Given the FBD of member AB, what is the correct equilibrium equation from summing moments about point A? You can neglect the thickness of the member.

Question

BlackTom AI · Accepted Answer

We start by restating what needs to be done: sum moments about point A for the given FBD of member AB, neglecting thickness, and set the sum equal to zero for rotational equilibrium. The moment contributions about A come from the couple MA at A, and from external forces that create a moment arm with respect to A. The force By at the far end (and any other forces with a lever arm from A) will contribute a moment term of (By) times its perpendicular distance to A, with signs determined by the sense of rotation that the force tends to produce.

Option 1: - 8 kN (2m) - By (4m) = MA
- This form presents moments about A in a mixed order, with MA on the right-hand side. In a standard sum about A, we typically move all moments to one side to equal zero, e.g., MA plus other contributions equals zero. Placing - 8 kN(2m) and - By(4m) on the left and MA on the right is algebraically acceptable if we consistently maintain sign conventions, but the minus signs before the force terms imply their rotation directions are opposite to the reference positive moment. If the actual applied moments from the 8 kN force and By produce a positive rotation about A, this form would misrepresent their sign. The lack of explicit equality to zero in the left-hand side (or right-hand side) and the ambiguous arrangement makes the conventional interpretation less clear.

Option 2: MA − 6 kN (2m) − By (4m) = 0
- Here the 6 kN is supplying the moment arm of 2 m. The presence of MA on the left and the other two terms subtracting on the left-hand side implies that the 6 kN force and the By force generate opposite sense moments relative to MA, and their coefficients (2 m and 4 m) reflect their perpendicular distances from A. The discrepancy from 8 kN(2m) in other forms suggests a possible mismatch in the stated force magnitude or in the assumed lever arm for the 6 kN component. If the actual problem geometry or force magnitudes assign 8 kN rather than 6 kN, this option would be inconsistent with the given data.

Option 3: - 8 kN (2m) - By (4m) = 0
- This option presents the sum of the two external moments about A on the left, set equal to zero (without MA on either side). However, the moment about A must include MA as part of the equation unless MA is explicitly defined as the reaction moment that balances the others. If MA is a free-end couple, omitting MA in the equilibrium sum would be incorrect. The conventional approach for a fixed-end moment problem is to include all couples about the chosen point, so dropping MA typically leads to an incorrect equilibrium condition unless MA is already zero or otherwise accounted for.

Option 4: MA − 8 kN (2m) − By (4m) = 0
- This form places MA on the left and subtracts the moments due to the 8 kN force (times 2 m) and the By force (times 4 m), setting the total to zero. This is a standard and consistent way to write the moment balance about point A: sum of all moments about A equals zero, with signs chosen according to the rotational convention (clockwise vs. counterclockwise). If MA is the reaction moment at A and the 8 kN and By forces produce moments that tend to rotate the member in the opposite sense to MA, then MA − [8 kN(2m) + By(4m)] = 0 correctly captures the equilibrium condition.

Across these options, we evaluate the consistency with typical sign conventions and the requirement to include all moment contributions about A. The key misconceptions to watch for are: omitting MA when it should be included, misassigning the sign of the moment produced by external forces, or using an incorrect lever arm distance for a given force. Option 4 provides a clean, conventional formulation that sums all moment contributions about A and sets the total to zero, aligning with standard static equilibrium practice, provided the force magnitudes and lever arms are as indicated in the problem.

Given the FBD of member AB, what is the correct equilibrium equation from summing moments about point A? You can neglect the thickness of the member. 单项选择题

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