Find the Shear and Moment on the Beam

[Northbysouth]

Homework Statement

Derive expressions for the shear force V and bending moment M as functions of x in the cantilever beam loaded as shown. Then answer the questions.

At x = 2 ft,
V =
M =

At x = 9.3 ft
V =
M =

I have attached an image of the question

The maximum (absolute value) shear force in the beamis 2109.19 lb
The maximum (absolute value) bending moment in the beam is lb·ft

Homework Equations

W = -dV/dx

dM = Vdx

The Attempt at a Solution

First I found the equation w = w₀+kx²

When x = 0
120 lb/ft = w₀ +k(0)²

Hence w₀ = 120 lb/ft

When x = 11.3'
320 lb/ft = 120 + k(11.3)²
k = 1.566

Hence:
w = 120 + 1.566x²\

To find the shear, V, I took the integral of w because:

w = -dV/dx hence
-∫w = V

V = -120x - 1.566x²

Thus when x = 2
V = -120(2) - 1.566(2)³
V = -244.176

When x = 9.3
V = -120(9.3) - 1.566(9.3)³
V = -1535.87

For the Moments I did the following

dm = Vdx
m = ∫V
m = ∫-120x - 0.522x³
m = -0.135x⁴-60x²

When x = 2
m = -242.

When x = 9.3
m = -6165.607

Then to try and calculate the maximum shear:

V = -120(11.3)-0.522(11.3)³
V = -2109.19

But it wants the absolute value, hence V = 2109.19

For the max bending moment:

m = -0.1325(11.3)³ - 60x²
m = -9789.17

But it wants the absolute value, hence V = 9789.17
Except for the max shear all of my values are wrong and I'm at a loss for what I should do. Help would be greatly appreciated. Thank you

 

[SteamKing]

No image attached.

 

[Northbysouth]

Sorry. I have now fixed it. Thank you for pointing that out.

 

[SteamKing]

In order to obtain the correct shear force and bending moment diagrams for this beam, you must first find the fixed end reaction and moment which keep the beam in equilibrium. Remember, at the free end, V = M = 0.

 

[pongo38]

Although steamking has wise words, in this case you CAN work from the right hand end without the reactions. Let e be the distance from the RH end, positive to the left... Ehen ev is .. and Mv is ... (use the definitions). When you have finished, the reactions should appear when v=11.3'. You then have a check on any errors made.Northbysouth, when you integrated, did you forget the arbitrary constant?