geometry and design of truss structures

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



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.


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.


Systems are essential for efficiency.



GEOMETRY IS A KEY COMPONENT OF STRUCTURAL SYSTEMS.


Day One

Structure75%

Non-Structure

25%

EFFICIENT STRUCTURES CONSUME LESS RESOURCES

TOPOLOGY

SHAPE

DOMAIN

SIZE

WHAT MATTERS



Where to look for guidance on systems?


Let’s start at the beginning of modern structural engineering:

The Mid‐19th Century.


UNDERSTANDING OF STRUCTURAL BEHAVIOR BY THE MID‐1800’S


A new problem: trusses.

First metal trusses (US: 1840 UK: 1845)


Will focus not on the Maxwell‐Betti reciprocal theorem

but the little known Theorem of Load Paths

Today’s Presentation


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


What does Maxwell’s Theorem on Load Paths tell us?


The longer the tension load path is, the longer the compression load path has to be

and vice versa.


Inefficiencies are paid for exactly twice: once in tension and once in compression.


Optimizing tension load path automatically optimizes compression load path and vice

versa.


The Maxwell constant and either the tension load path or

the compression load path determines total load path.


If only tension members oronly compression members then load path is equal to Maxell’s Constant for all possible

layouts.

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


Load Path → Tonnage

Usually


x

y

iP

irii LF ,

x

y

PROOF: MAXWELL’S THEOREM


iii

ii rPLF

ii

ncompressioiii

tensioniii rPLFLF

WorkExternalWorkInternal x

y

iP

iP

ir

ir2

ii LF ,

ii LF ,

x

y

PROOF: MAXWELL’S THEOREM


Cantilever with 3 to 1 span

MAXWELL’S THEOREM ON LOAD PATHS: AN EXAMPLE


0rP




03 rP P

PBrPp




PBrP ii




PBLF TT 10

Moment diagram for truss geometry:

PBLF CC 9

PBLFLF CCTT

PBLFLF CCTT 19EB19



PBLF TT 9

PBLF CC 8

PBLFLF CCTT

PBLFLF CCTT 17EB17


Pratt truss:


PBLF TT 8

PBLF CC 7

PBLFLF CCTT

PBLFLF CCTT 15EB15


Warren truss:


The most material efficient truss is also the stiffest truss!



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”


Can we find a benchmark for our designs?


How low can we go?

FT LT FC LC 13.92PB



PBLF TT 7.7

PBLF CC 7.6

PBLFLF CCTT

PBLFLF CCTT 47.14EB47.14

Bounded optimal truss:



PBLF TT 52.8

PBLF CC 52.7

PBLFLF CCTT

PBLFLF CCTT 04.16EB04.16

Cantilever with only compression chord:



PBLFLF CCTT 17.13


How low can we go?


PBLF TT 8

PBLF CC 7

PBLFLF CCTT

PBLFLF CCTT 15EB15

Within 14% of Benchmark


Warren truss:


Preliminary System Design Using Maxwell’s Theorem on Load Paths

EXCHANGE HOUSE, LONDON



APPLYING MAXWELL’S THEOREM: EXCHANGE HOUSE


APPLYING MAXWELL’S THEOREM: CONCEPTUAL DESIGN OF EXCHANGE HOUSE


Tension members: Hangers

FT LThangers 2 Wydy

0

z 1 2 xB

2

dx0

B/2

4

15WBz2



FT LTtie

WB3H8z

Tension members: Tie



FT LT FT LThangers FT LT

tie

415

WBz2 18zWB3H

Tension members: Total



FT LT z

0 8Bz15

B3H8z2 0 z 15B2H

643

Minimum total load path:



P r BHW

H2

BH 2W2


Maxwell’s constant can be found using column support only:


Use Basic Maxwell Theory Application

FLcolumns

only




FLcolumns

only

2 FT LThangersand ties



FLcolumns

only

2 FT LThangersand ties

FLtotal




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.



The Arch Load Path was calculated but not explicitly. How?



Consider a segment of an arch:




Total load path: FC LCarch FT LT

tie FT LT

hangers FC LC

columnsbelow arch


Michell Trusses (1904)


P

lN

(1±)l

P

Actual frame Virtually deformed frame

APPLICATION OF VIRTUAL WORK


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


1

2

2

n12

avg

n,max / 2y, xy / 2

x, xy / 2

2n


Mohr’s Circle:

Plane strain


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



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)


Minimum volume structures:

P allowable tensile stressQ allowable compressive stress

a AB a AC CBa AC CB



2a AB

P allowable tensile stressQ allowable compressive stressL moment of transmitted couple latitude of circles about pole


Minimum volume structures:


x3

y3

x2

y2

x1

y1

x

1.0 sym.

