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I'm a bit confused with the wording of the question. What does it mean by expressing the bending moment and the deflected shape of BC as a function of z? Thank you![enter image description here]1

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In this question, the axes x and y are in the plane of the cross-section of the beam and z is along the length of the beam pointing horizontally to the right.

The reaction forces are,

$$ \Sigma M_C=0 \quad 20kN*4*2+ 30kN*6-B_V*4=0 \\ B_v = 340kN/4 =85kN \\ C_v= 80+30-85=25kN $$

The moment on the cantilever part is $$ M= 30kN*( z+2) \ assuming\ z_B =0 $$ And I let you do the deflection,by either double integration or area moment.

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You need to find the right expression that explains the bending moment along the z dimension.

Something like for $z=2meters$ (your limit is $-2m<=z<=4m$) the bending will be max. And for $z=4meters$ the bending will be 0mm.

I'll not put the equation here because this looks like a homework question so you can find with your own material.

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1.) Find all the reaction forces first, then draw a SFD diagram.

2.) Express the shear force using z as variable, you will get your f(z).

3.) Then calculate the bending moment, then draw a BMD diagram.

3.) Integrate the shear force equation from 2.) into a bending moment equation f'(z) with a constant C at the back.

4.) Identify the constant C in the bending moment equation with the maximum moment.

5.) Integrate the bending moment equation from 4.) into a deflection equation f''(z) with a constant C at the back.

6.) Identify the constant C in the deflection equation, substitute back into f''(z).

Then you are done, a deflection equation f''(z) in terms of z.

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