Aircraft structures for engineering students - part 6 potx

9.2 General stress, strain and displacement relationships 291

3

Fig. 9.13 Distribution of direct stress in Z-section beam of Example 9.3.

deform the beam section into a shallow, inverted 's' (see Section 2.6). However, shear

stresses in beams whose cross-sectional dimensions are small in relation to their

lengths are comparatively low so that the basic theory of bending may be used

with reasonable accuracy.

In thin-walled sections shear stresses produced by shear loads are not small and

must be calculated, although the direct stresses may still be obtained from the basic

theory of bending so long as axial constraint stresses are absent; this effect is discussed

in Chapter 1 1. Deflections in thin-walled structures are assumed to result primarily

from bending strains; the contribution of shear strains may be calculated separately

if required.

e 6 Istress, ^st r a i'n an d-dEplace me nt re la t i o ns h i ps

for open and single cell closed section thin-walled

beams

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 Section 9.6. 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 constants.

292 Open and closed, thin-walled beams

(a) (bl

Fig. 9.14 (a) General stress system on element of a closed or open section beam; (b) direct stress and shear

flow system on the element.

The parameter s in the analysis is distance measured around the cross-section from

some convenient origin.

An element 6s x 6z x t of the beam wall is maintained in equilibrium by a system of

direct and shear stresses as shown in Fig. 9.14(a). The direct stress a, is produced by

bending moments or by the bending action of shear loads while the shear stresses are

due to shear and/or torsion of a closed section beam or shear of an open section beam.

The hoop stress us is usually zero but may be caused, in closed section beams, by inter￾nal 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 length 6s. Also, from Eqs (1.4), we deduce that

rrs = rsz = r say. 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. 9.14(b)

q = rt (9.21)

For equilibrium of the element in the z direction and neglecting body forces (see

and is regarded as being positive in the direction of increasing s.

Section 1.2)

(a, +z6r)*6s - azt6s + (2) q+-& sz - qsz = 0

which reduces to

a4 aaz

as az -+t-=O

Similarly for equilibrium in the s direction

(9.22)

(9.23)

The direct stresses a, and us produce direct strains E, and E,, while the shear stress r

induces a shear strain y(= T~~ = T,). 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. 9.15). Of these components v, is a tangential displacement in the xy plane and is

taken to be positive in the direction of increasing s; w,, is a normal displacement in the

9.2 General stress, strain and displacement relationships 293

X

z

Fig. 9.15 Axial, tangential and normal components of displacement of a point in the beam wall.

xy plane and is positive outwards; and w is an axial displacement which has been

defined previously in Section 9.1. Immediately, from the third of Eqs (1.18), we have

dW

az

& =- (9.24)

It is possible to derive a simple expression for the direct strain E, in terms of ut, wn, s

and the curvature 1/r in the xy plane of the beam wall. However, as we do not require

E, in the subsequent analysis we shall, for brevity, merely quote the expression

aV, vn & =-+-

as r

(9.25)

The shear strain y is found in terms of the displacements w and ut by considering the

shear distortion of an element 6s x Sz of the beam wall. From Fig. 9.16 we see that the

shear strain is given by

7 = 41 + 42

or, in the limit as both 6s and Sz tend to zero

(9.26)

Fig.

Distorted shape

of element due \---**-

to shear f:. 1 --._

L.

I .

4 -. -.._

9.16 Determination of shear strain y in terms of tangential and axial components of displacement.

294 Open and closed, thin-walled beams

Fig. 9.17 Establishment of displacement relationships and position of centre of twist of beam (open or

closed).

In addition to the assumptions specijied in the earlier part of this section, we further

assume that during any displacement the shape of the beam cross-section is main￾tained 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 and 21 and a rotation 6 (see Fig. 9.17).

At first sight this appears to be a rather sweeping assumption but, for aircraft struc￾tures of the thin shell type described in Chapter 7 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

vt of any point N in the wall of either an open or closed section beam is seen from Fig.

9.17 to be

v, = p6 + ucos $ + vsin $ (9.27)

where clearly u, w and B are functions of z only (w may be a function of z and s).

The origin 0 of the axes in Fig. 9.17 has been chosen arbitrarily and the axes suffer

displacements u, w and 0. 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. Thus, in Fig. 9.17

and

(9.28)

pR = p - xR sin 1(, + yR cos $

which gives

9.3 Shear of open section beams 295

and

dv, de de de - = p - - XR sin +- + yR cos +- dz dz dz dz

Also from Eq. (9.27)

de du dv . 3 = p- + -cos + -sm + dz dz dz dz

Comparing the coefficients of Eqs (9.29) and (9.30) we see that

dvldz duldz

dO/dz I YR =- dQ/dz

XR = --

(9.29)

(9.30)

(9.31)

The open section beam of arbitrary section shown in Fig. 9.18 supports shear loads S,

and Sy such that there is no twisting of the beam cross-section. For this condition to

be valid the shear loads must both pass through a particular point in the cross-section

known as the shear centre (see also Section 11.5).

Since there are no hoop stresses in the beam the shear flows and direct stresses

acting on an element of the beam wall are related by Eq. (9.22), i.e.

aq do, -+t-=o

as dz

We assume that the direct stresses are obtained with sufficient accuracy from basic

bending theory so that from Eq. (9.6)

-- acz - [(aM,/az)Ixx - (awaz)r.Xyl + [(aMx/wryy - (dMy/wx,l

dZ IxxI,, - I:, Ixxryy - I&

't

Fig. 9.18 Shear loading of open section beam.

296 Open and closed, thin-walled beams

Using the relationships of Eqs (9.11) and (9.12), i.e. aMy/az = S, etc., this expression

becomes

(SXZXX - SyZxy) + (SyZyy - SxZxy) Y -- -

az zx,zyy - z:y zxxzyy - Ey

Substituting for &Jaz in Eq. (9.22) gives

(9.32)

Integrating Eq. (9.32) with respect to s from some origin for s to any point around the

cross-section, we obtain

(9.33)

If the origin for s is taken at the open edge of the cross-section, then q = 0 when s = 0

and Eq. (9.33) becomes

For a section having either Cx or Cy as an axis of symmetry Zxy = 0 and Eq. (9.34)

reduces to

Example 9.4

Determine the shear flow distribution in the thin-walled 2-section shown in Fig. 9.19

due to a shear load Sy applied through the shear centre of the section.

-

2

Fig. 9.19 Shear-loaded Z-section of Example 9.4: