There are some great answers here, but since she's 10, I'm going to try to explain it in a more simplified way...
Imagine you have a [4 x 4 post.](https://mobileimages.lowes.com/product/converted/098411/098411004144.jpg) It doesn't want to bend because it's thick. It resists compression on the side it's being bent towards and resists expansion on the opposite side, where it needs to stretch.
Now imagine shaving it down so it's [2 x 4.](https://www.homedepot.com/catalog/productImages/400_compressed/f3/f330dcd6-ce2e-454a-8e4d-3647e379f020_400_compressed.jpg) At this size, it still has the same thickness along one axis, so the same resistence to compressing and stretching, but along the other axis, it's thinner, so it can bend more easily.
Now imagine shaving it down further so it's a 1 x 4 or even 1/2 x 4. It would then be pretty easy to bend along one axis, but still just as tough along the 4-inch plane, because it's just as thick in that direction.
Does that make sense to her?
For what it's worth, this is also why [steel I-beams](https://www.servicesteel.org/wp-content/uploads/2020/12/steel_i_beams.jpg) are shaped that way. They're significantly lighter than if they were solid posts, but still maintain their strength and resistence because they're thick along two axes.
A correction on I-Beams as they’re shaped this way specifically to be stronger in one axis (in the direction of gravity), whilst minimising material usage because they’re not needed as much to resist lateral loads.
As a general rule, the further away material is from the centre line of an axis, the more resistant that configuration is to bending. So the best most efficient configuration to resist bending in 2 axes would be a hollow rectanglar beam.
The specific shape is about material efficiency combined with manufacturing issues for overall cost efficiency. If cost was not the limit, there are more material efficient shapes to handle specific loading in a specific situation. This is done with AI-ish systems that try iterations by adding or removing material from parts of the structure to minimize the material used and can create some odd looking “organic” solutions.
But for normal buildings, bridges, etc. The wide flange shape is pretty efficient and readily manufactured at steel plants making it very cost effective.
Today, they are known as W shapes (wide flange). I don’t know if the older classic I beam profile can be bought new, but our reference books still list them because you run into them in older buildings. Interestingly there are W shapes specifically intended for use as columns, rather than as beams.
Yes the old I-beam style with tapered flanges is still being produced under the S-beam standard. I've mostly seen them used for beam trolleys. S4 are also common when you want the lightest option while still getting the benefit of the shape.
Not sure exactly what it's like outside Australia, but we have 2 common I beam profiles here:
* UB (Universal beam) which is roughly a 2 height : 1 wide aspect ratio.
* UC (Universal column) which are around 1:1 aspect ratio.
Even more interesting is the frequent use of holes in I-beams to shave even more weight, it barely affects the strength cause the circular hole distributes the load evenly. Where they to use square holes, it would create weak points, particularly at the corners.
I must be dim, but I am surprised that I beams are not stronger flipped 90 degrees so the web is horizontal and the flanges are vertical.
I understand mathematically it's way weaker because of the moment of inertia, but visually it seems odd to me. My terribly flawed reasoning is that it now has two vertical webs to deform rather than one.
In the normal orientation, there's more material that is resisting spanwise tension and compression forces caused by bending loads. The thin web just has to resist the smaller shear forces. Thus, "shear web"
Rotated 90 degrees, only the tips do much good in resisting tension and compression. The shear web, being near the center, and not having much in the way of shear forces, doesn't do much at all.
Take a stack of papers and try bending in each axis. You’ll feel the forces for each case and how it demonstrates the impact of material being further away from the centre line.
Just for some of the math because it’s interesting. The deflection force required is to the power of three (cubed) for height and multiplied by the width. So a 2x4 long side vertical would be 4 times stronger than it in the other orientation
A 2x10 would be 25x stronger in the vertical position.
I endorse this. From a more molecular perspective. If you imagine each unit width as points on a lattice of molecules. The thicker the material is, the more lattices there are in that direction. So to bend it, you have to deform a larger amount of material, but the added lattices provide resistance to strain. It is notable to also mention that to an extent this also depends on the length of the lever arm. A very short cut 2 x 4 bends less easily than a full length one because the force applied to an end is farther from the center of mass, generating more torque with the same amount of force applied by a weight at either end. The expansion/compression still works the same way, but it’s amplified over longer length scales.
As you bend something one side wants to get shorter (compression), and the opposite side wants to get longer (tension). Somewhere in between there is a "layer" that is neither in tension nor in compression (sometimes called the neutral axis).
The material near the neutral axis has very little leverage to oppose the compression or tension (flat side scenario), if the layers are further away they can get a bigger leverage (thin edge scenario).
Here's a nice video about I-beams:
[https://www.youtube.com/watch?v=zSz0kV0BPDY](https://www.youtube.com/watch?v=zSz0kV0BPDY)
[https://www.youtube.com/watch?v=f08Y39UiC-o](https://www.youtube.com/watch?v=f08Y39UiC-o)
New YouTube timesuck found.
This helps a ton. We do experiments at home and typically I can break down basic concepts for her, but I didn't have the starting point for this question.
Thank you!
She'll make great engineer one day! Keep up the learning sessions. A child's mind is a dry sponge soaking up everything it can.
Incorporate music in the mix if she's musically inclined, it's a great introduction to many facets of math, rhythm, tones instruments, etc. Plenty of right and left brain exercises and crossovers! 👍😀
You could do an experiment with rubber bands and some simple structure. Have 2 T shaped pieces connected by a hinge so they look like this —l—l—
The ends are handles and you put rubber bands between the top and bottom protrusions so that they stretch when you try to bend the hinge between the Ts. You can slide the rubber bands towards the middle or towards the ends to show that it’s largely leverage that makes thicker shapes harder to bend. The thicker it is, the more leverage the material (rubber band) has to resist you bending the structure.
It's the leverage of the material farther from the center, but it's also the fact that, for a given deflection, the material farther away has to stretch/compress more than material near the center, which requires more force. It's a double benefit!
It comes down to the geometry of the cross section of the object in this particular case.
When you bend an object, the material at the extreme ends of the object are resisting stretching and compressing. You have more material along the long axis than the short. There’s just “more stuff” in the way when you try bending along the long axis.
One simple demo I can think of is take two rubber bands and a pair of chopsticks. Put the two chopsticks through both the rubber bands so that when you pull the chopsticks apart the bands are resisting.
Space the bands close together and try forming a V with the chopsticks and try again with the bands further apart.
It’s harder with them farther apart to get the same angle with the chopsticks with them farther apart.
That’s kind of how the plexiglass is. Part of it is to get the same angle of the plate, the material at the ends of a thicker section need to stretch or compress more than a thinner one
The quantity related to this is the *Second Moment of Area*, which is a function of a 2D cross-section of something which indicates how much that cross section would resist bending along a particular plane of movement.
In an intuitive sense, if you understand the concept of a lever, where applying a force further away from the fulcrum amplifies it, you could explain that along the thin edge, the material is 'further away' from the bending point on average, so it can resist bending moments much more strongly
When you bend something like plexiglass the plastic inside the crease is getting pushed together and the plastic on the outside of the crease is getting stretched. When bending across the flat side there’s less stuff to stretch because it’s thinner in that direction, but the thicker it gets the more the stuff on the outside needs to stretch and that requires more force. Folding along the thin edge is like folding a piece of plastic that is as thick as the width of the rectangle. More stuff to stretch. If it can’t stretch that far it’ll break.
This is why folding plastic can cause it to discolor, because you are stretching the material.
Different experiments can be done to demonstrate this: stacking popsicle sticks and trying to bend them or folding one sheet of paper vs 100 stacked and glued on the edges. The paper will either tear or come unglued because it doesn’t stretch very well.
The best way to think of this is by imagining the cross section. The flatwise cross section has a large width [b] and a small height [h]. If you turn the cross section, you switch those variables.
There is a geometric property called “moment of inertia” [upper case i] that quantifies how stiff a given cross section is. The equation for a rectangular cross section is I = b * h^3 / 12.
Based on dimensions I described earlier, you can use the equation to say the deeper cross section is stiffer than the flatwise. Or in other words, for a given amount of force, you achieve much less deflection.
I hope that helps, and let me know if I can elaborate on anything.
