Transcript EML 4500 FINITE ELEMENT ANALYSIS AND DESIGN

CHAP 8 STRUCTURAL DESIGN USING
FINITE ELEMENTS
FINITE ELEMENT ANALYSIS AND DESIGN
Nam-Ho Kim
Edited and audio Raphael Haftka
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INTRODUCTION
• FEA: determining the response of a given structure for a
given set of loads and boundary conditions
– Geometry, material properties, BCs and loads are well defined
• Engineering design: a process of synthesis in which parts are
put together to build a structure that will perform a given set of
functions satisfactorily
• Creative design: creating a new structure or machine that
does not exist
• Adaptive design: modifying an existing design (evolutionary
process)
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INTRODUCTION – STRUCTURAL DESIGN
• Structural design: a procedure to improve or enhance the
performance of a structure by changing its parameters
• Performance: a measurable quantity (constraint and goal)
– the weight, stiffness or compliance; the fatigue life; noise and vibration
levels; safety
• Constraint: As long as the performance satisfies the criterion,
its level is not important
– Ex: the maximum stress should be less than the allowable stress
• Goal: the performance that the engineer wants to improve as
much as possible, usually called objective function
• Design variables: system parameters that can be changed
during the design process
– Plate thickness, cross-sectional area, shape, etc
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SAFETY MARGIN
• Factor of Safety (stress performance)
– Structures should satisfy a constraint: s i ( x ) £ s allowable
– si(x): the calculated stress at a point x in the i-th component
– sallowable: allowable stress from material strength (failure strength)
s allowable
sF
=
SF
Failure stress
Factor of safety
– Factor of safety: effect of material variability, experimental errors, etc
• For general type of performance, use response and capacity
– Response Ri: calculated value of performance
– Capacity Ci: allowed value of performance
Ri ( x ) £
Ci
SF
– When multiple loads are applied simultaneously Ri =
NL
å
j= 1
Ri (Fj )
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SAFETY MARGIN cont.
• Safety Margin: the excess capacity compared with the
response
Zi = Ci - Ri ³ 0
– The member can afford additional Zi stress before it fails
• Sufficiency Factor Si: ratio of the allowable capacity to the
response
Ci
= Si ³ 1
SF Ri
– Normalized in terms of the capacity
• Example: Failure stress is 100ksi, factor of safety is 1.25,
actual stress is 60 ksi. Safety margin is 40ksi, the sufficiency
factor is 80/60=1.333
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LOAD FACTOR
• Instead of dividing the capacity by SF, increase applied loads
• Load factor l: the factor by which the applied loads must be
multiplied just enough to cause the structure to fail
• Let a set of loads be F = {F1, F2, …, FN}T. For a given failure
strength Ci, the load factor is defined by
Ri (l F) = Ci
• For proportional loading with linear system
Ci
= l
Ri (F)
• Load factor is the ratio between capacity and response
• The structure should be designed in order to satisfy a given
level of load factor
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EXAMPLE
• Design the height h of cantilevered beam with SF = 1.5
– E = 2.9104 ksi, w = 2.25 in.
1) Allowable tip deflection Dallowable = 2.5 in. (No need SF)
–
–
FE equation after applying BCs
EI é 12 - 6L ùìï v 2 ïü ïì F ïü
ê
ú
ý= í ý
2 úíï
3 ê
ïîï 0 ïþ
L ë- 6L 4L ûîï q2 ïþ
ï
ï
FE solution
4FL3
6FL2
v2 =
, q2 =
3
Ewh
Ewh3
FL3
4FL3
v2 =
=
= Dallowable
3EI Ewh3
Þ
h=
3
4FL3
= 3.66 in
EwDallowable
L = 100 in
h
w
F = 2,000 lb
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EXAMPLE cont.
2) Failure strength = 40 ksi (Need SF)
–
Supporting moment at the wall
EI é
2
2 ù
6
Lv
+
4
L
q
6
Lv
+
2
L
q2 û= - FL
1
1
2
3 ë
L
Maximum stress at the wall
M h2
6FL
s max =
=
I
wh2
Height calculation with the factor of safety
C1 =
–
–
6FL s F
=
2
SF
wh
Þ
h=
6FLSF
= 4.47 in
ws F
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