Structures Notes Notes 1993NST_5g
Shear Shear in reinfor ced concrete piles and circular col umns Ian Felth Felth am, AR&D, London Ab st rac t This Note proposes a procedure for designing piles and circular columns for shear; it recommends 1 that the method given in the Highways Agency’s design manual should not be used. A paper , making the same proposals as this Note, has been published in The Structural Engineer . This Note proposes a procedure for calculating the capacity of spiral links and discusses the 2 recommendations of the British Cement Association (BCA) in a paper by Clarke and Birjandi . This 3 paper confirms that design for shear in circular sections can follow the approach given in BS5400 4 and BS8110 for rectangular sections with the following specific definitions: (1) The area of tension reinforcement, A st, should be taken as the area of the steel below the mid-depth of the section (2) The effective depth, d, should be taken as the distance from the extreme compression fibre to the centroid of the tension reinforcement (3) When determining the shear capacity of the concrete or the shear reinforcement, the term ‘bvd’, later referred to as A v, should be taken as the area of concrete from the extreme compression fibre down to depth d. However, equilibrium necessitates necessitates the consideration consideration of both the links’ curvature and the asymmetry of spirals, neither of which is considered by the BCA’s proposals (BCA method. Although rules rules compatible with the first two of the above recommenda recommendations tions have been been incorporated incorporated 5 6 into BD42/00 and BD74/00 (BD74 method), these procedures state that the width of the section for the calculation of shear stresses should be taken as the pile diameter. This definition results in the shear stress being calculated using an area much greater than that of the section down to the neutral axis, sometimes even greater than the full cross-sectional area. This seems most unlikely, a misgiving confirmed by the BCA’s tests, which are discussed later in this Note. The following method is recommended for calculating the area of circular and spiral links to BS8110. BS8110 . It considers the links’ curvature and the asymmetry of spirals and provides a safe solution compared with the test results. A corresponding approach can be used with BS5400. BS5400 . Circular links
circ circul ular ar link link
long longit itud udin inal al stee steell
failure plane
link cut by failure plane
link forces acting on section to right of failure plane
b
Fi cosψi Fi
r s
α r
V
d
ψi β
r sv
a
Fn
a
r sv
F1
b centroid of steel below centre line
1a Cross section failure plane
d
1b Elevation showing failure plane
shear force taken by li nks equals sum of Fi cosψi, for i = 1 to n
1c Link forces on
Fig. 1 Geometry of section Feedback Notes are copyright and published for distribution only within Arup Group Ltd. They are not intended for any third party.
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The effective depth, d, is taken as the depth to the centroid of the steel below the centre line of the section (a-a on Fig.1a). The position of the centroid depends on the orientation of the bars; for six bars, the minimum number for a circular section, its distance below the centre line is varies between 0.577r s and 0.667r s. For larger numbers of bars the distance tends to 0.637r s (= 2r s/π) below the centre line. Since the orientation of the cage is usually unknown, it seems reasonable to use this value for all sections, unless better information is known. The inaccuracy in d will always be less than 4%. d = r(1+sin α) where sin α = 2r s/πr
(0 < α < π/2)
The design shear stress is calculated by assuming a uniform stress over the area, A v, of the segment of the circle above the line a-a: 2
Av = r (π/2 + α + sin
α cos α)
v = V/Av where V is the total shear force on the section
∴v = V / [r 2 (π/2 + α + sin α cos α)] The design concrete shear strength, v c, should be obtained from BS8110, Table 3.8. A s should be taken as half the total area of longitudinal steel and ‘b vd’ should be taken as A v. The value of v c should be enhanced for axial compression in accordance with clause 3.4.5.12. If v exceeds v c, design reinforcement is required (if v exceeds 0.5v c nominal links should be provided). The equation for the area of design reinforcement in BS8110, Table 3.7 can be rearranged to show that the shear force taken by the links equals (0.95f yv)Asv.d/sv. Therefore, the number of links, intersected by a failure plane, that contribute to the shear strength of any section equals the effective depth divided by the spacing of the links. If a failure plane through the links in a circular section (b-b on Fig.1b) is considered, it will be apparent that the applied shear force can only be resisted by a component of the force in the links (see Fig.1c). It is not obvious over what depth intersections of the link with the failure plane should be considered, but it is assumed that only the portion of the link within the effective depth contributes towards the strength of the section. Hence, the stress that can be taken by the links equals the sum of the components of force above the plane a-a, over a length of failure plane equal to the effective depth, d. The maximum design shear stress that can be taken by the links is given by: (0.95f yv ) A sv r 1 + sin α ( π / 2 + β + sin β cos β) 2 A v s v 1 + sin β where
sin β
=
2r s
π r sv
0 < β< π/2
This stress must be sufficient to meet the deficiency in the shear capacity:
∴ A sv ≥ where
k
2k.r .s v ( v − v c ) 0.95f yv
=
( π / 2 + α + sin α cos α)(1 + sin β) ( π / 2 + β + sin β cos β)(1 + sin α )
For practical geometries k can be taken as unity. 2r .s v ( v v c ) A sv 0.95 f yv Note that Asv follows the definition given in BS8110. Since each link is cut twice by t he shear plane, Asv is twice the cross sectional area of the link.