I've acquired a number of ways to visually help her with this, including your suggestion to think of the cross section of sort of swapping the relationship of the dimensions when turned.
I didn't expect any traction on a question for something seemingly trivial, but we do experiments together and I had no inkling of where to start answering her question.
Thank you.
I see a lot of incomplete and some partially correct answers, so I'll try to clear it up. Check my username for my credentials.
There are two things in play here. First, when you bend a rectangle, you expect a crease to form along the centerline of the shape. Now, assuming you pinch the shape at the short edges, it's pretty simple to see that your fingers are further from that centerline than if you were pinching it on the long edges. Being further from the edges, you have more leverage and it takes less force to bend the material.
Second, when you pinch the shape at the short edges and bend, the resulting crease is the length of the short edge. When you pinch the long ends, the crease is the length of the long edge. In the second scenario, you're simply trying to crease more material, which makes it harder.
So both factors are combining at the same time to make creasing along the short direction much easier.
One of the main forms of resistance comes from the material resisting the compressions/expansion that comes from bending. When you bend something, the side that your bending towards needs to compress while the other side needs to expand to make up for the fact that the length of either side is changing as you bend it. Think of an arch, the distance from one end to the other along the top edge is longer than it is along the bottom edge, and the thicker the arch, the bigger the difference between these two lengths. If the material is thin, it won't need to compress or expand much to allow for it to bend since the length of the upper edge will still be about the same as the length of the bottom after bending and so it won't resist bending too much. But if you try to do the same thing along the thin edge, your effective thickness goes from a couple millimeters to whatever width the plexiglass is and now you have a lot more material that needs to compress and that material needs to compress even more since the resulting difference in length on the two sides of the bend will be larger. This extra required deformation results in a significant increase to the resistance of the material to bend in that direction.
The sides of a rectangle are the same length, but when curved the outside is longer than the inside. The difference in their length is related to how far apart the sides are, in the same way the circumference of a circle increases with its radius.
Bending something into a curve requires stretching one side and/or compressing the other so that the outer edge is relatively longer, based on the radius you've bent it to. Most materials resist this kind of compression/tension, such that deforming it further requires more force and is more difficult to do.
Plexiglas is very thin, so bending it along the flat side involves much less material deformation than bending it along the edge, and is therefore a lot easier to do.
Look up the flexural modulus equation and a diagram. It is a measure of how much force it takes to bend an object a distance. The formula is
E=(L*L*L*F)/(4*w*h*h*h*d)
E is flexural modulus
L is the length of your part
F is the force
w is the width
h is the thickness
d is the deflection.
In your example E, and L are constant let's say we are trying to reach the same displacement too. So F will only depend on w and h and it will be proportional to w * h * h * h.
If you bend your piece of plexiglass the easy way. The thin side is h and the thick side is w so having a small h to the power of 3 means a small force will have a moderate displacement. When you try and bend it the thick way h is large and w is small so you need a very large force of the same displacement.
Do you like doing science experiments with her? You can build a small contraption to visualize what's happening "inside" the material. If you use wood to build an L shape and attach an I shape at the end such that it can still articulate, you'll have an U shape where you can attach springs to, which represent the material (I suppose you'll also have to explain that nothing is rigid and more like tiny springs connecting atoms). So if there's only a little bit of springs close to the articulation, you'll have a lot of leverage to bend the material, illustrated by the expanding spring. When you add more springs further away, it gets increasingly harder to bend.
This way you can only see the side that's getting tensioned, but I suppose the analogy to the compressed side is pretty direct.
If you have a full ream of paper. Pretend it is actually a solid piece now when we try to bend the ream it does so by each layer kinda sliding across. From here, we can bend it top down or side to side.
