## How to design a steel beam? Solved problem-7-4-1.

### How to design a steel beam?-Solved problem-7-4-1.

It is required to design a steel beam with the lightest W or m section, under a uniformly distributed load and a given live load. The compression flange of the beam is fully supported against lateral movement. there are three cases according to the yield stress which will be dealt with.

### For Case#1- design a steel beam with Fy=36 ksi.

For the first case of fy=36 ksi. It is required to design a steel beam of W- section or M section for a simply supported beam of span =20′, for the three cases of Fy values.

Starting with *ASTM A36*, where the Fy=36 ksi. The first step of designing a steel beam is to estimate the Ultimate load value of 1.20D+1.6L of the steel beam. Wult=1.2*0.20+1.60*0.80=1.52kips/Ft. The ultimate moment for a supported beam, Mult= wult*L^2/8. Mult=1.52*L=20′, then Wult*l^2/8=1.52*20^2/8=76 Ft.kips. This is the value of Multimate.

We will estimate the slenderness ratios for both the flange and web and the flange lambda λF, lambda λw and compare these values with the limiting values of lambda λp, based on the comparison, we will determine if lambda λF or lambda λw< lambda λp – plastic.

Then we will be in zone 1, Mn=Mp=Zx*Fy, which is the plastic moment for the design of a steel beam. For the LRFD, the permitted Moment=Φb*Mn, Mult<=Φb*Mn=0.90*Fy*Zx. First, How do we design the section, without knowing the Zx value? Our first trial is to equate Mult/(0.90*Fy)=Zx and get the Zx then check the table.

The next slide shows the limiting slenderness parameters lambda λF and lambda λw, for ASTM A36, where Fy=36 ksi, lambda λf=0.38*sqrt*(E/Fy)=0.38*sqrt(29000/36) or 65/Fy=10.84, while lambda λw=640/sqrt(36)=106.666.

The Mult- is to be multiplied by 12 to convert to an inch. kips, Zx should be in inch3.

Check the units in the numerator, which is inch .kips/kips/inch^2. The final unit will be inch3.

Zx=76*12/(0.90*36)=28.15 inch3. The AISC has arranged sections based on Zx AISC- p3-26 as shown in the next slide.

The relevant table for the W-shaped Table 3-2, Selection by Zx. Starting with the biggest values of Zx. the tables in the AISC are sorted to the data of Zx and arranging the data in the form of Excel sheets.

For the beam section based on the calculation, Zx- required =28.15 inch3. We will select the upper top w- section in the table that has Zx >28.15 inch3. The following w section is W8x28 will give a lower Zx value.

The sections are arranged based on the lightest section, section w8x28 has a higher weight / Ft. Our Zx=28.15 inch3. The selection will be W12x22 with a little higher value of Zx.

We are still at the table, 3-22. Select a W12x22 but again we want table 1-1 for the properties for the dimensions of the section, width, and section of the flange, width of the web. I have included that table from the data of one manufacturer of steel sections. W12x22 has an area of 6.48 inch2. The overall depth is 12.31 inches.

The web thickness is 0.26 inches, the flange width =4.03 inches. the flange thickness is 0.425 “and all the other properties can be found.

Now we will be able to check the lambda of Flange and web for our selected section. Bf/2tf=4.03/(2*0.425), then λf=4.70. The limiting λf-p =65/sqrt(Fy)=10.83. λf is <λf-p for the flange, and the section is compact for the flange. the web is safe.

We will readjust the value of the ultimate moment by adding its weight. our section is W12x22, the weight is 22 lb/ft. The beam span =20′. Then, the adjusted W ult will be the previously estimated value, which was=1.52 kips/ft+1.20*(0.022), W ult =1.55 Ft/kips, then the new Mult=1.555*20^2/8=77.32 ft. kips.

Since our selected section is compact, then the Mn=Mp. Mp=Fy*Zx. For the LRFD the value of Φb*Mn=Φb*Fy*Zx, Which is=0.90*36 *29.30/12= the numerator is Lb*inch3*Ft, the denominator is inch2*12*inch, since 1 ft=12 inch. Φb*Fy*Zx=79.11 Ft.kips. This is the LRFD value of a moment.

Since the acting Multimate is only 77.32 ft. kips, then the section is safe, *since 79.11 is >77.32 ft. kips. *Finally, Φb*Mn >M-ult. This is the last step of designing a steel beam under Fy=36 ksi.

