Position and magnitude of the maximum bending moment

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Homework Statement

I need to calculate the position and magnitude of the maximum bending moment.

Knows are:
E=210GPa

Homework Equations

The Attempt at a Solution

I've calculated the following:
R1=33kN
R2=32kN

From the shear force diagram I know that at 2m from R1 at the concentrated load of 10kN the shear force is +3kN.
The max bending moment occurs when the shear force =0 when it changes from + to -, therefore the max bending moment will be just above 2m. The bending moment at 2m is +46kNm.

Can someone guide me on how to calculate the max bending moment and the distance?

 

[SteamKing]

Homework Statement

I need to calculate the position and magnitude of the maximum bending moment.

Knows are:
E=210GPa

Homework Equations

The Attempt at a Solution

I've calculated the following:
R1=33kN
R2=32kN

From the shear force diagram I know that at 2m from R1 at the concentrated load of 10kN the shear force is +3kN.
The max bending moment occurs when the shear force =0 when it changes from + to -, therefore the max bending moment will be just above 2m. The bending moment at 2m is +46kNm.

Can someone guide me on how to calculate the max bending moment and the distance?

OK, you've figured out what the shear force and bending moment are at 2 m from the left end of the beam. Keep going; the shear is still positive.

It's not clear to me why you haven't constructed the entire shear force curve for this beam. If you do that, the points at which the BM will possibly be a maximum can be determined by inspection, i.e., where the shear force = 0.

The value of the BM at these locations can be found by calculating the area under the SF curve from the end of the beam up to these locations.

 

[sponsoraw]

Thanks for the reply SteamKing. I've drawn the full shear force diagram, hence I know that the 0 shear force is not at the 2m from the left. I can draw a bending moment diagram below and read the distance from the graph and then calculate the bending moment at this distance, however I was hoping for a more accurate method.

Can you expand on the area method?

 

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Do you think that the graphical method will be accurate enough? I don't want to spend to much time if I don't have to.

 

[SteamKing]

Thanks for the reply SteamKing. I've drawn the full shear force diagram, hence I know that the 0 shear force is not at the 2m from the left. I can draw a bending moment diagram below and read the distance from the graph and then calculate the bending moment at this distance, however I was hoping for a more accurate method.

Can you expand on the area method?

It's not clear how you are constructing the BM curve for this beam if you are not calculating the area under the SF curve. If you can provide additional information on this point, that would be most helpful.

 

[sponsoraw]

M0=0
M1=(33*1)-(10*1*0.5)=33-5=+28kNm
M2=(33*2)-(10*2*1)=66-20=+46kNm
M3=(33*3)-(10*1)-(10*3*1.5)=99-10-45=+44kNm
M4=(33*4)-(10*2)-(10*4*2)=132-20-80=+32kNm
M5=(33*5)-(10*3)-(15*1)-(10*4*3)=165-30-15-120=0

Plotting bending moment values over 5m.

The 0 SF is at just past 2m from the left and that is where the max BM will be. How to calculate this accurately?

 

[SteamKing]

M0=0
M1=(33*1)-(10*1*0.5)=33-5=+28kNm
M2=(33*2)-(10*2*1)=66-20=+46kNm
M3=(33*3)-(10*1)-(10*3*1.5)=99-10-45=+44kNm
M4=(33*4)-(10*2)-(10*4*2)=132-20-80=+32kNm
M5=(33*5)-(10*3)-(15*1)-(10*4*3)=165-30-15-120=0

Plotting bending moment values over 5m.

The 0 SF is at just past 2m from the left and that is where the max BM will be. How to calculate this accurately?

Take a look at the SF curve. It's all straight lines. You can plot the curve of SF between x = 2 m and x = 4 m. The SF is a straight line between those two locations.

You can find out where the SF = 0 by using linear interpolation or simple trigonometry. Once you find the crossing point, you can calculate the value of the max. BM.

BTW, your calculations above for the values of the BM are the same as calculating the area under the SF curve, whether you realize it or not.

 

[sponsoraw]

I've used the linear interpolation method with equation (y-y0)/(x-x0)=(y1-yo)/(x1-x0). for y=0 I got x=0.3, therefore the max bending moment (at SF=0) is at x=2.3. Is that correct? I just need to calculate the bending moment at 2.3m.

 

[SteamKing]

I've used the linear interpolation method with equation (y-y0)/(x-x0)=(y1-yo)/(x1-x0). for y=0 I got x=0.3, therefore the max bending moment (at SF=0) is at x=2.3. Is that correct? I just need to calculate the bending moment at 2.3m.

Yes, this looks good. Calculate the max. BM now.

 

[sponsoraw]

Thanks for your help, much appreciated.