of 16 the diameter of the
9.3 SIMPLIFIED (APPROXIMATE) METHODS O F ANALYSIS
9.3.1 M o m e n t C o e f f i c i e n t s f o r C o n t i n u o u s B e a m s Under Gravity L o a d s
The use of (in lieu of rigorous of gravity
patterns) was introduced in Section 5.6.1, with reference to continuous one-way These 'moment coefficients' as well as related 'shear coefficients' 22.5.1 of the Code) are listed in 9.1. The of the moment coelficients has already been demonstrated in Example 5.3.
It be noted that the use of the coeflicients, as as the related 'shear for continuous beams and onc-way slabs is to the
conditions 22.5.1 of the Code):
9.3.2 S u b s t i t u t e F r a m e M e t h o d of F r a m e Analysis f o r Gravity L o a d s As explained earlier, the skeleton of a typical framed building is a three-dimensional 'rigid frame', which may be resolved, for the purpose of analysis, into a series of independent parallel 'plane frames' along the column lines in the longitudinal and transverse directions of the building. Each of these plane frames [Fig. in turn.
ANALYSIS FOR MOMENTS IN CONTINUOUS SYSTEMS 325 has to be analysed separately for a number of gravity load patterns as well as lateral loads.
Table 9.1 Factored moments and shears in continuous b e a m s using Code coefficients 22.5 of the Code)
'POSITIVE' MOMENTS:
End Spans
2. Interior Spans + DL
16 12
'NEGATIVE' MOMENTS:
End support (if partially restrained)
2. First interior support
3. Other interior supports
SHEAR FORCES:
1. End support unrestrained
partially restrained 1
2. First interior support exterior face interior face 3. Other interior supports
This is essentially a problem of structural analysis of indeterminate structures, and various techniques are available for this purpose [Ref. 9.4, For the detailed analysis of large frames, with high indeterminacy, computer-based matrix methods are ideally suitable [Ref. A large number of established 'finite element method' based software packages (such as ANSYS, NASTRAN, NISA, SAP) are available in the market and are increasingly being used by designers worldwide. However, the
good old manual methods such as the are still in vogue,
326 REINFORCED CONCRETE
and are ideally suited for analysing small frames under gravity loading.
Convenient tabular arrangements far performing distribution (up to cycles) are in Ref. 9.6 and
(a) typical plane frame for analysis
ROOF LEVEL
TYPICAL FLOOR LEVEL
(b) substitute frames
(c) substitute frames truncated
9.4 'Substitute frames' for gravity load analysis
ANALYSIS FOR DESIGN MOMENTS IN CONTINUOUS SYSTEMS 327 In the frames that are substantially or unsymmetrically and are otherwise not braced against sway, the effeca of sway also have to be considered. For such the of the Moment Distribution Method can be cumbersome, a s it requires lnore one distribution table to be constructed (and later combined): Method is better suited for such analyses, as it manages with
a table.
cases where the effects of are negligible, it is found that the problem of frame analysis can be considerably simplified by resolving the frame into partial frames ('substitute frames'), as indicated in Fig. 9.4 [Ref. This simplification is justified on the grounds that the bending moment or shear force at a particular section (of a beam or column) is not influenced significantly by gravity loads on spans far removed from the section under [refer (4) in Section Accordingly. an entire floor, comprising the and columns (abavc below)
be isolated, with the far of the above below considered 'fixed' [Fig. This frame can be conveniently analysed for various gravity load patterns by the moment distribution method; two cycles of distribution are generally adequate [Ref. 9.8, This method of 'substitute frame analysis' is permitted by Code If are a large number of bays in the substitute then a simplification is possible by truncating the beams appropriately, and treating truncated end as 'fixed' [Fig. In determining the support moment, the beam be assumed as fixed at any support two panels distant, provided the beam continues beyond that point. In analysing for maximum and minimum 'positive' moments in spans, the far ends of adjacent spans may similarly be considered as fixed.
