nanoparticles
4. Viscosity effects on the dynamical magnetic response of magnetic
4.2. Viscosity effects on the magnetism of the MNPs
Reinforced concrete structures are generally analysed by the conventional elastic theory (refer of the Code). In flexural members, this is tantamount to assuming a linear momcnt-curvature relationship, even under loads. For under-reinforced sections, this assumption is approximately true [refer Fig.
provided the reinforcing steel has not yielded at any section. Once yielding takes place (at any section), the behavionr of a statically indeterminate structure enters an inelastic phase, and linear elastic structural analysis is strictly no longer valid.
For a
. .
determination of the distribution of moments for beyond the yielding stage at any section, inelastic analysis is called for. This is generally referred to as limit analysis, when applied to reinforced concrete framed structures [Ref. and 'plastic analysis' when applied to steel stmctures. In the special case of reinforced concrete slabs, the inelastic analysis usually employed is the 'yield line analysis' due to Johansen [Ref. The assumption generally made in limit analysis is that the moment-curvature relation is an idealised bilinear plastic relation [Fig. This has validity only if the section is adequately reinforced and the reinforcing steel has a well-defined yield plateau. ultimate moment of resistance.
... of such sections, with area of steel. can be assessed, as described in Chapter 4.of steel Moment M
Curvature
Fig. 9.8 ldeaiised moment-curvature relation
ANALYSIS FOR DESIGN MOMENTS IN CONTINUOUS SYSTEMS 335 Plastic Hinge Formation
With the idealised M - the. ultimate moment of resistance is assumed to have been at a 'critical' section in a flexural member with the yielding of the tension steel [Fig. On further in curvature:
the moment at the section cannot increase. However, the section 'yields', and the curvature continues to increase under a constant moment In general (with bending moment varying along the length of the member), the zone of 'yielding' spreads ovei a region in the immediate neighbourhood of the section under consideration, continued. rotation, as though a 'hinge' is present at the section, but one that continues to resist a fixed moment Aplastic is said to have formed at the section. If the structure is statically indetenninate, it is still stable after the of a hinge, and for further loading, it behaves as a modified structure with a hinge at the plastic hinge location (and one less of indeterminacy). It can continue to additional loading (with of additional plastic hinges) until limit state of collapse is reached on account of one of the following reasons:
formation of sufficient number of plastic hinges, to convert the structure (or a part of it) into a 'mechanism';
.
limitation in ductile curvature reaching the value or, in other words a plastic hinge reaching its ultimate at any one plastic hinge locution, resulting in local crushing of that section.Example
of
AnalysisA simple application of limit analysis is demonstrated here, with reference to a span continuous beam subjected to an increasing uniformly distributed load w per unit
length [Fig. For it may be that the beam has uniform additional load the beam can take will now depend on the plastic rotation capacity of this 'plastic hinge'. For any additional loading, the beam as a two-span with a hinge at support two simply supported spans) and the span moment alone increases [Fig. while the support moment remains constant at Assuming that the support section is sufficiently such that it will break down prior to the of the next plastic this phase of behaviour will continue until the moment'in the span reaches [Fig.
Analysis of the structure for condition [Fig. indicates that this corresponds to a maximum span moment given by:
336 REINFORCED CONCRETE DESIGN
loading on beam
I
moments up to of
elastic (phase
= limit
2 =
= 0.41421 and = 1.46
11.656
bending moment In
limit analysis ("equilibrium method) the 'inelastic' phase (phase
variation of moment with loading
9.9 Limit analysis of a two-span continuous beam
ANALYSIS FOR DESIGN MOMENTS IN CONTINUOUS SYSTEMS
=
As the load is reached, two additional plastic hingest are fonned in the two spans at the peak moment locations, and the structure is transformed into an unstable hinged mechanism which deflect with no increase in load
Obviously, is the ultimate (collapse) load of the structure, even allowing for inelastic behaviour.
This indicates that the beam is capable of carrying additional loads up to 46 percent beyond the limit of elastic behaviour, thanks to the ductile behaviour of the beam section at the continuous support.
