bending, are not applicable. Such regions occur near force discontinuities arising from concentrated forces or reactions and near geometric discontinuities such as abrupt changes in cross section etc. The strut-and-tie method of design is based on the assumption that appropriate regions in concrete structures can be analysed and designed using hypothetical pin-jointed trusses consisting of struts and ties connected at nodes. Although IS 456:2000 recommends the strut-and-tie method for design of corbels, no guidelines are given for determination of concrete strut and node dimensions and for the allowable stresses for design. The ACI Code 318-02 on the other hand, gives explicit and practical recommendations related to the design of structural members using the strut-and-tie method. To amplify the application of these recommendations to the design of structural concrete a complete example on the analysis and design of a double corbel using the strut-and-tie method has been presented.

Keywords: corbel, strut, tie, node, truss, detailing

LIST OF NOTATIONS

Pv = Design Vertical Load on Corbel Ph = Design Horizontal Load on Corbel fck = Characteristic Cube Compressive Strength of Concrete f’

c = Specified Cylinder Compressive Strength of Concrete fy = Specified Yield Strength of Reinforcement fcu = Effective Compressive Strength of Concrete in a Strut or a Nodal Zone βn = Factor whose values are specified in Clauses A.5.2.1 through A.5.2.3 of ACI 318-02 ∗ Email-address of the corresponding author: [email protected]

Bhupinder Singh, Yaghoub Mohammadi and S.K. Kaushik

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Φ = Strength Reduction Factor a = Shear Span, Distance between Concentrated Load and Face of Support d = Distance from Extreme Compression Fibre to Centroid of Longitudinal Tension Reinforcement D = Overall Depth of Corbel at Column face h = Overall Thickness of Member Ws = Width of Strut b = Out-of-plane Dimension of Corbel Ast = Area of Tension Reinforcement Astmin. = Required Minimum Area of Tension Reinforcement βs = Factor whose values are specified in Clauses A.3.2.1 through A.3.2.4 of ACI 318-02 AH = Area of Shear Reinforcement Parallel to Flexural Tension Reinforcement An = Area of Reinforcement in Bracket or Corbel Resisting Tensile Force Nuc Nuc= Tension Force Applied at Top of Bracket or Corbel Asi = Area of Surface Reinforcement in the ith layer crossing a strut Si = Spacing of Reinforcement in the ith layer Adjacent to the Surface of the Member γ i = Angle Between the Axis of a Strut and the Bars in the ith layer of Reinforcement

Crossing the Stru

INTRODUCTION

The corbel geometry and the loads for which the corbel has to be designed are shown in Figure 1. The principal dimensions of the corbel may not be known a priori and the fixing of such dimensions is explained later on. The structure under consideration is a region of discontinuity since it encompasses abrupt changes in geometry and is in the vicinity of concentrated forces. The conventional flexure theory is not applicable to regions of discontinuities and hence the application of the strut-and-tie method for the design of the corbel is justified. The structure will be designed as per recommendations of Appendix-A of ACI 318-02, Ref. [1] (here-in-after referred to as the Code) and other applicable clauses of the same Code.

LOADS AND MATERIAL PROPERTIES

Design vertical load on each corbel= Pv = 325 kN (Figure 1) Design horizontal load on each corbel = Ph = 75 kN (Figure 1) Design axial load on column = 950 kN (Figure 1) Characteristic cube compressive strength of concrete (assumed) = fck = 25 MPa Take cylinder compressive strength of concrete = f’c = 0.80 fck = 20 MPa Yield strength (0.2% proof stress) of reinforcement bars = fy = 415 Mpa

DESIGN OF BEARING PLATES

The sizes of the bearing plates located at the loading points have to be determined. The

DESIGN OF A DOUBLE CORBEL USING THE STRUT-AND-TIE METHOD...

23

bearing plates at the load locations, as will be seen later on, are resting on the underlying CCT (Compression-Compression-Tension) nodes of the strut-and-tie model. The applied load on the bearing plates exerts a bearing stress on a ‘face’ of the hypothetical nodal zone, which is assumed to enclose the CCT node. This bearing stress should be within Code specified limits for CCT nodes.

Figure 1. Loads on the double corbel

Assume the size of the bearing plates as 300 x 200 mm, the longer dimension being along

the plane of the corbel.

Hence, the bearing stress in concrete at loading points = 200300

310325×

× = 5.42 MPa

As per Clause A.5.2 eq. A-8 [1], the effective compressive stress at the face of a node=fcu=0.85βn f′c .

