geometry and design of truss structures
TRANSCRIPT
![Page 1: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/1.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Bill Baker, PE, SE, NAE, FREngSkidmore, Owings & Merrill LLP
Geometry and Design of Truss
Structures
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
![Page 2: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/2.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
William F. Baker, Lauren L. Beghini, Arkadiusz Mazurek, Juan Carrion and Alessandro Beghini (2015). "Structural Innovation: Combining Classic Theories with New Technologies," Engineering Journal, American Institute of Steel Construction, Vol. 52, pp. 203‐217.
![Page 3: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/3.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Baker, W., McRobie, A., Mitchell, T., Mazurek, A., “Mechanisms and states of self‐stress of planar trusses using graphic statics, Part I: Introduction and background.” Proceedings of theInternational Association for Shell and Spatial Structures (IASS) Symposium 2015, 2015,Amsterdam, The Netherlands.
Mitchell, T., Baker, W., McRobie, A., “Mechanisms and states of self‐stress of planar trusses using graphic statics, Part II: The Airy stress function and the fundamental theorem of linear algebra.”Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium2015, 2015, Amsterdam, The Netherlands.
McRobie, A., Baker, W., Michell, T., Konstantatou, M., “Mechanisms and states of self‐stress of planar trusses using graphic statics, Part III: Applications and extensions.” Proceedings of theInternational Association for Shell and Spatial Structures (IASS) Symposium 2015, 2015,Amsterdam, The Netherlands.
![Page 4: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/4.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Systems are essential for efficiency.
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
![Page 5: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/5.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
GEOMETRY IS A KEY COMPONENT OF STRUCTURAL SYSTEMS.
![Page 6: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/6.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Day One
Structure75%
Non-Structure
25%
EFFICIENT STRUCTURES CONSUME LESS RESOURCES
![Page 7: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/7.jpg)
TOPOLOGY
SHAPE
DOMAIN
SIZE
WHAT MATTERS
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
![Page 8: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/8.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Where to look for guidance on systems?
![Page 9: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/9.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Let’s start at the beginning of modern structural engineering:
The Mid‐19th Century.
![Page 10: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/10.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
UNDERSTANDING OF STRUCTURAL BEHAVIOR BY THE MID‐1800’S
![Page 11: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/11.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
A new problem: trusses.
First metal trusses (US: 1840 UK: 1845)
![Page 12: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/12.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
![Page 13: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/13.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Will focus not on the Maxwell‐Betti reciprocal theorem
but the little known Theorem of Load Paths
Today’s Presentation
![Page 14: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/14.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
![Page 15: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/15.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
MAXWELL’S THEOREM ON LOAD PATHS
Maxwell’s theorem states that, for any truss, the following is true:
where
iiCCTT rPLFLF
cosiiii rPrP
iP
ir
![Page 16: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/16.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
What does Maxwell’s Theorem on Load Paths tell us?
![Page 17: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/17.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
The longer the tension load path is, the longer the compression load path has to be
and vice versa.
![Page 18: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/18.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Inefficiencies are paid for exactly twice: once in tension and once in compression.
![Page 19: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/19.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Optimizing tension load path automatically optimizes compression load path and vice
versa.
![Page 20: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/20.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
The Maxwell constant and either the tension load path or
the compression load path determines total load path.
![Page 21: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/21.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
If only tension members oronly compression members then load path is equal to Maxell’s Constant for all possible
layouts.
