CHAPTER 3 METHODOLOGY
3.7 Member Stiffness
3.7.1 Stiffness of RC members
Different moment of inertia was considered for RCC members depending on the purpose of calculation. Concrete is good in compression. But it easily cracks under tension. The tensile stress may be the result of direct tension or flexure. Before application of loads concrete members remains untracked. But after application of loads; concrete cracks and the moment of inertia of concrete member changes.
As per section 10.10.4 of ACI 318-08, for strength design of RCC members, the stiffness EI should represent the stiffness of the members prior to failure. At ultimate loads RCC members crack and their stiffness decreases. Section 10.10.4.1 suggests the following values for effective moment of inertia (Ieff) for strength design purpose:
Columns………0.70Ig Walls (Uncracked)………0.70Ig Walls (Cracked)……….0.35Ig Beams……….0.35Ig Flat plates and flat slabs……….0.25Ig
Where, Ig is the gross moment of inertia of corresponding members.
For deflection calculation it is important to know the effective moment of inertia at service loads. For all concrete members there is a certain cracking moment Mcr, up to which the sections remain un-cracked. When service load moment or applied moment Ma
exceeds the cracking moment than the section cracks and the inertia of the section reduces. The effective moment of inertia depends on the degree of cracking. Figure 3.11 shows the variation of effective moment of inertia with moment ratio Ma/Mcr.It is observed that for a value of Ma=Mcr, the effective moment of inertia Ie, of beam is equal to Iut. Here, Iut is the moment of inertia of uncracked transformed section.
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Figure 3.11 Variation of beam Ie with moment ratio (Nilson et. al 2009)
With the increase of applied moment, effective moment of inertia decreases. When the service load moment Ma is about 3 times the cracking moment, effective moment of inertia decreases to Icr. Here Icr is the moment of inertia cracked transformed section.
When applied moment is low i.e. Ma< Mcr, then immediate deflection of RCC beams can be calculated from the following equation:
Where f is a function of load, span and support arrangement (Nilson et. al 2009). Ec is the modulus of elasticity of concrete and Iut is the moment of inertia of untracked transformed section.
Equation 3.2 is valid only for a very small range of load. Typically, for a beam service load Ma ranges 1.5 times to 3 times of cracking moment, Mcr. At higher loads, flexural tension cracks are formed in the beams. It reduces the moment of inertia of the RC beams.
For higher range of load, deflection or RC beams can be calculated from the equation below:
𝛥 𝑓
𝐸 𝐼
(3.2)
𝛥 𝑓
𝐸 𝐼
(3.3)
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Where,
Here,
Mcr= Cracking moment of RC beam, which can be calculated from the following equation:
Icr= Moment of inertia of cracked transformed section of RC beam.
Ec is the modulus of elasticity of concrete which can be calculated from the following equation provided by ACI (2008):
Ec= 57000√f′c Psi
As long as the applied moment is smaller than the cracking moment, deflection is proportional to the moments. At large moments the effective moment of inertia become progressively smaller. So deflection increases with the increase in moment. The relationship between the applied moment, beam uncracked or cracked stiffness with the dead and live load deflection can be shown in a graphical method in Figure 3.12.
Figure 3.12 Deflection of Reinforced Concrete Beam (Nilson et. al 2009)
I 𝑀
𝑀 I 1 M
M I I (3.4)
𝑀 𝑓 𝐼 𝑦
(3.5)
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Most RC beams are continuous over spans or rigidly connected at the ends. So for RC beams some rotational restraint is available at the ends. As per section 9.5.2.4 of ACI 318-08, a simple average of values obtained from Equation 3.4 for the critical positive and negative moment sections can be used to calculate effective moment of inertia of the RC beams. This produces the equation below:
ACI committee 435 showed that, in case of prismatic members, improved results can be obtained by using the following equation:
In equation 3.6 and 3.7;
Iem is the effective moment of inertia for the mid span section and Ie1 and Ie2 are those for the negative moment sections at the end of the beams. Iem, Ie1 and Ie2 will be calculated using equation 3.4. A basic problem for continuous span is the uncertainty of calculating the moment on the beam. Deflection depends on the moment diagram and on the flexural rigidity EI of each member of the frame. Again the flexural rigidity depends on the degree of cracking. Cracking depends on the moments. So it is a circular process. The process is iterative and very time consuming. So an approximate approach is required.
For service load analyses of structure ACI (2008) suggests to use moment of inertia that represents the degree of cracking in concrete members at different load levels. However, in section 10.10.4 of ACI 318-08, a relatively simple method was given to calculate the member’s stiffness for service load analyses in absence of the accurate estimation of degree of cracking. For service load analysis or deflection calculation ACI (2008) permits to use 1.43 times the moment of inertia used for strength design.
So for deflection calculation the following moment of inertia was used:
For Columns………Ie= 0.70 Ig x 1.43= 1.0 Ig
For Beams………Ie= 0.35 Ig x 1.43= 0.5 Ig
This simplified consideration resembles with figure 3.11. When the applied moment is about 2 to 3 times of the cracking moment, the effective moment of inertia of RC beams reduces down to almost half of the uncracked section. Also it is easy to use and produces I 0.50I 0.25 I I (3.6)
I 0.70I 0.15 I I (3.7)
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a very close result which can be found by performing rigorous calculation for the effective RC beam stiffness.