The values of the parameter βn is specified in Clauses A.5.2.1, A.5.2.2 and A.5.2.3 [1]. For the CCT node which anchors one tie (the ‘T’ in CCT), Clause A.5.2.2 [1] is applicable

wherein βn = 0.80. Therefore, fcu= 0.85 x 0.80 x 20 = 13.60 MPa The allowable bearing stress is equal to Φfcu where Φ is the strength reduction factor [1].

As per Clause 9.3.2.6 [1], for strut-and-tie models, Φ= 0.75 Hence, Φfcu = 0.75 x 13.60 =10.20 > 5.42 MPa, ok. Hence, the assumed size of the bearing plates is adequate.

SELECTION OF CORBEL DIMENSIONS

As per Clause 11.9.1 of Ref. [1], for application of the recommendations related to the strut-and-tie method given in Appendix-A of Ref. [1] for the design of corbels, the span-to-depth


24

ratio of the corbel, da (Figure 2), should be less than 2. Further, Clause 11.9.2 of Ref. [1]

requires the corbel depth beyond the bearing area to be at least 0.5 d.

Figure 2. Structural action of a corbel

Select overall depth of corbel at column face, D, equal to 500 mm. Select a depth of 250

mm at the free end of the corbel. Allow a clearance of 100 mm from the outer edge of the bearing plate to the free edge of

the corbel. The selected dimensions of the double corbel are presented in Figure 3.

Figure 3. Selected dimensions of the double corbel


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SELECTION OF STRUT-AND-TIE MODEL

Multiple strut-and-tie models can be developed for a single load case. Some options for the double corbel being designed are shown in Figure 4. The moot question at this stage is how to select the optimal model. Usually, that model is the best in which the loads follow the path with the least force and the least deformation [2]. At the same time, since ties are more deformable than concrete struts, the model with the least number and the shortest ties is likely the best. This requirement can be quantified as ΣFIliεmi = minimum, where Fi is the force in the strut or tie, li is the length of the member i and εmi is the mean strain in member i Ref. [2]. In addition to the above requirements, the selected strut-and-tie model should be such that the angle between the axes of the struts and ties acting at a node should be as large as possible so as to mitigate possible cracking and to avoid incompatibilities due to shortening of the struts and lengthening of the ties occurring in almost the same directions. The ACI Code of Ref. [1], recommends that the angle between the axes of any strut and tie entering a single node shall not be less than 25º.

In light of the above recommendations, the strut-and-tie model of Figure 4 (a) is selected for modeling the double corbel.

Figure 4. Strut and tie models for a double corbel


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DETERMINATION OF TRUSS FORCES

The forces in the members of the strut-and-tie model are determined from conditions of static equilibrium. The location and orientation of the struts and ties is defined by the position of the nodes. The selected strut-and-tie model for the double corbel is shown in Figure 5. The center of the top horizontal tie BC (Figure 5), is assumed to be at a distance of 75 mm from the top of the corbel. Since the tie capacity is assumed to be furnished exclusively by the reinforcement bars, the assumed cover takes care of requirements for clear cover to the tie reinforcement bars and for the provision, if required, for two layers of reinforcement bars for the tie BC. Like wise, all ties are assumed to have an effective cover of 75 mm.

Figure 5. Selected strut and tie model for the double corbel

Hence, d = 500 –75 = 425 mm.

Therefore, da

= 425200

= 0.47 < 2, ok..

The strut AD is assumed to lie on an imaginary horizontal line running through the point of intersection of the sloping face of the corbel and the vertical face of the column.

As shown in Figure 5, the column axial load of 950 kN is resolved into two equal loads of 475 kN each acting in line with strut AG and DH respectively.

The position of the centerline of the strut AE can be found by calculating the width sw of strut AE (Figure 5).

Let FAE be the required compressive force in strut AE. As per Clause A.3.1 of Ref. [1], the nominal compressive strength of a strut = Fns = ΦfcuAc , where CA is the cross-sectional


27

area at one end of the strut and fcu is the effective compressive strength of concrete in the strut.

As per Clause A.3.2 eq. A-3 [1], fcu = 0.85βs fc . Assuming strut AE to have a uniform cross-sectional area (prismatic strut), the value of

βs as per Clause A.3.2.1 of Ref. [1], is equal to 1.0. AC = wsb, where b is the out-of-plane dimension of the corbel and is equal to 350 mm in

this case. Hence, FAE = 0.75 x 0.85 x 1.0 x 20 x 350 x sw --------------(i) By inspection of the conditions of loading symmetry in Figure 5, it can be easily

determined from conditions of equilibrium that the force in the strut

AE=FAE= 3252

950+ =800 kN

Substituting for FAE in (i) and solving, gives sw = 179.27, say 180 mm.

Hence, the centerline of the strut AE and the center of node A are located at 2

180 = 90

mm from the vertical face of the column. Therefore, the geometry of the strut-and-tie model is fixed.

The forces in all the members of the strut-and-tie model are determined from conditions of static equilibrium and are summarized in Table 1 and also illustrated in Figure 6.

Figure 6. Computed forces in the members of the strut-and-tie model


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Table 1. Computed truss forces

Member AB = CD BC AD AE = DF AG = DH

Force (kN) +386.77 - 284.68 +221.84 +800 +475

+Compression, -Tension

DESIGN OF TIES

The capacity for tie BC is furnished by steel reinforcement and concrete is not assumed to carry any tensile loads.

The area of reinforcement required for tie BC is equal to Ast = Y

BCF

σ, where FBC is the

tensile force in tie BC and σy is the permissible tensile stress in the steel reinforcement and is equal to Φfy yfΦ where Φ is the strength reduction factor for the reinforcement yield

stress fy yf and is taken as 0.75 as per recommendations of Clause 9.3.2.6 of Ref. [1].

Therefore, Ast = 41575.01068.284 3

××

= 914.63 mm2

As per Clause 11.9.5 [1], the required minimum area of tensile reinforcement =

bdff

y

c

'

04.0 = 0.04 42535041520

×

= 286.74 mm2 < 914.63 mm2, ok.

Clause A.4.2 of Ref. [1] requires tie reinforcement to be distributed approximately uniformly over the width of the tie. This may entail furnishing the tie reinforcement in several layers rather than concentrating the rebars in a section normal to the plane of the member being designed, as is usually done in the case of beams.

Provide 6 bars of 16 mm diameter in two layers of 3 bars each as tie reinforcement. Area of steel provided = 6 x 201 = 1206 > 914.63 mm2, ok.

CHECK ON NODAL ZONES AND ANCHORAGES

The nodal zones of interest are at nodes A, B, C and D. The dimensions of the nodal zones have to be such that the stresses acting on the faces of the nodal zones are within permissible limits.

The width sw of strut AE was so determined that the stresses in the strut are within permissible limits. Strut AE bears on one face of nodal zone A and Appendix-1 of Ref. [1] suggests that the faces of nodal zones loaded in compression shall have the same widths as that of the ends of the struts bearing on them. In consequence, the stresses in nodal zone A (and D) are deemed to be within permissible limits.


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Therefore, only nodal zones B (and C) need to be checked. Nodal zone B encompasses a CCT node. If any of the forces acting on a node is tensile, the required minimum width of a side of the nodal zone containing such a node is calculated from the width of a hypothetical bearing plate anchoring one end of the tie which is assumed to exert a uniform bearing pressure on the back side of the nodal zone, Figure 7. The width of the hypothetical bearing plate in turn is equal to the width of the tie anchored in the node.

Figure 7. Tie width in a CCT node

The permissible bearing pressure for a CCT node has already been computed as = cufΦ =

10.20 MPa

Therefore, the required width of the tie to be anchored in the CCT node = bf

F

cu

BCΦ

=

35020.101068.284 3

×× = 79.74 mm

Available tie width = 75 x 2 = 150 > 79.74 mm, ok. To provide positive anchorage to tie BC, weld the six 16 mm bars to a steel angle ISA

150 x 150 x12 located at the free end of the corbel, as shown in Figure 8.

CHECK ON STRUTS

The check on struts involves determination of strut widths required to shoulder the computed strut forces and to determine whether the required strut widths will fit in the geometry of the structure.


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Figure 8. Strut and tie dimensions

Consider strut AD The effective compressive strength of strut AD = Φfcu, where fcu = 0.85βsfc.

Strut AD is expected to have a uniform cross-sectional area throughout its length (prismatic strut). For such struts, as per Clause A.3.2.1 [1], the parameter βs =1.0.

Therefore, required width of strut AD = bf

F

cu

ADΦ

= 35020185.075.0

1084.221 3

××××× = 49.71 mm.

Select a width of 100 mm for strut AD. It can be verified that this width can be accommodated in the geometry of the structure, Figure 8.

Consider struts AE and DF The required widths of struts AE and DF have already been calculated as 180 mm each and it can be verified that this width can be accommodated in the geometry of the structure, Figure 8. Consider struts AG and DH The effective compressive strength of strut AG = Φfcu, where fcu = 0.85βsf′c .

Strut AG is expected to be a bottle-shaped strut as the width of the compressed concrete in the strut at its mid-length can spread laterally. If no transverse crack control reinforcement to resist the bursting forces is provided in this strut then as per recommendations of Clause A.3.2.2 [1], βs = 0.60.

Therefore, required width of struts AG (and DH)=bf

F

cu

AGΦ

=3502060.085.075.0

10475 3

×××××


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= 177.40 mm. Select a width of 180 mm for each of the struts AG and DH such that their widths are

also equal to the widths of the underlying struts AE and DF respectively. It can be verified that these widths can be accommodated in the geometry of the structure, Figure 8.

Consider struts AB and CD The effective compressive strength of struts AB and CD = Φfcu, where fcu = 0.85βsf′c .

Struts AB and CD are non-prismatic and assuming that the required transverse crack control reinforcement is provided in them, the parameter βs as per recommendations of Clause A.3.2.2 [1] can be taken equal to 0.75.

Therefore, required width of strut AB (and CD) =bf

F

cu

ABΦ

=3502075.085.075.0

1077.386 3

×××××

= 115.56 mm. The above required strut width is compared with the available strut width at the smaller

end of the strut AB. A perusal of the geometry of strut AB in Figure 8 reveals the width of the smaller end of the strut to be 147 mm> 115.56 mm. Hence, the available strut width is adequate.

All the strut widths fit within the geometry of the corbel and thus the proposed strut-and-tie model is acceptable.

CRACK CONTROL REINFORCEMENT

For adequate crack control in the vicinity of a tie, Clause 11.9.4 of Ref. [1] requires the provision of reinforcement in the form of horizontal stirrups, the plane of these stirrups being parallel to the reinforcement in the tie. The Clause further specifies this crack control

reinforcement to be distributed uniformly within 32 of the effective depth ( 3.283425

32

=× ,

say 280 mm) of the corbel adjacent to the tie BC. As per Clause 11.9.4 of Ref. [1], the area of these horizontal stirrups must not be less

than Ah = 0.5 (Ast – An), where An is the reinforcement area required to resist the horizontal load at the bearings and Ast is the reinforcement provided in the tie BC.

Therefore, Ah = 0.5

×

×−

41585.010751206

3

= 496.69 mm2

Provide 4 numbers of 10 mm diameter two legged horizontal stirrups within 280 mm

from the top edge of the corbel at an average spacing of 3

280= 93.33 mm, say 90 mm c/c.

Area of stirrups provided = 628.31 > 496.69 mm2, ok. Since βs has been assumed as 0.75 while computing the strength of struts AB and CD,

confining reinforcement is required for these struts to take care of the transverse tensile stresses. According to Clause A.3.3.1 of Ref. [1], for concrete strengths less than 42 MPa, the requirement of confining reinforcement shall be deemed to be satisfied if the axis of the


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strut under consideration is crossed by reinforcement that satisfies the following relationship:

∑ ≥ii

si

bsA

γsin 0.0030

where siA is the total area of reinforcement at a spacing is making an angle of iγ to the axis of the strut, iγ not being less than 40°.

For this double corbel, the crack control reinforcement provided above crosses the axis of the struts AB and CD at an angle of 55° (Figure 8).

Hence, ∑ γ ii

si sinbsA

= 55sin9035031.628

×° = 0.016 > 0.003, ok.

Therefore, the provided reinforcement serves the purpose of crack control as well as confinement.

Also provide three numbers of 12 mm diameter framing bars in the double corbel. The detailing of reinforcement in the double corbel is presented in Figure 9.

Figure 9. Detailing of reinforcement in the double corbel

Acknowledgement: Generous assistance has been derived from ACI SP-208 of Ref. [3] while preparing this manuscript which is gratefully acknowledged


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REFERENCES

1. ACI 318-02, “Building Code Requirements for Structural Concrete (ACI 318-02) and Commentary (ACI 318R-02) “, American Concrete Institute, Michigan, USA, 2002, 443 pp.

2. Schlaich, J., Schafer, K. and Jennewin, M., “Toward a Consistent Design of Structural Concrete”, PCI Journal, May-June 1987, pp. 75-146.

3. SP-208, “Examples for the Design of Structural Concrete with Strut-and-Tie Models”, American Concrete Institute, Michigan, USA, 2003, 242 pp.

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Corbel Design Paper