Chapter 1

– Review of Mechanics of Materials

1-1 Locate the centroid of the plane area shown

1-2 Determine the location of centroid of the composite area shown.

1-3 Verify that the radius of gyration for a circle of diameter d with respect to a

centroidal axis is d/4.

1-4 Determine the moment of inertia of the shaded area with respect to the x-

axis.

x

y

650 mm

1000 mm

650

300

300 m

m

150 mm

radius

300 mm

radius

300 m

m

600 mm

300 m

m

400 mm

400 m

m

400 m

m

200

400 m

m

600 mm

x

y

1-5 Determine the product moment of inertia of the triangle with respect to the

x and y axes.

1-6 Determine the product moment of inertia of the triangle in the previous

question with respect to the x’ and y’ axes. The centroid of the triangle is at

G.

Answers:

1-1 x = 762mm, y = 308 from the bottom left corner

1-2 x = 601mm, y = 300mm from bottom left corner

1-3 Hint: find the area moment of inertia and the area

1-4 0.0918m

4

1-5 (b

2

h

2

)/24

1-6 -(b

2

h

2

)/72

y

x

b

h

G

x’

y’

Chapter 2

– Basic Elasticity

2-1 The two dimensional stress state at a point of an element of a material is

given as shown.

Calculate (a) the axial and shear stress on a plane whose normal is 40

0

clockwise to the x-direction (b) the magnitude and directions of the

principal stresses and (c) the maximum shear stress.

2-2 A plane element is subjected to a constant axial stress of 50 MPa in the x-

direction and an axial stress varying from -50 MPa to 50 MPa in the y-

direction. Plot the maximum shear stress acting in the plane element with

respect to the axial stress in the y-direction. What is the largest shear stress

magnitude?

2-3 Determine the magnitude and directions of the principal strains and the

maximum shear strain on an element with the following strains:

ε

x

= 160 x 10

-6

;

ε

y

= -80 x 10

-6

;

γ

xy

= 120 x 10

-6

.

2-4 The principal strains have been found to be 0.000400 and -0.000050

respectively. Determine (a) the maximum shear strain and (b) the maximum

shear stress given that the shear modulus of elasticity is 26.3 GPa.

2-5 The element shown is subject to 50 MPa and 75 MPa compressive stresses

in the x and y directions respectively and a shear stress of unknown

magnitude but acting in the described sense. When this element is rotated

clockwise at 25

o

, the shear stress magnitude is equal but acts in the opposite

sense; while the axial stress magnitudes are unchanged. Determine the value

of the unknown shear stress.

75 MPa

30 MPa

40 MPa

50 MPa

75 MPa

unknown

2-6 The element shown is subject to an unknown axial stress in the x direction

and zero axial stress in the y direction. The shear stress is 30 MPa. When

this element is rotated around, the maximum shear stress recorded is 50

MPa. Determine (a) the axial stress in the x direction, and (b) the principal

stresses.

2-7 A pair of strain gages gave the following readings: with 0

o

gage = 500

microstrains, with 90

o

gage = –100 microstrains. The strain gages register

equal values after a 30

o

anti-clockwise rotation. Determine (a) the maximum

shear strain, and (b) the principal strains.

2-8 A beam of length l with a thin rectangular cross-section is clamped at the

end x = 0 and loaded at the tip with vertical force P. Show that the stress

distribution can be represented by

CyxxByAy ++=φ

33

Determine the coefficients A, B, and C.

2-9 The cantilever beam shown is in a state of plane strain and is rigidly

supported at x = L. Examine if the stress function given meets the

biharmonic equation and boundary conditions.

)2515(

20

5323222

3

yyhyxyxh

h

w

+−−=φ

unknown

0 MPa

30 MPa

Answers:

2-1

(a) 71 MPa -26 MPa -44.8 MPa (b) 88.5 MPa -43.5 MPa (c) 66 MPa

2-2

50 MPa when the axial stress = -50 MPa

2-3

176 x 10

-6

, -94 x 10

-6

, 272 x 10

-6

, 13

o

2-4

(a) 0.000450 (b) 11.84 MPa

2-5

5.83 MPa

2-6

(a) 80 MPa (b) 90 MPa, -10 MPa

2-7

(a) 680 microstrains (b) 540 microstrains, -140 microstrains

2-8

2Pl / td

3

, -2P/td

3

, 3P/2td

Chapter 3

– Principles of Aircraft Construction

3-1 The Ford Trimotor, nicknamed The Tin Goose, was a three engine civil

transport aircraft first produced in 1925 by Henry Ford and continued until

June 7, 1933. The structure of the plane consists of a truss-work of U-

shaped aluminum beams, with a thin skin of aluminum riveted on top, using

skin corrugations instead of wing ribs and fuselage stringers. Briefly discuss

the benefits and disadvantages with such a construction.

