Chapter 5_Bending - CHAPTER 5 BENDING OF THIN-WALLED...

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1 CHAPTER 5 BENDING OF THIN-WALLED STRUCTURES 5.1 Bending in 3-D space Kinematics of displacement Strain-displacement relationship Stress-strain relationship 5.2 Centroidal bending axes and principal axes Centrodal bending (CB) axes Centroidal bending and principal (CB/P) axes Determination of CB axes Parallel axis theorem Determination of CB/P axes
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2 5.1 Bending in 3-D Space Kinematics of Displacement In the absence of torsional moment, it can be assumed that the cross-sections of a slender structure translate and rotate around y and z axes as a rigid plane. Under the assumption of small rotation, the displacement of a point P ( x, y, z ) on the cross-section located at x can be expressed as ) ( ) ( ) ( ) , , ( 0 x y x z x u z y x u z y θ + + = ) ( ) , , ( 0 x v z y x v = ) ( ) , , ( 0 x w z y x w = where ) ( ), ( 0 0 x v x u and ) ( 0 x w : the translational displacements of the C-S in the x , y and z directions, ) ( x z y : the horizontal displacement of point P due to rotation of the C-S around the y axis, ) ( x y z : the horizontal displacement of point P due to rotation of the C-S around the z axis. z y P(x,y,z) y z y + x 0 u , x u z x 0 w x z x 0 v x y z + z,w y,v z y P(x,y,z) P(x,y,z)
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3 Strain-Displacement Relation The axial strain is related to displacement u as follows: xx u x ε = ( 1 ) Form the previous section, 0 yz uu z y θ =+ + ( 2 ) Placing equation (2) into equation (1), 0 y z xx u u zy x xxx == + + ∂∂ ( 3 ) Also, it can be shown that, for a slender body undergoing bending. x v z = 0 and x w y = 0 ( 4 ) Placing equation (4) into the strain-displacement relation in equation (3), 22 000 xx uwv u x xx x ( 5 ) or 12 3 () xx Cx z Cx y Cx + ( 6 ) where 00 0 12 3 , CC C x = (7)
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4 Stress-Strain relationship Axial stress σ xx is related to axial strain ε xx as follows: 12 3 () xx xx EE C C z C y ==+ + ( 1 ) Using the above expression, the resultant forces and moments can be expressed as follows: 3 xx xx F dA E dA E C C z C y dA σε == = + + ∫∫ 3 yx x x x M z d AE z d C C z C y z d A = + + (2) 3 zx x x x M yd A Ey d A E CC z C y y d A = + + For a section of single material, E is uniform over the section and equation (2) can be expressed as 3 FE Cd Cz d Cy d A =+ + 2 3 y M EC zdA EC z dA EC yzdA + ( 3 ) 2 3 z M EC ydA EC yzdA EC y dA +
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5 5.2 Centroidal Bending Axes and Principal Axes Centrodal Bending Axes : Centroidal bending (CB) axes are defined such that the following integrals are identically zero. 0 0 ydA zdA = = ( 1 ) Now let’s define the following cross-sectional properties: 2 y I zdA = : area moment of inertia around the y axis 2 z I ydA = : area moment of inertia around the z axis yz I yzdA = : area product of inertia Ad A = : cross-sectional area Then from equation (3) of the previous section 1 F EAC = 23 yy y z M EI C EI C = + zy z z M EI C EI C =+ ( 2 ) Then from equation (2), 1 F C EA = 2 2 1 () zy y zz yz y z CI M I M EII I =− 3 2 1 yz y y z y z M I M I + (3) Note : For CB axes, 1) Axial force induces 1 C only. It does not induce 2 C or 3 C .
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Chapter 5_Bending - CHAPTER 5 BENDING OF THIN-WALLED...

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