Aircraft Structures 3E Episode 10 pdf

Determine and sketch the distribution of direct stress, according to the basic theory of bending, along the length of the beam for the points 1 and 2 of the cross-section.. Show that the

Derive expressions for the vertical and horizontal components of the deflection of the beam midway between the supports B and D The wall thickness t and Young’s modulus E are constant throughout

Ans u = 0.186W13/Ea3t, v = 0.177Wl3/Ea3t

P.9.3 A uniform beam of arbitrary, unsymmetrical cross-section and length 21 is built-in at one end and simply supported in the vertical direction at a point half-way along its length This support, however, allows the beam to deflect freely in the horizontal x direction (Fig P.9.3)

For a vertical load W applied at the free end of the beam, calculate and draw the bending moment diagram, putting in the principal values

Ans Mc = 0, MB = WI, M A = - W1/2 Linear distribution

Yt

w l

Fig P.9.3

P.9.4 A beam, simply supported at each end, has a thin-walled cross-section

shown in Fig P.9.4 If a uniformly distributed loading of intensity w/unit length acts on the beam in the plane of the lower, horizontal flange, calculate the maximum

’t

per unitlength

Fig P.9.4

Problems 347

direct stress due to bending of the beam and show diagrammatically the distribution

of the stress at the section where the maximum occurs

The thickness t is to be taken as small in comparison with the other cross-sectional

dimensions in calculating the section properties Ixx, Iyy and Ixy

Ans uZ-;- = uz,3 = 13w12/384a2t, ui:l = w12/96a2t,

uzT2 =-w12/48a2t

P.9.5 A thin-walled cantilever with walls of constant thickness t has the cross-

section shown in Fig P.9.5 It is loaded by a vertical force W at the tip and a

horizontal force 2W at the mid-section, both forces acting through the shear

centre Determine and sketch the distribution of direct stress, according to the

basic theory of bending, along the length of the beam for the points 1 and 2 of the

cross-section

The wall thickness t can be taken as very small in comparison with d in calculating

the sectional properties I,,, Ixy etc

Ans ui:] (mid-point) = -0.05 WZ/td2,

uiF2 (mid-point) = -0.63 Wl/td2,

uz,l (built-in end) = -1.85 Wl/td2 (built-in end) = 0.1 Wl/td2

Fig P.9.5

P.9.6 A uniform cantilever of arbitrary cross-section and length I has section

properties, JYX, Iyy and lYy with respect to the centroidal axes shown in Fig P.9.6

It is loaded in the vertical (yz) plane with a uniformly distributed load of intensity

wlunit length The tip of the beam is hinged to a horizontal link which constrains

it to move in the vertical direction only (provided that the actual deflections

are small) Assuming that the link is rigid, and that there are no twisting effects:

calculate:

(a) the force in the link;

(b) the deflection of the tip of the beam

Ans (a) 3wZIxy/81xx; (b) w f / 8 E I x x

Fig P.9.6

P.9.7 A thin-walled cantilever has a constant cross-section of uniform thickness with the dimensions shown in Fig P.9.7 It is subjected to a system of point loads acting in the planes of the walls of the section in the directions shown

Calculate the direct stresses according to the basic theory of bending at the points 1 ,

2 and 3 of the cross-section at the built-in end and half-way along the beam Illustrate your answer by means of a suitable sketch

The thickness is to be taken as small in comparison with the other cross-sectional dimensions in calculating the section properties Ixx, IxY etc

Ans At built-in end, u2,] = -11.4N/mm2, uz,2 = -18.9N/mmZ,

the maximum direct stress for a bending moment M , = 3.5Nm applied about the

horizontal axis Cx Take r = 5 mm, t = 0.64mm

Ans a = 51.9", a,,,,, = 101 N/mm2

Problems 349

Fig P.9.8

P.9.9 A beam has the singly symmetrical, thin-walled cross-section shown in Fig

P.9.9 The thickness t of the walls is constant throughout Show that the distance of

the shear centre from the web is given by

P.9.10 A beam has the singly symmetrical, thin-walled cross-section shown in

Fig P.9.10 Each wall of the section is flat and has the same length a and thickness

t Calculate the distance of the shear centre from the point 3

Ans 5a cos a/8

P.9.11 Determine the position of the shear centre S for the thin-walled, open

cross-section shown in Fig P.9.11 The thickness t is constant

Ans m / 3

Fig P.9.11

P.9.12 Figure P.9.12 shows the cross-section of a thin, singly symmetrical

I-section Show that the distance ts of the shear centre from the vertical web is given by

