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 mm

150 mm radius

300 mm radius

300 mm 600 mm

300 mm

400 mm

400 mm 400 mm 200

400 mm 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 0.0918m4 1-5 (b2h2)/24 1-6 -(b2h2)/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 400 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 25o, 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 0o gage = 500

microstrains, with 90o gage = –100 microstrains. The strain gages register equal values after a 30o 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

53232223 yyhyxyxh

hw

+−−=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, 13o 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 / td3, -2P/td3, 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/mm2, and the tensile strength of the plate is 465 N/mm2, 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. Immediately 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) = 46.5m2; Wing mean cord (c) = 3m; CD = 0.01 + 0.05CL

2; CM,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 40o 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/s2. The moment of inertia of the aircraft for pitching is 204,000 kgm2. Find (i) the additional tail load required to check the angular velocity in pitch. The aircraft has wings 88.5m2 in area, mean cord of 1m, and the pitching moment coefficient for all parts excluding the tailplane through the CG is given by CM.CG.c = 0.427CL – 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/s2 193.3kN 29kNm (clockwise) 0.32m 19.5kN 59.6kNm 4-2 78.03o, 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 The stress function φ = k(r2 – a2) is applicable to the solution of a solid

circular section bar of radius a. Determine the stress distributions τyz, τzx in the bar in terms of the applied torque, 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 τyz, τzx, and (iii) the maximum shear stress.

5-3 The stress function φ = m(x2 – a2)(y2 – b2) is applicable to the solution of the rectangular section bar shown. Determine the stress distributions τyz, τzx dw/dx, dw/dy in the bar in terms of the applied torque.

Answers: 5-1 -2Tx/πa4, -2Ty/πa4, 0, 0, 0

5-2 3

)2( 3tba + , dzdGx θ2 , 0, 2)2(

3tba

T+

±

5-3 33

22

16)(9

babyTx

zy−−

=τ , 33

22

16)(9

baaxTy

zx−

=τ , 33

2222

32)(9

babyaxTy

dxdw +−−

= ,

33

2222

32)(9

babyaxTx

dydw +−−

=

x

y

2a

2b

Chapter 6 – Bending of Thin-Walled Beams 6-1 A bending moment of 3000Nm is applied on the section shown at 30o to the

vertical y axis. The sense of the bending moment is such that its components Mx and My 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/mm2 6-2 194.7N/mm2

6-3 Mxthz 21,

41.0=σ , Mx

thz 22,5.0−

=σ , Mxthz 23,5.0

=σ , Mxthz 24,

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

sin261cossinds 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 Sy applied at the apex. Calculate the shear flow over the cross section.

Answers: 7-1 -2Tx/πa4, -2Ty/πa4, 0, 0, 0

7-2 ⎥⎦

⎤⎢⎣

⎡+−

θay

axx

dzdG

23

23 22

, ⎥⎦⎤

⎢⎣⎡ +

θ−

axyy

dzdG 3 ,

dzd

axy θ

−3 ,

dzd

ay

ax θ

⎥⎦

⎤⎢⎣

⎡+−

23

23 22

,

dzdyxy

aθ

− )3(21 23

7-3 )2(

)3/3( 21

12 dhhdhdsS

q y

+

−−= ,

)2()66(

2

22

22

23 dhhhhssS

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 10-5m3. 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

PA 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 30mm2 in cross section and all other members are 20mm2 in cross section. The members AD, BC, and DC are 800mm long. E = 200,000N/mm2.

Answers: 8-1 250.8 MPa

8-2 P a bEIL

a b2 2 2

26( )+

8-3 2.07 J 8-4 40 kN

8-5 EI

WL16

2

8-6 EI

wL24576119 4

, EI

wL3845 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

PLv21 −= ,

AEPLv 293.0

3 −= , 21412Pss == , Pss 207.04323 −==

9-2 449.0=PH ,

AEPlv

)329(4

2+

−= , AE

Plv)329(

63

+−=

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/mm2

P

L