In order to determine the distribution of bending moments under the design loads, Structural analysis has to be performed. For convenience, the IS: 456 -2000 lists moment coefficients as well as shear coefficients that are close to exact values of the maximum load effects obtainable from rigorous analysis on an infinite number of equal spans on point supports. Table 15.5 gives the bending moment coefficients and Table 15.6 gives the shear coefficients according to IS: 456 - 2000. These coefficients are applicable to continuous beams with at least three spans, which do not differ by more than 75 percent of the longest. These values are also applicable for composite continuous beams.
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Tablel5.5: Bending moment coefficients according to IS: 456-2000
TYPE OF LOAD SPAN
IVI
[OMENTS SUPPORT] MOMENTS
Near middle of end span At middle of interior span At support next to the end support At other interior supports Dead load + Imposed
load (fixed)
+ 1/12 + 1/16 - 1/10 - 1/12
Imposed load (not fixed)
+ 1/10 + 1/12 - 1/9 - 1/9
For obtaining the bending moment, the coefficient shall be multiplied by the total design load and effective span.
Table 15.6: Shear force coefficients TYPE OF LOAD At end support At support n sup
ext to the end port
At all other interior supports Outer side Inner side
Dead load + Imposed load (fixed) 0.40 0.60 0.55 0.50 Imposed load (not fixed) 0.45 0.60 0.60 0.60
For obtaining the shear force, the coefficient shall be multiplied by the total design load 15.9.2 Lateral torsional buckling of continuous beams
The concrete slab prevents the top flange of the steel section (connected to concrete slab) from moving laterally. In negative moment regions of continuous composite beams the lower flange is subjected to compression. Hence, the stability of bottom flange should be checked at that region. The tendency of the lower flange to buckle laterally is restrained by the distortional stiffness of the cross section. The tendency for the bottom flange to displace laterally causes bending of the steel web, and twisting at top flange level, which is resisted by bending of the slab as shown in Fig. 22.6.
Fig 15.6 Inverted - U frame Action
Lateral Torsional Buckling of Continuous Beams can be neglected if following conditions are satisfied.
1. Adjacent spans do not differ in length by more than 20% of the shorter span or where there is a cantilever, its length does not exceed 15% of the adjacent span.
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2. The loading on each span is uniformly distributed and the design permanent load exceeds 40% of the total load.
3. The shear connection in the steel-concrete interface satisfies the requirements of
section
15.10 Serviceability
Composite beams must also be checked for adequacy in the Serviceability Limit State. It is not desirable that steel yields under service load. To check the composite beams serviceability criteria, elastic section properties are used.
IS: 11384-1985 limits the maximum deflection of the composite beam to The total
elastic stress in concrete is limited to while for steel, considering different stages of construction, the elastic stress is limited to Unfortunately this is an error made in the Code as the same limits are applied for steel in determining the ultimate resistance of the cross section. Since EC4 gives explicit guidance for checking serviceability Limit State, therefore the method described below follows EC 4.
15.10.1 Deflection
The elastic properties relevant to deflection are section modulus and moment of inertia of the section. Applying appropriate modular ratio m the composite section is transformed into an equivalent steel section. The moment of inertia of uncracked section is used for calculating deflection. Normally unfactored loads are used for for serviceability checks. No stress limitations are made in EC 4.
Under positive moment the concrete is assumed uncracked, and the moment of inertia is calculated as:
Where
is the ratio of the elastic moduli of steel to concrete taking into account creep.
is the moment of inertia of steel section.
Simply supported Beams: The mid-span deflection of simply supported composite beam under distributed load w is given by
Where, is the modulus of elasticity of steel and is the gross uncracked moment of inertia of composite section.
Influence of partial shear connection: Deflections increase due to the effects of slip in the shear connectors. These effects are ignored in composite beams designed for full shear connection. To take care of the increase in deflection due to partial shear connection, the following expression is used.
Where
are deflection of steel beam and composite beam respectively with proper serviceability load.