![Page 22: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/22.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017 Equivalent Load Paths
FAB
FBCFCA
FAB
FBCFCA
FAB
FBCFCA
FAB
FBC
FCA
Applied Loads
EQUIVALENT OPTIMAL TRUSSES
![Page 23: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/23.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Load Path → Tonnage
Usually
![Page 24: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/24.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
x
y
iP
irii LF ,
x
y
PROOF: MAXWELL’S THEOREM
![Page 25: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/25.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
iii
ii rPLF
ii
ncompressioiii
tensioniii rPLFLF
WorkExternalWorkInternal x
y
iP
iP
ir
ir2
ii LF ,
ii LF ,
x
y
PROOF: MAXWELL’S THEOREM
![Page 26: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/26.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Cantilever with 3 to 1 span
MAXWELL’S THEOREM ON LOAD PATHS: AN EXAMPLE
![Page 27: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/27.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
MAXWELL’S THEOREM ON LOAD PATHS: AN EXAMPLE
Cantilever with 3 to 1 span
![Page 28: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/28.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
0rP
Cantilever with 3 to 1 span
MAXWELL’S THEOREM ON LOAD PATHS: AN EXAMPLE
![Page 29: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/29.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
0rP
MAXWELL’S THEOREM ON LOAD PATHS: AN EXAMPLE
Cantilever with 3 to 1 span
![Page 30: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/30.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
03 rP P
PBrPp
MAXWELL’S THEOREM ON LOAD PATHS: AN EXAMPLE
Cantilever with 3 to 1 span
![Page 31: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/31.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
PBrP ii
MAXWELL’S THEOREM ON LOAD PATHS: AN EXAMPLE
Cantilever with 3 to 1 span
![Page 32: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/32.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
PBLF TT 10
Moment diagram for truss geometry:
PBLF CC 9
PBLFLF CCTT
PBLFLF CCTT 19EB19
MAXWELL’S THEOREM ON LOAD PATHS: AN EXAMPLE
![Page 33: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/33.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
PBLF TT 9
PBLF CC 8
PBLFLF CCTT
PBLFLF CCTT 17EB17
MAXWELL’S THEOREM ON LOAD PATHS: AN EXAMPLE
Pratt truss:
![Page 34: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/34.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
PBLF TT 8
PBLF CC 7
PBLFLF CCTT
PBLFLF CCTT 15EB15
MAXWELL’S THEOREM ON LOAD PATHS: AN EXAMPLE
Warren truss:
![Page 35: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/35.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
The most material efficient truss is also the stiffest truss!
MAXWELL’S THEOREM ON LOAD PATHS: AN EXAMPLE
![Page 36: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/36.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Moment Diagram Cantilever versus Warren Truss Cantilever
MAXWELL’S THEOREM ON LOAD PATHS: EQUAL DEFLECTION
12
60% More
27% More
Deflection
B
A
VV
Strength
Truss “A” Truss “B”
![Page 37: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/37.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Can we find a benchmark for our designs?
![Page 38: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/38.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
How low can we go?
FT LT FC LC 13.92PB
MAXWELL’S THEOREM ON LOAD PATHS: AN EXAMPLE
![Page 39: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/39.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
PBLF TT 7.7
PBLF CC 7.6
PBLFLF CCTT
PBLFLF CCTT 47.14EB47.14
Bounded optimal truss:
MAXWELL’S THEOREM ON LOAD PATHS: AN EXAMPLE
![Page 40: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/40.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
PBLF TT 52.8
PBLF CC 52.7
PBLFLF CCTT
PBLFLF CCTT 04.16EB04.16
Cantilever with only compression chord:
MAXWELL’S THEOREM ON LOAD PATHS: AN EXAMPLE
![Page 41: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/41.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
MAXWELL’S THEOREM ON LOAD PATHS: AN EXAMPLE
![Page 42: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/42.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
PBLFLF CCTT 17.13
MAXWELL’S THEOREM ON LOAD PATHS: AN EXAMPLE
How low can we go?