-

Es - 3 P U - PI

d - (1 + 12p) where p = d / h The thickness tis taken to be neghgibly small in comparison with the

other dimensions

Problems 351

Fig P.9.12

P.9.13 A thin-walled beam has the cross-section shown in Fig P.9.13 The thick-

ness of each flange varies linearly from tl at the tip to t2 at the junction with the web

The web itself has a constant thickness t 3 Calculate the distance Es from the web to

the shear centre S

A ~ S d 2 ( 2 t l + l2)/[3d(tl + t.) + ht3]

Fig P.9.13

P.9.14 Figure P.9.14 shows the singly symmetrical cross-section of a thin-walled

open section beam of constant wall thickness t , which has a narrow longitudinal slit at

the corner 15

4

Fig P.9.14

Calculate and sketch the distribution of shear flow due to a vertical shear force S,

acting through the shear centre S and note the principal values Show also that the

distance & of the shear centre from the nose of the section is tS = 1/2( 1 + a/b)

A m q2 = q4 = 3bSY/2h(b + a), q3 = 3SY/2h Parabolic distributions

P.9.15 Show that the position of the shear centre S with respect to the intersection

of the web and lower flange of the thin-walled section shown in Fig P.9.15, is given

by

5's = -45a/97, 7s = 46a/97

Fig P.9.15

P.9.16 Figure P.9.16 shows the regular hexagonal cross-section of a thin-walled

beam of sides a and constant wall thickness t The beam is subjected to a transverse shear force S , its line of action being along a side of the hexagon, as shown Find the rate of twist of the beam in terms oft, a, S and the shear modulus G Plot the shear flow distribution around the section, with values in terms of S and a

Fig P.9.16

Parabolic distributions, q positive clockwise

P.9.17 Figure P.9.17 shows the cross-section of a single cell, thin-walled beam

with a horizontal axis of symmetry The direct stresses are carried by the booms B1

to B4, while the walls are effective only in carrying shear stresses Assuming that

the basic theory of bending is applicable, calculate the position of the shear centre

S The shear modulus G is the same for all walls

Cell area = 135000mm2 Boom areas: B1 = B4 = 450mm 2 , B2 = B3 = 550mm 2

Wall Length (mm) Thickness (mm)

100 mm

0.8 mm

500 mm

Fig P.9.17

P.9.18 A thin-walled closed section beam of constant wall thickness t has the

cross-section shown in Fig P.9.18

Fig P.9.18

Assuming that the direct stresses are distributed according to the basic theory of bending, calculate and sketch the shear flow distribution for a vertical shear force

S,, applied tangentially to the curved part of the beam

Ans qol = S,,( 1.61 cos 8 - 0.80)/r

P.9.19 A uniform thin-walled beam of constant wall thickness t has a cross- section in the shape of an isosceles triangle and is loaded with a vertical shear force

Sy applied at the apex Assuming that the distribution of shear stress is according

to the basic theory of bending, calculate the distribution of shear flow over the cross-section

Illustrate your answer with a suitable sketch, marking in carefully with arrows the direction of the shear flows and noting the principal values

Problems 355

I

2 5 0 mm

P.9.21 A uniform, thin-walled, cantilever beam of closed rectangular cross-

section has the dimensions shown in Fig P.9.21 The shear modulus G of the top

and bottom covers of the beam is 18 000 N/mm2 while that of the vertical webs is

26 000 N / m '

The beam is subjected to a uniformly distributed torque of 20 Nm/mm along its

length Calculate the maximum shear stress according to the Bredt-Batho theory

of torsion Calculate also, and sketch, the distribution of twist along the length of

the cantilever assuming that axial constraint effects are negligible

P.9.22 A single cell, thin-walled beam with the double trapezoidal cross-section

shown in Fig P.9.22, is subjected to a constant torque T = 90 500 N m and is con-

strained to twist about an a x i s through the point R Assuming that the shear stresses

are distributed according to the Bredt-Batho theory of torsion, calculate the distribu-

tion of warping around the cross-section

Illustrate your answer clearly by means of a sketch and insert the principal values of

the warping displacements

The shear modulus G = 27 500 N/mm2 and is constant throughout

AFZS Wi = -Wg = - 0 5 3 m , W 2 = -W5 = O.O5mm, W3 = -W4 = 0 3 8 m

Linear distribution

P.9.23 A uniform thin-walled beam is circular in cross-section and has a constant

thickness of 2.5 mm The beam is 2000 mm long, carrying end torques of 450 N m and,

in the same sense, a distributed torque loading of 1 .O N m/mm The loads are reacted

by equal couples R at sections 500 mm distant from each end (Fig P.9.23)

