Solved Problem 5-2 For the plastic moment value.
It is required to estimate Plastic moment Mp for the given section, W10x60 of A992.
Using the first option by considering the W section as a group of plates.
I have considered, as a first option, the W section as consisting of three plates, for the sake of comparison, after evaluating the Zx and Mp. for the first option as an assembly of plates for the solved problem 5-2.
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The upper and lower plate dimension is (10.10″)*0.68″, while the intermediate plate dimension is 8.84″*0.42″. The following pictures show in detail the sequence for that calculation.
Due to the symmetry of the section, the P.N.A will be in the middle. for A1+A2 areas, where A1 is the upper plate area, its value=6.868 inch2, while A2 is the area of half of the web, adding them together will give AT/2, where AT is the total area. the value of the AT/2=8.7244 inch2.
The next step is to estimate the Cg distance For A1+A2, from the first moment of areas about the P.N.A.
We consider the formula AT/2*y bar=A1*y1+A2*y2, we have=Ybar=(6.868*4.76+1.8564*2.21)/8.7244, which will be Y bar =4.22″, as we can see from the next slide.
Zx can be estimated as Zx=AT/2*(Y1bar+Y2 bar), the AT/2=8.7244 inch2, while Y1 bar=y2 bar=4.22″, then Zx=8.7244*(2*4.22)=73.63 inch3.
For the formula Mp=Zx*Fy, then the Plastic moment can be calculated as Mp=73.63*50=3681.50 inch-kips, to convert that value to Ft-kips, just divide by 12, at the end the plastic moment value MP=307.0 Ft- kips for the solved problem 5-2 as an option- a).
Can we use Table 1.1 to get Zx and Mp?
Using table 1-1 to get the plastic moment will give us only the Zx value and the dimension of the sections as the W10x60 section from table1-1.
The flange width=10.10″ and the tf=0.68″ and the overall depth=10.20″ and web thickness=0.42″. The Zx value is 74.60 inch3, we need y plastic to estimate the Mp value, we cannot find the Y plastic in that table.
The only solution is to consider W10x60 as composed of two Wt sections. Each Wt section is Wt 5×30. We need to proceed to another table.
The plastic section modulus or Z value of w10x60 can be found in table 1-1. The Z value =74.60 inch3.
Using the second option by considering the W section as two Wt sections.
The data for the needed Wt section of WT5x30 can be found in table 1-8.
We can get the Y bar from the section property of WT5x30, the value Y bar=0.884″, which is the distance from the plastic neutral axis to the upper top flange, as we can see from the next slide.
The total area of the Wt 5×30 =8.84 inch2.
To get the plastic moment we join the two Wt 5×30 to create W10x60, the compression force C is acting in the plastic Neutral axis of Wt which is=0.884″ as we get from the previous data of WT5x30. The acting Tensile force T due to symmetry acts at the plastic neutral axis but at a distance =0.844″ from the bottom.
The distance between C and T which we call it Yct is the whole depth(-) 2*yp. the overall depth we can get from table1-1 fr W10x60, which is=10.20″. The compression force C=T=At/2*Fy=(17.70/2)*50==442.50 kips, yct=10.20″-2*0.884=8.432″.
The Plastic moment=442.50*8.432/12=310.93 Ft.kips and can be approximated to 311.0 Ft.kips.
There is another way by considering Mp=Zx*Fy=At/2*yct, which gives us the same result as Mp=311.0 ft. kips.
The following calculation, accompanied by a sketch from the next slide shows the value of Zx and Mp.
For the sake of comparison between options a and b, the values of Mp are written side by side.
This is a link to download the pdf file used for the illustration of this post.