![Page 43: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/43.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
PBLF TT 8
PBLF CC 7
PBLFLF CCTT
PBLFLF CCTT 15EB15
Within 14% of Benchmark
MAXWELL’S THEOREM ON LOAD PATHS: AN EXAMPLE
Warren truss:
![Page 44: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/44.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Preliminary System Design Using Maxwell’s Theorem on Load Paths
![Page 45: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/45.jpg)
EXCHANGE HOUSE, LONDON
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
![Page 46: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/46.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
![Page 47: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/47.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
APPLYING MAXWELL’S THEOREM: EXCHANGE HOUSE
![Page 48: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/48.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
APPLYING MAXWELL’S THEOREM: CONCEPTUAL DESIGN OF EXCHANGE HOUSE
![Page 49: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/49.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Tension members: Hangers
FT LThangers 2 Wydy
0
z 1 2 xB
2
dx0
B/2
4
15WBz2
APPLYING MAXWELL’S THEOREM: EXCHANGE HOUSE
![Page 50: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/50.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
FT LTtie
WB3H8z
Tension members: Tie
APPLYING MAXWELL’S THEOREM: EXCHANGE HOUSE
![Page 51: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/51.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
FT LT FT LThangers FT LT
tie
415
WBz2 18zWB3H
Tension members: Total
APPLYING MAXWELL’S THEOREM: EXCHANGE HOUSE
![Page 52: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/52.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
FT LT z
0 8Bz15
B3H8z2 0 z 15B2H
643
Minimum total load path:
APPLYING MAXWELL’S THEOREM: EXCHANGE HOUSE
![Page 53: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/53.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
P r BHW
H2
BH 2W2
APPLYING MAXWELL’S THEOREM: EXCHANGE HOUSE
Maxwell’s constant can be found using column support only:
![Page 54: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/54.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Use Basic Maxwell Theory Application
FLcolumns
only
APPLYING MAXWELL’S THEOREM: EXCHANGE HOUSE
![Page 55: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/55.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Use Basic Maxwell Theory Application
FLcolumns
only
2 FT LThangersand ties
APPLYING MAXWELL’S THEOREM: EXCHANGE HOUSE
![Page 56: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/56.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
FLcolumns
only
2 FT LThangersand ties
FLtotal
Use Basic Maxwell Theory Application
APPLYING MAXWELL’S THEOREM: EXCHANGE HOUSE
![Page 57: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/57.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
FLtotal 2 FT LT
P r
2 415
WBz2 WB3H
8z
WBH 2
2
BHW 815
z2
H
B2
4z
H2
Using the tension load path and the constant, the total load path can be computed:
Dividing by an average stress, the total tonnage of steel can now be estimated.
APPLYING MAXWELL’S THEOREM: EXCHANGE HOUSE
![Page 58: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/58.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
The Arch Load Path was calculated but not explicitly. How?
APPLYING MAXWELL’S THEOREM: EXCHANGE HOUSE
![Page 59: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/59.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Consider a segment of an arch:
APPLYING MAXWELL’S THEOREM: EXCHANGE HOUSE
![Page 60: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/60.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
APPLYING MAXWELL’S THEOREM: EXCHANGE HOUSE
Total load path: FC LCarch FT LT
tie FT LT
hangers FC LC
columnsbelow arch
![Page 61: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/61.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Michell Trusses (1904)
![Page 62: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/62.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
![Page 63: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/63.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
![Page 64: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/64.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
![Page 65: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/65.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
P
lN
(1±)l
P
Actual frame Virtually deformed frame
APPLICATION OF VIRTUAL WORK
![Page 66: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/66.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Mohr’s Circle:
n
2n
12
avg
n,max / 2y, xy / 2
Arbitrary strain
y
x
xy / 2
x, xy / 2
MICHELL’S OPTIMAL TRUSSES
![Page 67: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/67.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
1
2
2
n12
avg
n,max / 2y, xy / 2
x, xy / 2
2n
MICHELL’S OPTIMAL TRUSSES
Mohr’s Circle:
Plane strain
![Page 68: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/68.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
1
2
2
n12
avg
n,max / 2y, xy / 2
x, xy / 2
2n
MICHELL’S OPTIMAL TRUSSES
Mohr’s Circle:
Plane strain
![Page 69: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/69.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Under what conditions is for all members in a frame?
• Frames consisting of orthogonal curves such as• Systems of tangents and involutes• Equiangular spirals (systems of concentric circles, rectangular networks of straight lines)
a
MICHELL’S OPTIMAL TRUSSES
![Page 70: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/70.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Conclusion: All tension (resp. compression) members have similar curvature variations.
Φ(α1 ,β1)
means Φ(α1 ,β1) - Φ(α0 ,β1) = Φ(α1 ,β0) - Φ(α0 ,β0)
![Page 71: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/71.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Minimum volume structures:
P allowable tensile stressQ allowable compressive stress
a AB a AC CBa AC CB
MICHELL’S OPTIMAL TRUSSES
![Page 72: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/72.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
2a AB
P allowable tensile stressQ allowable compressive stressL moment of transmitted couple latitude of circles about pole
MICHELL’S OPTIMAL TRUSSES
Minimum volume structures:
![Page 73: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/73.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
x3
y3
x2
y2
x1
y1
x
1.0 sym.