The bending moment distributions in the inelastic phase are indicated in Fig. It is seen in this example that, a 'redistribution of moments' takes place, with the support moment remaining constant at while the moments continue to increase until they too reach The variation of support moment and maximum span moment with increasing loading is shown in Fig. T h e gain in moments is linear in the 'elastic phase' and corresponding to the formation of first plastic hinge (at w there is a discontinuity in each of the two M-w curves.
In deriving the expression for allowing full moment redistribution on to the spans, it was assumed abovethat the plastic hinge at the support section will continue to yield (rotate) without breakdown. If the rotation capacityt of plastic at gets exhausted prior to the span moment reaching the 'inelastic' phase will get terminated at .a w If the plastic hinge possesses adequate ductility, then the .maximum collapse load is reached at = corresponding to the formation of a
9.7.2 Moment Redlstrlbutlon
As seen in the previous the distribution of bending moments in a continuous beam (or frame) gets modified significantly in the inelastic phase. The term moment redistribution is generally used to refer to the transfer of moments to the less stressed sections as sections of peak moments yield on ultimate capacity being reached in the example above). From a design viewpoint, this behaviour can he taken advantage of by attempting to effect a redistributed bending moment diagram which achieves a reduction in the moment levels (and a corresponding increase in the lower moments at other locations).
In this example, two plastic hinges will - one in each span, due la symmetry in the geometry as well the loading.
'For a detailed calculation of plastic rotations, reader is advised to consult Ref.
338 REINFORCED CONCRETE DESIGN ANALYSIS FOR DESIGN MOMENTS IN CONTINUOUS SYSTEMS 3 3 9
factored unit length M
(a) two-span beam design moments of resistance
with uniform loading (with no
elastic bending moments redistributed moment's for design (under factored loads)
0.5 =
A 1.00
plastic hinge at 8, 0.25 0.5 0.75 1.0 and the second at
limit analysis (considering a between reduction in reduced support moment support moment with increase in
span moment (redistributed) 9.10 Moment redistribution in a two-span continuous beam
Such adjustment in the moment diagram often leads to the design of a more economical suucture with better balanced proportions, and less congestion of reinforcement at the critical sections.
Considering the earlier example of the two-span continuous beam [Fig. as a problem (rather than an analysis problem), it may be seen that the designer has several alternative factored moment diagrams to choose from, depending on the amount of redistribution to be considered. If the design [Fig. is to be o n a purely elastic moment distribution (without considering any redistrihution) then the bending moment diagram to be considered is as shown in Fig. and the corresponding design support moment and span moment are obtained as:
where and denote the support and span in the elastic solution:
the subscript here represents elastic analysist. Reduction in Peak 'Negative' Moments
The relatively high support moment may call for a large section (if singly reinforced); alternatively, for a given limited cross-section, large amounts of reinforcement may be required. Therefore, in such situations, it is desirable to reduce the design moment at the to a value, say (where the factor has a value less than unity), and to correspondingly increase the span (positive) moments which are otherwise relatively low. The percentage reduction in the design support moment is given by:
(9.2) Consequent to a reduction in the support moment to there is an increase the design ('positive') moment in the span region from to
where the factor obviously is greater than unity. Accordingly, as indicated in Fig.
where the subscript limit analysis. The factor (indicating the increase in the elastic span moment depends on factor The factor is fixed (based on the percentage reduction desired), and the factor has to b e determined for design
-
by considering analysis' [Fig. It can be shown easily, by applying static equilibrium, that:Introducing Eq. 9.1 in 9.4, the following quadratic relationship between the constants and can be established:
This relation is depicted in Fig. It is seen that, for instance, a 25 percent reduction in the elastic support results in a 17.3 percent increase the span and a 50 percent reduction in in a 36.1 percent increase in However, it should be noted that a large amount of moment redistribution requires a large amount of plastic rotation of 'In this example, it is tacitly assumed that the gravity loads indicated in Fig. 9.10 are entirely due to dead loads, and that there are no live
ANALYSIS FOR DESIGN MOMENTS IN CONTINUOUS SYSTEMS 341
the plastic (at in this example) which is often not practically The span from elastic analysis [Fig. can be
feasible. If the desired ductility is not available, a failure likely (due to redistributed by allowing the first plastic hinge to form the span region. The crushing of the concrete in compression zone at the plastic hinge forming region) reduction in span inomcnt is accompanied by a increase support
at a load that is than the load moment [Fig. to maintain equilibrium at the state. Such a
redistribution may be desirable if the elastic span moment is relatively high, as would be the case if the live load component in the loading is high.
