1. **Problem Statement:** We have a simply supported beam of length 10 m with two downward forces: a 10 kN force at 3 m from the left support acting at an angle downward-left, and an 8 kN vertical downward force at 7 m from the left support (3 m from the right support). We need to find the shear force diagram, axial (normal) force diagram, and bending moment diagram. 2. **Calculate support reactions:** Let the left support reaction be $R_A$ (vertical) and the right support reaction be $R_B$ (vertical). Since the beam is simply supported, horizontal reactions are zero (no horizontal loads given). Sum of vertical forces: $$R_A + R_B = 10 + 8 = 18 \text{ kN}$$ Taking moments about left support (point A): $$\sum M_A = 0 = -10 \times 3 - 8 \times 7 + R_B \times 10$$ $$-30 - 56 + 10 R_B = 0$$ $$10 R_B = 86$$ $$R_B = 8.6 \text{ kN}$$ Then, $$R_A = 18 - 8.6 = 9.4 \text{ kN}$$ 3. **Shear force diagram (V):** - Start at $x=0$, $V=R_A=9.4$ kN upward. - At $x=3$ m, apply 10 kN downward force (component vertical assumed full 10 kN for shear since no horizontal shear given), shear drops: $$V = 9.4 - 10 = -0.6 \text{ kN}$$ - At $x=7$ m, apply 8 kN downward force, shear drops: $$V = -0.6 - 8 = -8.6 \text{ kN}$$ - At $x=10$ m, reaction $R_B=8.6$ kN upward, shear returns to zero. 4. **Axial force diagram:** No horizontal forces or axial loads given, so axial force is zero along the beam. 5. **Bending moment diagram (M):** - Moment at left support $M_0=0$. - Between 0 and 3 m: $$M(x) = R_A \times x = 9.4 x$$ - At $x=3$ m: $$M(3) = 9.4 \times 3 = 28.2 \text{ kN}\cdot\text{m}$$ - Between 3 and 7 m: $$M(x) = R_A x - 10 (x - 3) = 9.4 x - 10 (x - 3) = 9.4 x - 10 x + 30 = -0.6 x + 30$$ - At $x=7$ m: $$M(7) = -0.6 \times 7 + 30 = -4.2 + 30 = 25.8 \text{ kN}\cdot\text{m}$$ - Between 7 and 10 m: $$M(x) = R_A x - 10 (x - 3) - 8 (x - 7) = 9.4 x - 10 (x - 3) - 8 (x - 7)$$ $$= 9.4 x - 10 x + 30 - 8 x + 56 = (9.4 - 10 - 8) x + 86 = -8.6 x + 86$$ - At $x=10$ m: $$M(10) = -8.6 \times 10 + 86 = -86 + 86 = 0$$ **Final answers:** - Support reactions: $R_A=9.4$ kN, $R_B=8.6$ kN - Shear force diagram: starts at 9.4 kN, drops to -0.6 kN at 3 m, -8.6 kN at 7 m, returns to 0 at 10 m - Axial force diagram: zero everywhere - Bending moment diagram: linear segments with values $M(0)=0$, $M(3)=28.2$, $M(7)=25.8$, $M(10)=0$