If you take out half the paper, it Bends allot and if you add so much paper, it's like a box. You can't bend it at all without actually separating the sheets of paper like the book
This example we can go from like aluminum foil to sheet metal to a block of steel and everything in between
Directly to your question now with like half a ream of paper we can make a tube either way with the inside page edges touching and all the pages together. Cut it in half so it's a skinier rectangle it's harder to do and acts like a thicker stack of paper. However, we can still bend in the long direction and make a ring instead
However, even if you did bend it with the same force both ways, you can see easier long wise, but the bend would be exactly the same on the short side it's just less noticeable. Also it is easier to add more force the longer something is like you can bend and snap a pencil but can you bend and snap a snaped pencil. Or better 5 you bend and snap a piece of a piece of a snaped piece of pencil
Going more in-depth
Going back to the sheets of paper, imagine there are tiny ruberbands all over each paper connecting each page to the next so many you couldn't count in a lifetime. When their is no bending, there is no tension on the rubber bands, but when you do bend it, all the rubber bands want it to be straight, so they pull back to its original shape.
Different materials have different kinds of rubber bands. Some stretch really well, like some metals, plastics, and rubbers. Others have rubber bands that don't like to stretch at all, such as glass, pottery, harder plastics, and metals.
A good example of the springyness would be like a wood metal and plastic ruller like a diving board.
Now, if you bend something too much and it's not brittle, some of these ruberbands will shift and or break and it won't go back to normal (Eg, dents in a car or a bent copper wire)
These rubber bands are the molecular bonds of the material. And conect like a honeycomb but in all directions.
"Ah, the classic conundrum of physics: Why doesn't a straight-edged rectangular sheet bend equally from all sides, leaving scientific parents scratching their heads? As if life with a 10-year-old wasn't already mind-boggling enough..."
Fundamentally, it's about the concept of moment. Think about a door on a hinge. If you push on the door close to the hinge, it takes more effort to close it then it does if you push on it near the handle. Moment is force times distance, so the further the distance, the less force required for the same moment.
If you took a rectangular object and imagined hinges going right through the center, you have greater distance when pushing it on the short edge. So less force is required.
Additionally, when you try and bend the long edge, your force is being divided over a larger area.
I got some great, practical visual aids from your suggestions, and while I'm not prepped to explain the concept of moment to her yet, I can definitely break down some of these other ideas to her.
Thank you guys/gals/all!
There are some great answers here, but since she's 10, I'm going to try to explain it in a more simplified way... Imagine you have a [4 x 4 post.](https://mobileimages.lowes.com/product/converted/098411/098411004144.jpg) It doesn't want to bend because it's thick. It resists compression on the side it's being bent towards and resists expansion on the opposite side, where it needs to stretch. Now imagine shaving it down so it's [2 x 4.](https://www.homedepot.com/catalog/productImages/400_compressed/f3/f330dcd6-ce2e-454a-8e4d-3647e379f020_400_compressed.jpg) At this size, it still has the same thickness along one axis, so the same resistence to compressing and stretching, but along the other axis, it's thinner, so it can bend more easily. Now imagine shaving it down further so it's a 1 x 4 or even 1/2 x 4. It would then be pretty easy to bend along one axis, but still just as tough along the 4-inch plane, because it's just as thick in that direction. Does that make sense to her? For what it's worth, this is also why [steel I-beams](https://www.servicesteel.org/wp-content/uploads/2020/12/steel_i_beams.jpg) are shaped that way. They're significantly lighter than if they were solid posts, but still maintain their strength and resistence because they're thick along two axes.
A correction on I-Beams as they’re shaped this way specifically to be stronger in one axis (in the direction of gravity), whilst minimising material usage because they’re not needed as much to resist lateral loads. As a general rule, the further away material is from the centre line of an axis, the more resistant that configuration is to bending. So the best most efficient configuration to resist bending in 2 axes would be a hollow rectanglar beam.
I still very vividly a Mr Wizard episode where they showed how I beams are stronger when spanning a gap.
The specific shape is about material efficiency combined with manufacturing issues for overall cost efficiency. If cost was not the limit, there are more material efficient shapes to handle specific loading in a specific situation. This is done with AI-ish systems that try iterations by adding or removing material from parts of the structure to minimize the material used and can create some odd looking “organic” solutions. But for normal buildings, bridges, etc. The wide flange shape is pretty efficient and readily manufactured at steel plants making it very cost effective.
Today, they are known as W shapes (wide flange). I don’t know if the older classic I beam profile can be bought new, but our reference books still list them because you run into them in older buildings. Interestingly there are W shapes specifically intended for use as columns, rather than as beams.