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Spiral links The shear plane will usually cut a spiral link twice in each revolution, but one side the spiral will be less beneficial than the other at resisting shear because of its inclination to the shear plane (see Fig.2). However, to prevent torsion being applied to the concrete section, the two sides of the spiral must take the same force. It is therefore proposed to consider twice the capacity of the less beneficial side rather than the net capacity of the spiral. Asvh, the required sectional area of the spiral link, is given by: A svh
≥
where
k.r .p( v − v c ) 0.95f yv
k
⎛ p ⎞ ⎟⎟ 1 + ⎜⎜ π 2 r ⎝ sv ⎠
=
2
× (π / 2 + α + sin α cos α )
⎡ ⎛ 1 + sin α ⎞ ⎛ sin α ⎞⎤ p ⎟⎟ − ⎟⎟⎥ (1 + sin β)⎜⎜ ⎢( π / 2 + β + sin β cos β)⎜⎜ + β π β 1 sin r sin ⎝ ⎠ ⎝ ⎠ sv ⎣ ⎦
and p is the pitch of the helix To ensure at least three intersections of the failure plane with the spiral, which are necessary for equilibrium, the contribution of spiral links to the shear resistance should only be included when p 0.5d. For link geometry complying with this limit, the above expression can be approximated by: 1 k = 1 − 0.225p / r A svh
r .p (v
≤
vc )
0.95f yv .(1 0.225p / r ) link forces acting on section to right of failure plane
failure plane p b
F’i cosψi
F’m F’i
V
d
ψi r sv
F’1
b d Link cut by failure plane on side where inclination of link is: less beneficial more beneficial
shear force taken by links equals sum of 2F’i cosψi, for i = 1 to m, where F’i=Fi / √[1 + (p / 2πr sv)2] and m is the number of intersections of the spiral with the failure plane where inclination of link is less beneficial
2a Elevation showing failure plane
2b Link forces on failure plane
Fig.2 Failure plane through spiral li nk The Britis h Cement Association t ests The BCA tested 50 specimens and generally obtained two results from each specimen as shear failure was tested at both ends. Most of the specimens were of 300mm diameter, although some were of 152mm and 500mm diameter. High yield longitudinal reinforcement was placed at eight locations around the section; for the 500mm specimens two or three bars were bunched at each location. Feedback Notes are copyright and published for distribution only within Arup Group Ltd. They are not intended for any third party.
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Shear reinforcement, where provided, was 6mm or 8mm mild steel circular links but six of the specimens contained 6mm spiral links of 150mm pitch. Nominal concrete strengths of 25MPa, 35MPa and 50MPa were used. The specimens were tested horizontally, supported and loaded though concrete saddles. Loads were applied by a hydraulic jack. The shear span was varied for the 300mm diameter specimens. Figs.3a, 3b and 3c compare the test results with calculated resistances, removing the partial factor for materials in the design equations. The basic shear strength, v c, has been factored by (3d/a v) for loads 7 applied at a distance, a v, closer than 3d to the support. This approach, used in BD44/95 , the standard for bridge assessment, was used in the BCA paper as it was considered a more accurate representation than the factor (2d/a v) given in both BS5400 and BS8110. Test specimens with no shear reinforcement
Fig.3a shows that for piles not containing shear reinforcement, the experimental resistances are generally a little greater than the theoretical ones calculated using the first three recommendations of the BCA paper. The mean of the ratio of test resistance to theoretical resistance is 1.11, with a standard deviation of 0.13. This theoretical resistance is referred to as V c in this paper. Using the BD74 method, the mean ratio reduces to 0.99, with a standard deviation of 0.11. These figures support the observation that the BD74 method is too optimistic in calculating the shear strength of the section without shear reinforcement.