### For Case#2- design a steel beam with Fy=50 ksi.

The second case for ASTM A 992, where Fy=50 ksi. Mult without the superimposed load was =76 ft. kips.

We need Zx to start with table, 3-2. Zx=Mult/(0.90*50)=20.22 inch3.

*We will proceed* to table 3-2, but select Zx>20.22 inch3, which will be W10x19 with Zx=21.60 inch3. Zx selected=21.60 inch3. We proceed to get the properties of our section. W10x19.

This is a new slide for the W section W10x19 properties. The second case for ASTM A 992, where Fy=50 ksi. M ult without the superimposed load was = 76 ft. kips.

We need Zx to start with through table 3-2, Zx=Mult/(0.90*50)=20.22 inch3, we will proceed to table 3-2, but select Zx >20.22 inch3, which will be W10x19 with Zx=21.60 inch3. Zx selected=21.60 inch3.

We proceed to get the properties of our section W10x19. This is a new slide for the W section W10x19 properties of bf=4.02″, tf=0.395″, as for Bf/2tf=4.02/(2*0.395).

λF=5.10, the criteria λF-p=65/sqrt(Fy)=65/(50)^0.50=9.19. Then λF=Bf/2Tf< 9.19, the section is compact and the web is also compact for the w section. For the LRfd Φb*Mn= 0.90*Mn=Fy*Zx =50*21.60 /12 to convert to Ft.kips.=81.00 ft. kips.

For the Mult we will adjust due to the superimposed load, the weight of the beam, which is (19 lb/ft), the extra Mult=1.20*(19/1000*(20)^2/(8))=77.14 ft. kips.Φb*Mn=81.00 Ft.kips >Mult. The section is compact, 81.00 ft. kips>77.14 Ft.kips, the section is safe. This is the last step of designing a steel beam under Fy=50 ksi.

### For Case#3- design a steel beam with Fy=65 ksi.

The next slide includes the third part for ASTM A 572, for the design of a steel beam, Fy=65 Ksi.Φb*Mn=0.90*Fy*Zx=>Mult, Zx=76*12/(0.90*65)=15.60 inch3. We use Table 3-2.

The highest section is W12x14, with Zx value=17.40 inch3, Zx selected > Zx required.

We get the properties for W12x14. We have Bf=3.97″, tf=0.225″, for the flange Bf/2Tf =3.97/(2*0.225)=8.80, lambda λF=Bf/2Tf.

λF-p=65/sqrt(65)=8.10, λF=Bf/2Tf>λF-p, so the slenderness is between λF-p and λF-r, since 8.8>8.10, for tweb=0.20″, the overall depth =11.91″.

For λw is approximately=(11.91-0.45)/0.20)=57.20, λw=54.30 from table 1-1, λw-p= 640/sqrt(65)=79.38, λw<λw-p, but λF>λF-p.

The section is called non-compact. The section is in the second zone, Then Mn is not= Zx*Fy, but its value is between Mp and 0.70**Fy*Sx.

For the next slide, for the flange λF =8.10, λf-r=170/sqrt(65)=21.10.

Our λf=8.80, in between λf-p and λf-r. Then Mn is in between Mp and 0.70*Fy*Sx, it is required to get the Mn value.From Equation F3-1, Mn=(Mp-(Mp-0.70*Fy*Sx)/(λf-r-λf-p)). This is the last step of designing a steel beam under Fy=65 ksi.

Mn is the same as the equation for a straight line y=m*x. We will estimate the Mn for the upper Point=Mp=Fy*Zx=65*17.40=1131 inch. Kips. The second point Mn=0.70*Fy*Sx, Sx for the section=14.90 inch2.

0.70*Fy*Sx=0.70*65*14.90= 678.00 inch. kips.

For the LRFD, Φb*Mn =0.90*92.22=83.00 Ft.kips. Mult =76+Moment due to own weight, 1.20*(14/1000)*20^2/8. For a simple beam, Mult =76.84 Ft.kips, while Φb*Mn=83.00 ft. kips. The section is safe since Φb*Mn>Mult. This is the end of the post.

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

This is the link for **Chapter 8 – Bending Members**.

This links to the next post, 10-lateral-torsional buckling for steel beams.