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• A. J. Clark School of Engineering •Department of Civil and Environmental Engineering• A. J. Clark School of Engineering •Department of Civil and Environmental Engineering• A. J. Clark School of Engineering •Department of Civil and Environmental Engineering
Third EditionLECTURE
32.8
Chapter
byDr. Ibrahim A. Assakkaf
SPRING 2003ENES 220 – Mechanics of Materials
Department of Civil and Environmental EngineeringUniversity of Maryland, College Park
RODS: AXIAL LOADING AND DEFORMATION
LECTURE 3. RODS: AXIAL LOADING AND DEFORMATION (2.8) Slide No. 1ENES 220 ©Assakkaf
Deformations of Members under Axial Loading
Uniform Member– Consider a homogeneous rod as shown in
the figure of the next viewgraph.– If the resultant axial stress σ = P/A dos not
exceed the proportional limit of the material, then Hooke’s law can applied, that is
elasticity of modulusstrain axial stress, axial
===
=
E
Eεσ
εσ
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LECTURE 3. RODS: AXIAL LOADING AND DEFORMATION (2.8) Slide No. 2ENES 220 ©Assakkaf
Deformations of Members under Axial Loading
Uniform Member
P
δ
L
LECTURE 3. RODS: AXIAL LOADING AND DEFORMATION (2.8) Slide No. 3ENES 220 ©Assakkaf
Deformations of Members under Axial Loading
Uniform Member– From Hooke’s law, it follows that
EAPL
LE
AP
L
EAP
E
=⇒=
=
=
=
δδ
δε
ε
εσ
, therefore,But
law)s(Hooke'
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LECTURE 3. RODS: AXIAL LOADING AND DEFORMATION (2.8) Slide No. 4ENES 220 ©Assakkaf
Deformations of Members under Axial Loading
Uniform Member– The deflection (deformation),δ, of the
uniform member subjected to axial loading P is given by
EAPL
=δ (1)
LECTURE 3. RODS: AXIAL LOADING AND DEFORMATION (2.8) Slide No. 5ENES 220 ©Assakkaf
Deformations of Members under Axial Loading
Multiple Loads/Sizes– The expression for the deflection of the
previous equation may be used only if the rod or the member is homogeneous (constant E) and has a uniform cross sectional area A, and is loaded at its ends.
– If the member is loaded at other points, or if it consists of several portions of various cross sections, and materials, then
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LECTURE 3. RODS: AXIAL LOADING AND DEFORMATION (2.8) Slide No. 6ENES 220 ©Assakkaf
Deformations of Members under Axial Loading
Multiple Loads/Sizes– It needs to be divided into components
which satisfy individually the required conditions for application of the formula.
– Denoting respectively by Pi, Li, Ai, and Ei, the internal force, length, cross-sectional area, and modulus of elasticity corresponding to component i,then
∑∑==
==n
i ii
iin
ii AE
LP11
δδ
LECTURE 3. RODS: AXIAL LOADING AND DEFORMATION (2.8) Slide No. 7ENES 220 ©Assakkaf
Deformations of Members under Axial Loading
Multiple Loads/Sizes
E1 E2 E3
L1 L2 L3
∑∑==
==n
i ii
iin
ii AE
LP11
δδ
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LECTURE 3. RODS: AXIAL LOADING AND DEFORMATION (2.8) Slide No. 8ENES 220 ©Assakkaf
Deformations of Members under Axial Loading
Multiple Loads/Sizes– The deformation of of various parts of a rod
or uniform member can be given by
∑∑==
==n
i ii
iin
ii AE
LP11
δδ
E1 E2 E3
L1 L2 L3
(2)
LECTURE 3. RODS: AXIAL LOADING AND DEFORMATION (2.8) Slide No. 9ENES 220 ©Assakkaf
Deformations of Members under Axial Loading
Example 4– Determine the deformation of the steel rod
shown under the given loads. Assume that the modulus of elasticity for all parts is
– 29×106 psi
75 kips45 kips
12 in 12 in 16 in
30 kips
