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Determine expressions for internal shear and bending moment as a function of $x$ for a beam with two vertical loads.

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{diagram}

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The $\\var{qty(12, 'm')}$ long beam is loaded with $\\var{qty(B, 'kN')}$ force $B$ and $\\var{qty(C, 'kN')}$ force $C$, as shown.

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1.  Draw a FBD of the whole beam and take moments at $D$ to find $A = \\var{A'}$.

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2.  Take cuts in each of the three segments, and draw three free-body diagrams.  Show internal shear and moment at the cut with the standard sign convention.

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3.  For each FBD, apply $\\Sigma F_y = 0$ and $\\Sigma M_\\text{cut}=0$ to determine expressions for $V(x)$ and $M(x)$ as  functions of the forces and $x$.

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Find the shear and bending moment as functions of $x$ on the range:  $0 \\text{ m}< x < \\var{d1}$ m

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$V_1(x)$ = [[0]] kN


$M_1(x)$ = [[1]]  kN-m 

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Find the shear and bending moment as functions of $x$ on the range: $\\var{d1}\\text{ m} < x < \\var{d2}$ m

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$V_2(x)$ = [[1]] kN

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$M_2(x)$ = [[0]] kN-m

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Find the shear and bending moment as functions of $x$ on the range:  $\\var{d2}\\text{ m} < x < 12$ m

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$V_3(x)$ = [[0]] kN

$ M_3(x)$ =[[1]]kN-m

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