Lecture 6 part II

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

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9 LECTURE 6 SECOND MOMENT OF AREA PART II CONTENTS SECOND MOMENT OF PLANE AREAS (CONT’)
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10 Radius of gyration of areas Since we have 2 x A I ydA = 2 y A I xdA = 2 zx y A I rdA I I = =+ (8) According to the second mean value of theorem for integration , we can have the integrals evaluated as ( ) 2 2* 2 x x A I y A Ak === ( ) 2 2 yy A I x ( ) 2 2 zz A I rdA z (9) With x x I k A = y y I k A = z z I k A = (10) And 222 zxy kkk = + (11) The distance k is called the radius of gyration . It is the location where the area can be lumped to produce the same second moment of area as does the actual area . It is also used in engineering to determine the buckling strength of columns, as you will be seeing a lot the later-on classes, especially in CVEN446 Structural Steel Design. The corresponding relation exists between the radii of gyration of the area with respect to two parallel axes, one of which passes through the centroid of the area. Thus, we have
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11 22 2 x xC x kk d = + 2 since xx Cx I Id A =+ 2 then x I AI Ad ′′ = + 2 and I kA = 2 I = (12) Similarly 2 yy Cy d = + (13) for polar second moment of area and radii of gyration, we also have 222 2 z z Cx yz Cz ddk d + (14) Solving Eqs. (12) to (14) for rectangular and polar radii of gyration for arbitrary and centroidal x -, y -, and z
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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 II - LECTURE 6 SECOND MOMENT OF AREA PART II...

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