Strong axis bending question

[steve321]

Homework Statement

there's a composite beam that's wood on top (150mm along the x axis, 250 along the y) and a little piece of steel along the bottom (150mm along the x, only 10 along the y). Ew is 10,000MPa, Es is 200,000MPa (these are the elastic moduluseseses). the beam is to be analyzed for strong axis bending using the principle of transformed sections.

1. if the transformed section is considered as being wood, what is the transformed width of the steel plate.

2. in the transformed section, what is the distance between the centroid and the bottom surface of the beam.

3. what is the moment of inertia of the transformed section.

4. if this beam is sub ject to a strong axis bending moment of 30kn.m, what is the max tensile stress in the wood

5. if this beam is subjected to a strong axis bending moment of 30kn.m, what is the max tensile stress in the steel.

Homework Equations

1. i have absolutely no idea. i also don't understand what exactly the 'neutral axis' is, or why it's important in figuring this out. if I'm understanding this chapter correctly, the strain on a composite material is the same, but the stress is different. does the 'netural axis' run through the centroid of a single material? does it run through the centroid of a composite material?

2. also drawing a blank here.

3. i think this variable is 'I'. so the little weird 'o'x thing = -My/I
I = -My / o

the o thing is stress, but i don't know why there's a little 'x' to it, and then the book starts talking about obtaining stress o1 and o2, which haven't been mentioned previously whatsoever so that's pretty useless.

i just looked up o1 in the back of the book, and it says it's equal to - E1y/p. so that's -200,000 (assume E1 is wood, i have no idea if i can arbitrarily assign either material to it)(y) [i have no idea what 'y' is], / p, which apparently is the radius of an arc, something that i highly doubt is in the square diagram I've been presented with. I'm going to skip to 4 because i have no idea what's going on.

4. now we're talking - there's actually an example of this in the book! hopefully i can follow along.

weird 'o' of wood = Mc2/I = (30 x 10^3 N.m)(apparently this wood farthest from the neutral axis) / I

the neutral axis is apparently Y = EyA / EA and there's a bunch of lines over some of the letters. I'm totally lost again.

5. i assume this is going to be a lot like 4.

i'm going to read up some more on this and see if i can't get a little further in the next 25 minutes, but if anyone can shed some light on these ridiculous questions or give me some firm ground to stand on by way of basic explanation of what I'm even trying to find out, i'd appreciate it.

thanks!

 

[steve321]

i googled 'transformed section' and pulled up this gem

http://www.bgstructuralengineering.com/BGSMA/BGSMA_Itr/BGSMA_ITR03.htm

opening statement: "A Moment of Inertia, I, is a section property (i.e. solely depended on cross sectional dimensions) taken about a specific axis."

so it's a property taken about a specific axis. now what exactly are some characteristics of this property? what axis specifically are we 'taking it about'? what does 'taken about' actually mean? do we take this property and rotate it around an axis, and if so, why? what does the 'moment of inertia' tell me?

wikipedia is a bit more succinct and says it's the measure of an object's resistance to rotation. I'm guessing a higher moment of inertia means it's more resistant to rotating. how is a piece of wood that's being bent also being rotated?

 

[steve321]

do i have this correct:

if you bend something, like a beam, there is a 'neutral plane' within in where the stress & the strain is 0. I'm not sure why this exists because it would seem that if the entire beam is being bent, the entire beam is under stress and strain, but i think that's what the neutral plane is and will take someone's word for it.

if you do a section of the neutral plane at any given point, you will be given the 'neutral axis', which is a straight line across the section. the section is also technically called the 'transformed section'.

is that accurate?

 

[steve321]

okay, so if you have two materials smushed together, and you want to analyze the properties of it's section, you have to do a few things to pretend that it's just one material. so in my example above, if you have a piece of wood and a piece of steel, and you want to pretend the wood is steel, what you do is multiple the horizontal distances by 'n', which is Ew/Es, right?

so to answer my first question, if i want to pretend the whole beam is made of wood, i need to multiply the horizontal distance of the steel (150mm) by 200,000/10,000. so is the answer to # 1 150x20 = 3000mm?

or do i have that one backwards?

 

[rwisz]

It seems like you are struggling with understanding the concept of transformed sections and the neutral axis in composite materials. Let me break it down for you.

1. The transformed width of the steel plate is the equivalent width of the steel plate if it were to be made of the same material as the wood. This is important because it allows us to analyze the composite beam as if it were made of one material, simplifying the calculations. The neutral axis is the imaginary line that divides the composite beam into two sections, and the transformed width is measured from this line to the outer edge of the steel plate. It is calculated by taking the ratio of the elastic moduli of the two materials (Ew and Es) and multiplying it by the actual width of the steel plate. So, the transformed width of the steel plate would be 150mm x (200,000MPa/10,000MPa) = 300mm.

2. The distance between the centroid and the bottom surface of the beam is called the depth of the neutral axis. It is calculated by taking the ratio of the elastic moduli of the two materials and multiplying it by the actual depth of the beam. So, in this case, the depth of the neutral axis would be 250mm x (200,000MPa/10,000MPa) = 5,000mm.

3. The moment of inertia of the transformed section is a measure of its resistance to bending. It is calculated by taking the moment of inertia of the actual section and multiplying it by the ratio of the elastic moduli of the two materials. So, the moment of inertia of the transformed section would be (150mm x 250mm^3) x (200,000MPa/10,000MPa) = 7.5 x 10^6 mm^4.

4. To find the maximum tensile stress in the wood, we can use the equation you mentioned (o = My/I). In this case, we are looking for the stress in the wood farthest from the neutral axis, which is at the bottom of the beam. So, the stress would be (30 x 10^3 N.m)(250mm/7.5 x 10^6 mm^4) = 1 MPa.

5. Similarly, to find the maximum tensile stress in the steel, we can use the same equation. In this case, the stress would be (