Elastic And Plastic Section Moduli.
The video I used for illustration.
The video includes an introduction to the plastic moment and how to find an expression for the plastic section modules Zx in its first part from time 0:00 till 5:36.
From time 5:40 till the end, it covers the solved problem 5.1 in post 5; here is the link to the post.
You can click on any picture to enlarge, then press the small arrow at the right to review all the other images as a slide show.
The topics included in our discussion are shown in the next slides. This is how we estimate the Sx and Zx for a rectangular section.
To estimate the elastic section modulus, the acting moment should be equal My, where My is the yielding moment.
Estimate the stress on the upper fiber by using the formula Fy=M*y/Ix.
Y distance= h/2, where h is the overall height. Divide My/Fy will give us the expression of Sx, which is the elastic section modulus.
If the load acting on a beam is increased, more than the load creates the first yield in the upper fiber until the stress Fy propagates from the upper fiber to a lower one and the section becomes fully yielded and thus, we get our plastic neutral axis.
The stress profile has changed from a triangle to a trapezium shape. In the end, it will become a rectangular shape, as shown in the sketch.
If we consider the symmetry, the area above the Plastic neutral axis will be equal to half the area or A/2, where A is the total area of the whole section. A compression force will act on the upper section.
This force at plastic stage Cp=Fy* area of a rectangle (b*h/2),or Fy* area of a rectangle (b*h/2), there will be a tension force Tp equal to compression force Cp, for the lower portion below the P.N.A, Cp =Tp =Fy*(b*h/2).
Both forces will act on the Cgs, or the center of gravities, at h/4 above and below the neutral axis. The term yct is the distance between the tension and compressive forces, which will be=(h/4)+(h/4)=h/2.
Let us write the following information in an equation, where Cp=Tp=Fy*A/2, while yct=h/2. The plastic moment Mp=Fy*(A/2)*(h/2).
A new term, Zx, or the plastic section modulus, will appear. The value of Zx=(A/2)*(h/2) =(b*h)/2*(h/2)=b*h^2/4 or Zx=(A/2)*2 y bar. This is one y bar distance, measured from the Cg of the upper area to the P.N.A. This why 2*y bar=2*(h/4)=h/2.
In the case of a rectangle, the plastic section modulus Zx=b*h/4. What is the shape factor, the shape factor? The shape factor is the ratio= Zx/Sx or (plastic section modulus/ elastic section modulus).
Section modulus Sx for a rectangle shape-part 1.
This is a general way to estimate the section modulus for a rectangle at any axis, located at a distance Kd from the bottom of the rectangular section.
1-At first, it is required to get the Y bar for the section, by summing the first moment of areas, consider the datum line is at the bottom, the product of A1y1+A2*y2= At*y bar since the total area is known, so we can get- Ybar.
2- we estimate the inertia about the neutral axis, we have just evaluated at a distance y bar from the datum.
-3- we estimate y-max value, for a rectangle as y max=d/2, Sx is simply the division of Inertia Ix/ymax.
How to evaluate Zx, Plastic section modulus for a rectangle?
A-To estimate the Zx, the plastic section modulus assumes there is an axis that divides the whole section into two equal areas and assumes that it is apart by Kd distance from the datum line.
B-equate A1 and A2 and get the value of k, which in the case of a rectangle=1/2.
B-The product of (At/2 )*(y1+y2) will give us the plastic section modulus, where y1 is the distance from the P.N.A to CG of Area A1, while y2 is the distance from the P.N.A to CG of Area A2.
Section modulus Sx for any shape.
The same procedure which we have used in the case of the rectangular shape will be used.
1-We estimate the value of y- bar for any area by estimating to get the Y bar for the section, by summing the first moment of areas, consider the datum line is at the bottom, the product of A1y1+A2*y2= At*y bar, since the total area is known, so we can get Y-bar.
2- We estimate the inertia about the neutral axis, we have just evaluated at a distance y bar from the datum.
3- we estimate y max value, which will be the maximum value of (y1,y2).
4- Estimate Sx which is the Ix/ymax.
How to evaluate Zx, Plastic section modulus for any shape?
A-To estimate Zx, the plastic section modulus assumes an axis divides the whole section into two equal areas.
We call it P.N.A, or the plastic neutral axis, then estimate the value of y bar from the first moment of areas for y1 bar and Y2 bar.
B-The product of (At/2 )*(Y1+y2) will give us the plastic section modulus, The product of (At/2 )*(y1+y2) will give us the plastic section modulus, Where y1 is the distance from the P.N.A to CG of Area A1.
While y2 is the distance from the P.N.A to CG of Area A2.
This is the pdf data used for the illustration of this post.