3-2 The Gossamer Albatross is a human-powered aircraft built by American

aeronautical engineer Paul B. MacCready. Briefly discuss the merits of the

external wire bracing construction used over truss-work or monocoque

construction.

3-3 Briefly explain why composite materials have led to huge advances in the

monocoque construction of aircrafts.

3-4 The double riveted joint shown connects two plates. If the failure strength of

the rivets in shear is 370 N/mm

2

, and the tensile strength of the plate is 465

N/mm

2

, determine the rivet pitch if the joint is to be designed so that failure

due to shear in the rivets and failure due to tension in the plate occur

simultaneously. Find also the joint efficiency.

Answers:

3-4 12mm, 75%

Chapter 4

– Airframe Loads

4-1 The aircraft shown weighs 135kN and has landed such that at the instant of

impact the ground reaction on each main undercarriage wheel is 200kN and

its vertical velocity is 3.5m/s. Find (i) the acceleration experienced.

Each undercarriage wheel weighs 2.25kN and is attached to a strut.

Calculate the (ii) axial load, and (iii) bending moment in the strut.

At section AA the wing outboard of this section weighs 6.6kN and the

center of gravity is 3.05m from AA. Calculate the (iv) shear force and (v)

bending moment at section AA.

4-2 An aircraft makes a correctly banked turn at radius 610m at a speed of

168m/s. Find (i) the angle of bank, and (ii) load factor.

After making the turn and restoring to symmetric flight, the figure shows the

relative positions of the center of gravity, aerodynamic center of the

complete aircraft less the tailplane, and the tailplane center of pressure at

zero lift incidence. The specifications are:

Weight (W) = 133,500N; Wing area (S) = 46.5m

2

; Wing mean cord (c) =

3m; C

D

= 0.01 + 0.05C

L

2

; C

M,O

= -0.03.

Find (iii) the lift coefficient, (iv) drag force, and (v) pitching moment. If the

change in lift coefficient per wing incidence is 4.5/rad. Determine (vi) the

tail load.

4-3 During pullout from a dive with zero thrust at 215m/s, an aircraft weighing

238,000N has the flight path at 40

o

to the horizontal with radius of curvature

1525m. The distance between the CG and tail is 12.2m. The angular

velocity of pitch is checked by applying an angular retardation of 0.25

rad/s

2

. The moment of inertia of the aircraft for pitching is 204,000 kgm

2

.

Find (i) the additional tail load required to check the angular velocity in

pitch.

The aircraft has wings 88.5m

2

in area, mean cord of 1m, and the pitching

moment coefficient for all parts excluding the tailplane through the CG is

given by C

M.CG

.c = 0.427C

L

– 0.061. Find (ii) the amount of lift, (iii) the lift

coefficient, and (iv) pitching moment, and (v) tail load. (Hint: neglect the

tail loads for the first approximation of lift, 2 iterations is sufficient)

Answers:

4-1

19.23m/s

2

193.3kN 29kNm (clockwise) 0.32m 19.5kN 59.6kNm

4-2

78.03

o

, 4.82, 0.80, 33,707N, -72,229Nm, 73,160N

4-3

4180N, 898779N, 0.359, 230880Nm, 18925N

Chapter 5

– Torsion of Solid Sections

5-1 Show that the stress function φ = k(r

2

– a

2

) is applicable to the solution of a

solid circular section bar of radius a. Determine the stress distributions τ

yz

,

τ

zx

in the bar in terms of the applied torque, the angle of twists dw/dx,

dw/dy, and warping of the cross section.

5-2 A torque T is applied on the section comprising narrow rectangular strips

shown. Determine (i) the torsional constant, (ii) the stress distributions τ

yz

,

τ

zx

, and (iii) the maximum shear stress.

Answers:

5-1

-2Tx/πa

4

, -2Ty/πa

4

, 0, 0, 0

5-2

3

)2(

3

tba +

,

dz

d

Gx

θ

2, 0,

2

)2(

3

tba

T

+

±

Chapter 6

– Bending of Thin-Walled Beams

6-1 A bending moment of 3000Nm is applied on the section shown at 30

o

to the

vertical y axis. The sense of the bending moment is such that its components

M

x

and M

y

both produce tension in the positive xy quadrant. Find the

distances of C from edges BC and AB. Deduce the point where the flexural

stress is maximum and calculate the amount.

6-2 A thin-walled cantilever beam of unsymmetrical cross-section supports the

shear forces at the free end of the section shown. Calculate the flexural

stress midway along A on the beam. It can be assumed that no twisting of

the beam occurs.

6-3 A thin walled beam has the cross-section shown. If the beam is subjected to

a bending moment Mx in the plane of web 23, calculate the distribution of

flexural stress in the beam cross section.

