Maximum stress and bending moments

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• Sep 5th 2013, 11:38 AM
Jock
Maximum stress and bending moments
A beam of rectangular cross section 200 mm deep and 100 mm wide. If
the beam is 3m long, simply supported at either end and carries point
loads at 1m from the left end 5kN and 1m from the right end 10kN. Now I think I have worked out the maximum bending moment to be 8.4kNm at 2m from the left hand end correctly. I now need to find (a) the maximum stress in the beam (b) draw a sketch graph of the stress distribution through the thickness of the beam indicating which are tensile and compressive. I have drawn a rough sketch but don't know the values for it or how to achieve them. (c)

Determine the dimensions of the cross section which will minimise
the maximum stress value if:
• the cross sectional area of the beam can be increased by 20%
• the beam section is to remain a solid rectangle
• neither the breadth or depth of the beam section can be reduced
below their original dimensions.
Show the dimensions of the proposed beam cross section with the aid
of a sketch.

I have for this 240mmx120mm is that correct this would give a percentage reduction in stress value of 34.72%. Any help with this little lot I would be very grateful
• Sep 6th 2013, 05:50 AM
Jock
Re: Maximum stress and bending moments
(a)= 12.5nmm^2 (b) Don't know how to get the tensile force for the diagram if the compressive is 12.5 Nmm^2????? (c)120mmx240mm (d)57.6% Hope this is all correct so far and a little help with C wouldn't go a miss anyone.....
• Sep 6th 2013, 11:49 AM
ebaines
Re: Maximum stress and bending moments
Your answer for (a) is correct, but you should put it in units of Pa, which is N/m^2, not N/mm^2.

For b, the stress in a rectangular beam is My/I, where y is the distance from the centerline of the beam. It's negative (compressive) at the top edge of the beam and positive (tensile) at the bottom edge. So your sketch should be simply a linear function going from -Mc/I at the top to +Mc/I at the bottom.

For c the most efficient way to make a beam stronger is to make it deeper. One of the constraints is that the area is allowed to grow by 20%, but using your dimensions of 240mm x 120mm the new area is 44% larger than the original. Try simply increasing the depth by 20% and see what you get.
• Sep 6th 2013, 12:06 PM
Jock
Re: Maximum stress and bending moments
what does mc/I letters stand for? just to make sure I work it out right then Ill post my answer. Did the last one again and increased the depth to 240mm and got a final answer of 8.67x10^-21 which is 8.67Pa and gave a different percentage of 69.36% is that more like it?......
• Sep 6th 2013, 12:47 PM
ebaines
Re: Maximum stress and bending moments
Quote:

Originally Posted by Jock
what does mc/I letters stand for?

Hmmm.. How did you calculate the answer to (a)? I figured you used the well known formula: stress = Mc/I, where M= moment, c = distance from the centerline to the top or bottom edge of the beam, and I = area moment of inertia.

Your new answer is good except for the powers of ten. You should have 8.67 x 10^9 Pa. How do you get the power of 10^-21? Also the percent difference is not 69.36% but rather one minus that. The percent reduction in something is calculated using: (original value - new value)/original value.
• Sep 6th 2013, 12:49 PM
Jock
Re: Maximum stress and bending moments
I got the negative compressive force to be -6.29Pa and the positive tensile force to be +6.29 that correct?.....
• Sep 6th 2013, 12:52 PM
Jock
Re: Maximum stress and bending moments
I used o-=my/I guess its the same thing but just used my letters as that's the way I had been taught which was 8.4x100x10^-3/6.67x10^-5 which gave me 12.5 Pa
• Sep 6th 2013, 12:55 PM
ebaines
Re: Maximum stress and bending moments
Quote:

Originally Posted by Jock
I got the negative compressive force to be -6.29Pa and the positive tensile force to be +6.29 that correct?.....

No. As noted previously you were on the right track with your answer of 8.67, except the powers of ten is incorrect. And we're talking about stress, not force.
• Sep 6th 2013, 01:00 PM
Jock
Re: Maximum stress and bending moments
8.67 cant be correct though as I put in a wrong dimension it should have been 100x10^-3 and not 120x10^-3 so the right answer I believe is 6018.786127 which is 6.19x10^8 I think.....
• Sep 6th 2013, 01:02 PM
ebaines
Re: Maximum stress and bending moments
Quote:

Originally Posted by Jock
8.4x100x10^-3/6.67x10^-5 which gave me 12.5 Pa

The max moment is 8333 N-m, not 8.33 N-m - remember the point forces were gibven in kN, or thousands of newtons. So your calculation should be:

(8.33 x 10^3 N-m)(0.1 m)/( 6.67x 10^-5 m^4) = 12.5 x 10^6 Pa.
• Sep 6th 2013, 01:05 PM
Jock
Re: Maximum stress and bending moments
Since we are almost there can you list all the details in one post so I can read through it please as I have lost where we are and have bits of paper everywhere now please.....
• Sep 6th 2013, 01:07 PM
ebaines
Re: Maximum stress and bending moments
Quote:

Originally Posted by Jock
8.67 cant be correct though as I put in a wrong dimension it should have been 100x10^-3 and not 120x10^-3 so the right answer I believe is 6018.786127 which is 6.19x10^8 I think.....

Jock - please slow down and take a deep breath. The dimensions you should be using for the last part of this problem are width = 0.1m and depth = 0.24m. What do you get for I using these new numbers? And this part:

Quote:

Originally Posted by Jock
6018.786127 which is 6.19x10^8

makes no sense at all.
• Sep 6th 2013, 01:11 PM
Jock
Re: Maximum stress and bending moments
Maximum bending moment for the original 100x200mm beam is 8.4 kNm
Maximum stress in the beam 100x200mm is 12.5x10^6
Havent worked out the compressive and tensile
New dimensions are 100x240mm
The new maximum stress is 6.19x10^8
That's what I have so far............
• Sep 6th 2013, 01:14 PM
ebaines
Re: Maximum stress and bending moments
Quote:

Originally Posted by Jock
Maximum bending moment for the original 100x200mm beam is 8.4 kNm
Maximum stress in the beam 100x200mm is 12.5x10^6

Good.

Quote:

Originally Posted by Jock
Havent worked out the compressive and tensile
New dimensions are 100x240mm
The new maximum stress is 6.19x10^8

I disagree with this - please show your calculation for stress with the new dimensions.
• Sep 6th 2013, 01:19 PM
Jock
Re: Maximum stress and bending moments
8.33x100x10^-3/1.384x10^-4
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