Position and magnitude of the maximum bending moment

In summary, to calculate the position and magnitude of the maximum bending moment, one can use the shear force diagram to determine where the shear force equals zero. This point can be found by using linear interpolation or simple trigonometry. The maximum bending moment will occur at this point and can be calculated by finding the area under the shear force curve or by using the equation (y-y0)/(x-x0)=(y1-yo)/(x1-x0) and solving for the value of x when y = 0.
  • #1
sponsoraw
43
1

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?
 

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  • #2
sponsoraw said:

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.
 
  • #3
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?
 
  • #4
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.
 
  • #5
sponsoraw said:
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.
 
  • #6
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?
 
  • #7
sponsoraw said:
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.
 
  • #8
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.
 

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  • #9
sponsoraw said:
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.
 
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  • #10
Thanks for your help, much appreciated.
 

FAQ: Position and magnitude of the maximum bending moment

What is the maximum bending moment?

The maximum bending moment is the maximum value of the internal bending moment that occurs at a specific point along a beam or structural element. It is a measure of the maximum amount of stress that the beam experiences at that point.

How is the position of the maximum bending moment determined?

The position of the maximum bending moment is determined by calculating the bending moment at different points along the beam and finding the point with the highest value. This can be done using equations or by creating a bending moment diagram.

What factors affect the position of the maximum bending moment?

The position of the maximum bending moment is affected by the type of load applied to the beam, the shape and size of the beam, and the support conditions. A concentrated load, for example, will result in a different position of the maximum bending moment compared to a distributed load.

How is the magnitude of the maximum bending moment calculated?

The magnitude of the maximum bending moment is calculated by multiplying the maximum value of the bending stress by the moment arm, which is the distance from the point of interest to the point of zero stress. This can be determined using the moment of inertia and the distance from the neutral axis.

Why is the position and magnitude of the maximum bending moment important?

The position and magnitude of the maximum bending moment are important because they help engineers and designers determine the strength and stability of a structure. By understanding where and how much stress is being applied to a beam, they can ensure that the structure can support the load without failing or deforming. This information is crucial in the design and construction of buildings, bridges, and other structures.

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