Table H3.4 Design parameters for rectangular sections [
3.4.5 Design shear resistance of beams
Calculations for the strength of reinforced concrete beams are based on comparing the average shear stress on a section calculated from equation 3 with a nominal value of ultimate shear stress v~given in Table 3.9. The shear stresses given in Table 3.9 are based on an extensive study of test data (Figure H3.12). When the average shear stress is treater than the nominal stress from the Table, shear reinforcement is provided in proportions calculated on the assumption that the reinforcement forms the tension members of one or more series of pin-jointed trusses. This approach, commonly called the truss analogy, has been shown in tests to be conservative and the contribution to the shear strength from the concrete. v~bd, is therefore assumed not to be lost ifv>v~.
If v is only just greater than v~, it will be necessary to increase the amount of shear reinforcement required from the truss analogy to the Code minimum permitted amount.
When v is less than v~, minimum reinforcement is still required in many cases: see Table 3.8.
3—
2
oe~ 10
Figure H3.12. Shear strength of beams without shear reinforcement.
Where large shears are carried, it is possible for the diagonal compressive stresses to cause crushing of the web concrete. For this reason. the maximum shear carried by the beam must be limited: in the Code this limitation is given as 0.8Vf~~ or 5 N/mm2 whichever is lesser.
3.4.5.1 Symbols
3.4.5.2 Shear stress in beams
The actual way in which shear is carried by a section is highly complex and the value given by equation 3 should not be viewed as a stress which actually exists within a section:
it is merely a mathematical device used for the interpretation of test results in the derivation of an empirical design method.
3.4.5.3 Shear reinforcement: form, area and stress
The intentions of Table 3.8 may be summarised diagramatically in Figure H3. 13.
2 3 4 5 6 7 8
39
Handbook to BS8IO:198S
Miniumum shear reinforcement may be omitted in some members of minor importance
Figure H3.13: Provision of shear reinforcement in beams.
3.4.5,4 Concrete shear stress
Since the shear stress is a function of the amount of tension reinforcement at the section considered, it is clearly important to establish just what reinforcement can be included.
This caused some confusion in CP1 10 and an attempt has been made to give clearer rules. In general. any bar which extends more than an effective depth on either side of the section being considered can be included. This definition will not be satisfactory at a single support but there is ample evidence, to show that the full area of bottom steel at the support may be used in this case provided it is anchored according to the normal rules. The final paragraph of3.4.5.4aims to clarify which reinforcement should be used when nominal top steel is provided at a notional simple support. It is believed that there is a misprint in the final sentence which should read “This steel should extend into the span for a distance of at least three times the effective depth.” Clearly, it should also be fully anchored into the support.
3.4.5.5 Spacing of links
The truss analogy would suggest a maximum spacing equal to the lever arm: 0.75diS a lower bound value for this. The logic behind the limit on lateral spacing of the legs of stirrups is less clear but experimental evidence suggests a reason why such a limit is valuable. One of the major functions of stirrups is to inhibit ~dowel’failure of the tension steel. This is a tearing out of the bars in the way sketched in Figure H3.14.
Figttre H3. 14: Normal mode of shear failure.
a
_ -- . - - - i.—.. _-NT--’...- --~-:--:.---- ---- -- - ---- -- ,-. - -- -.-
.71
Part I: Sectwti S
Clearly the effectiveness of the stirrup in achieving this will reduce ‘~‘ith increasing distance of the vertical leg from the bar considered. Clearly, if a bar is placed further than 150mm from a stirrup leg, it can still be used to provide flexural strength but should be ignored in assessingVc.The situation in slabs can be used to extend the interpretation of this clause further. In slabs, stirrups are not required until v exceeds v~. It seems logical to argue from this that the requirement relating to the spacing of stirrup legs is to ensure that ~c can be maintained in circumstances where v exceeds v~. It therefore seems reasonable to conclude that the limitations on lateral spacing may be ignored where vis less thanv~both in beams and slabs.
3.4.5.6 Shear resistance of bent-up bars
The truss analogy assumes that the tensions in the truss are carried by longitudinal and stirrup reinforcement and the concrete carries the thrust in the compression zone and the diagonal thrust across the web.
Equation 4 is derived directly from consideration of the equilibrium of this system (Figure H3A5).
/
Ii .
I, ,
A A
2/I-I
a A
A. ~ /~~- ~1
Figure H3. iS: Truss systems for shear.
3.4.5.7 Anchorage and bearing of bent-up bars
3.4.5.8 Enhanced shear strength of sections close to supports
When the ratio of shear span to effective depth of a beam is reduced below 2, the shear capacity is considerably increased.
