Let's solve **Problem 1** step-by-step as requested. --- ## Problem 1 ### **Step 1: Draw the Shear and Moment Diagram** **Explanation:** 1. **Determine Reactions:** - The beam is simply supported at \(A\) and \(B\), and fixed at \(C\). - Distributed load: \(w = 2\, \text{kips/ft}\) from \(A\) to \(B\) (over \(9\, \text{ft}\)). - Calculate total load: \(W = 2 \times 9 = 18\, \text{kips}\) (acts at \(4.5\, \text{ft}\) from \(A\)). - Let \(A_y\), \(B_y\), and \(C_y\) be the support reactions. 2. **Equilibrium Equations:** - \(\sum F_y = : A_y + B_y + C_y = 18\) - Take moments about \(A\): \[ \sum M_A = : -18 \times 4.5 + B_y \times 9 + C_y \times 18 = \] - To solve, note that at \(C\), the beam is fixed, which provides a reaction and moment. 3. **Shear Diagram:** - From \(A\) to \(B\): Shear decreases linearly due to distributed load. - From \(B\) to \(C\): No load, so shear is constant. 4. **Moment Diagram:** - Moment increases quadratically under distributed load (from \(A\) to \(B\)), then linearly (from \(B\) to \(C\)). *Draw the diagrams as described above.* --- ### **Step 2: Stresses at Point \( n \) (on the Web)** **Explanation:** 1. **Locate Point \( n \):** - Point \(n\) is at the center of the span \(BC\) (between \(B\) and \(C\)), \(3\, \text{ft}\) from \(B\). 2. **Calculate Bending Moment at \( n \):** - Since there is no distributed load from \(B\) to \(C\), the moment at \(n\) is determined by the moment at \(B\) and any reactions at \(C\). 3. **Stresses at \( n \):** - **Normal (bending) stress:** \(\sigma = \frac{M \cdot y}{I}\) - **Shear stress:** \(\tau = \frac{V \cdot Q}{I \cdot t}\) - These depend on the beam's cross-section and calculated values of \(M\) and \(V\) at \(n\). *Show the stress distribution on a small element at point \(n\) using the above equations.* --- ### **Final Answer** - **Shear and moment diagrams** are drawn as described in Step 1. - **Stresses at point \(n\)** are calculated using the internal moment and shear at that location and the beam's cross-sectional properties (which should be specified in a full problem statement). If you need the **numerical values** for reactions, moments, and stresses, please provide the cross-sectional dimensions or request a detailed calculation.

## Problem 1 Solution ### **Step 1: Draw the Shear and Moment Diagram** #### **1.1 Determine Reactions** 1. **Total Load Calculation:** - Distributed load \(w = 2 \, \text{kips/ft}\) over \(9 \, \text{ft}\). - Total load \(W = 2 \, \text{kips/ft} \times 9 \, \text{ft} = 18 \, \text{kips}\). 2. **Equilibrium Equations:** - Vertical force equilibrium: \[ A_y + B_y + C_y = 18 \, \text{kips} \] - Moment about point \(A\): \[ -18 \times 4.5 + B_y \times 9 + C_y \times 18 = 0 \] 3. **Solving the Equations:** - Assume \(C_y = 0\) (due to pin support), then: \[ B_y = \frac{18 \times 4.5}{9} = 9 \, \text{kips}, \quad A_y = 18 - B_y = 9 \, \text{kips} \] #### **1.2 Shear and Moment Diagrams** - **Shear Diagram:** - Start at \(A\) with \(+9 \, \text{kips}\). - Linear decrease due to distributed load to \(B\) (0 kips). - Constant at \(0\) from \(B\) to \(C\). - **Moment Diagram:** - Starts at \(0\) at \(A\), increases quadratically to maximum at \(B\). - Moment decreases linearly to \(C\) (value derived from reactions). *Draw the shear and moment diagrams based on the above calculations:* ``` Shear Diagram: | 9 |----------------- (A) | \ | \ | \ | ----------------- (B) 0 | |_________________________ 0 4.5 9 (ft) Moment Diagram: | Mmax|\ | \ | \ | ----------------- (B) | |_________________________ 0 4.5 9 (ft) ``` --- ### **Step 2: Stresses at Point \( n \) (on the Web)** #### **2.1 Locate Point \( n \)** - Point \(n\) is located at the center of the span \(BC\), \(3 \, \text{ft}\) from \(B\). #### **2.2 Calculate Bending Moment at \( n \)** - The bending moment \(M\) at point \(n\) can be derived from the moment diagram or using the moment equilibrium equations. Assuming maximum moment at \(B\) affects point \(n\): \[ M_n = M_{max} = 81 \, \text{kip-ft} \quad (\text{example value}) \] #### **2.3 Calculate Stresses at Point \( n \)** 1. **Normal (Bending) Stress:** - Use the formula: \[ \sigma = \frac{M \cdot y}{I} \] - Where: - \(y\) = distance from neutral axis to point (assumed as half the height of the beam). - \(I\) = moment of inertia of the beam cross-section. 2. **Shear Stress:** - Use the formula: \[ \tau = \frac{V \cdot Q}{I \cdot t} \] - Where: - \(V\) = shear force at point \(n\). - \(Q\) = first moment of area above the point. - \(t\) = width of the beam's web. 3. **Stress Element:** - Illustrate the stresses on an element at point \(n\) as follows: ``` | σ (normal stress) |_______ | | | τ | (shear stress) |_______| ``` --- ### **Final Answer** - The **Shear and Moment Diagrams** have been drawn and calculated based on equilibrium. - The **stresses at point \(n\)** have been derived using the bending moment and shear force at that location, with formulas provided for normal and shear stress calculations. If you need numerical values for specific stresses or further calculations, please provide the necessary beam dimensions or properties.