 | | From: | Philip | | Subject: | Question on mechanics of material | | Date: | Fri, 07 Jan 2005 03:08:41 -0800 |
|
|
 | It has been several years since i last worked on strength of materials,
and i can't be certain of what seem to be a simple problem:
Suppose i have a rectangle block. Inside the block is a rectangle cavity
with the long side horizontal. Consider the secanarios whereby the
interior cavity is rotated 30, 45 , 60 and 90 degrees. In which case is
the compressive strength the highest?
Since the max shear is at 45 degrees, the weakest block should be the
one with the cavity align at 45. Right? And 30 and 60 should be the same??
philip
|
|
| | From: | Philip | | Subject: | Re: Question on mechanics of material (Clarification) | | Date: | Fri, 07 Jan 2005 10:39:01 -0800 |
|
|
| Philip wrote:
> It has been several years since i last worked on strength of materials,
> and i can't be certain of what seem to be a simple problem:
>
> Suppose i have a rectangle block. Inside the block is a rectangle cavity
> with the long side horizontal. Consider the secanarios whereby the
> interior cavity is rotated 30, 45 , 60 and 90 degrees. In which case is
> the compressive strength the highest?
>
> Since the max shear is at 45 degrees, the weakest block should be the
> one with the cavity align at 45. Right? And 30 and 60 should be the same??
>
> philip
To add on, The only applied force is acting on the top surface in a
downward direction so that there is only compression and no shear.
I was assuming that the failure mode would be shear failure. What I
really want to ask if its stupid to put the cavity at a 45 degree angle
if tilting it up or down a little will prevent premature shear failure.
|
|
| | From: | dmartin at newarts.com | | Subject: | Re: Question on mechanics of material | | Date: | 7 Jan 2005 09:25:37 -0800 |
|
|
|
Philip wrote:
> It has been several years since i last worked on strength of
materials,
> and i can't be certain of what seem to be a simple problem:
>
> Suppose i have a rectangle block. Inside the block is a rectangle
cavity
> with the long side horizontal. Consider the secanarios whereby the
> interior cavity is rotated 30, 45 , 60 and 90 degrees. In which case
is
> the compressive strength the highest?
>
> Since the max shear is at 45 degrees, the weakest block should be the
> one with the cavity align at 45. Right? And 30 and 60 should be the
same??
>
> philip
It depends a lot on how you define failure.
A rectangular void is problematic because of unbounded logrithmic
stress concentrations at the corners; this leads to local plastic
deformation for any load.
Would an elliptically shaped slot be ok? That's a geometry for which
there is an analytic answer.
Are you sure it won't fail due to local tension rather than local
shear? In the tensile failure case the elliptical void's long axis in
the loading direction results in failure at the lowest applied load.
A fracture mechanics approach may not care about details of defect tip
geometry but you'll still have to worry about failure mode.
Dave
|
|
| | From: | dmartin at newarts.com | | Subject: | Re: Question on mechanics of material (Clarification) | | Date: | 8 Jan 2005 10:19:08 -0800 |
|
|
| I think I understand.
First let me say that you should *always* avoid sharp corners. A true
rectangle will be problematic in any geometry. At least use the biggest
fillets you can at the corners; best would be to use a slot with
semicircular ends. (The best slot of all would be a cylindrical hole).
Second the bottom line of what follows is that the slot would best be
in the x-y plane as you've defined the geometry; ie. the compressive
load should be perpendicular to the slot.
Third, a confusing attempt to describe the geometry as I understand it
follows:
There is a plate in the y-z plane of thickness x. There is a
compressive load in the z direction.
You are going to stab through the plate with a knife.
Shear stresses leading to shear failure of the plate will be a minimum
when the knife blade's two axes are perpendicular to the plate's z
axis.
Other geometries are hard to describe in a crystal clear fashion as
there are six axes involved (3 for the plate & 3 for the knife.)
Perhaps the simplest way to think about it is to imagine the knife
blade (or stab wound) like a long envelope lying face up on the desk;
the short direction is the width of the blade and the length is the
depth of the stab wound (length varies depending on how the blade is
tilted with respect to the plate when the stab is made.)
Now put a pencil point in the center of the envelope and hold it in the
vertical position with your finger on the eraser; this represents the
loading direction. You can tilt the loading direction along the
envelope's long axis or along it's short axis.