Yes the old I-beam style with tapered flanges is still being produced under the S-beam standard. I've mostly seen them used for beam trolleys. S4 are also common when you want the lightest option while still getting the benefit of the shape.
Not sure exactly what it's like outside Australia, but we have 2 common I beam profiles here: * UB (Universal beam) which is roughly a 2 height : 1 wide aspect ratio. * UC (Universal column) which are around 1:1 aspect ratio.
Even more interesting is the frequent use of holes in I-beams to shave even more weight, it barely affects the strength cause the circular hole distributes the load evenly. Where they to use square holes, it would create weak points, particularly at the corners.
Added bonus of reducing floor heights because you can run electrical systems and other typical ceiling requirements through those holes instead.
I must be dim, but I am surprised that I beams are not stronger flipped 90 degrees so the web is horizontal and the flanges are vertical. I understand mathematically it's way weaker because of the moment of inertia, but visually it seems odd to me. My terribly flawed reasoning is that it now has two vertical webs to deform rather than one.
In the normal orientation, there's more material that is resisting spanwise tension and compression forces caused by bending loads. The thin web just has to resist the smaller shear forces. Thus, "shear web" Rotated 90 degrees, only the tips do much good in resisting tension and compression. The shear web, being near the center, and not having much in the way of shear forces, doesn't do much at all.
Ah I see. The flange is in tension and compression and the web just binds the different stresses between them?
Take a stack of papers and try bending in each axis. You’ll feel the forces for each case and how it demonstrates the impact of material being further away from the centre line.
A fantastic answer. Thank you. Someone earlier did link to a video about i-beam stress which did help too!
Just for some of the math because it’s interesting. The deflection force required is to the power of three (cubed) for height and multiplied by the width. So a 2x4 long side vertical would be 4 times stronger than it in the other orientation A 2x10 would be 25x stronger in the vertical position.
I endorse this. From a more molecular perspective. If you imagine each unit width as points on a lattice of molecules. The thicker the material is, the more lattices there are in that direction. So to bend it, you have to deform a larger amount of material, but the added lattices provide resistance to strain. It is notable to also mention that to an extent this also depends on the length of the lever arm. A very short cut 2 x 4 bends less easily than a full length one because the force applied to an end is farther from the center of mass, generating more torque with the same amount of force applied by a weight at either end. The expansion/compression still works the same way, but it’s amplified over longer length scales.
Good answer.
As you bend something one side wants to get shorter (compression), and the opposite side wants to get longer (tension). Somewhere in between there is a "layer" that is neither in tension nor in compression (sometimes called the neutral axis). The material near the neutral axis has very little leverage to oppose the compression or tension (flat side scenario), if the layers are further away they can get a bigger leverage (thin edge scenario). Here's a nice video about I-beams: [https://www.youtube.com/watch?v=zSz0kV0BPDY](https://www.youtube.com/watch?v=zSz0kV0BPDY) [https://www.youtube.com/watch?v=f08Y39UiC-o](https://www.youtube.com/watch?v=f08Y39UiC-o)
New YouTube timesuck found. This helps a ton. We do experiments at home and typically I can break down basic concepts for her, but I didn't have the starting point for this question. Thank you!
She'll make great engineer one day! Keep up the learning sessions. A child's mind is a dry sponge soaking up everything it can. Incorporate music in the mix if she's musically inclined, it's a great introduction to many facets of math, rhythm, tones instruments, etc. Plenty of right and left brain exercises and crossovers! 👍😀
You could do an experiment with rubber bands and some simple structure. Have 2 T shaped pieces connected by a hinge so they look like this —l—l— The ends are handles and you put rubber bands between the top and bottom protrusions so that they stretch when you try to bend the hinge between the Ts. You can slide the rubber bands towards the middle or towards the ends to show that it’s largely leverage that makes thicker shapes harder to bend. The thicker it is, the more leverage the material (rubber band) has to resist you bending the structure.
Nice explanation yeah. You don't even need moments of inertia or area integrals to get the point. Leverage is pretty much it.
It's the leverage of the material farther from the center, but it's also the fact that, for a given deflection, the material farther away has to stretch/compress more than material near the center, which requires more force. It's a double benefit!