9.3.3 M e t h o d s f o r Load Analysis
Multi-storeyed have to be designed to the effects of lateral loads due to wind or earthquake, in combination gravity loads. The lateral load transfer mechanism of a framed building has been explained briefly in Chapter (refer Fig. 1.8). It is seen that the lateral are effectively resisted by the various plane frames aligned in direction of the loads. The wind loads seismic loads are to be estimated in accordance with IS and
They are assumed to act at the floor levels and are appropriately distributed the various resisting (in proportion to the relative translational stiffnesst).
Although these loads are essentially in nature, the loading Codes prescribe equivalent static loads to facilitate static analysis in lieu of the dynamic analysis [Ref. if the building is very tall unsymmetrically
A of the of a plane frame in a building is
approximately given by of moments of inertia of the (with axis of that part of the frame. However, the presence of all masonry can substantially (up to two times and
this also he accounted for. if the building is of
walls (on of
.
. the adverse effects of reduction instiffness this critical storey should be specially accounted far in design, as advocated in IS 1893 (2002).
328 REINFORCED CONCRETE DESIGN
proportioned, rigorous dynamic analysis is called for. Otherwise, in most ordinary situations, simplified static analysis is permitted by the Code In fact, the Explanatory Handbook to the Code [Ref. states:
The simple methods of load analysis in vogue arc the Porral Method and the Cantilever Method [Ref. wherein the static indeterminacy in the frame is eliminated by reasonable assumptions. The Portal Method, in particular, is very simple and easy to apply. It is based on the following assumptions:
The inflection points of all columns and beams are located at their respective.
middle points.
For any given storey, the 'storey shear' (which is equal and opposite to the sum of all lateral loads acting above the storey) is apportioned among the various columns in such a way that each interior twice the shear that is carried by each exterior column.
The of the Portal Method (as as Method) is that it does not account for the relative stiffnesses of the various beams and columns. An improved method of analysis, which takes into account these relative stiffnesses (in locating the points of inflection and apportioning the storey shear to the various columns) is the so-called Factor Method, developed by Wilbur [Ref. This method works out to fairly accurate, although it requires more computational effort.
9.4 PROPORTIONING O F MEMBER FOR PRELIMINARY DESIGN
As mentioned earlier, it is to estimate the cross-sectional dimensions of beams and columns prior to frame analysis and subsequent design in order to
assess the dead loads due to self-weight;
determine the various member stiffnesses for analysis.
This requires a preliminary design, whereby the design values of bending shear forces and axial forces in the various members may be approximately computed.
Gravity Load Effects
The apportioning of gravity loads to the various and columns may bc done by the shown in Fig. 9.5. These areas are based on the assumption that the distributed) gravity loads in any panel are divided among the supporting beams by lines midway the lines of support, the load in each area is transferred to support. In the general case of a 'two-way' rectangular slab panel, this a triangular shaped tributary area on each of the two short span beams and a area on each of the two
ANALYSIS DESIGN MOMENTS IN CONTINUOUS SYSTEMS 329 long span beams, as shown Figs 9.5 (a). [refer 24.5 of the When slab is 'one-way', the trapezoidal tributary area on each long may be approximated as a simple rectangle, as indicated.
TWO-WAY SLAB ONE-WAY SLAB (a) slabs
slabs with primary beams load secondary
two-wa slab one-way slab
ONDARY BEAM
slab loads on primary beam
AA) A
beam
I I slab loads on secondary beam
slabs primary and secondary beams
slab loads an secondary beam
column tributary areas
Fig. 9.5 Tributary areas for beams and columns
When the floor system is made up of a combination primary beams and secondary' beams, the tributary areas may be formed in a similar fashion
lines farming the two of the triangle may assumed be at 45" to the
'Primary' beams (or girders) are those which frame into the whereas 'secondary' are those which are by the primary beams as explained is Section
330 REINFORCED CONCRETE DESIGN
[Figs. One-way slabs are assumed to transfer the loads only on to the longer supporting sides [Fig. The loads from the 'secondary beams' are transferred as concentrated loads to the supporting 'primary beams' [Fig.
