Chapter 13 (cont..12) - Calculation of spacing of stirrups

In the previous section we completed the discussion on 'bent-up bars'. With that, we know how any of the steel reinforcements (vertical stirrups, inclined stirrups, or bent-up bars) will contribute towards the shear capacity of a beam section. We also know the contribution V_uc from the concrete. So the sum of these two contributions will give the total shear capacity of the beam section.

Both V_uc and V_us are the values at the 'ultimate state'.
• V_uc is the shear resistance offered by the concrete when the beam section is at the 'point of impending failure' due to it's shear capacity being exceeded.
• V_us is the shear resistance offered by the shear reinforcements at this 'point of impending failure'.

This is shown in the fig. below:

If we are given a beam section and it's properties, and also the details of the shear reinforcements, we can calculate both V_uc and V_us. Their sum will be the total 'ultimate shear resistance' that the beam section can offer. We denote this ultimate shear resistance as V_uR. Thus we can write:

Eq.13.53:

V_uR =V_uc +V_us

This Eq.13.53 can be used for analyzing the shear capacity of a beam. That is, if we are given a beam section, and asked to calculate the ultimate shear resistance, we can use this Eq.

If we look at the Eq. from another point of view, we can use it for design purpose also. This can be explained as follows: When we design a new beam section for shear, we have to make sure that the factored shear force V_u applied at the section is less than or equal to the ultimate shear resistance V_uR of the section. That is, the following relation should be satisfied:

13.54:

V_u ≤  V_uR

Substituting for V_uR from Eq.13.53, we get

13.55:

V_u ≤ V_uc +V_us

While designing, we can first assume the left side to be equal to the right side. That is:

Eq.13.56:

V_u = V_uc + V_us

Here

• V_u is a known value given to us.

• V_uc can be calculated by using Eq.13.34.: V_uc = τ_cbd

• V_us is calculated based on the type of shear reinforcement that we intend to use for the beam. Thus we have:

  ♦ Eq.13.39 (for vertical stirrups):

  ♦ Eq.13.47 (for inclined stirrups):

  ♦ Eq.13.49 (for  single or parallel bent-up bars):

         V_us = 0.87f_y A_sv sinα  

  ♦ Eq.13.52 (for bent-up bars in series):

From the above equations for V_us, we can see that, in Eq.13.56, there will be only one unknown, and that is S_v. [A_sv is not an unknown because the diameter of stirrups and no. of legs will be already assumed.]

We have seen that when bent-up bars are used to resist shear, stirrups should also be provided, and these stirrups should take up equal to or greater than 50% of the factored shear force. Thus, when bent-up bars are provided, Eq.13.56 can be modified as:

Eq.13.57:

V_u = V_uc + V_us(stirrups) + V_us(bent-up)

When single or parallel bent-ups are used, Eq.13.49 above can be used for V_us(bent-up). But in Eq.13.49, S_v is not coming into the picture at all. A_sv of the bent-up bars will already be fixed while designing the beam for flexure. So in this case also S_v, which is coming in the second term of Eq.13.57 is the only unknown.

When a series of bent-up bars are provided, Eq.13.52 is to be used in Eq.13.57. In this case, S_v can be assumed to be the same for both stirrups and the bent-up bars, and so it becomes the only unknown.

So in Eq.13.56, we can bring V_us to the left side because it is the term which contains the unknown. Thus we get:

Eq.13.58:

V_us = V_u - V_uc ⇒ V_us = V_u -τ_cbd

Once we do the above subtraction and obtain V_us, S_v can be calculated. We will have to round off the spacing S_v to multiples of 5 or 10 mm. When this rounding off is done, the final S_v provided should be less than the S_v calculated. This will give more area of steel to resist the shear. Because of providing more area than required, the V_us  part of Eq.13.56 or 13.57 will be more than what is required to resist V_u. Thus 13.56 will become an inequality as shown below:

13.59:

V_u <  V_uc + V_us

But the right side of 13.59 is the ultimate shear resistance V_uR provided. So we can write:

13.60: V_u < V_uR

This satisfies our basic requirement given in 13.54.

We will now do a problem just to demonstrate the calculation of S_v. [It may be noted that the result obtained in this problem is not a valid result, as it does not do the various checks and code provisions. Here we are interested in the 'steps involved in the calculation' only. we will discuss about the checks and code provisions the next sections.]

Problem:

Find the spacing of stirrups in a beam section, to resist a factored shear force of 75 kN. The beam has a width of 230 mm and an effective depth of 400 mm. The tensile steel consists of 3 - #16. Assume Fe 415 steel and M20 concrete.

Solution:

We will write the given data:

b =230 mm, d =400 mm, A_st =603.186 mm², f_y =415 N/mm², f_ck =20 N/mm², V_u =75 kN

First we calculate V_us using Eq.13.58:

V_us = V_u - τ_cbd

So we must calculate τ_c. (Details here). For this we look at table 19 of the code.

pt = 100Ast/bd =0.6556

0.5000  → 0.4800

0.6556  → 0.5298
0.7500  → 0.5600. So we get τ_c =0.5298 N/mm².

Substituting all the known values in Eq.13.58 we get V_us =26258.055 N

V_us for vertical stirrups is given by Eq.13.39. Assuming 2-legged stirrups of  8 mm dia., we get A_sv =100.53 mm². 

Substituting all the known values in Eq.13.39 we get, S_v = 552.92 mm. Rounding this off to multiples of 5 or 10 mm, we get S_v =550 mm.

This completes the design part. We can now do an analysis:

• The actual V_us obtained at the section can be obtained by using Eq.13.39 again. But this time, V_us is the 'unknown'. We get V_us =26397.350 N

• V_uc is obtained using Eq.13.34: V_uc = τ_cbd. We get V_uc = 48741.945 N.

Adding the above two we get V_uR =75139.295 N =75.13 KN. This is greater than the applied factored shear force. So the section is safe.

But it should be noted that the spacing of 550 mm exceeds the maximum spacing allowed by the code. Also various checks have to be carried out in the design process. The above problem is just a demonstration of the method of calculation of S_v. We will discuss about the code provisions and various checks in the next sections.

PREVIOUS

NEXT                                          

Copyright ©2015 limitstatelessons.blogspot.com - All Rights Reserved