DISCRETE OPTIMAL TRUSSES
![Page 74: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/74.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
DISCRETE OPTIMAL USING MathCAD
![Page 75: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/75.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
1.0
CHARACTERISTICS OF DISCRETE OPTIMAL TRUSSES
Mazurek, A., Baker, W. F., Tort, C. “Geometrical Aspects of Optimum Truss‐Like Structures.” Structural and Multidisciplinary Optimization, 2011, Vol. 43, No. 2, pp. 231‐242.
![Page 76: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/76.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
The entire geometry can be described by only one angle!
Mazurek, A., Baker, W. F., Tort, C. “Geometrical Aspects of Optimum Truss‐Like Structures.” Structural and Multidisciplinary Optimization, 2011, Vol. 43, No. 2, pp. 231‐242.
DISCRETE OPTIMAL TRUSSES
![Page 77: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/77.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Circular quads + discrete Michell turning condition = all quads have same angles.
MAZUREK’S CIRCULAR QUADS
![Page 78: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/78.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Circular quads.
MAZUREK’S CIRCULAR QUADS
![Page 79: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/79.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
We need to benchmark our designs.
![Page 80: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/80.jpg)
Benchmarks α·P·L
π/2=1.57
0.5+π/4=1.29
~1.0
~0.76
1. Michell, 1904, Phil Mag.2. Beghini et al, 2013 Struct. Mult. Opt.
1.
2.
1.
2.
~0.9846
~0.7567
3.
3.
L
d
q
L
d
q
BENCHMARKS α∙P∙L
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
![Page 81: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/81.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Tools for finding optimal geometries
![Page 82: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/82.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Material distribution using density methods
Talischi, C, G.H. Paulino, A. Pereira, I.F.M. Menezes. "PolyMesher: A general‐purpose mesh generator for polygonal elements written in Matlab." Structural and Multidisciplinary Optimization. Vol. 45, No. 3, pp. 309‐328, 2012.
Talischi, C., G.H. Paulino, A. Pereira, I.F.M. Menezes. "PolyTop: a Matlab implementation of a general topology optimization framework using unstructured polygonal finite element meshes." Structural and Multidisciplinary Optimization. Vol. 45, No. 3, pp. 329‐357, 2012.
?
TOPOLOGY OPTIMIZATION
![Page 83: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/83.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Material distribution using density methods
TOPOLOGY OPTIMIZATION
Talischi, C, G.H. Paulino, A. Pereira, I.F.M. Menezes. "PolyMesher: A general‐purpose mesh generator for polygonal elements written in Matlab." Structural and Multidisciplinary Optimization. Vol. 45, No. 3, pp. 309‐328, 2012.
Talischi, C., G.H. Paulino, A. Pereira, I.F.M. Menezes. "PolyTop: a Matlab implementation of a general topology optimization framework using unstructured polygonal finite element meshes." Structural and Multidisciplinary Optimization. Vol. 45, No. 3, pp. 329‐357, 2012.
![Page 84: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/84.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Sokol, T. “A 99 line code for discretized Michell truss optimization written in Mathematica.” Structural and Multidisciplinary Optimization, Vol. 43, pp. 181‐190, 2011.
Ground structures approach:
Let’s assume this to be our benchmark solution. How do other designs compare?