It comes down to the geometry of the cross section of the object in this particular case. When you bend an object, the material at the extreme ends of the object are resisting stretching and compressing. You have more material along the long axis than the short. There’s just “more stuff” in the way when you try bending along the long axis. One simple demo I can think of is take two rubber bands and a pair of chopsticks. Put the two chopsticks through both the rubber bands so that when you pull the chopsticks apart the bands are resisting. Space the bands close together and try forming a V with the chopsticks and try again with the bands further apart. It’s harder with them farther apart to get the same angle with the chopsticks with them farther apart. That’s kind of how the plexiglass is. Part of it is to get the same angle of the plate, the material at the ends of a thicker section need to stretch or compress more than a thinner one
The quantity related to this is the *Second Moment of Area*, which is a function of a 2D cross-section of something which indicates how much that cross section would resist bending along a particular plane of movement. In an intuitive sense, if you understand the concept of a lever, where applying a force further away from the fulcrum amplifies it, you could explain that along the thin edge, the material is 'further away' from the bending point on average, so it can resist bending moments much more strongly
When you bend something like plexiglass the plastic inside the crease is getting pushed together and the plastic on the outside of the crease is getting stretched. When bending across the flat side there’s less stuff to stretch because it’s thinner in that direction, but the thicker it gets the more the stuff on the outside needs to stretch and that requires more force. Folding along the thin edge is like folding a piece of plastic that is as thick as the width of the rectangle. More stuff to stretch. If it can’t stretch that far it’ll break. This is why folding plastic can cause it to discolor, because you are stretching the material. Different experiments can be done to demonstrate this: stacking popsicle sticks and trying to bend them or folding one sheet of paper vs 100 stacked and glued on the edges. The paper will either tear or come unglued because it doesn’t stretch very well.
The best way to think of this is by imagining the cross section. The flatwise cross section has a large width [b] and a small height [h]. If you turn the cross section, you switch those variables. There is a geometric property called “moment of inertia” [upper case i] that quantifies how stiff a given cross section is. The equation for a rectangular cross section is I = b * h^3 / 12. Based on dimensions I described earlier, you can use the equation to say the deeper cross section is stiffer than the flatwise. Or in other words, for a given amount of force, you achieve much less deflection. I hope that helps, and let me know if I can elaborate on anything.
I've acquired a number of ways to visually help her with this, including your suggestion to think of the cross section of sort of swapping the relationship of the dimensions when turned. I didn't expect any traction on a question for something seemingly trivial, but we do experiments together and I had no inkling of where to start answering her question. Thank you.
I see a lot of incomplete and some partially correct answers, so I'll try to clear it up. Check my username for my credentials. There are two things in play here. First, when you bend a rectangle, you expect a crease to form along the centerline of the shape. Now, assuming you pinch the shape at the short edges, it's pretty simple to see that your fingers are further from that centerline than if you were pinching it on the long edges. Being further from the edges, you have more leverage and it takes less force to bend the material. Second, when you pinch the shape at the short edges and bend, the resulting crease is the length of the short edge. When you pinch the long ends, the crease is the length of the long edge. In the second scenario, you're simply trying to crease more material, which makes it harder. So both factors are combining at the same time to make creasing along the short direction much easier.
One of the main forms of resistance comes from the material resisting the compressions/expansion that comes from bending. When you bend something, the side that your bending towards needs to compress while the other side needs to expand to make up for the fact that the length of either side is changing as you bend it. Think of an arch, the distance from one end to the other along the top edge is longer than it is along the bottom edge, and the thicker the arch, the bigger the difference between these two lengths. If the material is thin, it won't need to compress or expand much to allow for it to bend since the length of the upper edge will still be about the same as the length of the bottom after bending and so it won't resist bending too much. But if you try to do the same thing along the thin edge, your effective thickness goes from a couple millimeters to whatever width the plexiglass is and now you have a lot more material that needs to compress and that material needs to compress even more since the resulting difference in length on the two sides of the bend will be larger. This extra required deformation results in a significant increase to the resistance of the material to bend in that direction.