Area = 0.9 in2
Area = 0.3 in2
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LECTURE 3. RODS: AXIAL LOADING AND DEFORMATION (2.8) Slide No. 10ENES 220 ©Assakkaf
Deformations of Members under Axial Loading
Example 4 (cont’d)– Analysis of internal forces
75 kips45 kips 30 kips
1 2 3
30 kipsP3
kips 30
030 ;0
3
3
=∴
=+−=+→ ∑P
PFx
LECTURE 3. RODS: AXIAL LOADING AND DEFORMATION (2.8) Slide No. 11ENES 220 ©Assakkaf
Deformations of Members under Axial Loading
Example 4 (cont’d)– Analysis of internal forces
75 kips45 kips 30 kips
1 2 3
30 kipsP2
kips 1503045 ;0
2
2
−=∴
=+−−=+→ ∑P
PFx
45 kips
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LECTURE 3. RODS: AXIAL LOADING AND DEFORMATION (2.8) Slide No. 12ENES 220 ©Assakkaf
Deformations of Members under Axial Loading
Example 4 (cont’d)– Analysis of internal forces
75 kips45 kips 30 kips
1 2 3
75 kips45 kips 30 kipsP1
kips 600304575 ;0
1
1
=∴
=+−+−=+→ ∑P
PFx
LECTURE 3. RODS: AXIAL LOADING AND DEFORMATION (2.8) Slide No. 13ENES 220 ©Assakkaf
Deformations of Members under Axial Loading
Example 4 (cont’d)– Deflection
• Input parameters
lb 30,000kips 30lb 15,000kips 15
lb 60,000kips 60forces, internal of analysis From
in 3.0 in 9.0 in 9.0
in 16 in 12 in 12
3
2
1
23
22
21
321
==−=−=
==
===
===
PPP
AAALLL
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LECTURE 3. RODS: AXIAL LOADING AND DEFORMATION (2.8) Slide No. 14ENES 220 ©Assakkaf
Deformations of Members under Axial Loading
Example 4 (cont’d)– Carrying the values into the deformation
formula:
in 0759.03.0
)16(000,309.0
)12(000,159.0
)12(000,601029
1
1
6
3
33
2
22
1
113
1
=
+
−+
×
++== ∑
=
δ
δALP
ALP
ALP
EAELP
i ii
ii
LECTURE 3. RODS: AXIAL LOADING AND DEFORMATION (2.8) Slide No. 15ENES 220 ©Assakkaf
Deformations of Members under Axial Loading
Relative DeformationA A
L
δA
δBB
P
DC
AEPL
ABAB =−= δδδ /
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LECTURE 3. RODS: AXIAL LOADING AND DEFORMATION (2.8) Slide No. 16ENES 220 ©Assakkaf
Deformations of Members under Axial Loading
Relative Deformation– If the load P is applied at B, each of the
three bars will deform.– Since the bars AC and AD are attached to
the fixed supports at C and D, their common deformation is measured by the displacement δA at point A.
LECTURE 3. RODS: AXIAL LOADING AND DEFORMATION (2.8) Slide No. 17ENES 220 ©Assakkaf
Deformations of Members under Axial Loading
Relative Deformation– On the other hand, since both ends of bars
AB move, the deformation of AB is measured by the difference between the displacements δA and δB of points A and B.
– That is by relative displacement of B with respect to A, or
EAPL
ABAB =−= δδδ /(3)
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LECTURE 3. RODS: AXIAL LOADING AND DEFORMATION (2.8) Slide No. 18ENES 220 ©Assakkaf
Deformations of Members under Axial Loading
Example 5The rigid bar BDE is supported by two links AB and CD.
Link AB is made of aluminum (E = 70 GPa) and has a cross-sectional area of 500 mm2. Link CD is made of steel (E = 200 GPa) and has a cross-sectional area of (600 mm2).
For the 30-kN force shown, determine the deflection a) of B, b) of D, and c) of E.
LECTURE 3. RODS: AXIAL LOADING AND DEFORMATION (2.8) Slide No. 19ENES 220 ©Assakkaf
Deformations of Members under Axial Loading
Example 5 (cont’d)
SOLUTION:
• Apply a free-body analysis to the bar BDE to find the forces exerted by links AB and DC.
• Evaluate the deformation of links ABand DC or the displacements of Band D.
• Work out the geometry to find the deflection at E given the deflections at B and D.