TOPOLOGY OPTIMIZATION
![Page 85: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/85.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
VV0
105.3%
Vol. Ratio for Const. Stress
Deflection for Const. Stress
Vol. Ratio for Equal Deflection
VV0
102.6% 0
102.6%
Discretized optimal truss:
TOPOLOGY OPTIMIZATION
![Page 86: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/86.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
VV0
124.7%VV0
111.6% 0
111.6%
Lattice truss:
TOPOLOGY OPTIMIZATION
Vol. Ratio for Const. Stress
Deflection for Const. Stress
Vol. Ratio for Equal Deflection
![Page 87: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/87.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
VV0
124.7%VV0
111.6% 0
111.6%
TOPOLOGY OPTIMIZATION
Warren truss:
Vol. Ratio for Const. Stress
Deflection for Const. Stress
Vol. Ratio for Equal Deflection
![Page 88: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/88.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
VV0
129.2%VV0
113.7% 0
113.7%
Vol. Ratio for Const. Stress
Deflection for Const. Stress
Vol. Ratio for Equal Deflection
TOPOLOGY OPTIMIZATION
Combined Warren/Pratt truss:
![Page 89: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/89.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
VV0
143.3%VV0
119.7% 0
119.7%
Vol. Ratio for Const. Stress
Deflection for Const. Stress
Vol. Ratio for Equal Deflection
TOPOLOGY OPTIMIZATION
Compression diagonal Pratt (Howe) truss:
![Page 90: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/90.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
VV0
168.4%VV0
129.8% 0
129.8%
Vol. Ratio for Const. Stress
Deflection for Const. Stress
Vol. Ratio for Equal Deflection
TOPOLOGY OPTIMIZATION
Tension diagonal Pratt truss:
![Page 91: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/91.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Stocky Members Story Deep Truss Slender Members
![Page 92: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/92.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
“Stocky Members”
100%
109%
109%
110%
117%
122%
Discrete optimal truss
Lattice truss
Warren truss
Combined Warren/Pratt truss
Tension diagonal Pratt truss
Compression diagonal Howe truss
![Page 93: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/93.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
100%
112%
113%
114%
128%
128%
“Stocky Members”Discrete optimal truss
Lattice truss
Warren truss
Combined Warren/Pratt truss
Tension diagonal Pratt truss
Compression diagonal Howe truss
![Page 94: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/94.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
100%
165%
178%
173%
254%
158%
“Slender Members”Discrete optimal truss
Lattice truss
Warren truss
Combined Warren/Pratt truss
Tension diagonal Pratt truss
Compression diagonal Howe truss
![Page 95: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/95.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017 Wind load case Unitary virtual load case
1
0 1
2
3
4
1
2
3
1
1
1
2
2
2
2
1
iii FAL , ,
VR
iii ii
iii
E
VLAAELfF
1
Minimum Tip Deflection
Optimize w.r.t. Volume
PRINCIPLE OF VIRTUAL WORK
![Page 96: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/96.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017 Wind load case Unitary virtual load case
1
0 1
2
3
4
1
2
3
1
1
1
2
2
2
2
1
iii FAL , ,
PRINCIPLE OF VIRTUAL WORK
E
VLAAELFFLF ii
ii
iiiii
2
)(
Minimum Compliance
Optimize w.r.t. Volume
![Page 97: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/97.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Reciprocal Frames & Graphic Statics
![Page 98: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/98.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
MAXWELL 1864
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
![Page 99: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/99.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
MAXWELL 1864
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
![Page 100: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/100.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Reciprocal diagrams of a gable truss:
Form Diagram Force Diagram
GRAPHIC STATICS
![Page 101: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/101.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
GRAPHIC STATICS
Form Diagram Force Diagram
How is the force diagram constructed?
Step 1 – Create the force polygon by drawing the external forces end to end
![Page 102: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/102.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
GRAPHIC STATICS
Force Diagram
How is the force diagram constructed?
Step 2 – identify Node 1 by drawing parallel lines on the force diagram corresponding to members A‐1 and G‐1
Form Diagram
![Page 103: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/103.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
GRAPHIC STATICS
Force Diagram
How is the force diagram constructed?
Step 2 – identify Node 1 by drawing parallel lines on the force diagram corresponding to members A‐1 and G‐1
Form Diagram
![Page 104: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/104.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
GRAPHIC STATICS
Force Diagram
How is the force diagram constructed?
Step 2 – identify Node 1 by drawing parallel lines on the force diagram corresponding to members A‐1 and G‐1
Form Diagram
![Page 105: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/105.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
GRAPHIC STATICS
Force DiagramForm Diagram
How is the force diagram constructed?
