Inflection Points of Point Load on Continuous Beam

أكتوبر 22, 2022 إرسال تعليق

In the previous section we saw the development length required (at the support) for the bottom bars of a simply supported beam. Now we will see the bottom bars of a continuous beam. Fig.15.38 below is a part elevation showing an intermediate span CD of a continuous beam.

Fig.15.38

Intermediate span of a continuous beam

The fig. also shows the bending moment diagram. The points of inflection are marked as R and S. [T he bending moment at the 'points of inflection' is equal to zero. So the portion between the R and S is similar to the portion between the supports of a simply supported beam] The bottom bars provided for the sagging moment between R and S will be under tension, and so will be stretched. They will be trying to contract to their original length. So we can say that, the ends of the bars will be trying to pull inwards (towards the center of the span) from R and S.

• In the case of simply supported beams, we calculated the minimum length that is available to prevent the pull out from the support.

• In the above continuous beam, our aim is to calculate the minimum length that is available to prevent the contraction of the bottom bars between R and S. Or in other words, the 'pulling out' from R and S.

So both cases are similar. Just as we did in the case of a simply supported beam, here also we can take a small segment pq of the beam for analysis. This small segment is at a distance of l _x from the point of inflection R as shown in the fig.15.39 below:

Fig.15.39

Small segment pq of the beam

The forces on the segment pq will be exactly similar to what we saw in fig.15.30 for the simply supported beam. So the calculations will also be the same, and the available length which will resist the pull out will be derived as M _uR ⁄ V _u .

• In the case of simply supported beam, V _u is the maximum shear force that occur between the supports. This maximum occurs at the supports.

• In the case of the above continuous beam, V _u is the maximum shear force that occurs between R and S. From the topic on 'Analysis of continuous beams', we know that this maximum occurs exactly at R or S as shown in fig.13.40 below:

Fig.15.40

Shear force diagram

Values immediately to the left of R and to the right of S in the above shear force diagram are higher. But those values are related to the hogging moment at supports. They are not related to the bottom bars. So we can ignore those higher values.

Thus the length M _uR ⁄ V _u is a part of 'that length of the bar which is within R and S'. This is the least available length that will help to prevent the pull out.

But the length L ₀ if any, beyond R and S will also contribute to prevent the pull out. So, just as the simply supported beam, the total length available to resist the pull out is equal to M _uR ⁄ V _u , + L ₀ . It must be greater than or equal to L _d (unique value) . So we can write:

L _d (unique value) ≤ M _uR ⁄ V _u + L ₀
This expression is same as 15.6 that we derived earlier for simply supported beams.

[It may be noted that, in the fig.15.38 above, the BM diagram shown is that for a uniform loading. There is continuity between sagging and hogging parts. But if Live loads are present, we must consider the 'envelope'. This topic was discussed in a previous section of this chapter, with the help of fig.15.17 . In such a case, the points of inflection to be taken are those marked as p₀ ^s in fig.15.17]

But this L ₀ in the case of a point of inflection has an important difference from that in a simply supported beam. According to cl.26.2.3.3 (c) of the code, L ₀ at a point of inflection cannot exceed the larger of the following:

(i) d

(ii) 12Φ

This requirement can be detailed as follows:
At an intermediate support, there is enough space to extend the bottom bars to a longer distance. But only a certain length (which is the larger of d or 12Φ ) measured from the point of inflection will be eligible to be considered as L ₀ . This is shown in fig.15.41 below:

Fig.15.41

Restriction on L ₀ at point of inflection

Anchorage and development length requirements at inflection points in continuous beams. Bars of low diameters should be reduced if necessary.

In a simply supported beam, there is no such restrictions. The bar can be extended to any distance. We can also give bends (fig.15.36) and even extension beyond bends (fig.15.37). But such measures to increase L ₀ should satisfy cover requirements. So it may not be possible to extend the bars to the required length. So in effect, there are restrictions on the availability of L ₀ in the cases of both simply supported beams and continuous beams.

We have to satisfy the relation:

L _d (unique value) ≤ M _uR ⁄ V _u + L ₀

To increase the right side, we can increase L ₀ . But we have seen that L ₀ has upper limits. Even with the upper limiting value, it may not be possible to satisfy the relation. In such a situation, we can increase M _uR . For this, we will have to reduce the curtailments done to the bottom bars so that more bars reach the support. This method is effective even though it will result in increased costs.

The best solution that the code recommends is to decrease the left side. ie., decrease L _d . Let us see how this can be done:

We have seen the details about L _d , the unique value of development length for a bar of a particular diameter. We have:

In this equation, the diameter Φ is in the numerator. So when Φ increases, L _d also increases and vice versa. This means that, for a larger diameter bar, more length will be required to exert the necessary gripping force to keep it in position with out causing a pull out. And for a bar of lesser diameter, lesser length will be sufficient.

We have:
L _d (unique value) ≤ M _uR ⁄ V _u + L ₀
Thus by using bars of lower diameter, the left side of the above expression can be reduced. By reducing the left side, we have a better chance of satisfying the condition. Thus, if even after providing the maximum allowable value of L ₀ , the condition cannot be satisfied, we must reduce the diameter of the bars and check again. However, we must remember that we cannot use very low diameter bars. Diameters less than 12 mm are not generally used for bottom bars of beams.

So now we know how to ensure that 'the development length requirements are satisfied at the simple supports and at the points of inflection'. The required length should be provided for all the bars at the support or at the inflection point.