Lecture 6 part III

Lecture 6 part III - LECTURE 6 SECOND MOMENT OF AREA PART...

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21 LECTURE 6 SECOND MOMENT OF AREA PART III CONTENTS PRINCIPLE SECOND MOMENT USING VERTICAL AREA TRIP TO COMPUTE THE SECOND MOMENT OF AREA OF COMPLEX GEOMETRY
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22 PRINCIPLE SECOND MOMENTS The second moment of area A with respect to x - and y -axes depends on the angle θ as shown in Fig. 10-7. From Eq. (2), we have Fig. 10-7 Element of area d A in the xy -plane zxyxy JIIII =+=+ (22) There must be a particular angle at which x I reaches its maximum and y I is at the minimum. We denote the x - and y -axes at this particular angle as u - and v -axes. Then the second moments of the area are called the principle second moments of area , designated as u I and v I . For the same point in different coordinate systems in xOy and x ' O y ', we have the transformation from ( x ', y ' ) to ( x , y ) as cos sin cos sin xx y yy x θθ ′′ =+ =− (23) Starting from
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23 () 2 2 cos sin x dI y dA y x dA θθ == (24) We have 2 2 22 cos sin cos sin 2sin cos cos sin 2 sin cos 1c o s 2 o s 2 sin 2 11 cos 2 sin 2 xx AA A A xy x y xyx y x y I dI y dA y x dA y dA x dA xydA II I III I θθθ θ ′′ = =+ =+− +− =⋅ +⋅ − ⋅ ∫∫ (25) Take the first derivative of x I with respect to and let it vanish, we have sin 2 2 cos2 0 x x y dI I d =− = (26) Then this particular angle 2 tan 2 xy p x y I I I = (27) There are values satisfying Eq. (27). These two values of p are 180° apart . The maximum and minimum second moments of area are 1/2 2 , uv x y x y I I I I ± + (28) The mixed second moment of area at angle is
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24 () ( ) 22 2 2 2 sin cos cos sin cos sin sin 2 sin cos sin cos cos sin 1 cos2 sin 2 2 xy A AA x y x y xy x y Id I y x dA xydA II I I I θθ θ ′′ == =− + =+ + ∫∫ (29)
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This note was uploaded on 04/20/2009 for the course CVEN 221 taught by Professor - during the Summer '08 term at Texas A&M.

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Lecture 6 part III - LECTURE 6 SECOND MOMENT OF AREA PART...

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