Note: For this additional simplification can usually beignored
Shrinkage induced deflections: For simply supported beams, when the span to depth ratio of beam exceeds 20, or when the free shrinkage strain of the concrete exceeds
shrinkage, deflections should be checked. In practice, these deflections will only be significant for spans greater than 12 m in exceptionally warm dry atmospheres. The shrinkage-induced deflection is calculated using the following formula:
is the effective span of the beam and is the curvature due to the free shrinkage strain, given by
modular ratio appropriate for shrinkage calculations
Note: This formula ignores continuity effects at the supports.
Continuous Beams: In the case of continuous beam, the deflection is modified by the influence of cracking in the hogging moment regions (at or near the supports). This may be taken into account by calculating the second moment of area of the cracked section under negative moment (ignoring concrete). In addition to this there is a possibility of yielding in the negative moment region. To take account of this the negative moments may be further reduced. As an approximation, a deflection coefficient of 3/384 is usually appropriate for determining the deflection of a continuous composite beam subject to uniform loading on equal adjacent spans. This may be increased to 4/384 for end spans. The second moment of area of the section is based on the uncracked value.
Crack Control: Cracking of concrete should be controlled in cases where the functioning of the structure or its appearance would be affected. In order to avoid the presence of large cracks in the hogging moment regions, the amount of reinforcement should not exceed a minimum value given by,
Where
is the percentage of steel
is a coefficient due to the bending stress distribution in the section
is a coefficient accounting for the decrease in the tensile strength of concrete
is the effective tensile strength of concrete. A value of 3 is the minimum adopted.
is the maximum permissible stress in concrete
Generally the span/depth ratios specified by codes take care of the shrinkage deflection. However, a check on shrinkage deflection should be done in case of thick slabs resting on small steel beams, electrically heated floors and concrete mixes with high "free
shrinkage". Eurocode 4 recommends that the effect of shrinkage should be considered
when the span/depth ratio exceeds 20 and the free shrinkage strain exceeds 0.04%. For dry environments, the limit on free shrinkage for normal- weight concrete is 0.0325% and for lightweight concrete 0.05%.
15.10.2 Vibration
Generally, human response to vibration is taken as the yardstick to limit the amplitude and frequency of a vibrating floor. The present discussion is mainly aimed at design of an office floor against vibration. To design a floor structure, only the source of vibration near or on the floor need be considered. Other sources such, as machines, lift or cranes should be isolated from the building. In most buildings following two cases are considered-
Fig. 15.7. Curves of constant human response to vibration, and Fourier component factor
The root mean square acceleration of the floor is plotted against its natural frequency for acceptable level R based on human response for different situations such as, hospitals, offices etc. The human response R-l corresponds to a "minimal level of adverse comments
from occupants" of sensitive locations such as hospital, operating theatre and precision
laboratories. Curves of higher response (R) values are also shown in the Fig. 15.7. The recommended values of R for other situations are
R = 4 for offices R = 8 for workshops
These values correspond to continuous vibration and some relaxation is allowed in case the vibration is intermittent (see BS6472 for further information).
Natural frequency of beam and slab: The most important parameter associated with vibration is the natural frequency of floor. For free elastic vibration of a beam or one way slab of uniform section the fundamental natural frequency is,
Where,
for simple support; and for both ends fixed.
= Flexural rigidity (per unit width for slabs) = span
= vibrating mass per unit length (beam) or unit area (slab).
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i) People walking across a floor with a pace frequency between 1.4 Hz and 2.5Hz. ii) An impulse such as the effect of the fall of a heavy object.
The effect of damping (being negligible) has been ignored.
Un-cracked concrete section and dynamic modulus of elasticity should be used for concrete. Generally these effects are taken into account by increasing the value of by
10% for variable loading. In absence of an accurate estimate of mass (m ), it is taken
as the mass of the characteristic permanent load plus 10% of characteristic variable load.
The frequencies for slab and beam (each considered alone) and are given by
Where is the Fourier component factor. It takes into account the differences between the frequency of the pedestrians' paces and the natural frequency of the floor. This is given in the form of a function of in Fig. (15.7):
= magnification factor at resonance
=0.03 for open plan offices with composite floor
To check the susceptibility of the floor to vibration after finding from Eqn.15.22 and the value of R from Eqn. 15.23 compare the result with the target response curve as in Fig. (15.7).
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