400 ) 350 N k ( e 300 c n a t s 250 i s e r r a 200 e h s l a 150 t n e m i r 100 e p x E
recomended & BCA
50
BD74
0 0
50
100
150
200
250
300
350
400
Theoretical shear resistance (kN)
3a Sections with no shear reinforcement Test specimens with spirals
Test specimens w ith links
2.0
2.0
c V / 1.8 ) e c n a t s i s 1.6 e r r a e h s l 1.4 a t n e m i r e p 1.2 x E (
c V / 1.8 ) e c n a t s i s 1.6 e r r a e h s l a 1.4 t n e m i r e p 1.2 x E (
recommended BCA BD74
recommended BCA BD74
1.0
1.0 1.0
1.2
1.4
1.6
1.8
2.0
1.0
1.2
1.4
1.6
1.8
2.0
(Theoretical shear resistance) / Vc
(Theoretical shear resistance) / Vc
3b Sections with link reinforcement
3c Sections with spiral reinforcement
Fig.3 Comparison of experimental with theoretical shear capacities
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Figs.3b and 3c compare results for piles with circular links and spiral reinforcement respectively. The theoretical resistances have been calculated using the recommended, BCA and BD74 methods. Both the experimental and theoretical resistances have been normalised by V c, the theoretical resistance calculated using the first three recommendations of the BCA paper. Presenting the results in this way emphasises that the contribution by the shear reinforcement in all the tests is modest, in all cases being less than Vc for the links and 0.5V c for the spirals. There is considerable scatter in these results and it is difficult to conclude which calculation method represents the test results best. However, there are no tests for situations where the reinforcement has to take a shear force equivalent to many times Vc, where the difference between the recommended approach, which considers curvature, and the other two methods will be more significant. The BCA method uses a simple reduction factor for spiral links, based on the sine of the angle between the spiral and the longitudinal axis. The theory to obtain this factor again ignored the curvature of the links and averaged the contribution of the two sides of the spiral. This may not be a safe assumption and cannot be confirmed by the experimental tests as only six specimens contained spiral links and they were all the same size with links of the same diameter and pitch. References (1) FELTHAM, I. Shear in reinforced concrete piles and circular columns. The Structural Engineer , 82(11), pp.27-31, 2 June 2004. (2) CLARKE, J.L. and BIRJANDI, F.K. The behaviour of reinforced concrete circular sections in shear. The Structural Engineer , 71(5), pp.73-78 & 81, 2 March 1993. (3) BRITISH STANDARDS INSTITUTION. BS5400: Part4: 1990. Steel, concrete and composite bridges. Part 4. Code of practice for design of concrete bridges. BSI, 1990 (4) BRITISH STANDARDS INSTITUTION. BS8110: Part 1: 1997. Structural use of concrete. Part 1: Code of practice for design and construction. BSI, 1997. (5) HIGHWAYS AGENCY. BD 42/00. Design manual for roads and bridges: design of embedded retaining walls and bridge abutments. HMSO, 2000. (6) HIGHWAYS AGENCY. BD 74/00. Design manual for roads and bridges: foundations. HMSO, 2000. (7) HIGHWAYS AGENCY. BD 44/95. Design manual for roads and bridges: the assessment of concrete highway bridges and structures. HMSO, 1995. Related Note 'Shear in piles and columns', Notes on Structures 1991NST_1 (incorporated in this Note) Origin ally pu blis hed in June 1993 as NST40/4 Revised (a) February 2001 - references to 'bd' replaced by A v to avoid possible confusion over definition of b; notes concerning BS8110 equations 6a and 6b added; editorial changes and corrections Revised (b) Apri l 2002 - 0.87 factors on steel strength replaced by 0.95 in accordance with BS8110: Part 1: 1997 Revised (c) Jul y 2002 - Titles to Fig.4 added Revised (d) September 2002 - remaining references to ‘bd’ replaced by A v and to 0.87 by 0.95 Revised (e) July 2003 - Failure plane defined to cut links over a length d, rather than to be at 45 °; method of determining spiral resistance amended; reference made to Highway’s Agency formulae; figures redrawn, editorial changes Revised (f) Jul y 2004 - reference made to paper, making the same proposals as this Note; simplified expression for A svh modified to be consistent with paper Revised (g) Jul y 2004 - reference updated for BS8110: Part 1: 1997 and Table 3.8
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