Why is the roadster lighter than the coupe?

[Bammered]

Originally Posted by The HACK

Just because the part numbers are different doesn't mean they're actually different. The strut-hat from the MZ4 is identical to the E36 M3 strut hat but it received a different part number. And the MZ4R and MZ4C have the same identical suspension parts EXCEPT for the front swaybar. But not 15kg worth different.

The only thing different between the MZ4R and MZ4C, hardware wise under the skin, are the steering rack and the front swaybars. On the surface the rear quarter panel, the roof, and the truck/hatch are different.

The reason the coupes are heavier than the roadsters, is because the coupes are built from the roadsters using roadster designs but adding the roof and hatch. The structural rigidity difference between the coupe and roadster, is 32000Nm vs. 12500Nm. That's almost 2.5X more stiff. Compare to equivalent roadsters built from coupes, like the E46, coupe stiffness is 12000Nm while the convertible stiffness is 10500Nm. Simply put, the Z4 roadster was designed to be as stiff as an E46 Coupe, hence you add all the roof and hatch to the car to make it nearly 2.5X as stiff it's going to be heavier. The counterpart, E46 convertibles were made from an existing coupe body that's only so stiff, to cut the roof off of that car would likely result in a convertible that's as stiff as spaghetti boiled for 24 hours, so they had to add a ton of braces and reinforcement to stiffen up the chassis to a respectable number, resulting in a convertible being much heavier than its coupe counterpart.

+1 i concure

[Dammmittt]

Originally Posted by Bammered

+1 i concure

I'll go back to my research now

[The HACK]

Originally Posted by Dammmittt

I think the best way to look at the car is a finite element model of beams welded together...but since we don't have access to that, we can look at it like a series of beams. If you look at the skeletons in my previous post alone, you see that there are plenty of horizontal supports already there..and the notion that the thin roof panel and supports are enough to increase the torsional stiffness by double just doesn't make any sense to me...especially when you see it decreasing only a 15-20% percent when hard top models are converted to convertibles.

This is going to be my last post on the subject. When you take a hard top model and convert them to convertibles, their structural rigidity doesn't just drop 15-20%. They drop about 50%. The reason why BMW convertible versions of their hard-top counter-parts are only about 15% less stiff than their coupe counterpart, is because BMW put a ton of structural reinforcement in the A pillar, window frame, various cross braces behind the rear seat, cowl, subframe reinforcement, cross brace on the front firewall, and rear subframe reinforcement. That's why the typical BMW 3 series convertible weights about 200lbs more than their coupe counterpart, while the Z4 roadster is actually 50lbs or so LIGHTER.

Case in point, if you just compare what is different between the E46 coupe and convertible, convertible rear seats are locked down and not fold-able (more structural rigidity), convertible rear subframe has an additional cross brace, convertible front subframe uses a heavier and stronger steel cross brace compared to the aluminum subframe brace used in the coupes, convertible A pillar design is completely different with significantly more mass added, and the E46 convertible is where BMW started to weld a cross brace from the top of the front strut towers to the chassis bulkhead.

But it's pointless to continue this discussion now. I'll continue to believe what I think I know, and you'll continue to believe what you think you know. Nothing I can say at this point will convince you so I'll just go on my gouchy way.

[RichardTS]

Originally Posted by Dammmittt

The roof supports can't be what makes the car twice as stiff...doesn't add up.

They are. Adds up fine if you analyze the forces.

[Dammmittt]

Prove it.

[RichardTS]

Originally Posted by Dammmittt

Prove it.

The car is in bending due to the vertical forces upwards from the wheels. This bending moment can be imagined as being in the middle of the body between the two wheels. In addition to the forces within the body that resist the bending moment, the coupe has additional forces down the roof pillars that resist this bending. The roadster does not have the benefit of these forces because there is no roof to support those forces.

[Dammmittt]

That's not exactly proof and your mechanics aren't exactly great. The only thing that could make the bending stiffness double the original would be to add supports that double the young's modulus*bending moment of inertia. The young's modulus is likely the same, so that leaves the bending moment of inertia. Now if you can prove to me that the shape of the roof pillars doubles the bending moment of inertia, i'll go along with it, but I don't see it happening.

If I get bored later I may build a simple FEA model of the car's skeleton and see what happens.

[RichardTS]

And the FEA model is going to show you exactly what I said. It is all about the forces down the pillars - they act as columns. The structure of the coupe is inherently more stable than that of the roadster. When you build the FEA model make sure you include these forces - if you don't you will probably just justify your own position.

[RichardTS]

Originally Posted by vachss

Think about a car like torsional springs in series not parallel. The center section without the roof is much weaker in torsion than the roofed version (here the 3 sided vs. 4 sided box analogy is good). The torsional stiffness of the whole car is limited by the reduced torsional stiffness of the center section in the open roofed version.

Spot on!

[murellus]

Why is it so hard to believe?

Take 4 sticks, connect them at the ends to form a rectangle, and flex it...flexes easily.

Now more sticks and give the 4 sticks a roof like shape (so make a smaller rectangle and connect the corners of each rectangle to each corresponding corner). Now try to flex it. Takes more effort.

