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

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The displacement in direction F D is thenδ D = 41 2 2 2( − ν ) qDa b ( a+b)3 2 2= q D I w3 /(tK) , (44)Et ( a + b )from which the spring constantp 2 = q D /δ D = tK/I w3 (45)The twisting component q T of the corner load does not bend the walls, but it has a componentbending the beam. So the distorting component has the direction q D . The spring constantcalculated in direction q D is valid in direction v, the w-component of q D − p 2 δ D constitutesthe reaction force in the lateral support.The bending load q v (x) and the twisting moment load m 5 (x) are obtained form the cornerforces:q v (x) = q D (x)cos(α − β) − q T (x)sin(α − β) (46)q w (x) = q D (x)sin(α − β) + q T (x)cos(α − β) (47)m 5 (x) = q v (x)d 1 − q w (x)2h = − q D (x)d 1 cos(α − β) − q T (x)2h (48)Normal stresses are obtained from the bending moment of the solving system.σ x( xs , )DiscussionM( x) M ( x) ( y03 )(49)=w5wy =53y −Iw3Iw3The idea of the theory of guided beams is the fact that for thin-walled beams, the deformationsof which are restricted by external continuous lateral or longitudinal restraints, themutually independent effective force quantities, which are determined by only externalloads, are defined in a solving system of axes and special points, which does not coincidewith the fundamental system with the origin at the centroid, the pole at the shear centre,and the axes parallel to the (central) principal directions. Besides they can be solved, theyeven ought to be solved in the solving system, if they are solved right. The choice of thecorrect system is analogous to the separating correctly a group of mutually dependent differentialequations to mutually independent equations.The group of equations (33) is a model of a Vlasov BEF beam. It has solutions in elementaryfunctions, if the load deviations are simple enough functions of x. Maybe this makes itpossible to develop for a thin-walled rectangle beam an exact stiffness matrix includingthe torsional and distortional degrees of freedom, which are mutually coupled, because theline of influence of the spring reaction does not pass through the guided torsional axis A 5of the segment cross-section. If d 1 = 0, as for a square cross-section the wall thickness tbeing constant, the group is separated into two separate differential equations.The theory of guided beams is now extended to the field of closed thin-walled crosssections.The cases with two or several non-compressible axes constitute an intermediatelink between the theories of open and closed thin-walled cross-sections.
References:/1/ Koivula Risto: Avoprofiilisen sauvan ohjattu vääntö (Guided torsion of Vlasovbeams, in Finnish), Rakenteiden mekaniikka, Vol. 18 No 4 1985, pp 57 - 94/2/ Koivula Risto: Transformations of force and displacement quantities between differentsystems of axes and special points, Proceeding of 2 nd International Conference of Thin-Walled Structures, Singapore 1998/3/ Kähönen, A., Niemi, E.: Distortion of a double symmetric box section subjected toeccentric loading using the beam on elastic foundation approach, Lappeenranta Universityof Technology 1986
Page 1 and 2: ANALYSIS OF DISTORTION OF A THIN-WA
Page 3 and 4: Our basic element is an angle cross
Page 5 and 6: q D (x)ba y5 − a y0v 1 w ~ w z z3
Page 7: surface or volume loadings:∂σx(

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

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