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**Problem F6–20: Determine the Reactions at Point D** In this problem, a structural frame is depicted consisting of a beam ABCD. The beam is supported with a hinge at point A and a roller at point D. - **Points and Dimensions:** - Point A is positioned at the base left of the structure with a height of 4 meters leading to point B. - The horizontal section BC is 12 meters long, divided into four equal segments of 3 meters each with supports or points at B and D. - Point D is at the same horizontal level as point C, at the far right end of the beam. - **Forces Acting on the Beam:** - A downward force of 10 kN is applied directly downward at point B. - Another downward force of 15 kN is applied directly at point C. The problem requires determining the reactions at point D due to these applied forces. The diagram suggests static equilibrium analysis where the sum of moments and forces needs to be balanced around the supports.

**Problem F6–20: Determine the Reactions at Point D** In this problem, a structural frame is depicted consisting of a beam ABCD. The beam is supported with a hinge at point A and a roller at point D. - **Points and Dimensions:** - Point A is positioned at the base left of the structure with a height of 4 meters leading to point B. - The horizontal section BC is 12 meters long, divided into four equal segments of 3 meters each with supports or points at B and D. - Point D is at the same horizontal level as point C, at the far right end of the beam. - **Forces Acting on the Beam:** - A downward force of 10 kN is applied directly downward at point B. - Another downward force of 15 kN is applied directly at point C. The problem requires determining the reactions at point D due to these applied forces. The diagram suggests static equilibrium analysis where the sum of moments and forces needs to be balanced around the supports.

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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**Problem F6–20: Determine the Reactions at Point D** In this problem, a structural frame is depicted consisting of a beam ABCD. The beam is supported with a hinge at point A and a roller at point D. - **Points and Dimensions:** - Point A is positioned at the base left of the structure with a height of 4 meters leading to point B. - The horizontal section BC is 12 meters long, divided into four equal segments of 3 meters each with supports or points at B and D. - Point D is at the same horizontal level as point C, at the far right end of the beam. - **Forces Acting on the Beam:** - A downward force of 10 kN is applied directly downward at point B. - Another downward force of 15 kN is applied directly at point C. The problem requires determining the reactions at point D due to these applied forces. The diagram suggests static equilibrium analysis where the sum of moments and forces needs to be balanced around the supports.
Transcribed Image Text:**Problem F6–20: Determine the Reactions at Point D** In this problem, a structural frame is depicted consisting of a beam ABCD. The beam is supported with a hinge at point A and a roller at point D. - **Points and Dimensions:** - Point A is positioned at the base left of the structure with a height of 4 meters leading to point B. - The horizontal section BC is 12 meters long, divided into four equal segments of 3 meters each with supports or points at B and D. - Point D is at the same horizontal level as point C, at the far right end of the beam. - **Forces Acting on the Beam:** - A downward force of 10 kN is applied directly downward at point B. - Another downward force of 15 kN is applied directly at point C. The problem requires determining the reactions at point D due to these applied forces. The diagram suggests static equilibrium analysis where the sum of moments and forces needs to be balanced around the supports.
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In my textbook these are the correct answer but all are positive, why is that I thought we can assume the the forces.

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This is how I solved for reaction D, is this also correct?

**Problem (6-20)** ### Diagram: - A horizontal beam subjected to two downward forces (10 kN and 15 kN) at equal 3-meter intervals. - A point A with force FAB at an angle using components FAB(3/5) in the x-direction and FAB(4/5) in the y-direction. - Support at point C with unknown components Cx and Cy. - Axes: Positive x-direction to the right, y-direction upwards. ### Equations and Calculations: #### Equilibrium in x-direction: \[ \sum F_x = FAB\left(\frac{3}{5}\right) + C_x = 0 \] #### Equilibrium in y-direction: \[ \sum F_y = FAB\left(\frac{4}{5}\right) - 10\, \text{kN} - 15\, \text{kN} + C_y = 0 \] #### Moment about C: \[ \sum M_C = (3\, \text{m} \cdot 15\, \text{kN}) + (6\, \text{m} \cdot 10\, \text{kN}) - (9\, \text{m} \cdot FAB\left(\frac{4}{5}\right)) = 0 \] ### Solving: - \( FAB = 14.583 \, \text{kN} \) #### Solve for Cx: \[ C_x = -\left(14.583\, \text{kN}\right)\left(\frac{3}{5}\right) = -8.749 \, \text{kN} \] #### Solve for Cy: \[ C_y = -14.583\left(\frac{4}{5}\right) + 10 + 15 = 13.333 \, \text{kN} \] ### Vertical Beam Diagram: - Vertical beam 4 meters high with loads: - Horizontal force = 8.749 kN - Vertical force = 13.333 kN at the top. #### Moment about D: \[ \sum M_D = M - (4\, \text{m} \cdot 8.749\, \text{kN}) = 0 \] #### Equilibrium in x-direction: \[ \sum F_x = D_x + 8.749\, \text{kN} = 0 \] #### Equilibrium in
Transcribed Image Text:**Problem (6-20)** ### Diagram: - A horizontal beam subjected to two downward forces (10 kN and 15 kN) at equal 3-meter intervals. - A point A with force FAB at an angle using components FAB(3/5) in the x-direction and FAB(4/5) in the y-direction. - Support at point C with unknown components Cx and Cy. - Axes: Positive x-direction to the right, y-direction upwards. ### Equations and Calculations: #### Equilibrium in x-direction: \[ \sum F_x = FAB\left(\frac{3}{5}\right) + C_x = 0 \] #### Equilibrium in y-direction: \[ \sum F_y = FAB\left(\frac{4}{5}\right) - 10\, \text{kN} - 15\, \text{kN} + C_y = 0 \] #### Moment about C: \[ \sum M_C = (3\, \text{m} \cdot 15\, \text{kN}) + (6\, \text{m} \cdot 10\, \text{kN}) - (9\, \text{m} \cdot FAB\left(\frac{4}{5}\right)) = 0 \] ### Solving: - \( FAB = 14.583 \, \text{kN} \) #### Solve for Cx: \[ C_x = -\left(14.583\, \text{kN}\right)\left(\frac{3}{5}\right) = -8.749 \, \text{kN} \] #### Solve for Cy: \[ C_y = -14.583\left(\frac{4}{5}\right) + 10 + 15 = 13.333 \, \text{kN} \] ### Vertical Beam Diagram: - Vertical beam 4 meters high with loads: - Horizontal force = 8.749 kN - Vertical force = 13.333 kN at the top. #### Moment about D: \[ \sum M_D = M - (4\, \text{m} \cdot 8.749\, \text{kN}) = 0 \] #### Equilibrium in x-direction: \[ \sum F_x = D_x + 8.749\, \text{kN} = 0 \] #### Equilibrium in
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