1. **Problem Statement:** Calculate the shear forces and bending moments for a cantilever beam fixed at A with a downward reaction of 24 kN at A, a uniformly distributed load (UDL) of 8 kN/m over the middle 3 m segment, and a downward point load P = 40 kN at the free end. 2. **Given:** - Reaction at A: $R_A = 24$ kN downward - UDL: $w = 8$ kN/m over 3 m - Point load at free end: $P = 40$ kN downward - Beam segments: 1 m (left), 3 m (middle), 1 m (right) 3. **Step 1: Calculate total UDL load** $$W = w \times 3 = 8 \times 3 = 24 \text{ kN}$$ 4. **Step 2: Calculate net vertical force at A (By)** Sum of downward loads: $24 + 24 + 40 = 88$ kN Since $R_A = 24$ kN downward, the vertical reaction at A to balance is: $$B_y = 88 - 24 = 64 \text{ kN upward}$$ 5. **Step 3: Shear force at 1 m (just right of fixed end)** Shear force just right of A: $$V_{1m} = R_A - 0 = 24 \text{ kN downward}$$ 6. **Step 4: Shear force at 1 m (just left of UDL start)** Shear force just before UDL: $$V_{1m-} = 24 \text{ kN downward}$$ 7. **Step 5: Shear force at 1 m (just right of UDL start)** Shear force after 1 m of UDL: $$V_{1m+} = 24 - (8 \times 1) = 24 - 8 = 16 \text{ kN downward}$$ 8. **Step 6: Shear force at B (free end)** Shear force at free end: $$V_B = 24 - 24 - 40 = -40 \text{ kN downward}$$ 9. **Step 7: Moment at B (1B)** Moment at free end is zero since it is free: $$M_B = 0$$ 10. **Step 8: Calculate max and min shear and moments** - Max shear $V_{max} = 24$ kN downward at A - Min shear $V_{min} = -40$ kN downward at free end - Max moment occurs at fixed end A due to loads: $$M_{max} = 24 \times 0 + 8 \times 3 \times \frac{3}{2} + 40 \times 5 = 0 + 36 + 200 = 236 \text{ kNm}$$ - Min moment $M_{min} = 0$ at free end 11. **Summary of answers:** - $\Delta Y = 0$ (no vertical displacement given) - $B_y = 64$ kN upward - $V_{POS 1m} = 24$ kN downward - $V_{NEG 1m} = 16$ kN downward - $V_{NEG B} = 40$ kN downward - $M_B = 0$ kNm - $V_{max} = 24$ kN downward - $V_{min} = -40$ kN downward - $M_{max} = 236$ kNm - $M_{min} = 0$ kNm All values rounded to 3 significant digits.