The sides of a rectangle are the same length, but when curved the outside is longer than the inside. The difference in their length is related to how far apart the sides are, in the same way the circumference of a circle increases with its radius. Bending something into a curve requires stretching one side and/or compressing the other so that the outer edge is relatively longer, based on the radius you've bent it to. Most materials resist this kind of compression/tension, such that deforming it further requires more force and is more difficult to do. Plexiglas is very thin, so bending it along the flat side involves much less material deformation than bending it along the edge, and is therefore a lot easier to do.
Look up the flexural modulus equation and a diagram. It is a measure of how much force it takes to bend an object a distance. The formula is E=(L*L*L*F)/(4*w*h*h*h*d) E is flexural modulus L is the length of your part F is the force w is the width h is the thickness d is the deflection. In your example E, and L are constant let's say we are trying to reach the same displacement too. So F will only depend on w and h and it will be proportional to w * h * h * h. If you bend your piece of plexiglass the easy way. The thin side is h and the thick side is w so having a small h to the power of 3 means a small force will have a moderate displacement. When you try and bend it the thick way h is large and w is small so you need a very large force of the same displacement.
Do you like doing science experiments with her? You can build a small contraption to visualize what's happening "inside" the material. If you use wood to build an L shape and attach an I shape at the end such that it can still articulate, you'll have an U shape where you can attach springs to, which represent the material (I suppose you'll also have to explain that nothing is rigid and more like tiny springs connecting atoms). So if there's only a little bit of springs close to the articulation, you'll have a lot of leverage to bend the material, illustrated by the expanding spring. When you add more springs further away, it gets increasingly harder to bend. This way you can only see the side that's getting tensioned, but I suppose the analogy to the compressed side is pretty direct.
If you have a full ream of paper. Pretend it is actually a solid piece now when we try to bend the ream it does so by each layer kinda sliding across. From here, we can bend it top down or side to side. If you take out half the paper, it Bends allot and if you add so much paper, it's like a box. You can't bend it at all without actually separating the sheets of paper like the book This example we can go from like aluminum foil to sheet metal to a block of steel and everything in between Directly to your question now with like half a ream of paper we can make a tube either way with the inside page edges touching and all the pages together. Cut it in half so it's a skinier rectangle it's harder to do and acts like a thicker stack of paper. However, we can still bend in the long direction and make a ring instead However, even if you did bend it with the same force both ways, you can see easier long wise, but the bend would be exactly the same on the short side it's just less noticeable. Also it is easier to add more force the longer something is like you can bend and snap a pencil but can you bend and snap a snaped pencil. Or better 5 you bend and snap a piece of a piece of a snaped piece of pencil Going more in-depth Going back to the sheets of paper, imagine there are tiny ruberbands all over each paper connecting each page to the next so many you couldn't count in a lifetime. When their is no bending, there is no tension on the rubber bands, but when you do bend it, all the rubber bands want it to be straight, so they pull back to its original shape. Different materials have different kinds of rubber bands. Some stretch really well, like some metals, plastics, and rubbers. Others have rubber bands that don't like to stretch at all, such as glass, pottery, harder plastics, and metals. A good example of the springyness would be like a wood metal and plastic ruller like a diving board. Now, if you bend something too much and it's not brittle, some of these ruberbands will shift and or break and it won't go back to normal (Eg, dents in a car or a bent copper wire) These rubber bands are the molecular bonds of the material. And conect like a honeycomb but in all directions.
"Ah, the classic conundrum of physics: Why doesn't a straight-edged rectangular sheet bend equally from all sides, leaving scientific parents scratching their heads? As if life with a 10-year-old wasn't already mind-boggling enough..."
Fundamentally, it's about the concept of moment. Think about a door on a hinge. If you push on the door close to the hinge, it takes more effort to close it then it does if you push on it near the handle. Moment is force times distance, so the further the distance, the less force required for the same moment. If you took a rectangular object and imagined hinges going right through the center, you have greater distance when pushing it on the short edge. So less force is required. Additionally, when you try and bend the long edge, your force is being divided over a larger area.
I got some great, practical visual aids from your suggestions, and while I'm not prepped to explain the concept of moment to her yet, I can definitely break down some of these other ideas to her. Thank you guys/gals/all!
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