# Chapter 1 – Review of Mechanics of Materials 1-1 ... - Biofuturex.com

Mechanics

Oct 29, 2013 (4 years and 8 months ago)

128 views

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

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