12- Easy introduction to plastic bracing length Lp, Lr, Fcr.

Last Updated on August 11, 2024 by Maged kamel

How to estimate Lp?

Lp is the maximum un-braced length that will let the shape reach its plastic moment strength. The relevant equation for lp is introduced from Lp=300*ry/sqrt(Fy).

This equation gives Lp the bracing length at the plastic stage for any given section. We need to find the radius of gyration at the y-direction and the corresponding Fy or yield stress for the section in question.

Lp can be obtained if Fy=50 ksi from table 3-2 for the sorted w section by Zx.

How do we estimate limiting laterally unbraced length Lr?

Lr is the limiting laterally unbraced length for the limit state. The next slide image shows the relevant equation for limiting the laterally unbraced length for the limit state.

All requirements for this equation can be obtained from Table 1-1 for the W sections. Sx is the elastic section modulus at the x-direction, Fy is the yield stress, and E is the Modulus of elasticity.

Lr is the Limiting laterally unbraced length for the limit state. The general equation for the value of lr is included in the slide image.

Bracing length conditions.

The next slide shows the different values of Mn, or the nominal moment of a section, based on the bracing length, whether Lp, between Lp and Lr, or >Lr. Based on these conditions, we have three zones.

The relation between Nominal moment Mn and bracing length Lb.

The next slide shows the equations used in the AISC to determine the value of a nominal moment for a steel beam. What does lb stand for?

The stress values are based on the lb, whether bigger or smaller than Lp, Lr.

Practice problem for Lp & Lr and Fcr.

The bracing length at the plastic stage Lp, the limiting laterally un-braced length, and the value of Mp and Mr for a given steel W24x176 at Fy=36 ksi are required. The steps will be as follows:
1-From AISC table 1-1, we get the values ry to get the Lp from the equation Lp=300*ry/sqrt Fy. The ry value can be found in AISC table 1-1 for the given section.
2-Substitute for Fy=36 ksi, ry=3.04 inch in the previous formula.

3- Apply in the formula of Lp=300*(3.04)/sqrt(36)=152″, we convert into feet by dividing /12.The value for Lp=12.70′ approximately.

How do we get limiting laterally unbraced length for the limit state value?

1-If we refer to the next slide image, we know to estimate the limiting laterally unbraced length for the limit state value. From AISC 1-1, we get the values ry, rts, Sx, ho, and J, considering C =1. These are the parameters for estimating the limiting laterally unbraced length Lr for the limit state from formula F2-6.

The radius of gyration about y, ry=3.04 inch2,rts=3.57″ and the section modulus=450 inch3, ho=23.90″ n j polar=23.90 inch4, cw=68400.

2-Substitute for Fy=36 ksi and all the needed items. The relevant calculation is shown in the coming slide image. The value for limiting laterally unbraced length for the limit state=49.01.’

How to estimate MP and Mr.

3—The calculations are shown in the next slide images. When we substitute the equation Mp=Fy*Zx with the equation Mp=Fy*Zx, we can get the Mp, the plastic moment value. The z value is 511 inch3, while Fy=36 ksi.

The value of Mp is 1533.0 Ft.kips. We can get the Mr, which is the inelastic moment value, from the equation Mr=0.70Fy*Sx, where Sx is the statical section modulus for W24x176 and can be found to be 450 inch3.

We can get the slope S between Mp and 0.70Fy*Sx. Please refer to the next slide image for more information.

How to estimate critical stress Fcr?

For a given bracing length Lb=50′, which is >Limiting laterally unbraced length for the limit state, we can find the value of Fcr by substituting it in the shown equation in the next slide image. The critical stress Fcr value is 24.60 KSI.

Using the following equation, Mrx =Fcr*Sx. When we substitute, we will get the value of Mrx=922.55 Ft.kips.

Use Excel graph to plot Lb virus Mn.

We can use an Excel graph to plot the bracing length Lb versus the nominal moment; please refer to the next slide image.