Tilting the loading direction around the blade's long axis increases
the shear stress component which reaches a maximum at 45 degrees. Such
tilts don't affect the depth of the stab wound (the length of the
envelope.) In a fracture mechanics sense such tilts result in Mode II
loading.
Tilting around the blade's short axis also increases the effective
shear stress component (but not as rapidly because the depth of the
stab wound increases as the angle increases). In a fracture machanics
sense such tilts result in Mode III loading.
I'm sorry for the complex description but the geometry is difficult to
describe in words and precise mathematical descriptions aren't very
useful in visualizing things for me.
I hope the above describes your situation in an understandable way. Are
there other constraints on the allowed orientation of the stab?
The ratio of the plate thickness to knife width is a detail that might
matter. I've assumed the knife width is big compared to the plate
thickness.
Dave
|
|
| | From: | Philip | | Subject: | Re: Question on mechanics of material (Clarification) | | Date: | Sat, 08 Jan 2005 12:27:30 -0800 |
|
|
| dmartin@newarts.com wrote:
> Tilting the loading direction around the blade's long axis increases
> the shear stress component which reaches a maximum at 45 degrees. Such
> tilts don't affect the depth of the stab wound (the length of the
> envelope.) In a fracture mechanics sense such tilts result in Mode II
> loading.
>
> Tilting around the blade's short axis also increases the effective
> shear stress component (but not as rapidly because the depth of the
> stab wound increases as the angle increases). In a fracture machanics
> sense such tilts result in Mode III loading.
>
> I'm sorry for the complex description but the geometry is difficult to
> describe in words and precise mathematical descriptions aren't very
> useful in visualizing things for me.
>
> I hope the above describes your situation in an understandable way. Are
> there other constraints on the allowed orientation of the stab?
>
> The ratio of the plate thickness to knife width is a detail that might
> matter. I've assumed the knife width is big compared to the plate
> thickness.
>
> Dave
>
Dave
your description and assumptions are correct. So bottomline is the
slot/stab has to be as flat as possible.
jbuch
Yeah... a slot, a slot...
|
|
| | From: | dmartin at newarts.com | | Subject: | Re: Question on mechanics of material (Clarification) | | Date: | 7 Jan 2005 19:35:41 -0800 |
|
|
| "The cavity really runs all the way through the block
in the x-dir and most of the compression is taken by material in the
out-of-plane direction. ..."
I'm sorry, I still don't get it.
Is the cavity in the x-y plane and the loading in the z direction?
|
|
| | From: | Philip | | Subject: | Re: Question on mechanics of material (Clarification) | | Date: | Fri, 07 Jan 2005 20:12:43 -0800 |
|
|
| dmartin@newarts.com wrote:
> "The cavity really runs all the way through the block
> in the x-dir and most of the compression is taken by material in the
> out-of-plane direction. ..."
>
> I'm sorry, I still don't get it.
> Is the cavity in the x-y plane and the loading in the z direction?
>
Yes. Considering that the rectangular cavity can rotate about the
y-axis, at 0 degree (laying flat), it will be in the xy plane. At 90
degrees, it will be in the yz plane. It extends all the way in the -x
and +x direction making two holes in the enclosing block. But it does
not extend in the -y and +y direction. And yes, loading is in the z
direction.
|
|
| | From: | jbuch | | Subject: | Re: Question on mechanics of material (Clarification) | | Date: | Sat, 08 Jan 2005 09:47:04 -0600 |
|
|
| Philip wrote:
> dmartin@newarts.com wrote:
>
>> "The cavity really runs all the way through the block
>> in the x-dir and most of the compression is taken by material in the
>> out-of-plane direction. ..."
>>
>> I'm sorry, I still don't get it.
>> Is the cavity in the x-y plane and the loading in the z direction?
>>
>
> Yes. Considering that the rectangular cavity can rotate about the
> y-axis, at 0 degree (laying flat), it will be in the xy plane. At 90
> degrees, it will be in the yz plane. It extends all the way in the -x
> and +x direction making two holes in the enclosing block. But it does
> not extend in the -y and +y direction. And yes, loading is in the z
> direction.
You really meant a SLOT, didn't you? Not a cavity.
So, you are concerned with the compression weakening aspects of a slot
and the orientation of the slot.
This cavity stuff boggles the mind.
|
|