Calculate the maximum shear stress in the beam and sketch the distribution of twist along its length Take G = 30 000 N/mm2 and neglect axial constraint effects

twist about a longitudinal axis through the centre C of the semicircular arc 12 For

the curved wall 12 the thickness is 2 mm and the shear modulus is 22 000 N/mm2

For the plane walls 23, 34 and 41, the corresponding figures are 1.6mm and

27 500 N/mm2 (Note: Gt = constant.)

Calculate the rate of twist in radians/mm Give a sketch illustrating the distribution

of warping displacement in the cross-section and quote values at points 1 and 4

Problems 357

Fig P.9.24

A m de/& = 29.3 x rad/mm, w 3 = -w4 = -0.19 mm,

w z = - ~1 = - 0 0 5 6 m P.9.25 A uniform beam with the doubly symmetrical cross-section shown in Fig

P.9.25, has horizontal and vertical walls made of different materials which have shear

moduli G , and Gb respectively If for any material the ratio mass density/shear

modulus is constant find the ratio of the wall thicknesses tu and tb, so that for a

given torsional stiffness and given dimensions a, b the beam has minimum weight

per unit span Assume the Bredt-Batho theory of torsion is valid

If this thickness requirement is satisfied find the a / b ratio (previously regarded as

fixed), which gives minimum weight for given torsional stiffness

Ans tb/ta = Gu/Gb, b / a = 1

Fig P.9.25

P.9.26 Figure P.9.26 shows the cross-section of a thin-walled beam in the form of

a channel with lipped flanges The lips are of constant thickness 1.27 mm while the

flanges increase linearly in thickness from 1.27mm where they meet the lips to

2.54mm at their junctions with the web The web has a constant thickness of

2.54 mm The shear modulus G is 26 700 N/mmz throughout

The beam has an enforced axis of twist RR' and is supported in such a way that

warping occurs freely but is zero at the mid-point of the web If the beam carries a

torque of 100Nm, calculate the maximum shear stress according to the St Venant

h s Tma = f 2 9 7 4 N / m 2 , W1 = - 5 4 8 m = -Wg,

w 2 = 5.48mm = -w5, w 3 = 17.98mm = -w4 P.9.27 The thin-walled section shown in Fig P.9.27 is symmetrical about the x

axis The thickness to of the centre web 34 is constant, while the thickness of the

other walls varies linearly from to at points 3 and 4 to zero at the open ends 1, 6, 7 and 8

Determine the St Venant torsion constant J for the section and also the maximum value of the shear stress due to a torque T If the section is constrained to twist about

an axis through the origin 0, plot the relative warping displacements of the section per unit rate of twist

Problems 359

1

X

6

P.9.28 A uniform beam with the cross-section shown in Fig P.9.28(a) is sup-

ported and loaded as shown in Fig P.9.28(b) If the direct and shear stresses are

given by the basic theory of bending, the direct stresses being carried by the booms

and the shear stresses by the walls, calculate the vertical deflection at the ends of

the beam when the loads act through the shear centres of the end cross-sections,

allowing for the effect of shear strains

Problems 361

faces of the wedge Find the vertical deflection of point A due to this given loading

If G = 0 4 E , t/c=0.05 and L = 2 c show that this deflection is approximately

5600p0c2/Et0

P.9.30 A rectangular section thin-walled beam of length L and breadth 3b, depth

b and wall thickness t is built in at one end (Fig P.9.30) The upper surface of the

beam is subjected to a pressure which vanes linearly across the breadth from a

value p o at edge AB to zero at edge CD Thus, at any given value of x the pressure

is constant in the z direction Find the vertical deflection of point A

Stress analysis of aircraft components

In Chapter 9 we established the basic theory for the analysis of open and closed section thin-walled beams subjected to bending, shear and torsional loads In addi- tion, methods of idealizing stringer stiffened sections into sections more amenable

to analysis were presented We now extend the analysis to actual aircraft components including tapered beams, fuselages, wings, frames and ribs; also included are the effects of cut-outs in wings and fuselages Finally, an introduction is given to the analysis of components fabricated from composite materials

