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

surface or volume loadings:∂σx( xs , ) ∂[ ts ( ) τxs( xs , )]ts () + = 0 (29)∂x∂sThe axial equilibrium condition of a beam element dx with equation (27) givesEI w3 v 5 (4) (x) = M w5´´(x) = F v (x) = − q v (x) (30)The presupposition of eq. (30) to be valid is, that at the edges either the shear stressesτ xs (s) or the unit warping y(s) − y 03 (in terms of the generalized beam theory of R. Schardt)is zero at the edges. Then this force quantity of the solving system is determined by externalloadings only. For the StVenant torsionMu( x)= GI t θ u´(x) (31)GI t θ u´´(x) = Mx (32)uWhen to this system at point A 4 is applied a spring resisting the deflection v 4 (x) = v 5 (x) +d 1 θ u (x) with a spring constant p 2 , for displacement quantities v 5 (x) ja θ u (x) is obtained acoupled group of differential equationsEI w3 v 5(4)(x) = q v (x) − p 2 [v 5 (x) + d 1 θ u (x)] (33)− GI t θ u´´(x) = m 5 (x) − p 2 d 1 [v 5 (x) + d 1 θ u (x)] , eli (34)4 2 2⎡EI D p p d v xw3+1⎤ 5⎢ 22 2 2 ⎥ ⎧ ( ) ⎫⎨ ⎬⎭ =⎣ pd GID pd x1−t+1 ⎦ ⎩ θu( )d 1 = a − ba+b cThe determinant of the differentiation matrix is⎧qv ( x)⎫⎨ ⎬⎩m5( x)⎭(Ei w3 D 4 + p 2 )(− GI t D 2 + p 2 d 2 1 ) − pd4 2= − D 2 (EI1w3 GI t D 4 − EI w3 p 2 d 2 1 D 2 + GI t p 2 ) (37)D 2 (D 4 − q 2 d 2 1 D 2 + q 2 k 2 ) = 0 (38)k 2 = GI2 2 2 22t 1 4ta b 3( a + b )=2 2EI 21 ( + ν)a+b ab( a+b )= 6 c2(39)1+ ν ( a + b)w322pq = (40)GI tThe roots of equation (38) are2 2 4 4 2 2D 1 = D 2 = 0 , D 3,4,5,6 = ± qd ± qd −4 kq / 2 (41)1The spring constant can be calculated for instance using the Mohr integrals. At the cornerA of the cross-section the plate bending moment in obtained fromm sA (x) = q D (x)ab/c (42)The plate stiffness constant of the wall is1(35)(36)K =Et 3 212( 1− ν )(43)