Given the FBD of member AB, what is the correct y-force equilibrium equation?

Question

BlackTom AI · Accepted Answer

First, I restate the question and options to ensure clarity about what we are evaluating. Question: Given the FBD of member AB, what is the correct y-force equilibrium equation?

Answer options:
1) Ay - 6 kN - By = 0
2) Ay - 8 kN - By = 0
3) Ay - 8 kN - By - MA = 0
4) Ay - 8 kN = - By

Now, analyze each option in terms of the vertical force balance (y-direction):

Option 1: Ay - 6 kN - By = 0.
- In the FBD, the vertical forces include Ay (the reaction at A in y, assumed upward), the vertical component of the external load, and By (the vertical component at B, direction shown downward). The external 10 kN load is given with a 4-5-3 triangle; its vertical component is 10*(4/5) = 8 kN downward, not 6 kN. Therefore, subtracting 6 kN does not reflect the actual vertical load, so this option misstates the vertical loading.

Option 2: Ay - 8 kN - By = 0.
- This matches the correct vertical-force balance: upward Ay, downward 8 kN from the external load, and downward By at the right end. Setting the sum of vertical forces to zero gives Ay (up) minus 8 kN (down) minus By (down) equals zero. This aligns with the shown 8 kN vertical component and the direction of By.

Option 3: Ay - 8 kN - By - MA = 0.
- This introduces an MA term, which would be a moment (torque) term about a point, not a force in the y-equilibrium equation. In a pure y-force equilibrium equation, moments do not appear unless we are writing a moment equilibrium equation. Including MA here mixes types of equilibrium equations, making it incorrect for a straightforward vertical-force balance.

Option 4: Ay - 8 kN = - By.
- This is algebraically equivalent to Ay - 8 kN + By = 0 if you move terms around, but as written it implies Ay - 8 kN is equal to negative By. While mathematically rearrangeable to the same balance, the standard convention for a vertical-force sum is to write all upward forces minus all downward forces equals zero. Writing equals negative By is less conventional and can be confusing. It does reflect the same physics if interpreted correctly, but as presented, it does not clearly match the standard summation convention used for FBDs.

In summary, evaluating the vertical-force balance with proper sign convention and the given load components shows that the vertical equilibrium should account for Ay upward, the 8 kN downward from the external load, and By downward. Therefore, the equation Ay - 8 kN - By = 0 correctly represents the y-force equilibrium, while the other options either misstate the magnitude of the vertical component, introduce an inappropriate moment term, or present the balance in a less conventional form.

Given the FBD of member AB, what is the correct y-force equilibrium equation? 单项选择题

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