Answers:

6-1

25.9mm, 38.4mm, C, 63.3N/mm

2

6-2

194.7N/mm

2

6-3

Mx

th

z

2

1,

41.0

=σ,

Mx

th

z

2

2,

5.0−

=σ,

Mx

th

z

2

3,

5.0

=σ,

Mx

th

z

2

4,

04.0

=σ

Chapter 7

– Shear of Thin-Walled Beams

7-1 A beam has singly symmetrical thin-walled cross section shown. The

thickness of the walls is constant throughout. Show that the distance of the

shear centre from the web is given by

αρ+ρ+

ααρ

−=ξ

23

2

sin261

cossin

d

s

for ρ = d / h

7-2 A beam has singly symmetrical thin-walled cross section shown. Each wall

of the section is flat and has the same length a and thickness t. Calculate the

distance of the shear centre from point 3.

7-3 A uniform thin walled beam of thickness t has a cross-section in the shape

of an isosceles triangle. It is loaded by a vertical shear force S

y

applied at the

apex. Calculate the shear flow over the cross section.

Answers:

7-1

-2Tx/πa

4

, -2Ty/πa

4

, 0, 0, 0

7-2

⎥

⎦

⎤

⎢

⎣

⎡

+−

θ

a

y

a

x

x

dz

d

G

2

3

2

3

22

,

⎥

⎦

⎤

⎢

⎣

⎡

+

θ

−

a

xy

y

dz

d

G

3

,

dz

d

a

xy

θ

−

3

,

dz

d

a

y

a

x θ

⎥

⎦

⎤

⎢

⎣

⎡

+−

2

3

2

3

22

,

dz

d

yxy

a

θ

− )3(

2

1

23

7-3

)2(

)3/3(

2

1

12

dhh

dhdsS

q

y

+

−−

=,

)2(

)66(

2

2

2

2

2

23

dhh

hhssS

q

y

+

−+−

=

Chapter 8

– Virtual Work & Energy Methods

8-1 During a routine manufacturing operation, rod AB must acquire an elastic

strain energy of 12 J. Determine the yield strength of the steel if the factor

of safety = 5 and E = 200 GPa.

8-2 Evaluate the strain energy of the prismatic beam for the loading shown.

8-3 The element shown is taken from part of a bar subjected to axial stresses

in x and y axis. The shear stress is zero. Find the strain energy stored in

the bar of volume 3.75 x 10

-5

m

3

. The modulus of elasticity is 200 GPa and

the Poisson’s ratio is 0.28.

8-4 Determine the force in member AB in the truss shown in (a) using the

principle of virtual work given the deformation described in (b).

1.5

m

P

B A

18 mm diameter

a

b

P

A

B

D

L

120 MPa

60 MPa

x

y

8-5 Determine the slope A of the beam ABC at A using the principle of virtual

work.

8-6 Calculate the vertical displacements of B and C in the simply supported

beam of length L and flexural rigidity EI using the energy method.

8-7 Calculate the loads in the members of the singly redundant pin-jointed

framework using the energy method. The members AC and BD are 30mm

2

in cross section and all other members are 20mm

2

in cross section. The

members AD, BC, and DC are 800mm long. E = 200,000N/mm

2

.

Answers:

8-1

250.8 MPa

8-2

P a b

EIL

a b

2 2 2

2

6

( )+

8-3

2.07 J

8-4

40 kN

8-5

EI

WL

16

2

8-6

EI

wL

24576

119

4

,

EI

wL

384

5

2

8-7

R = 2.1 N

Chapter 9

– Matrix Methods

9-1 The square symmetrical pin-jointed truss is pinned to rigid supports at 2 and

4; whilst loaded at 1. The axial rigidity for all members is EA. Use the

matrix method to (a) find the displacements in 1 and 3 and (b) solve for all

internal member forces and support reactions.

9-2 The displacement at node 4 of the pin-jointed frame is zero. Use the matrix

method to find (a) the ratio H/P and the (b) displacements of nodes 2 and 3.

Answers:

9-1

AE

PL

v

2

1

−=,

AE

PL

v

293.0

3

−=

,

2

1412

P

ss ==

, Pss 207.0

4323

−

=

=

=

㤭9

449.0=

P

H

,

AE

Pl

v

)329(

4

2

+

−=

,

AE

Pl

v

)329(

6

3

+

−=

Chapter 10

– Stress/Strain Measurement

10-1 A cantilever bar is to be loaded as shown and the strain axial strain

measured at midspan with strain gages. Briefly suggest a readout scheme

wherein the highest voltage is obtained for the load applied.

10-2 In certain strain gage applications, it is necessary to record strains over a

long period of time without having the opportunity to recheck the zero

reading. The strain indicator will have an effect of the zero position drifting.

Suggest how the measuring method can be done in order to eliminate the

strain indicator drifting effect and how the instrumentation drift amount can

be determined.

10-3 A birefringent disk of thickness of 5mm and material fringe value of 12.5

N/mm is viewed under a circular polariscope. Along a horizontal section in

the middle, the outer ends have zero relative retardation. Find the principal

stress difference at the middle of the disk.

Answers:

10-3 15N/mm

2

P

L

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