On the Insurmountable Size of Truss-like Structures

On the Insurmountable Size of

Truss-like Structures

Carlos Olmedo Rojas; Mariano Vázquez Espí;

Jaime Cervera Bravo

Contents

1 Galileo’s problem and structural efficiency

2

2 Truss-like Structures and self-weight

6

3 The scope of structural schema

10

4 Use of Galileo’s rule in practise

17

5 Conclusion: future researches

18

1

Galileo’s problem and structural efficiency

Efficiency and Cost

Thermodynamic efficiency:

useful quantity

total quantity

=

exergy

energy

< 1

“Load” efficiency:

r =

useful load

useful load

+

structural weight

=

Q

Q + P

< 1

Thermodynamic cost:

1

thermodynamic efficiency

> 1

“Load” cost: k =

useful load

+

structural weight

useful load

=

Q + P

Q

> 1

Galileo

H < L

H

H = L

H

H > L

H

Cylindrical columns of given

material:

ρ(kN/m

3

_{) and f (N/mm}

2

₎

H ≤ L (insurmountable size)

relative size:

H

L

Galileo’s rule:

r = 1 −

H

L

=

1

k

P = Q · (k − 1)

L = A =

f

ρ

(material scope)

Maxwell and Michell

Q

Q

Q

L

L

L

R

R

R

Maxwell’s problem

Q

Q

Q

−

Q

−

Q

Q

−

Q

Q

Q

−

Q

υ =

3

2

L

L

L

L

L

A Maxwell structure

Size measure: L. Useful load: Q

Maxwell number:

M =

P

F

~

i

· ~

R

i

= −

1

₄

QL

Stress volume:

V =

P

|N

i

| · ℓ

i

≥ |M|

Michell number:

_{υ = V/QL}

Aroca

If L = 0 then P =

υ · QL

A

If L > 0 then P >

υ · QL

A

Aroca rule: If {P} ≈ {Q} then P ≈

υ · P L

A

, hence:

L ≈

A

υ

GA rule:

_{r ≈ 1 − υ}

L

A

2

Truss-like Structures and self-weight

Model assumptions:

• bars are of constant cross-section

• identical tension-compression patterns on bars due to

both useful load and self-weight (i.e., equal sign of the

internal force in each bar in these two independent

load conditions)

• the material has the same absolute value of allowable

stress in tension than in compression

• buckling of compressed bars is not taken into

consideration

Our questions:

scope L? efficiency r? load cost k?

The model for Maxwell’s structures

Q → N

Q

, P → N

P

, subject to equilibrium condition

Q = HN

Q

P = HN

P

Q + P = HN

χ

N

i

χ

N

i

χ

P

0

= ω

0

i

N

χ

i

P

1

= ω

1

i

N

χ

i

P

0

+ P

1

(generic bar: ρ, A)

ℓ

x

ℓ

β

The model for Maxwell’s structures

P = Ω

L

N

χ

Q = (H − Ω

L

) N

χ

(H − Ω

L=L

) N

χ

= 0

|H − Ω

L

| = 0

N

i

χ

N

i

χ

P

0

= ω

0

i

N

χ

i

P

1

= ω

1

i

N

χ

i

P

0

+ P

1

(generic bar: ρ, A)

ℓ

x

ℓ

β

The model for Maxwell’s structures

bar type

ω

ij

straight beam

ω

0

j

= ω

1

j

= f (h/ℓ, ℓ, β, A, . . .)

catenary arc

ω

ij

= g

i

(ℓ, β, A)

N

i

χ

N

i

χ

P

0

= ω

0

i

N

χ

i

P

1

= ω

1

i

N

χ

i

P

0

+ P

1

(generic bar: ρ, A)

ℓ

x

ℓ

β

3

The scope of structural schema

j

t

p

ci

cs

a

a

2a

2a

L

h

b

Geometry Definition

λ = 8

λ = 16

λ = 4

Affine transformations: schema

Slenderness, size and structural scope

Q/2

Q

Q

Q/2

N

=

Q

N

=

Q

N

= −

Q

N

= −

Q

N

=

Q

Useful Load

Slenderness, size and structural scope

Q/2

Q

Q

Q/2

N

=

Q

N

=

Q

N

= −

Q

N

= −

Q

N

=

Q

Useful Load

P

P

P

P

P

P

N

=

(P

+ 2P

)

N

=

(P

+ P

)

N

= −

(P

+ 2P

)

= −

(P

+ P

)

N

=

P

Self Weight

Slenderness, size and structural scope

Q/2

Q

Q

Q/2

N

=

Q

N

=

Q

N

= −

Q

N

= −

Q

N

=

Q

Useful Load

Layout with Catenary Bars for the Insurmountable Size

Case λ = 8, L(λ = 8) = 0.66A

Slenderness, size and structural scope

Direct application of GA rule (L → 0)

υ(λ) =

5λ

27

+

12

9λ

λ

opt

= min

λ

υ(λ) =

6

√

5

= 2.6833

GA rule Present Method

λ

opt

2.6833

2.455

scope L/A

1.006

1.1851

Slenderness, size and structural scope

: Numerical solution

: GA rule

0

.1

0

.2

0

.3

0

.4

0

.5

0

.6

0

.7

0

.8

0

.9

1

.0

1

.1

1

.2

0

.1

0

.2

0

.3

0

.4

0

.5

0

.6

0

.7

0

.8

0

.9

1

.0

L/A

r

λ = 2.455

8

Slenderness, size and structural scope

Catenary Arcs

: Numerical Results

: GA Rule

Rectangular Cross-Section Beams, Numerical Results

: k

= 0

.1, ℓ/h = 10

: k

= 0

.2, ℓ/h = 5

10

20

30

40

50

0

.1

0

.2

0

.3

0

.4

0

.5

0

.6

0

.7

0

.8

0

.9

1

.0

1

.1

1

.2

λ

L/A

λ = 2.455

λ = 2.6833

4

Use of Galileo’s rule in practise

Design examples

A = 2300 m, rectangular cross-section, λ = 8

h/ℓ

0.1

0.2

L (m)

407

688

size (m)

10

50

150

relative size

2.5% 12%

37%

22%

r

98%

88%

63%

78%

k

1.02

1.14

1.58

1.28

P/Q

2%

14%

58%

28%

quality

OK

Ok

not good

ok

5

Conclusion: future researches

The proposed method when applied to well definite Maxwell problems can be

used to resolve concrete structural form determining the thickness of its bars,

its total self-weight and its structural efficiency. Besides, the method can be

used to explore the space of solutions associated to the Maxwell problem and

to a definite structural schema, determining the relationships between size,

slenderness, insurmountable size, and efficiency of each solution in the search

space.

It could be improved in several ways:

• incorporating a more realistic model of the stress

distribution for beams, e.g., Timoshenko beam theory

• managing different values for tension and compression

allowable stress

• considering the local buckling of beams and

compressed catenary arcs.

On the Insurmountable Size of Truss-like Structures

Carlos Olmedo Rojas; Mariano Vázquez Espí; Jaime Cervera Bravo

Departamento de Física y Estructuras de Edificación

Grupo de Investigación en Arquitectura, Urbanismo y Sostenibilidad

Universidad Politécnica de Madrid

Edición del 29 de junio de 2015

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