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CHINE DESIGN - An Integrated Approach, 2ed by Robert L. Norton, Prentice-Hall 2000
ading Loading
ear Shear
oment Moment
ope Slope
eflection Deflection
yFa
EI
a lmax = ( )2
6
3 yw
EI
l a l amax = + ( )24
3 44 3 4
when :a ywl
EImax= = 0
8
4
M Mw
l amax = = ( )1 2 22
when :a Mwl
max= =02
2
when :a l M Flmax= =
R F1 = R w l a1 = ( )
M Famax =
= + ( )FEI
ax x x a2
22 2
M M R x F x a
F a x x a
= +
= + ( )1 1
1
1
= +
1 2
2
11 2
2EI
M xR
x
Fx a
yEI
Mx
Rx
Fx a
F
EIx ax x a
= +
= ( )
1
2 6 6
63
1 2 1 3 3
3 2 3
when :a l yFl
EImax= =
3
3
M M R x w x a
wl a x l a x a
= +
= ( ) ( ) [ ]1 1
2
2 2 2
22 2
= ( ) ( ) ( )wEI
l a x l a x x a6
3 32 2 2 3
= +
1 2
6
11 2
3EI
M xR
x
wx a
yEI
Mx
Rx
wx a
w
EIl a x l a x x a
= +
= ( ) ( ) ( )
1
2 6 24
244 6
1 2 1 3 4
3 2 2 2 4
M
w
l a1
2 2
2= ( )M Fa
1 =
q M x R x F x a= + 12
1
1 1q M x R x w x a= + 1
2
1
1 0
V M x R F x a
F x a
= +
= ( )
1
1
1
0
0
1
V M x R w x a
w l a x a
= +
= ( ) [ ]
1
1
1
1
1
V R Fmax = =1 V R w l amax = = ( )1
G U R E D - 1
tilever Beams with Concentrated or Distributed Loading. Note: < > Denotes a Singularity Function
) Cantilever beam with concentrated loading (b) Cantilever beam with uniformly distributed loading
Fx a 1
x
R1
l
a wxa 0
x
M1
R1
l
a
x
Vmax
x
Mmax
x
max
x
ymax
x0
Vmax
V
x0
MMmax
x0
max
x0
yymax
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CHINE DESIGN - An Integrated Approach, 2ed by Robert L. Norton, Prentice-Hall 2000
(b) Simply supported beam with uniformly distributed loadingSimply supported beam with concentrated loading
pe Slope
Momentoment
ear Shear
ading Loading
flection Deflection
q R x w x a R x l= + 11 0
21
Rw
ll a1
2
2= ( )
Rw
ll a22 2
2= ( )
V R Rmax = MAX( , )1 2
when = :a Mwx
l x02
= ( )
when = :
a
ywx
EIlx x l
0
242
2 3 3= ( )
R Fa
l1 1=
R Fa
l2 =
V R Rmax = MAX( , )1 2
q R x F x a R x l= + 11 1
21
V R F x a R x l
Fa
lx a
= +
=
10
20
01
M R x F x a R x l
Fa
lx x a
= +
=
11
21
11
yF
EIa
a
llamax =
32
34
2
M Faa
lmax =
1
when = :al
MFl
max2 4
=
=
+ + ( )
F
EI
a
lx x a
a
la al l
2
1
33 2
2 2
2 2
yF
EI
a
lx x a
a
la al l x
=
+ + ( )
6
1
3 2
3 3
2 2
=( )
+ ( ) ( )[ ]
w
EI
x
ll a x a
ll a l l a
24
64
12
22 3
4 2 2
M R xw
x a R x l
w x
ll a x a
= +
= ( )
12
21
2 2
2
2
V R w x a R x l
wl
l a x a
= +
= ( )
11
20
2 11
2
yw
EI
xl
l a x a
x
ll a l l a
=( )
+ ( ) ( )[ ]
24
2
2
3
2 4
4 2 2
U R E D - 2
ly Supported Beams with Concentrated or Distributed Loading. Note: < > Denotes a Singularity Function
x
x
ymax
x0
V
x0
y
ymax
l
a
x
R1 R2
wxa0Fxa 1l
a
x
R1 R2
x
Mmax
x
max
x0
max
x0
M Mmax
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CHINE DESIGN - An Integrated Approach, 2ed by Robert L. Norton, Prentice-Hall 2000
ope Slope
Momentoment
ear Shear
ading Loading
eflection Deflection
q R x w x w x a R x b= + + 11 0 0
2
R Fa
b2 =
q R x F x a R x b= + 11 1
2
1
V Fb a
b xa
b x b x a=
+ 0 0 0
Rwa
b2
2
2=
Mw
aa
bx x
x aa
bx b
=
+ +
2
2
22
22
1
=
+
+
+ +
w
EI
a b ab bb
b a
a a
bx x
a
bx b x a
1
242 4
1
2 4
1
6
4
1
6
2 2 3 4
22 3
22 3
yw
EI
a b ab bb
b a x
aa
bx x
x aa
bx b
=
+
+
+ +
24
2 41
42
2
2 2 3 4
23 4
42
3
=
+
+ ( )
F
EI
b a
bx
a
bx b x a
ba b
2
3
2 2 2
V w a x x a
a
b x b= + + ( )
1
20
21
R waa
b1 1
2=
M Fb a
bx
a
bx b x a=
+
1 1 1
R Fb a
b1 =
yF
EI
b a
bx
a
bx b x a
b a b x
=
+
+ ( )
6
3 3 3
G U R E D - 3
rhung Beams with Concentrated or Distributed Loading. Note: < > Denotes a Singularity Function
x
x
ymax
wxa 0
x
Mmax
R1
l
x
R2
b
a
Fxa1
R1
l
x
R2
b
a
Overhung beam with concentrated loading (b) Overhung beam with uniformly distributed loading
x0
V
x0
yymax
x
0
max
min
0
M
Mmax
x
max
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Biaxial stresses
Locate critical point (max stresses)
P
a
b
M=Pb
T=Pa
=Mc/I
=
Td/J
x
xz
x
xz
Mohrs circle
Determine maximum shear stress
(x,
xz)
12
max)
max)
2
How to Predict Failure
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Maximum Shear Stress Theory (MSST) (A.k.a Tresca)
Tension test specimenyields whensy
(0,0)
max= -sy/2
(sy,0) 2
max
ys=
In real applications,2
21max
= 2 1
max
So yielding occurs when ys= 21
Safety factor is21
=y
s
sn
3-D Mohr Max. Shear Stress Theory (MSST)
Be careful when z
= 0 is outside 2-D Mohrs circle.
