# 17- Introduction to Cb-Bending coefficient part-1 for steel.

## Cb-Bending coefficient-moment gradient factor part-1 for steel beams.

### How do we estimate the CB-bending coefficient values for different loading types and bracing locations?

We are going to talk about a new subject, which is how to determine the Cb-bending coefficient. The Cb is also called the moment gradient. A discussion of the calculation for different loading and bracing is provided. For all the previous examples we take the value of Cb=1 whether for the estimation of nominal design strength or the design of a given section.

The different values of the CB-bending coefficient or moment gradient factor Cb are shown for the different types of loading for a simple beam, as quoted from Prof. Mccormack’s book.
The first figure shows a simply supported beam with a uniformly distributed load W-ult k/ft.
The bracings are at the support left end and the right support; each bracing is shown as an X mark.
For the unbraced length =l, the value of the Cb=1.14 for the first case.

The next figure shows a supported beam similar to the first case, but this time, a bracing is added in the middle of the span.

So, in this case, we have three bracings: on the left side, middle side, and right edge. The middle bracing is located at a distance =L/2 from the left support.

The uniformly distributed Load =W as the Ultimate load Kips/Ft; in that case, the cb-bending coefficient value is =1.30.

The third case is for a simply supported beam with a concentrated load acting in the middle. The cb-bending coefficient value for this case is =1.32 for two bracings only.
The fourth case is for a supported beam, similar to the third case, but with three bracings. The acting concentrated load is Pult. The Cb-bending coefficient-moment gradient for this case is 1.67.

There are several cases of two concentrated loads and also bending moments acting at the supports. The Cb-bending coefficient value for the last case is 2.27.

For statically indeterminate structures, we have a Fixed end beam under W-ult as uniform load, with two bracings at the two edges. The CB-bending coefficient value is 2.38.

We have the CB-bending coefficient value =2.38 for the same beam but with three bracings.
The third case is for the fixed-end supported beam with a Pu load acting in the middle; the CB-bending coefficient value is =1.92. The last case is for two fixed-end supported beams with a Pu load acting in the middle but with three bracings; the CB-bending coefficient value is = 2.27.

### How do you estimate moment gradient cb for a simply supported beam under a uniform loading?

The equation we use to evaluate the moment gradient factor Cb is cb=12.50*M-max/(2.50* Max+3*MA+4*Mb+3*Mc); the sum of the denominator values is 2.50+3+4+3=12.50.
A is the first quarter-point. B is the second quarter-point. C is the third quarter-point.

In the case of a simply supported beam under uniformly distributed W-ult k/Ft, for which we have two bracings, the first bracing is located at the left edge, while the second bracing is at the right support.

The M-max value, as we are familiar with a simply supported beam under uniform loading, is Mmax= W-Ult*L2/8, where L is the span distance.

This is the M-max for Ma value at the first-quarter point. First, estimate the reaction is W*L/2.

The moment caused by the reaction = Wult*L/2*L4, opposed by a moment for loading =Wul*(L/4)*1/2*(L/4)= the moment value Wul*l2/8-Wul*L2*1/32.

Finally, MA=3*Wult*L^2/32, due to symmetry Mc value is the same as the Ma, what is left is the Mb value.
For Mb value, Wult*L^2/8, now we have all the information to estimate the Cb value, as 12.50 *(Wul*L^2/8)/(2.50*(Wul*L^2/8)+(3+3)* Wult*3*L^2/32.

For the numertaor 12.50*Wult*L^2/8/(2.5*Wult*L^2/8)+3* (3*Wult*L^2/32) +(4*3*Wult*L^2/8) +(3*Wul*L^2/32).
Take Wult*l^2/8 as a common factor; we have (2.50+ 9/4+9/4+4) for the denominator. The denominator sum=2.50+4+18/4)=11, cb=12.50/11=1.14, the same result as the first case.

### How do we estimate the moment gradient factor Cb for a fixed ends beam under a uniform loading?

Let us check another case. L is the span length under a uniform loading, with two bracings at the two edges for a beam with two fixed supports.
For the fixed beam, the Fixed end moment M-Fe=Wul*L2/12 if the loading is a uniform load = W-ult k/Ft.
We will divide it into A, B, and C, where A at a distance = L/4, B in the middle, and C at L/4 from the right support.

We draw the bending moment diagram but must estimate the reaction at the supports. We will perform the same exercise as before.

For the first point, A. The total load =Wul*L, then the R-a, the reaction at A=WUlt*L/2=R-b, while the two fixed end moments are W*ult*L^2/12. Ma=moment from reaction-Moment from the W ult for a span of L/4-W*L^2/12 as resisting moments.

All resisting moments =Wult*(L/4)*1/2*(L/4)=Wult*L^2/32+W*l^2/12, we sum all moments, as shown, we get W-ult as a common factor.
We have wult*(-L^2/12+ L^2/8-L^2/32), make a common denominator as (32*12), so we have MA=-Wu(-32*L^2)+ +4*12*L^2 -12*L^2). So as a sum=(-32+48-12)=Ma=Wult*L^2(+4/(12*32)=+Wult*L^2/96, with a positive sign.

Due to symmetry, the Mc = +W ult*L^2/96.

At the middle Mb=Wul*L^2/24, estimated as (Wult*L^2/8-W*L^2/12), we give the same value of Wult*L^2/24.
For cb=12.5*Mmax/(2.50*Mmax+3*MA+4*Mb+3*Mc) . Max moment. In this case W*L^2/12>Wult*L^2/24.

The  numerator=12.50*Wul*L^2/12/(2.50*W*L^2/12+3*(1/96)*Wul*L^2+4*Wult*L^2/24+3*(Wul*L^2/96).So we have in the denominator: WulL^2(2.50/12)+(3/96)+(4/24+3/96).

We will sum all the values as (2.50/12)+(6/96)+(4/24), delete the item Wult*L^2, we are left with(12.50/12)/(make a common factor as 96*(8*2.50+6+ 16)/96. CB value =(12.50/12)/(42/96).cb=9612.50)/(4212)=8*12.50/42=2.38. The moment gradient factor Cb or the coefficient of moment equals 2.38 for the fixed end beam.

This is the PDF file used to illustrate this post.

For more detailed illustrations of the CB, please follow this Flexural Limit State Behavior.

This is the complete list of all posts related to Cb:

1-Introduction to Cb-Bending coefficient part-1 for steel.-post 17-this post.
2- Cb-The coefficient of bending part 2 for steel beams-post 18-next post.

3-Cb-The coefficient of bending part-3 for steel beams-post 18a.

4-Cb value-bracing at the midpoint of a beam-uniform load-Post 18b.

5-Cb value bracing at third points of a beam-U load-Post18C.

This is a brief illustration of the content of all the previous posts.

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