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In this problem(attached as a picture), the authors have chosen the horizontal element to calculate the shear and I understand the solution. But, the thing I don't understand is: Why don't they take the vertical element (shaded by me, in the blue color) if we take the horizontal element we find a smaller value for shear, which is in our interest cause we are looking for the smallest shear.

retrieved from Mechanics of materials P.Beer

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First of all let it be clarified that the smallest shear force is not our point of interest here. Here the question is to find the shear acting at the two joints,A and B.You can conclude by the first glance itself that the shear on these two joints are of different magnitudes as A and B are not symmetrical about any axis.

Now by the method followed in the given text, the average shear stress along a given plane is calculated using the Q of the element on one side of an axis parallel to our plane of interest, ie V.

Now, in the given problem the joint A is horizontal. So we should consider a line parallel to this and passing through A, and calculate the Q of the element on either side of the line(in this case the horizontal element AB). Now by applying the formula we can find out the average shear stress.

Now as we come to the second part of the question, the joint B is vertical. Hence we should choose a vertical element. You might have done it that way.

Hope everything is clear.

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