ANALYSIS OF DISTORTION OF A THIN-WALLED ... - students.tut.fi

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q Dq DvA P 1P 2 B P 4P 3q Dq DFigure 1. Distortion of a thin-walled rectangular cross-sectionThe mid-axes P i of the webs are non-compressible hinges. The hinged “planesupports“ BP i are rigid. The “deflection“ v of the part beam is normal to the line P 1 P 2 .Angle cross-section, guided by non-compressible axes at the edges
Our basic element is an angle cross-section, the deformation of which is restricted by anon-compressible axis, a continuous axial support at both edges. Its degrees of freedom,until further notice neglecting the shear deformation, are bending around the line P 1 P 2 , thedeflection being normal to this line, and the angle of twist around the torsional axis, thecorner axis A. The indices refer to corresponding systems of axes and special points. As tothe transformations between these systems I refer to the reference /1/.v 1 ,v 2 w,w 3~w z 3v,v 3h~v b/2 z P 1 αz 1 ,w 1 ,w 2A,O 1~ zy 01y 01 = a 2 /(a + b)/2 = S w1 /A (1)a/2 O z 2φ 0 z 01 = b 2 /(a + b)/2 = S v1 /A (2)t~ y2 2O 3 c = a + b(3)y 2 y, y 3 aβ = arctan(b/a) = arcsin(b/c)= arccos(a/c) = π/2 − α (4)βα = arcsin(a/c) = arccos(b/c)(5)P 2 h = bsinα = acosα = ab/c (6)z 01by 1Figure 2. An angle cross-section with non-compressible axes at the edgesThe displacement and force quantities like deflections v(x) and w(x), rotations θ v (x) =− w′( x) = − dw(x)/dx and θ v (x) = v´(x), bending moments and shear forces are given in aright-handed (anti-clockwise) system of axes, but the measure quantities like x, y, z and thesectorial area ω(s) in a left-handed (clockwise) system. Then some important matrices aresymmetric, and the force and displacement quantity “vectors“ {F} and {u} are in the sametime without minus signs.In <strong>fi</strong>gure 2 <strong>fi</strong>ve systems of axes and special points are shown. Each of them has the sectorialpole at point A.- AO 1 xy 1 z 1 , the axes of which coincide with the mid-lines of the webs, is a generic system,in which the cross-section integrals are easy to calculate.- AOxy 2 z 2 is a parallel to the web mid-lines centroidal system- AOx ~~ yz is the fundamental system with the origin at the centroid and axes parallel to theprincipal directions. In this case it has not much use.- AOxyz is a centric system with the axes parallel to those of the solving system. Theguided (eccentric) special points are mostly determined using this system.
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Page 5 and 6: q D (x)ba y5 − a y0v 1 w ~ w z z3
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Page 9: References:/1/ Koivula Risto: Avopr

ANALYSIS OF DISTORTION OF A THIN-WALLED ... - students.tut.fi

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