9.7.3 Code Recommendations for Moment Redistribution Through design and detailing, it may be possible to muster the ductility
for significant amounts of moment redistribution. However, excessive moment redistribution can be undesirable if it results in plastic hinge formation at low loads (less than the service loads), and the consequent crack-widths and deflections are likely to violate serviceability Codes generally attempt to preclude such a situation by ensuring that plastic hinges are not allowed to form under normal service loads. In general, codes allow only a limited amount of in reinforced concrete structures.
Reduction in Peak 'Positive' Moments
redistributiont may also be advantageously to situations where 'positive' moments are high and need to be for economy and less of reinforcement.
For instance, with reference to the earlier example of the two-span continuous beam, if part of the total factored load w,, is dm to live load then the arrangement of loads for maximum span moment is as shown in Fig. 9.1
A loading diagram for
maximum +ve span moment in
elastic factored moment diagram
moments
Fig. 9.1 1 Moment redistribution: reduction in peak positive moment It may noted that refers to of effects from heavily stressed locations to (or lightly) stressed locations, of whether the peak
are 'positive' 'negative'.
The Code 37.1.1) permits the designer to select the envelope of redistributed factored moment diagrams for design, in lieu of the envelope of elastic factored moments, provided the following conditions are satisfied:
Limit Equilibrium: The redistributed moments must be i n a state of static equilibrium with the factored loads at the limit state.
Serviceability: The ultimate moment of resistance at any section should not be less than 70 percent of the factored moment at that section, as obtained from the elastic envelope (considering all loading combinations). In other words, the flexural strength at any section should not be less than that given by the elastic factored moment envelope, scaled by a factor of 0.7:
This restriction is aimed at ensuring that plastic hinge formation does not take place under normal service loads, and even if it does take place, the yielding of the steel will not be so significant as to result in excessive crack-widths and deflections. It is mentioned in the Explanatory Handbook to the Code that the of 70% is arrived at as the ratio of service loads to ultimate loads with to load combinations involving a uniform load factor of 1.5, as 111.5 = 0.67
3. Demand for High Plastic Hinge Rotation Capacities: The reduction in the elastic moment ('negative' or 'positive') at any section due to a particular combination of factored loads should not exceed 30 percent of the absolute maximum factored moment as obtained from the of factored elastic moments (considering all loading combinations). Although the basis for this clause in the Code (CI. is different from the previous clause, which is based on the idea of preventing the formation of plastic at service loads; for the case of gravity loading, in effect, this is no different. However, in the design of lateral load resisting frames (with of storeys exceeding four), the Code 37.1.1.e) imposes an additional over- riding rcstriction. The reduction in the elastic factored moment is restricted to 1 0 percent of M Thus,
342 REINFORCED CONCRETE DESIGN
This restriction is intended to ensure that the ductility requirements at the plastic hinge locations are not excessive.
4. Adequate Plastic Hinge Rotation Capacity: The design of the critical section (plastic hinge location) should be that it is sufficiently under-reinforced.
a low neutral axis depth factor satisfying:
where denotes the percentage reduction in the maximum factored elastic
moment at the section:
In practice, it is sometimes more convenient to Eq. 9.8 alternatively as:
For singly reinforced rectangular beam sections, the expression for is given by Eq. 5.1 1, which is repeated here for with = M,,,
M o m e n t Redistribution i n Beams
Low values of (and, thus large values of are generally not possible in beams without resorting to very large sections, which may be uneconomical. However, even with extreme case of a balanced section (with it can be shown, by applying 9.10 and Eq. 4.50 (or Table 4.3). that
6.9 for
12.1 for with = x (9.12)
14.4 for
Thus, it is seen that a distribution (for example, up to 12.1 in the case of Fe 415 steel) is possible, even with the limiting neutral axis depth permitted for design [refer Chapter
Inelastic Analysis o f Slabs
As discussed earlier (in the thicknesses of reinforced concrete slabs are governed by deflection control criteria, with the result that the sections
under-reinforced, with low values. Hence, significant inelastic action is possible in such cases.