DISCRETE OPTIMAL TRUSSES


DISCRETE OPTIMAL USING MathCAD


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.


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


Circular quads + discrete Michell turning condition = all quads have same angles.

MAZUREK’S CIRCULAR QUADS


Circular quads.

MAZUREK’S CIRCULAR QUADS


We need to benchmark our designs.

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



Tools for finding optimal geometries


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


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.



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?



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:



VV0

124.7%VV0

111.6% 0

111.6%

Lattice truss:






VV0

124.7%VV0

111.6% 0

111.6%


Warren truss:





VV0

129.2%VV0

113.7% 0

113.7%





Combined Warren/Pratt truss:


VV0

143.3%VV0

119.7% 0

119.7%





Compression diagonal Pratt (Howe) truss:


VV0

168.4%VV0

129.8% 0

129.8%





Tension diagonal Pratt truss:


Stocky Members Story Deep Truss Slender Members


“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


100%

112%

113%

114%

128%

128%

“Stocky Members”Discrete optimal truss

Lattice truss

Warren truss





100%

165%

178%

173%

254%

158%

“Slender Members”Discrete optimal truss

Lattice truss

Warren truss




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

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


Reciprocal Frames & Graphic Statics


MAXWELL 1864



Reciprocal diagrams of a gable truss:

Form Diagram Force Diagram

GRAPHIC STATICS


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


GRAPHIC STATICS

Force Diagram


Step 2 – identify Node 1 by drawing parallel lines on the force diagram corresponding to members A‐1 and G‐1

Form Diagram


GRAPHIC STATICS

Force DiagramForm Diagram


Step 3 – identify Node 2 by drawing parallel lines on the force diagram corresponding to members 1‐2 and G‐2


GRAPHIC STATICS



Step 4 – identify Node 3 by drawing parallel lines on the force diagram corresponding to members 2‐3 and B‐3


GRAPHIC STATICS



Step 5 – identify Node 4 by drawing parallel lines on the force diagram corresponding to members 3‐4 and G‐4


GRAPHIC STATICS



Step 6 – identify Node 5 by drawing parallel lines on the force diagram corresponding to members 4‐5 and C‐5


GRAPHIC STATICS



Step 7 – identify Node 6 by drawing parallel lines on the force diagram corresponding to members 5‐6 and D‐6


GRAPHIC STATICS



Step 8 – Exploiting the symmetry of the truss, the rest of the force polygon can be drawn.


GRAPHIC STATICS

How can we make the force in the top chord constant?

Modify the force diagram and work backwards!



Graphic Staticsas a Design Tool.



GRAPHIC STATICS




Lines a‐1, b‐3, c‐5, d‐6, e‐8 and f‐10 must be the

same length

GRAPHIC STATICS





CONSTANT‐FORCE GABLE TRUSS

Magazzini Generali WarehouseRobert Maillart, 1924


Shaping Structures: Statics, 1997Form and Forces: Designing Efficient, Expressive Structures ,2012

EDWARD ALLEN & WACLAW ZALEWSKI


Wolfe, William S. “Graphical Analysis: A Text Book on Graphic Statics.” McGraw‐Hill book Company, Incorporated, 1921


Topology Optimization Plus Graphic Statics


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.


Discrete Optimal Design


TOPOLOGY OPTIMIZATION AND GRAPHIC STATICS: BRIDGE DESIGN PROBLEM


What about unequal stresses or member buckling?


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


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


For a constant force in the bottom chord and the compression stresses are constant, the optimal geometry is:

TOPOLOGY OPTIMIZATION AND GRAPHIC STATICS


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!


On the Geometric Nature of Truss Forms and Forces


E

12

4 63

A C DB

5

e

1

2

4

6

3

a

c

d

b

5


E

12

4 63

A C DB

5

e

1

2

4

6

3

a

c

d

b

5

f


E

12

4 63

A C DB

5

e

1

2

4

6

3

a

c

d

b

5

F

f


Mechanisms and states of self‐stress


THREE VERSUS FOUR LEGGED STOOL.


32 bVN

13862 N 03962 N 131062 N

smN


032 bV

0m

0s

032 bV

1m

1s

TWO STRUCTURES WITH N = M – S = 0


ApplicationsIs a structure stiff?


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?


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?


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.


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.

THANK YOU!


geometry and design of truss structures

Documents