Probably oversimplistic but I'm a physicist

[RichardTS]

Exactly. And I was thinking of a thought experiment with a sheet of plywood, casters at the corner and a table with the legs screwed to the plywood...

[Dammmittt]

Originally Posted by RichardTS

And the FEA model is going to show you exactly what I said. It is all about the forces down the pillars - they act as columns. The structure of the coupe is inherently more stable than that of the roadster. When you build the FEA model make sure you include these forces - if you don't you will probably just justify your own position.

OMG dude. Stop saying forces. The only forces acting on the car are external. The roof supports are load bearing members...

[Dammmittt]

Originally Posted by murellus

Why is it so hard to believe?

Take 4 sticks, connect them at the ends to form a rectangle, and flex it...flexes easily.

Now more sticks and give the 4 sticks a roof like shape (so make a smaller rectangle and connect the corners of each rectangle to each corresponding corner). Now try to flex it. Takes more effort.

Probably oversimplistic but I'm a physicist

Very oversimplistic.

[RichardTS]

Originally Posted by Dammmittt

OMG dude. Stop saying forces. The only forces acting on the car are external. The roof supports are load bearing members...

So we can make the roof supports out of paper?

[Dammmittt]

Originally Posted by RichardTS

So we can make the roof supports out of paper?

You don't understand what I'm saying. The car is acted on by external forces. The stress distribution in the car is determined by the shape and materials of the car's structure. Sure there are internal forces, but those internal forces cancel each other out in a statics problem and it's not really key to solving the problem.

In a finite element model, the most important things are to get the external loading and boundary conditions correct. Then you mesh it using a element type. The more sophisticated (higher order) element types do a good job, but when you use a fine mesh it can take forever for the computer to solve the problem. Usually what you do is use at least a quadratic (curved side) element (for 3D solids you'd use a quadtratic tetrahedral element) and refine the mesh until the solution converges.

Now if I simplified the problem and made it a beam made of a certain kind of material, the shape gives it a certain moment of inertia. Adding the roof supports would change the shape, and keeping the material the same would change the moment of inertia. (Stiffness is proportional to the Young's Modulus of the material times the Bending Moment of Inertia)

What I'm trying to convey to you is that I seriously doubt the roof supports DOUBLE the moment of inertia of the car's skeleton as a beam. I'm sure it'll increase but a 200% change is very large for the amount of material that is added to distribute the stresses across.

So no. You can't make em out of paper. But there's more to doubling the stiffness of the car than adding the roof. If you look at the skeleton closely you'll see that the roadster and coupe's differ significantly by more than just the roof supports.

If you really want to know more about what I'm talking about read a strength of materials book.

[Dammmittt]

To appease myself and the rest of you I did a quick FEA model of the roadie skeleton today. Then I added the roof.

Roadie No Roof


Roadie with Roof Supports


The results show that with the roof supports the stiffness is 138% of the original...not over 200%. Sounds much more reasonable to me. So...like I said...they had to have added more than just roof supports.

[The HACK]

Originally Posted by Dammmittt

To appease myself and the rest of you I did a quick FEA model of the roadie skeleton today. Then I added the roof.

The results show that with the roof supports the stiffness is 138% of the original...not over 200%. Sounds much more reasonable to me. So...like I said...they had to have added more than just roof supports.

How about you try filling that hole in on the top and make the C pillar about 5 times thicker? Also add in an additional B pillar support (which the roadster lacks.

[Dammmittt]

Originally Posted by The HACK

How about you try filling that hole in on the top and make the C pillar about 5 times thicker? Also add in an additional B pillar support (which the roadster lacks.

I knew that someone would have issues. The point of this wasn't to make it perfect but show you the magnitude of the stiffness increase. It's much less than double. All I had to go off of was the realoem.com picture of the skeleton. I think I've done enough.

As far as the roof goes, it's glued bolted into place, so it's not adding as much stiffness as the parts that are welded/riveted to the frame.

If you want to keep doubting me that's fine...I don't really care anymore.

[RichardTS]

Stiffness is the ratio of force applied to deflection. Since deflection can only occur as a result of force applied, internal forces are an integral part of stiffness. You very much missed my point about making the pillars out of paper - I was being sarcastic. And, as you said, we can't make the pillars out of paper BECAUSE THE STIFFNESS OF THE STRUCTURE WILL BE COMPROMISED. Therefore, any discussion of stiffness MUST include analyzing the internal stresses. As you have actually done in the FEA. And, as long as we are talking about the steel members of the car, we do not need to worry about changes in Young's modulus because, more than likely, it will remain constant as it only varies by a very small amount across different steels.

Very interesting FEA presented and probably not near as simplistic as you have stated. I note that you have simply divided the max stress area in the roadster by the max stress area in the coupe. However, this maximum occurs OUTSIDE the passenger compartment. The passenger compartment should be the only part of the structure that should benefit from the additional stiffness.

As usual, the real answer is in the eyes of the beholder. Further investigation of the FEA shows that, in the area of the left sill, the factor is closer to 2.7. In fact, we could probably find almost any factor we were looking for, along with the accompanying error involved with correlating colors with deflection. Where did BMW measure the stiffness difference? We don't know this and all of us are only fishing in the dark.