It may be noted, however, that, in the case of one way continuous (and continuous no moment redistribution is permitted by the Code 22.5.1) if
ANALYSIS FOR DESIGN MOMENTS IN CONTINUOUS SYSTEMS 343 inelastic analysis line is often resorted to 9.121, and, in fact, the
moment coefficients given in (Table 26) for slabs with
various edge conditions are based on such analvses
. .
'Yield line analysis' is the equivalent for a two-dimensional flexural member @late or slab) of limit analysis of a onedimensional member (continuous beam), explained in Section It is based the relation [Fig.
according to which, as the moment at a section reaches a plastic hinge is formed, and therefore rotation place at constant In slabs, moments occur
along lines along support lines 'positive' moments
along lines near midspan), and hence the yielding (plastic formation) occurs along lines ("yield lines"), and at sections, as in beams. In a skeletal structure (continuous beam, grid, plane frame, space frame), the ultimate (collapse) load i s reached when sufficient number of plastic are to transform the structure into a In a similar way, the ultimate load is reached in plates sufficient number of yield lines are formed to transform the slab into a series of plate segmentst connected by 'yield lines', resulting in mechanism type behaviour.
As in the case of analysis of beams and frames, it is assumed in 'yield line analysis' [Fig. that the plastic hinges which form (along the 'yield lines') possess adequate plastic rotation capacities to on' till a complete set of yield lines are formed, leading to a type of collapse. This is justifiable in view of the relatively low values in slabs in general. of line analysis discussed further in Chapter 11. For a more study, reference be made to Refs.
M o m e n t Redistribution in C o l u m n s
Reduction of moments on account of moment redistribution is generally not applied to columns, which are essentially compression members that are also subjected to bending (due to frame action). In general, the neutral axis location' at the limit state is such that the Code requirements [Eq. cannot be satisfied by a column section
-
unless the is very lightly loaded axially and the eccentricity in loading is very large. Furthermore, in the case of a typical beam-column joint in a reinforced concrete building, it is desirable that the formation of plastic hinge occurs in the beam, rather than in column, because the subsequent collapse is likely to be less catastrophic. This is particularly necessary in design [refer Chapter'Such plate segment can about the line', in much the way a door can about a line hinge.
the loading on the column is not eccentric, the neutral axis will lie the
REINFORCED CONCRETE DESIGN
two-way slab system uniformly distributed collapse load
moment steel
. .. .. . . . . . .. . .. . .. .. . .. . . . . .. . . . .
moment Steel deflected shape at collapse
(section A-A)
Fig. 9.12 Concept underlying yield line analysis of slabs
It may be noted that when redistribution is applied to with the objective of reducing the peak moments in beams, this will also result in changes in the elastic factored moments in columns. These changes moments may be ignored in design, if the redistribution results in a reduction in column moments
adequate torsional has a similar torque-twist
relation [see Fig. In such beams, the of a hinge
and subsequent redistribution of [Ref.
Analysis can also be extended to structures with subjected to significant
ANALYSIS FOR DESIGN MOMENTS IN CONTINUOUS SYSTEMS
torsion, such as transversely loaded grid structures (or bridges), where the collapse mechanisms may involve torsional hinges as well [Ref. Recognising this, some codes [Ref. permit the limiting of the maximum design torque in spandrel beams to T , is the cracking torque of the spandrel beam, and reprcscnts a torque corresponding to a 'plastic torsional hinge' formation and consequent cracking and reduction in torsional stiffness.
9.8 DESIGN EXAMPLES