Maximum and minimum tensile/compressive strain in beam

[xJJx]

Hi, I'm quite stuck on this past paper question and would really appreciate some help on it. Below I have attached the questions and solutions; it asks me to determine where the maximum tensile and compressive strain in the beam using the bending moment diagram (attached as separate figure below). I'm confused at how a bending moment can let you know where the max tensile/compressive stress/strain would be, i thought a sagging moment causes both tensile and compressive stress (tensile at bottom, compressive at top)? same with a hogging moment (tensile at top, compressive at bottom). Can anyone explain why they did this? thank you.

 

[haruspex]

how a bending moment can let you know where the max tensile/compressive stress/strain would be, i thought a sagging moment causes both tensile and compressive stress (tensile at bottom, compressive at top)? same with a hogging moment

Aren't sagging and hogging moments just special cases of a bending moment?

 

[Chestermiller]

Please articulate your understanding of how the radius of curvature of the beam is related to the distribution of tensile strain over the cross section of a beam.

 

[xJJx]

Please articulate your understanding of how the radius of curvature of the beam is related to the distribution of tensile strain over the cross section of a beam.

I'm guessing the strain increases with R because the distance from the NA increases?

 

[xJJx]

Aren't sagging and hogging moments just special cases of a bending moment?

I'm not sure

 

[xJJx]

Please articulate your understanding of how the radius of curvature of the beam is related to the distribution of tensile strain over the cross section of a beam.

Also I don't get how the largest hogging moment is shown as positive on the diagram when hogging BM's are negative

 

[Chestermiller]

Also I don't get how the largest hogging moment is shown as positive on the diagram when hogging BM's are negative
View attachment 217854

You really just need to review your textbook sections on beam bending. We can't teach it all to you here. This is pretty straightforward stuff. If you have any specific questions on the material, we can help you with that.

 

[xJJx]

You really just need to review your textbook sections on beam bending. We can't teach it all to you here. This is pretty straightforward stuff. If you have any specific questions on the material, we can help you with that.

I have, I looked it up online too and I can't find anywhere that explains it. The main thing I'm confused by is the fact that my textbook and lecture notes say that a positive BM causes sagging and a negative BM causes hogging, yet when stating where each of them are on the BM diagram, the maximum negative BM on the diagram is labelled as 'sagging', which seems contradicting. Seems simple but the fact that i got this wrong caused my further calculations to be incorrect too.

 

[Chestermiller]

I have, I looked it up online too and I can't find anywhere that explains it. The main thing I'm confused by is the fact that my textbook and lecture notes say that a positive BM causes sagging and a negative BM causes hogging, yet when stating where each of them are on the BM diagram, the maximum negative BM on the diagram is labelled as 'sagging', which seems contradicting. Seems simple but the fact that i got this wrong caused my further calculations to be incorrect too.

There are different sign conventions used in different developments for the shear force and the bending moment. So you need to get an understanding of the particular sign convention being used in the material you are reading about at that time.

 

[PhanthomJay]

Yes the book used a different signage convention for bending moment that is opposite the convention you are familiar with. The convention you used is the generally used one, that is to say, the area under the shear curve between chosen points along the beam represents the change in moment between those points. Clearly starting at the left, the area under the shear curve is positive, so the moment change should be positive, in agreement with the calculus dM/dx = V. Or another way of viewing this is that the slope of the moment diagram at a point is the shear at that point. You have to be aware that the author used a different convention, whereby sagging moments are assumed negative. I know this is a cause for confusion.

 

[xJJx]

Yes the book used a different signage convention for bending moment that is opposite the convention you are familiar with. The convention you used is the generally used one, that is to say, the area under the shear curve between chosen points along the beam represents the change in moment between those points. Clearly starting at the left, the area under the shear curve is positive, so the moment change should be positive, in agreement with the calculus dM/dx = V. Or another way of viewing this is that the slope of the moment diagram at a point is the shear at that point. You have to be aware that the author used a different convention, whereby sagging moments are assumed negative. I know this is a cause for confusion.

Ahh, I see. I guess I'm safe with stating sagging moments as positive and hogging as negative. Do you know how he knew that the max compressive strain is at the largest sagging moment? i thought both sagging and hogging BM's cause both max compressive and max tensile stress (for sagging its max compressive at the top and max tensile at the bottom and viseversa for hogging). Thank you so much for replying to my question

 

[PhanthomJay]

Ahh, I see. I guess I'm safe with stating sagging moments as positive and hogging as negative. Do you know how he knew that the max compressive strain is at the largest sagging moment? i thought both sagging and hogging BM's cause both max compressive and max tensile stress (for sagging its max compressive at the top and max tensile at the bottom and viseversa for hogging). Thank you so much for replying to my question

you have missed sight of the fact that the strain gauges are placed on the top of the beam only, not on the top and bottom. The problem is asking you to determine the max compressive and tensile strains as recorded by the top strain gauges. Does this answer your question?

 

[xJJx]

you have missed sight of the fact that the strain gauges are placed on the top of the beam only, not on the top and bottom. The problem is asking you to determine the max compressive and tensile strains as recorded by the top strain gauges. Does this answer your question?

Not really, if the strain gauges are fitted at the top then shouldn't the value for either the max tensile or max compressive strain be 0? since at the top of the beam there's either only maximum tensile stress (and zero compressive stress) or max compressive (and zero tensile stress), depending on whether the beam is sagging or hogging? i don't get why both of the values are non-zero

 

[PhanthomJay]

Not really, if the strain gauges are fitted at the top then shouldn't the value for either the max tensile or max compressive strain be 0? since at the top of the beam there's either only maximum tensile stress (and zero compressive stress) or max compressive (and zero tensile stress), depending on whether the beam is sagging or hogging? i don't get why both of the values are non-zero

At the top outer fibers of the beam, maximum tensile stress occurs at the point of maximum hogging moment, and maximum compressive stress occurs at the point of maximum sagging moment, as you have noted. So what is the relationship between maximum stress and maximum strain? Strain is only 0 at the top fibers where the Moment (and stress) is 0.

 

[xJJx]

At the top outer fibers of the beam, maximum tensile stress occurs at the point of maximum hogging moment, and maximum compressive stress occurs at the point of maximum sagging moment, as you have noted. So what is the relationship between maximum stress and maximum strain? Strain is only 0 at the top fibers where the Moment (and stress) is 0.

Just wanted to say thank you for replying to my threads! You’ve been a big help and i understand statics more now haha. Had my exam too and it went okay!

 

[PhanthomJay]

Oh you are most welcome. Thank you for your courtesy. I am glad your exam went well.