1. **Stating the problem:** Calculate the moments $M_0$, $M_a$, $M_b$, $M_c$, $M_d$, and the maximum moment $M_{max}$ for a beam with given loads and distances. Given: $A=9$, $B=8$, $C=2$, $D=7$. 2. **Understanding the loads and distances:** - Point load at 0: $(B+C) = 8+2=10$ kg downward. - Point load at $a$: $(A+B) = 9+8=17$ kg downward. - Uniform distributed load from $b$ to $c$: $(A+C) = 9+2=11$ kg/m. - Distances: - $0$ to $a = \frac{A}{2} = \frac{9}{2} = 4.5$ m - $a$ to $b = \frac{A}{2} = 4.5$ m - $b$ to $c = A+B = 9+8=17$ m - $c$ to $d = \frac{A}{2} = 4.5$ m 3. **Calculate positions:** - $a = 4.5$ m - $b = a + 4.5 = 9$ m - $c = b + 17 = 26$ m - $d = c + 4.5 = 30.5$ m 4. **Calculate reactions and moments:** **Moment at 0 ($M_0$):** Assuming moment at left end due to loads: - Moment from point load at 0 is zero (at the point). - Moment from point load at $a$: $17 \times 4.5 = 76.5$ - Moment from uniform load between $b$ and $c$: - Load magnitude: $11$ kg/m over $17$ m = $11 \times 17 = 187$ - Load acts at midpoint of $b$ to $c$: $9 + \frac{17}{2} = 17.5$ m from 0 - Moment: $187 \times 17.5 = 3272.5$ - Moment from point load at $d$ is not given, so zero. Sum moments at 0: $$M_0 = 0 + 76.5 + 3272.5 = 3349$$ 5. **Moment at $a$ ($M_a$):** Moments to the left of $a$: - Point load at 0: $10 \times 4.5 = 45$ - Moment at $a$ itself is zero. Moments to the right of $a$: - Point load at $a$ is at $a$, so zero moment at $a$. - Uniform load from $b$ to $c$ is $187$ kg acting at $17.5$ m from 0, distance from $a$ is $17.5 - 4.5 = 13$ m - Moment from uniform load at $a$: $187 \times 13 = 2431$ Sum moments at $a$: $$M_a = 45 + 0 + 2431 = 2476$$ 6. **Moment at $b$ ($M_b$):** - Point load at 0: $10 \times 9 = 90$ - Point load at $a$: $17 \times (9 - 4.5) = 17 \times 4.5 = 76.5$ - Uniform load from $b$ to $c$ acts at midpoint $17.5$ m, distance from $b$ is $17.5 - 9 = 8.5$ m - Moment from uniform load at $b$: $187 \times 8.5 = 1589.5$ Sum moments at $b$: $$M_b = 90 + 76.5 + 1589.5 = 1756$$ 7. **Moment at $c$ ($M_c$):** - Point load at 0: $10 \times 26 = 260$ - Point load at $a$: $17 \times (26 - 4.5) = 17 \times 21.5 = 365.5$ - Uniform load from $b$ to $c$ acts at midpoint $17.5$ m, distance from $c$ is $26 - 17.5 = 8.5$ m - Moment from uniform load at $c$: $187 \times 8.5 = 1589.5$ Sum moments at $c$: $$M_c = 260 + 365.5 + 1589.5 = 2215$$ 8. **Moment at $d$ ($M_d$):** - Point load at 0: $10 \times 30.5 = 305$ - Point load at $a$: $17 \times (30.5 - 4.5) = 17 \times 26 = 442$ - Uniform load from $b$ to $c$ acts at midpoint $17.5$ m, distance from $d$ is $30.5 - 17.5 = 13$ m - Moment from uniform load at $d$: $187 \times 13 = 2431$ Sum moments at $d$: $$M_d = 305 + 442 + 2431 = 3178$$ 9. **Maximum moment $M_{max}$:** From the calculated moments, the maximum is: $$M_{max} = \max(3349, 2476, 1756, 2215, 3178) = 3349$$ **Final answers:** $$M_0 = 3349$$ $$M_a = 2476$$ $$M_b = 1756$$ $$M_c = 2215$$ $$M_d = 3178$$ $$M_{max} = 3349$$