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Architecture Engineering QUESTION #2369
Question 1
In the Slope-Deflection Method for analysis of indeterminate beams and frames, the slope-deflection equation for the near-end moment \(M_{AB}\) of member \(AB\) with near-end rotation \(\theta_A\), far-end rotation \(\theta_B\), chord rotation \(\psi = \Delta/L\), and Fixed-End Moment \(FEM_{AB}\) is:
  • \(M_{AB} = \dfrac{2EI}{L}(2\theta_A + \theta_B - 3\psi) + FEM_{AB}\)✔️
  • \(M_{AB} = \dfrac{EI}{L}(4\theta_A + 2\theta_B - 6\psi) + FEM_{AB}\)
  • \(M_{AB} = \dfrac{2EI}{L}(\theta_A + 2\theta_B - 3\psi) + FEM_{AB}\)
  • \(M_{AB} = \dfrac{4EI}{L}(\theta_A - \theta_B) + FEM_{AB}\)
Correct Answer Explanation
The slope-deflection equation is \(M_{AB} = \dfrac{2EI}{L}(2\theta_A + \theta_B - 3\psi) + FEM_{AB}\), where \(\theta_A\) is near-end rotation, \(\theta_B\) is far-end rotation, and \(\psi = \Delta/L\) is the chord rotation due to relative settlement or sway. The near-end joint rotation carries a coefficient twice that of the far-end (reflecting the carry-over factor of 1/2). The chord rotation term \(-3\psi\) accounts for sidesway. This equation is the foundation of the Slope-Deflection Method for all prismatic members.