Figure H3>16 shows a plot of test results reported in references 3,7. 3.8 and 3,9 illustrating the relationship between a/d and v. for beams without stirrups. The line shown on the graph is straight for all values ofa~Id greater than 2 when v/vt is 1. This clause defines the parabolic line shown in the Figure.
The strength of short beams depends to a large extent upon the detailing of the reinforcement. Adequate anchorage must be provided to the main tensile reinforcement, Vertical stirrups are not very effective in beams in which a.,./d is less than 0,6. in which case horizontal stirrups parallel to the main tension reinforcement are recommended,
The results sho~vn in Figure H3.16 derive from tests on short-span. point-loaded beams but the results will hold for any failure where the failure plane is constrained to form at an angle greater than tan’ (1/2) to the horizontal, The enhancement in strength can therefore be applied for any section closer to a support than 2d.
3.4.5.9 Shear reinforcement for sections close to supports
Equation 5 derives from the assumption that the effect of the enhancement is only on v,,and does not affect the efficiency of shear reinforcement. Application of truss analogy might be seen to suggest that the increased truss angle implied by a failure close to a support would increase the efficiency of shear reinforcement. This may be so, in which case, equation 5 is conservative, but, while the experimental evidence for the enhancement of v~ is clear (Figure H3.16), it is doubtful if sufficient exists to show the
effect of a~.Id on shear reinforcement unequivocally. 41
Handbook toBSSIIO:]98S 10
9
8
7
6 v
vc 5
Figure H3. 16: Ultimate shear stresses for beams loaded close to supports:Urtaken from Code
3.4.5.10 Enhanced shear strength near supports (simplified approach)
At a distance d from the support. Figure H3. 16 will show that the capacity of the section is increasing very rapidly. So much so that it is most unlikely that the shear force will be increasing more rapidly. The rule given in this clause will thus normally give a safe way of gaining the advantage of the strength enhancement for minimal effort.
3.4.5.11 Bottom loaded beams
A load applied near the bottom of a beam could break the bottom out as sketched in Figure H3. 17. The load. effectively, has to be transferred to the top by links before the design method is valid.
4,
possible mode of failureFigure H3.!7: Loads on the bottotn of beams, 3.4,5.12 Shear and axial compression
In dealing with the comments on the draft of BS 8110 circulated for public commentit became apparent that there was a considerable demand for guidance on the treatment ofshear and compression. particularly in columns. Equation 6 in this clause is an entirely empirical attempt to make allowance for the increased shear capacity given by compressive axial load. Truly applicable test data are not easy to find but Figure 1-13.18 compares test results from several series of tests ‘vith the proposed formula- -Inthe comparison. Vm has been taken as 1.0 when calculatingv~.
[I
F U [ I!
I
C
El
4
3
2
0
a’,
d
U I
I C
C C I [ U
42
- - - .
-. . ... . . -- ..-- - -- - -. - .
--J .1
I-a
LU~
4LU LLU~
LU CO CO—
0
Figure H3. 18: Shear and axial compression.
3.4.5.13 Torsion
When the system is statically determinate, ultimate torsional moments must be calculated and provided for. In indeterminate structures. it will in most cases not be necessary to consider torsion at the ultimate limit state. but itmay be necessary to considerit at the limit state of cracking. Figure H3.19 shows an example with edge beams in which torsion is statically indeterminate, i.e. it arises because of an imposed deformation and its magnitude will depend on the relative torsional and flexural stiffnesses in the structure.
Lack of torsional strength in such cases will not cause collapse since the members will crack and deform without developing the ultimate torsional condition.
However, in some cases the twist imposed on a member max’ cause excessive cracking.
This can happen, for example, in an edge beam where the span of the slab at right-angles to the beam is large, or as in Figure H3.19 in the short length of edge beam between the trimming beam and column where the imposed rotation from the secondary beam
I
Parr1: Section3
2 cALcULATED SHEAR STRESS (N/mm2)
not usually serious Serious torsional cracking likely
beam
FigureH3J9: Examples of torsion due to imposed rotation.
43
Hwtdbook ro BS8IIO:1985
[I
has to be accommodated in quite a short length. For the purpose of designing the reinforcement and determining the forces exerted on adjoining members the method ft)r calculating the imposed torque is based on tests on cracked reinforced beams’3”” and is generally conservative,
Torsional stiffnesses are generally small with respect to flexural stiffnesses and can therefore. be ignored in assessing the imposed twist. This will give an upper limit and one which will not be too conservative.
Explicit design for torsion is dealt with in Part 2 of the Code and will be discussed in the related section of the Handbook.
3.4.6 Deflection of beams