T ORSION OF T HIN -W ALLED B EAMS
Chapter 16 Bending of open and closed,
17.1 General stress, strain and displacement relationships for open and single cell closed
We shall establish in this section the equations of equilibrium and expressions for strain which are necessary for the analysis of open section beams supporting shear loads and closed section beams carrying shear and torsional loads. The analysis of open section beams subjected to torsion requires a different approach and is discussed separately in Chapter 18. The relationships are established from first principles for the particular case of thin-walled sections in preference to the adaption of Eqs (1.6), (1.27) and (1.28) which refer to different coordinate axes; the form, however, will be seen to be the same.
Generally, in the analysis we assume that axial constraint effects are negligible, that the shear stresses normal to the beam surface may be neglected since they are zero at each surface and the wall is thin, that direct and shear stresses on planes normal to the beam surface are constant across the thickness, and finally that the beam is of uniform section so that the thickness may vary with distance around each section but is constant along the beam. In addition, we ignore squares and higher powers of the thickness t in the calculation of section properties (see Section 16.4.5).
The parameter s in the analysis is distance measured around the cross-section from some convenient origin.
An elementδs×δz×t of the beam wall is maintained in equilibrium by a system of direct and shear stresses as shown in Fig. 17.1(a). The direct stressσzis produced by bending moments or by the bending action of shear loads while the shear stresses are due
Fig. 17.1(a) General stress system on element of a closed or open section beam; (b) direct stress and shear flow system on the element.
to shear and/or torsion of a closed section beam or shear of an open section beam. The hoop stressσsis usually zero but may be caused, in closed section beams, by internal pressure. Although we have specified that t may vary with s, this variation is small for most thin-walled structures so that we may reasonably make the approximation that t is constant over the lengthδs. Also, from Eq. (1.4), we deduce thatτzs=τsz=τsay.
However, we shall find it convenient to work in terms of shear flow q, i.e. shear force per unit length rather than in terms of shear stress. Hence, in Fig. 17.1(b)
q=τt (17.1)
and is regarded as being positive in the direction of increasing s.
For equilibrium of the element in the z direction and neglecting body forces (see Section 1.2)
σz+∂σz
∂zδz
tδs−σztδs+
q+∂q
∂sδs
δz−qδz=0 which reduces to
∂q
∂s +t∂σz
∂z =0 (17.2)
Similarly for equilibrium in the s direction
∂q
∂z +t∂σs
∂s =0 (17.3)
The direct stressesσzandσsproduce direct strainsεzandεs, while the shear stress τinduces a shear strainγ(=γzs=γsz). We shall now proceed to express these strains in terms of the three components of the displacement of a point in the section wall (see Fig. 17.2). Of these componentsvt is a tangential displacement in the xy plane and is taken to be positive in the direction of increasing s;vnis a normal displacement in the xy plane and is positive outwards; and w is an axial displacement which has been defined previously in Section 16.2.1. Immediately, from the third of Eqs (1.18), we have
εz= ∂w
∂z (17.4)
17.1 General stress, strain and displacement relationships 505
Fig. 17.2Axial, tangential and normal components of displacement of a point in the beam wall.
Fig. 17.3Determination of shear strainγin terms of tangential and axial components of displacement.
It is possible to derive a simple expression for the direct strainεsin terms ofvt,vn, s and the curvature 1/r in the xy plane of the beam wall. However, as we do not require εsin the subsequent analysis we shall, for brevity, merely quote the expression
εs= ∂vt
∂s + vn
r (17.5)
The shear strainγis found in terms of the displacements w andvt by considering the shear distortion of an elementδs×δz of the beam wall. From Fig. 17.3 we see that the shear strain is given by
γ=φ1+φ2 or, in the limit as bothδs andδz tend to zero
γ = ∂w
∂s +∂vt
∂z (17.6)
In addition to the assumptions specified in the earlier part of this section, we further assume that during any displacement the shape of the beam cross-section is maintained
by a system of closely spaced diaphragms which are rigid in their own plane but are perfectly flexible normal to their own plane (CSRD assumption). There is, therefore, no resistance to axial displacement w and the cross-section moves as a rigid body in its own plane, the displacement of any point being completely specified by translations u andvand a rotationθ(see Fig. 17.4).
At first sight this appears to be a rather sweeping assumption but, for aircraft structures of the thin shell type described in Chapter 12 whose cross-sections are stiffened by ribs or frames positioned at frequent intervals along their lengths, it is a reasonable approximation for the actual behaviour of such sections. The tangential displacement vtof any point N in the wall of either an open or closed section beam is seen from Fig.
17.4 to be
vt =pθ+u cosψ+vsinψ (17.7) where clearly u,vandθare functions of z only (w may be a function of z and s).
The origin O of the axes in Fig. 17.4 has been chosen arbitrarily and the axes suffer displacements u,vandθ. These displacements, in a loading case such as pure torsion, are equivalent to a pure rotation about some point R(xR,yR) in the cross-section where R is the centre of twist. Therefore, in Fig. 17.4
vt =pRθ (17.8)
and
pR=p−xRsinψ+yRcosψ which gives
vt =pθ−xRθsinψ+yRθcosψ
Fig. 17.4Establishment of displacement relationships and position of centre of twist of beam (open or closed).