Finding maximum bending moment when shear force equation is known

[Kasthuri]

Homework Statement

Calculate the Maximum Bending Moment of the bridge section using the values given.
R_a = 61837.91667 N
R_b = 78304.5833 N
(please refer to attached diagram)

Homework Equations

shear force is = 0 when x = 13.755998 m
(I worked this distance and the shear force equation out and found it to be correct)

Shear force equation is: v = (950/13)x² + 3490.096154x - 61837.91667

The Attempt at a Solution

The Attempt at a Solution

Integrate shear force equation:
∫(950/13)x² + 3490.096154x - 61837.91667

= (950/39)x³ + (3490.096154/2)x² - 61837.91667x + C

when x = 0, c = 0

Maximum bending moment = (950/39)x³ + (3490.096154/2)x² - 61837.91667x when x = 13.755998 m

∴ Maximum bending moment = -457024.4582 Nm

BUT, I know that the max bending moment is definitely not the answer I calculated.

Any help you can offer me will be greatly appreciated!
Thanks

 

[pongo38]

First, restrict your calculations to four significant figures, to reduce errors (ironically). Then check with standard formula WL/6 where W is the total load. That is for the triangular load. Add WL/8 for the UDL. Since first writing this I have found an error in your work The coefficient 950/13 for the x^2 term needs checking. In general you would do better to work in kN units.

 

[SteamKing]

Your shear force equation is wrong. To check, substitute x = 0 and x = 26, and get the corresponding shear values. Check these with the reactions to see if the bridge is in equilibrium.

 

[nvn]

Kasthuri: Your shear force equation appears correct. Your maximum bending moment in post 1 appears correct. Nice work. I currently do not know why you, and others, currently seem to think it is wrong. Why do you think your answer is wrong?

 

[SteamKing]

I reiterate, your shear force equation is incorrect.
Where does the factor 950/13 come from? The load diagram clearly states that q at the right end of the bridge is 3800 N/m.
Why do you take Ra as negative in the shear force equation?
Clearly, if you evaluate the shear force equation at x = 0, you should obtain a shear force equal to the reaction at A.
Your bending moment calculation won't be correct until you get the correct shear force equation.

 

[Kasthuri]

This is the shear force equation: (I missed the '2' initially, I'm sorry for that!)

v = (9250/13)x² + 3490.096154x - 61837.9166

So through integrating from x = 0 to x = 13.756m I get that bending moment:
MAX BM = 885754.75 Nm

 

[Kasthuri]

I reiterate, your shear force equation is incorrect.
Where does the factor 950/13 come from? The load diagram clearly states that q at the right end of the bridge is 3800 N/m.
Why do you take Ra as negative in the shear force equation?
Clearly, if you evaluate the shear force equation at x = 0, you should obtain a shear force equal to the reaction at A.
Your bending moment calculation won't be correct until you get the correct shear force equation.

I just now realized that it isn't in equilibrium, thanks.
Ignore my earlier comment. I looked back on my working and found the 950/13 comes from the weight of the triangular loading:
W = 1/2(base)(height)
= 1/2(x)(3800/26)(x)
= 950/13(x^2)

 

[Kasthuri]

I have fixed my shear force equation and integrated it:
∫ - (950/13)x2 - 3490.096154x + 61837.91667 from x = 0 to x = 13.756 metres:

BM = 61837.91667x - (3490.096154x²)/2 - [(950/13)]x³/3

so integrating from x=0 to x=13.756m:

BM max = 457024.483 Nm

 

[SteamKing]

Sorry, but 1/2(x)*(3800/26)*x is NOT equal to (950/13)*x^2

 

[SteamKing]

Cancel last post. Sorry.

 

[Kasthuri]

Thank you so much for your help SteamKing! :D
I got it right :)