1. **Problem Statement:** Draw the influence lines for the vertical reactions at supports A and E, and the reaction moment at support E for the given beam. 2. **Beam Description:** - Beam supports: Roller at A, fixed at E. - Lengths: AB = 6 m, BC = 6 m, CD = 3 m, DE = 3 m. - Hinge located between C and D. 3. **Key Concepts:** - Influence lines show how a moving unit load affects a response (reaction or moment) at a specific point. - For vertical reactions at supports, the influence line is the deflection shape of the beam when a unit load moves across it. - For the moment at a fixed support, the influence line is the moment at that support due to a unit load moving along the beam. 4. **Step 1: Influence Line for Vertical Reaction at A ($R_A$):** - Apply a unit load moving from A to E. - At A, the influence line value is 1 when the load is exactly at A. - Between A and the hinge at C, the influence line decreases linearly from 1 at A to 0 at C. - Between C and E, due to the hinge, the influence line is zero because the hinge releases moment and vertical reaction influence beyond it. Mathematically: $$ IL_{R_A}(x) = \begin{cases} 1 - \frac{x}{12}, & 0 \leq x \leq 12 \\ 0, & 12 < x \leq 18 \end{cases} $$ where $x$ is the distance from A, with $x=12$ m at hinge C. 5. **Step 2: Influence Line for Vertical Reaction at E ($R_E$):** - At E, the influence line value is 1 when the load is at E. - Between D and E (last 3 m), the influence line increases linearly from 0 at D to 1 at E. - Between A and C, the influence line is zero due to the hinge. - Between C and D, the influence line is constant zero because the hinge separates the beam. Mathematically: $$ IL_{R_E}(x) = \begin{cases} 0, & 0 \leq x \leq 15 \\ \frac{x - 15}{3}, & 15 < x \leq 18 \end{cases} $$ where $x=15$ m at D and $x=18$ m at E. 6. **Step 3: Influence Line for Moment at E ($M_E$):** - Moment at fixed support E due to a unit load moving along the beam. - For $0 \leq x \leq 15$ (up to D), the moment influence line is zero because the hinge at C-D releases moment. - For $15 < x \leq 18$, the moment influence line increases linearly from 0 at D to maximum at E. Mathematically: $$ IL_{M_E}(x) = \begin{cases} 0, & 0 \leq x \leq 15 \\ (x - 15)(18 - x), & 15 < x \leq 18 \end{cases} $$ 7. **Summary:** - Influence lines are piecewise linear or zero due to hinge and support conditions. - The hinge at C-D divides the beam into two segments affecting influence lines.