We also have already noted that the coupe weighs 33 lbs. more than the roadster. We will be hard pressed to come up with more structural material elsewhere in the coupe. If we might take a blind guess at the actual amount of steel added to the car by adding the roof and say that it is equivalent to, say, a sheet of 20 guage steel 4 ft. square. At 490lbs/cu-ft this sheet would weigh, you guessed it, 32.67 lbs. So, if they added more, where is it?

I also will not take offense at your sarcastic little comment about strength of materials. And, for consistency, you really need to ensure that the forces are applied the same way in both models.

[CalRick]

If the additional stiffness that's being seen is not due to the roof and related changes, where is it coming from? Why would BMW provide additional stiffening elements to the coupe and not the roadster?

Considering all the additional structure that is typically added to a chop-top coupe to bring its stiffness back up into the neighborhood of an original coupe structure, (and how flexible most convertibles remain,) it's not incredible to figure that a simple roof-ectomy results in a 50ish % loss in stiffness prior to reinforcement.

[Dammmittt]

Originally Posted by RichardTS

Stiffness is the ratio of force applied to deflection. Since deflection can only occur as a result of force applied, internal forces are an integral part of stiffness. You very much missed my point about making the pillars out of paper - I was being sarcastic. And, as you said, we can't make the pillars out of paper BECAUSE THE STIFFNESS OF THE STRUCTURE WILL BE COMPROMISED. Therefore, any discussion of stiffness MUST include analyzing the internal stresses. As you have actually done in the FEA. And, as long as we are talking about the steel members of the car, we do not need to worry about changes in Young's modulus because, more than likely, it will remain constant as it only varies by a very small amount across different steels.

Very interesting FEA presented and probably not near as simplistic as you have stated. I note that you have simply divided the max stress area in the roadster by the max stress area in the coupe. However, this maximum occurs OUTSIDE the passenger compartment. The passenger compartment should be the only part of the structure that should benefit from the additional stiffness.

As usual, the real answer is in the eyes of the beholder. Further investigation of the FEA shows that, in the area of the left sill, the factor is closer to 2.7. In fact, we could probably find almost any factor we were looking for, along with the accompanying error involved with correlating colors with deflection. Where did BMW measure the stiffness difference? We don't know this and all of us are only fishing in the dark.

We also have already noted that the coupe weighs 33 lbs. more than the roadster. We will be hard pressed to come up with more structural material elsewhere in the coupe. If we might take a blind guess at the actual amount of steel added to the car by adding the roof and say that it is equivalent to, say, a sheet of 20 guage steel 4 ft. square. At 490lbs/cu-ft this sheet would weigh, you guessed it, 32.67 lbs. So, if they added more, where is it?

I also will not take offense at your sarcastic little comment about strength of materials. And, for consistency, you really need to ensure that the forces are applied the same way in both models.

Where do I begin. Ok, first of all let me explain to you what you're looking at. You're not looking at a stress color plot, you're looking at a displacement color plot. So remember that F = k*x and similarly M = k*phi, and since I'm torquing the car with the same forces and boundary conditions in both cases the ratio of the displacements is going to be equal to the ratio of the stiffness in the two cases. Got it?

Next, realize that the Finite Element Method takes care of all the internal forces stresses and strains by way of the interpolation function (element type) and by using any of a variety of methods, either the direct method (force balancing) or by using the method of minimum potential energy or galerkin's method, etc. For the end user it doesn't really matter, as long as your boundary conditions and your external loading is right, it'll all work out assuming you have no issues with the mesh. This why you usually mesh the thing multiple times until your solution converges. So, unless you're trying to build an FEA model by hand or you're looking at it analytically, like I said...IT DOESN'T MATTER. The software, provided you're using it correctly, does it for you.

As far as the amount of material added and extra stiffness goes, like I said it probably has more to do with a change in bending moment of inertia. Just from looking at the pictures of the skeleton you can see that the frame has box beams in the coupe and a tube or cylindrical frame in the roadie. That can make a big difference in moment of inertia without changing the mass of the frame significantly.

Why do you think most buildings are made with I-beams? They have a high moment of inertia and they save weight.

Any other questions? Or should I give up?

Oh and sorry for the sarcasm...I tend to get that way when people argue with me and they don't really know what they're talking about. And if that offends you fine...but I'm an engineer and I know what I'm talking about it. And I get a little tired of defending my opinions from people who have a limited understand of mechanics.

[Dammmittt]

Originally Posted by CalRick

If the additional stiffness that's being seen is not due to the roof and related changes, where is it coming from? Why would BMW provide additional stiffening elements to the coupe and not the roadster?

Considering all the additional structure that is typically added to a chop-top coupe to bring its stiffness back up into the neighborhood of an original coupe structure, (and how flexible most convertibles remain,) it's not incredible to figure that a simple roof-ectomy results in a 50ish % loss in stiffness prior to reinforcement.

See above.

And as far as why BMW does anything, I dunno, ask them.