Aircraft structural components are, as we saw in Chapter 7, complex, consisting usually of thin sheets of metal stiffened by arrangements of stringers These structures are highly redundant and require some degree of simplification or idealization before they can be analysed The analysis presented here is therefore approximate and the degree of accuracy obtained depends on the number of simplifying assumptions made A further complication arises in that factors such as warping restraint, structural and loading discontinuities and shear lag significantly affect the analysis;

we shall investigate these effects in some simple structural components in Chapter

11 Generally, a high degree of accuracy can only be obtained by using computer- based techniques such as the finite element method (see Chapter 12) However, the simpler, quicker and cheaper approximate methods can be used to advantage in the preliminary stages of design when several possible structural alternatives are being investigated; they also provide an insight into the physical behaviour of structures which computer-based techniques do not

Major aircraft structural components such as wings and fuselages are usually tapered along their lengths for greater structural efficiency Thus, wing sections are reduced both chordwise and in depth along the wing span towards the tip and fuselage sections aft of the passenger cabin taper to provide a more efficient aerodynamic and structural shape

The analysis of open and closed section beams presented in Chapter 9 assumes that the beam sections are uniform The effect of taper on the prediction of direct stresses produced by bending is minimal if the taper is small and the section properties are

10.1 Tapered beams 363

calculated at the particular section being considered; Eqs (9.6)-(9.10) may therefore

be used with reasonable accuracy On the other hand, the calculation of shear stresses

in beam webs can be significantly affected by taper

Consider first the simple case of a beam positioned in the y z plane and comprising two

flanges and a web; an elemental length Sz of the beam is shown in Fig 10.1 At the

section z the beam is subjected to a positive bending moment M y and a positive

shear force Sy The bending moment resultants Pz,l and P3:2 are parallel to the z

axis of the beam For a beam in which the flanges are assumed to resist all the

direct stresses, Pz,l = M x / h and Pz,2 = - M x / h In the case where the web is assumed

to be fully effective in resisting direct stress, P Z ; ~ and PQ are determined by multiply-

ing the direct stresses oZ,] and found using Eq (9.6) or Eq (9.7) by the flange areas

B1 and B2 PZ,] and Pz,2 are the components in the z direction of the axial loads PI and

P2 in the flanges These have components Py,l and Py,2 parallel to the y axis given by

Again we note that Sy2 in Eqs (10.4) and (10.5) is negative Equation (10.5) may be

used to determine the shear flow distribution in the web For a completely idealized beam the web shear flow is constant through the depth and is given by Sy,,/h For

a beam in which the web is fully effective in resisting direct stresses the web shear flow distribution is found using Eq (9.75) in which Sy is replaced by SY,+,, and which, for the beam of Fig 10.1 , would simplify to

10.1 Tapered beams 365

2 mm and is fully effective in resisting direct stress The beam tapers symmetrically

about its horizontal centroidal axis and the cross-sectional area of each flange is

400 mm2

The internal bending moment and shear load at the section A A produced by the

externally applied load are, respectively

M x = 20 x 1 = 20kNm, S, = -2OkN The direct stresses parallel to the z axis in the flanges at this section are obtained either

from Eq (9.6) or Eq (9.7) in which M,, = 0 and Zx, = 0 Thus, from Eq (9.6)

Zxx = 22.5 x 1 0 6 m 4 Hence

The components parallel to the z axis of the axial loads in the flanges are therefore

sz 2 x 103 - -0.05, - 6, - 2 - x 103 Hence

S,.:w, = -20 x lo3 + 53 320 x 0.05 + 53 320 x 0.05 = -14668N

The shear flow distribution in the web follows either from Eq (10.6) or Eq (10.7) and

is (see Fig 10.2(b))

412 = 22.5 14''' x lo6 ([q150-s)ds+400 x 150 i.e

412 = 6.52 x + 300s + 60000) (ii)

The maximum value of q12 occurs when s = 150mm and q12 (max) = 53.8 N/mm

The values of shear flow at points 1 (s = 0) and 2 (s = 300mm) are q1 = 39.1 N/mm