Whereas the previous section (Section 4.6) dealt with the 'analysis at loads', the present section deals with the 'analysis at loads'. The former is based the method (WSM) and is also applicable to the analysis of 'serviceability limit states', the latter is based on 'ultimate limit state' the limit answer to this question was given in Chapter 3, where it was explained that a structure has to be both safe (at various ultimate limit states) serviceable (at various limit states). At ultimate iimit states, the loads are those corresponding to impending failure of structure, whereas at serviceability states, the loads and stresses are those applicable in the day-to-day service of the structure. This section investigates 'safety' of flexural members (of given design) at ultimate limit state in flexure. The previous section discussed the of stresses under service loads required for serviceability analysis (described Chapter and also the calculation of 'allowable bending moment' based on the WSM concept of permissible stresses. The latter was included to the student to gain a first-hand understanding of the traditional (and, earlier much-used) working stress method - which retains a place in the Code, albeit as an Appendix, and is used in the design of special structures such as water tanks and mad bridges.
Therefore, the will d o well to keep in perspective the background of the present section, dealing with the analysis at the 'ultimate limit state in flexure'. The expressions derived here will find use again in the next chapter (Chapter
deals with the design of reinforced concrete beams at the limit state in flexure.
In this section, the Code procedure for analysis is discussed. The calculations are based on the curves for concrete and steel, as specified the Code. Moreover, the design stress-strain curves (involving partial safety factors
, are used, as explained in Section 3.6.
4.7.1 Assumptions in Analysis
The behaviour of reinforced concrete beam sections at ultimate loads has been explained in detail in Section The basic assumptions involved in the analysis at ultimate limit state of flexure (CI. 38.1 of the Code) are listed here [see also Fig. (Most of these assumptions already been explained earlier.)
a) Plane sections normal to the beam axis aiter bending, in an initially straight beam, strain over the depth of the section.
b) The maximum compressive strain in concrete (at the outermost shall be taken as 0.0035 [Fig. This is so, because regardless of whether the
IN FLEXURE 135
beam is or over-reinforced, collapse invariably occurs by the crushiug of concrete explained 4.5.3).
stress-strain curve concrete flexural compression (recommended by the Code) is as depicted in Fig. 3.5. [The also permits of any of the stress-strain curve which in substantial agreement with results of tests.] The partial safety factor 1.5 is to be considered.
The tensile strength of the concrete is ignored.
The design stress-strain for mild steel and cold-worked bars are as depicted in Fig. 3.6 and Fig. 3.7 respectively. The partial safety factor
1.15 is to be considered.
The strain in reinforccmcnt (at its at the ultimate limit
state shall be less [Fig. as:
(0.87
+
0.002This is to defining the yield stress of as stress
to 0.002 strain offset (0.2% proof stress) regardless of whether steel has a well- defined yield point or not. The yield strain to is then given by
.
rhe partial safety factor 1.15 to allow variability in the steel strength, the designt yield strength, = = and using this in lieu o f f , , the yield strain [refer Fig. 3.71 is given by: state, so failure is in nature, providing ample warning of the impending collapse.characteristic yield results in a slightly (and hence, less conservative!) value of
yield strain Pigs 3.6,
136 REINFORCED CONCRETE DESIGN
beam section strains stresses
Fig. 4.17 Behaviour of singly reinforced rectangular section at ultimate limit state in flexure
According to Code [rcquircment (f) in Section 4.7.11, , that is a limiting value of the neutral axis corresponding to
.
This is obtained by substituting the expression for [Eq. in Eq. 4.49:The values for different grades of steel, obtained by applying Eq. 4.50, are listed in Table4.3. It may be noted that the given in TabIe4.3 are applicable to all cross-sectional shapes, and remain valid for doubly reinforced sections as well.
Table 4.3 depth of neutral tor diflerent grades of steel
limiting depth of neutral axis to the so-called balanced
. , section, a section that is expected to result in a 'balanced' at the limit state in [refer Section axis depth is less than
then the section is (resulting in a 'tension' failure); whereas if exceeds it is (resulting in a 'compression' failure).
IN FLEXURE 137
4.7.3 Analysis of Singly Rectangular Sections
Analysis of a given reinforced concrete section at the ultimate limit state of
implies the determination of the of of the section.
This is obtained from the couple resulting from the flexural stresses [Pig.
M,,, = = (4.51)
and are the resultant (ultimate) forces in and tension respectively, and is the lever arm.
(4.52) where
=
and the line of action of to the level of the centroid of the tension steel.
Concrete Stress Block In Compression
In order to determine the magnitude of and its line of action, it is necessary to analyse the block in compression. As ultimate failure of a reinforced concrete beam in flexure occurs by the crushing of concrete, for both under- and over- reinforced beams, the shape of the compressive stress distribution ('stress at failure will be, in both cases, as shown in Fig. 4.18. [also refer assumptions (b) and in The value of can be computed knowing that the compressive stress in concrete is uniform 0.447 for a depth of 7, and below this it varies parabolically over a depth of to zero at the axis [Fig.
BEAM SECTION STRAINS STRESSES (truncated)
Fig. 4.18 Concrete stress-block parameters in compression
For a section of width b.
REINFORCED CONCRETE DESIGN
Also, the of action of is the centroid of the stress block, located at a distance from the concrete fibres subjected to the maximum compressive strain Accordingly, moments of compressive forces C,,, and [Fig. about maximum compressive strain location,
Solving,
(4.54) Depth of Neutral Axis
For any given section, the depth of the neutral axis should be such that =
.
satisfying equilibrium of Equating = , with for C,, and , given hy Eq. 4.53 and Eq. 4.52 respectively:
valid only if resulting
For the condition , E [Fig. that, at the ultimate
limit state, the steel would not have 'yielded' (as per the proof stress definitiont 'and the steel stress cannot be taken as = Hence Eq. 4.52 and therefore
Eq. 4.55 are not applicable. When the steel has not yielded, the true location of the neutral axis is obtained by a method, called
method, involving the following steps:
1) Assume a suitable initial (trial) value of x,,
2) Determine by considering strain 4.491:
0.0035
-
(4.56)3) Determine the design to using the
stress-strain curve [Fig. 3.7,
4) Derive the value of corresponding by considering A,, and applying the force equilibrium condition whereby
5 ) this value of x,, with the value used in step If the difference between the two values is acceptably small, accept the value In the case grade steel which has a yield point, steel would have yielded reached even at a strain slightly Lower
.
In cases, one may find = far values ofx,, slightly in excessrepeat steps to with an average) value of , until convergence.
Ultimate-Moment of Resistance
The of of a given from
Eq. 4.51. The lever arm for the case of the singly reinforced rectangular section
[Fig. Fig. is by
=
Accordingly, in terms of the concrete compressive strength,
=
-
for (4.59). , , ... . ... , Alternatively, in of steel tensile stress,
for all (4.60)
= for x,, Moment of Resistance
The of resistance of a given (singly reinforced, rectangular) section, according to the Code corresponds to the
defined by Eq. 4.50. From Eq. 4.59, it follows that:
.-
,,,mu (4.61)
The values of the non-dimensional parameter different grades of steel [refer Table 4.31 are obtained as 0.1498. 0.1389 and 0.1338 for Fe 250, 415 and 500 respectively.
Limiting Percentage Tensile Steel
Corresponding to the moment of resistance there is a limiting percentage tensile steel = An expression for is obtainable
from Eq. 4.55 :
140 REINFORCED CONCRETE DESIGN
values of and M (in units) for, different combinations of steel and concrete grades are listed in Table 4.4. These values correspond to the so- called 'balanced' section [refer Section for a singly rectangular section
Table 4.4 Limiting values of and for singly reinforced rectangular beam sections for various grades of steel and concrete.
values
Safety Ultimate Limif State Flexure
The bending expected at a beam section at the limit state due to the loads is called M,,. For the consideration of combinntions of loads (dead loads, live loads, wind loads, etc.), appropriate load factors should be applied to the specified loads (as explained in
and the factored moment M,, is determined by structural analysis.
The beam scction will bc considered to be 'safe', according to the Code, if its ultimate moment of greater or to the factored moment In other words, for such a design, of is acceptably low. It is also the intention of the Code to that nt ultimate failure in flexure, the type of failure should bc a failure - as explained earlier. For this reason, Code requires the designer to ensure that [Table whereby it follows that, for reinforced section, the reinforcement
percentage not exceed a n d ultimate moment of resistance should not exceed [Table The Code clearly states:
The topic of design is covered in detail in Chapter 5. The present chapter deals with analysis and, in analysis, it is not unlikely to encounter beam sections
(already constructed) in which , whereby and
.
Evidently, in such 'over-reinforced' sections, the strength requirement may be satisfied, but not ductilityt requirement. The question arises: are such sections.
acceptable ? The answer, in general, would be in the negative, except in certain special situations where the section itself is not 'critical' in terms of ductility, and will not lead to a brittle failure of the under the given factored loads. In such exceptional cases, where and inelastic flexnral response* is never expected to under the given factored loads, over-reinforced sections cannot be strictly objected to.
It may be noted that the exact determination of of an over-reinforced section generally involves considerable computational effort, as explained in the next section.
An approximate (but conservative) estimate of ultimate capacity of such a section is given by limiting moment of resistance, which can be computed.
Variation of M (for singly reinforced rectangular sections)
For
.
it is possible to arrive at a simple expression lor the ultimate moment of resistance of a given section with a specified First, expressing A,, in terms of :and then substituting ih Eq. 4.55.
Further Eq. 4.63 and Eq. 4.64 in Eq. 4.60,
The ductility may be satisfied in the case of steel if x,, slightly exceeds is explained later with Fig. 4.19. [See also footnote
136.1
details on 'plastic hinge' at the state, refer Chapter 9.
142 REINFORCED CONCRETE DESIGN
percentage tension 400
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 percentage reinforcement