2 1
max
3
Always order principalnormal stresses according to
321 >>
22
31max
==
ys Now yielding occurs when
max
Safety factor is
31 =
y
s
sn
ys= 31
Graphical representation
ys= 21
2
1
sy
sy-sy
-sy
yieldor 21 >> yy ss
yieldor 21
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DET vs. MSST
2
1
sy
sy-sy
-syShear diagonal
2 1
Different especially when 1 = -2(e.g. pure torsional loading)
3-D Distortion Energy Theory (DET)
Be careful when z
= 0 is outside 2-D Mohrs circle.
2 13
Now von Mises stress is
( ) ( ) ( )[ ]12232132122
1 ++=e
Brittle Materials
Brittle materials dont yield, they fracture.
Strength in compression >> Strength in tension Three theories presented
Material strength
5 Sut= Ultimate (fracture) strength in tension.
5 Suc = Ultimate (fracture) strength in compression.
5 Strengths are always positive numbers
5 Stresses 1, 2, and 3 can be negative or positive.
Maximum Normal Stress Theory (MNST)
whichever is smaller
fractureor 21 >> yy ss
fractureor 21
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Internal Friction Theory (IFT)
fractureor 21 >> yy ss
fractureor 21 > yy ss
fractureor 21
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Iron is taken from the earth and copper is smelted from ore.Man puts an end to the darkness;
he searches the farthest recesses for ore in the darkness.The Bible (Job 28:2-3)
Image: Iron flows from a blast furnace. Source:American Iron and Steel Institute.
Figure 3.5 Stress-strain diagram for a ductile material.text reference: Figure 3.5, , page 96
A
P
P
(
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text reference: Figure 3.6, page 97
Figure 3.1 Ductile material from a standardtensile test apparatus. (a) Necking; (b) failure.
text reference: Figure 3.1, page 90
0
0
=
l
llEL
fr
%EL
Manifest
danger
stress concentrations
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Figure 3.2 Failure of a brittlematerial from a standard
tesiletest apparatus.
text reference: Figure 3.2, page 91
%EL%EL
text reference: Figure 3.7, page 98
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text reference: Figure 3.8, page 99
Figure 3.10 Stress-strain diagram for polymer below, at, and above its glass transitiontemperature Tg.
text reference: Figure 3.10, page 101
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=E
uniaxial
linear
Esteel
Ealum
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yx yx
torsion
=G
yx
x
y
steel
alum v
rubber vG E
transverse
axial
=
( )+=
EG
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yield
Allowable building
yallowy SS
yallow S=
yallowy SS
yallow S=
=
y
dUr
0rupture
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Class Members Short nameEnginering alloys(themetals and alloys ofengineering)
AluminumalloysCopper alloysLead alloysMagnesiumalloysMolybdenumalloysNickel alloysSteels
Tin alloysTitaniumalloys
Tungsten alloysZinc alloys
Al alloysCu alloysLead alloysMg alloysMo alloysNi alloysSteels
Tin alloysTi alloys
W alloysZn alloysEngineering polymers(thethermoplastics andthermosets of engineering)
EpoxiesMelaminesPolycarbonatePolyesterPolyethylene, high densityPolyethylene, low densityPolyformaldehydePolymethylmethacrylatePolypropylenePolytetrafluoroethylenePolyvinyl chloride
EPMELPCPESTHDPELDPEPFPMMAPPPTFEPVC
Engineering ceramics(fineceramics capableofload-bearing application)
AluminaDiamondSialonsSilicon carbideSilicon nitrideZirconia
Al2O3CSialonsSiCSi3N4ZrO2
Table 3.7 Material classesand members and shortnames of each member.[From Ashby (1992)].
text reference: Table 3.7, page 123
RA RB RC
SU HB HBSU HB
HB
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Charpy Izod
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quenching
critical temperature
below critical temperatureand
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carburizing
hardness
strength
hardness toughness
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Expensive!
y
y
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Brass zinc
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Figure 3.4 Cross section of fiber reinforcedcomposite material.
text reference: Figure 3.4, page 95
Figure33 Strength/density for variousmaterials