Step 3 – identify Node 2 by drawing parallel lines on the force diagram corresponding to members 1‐2 and G‐2
![Page 106: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/106.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
GRAPHIC STATICS
Force DiagramForm Diagram
How is the force diagram constructed?
Step 3 – identify Node 2 by drawing parallel lines on the force diagram corresponding to members 1‐2 and G‐2
![Page 107: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/107.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
GRAPHIC STATICS
Force DiagramForm Diagram
How is the force diagram constructed?
Step 3 – identify Node 2 by drawing parallel lines on the force diagram corresponding to members 1‐2 and G‐2
![Page 108: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/108.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
GRAPHIC STATICS
Force DiagramForm Diagram
How is the force diagram constructed?
Step 4 – identify Node 3 by drawing parallel lines on the force diagram corresponding to members 2‐3 and B‐3
![Page 109: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/109.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
GRAPHIC STATICS
Force DiagramForm Diagram
How is the force diagram constructed?
Step 4 – identify Node 3 by drawing parallel lines on the force diagram corresponding to members 2‐3 and B‐3
![Page 110: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/110.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
GRAPHIC STATICS
Force DiagramForm Diagram
How is the force diagram constructed?
Step 4 – identify Node 3 by drawing parallel lines on the force diagram corresponding to members 2‐3 and B‐3
![Page 111: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/111.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
GRAPHIC STATICS
Force DiagramForm Diagram
How is the force diagram constructed?
Step 5 – identify Node 4 by drawing parallel lines on the force diagram corresponding to members 3‐4 and G‐4
![Page 112: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/112.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
GRAPHIC STATICS
Force DiagramForm Diagram
How is the force diagram constructed?
Step 5 – identify Node 4 by drawing parallel lines on the force diagram corresponding to members 3‐4 and G‐4
![Page 113: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/113.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
GRAPHIC STATICS
Force DiagramForm Diagram
How is the force diagram constructed?
Step 5 – identify Node 4 by drawing parallel lines on the force diagram corresponding to members 3‐4 and G‐4
![Page 114: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/114.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
GRAPHIC STATICS
Force DiagramForm Diagram
How is the force diagram constructed?
Step 6 – identify Node 5 by drawing parallel lines on the force diagram corresponding to members 4‐5 and C‐5
![Page 115: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/115.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
GRAPHIC STATICS
Force DiagramForm Diagram
How is the force diagram constructed?
Step 6 – identify Node 5 by drawing parallel lines on the force diagram corresponding to members 4‐5 and C‐5
![Page 116: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/116.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
GRAPHIC STATICS
Force DiagramForm Diagram
How is the force diagram constructed?
Step 6 – identify Node 5 by drawing parallel lines on the force diagram corresponding to members 4‐5 and C‐5
![Page 117: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/117.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
GRAPHIC STATICS
Force DiagramForm Diagram
How is the force diagram constructed?
Step 7 – identify Node 6 by drawing parallel lines on the force diagram corresponding to members 5‐6 and D‐6
![Page 118: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/118.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
GRAPHIC STATICS
Force DiagramForm Diagram
How is the force diagram constructed?
Step 7 – identify Node 6 by drawing parallel lines on the force diagram corresponding to members 5‐6 and D‐6
![Page 119: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/119.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
GRAPHIC STATICS
Force DiagramForm Diagram
How is the force diagram constructed?
Step 7 – identify Node 6 by drawing parallel lines on the force diagram corresponding to members 5‐6 and D‐6
![Page 120: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/120.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
GRAPHIC STATICS
Force DiagramForm Diagram
How is the force diagram constructed?
Step 8 – Exploiting the symmetry of the truss, the rest of the force polygon can be drawn.
![Page 121: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/121.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
GRAPHIC STATICS
Force DiagramForm Diagram
How is the force diagram constructed?
Step 8 – Exploiting the symmetry of the truss, the rest of the force polygon can be drawn.
![Page 122: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/122.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
GRAPHIC STATICS
How can we make the force in the top chord constant?
Modify the force diagram and work backwards!
Force DiagramForm Diagram
![Page 123: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/123.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Graphic Staticsas a Design Tool.
![Page 124: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/124.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Force DiagramForm Diagram
GRAPHIC STATICS
How can we make the force in the top chord constant?
Modify the force diagram and work backwards!
![Page 125: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/125.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Lines a‐1, b‐3, c‐5, d‐6, e‐8 and f‐10 must be the
same length
GRAPHIC STATICS
How can we make the force in the top chord constant?
Modify the force diagram and work backwards!
Force DiagramForm Diagram
![Page 126: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/126.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Lines a‐1, b‐3, c‐5, d‐6, e‐8 and f‐10 must be the
same length
GRAPHIC STATICS
How can we make the force in the top chord constant?
Modify the force diagram and work backwards!
Force DiagramForm Diagram
![Page 127: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/127.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Lines a‐1, b‐3, c‐5, d‐6, e‐8 and f‐10 must be the
same length
GRAPHIC STATICS
How can we make the force in the top chord constant?
Modify the force diagram and work backwards!
Force DiagramForm Diagram
![Page 128: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/128.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Lines a‐1, b‐3, c‐5, d‐6, e‐8 and f‐10 must be the
same length
GRAPHIC STATICS
How can we make the force in the top chord constant?
Modify the force diagram and work backwards!
Force DiagramForm Diagram
![Page 129: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/129.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Lines a‐1, b‐3, c‐5, d‐6, e‐8 and f‐10 must be the
same length
GRAPHIC STATICS
How can we make the force in the top chord constant?
Modify the force diagram and work backwards!
Force DiagramForm Diagram
![Page 130: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/130.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Lines a‐1, b‐3, c‐5, d‐6, e‐8 and f‐10 must be the
same length
GRAPHIC STATICS
How can we make the force in the top chord constant?
Modify the force diagram and work backwards!
Force DiagramForm Diagram
![Page 131: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/131.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Lines a‐1, b‐3, c‐5, d‐6, e‐8 and f‐10 must be the
same length
GRAPHIC STATICS
How can we make the force in the top chord constant?
Modify the force diagram and work backwards!
Force DiagramForm Diagram
![Page 132: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/132.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Lines a‐1, b‐3, c‐5, d‐6, e‐8 and f‐10 must be the
same length
GRAPHIC STATICS
How can we make the force in the top chord constant?
Modify the force diagram and work backwards!
Force DiagramForm Diagram
![Page 133: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/133.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Lines a‐1, b‐3, c‐5, d‐6, e‐8 and f‐10 must be the
same length
GRAPHIC STATICS
How can we make the force in the top chord constant?
Modify the force diagram and work backwards!
Force DiagramForm Diagram
![Page 134: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/134.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
CONSTANT‐FORCE GABLE TRUSS
Magazzini Generali WarehouseRobert Maillart, 1924
![Page 135: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/135.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Shaping Structures: Statics, 1997Form and Forces: Designing Efficient, Expressive Structures ,2012
EDWARD ALLEN & WACLAW ZALEWSKI
![Page 136: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/136.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Wolfe, William S. “Graphical Analysis: A Text Book on Graphic Statics.” McGraw‐Hill book Company, Incorporated, 1921
![Page 137: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/137.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Topology Optimization Plus Graphic Statics
![Page 138: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/138.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
GRAPHIC STATICS PROVIDES THE INFORMATION NEEDED TO MINIMIZE THE LOAD PATH.
• The Force Diagram provides the member forces.
• The Form Diagram provides the member lengths.
![Page 139: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/139.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Discrete Optimal Design
Talischi, C., G.H. Paulino, A. Pereira, I.F.M. Menezes. "PolyTop: a Matlab implementation of a general topology optimization framework using unstructured polygonal finite element meshes." Structural and Multidisciplinary Optimization. Vol. 45, No. 3, pp. 329‐357, 2012.
TOPOLOGY OPTIMIZATION AND GRAPHIC STATICS: BRIDGE DESIGN PROBLEM
![Page 140: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/140.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
What about unequal stresses or member buckling?
![Page 141: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/141.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
But for the bridge problem, the Maxwell tells us:
CCTT LFLF
TT
CT
LPV11minmin
xx
The optimal geometry does NOT change if the compressive stress is a constant even if it is lower than the tensile stress!