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LECTURE 3. RODS: AXIAL LOADING AND DEFORMATION (2.8) Slide No. 20ENES 220 ©Assakkaf
Deformations of Members under Axial Loading
Example 5 (cont’d)Free body: Bar BDE
( )
( )ncompressioF
F
tensionF
F
M
AB
AB
CD
CD
B
kN60
m2.0m4.0kN300
0M
kN90
m2.0m6.0kN300
0
D
−=
×−×−=
=
+=
×+×−=
=
∑
∑
SOLUTION:
LECTURE 3. RODS: AXIAL LOADING AND DEFORMATION (2.8) Slide No. 21ENES 220 ©Assakkaf
Deformations of Members under Axial Loading
Example 5 (cont’d)
Displacement of B:
( )( )( )( )
m10514
Pa1070m10500m3.0N1060
6
926-
3
−×−=
××
×−=
=AEPL
Bδ
↑= mm 514.0Bδ
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LECTURE 3. RODS: AXIAL LOADING AND DEFORMATION (2.8) Slide No. 22ENES 220 ©AssakkafDeformations of Members under
Axial LoadingExample 5 (cont’d)
Displacement of D:
( )( )( )( )
m10300
Pa10200m10600m4.0N1090
6
926-
3
−×=
××
×=
=AEPL
Dδ
↓= mm 300.0Dδ
LECTURE 3. RODS: AXIAL LOADING AND DEFORMATION (2.8) Slide No. 23ENES 220 ©Assakkaf
Deformations of Members under Axial Loading
Example 5 (cont’d)Displacement of D:
( )
mm7.73
mm 200mm 0.300mm 514.0
=
−=
=′′
xx
xHDBH
DDBB
( )
mm 928.1mm 7.73
mm7.73400mm 300.0
=
+=
=′′
E
E
HDHE
DDEE
δ
δ
↓= mm 928.1Eδ
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LECTURE 3. RODS: AXIAL LOADING AND DEFORMATION (2.8) Slide No. 24ENES 220 ©Assakkaf
Deformations of Members under Axial Loading
Nonuniform Deformation– For cases in which the axial force or the
cross-sectional area varies continuously along the length of the bar, then Eq. 1
– (PL / EA) is not valid.– Recall that in the case of variable cross
section, the strain depends on the position of point Q, where it is computed from
dxdδε =
LECTURE 3. RODS: AXIAL LOADING AND DEFORMATION (2.8) Slide No. 25ENES 220 ©Assakkaf
Deformations of Members under Axial Loading
Nonuniform Deformation
xL
P
dxdδε =
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LECTURE 3. RODS: AXIAL LOADING AND DEFORMATION (2.8) Slide No. 26ENES 220 ©Assakkaf
Deformations of Members under Axial Loading
Nonuniform Deformation– Solving for dδ and substituting for ε
– But ε = σ / E, and σ = P/A. therefore
dxd εδ =
dxEAPdx
Edxd ===
σεδ (4)
LECTURE 3. RODS: AXIAL LOADING AND DEFORMATION (2.8) Slide No. 27ENES 220 ©Assakkaf
Deformations of Members under Axial Loading
Nonuniform Deformation– The total deformation δ of the rod or bar is
obtained by integrating Eq. 4 over the length L as
∫=L
x
x dxEAP
0
δ (5)
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LECTURE 3. RODS: AXIAL LOADING AND DEFORMATION (2.8) Slide No. 28ENES 220 ©Assakkaf
Deformations of Members under Axial Loading
Example 6– Determine the deflection of point a of a
homogeneous circular cone of height h, density ρ, and modulus of elasticity E due to its own weight.
h
a
LECTURE 3. RODS: AXIAL LOADING AND DEFORMATION (2.8) Slide No. 29ENES 220 ©Assakkaf
Deformations of Members under Axial Loading
Example 6 (cont’d)– Consider a slice of thickness dy– P = weight of above slice– = ρg (volume above)
hyδy r
( ) ydyEg
rE
yrg
EAPdyd
yrgP
331
31
2
2
2
ρπ
πρδ
πρ
=
==
=
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LECTURE 3. RODS: AXIAL LOADING AND DEFORMATION (2.8) Slide No. 30ENES 220 ©Assakkaf
Deformations of Members under Axial Loading
Example 6 (cont’d)
Egh
yEgydy
Eg
ydyEgd
hh
6
233
3
20
2
0
ρδ
ρρδ
ρδ
=
==
=
∫
LECTURE 3. RODS: AXIAL LOADING AND DEFORMATION (2.8) Slide No. 31ENES 220 ©Assakkaf
Deformations of Members under Axial Loading
Normal Stresses in Tapered Bar– Consider the following tapered bar with a
thickness t that is constant along the entire length of the bar.
