Theory of Structures - Theory of Structures Mohrs circle...

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Theory of Structures Prepared by: Amr A. El-Sayed, Civil Eng. Dept., El-Minia Univ., Eg. 1 Mohr’s circle for principle moment of inertia: (Ix+Iy)/2 Ix I x y Ix (Ix+Iy)/2 x u x u Ix > Iy & Ixy is positive Ix > Iy & Ixy is negative ( + ) / 2 Ixy Iy > Ix & Ixy is negative Iy > Ix & Ixy is positive Ixy u -ve +ve u φ u x v y y x u v y y
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Theory of Structures Prepared by: Amr A. El-Sayed, Civil Eng. Dept., El-Minia Univ., Eg. 2 General form of Normal Stress : For symmetrical section, I xy = 0, but for general cases, if the cross section is unsymmetrical, the product of inertia I xy will have a value. The general equation of stress is: Y I I I I M I M X I I I I M I M A N XY Y X XY Y Y X XY Y X XY X X Y 2 2 + + = σ Y I I I I e N I e N X I I I I e N I e N A N XY Y X XY X Y Y XY Y X XY Y X X 2 2 ) . ( ) . ( ) . ( ) . ( + + = [ ] [ ] Y I I I I e I e N X I I I I e I e N A N XY Y X XY X Y Y XY Y X XY Y X X 2 2 . . . . + + = [] [ ] + + = Y I I I I e I e A X I I I I e I e A A N XY Y X XY X Y Y XY Y X XY Y X X 2 2 . . . . 1 For the neutral axis, the stresses are equal to zero; i.e σ = 0.0 [ ] 0 . . . . 1 2 2 = + + Y I I I I e I e A X I I I I e I e A XY Y X XY X Y Y XY Y X XY Y X X To find the intersection of the N.A. with X-axis, put Y=0 1 . . 2 = X I I I I e I e A XY Y X XY Y X X XY Y X X Y X XY I e I e A I I I X . . 2 = ………. (1-1) XY X Y Y Y X XY I e I e A I I I Y . . 2 = ………. (1-2) Equations (1-1) and (1-2) give the intersection of the N.A. with the X and Y axis respectively. By analyzing those two equations, one can notice that the position of the N.A does not depend on the magnitude of the force, but on its position “e x & e y ”.
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Theory of Structures Prepared by: Amr A. El-Sayed, Civil Eng. Dept., El-Minia Univ., Eg. 3 If we but I XY = 0.0 in both equations “for the case of sections have an axis of symmetry”, equations (1-1) and (1-2) will be reduced to X y X Y e i e A I X 2 . = = ………. (1-3) Y x Y X e i e A I Y 2 . = = ………. (1-4) Equations (1-3) and (1-4) are valid for sections having one ore more axis of symmetry. As y x i i & are constants, the position of the N.A depends only on the eccentricity of the force. The N.A lies in the opposite quarter of the applied force, as shown in the following figure. N . A Force Force The previous fact is valid for shapes with one or more axes of symmetry, thus the two perpendicular axes X and Y though the C.G of the section are the principal axes, and the product moment of inertia I xy = 0.0 For unsymmetrical shape, such as L-sections, the N.A may not lie in the opposite quarter to the external force.
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Theory of Structures Prepared by: Amr A. El-Sayed, Civil Eng. Dept., El-Minia Univ., Eg. 4 EX. (1) 4.0 3 . 0 7 5.0 5.0 Area = 12 * 2 * 2 + 12 * 2 = 72.0 Cm 2 . I x = 2656.0 2 * 7 * ) 2 * 12 ( 12 12 * 2 2 * 12 2 * 12 2 3 3 + + = Cm 4 . I y = 1784.0 = 2 * 5 * ) 12 * 2 ( 12 2 * 12 2 * 12 12 * 2 2 3 3 + + Cm 4 . I xy = (2*12)*5*(-7) + (2*12)*(-5)*7 = -1680.0 Cm 4 .
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Theory of Structures - Theory of Structures Mohrs circle...

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