So, the problems can be rewritten
TOPOLOGY OPTIMIZATION AND GRAPHIC STATICS: BRIDGE DESIGN PROBLEM WITH CONSTANT BUT DIFFERENT TENSION AND COMPRESSION STRESSES
![Page 142: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/142.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
CCTT LFLF
CCb
CTT
T
LPLLLPV 20 )/(11minmin
xx
Once again, for the bridge problem, the Maxwell tells us:
The optimal geometry DOES change if the compressive stress varies with length!
So, the problems can be rewritten
TOPOLOGY OPTIMIZATION AND GRAPHIC STATICS: BRIDGE DESIGN PROBLEM WITH STRESSES VARYING WITH LENGTH
![Page 143: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/143.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
For a constant force in the bottom chord and the compression stresses are constant, the optimal geometry is:
TOPOLOGY OPTIMIZATION AND GRAPHIC STATICS
![Page 144: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/144.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
CCb
CTT
T
LPLLLPV 20 )/(11minmin
xx
TOPOLOGY OPTIMIZATION AND GRAPHIC STATICS
But if the compressive stresses are not constant but vary with the unbraced lengths, the truss becomes shallower!
![Page 145: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/145.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
On the Geometric Nature of Truss Forms and Forces
![Page 146: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/146.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
E
12
4 63
A C DB
5
e
1
2
4
6
3
a
c
d
b
5
![Page 147: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/147.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
E
12
4 63
A C DB
5
e
1
2
4
6
3
a
c
d
b
5
f
![Page 148: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/148.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
E
12
4 63
A C DB
5
e
1
2
4
6
3
a
c
d
b
5
F
f
![Page 149: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/149.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
E
12
4 63
A C DB
5
e
1
2
4
6
3
a
c
d
b
5
F
f
![Page 150: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/150.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
![Page 151: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/151.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
![Page 152: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/152.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
![Page 153: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/153.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
![Page 154: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/154.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
Mechanisms and states of self‐stress
![Page 155: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/155.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
THREE VERSUS FOUR LEGGED STOOL.
![Page 156: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/156.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
32 bVN
13862 N 03962 N 131062 N
smN
![Page 157: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/157.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
032 bV
0m
0s
032 bV
1m
1s
TWO STRUCTURES WITH N = M – S = 0
![Page 158: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/158.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
ApplicationsIs a structure stiff?
![Page 159: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/159.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
A structure with N=0 is stiff if m = 0m =0 if s=0If not a projection of polyhedron, s=0If the reciprocal diagram cannot be drawn, s=0
N = 2 v – b – 3 = m- s = 0
This structure is stiff
IS THE STRUCTURE STIFF?
![Page 160: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/160.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
N = 2 v – b – 3 = m- s = 0
A structure with N=0 is stiff if m = 0m =1 if s=1If it is a projection of polyhedron, s>=1, m>=1If the reciprocal diagram can be drawn, s>=1,m>=1
This structure is stressable and has a mechanism.If prestressed, it will have some stiffness but of a lower order.
IS THE STRUCTURE STIFF?
![Page 161: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/161.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
TAKE‐AWAYS
• Maxwell’s theorem on load paths is simple and powerful.• Inefficiencies must be paid for twice• Minimize one – the other is also minimized• Useful tool for systems design
• Discrete Michell trusses are regular and ordered.
• Graphic Statics is a powerful design tool.
• Topology optimization tools (plus Graphic Statics or other analysis methods) make efficient layouts for complex problems accessible to the designer.
![Page 162: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/162.jpg)
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017
WORDS OF CAUTION
• Check other load cases particularly non‐uniform load cases.
• Look out for structural mechanisms.
• Consider redundancy.
• If there is more than one dominated load case, try to develop a geometry that is appropriate.
• In the end, a structure only has one geometry; try to get the best.
![Page 163: Geometry and Design of Truss Structures](https://reader031.vdocuments.site/reader031/viewer/2022012016/61da91ae11d89a5fb03799a8/html5/thumbnails/163.jpg)
THANK YOU!
GEOMETRY AND THE DESIGN OF TRUSS STRUCTURES© SKIDMORE, OWINGS & MERRILL LLP 2017