Conjugate beam: slope and deflection at free end

Given data

Beam ABC: A hinge, B roller, $AB = 6$ m ($2EI$) with UDL 40 kN/m, overhang $BC = 2$ m ($EI$) with 30 kN at C.

   40 kN/m (2EI)              30 kN
 A =============== B ---------v C
 ^        6 m      ^  2 m (EI)

Step 1: Reactions

$R_{A} = 110.00$ kN, $R_{B} = 160.00$ kN (positive = upward; check: $\sum F_y = 270.00$ kN equals the total load 270.00 kN).

Step 2: Bending moment diagram (sagging +), as $M/EI$ pieces

(Stiffness is in terms of $EI$; "Area" is $\int M/EI\,dx$ in units of $1/EI$; $\bar x$ is the centroid distance from A.)

Step 3: Conjugate beam

The real beam has a hinge at A, a roller at the second support and a free end at the overhang tip, so the conjugate beam has a hinge at A, an internal hinge at the support and a fixed end at the tip. It carries the $M/EI$ diagram of Step 2 as load (sagging $M/EI$ acts downward, hogging acts upward). Conjugate shear $V'$ = slope (positive = clockwise) and conjugate moment $M'$ = deflection (positive = downward).

Step 4: Conjugate reaction at A (slope at A)

Taking moments of the conjugate beam about the support at $x = 6$ m (conjugate moment is zero there):

$$R'_A\times6 = (270.00)(3.333) = 900.00$$ $$R'_A = \frac{150.000}{EI}\ \text{(slope at A, clockwise)}$$

At the point 2 m from A ($x = 2$ m):

$$V' = R'_A - \sum A = 150.000 - [(83.33)] = \frac{66.667}{EI}\ \text{(slope, clockwise)}$$ $$M' = R'_A\,x - \sum A(x-\bar x) = 300.000 - [(83.33)(0.720)] = \frac{240.000}{EI}\ \text{(deflection, downward)}$$

At C (free end) ($x = 8$ m):

$$V' = R'_A - \sum A = 150.000 - [(270.00) + (-60.00)] = \frac{-60.000}{EI}\ \text{(slope, anticlockwise)}$$ $$M' = R'_A\,x - \sum A(x-\bar x) = 1200.000 - [(270.00)(5.333) + (-60.00)(1.333)] = \frac{-160.000}{EI}\ \text{(deflection, upward)}$$

Answer: at 2 m from A: slope $= \dfrac{66.67}{EI}$ (clockwise), deflection $= \dfrac{240}{EI}$ downward. At the free end C: slope $= \dfrac{60}{EI}$ (anticlockwise), deflection $= \dfrac{160}{EI}$ upward (kNm$^2$/EI and kNm$^3$/EI).