h(x)h1 h2x
L
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LECTURE 3. RODS: AXIAL LOADING AND DEFORMATION (2.8) Slide No. 32ENES 220 ©Assakkaf
Deformations of Members under Axial Loading
Normal Stresses in Tapered Bar
– The following relationship gives the height h of tapered bar as a function of the location x
( )Lxhhhhx 121 −+= (6)
h(x)h1 h2x
L
LECTURE 3. RODS: AXIAL LOADING AND DEFORMATION (2.8) Slide No. 33ENES 220 ©Assakkaf
Deformations of Members under Axial Loading
Normal Stresses in Tapered Bar
– The area Ax at any location x along the length of the bar is given by
( )
−+==
LxhhhtthA xx 121 (7)
h(x)h1 h2x
L
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LECTURE 3. RODS: AXIAL LOADING AND DEFORMATION (2.8) Slide No. 34ENES 220 ©Assakkaf
Deformations of Members under Axial Loading
Normal Stresses in Tapered Bar
– The normal stress σx as a function of x is given
( )
−+
==
Lxhhht
PAP
xx
121
σ(8)
h(x)h1 h2x
L
LECTURE 3. RODS: AXIAL LOADING AND DEFORMATION (2.8) Slide No. 35ENES 220 ©Assakkaf
Deformations of Members under Axial Loading
Example 7– Determine the normal stress as a function of x
along the length of the tapered bar shown if– h1 = 2 in– h2 = 6 in– t = 3 in, and– L = 36 in– P = 5,000 lb h(x)h1 h2
x
L
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LECTURE 3. RODS: AXIAL LOADING AND DEFORMATION (2.8) Slide No. 36ENES 220 ©Assakkaf
Deformations of Members under Axial Loading
Example 7 (cont’d)– Applying Eq. 8, the normal stress as a
function of x is given by
( )
( )
x
xx
Lxhhht
PAP
x
x
xx
+=
+=
−+
=
−+
==
1815000
36
5000
36)2623
5000
121
σ
σ
σ
LECTURE 3. RODS: AXIAL LOADING AND DEFORMATION (2.8) Slide No. 37ENES 220 ©Assakkaf
Deformations of Members under Axial Loading
Example 7 (cont’d)– Max σ = 833.3 psi
• At x =0– Min σ = 277.8 psi
• At x =36 in
h(x)h1 h2x
L
x (in) σ (psi)0 833.33 714.36 625.09 555.6
12 500.015 454.518 416.721 384.624 357.127 333.330 312.533 294.136 277.8
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LECTURE 3. RODS: AXIAL LOADING AND DEFORMATION (2.8) Slide No. 38ENES 220 ©Assakkaf
Deformations of Members under Axial Loading
Deflection of Tapered Bar– Consider the following tapered bar with a
thickness t that is constant along the entire length of the bar.
h(x)h1 h2x
L
dx
LECTURE 3. RODS: AXIAL LOADING AND DEFORMATION (2.8) Slide No. 39ENES 220 ©Assakkaf
Deformations of Members under Axial Loading
Deflection of Tapered Bar– Recall Eq. 5
– Substitute Eq. 7
– into Eq. 5, therefore
∫=L
x
x dxEAP
0
δ
( ) dxxhhLhEt
PL L
1
0 121∫ −+
=δ (9)
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LECTURE 3. RODS: AXIAL LOADING AND DEFORMATION (2.8) Slide No. 40ENES 220 ©Assakkaf
Deformations of Members under Axial Loading
Deflection of Tapered BarIntegrating Eq. 9, the deflection of a tapered bar is given by
( )[ ]LhhhhEt
PL12
12
ln1−
−
=δ (10)