However, for preliminary calculations, a uniformly distributed load may be assumed on each primary beam, with the lributary areas appropriately taken.
The axial loads on the columns at each floor level are obtained from the tributary areas the primary the load on each primary being shared equally by the two supporting columns. The tributary areas for the columns are indicated in Fig. In addition to the gravity loads the floor slabs, loads from masonry walls (wherever applicable) as well as self-weight of the
should be considered.
For a preliminary design, the influence of different possible live load patterns may be ignored, and all panels may be assumed to he fully loaded with dead loads plus live loads: Substitute frame analysis may be done to determine the bending moments and forces in the primary beams and columns; the secondary beams may be analysed as continuous beams using moment coefficients (except when they discontinuous at both ends). A of the maximum design moment in a
may be taken as where W,, is the total factored load on the and I its span: similarly the design shear force may he approximately taken as
Sizes of interior columns are primarily determined the axial loads coming from the tributary areas of all the floors above the floor under consideration. However, exterior columns are subjected to significant bending moments (on account of unbalanced beam end in addition to axial forces. Accordingly, these must be designed for the combined effect of axial compression and bending moment; this can be conveniently done using appropriate interaction curves (design aids), as explained in Chapter 13.
it should be noted that if the frames are unsymmetrical, the additional moments induced sway should also be accounted for.
Lateral Load Effects
In the case of tall buildings, the effects of wind or earthquake moments are likely to influence the design bending moments in primary beams and especially in the lower floors. These load effects may, for the purpose of preliminary design, be Portal Method described in Section The
,
.
for special design and detailing, as discussed in detail in Chapter 16 9.5 ESTIMATION OF OF FRAME ELEMENTS
A typical building frame - even a 'substitute frame' a statically indeternunate To enable its analysis, whether approximate methods or rigorous methods, it is necessary to know the (flexural, torsional, axial) of [refer Fig. In load transfer scheme shown in Fig. it is assumed that
offer 'rigid' to the secondary beams.
ANALYSIS FOR DESIGN MOMENTS IN CONTINUOUS SYSTEMS 331
various members that constitute the frame. Frequently, it is only
that needs to be known. Axial stiffness is generally high, resulting in axial deformations. as explained in 7, the torsional stiffness of a reinforced concrete is drastically reduced following torsional cracking and hence can be ignored altogether. for computing torsional stiffness, wherever required, are given in Section
The 'flexural stiffness' of a element (with the far end 'fixed') is given where EI is the 'flexural rigidity', obtained as a product of modulus of elasticity E and the second moment of area I, and is the length of the member. As the analysis is generally a 'linear static' analysis, the appropriate value of E is given by the static modulus of elasticity defined in Section The problem lies with specifying the value of I, must ideally reflect the degree of cracking, the amount of reinforcement and the participation of flanges (in beam-slab members).
The Code the calculation of flexural stiffness based on the 'gross' concrete section, 'uncracked-transformed' section or the
transformed'section [refer Chapter however, the same basis is to bc applied to all the elements of the frame to analysed. This is reasonable, because what really matters is the stiffness, and not the absolute stiffness. The most common (and simplest) procedure is to consider the 'gross' section ignoring both the of reinforcement and the of cracking) for calculating the second moment of
An alternative procedure, which better reflects the higher degree of flexural cracking in beams relative to columns, i s to use for columns and 0.5 for beam stems [Ref. 9.111.
In slab-beam systems, the presence of flange enhances the stiffness of the beam. However, the flanged action (with effective width as described in Section is not fully when the flanges are subjected to flexural tension, as in the regions of 'negative' moment. Some ignore contribution of the flanges (mainly for and treat the beam section as being rectangular. An procedure, suggested in 9.8, is to use twice the moment of inertia of the gross section This corresponds to an effective flange with = 0.2 to 0.4. The use of such a procedure eliminates consideration of the flanges; it gives reasonable results and is simple to apply, and hence is by the authors of this book.
9.6 ADJUSTMENT OF DESIGN MOMENTS AT BEAM-COLUMN