## local buckling parameters for steel beams.

### The relation between Nominal Moment Mn and the λ Values and the three regions.

The topics included in our discussion are shown in the next slide.

The relation between λ and Mn can is as three regions. The first region is the compact region, which we call F2-1, which states that Mn=Mp=Zx*Fy.

To determine Zx we have to evaluate the lambda and its value to reach this first region.

To determine Zx, we have to evaluate the lambda and its value to reach this first region. For the first zone, The lambda value λ should be from 0 to λp.

For the second region represented by the equation, for the Lateral-Torsional buckling region, F2-2, the value of the line for the second region starts from Mn=Mp, and at the end, the value is Mn=0.70 *Fy**Sx.

The value of 0.70 Fy at the other end is due to the residual stresses during fabrication. The residual stress will cause Internal stresses in opposite directions. So 0.30 Fy*Sx is reduced from the Mn.

We have a linear equation for the second region.

If your lambda value λ is between λp and λr, then the line equation Y=m*x will be Mn=Mp- (the difference between the x’s)*(λ-λp).

The final form will be Mn=Mp-(Mp-0.70*Fy*Sx)*((λf-λpf) /(λrf-λpf), where λrf for limiting slenderness for Flange as an upper limit on non-compact.**What are the values for λp?****For the Flange**, our lambda is λF= (Bf/2*Tf) used for the estimation of λF. *

*While in the case of the web, λw= (h/tw), is used for the estimation of λw. *

The first local buckling parameter for flange is λpf which is=0.38*sqrt(Et/Fy) or sometimes written as =65/sqrt(Fy), this is derived if E=29000 /sqrt of Fy in KSI.

We estimate the numerator as if multiplying 0.38*sqrt(2900),=0.38*170.29=64.711, which will be approximated as 65, λpf=65/sqrt(Fy) in ksi .

For instance, if Fy=36 ksi, then put the value of 36 inside the sqrt. λpf for Fy=36ks=65/6=10.833.

The first local parameter λpf for Fy=50 ksi =65/sqrt(50)=9.19.

For the web, the first local parameter λpW for web =3.76sqrt(E/Fy). Again the numerator will be set =3.76*sqrt(29000)=640.0 approximately.

λpw =3.76*sqrt(E/Fy)=640/sqrt(Fy). For Fy=36ksi, λpw =640/6=106.66=107.While for Fy=50 ksi, λpw =640/sqrt(50)=90.501. We can use the sketch of the three regions in the two cases of Flange and Web, but consider the different value λp for each case for Flange and Web.

### The slenderness ratio values for the non-compact section.

The non-compact section slenderness ratio starts from λp till λr.

λr will be our **second local parameter**.

For the value of λr for flange λrf, for the case of flange λrf=1.0*sqrt(E/Fy)=170/sqrt(Fy), there is a quick way to estimate to check the web whether compact or not if we equate 640/sqrt(Fy)=h/tw, then we can estimate the corresponding Fy. For any section, with given hw and tw, we get the corresponding Fy for the material used

While the **second local parameter**. λr equation for the web equals v5.70*sqrt(E/Fy)=970/sqrt(Fy).

### What are the unstiffened -elements?

Unstiffened – elements are elements that are supported only at one edge, for example, the flange of the W section, and the flange of C -the channel.

### How to estimate bf for the un-stiffened- elements?

The code has defined the value of b, used in the slenderness estimation for four items, the first item is for the case of an I-shaped member where b is 1/2 the full flange width bf.

For the second case where the section for legs of angles and flanges of channels and zees, b is the full leg width-refer to item 12 table B4.10.

For the third case where the section is a plate, b is the distance from the free edge to the first row of fasteners.

For the fourth case where the section is the stem of tees, d is the full depth of the section item no.14.

If we check Table B4.1b, which gives the ratio between width to thickness for members subject to flexure, item 10-For flanges or rolled I beam, item 11-Flanges of doubly or singly Symmetric I shaped built-up section.

The list from No.10 to no.14 for the un-stiffened elements and the λp, the first local buckling parameter for members **subject to flexure**. The limiting slenderness of the compact flange. The second local buckling parameter λr, is the limiting slenderness for the non-compact flange.

### What are the stiffened elements?

Stiffened – elements are elements that are supported at two edges, for example, the web of W section, the web of C -channel, and the tube sections.

### How to estimate h for stiffened elements?

B4.1 table gives the different elements and how to find h value.

There is also a graphical table for the unstiffened elements, check items from 10-14. *The items from 1-9 are for the elements subjected to axial compression.*

The list from No.15 to No. 20 for the stiffened elements in Aisc-360-10.

The values of λp, the limiting slenderness for compact flange, and λr, the limiting slenderness for non-compact flange, are shown in these tables. There is a total of 21 items in the AISC360-16. The last three items are shown in the next slide image.

This is the pdf file used for the illustration of this post.

For the buckling concept, please refer to this link from Prof T. Bart Quimby, P.E., Ph.D., F.ASCE site.

This is a link for the next post, Analysis of steel beam-Solved problems.