The compound bar containing steel, bronze, and aluminum segments carries the axial loads shown below. The properties of the segments and the working stresses are listed in the table. Determine the maximum allowable value of P in kips if the change in length of the entire bar is limited to 0.08 in. and the working stresses are not to be exceeded. Note: Draw the Free Body Diagram, include the units/dimensions, use the proper formula and round-off all the answers and final answers to 3 decimal places.

Structural Analysis
6th Edition
ISBN:9781337630931
Author:KASSIMALI, Aslam.
Publisher:KASSIMALI, Aslam.
Chapter2: Loads On Structures
Section: Chapter Questions
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The compound bar containing steel, bronze, and aluminum segments carries the axial loads shown below. The properties of the segments and the working stresses are listed in the table.

Determine the maximum allowable value of P in kips if the change in length of the entire bar is limited to 0.08 in. and the working stresses are not to be exceeded.

Note: Draw the Free Body Diagram, include the units/dimensions, use the proper formula and round-off all the answers and final answers to 3 decimal places.

 

Bronze
Steel
Aluminum
4 ft
2 ft
3 ft
2P
3P
4P
A (in.²)
E (psi)
Ow (psi)
Steel
0.75
30 x 106
20 000
Bronze
1.00
12 x 106
18 000
Aluminum
0.50
10 x 106
12 000
Transcribed Image Text:Bronze Steel Aluminum 4 ft 2 ft 3 ft 2P 3P 4P A (in.²) E (psi) Ow (psi) Steel 0.75 30 x 106 20 000 Bronze 1.00 12 x 106 18 000 Aluminum 0.50 10 x 106 12 000
Fundamental Equations of Mechanics of Materials
Axial Load
Shear
Normal Stress
Average direct shear stress
V
A
Tavg
A
Displacement
P(x)dx
Transverse shear stress
8 =
A (x)E
VQ
It
PL
AE
Shear flow
VQ
q = Tt =
δ- α ΔΤL
= a
Torsion
Stress in Thin-Walled Pressure Vessel
Shear stress in circular shaft
Тр
Cylinder
pr
pr
2t
where
Sphere
pr
= solid cross section
01 = 02 =
2t
J =
- (c,“ – c,")
Stress Transformation Equations
tubular cross section
0x + 0y
O, – 0,
+
y
cos 20 +
Power
sin 20
2
P = Tw = 2™fT
Ox – 0y
Angle of twist
sin 20 + Try cos 20
2
T(x)dx
Principal Stress
J(x)G
Txy
tan 26,
TL
$ = E
JG
(ox – 0,)/2
Ox + 0y
1,2
0,- 0,
y
Average shear stress in a thin-walled tube
+ Tây
2
T
Tavg
21A m
Maximum in-plane shear stress
(0, – 0,)/2
Shear Flow
tan 20,
T
Txy
q = Tavg!
2A m
Tmax
+ Txy
2
Bending
Ox + 0y
O avg
Normal stress
Мy
Absolute maximum shear stress
I
σ.
max
Unsymmetric bending
Tabs
max
for ởmax, O min same sign
2
Mạy
max
Ở min
tan a
Tabs
max
for omax, Omin Opposite signs
tan 0
2
Geometric Properties of Area Elements
Material Property Relations
Poisson's ratio
-A = bh
Elat
1, = bh
!, = hb
long
C
%3D
Generalized Hooke's Law
:[0, - vo, + o.)]
Rectangular area
%3D
Ex
E
[0, – v(o, + 0,)]
-A = bh
E
Ez
[0. - v(0, + 0,)]
E
1
Tyz, Yzx
1
%3D
Yxy
Txy Yyz
Triangular area
where
E
G =
2(1 + v)
A =
a
(9 + D)w :
Relations Between w, V, M
h
(2a + b\
a + b
dM
= V
dx
dV
b
w(x),
dx
Trapezoidal area
Elastic Curve
1
M
-A =
EI
d*v
EI
=w(x)
!, = }ar*
%3D
d'v
El
= V(x)
Semicircular area
dv
El
dx?
M(x)
-A = Tr²
Buckling
Critical axial load
1, = fart
!, = fart
T²EI
Per
(KL)²
Critical stress
VIJA
O cr
Circular area
(KL/r)²*
Secant formula
ес
P
Ở max
1 +
sec
A
V EA
žab
A =
Energy Methods
Conservation of energy
U. = U;
zero slope
Strain energy
N²L
Semiparabolic area
U; =
2AE
constant axial load
"M²dx
U, =
bending moment
%3D
2EI
A =
U; =
transverse shear
%3D
2GA
zero slope -
pLT°dx
torsional moment
%3D
2GJ
Exparabolic area
Transcribed Image Text:Fundamental Equations of Mechanics of Materials Axial Load Shear Normal Stress Average direct shear stress V A Tavg A Displacement P(x)dx Transverse shear stress 8 = A (x)E VQ It PL AE Shear flow VQ q = Tt = δ- α ΔΤL = a Torsion Stress in Thin-Walled Pressure Vessel Shear stress in circular shaft Тр Cylinder pr pr 2t where Sphere pr = solid cross section 01 = 02 = 2t J = - (c,“ – c,") Stress Transformation Equations tubular cross section 0x + 0y O, – 0, + y cos 20 + Power sin 20 2 P = Tw = 2™fT Ox – 0y Angle of twist sin 20 + Try cos 20 2 T(x)dx Principal Stress J(x)G Txy tan 26, TL $ = E JG (ox – 0,)/2 Ox + 0y 1,2 0,- 0, y Average shear stress in a thin-walled tube + Tây 2 T Tavg 21A m Maximum in-plane shear stress (0, – 0,)/2 Shear Flow tan 20, T Txy q = Tavg! 2A m Tmax + Txy 2 Bending Ox + 0y O avg Normal stress Мy Absolute maximum shear stress I σ. max Unsymmetric bending Tabs max for ởmax, O min same sign 2 Mạy max Ở min tan a Tabs max for omax, Omin Opposite signs tan 0 2 Geometric Properties of Area Elements Material Property Relations Poisson's ratio -A = bh Elat 1, = bh !, = hb long C %3D Generalized Hooke's Law :[0, - vo, + o.)] Rectangular area %3D Ex E [0, – v(o, + 0,)] -A = bh E Ez [0. - v(0, + 0,)] E 1 Tyz, Yzx 1 %3D Yxy Txy Yyz Triangular area where E G = 2(1 + v) A = a (9 + D)w : Relations Between w, V, M h (2a + b\ a + b dM = V dx dV b w(x), dx Trapezoidal area Elastic Curve 1 M -A = EI d*v EI =w(x) !, = }ar* %3D d'v El = V(x) Semicircular area dv El dx? M(x) -A = Tr² Buckling Critical axial load 1, = fart !, = fart T²EI Per (KL)² Critical stress VIJA O cr Circular area (KL/r)²* Secant formula ес P Ở max 1 + sec A V EA žab A = Energy Methods Conservation of energy U. = U; zero slope Strain energy N²L Semiparabolic area U; = 2AE constant axial load "M²dx U, = bending moment %3D 2EI A = U; = transverse shear %3D 2GA zero slope - pLT°dx torsional moment %3D 2GJ Exparabolic area
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