An engineer came up with three cross sections they intend to use for a beam. The...

 

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An engineer came up with three cross sections they intend to use for a beam. The first (a) is a circle with radius r. The second (b) has a height h, and z(y) is a parabola symmetric about the y and z-axis and defined by wmin and wmax. The third (c) also has a height h and is symmetric about the y-axis. The profile is described by ze(y)=sin(y) +2 if the axis is placed at the base of the cross section. Can you find the area A, the geometric center dNA, and second moment of area I, for these? a N A r b y Zb(y) at min h ze(y) h y VNP max 1. Find the area of a,b, and c using the area integral J dA. 2. Find the geometric center of dNA for cross section c using the formula A 3. Find the second moment of area I, of a,b, and c about the neutral axis using the integral JydA. An engineer came up with three cross sections they intend to use for a beam. The first (a) is a circle with radius r. The second (b) has a height h, and z(y) is a parabola symmetric about the y and z-axis and defined by wmin and wmax. The third (c) also has a height h and is symmetric about the y-axis. The profile is described by ze(y)=sin(y) +2 if the axis is placed at the base of the cross section. Can you find the area A, the geometric center dNA, and second moment of area I, for these? a N A r b y Zb(y) at min h ze(y) h y VNP max 1. Find the area of a,b, and c using the area integral J dA. 2. Find the geometric center of dNA for cross section c using the formula A 3. Find the second moment of area I, of a,b, and c about the neutral axis using the integral JydA.