Introduction to the S[tr/c]u[c/lp]ture workshop held at the Sapienza University of Rome on 11 04 2015

1. April 11 2015April 11 2015
Sapienza University of RomeSapienza University of Rome
School of EngineeringSchool of Engineering
//// Sapienza and DTU workshopSapienza and DTU workshop ////
Sapienza University of RomeSapienza University of Rome,, School of EngineeringSchool of Engineering,, AprilApril 11 201511 2015
Structural optimization in parametricsStructural optimization in parametrics
Optimal lines on free-form surfacesOptimal lines on free-form surfaces
IntroductionIntroduction
Konstantinos GkoumasKonstantinos Gkoumas
StroNGER srl co-founder and partnerStroNGER srl co-founder and partner

2. April 11 2015April 11 2015
Sapienza University of RomeSapienza University of Rome
School of EngineeringSchool of Engineering

3. April 11 2015April 11 2015
Sapienza University of RomeSapienza University of Rome
School of EngineeringSchool of Engineering
THE UNIVERSITY
• Founded in 1303 by Pope Boniface
• 63 Departments
• 11 Faculties
• 2 University Hospitals
• 154 Bachelor courses
• 120 Master courses
• 243 Prof. Master courses
• 86 PhD courses
• 128,963 students
• 3997 Professors (including
assistant-associate)
• 914 International agreements
• 20 double degrees
• 100 visiting professors
• 183 FP7 programs in 2007-2013

4. April 11 2015April 11 2015
Sapienza University of RomeSapienza University of Rome
School of EngineeringSchool of Engineering
THE FACULTY OF CIVIL AND INDUSTRIAL
ENGINEERING
• Founded in 1817
by Pope Pius VII
• In 1935, the
School became
the Faculty of
Engineering
• Nowadays:
“Faculty of Civil
and Industrial
Engineering”

5. April 11 2015April 11 2015
Sapienza University of RomeSapienza University of Rome
School of EngineeringSchool of Engineering
THE ST. PETER IN CHAINS
BASILICA
• first rebuilt on
older foundations
in 432–440 to
house the relic of
the chains that
bound Saint Peter
• It houses
Michelangelo's
Moses statue
(completed in
1515)

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Sapienza University of RomeSapienza University of Rome
School of EngineeringSchool of Engineering
THE ST. PETER IN CHAINS
BASILICA

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Sapienza University of RomeSapienza University of Rome
School of EngineeringSchool of Engineering
THE CLOISTER
Giuliano da Sangallo
15th century

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Sapienza University of RomeSapienza University of Rome
School of EngineeringSchool of Engineering
THE CLOISTER

9. April 11 2015April 11 2015
Sapienza University of RomeSapienza University of Rome
School of EngineeringSchool of Engineering
THE CLOISTER

10. April 11 2015April 11 2015
Sapienza University of RomeSapienza University of Rome
School of EngineeringSchool of Engineering
THE CLOISTER

11. April 11 2015April 11 2015
Sapienza University of RomeSapienza University of Rome
School of EngineeringSchool of Engineering
THE GROUP OF STRUCTURAL ANALYSIS AND
DESIGN
Franco Bontempi, PhD
Prof. of Structural Analysis and Design
Sapienza University of Rome
Str
o N
GER
www.stronger2012.com
Academic research Industry research - R&D
University courses Professional courses
Big group Small group

12. April 11 2015April 11 2015
Sapienza University of RomeSapienza University of Rome
School of EngineeringSchool of Engineering
THE GROUP OF STRUCTURAL ANALYSIS AND
DESIGN

13. April 11 2015April 11 2015
Sapienza University of RomeSapienza University of Rome
School of EngineeringSchool of Engineering
THE TECHNICAL UNIVERSITY OF
DENMARK
• Founded in 1829
• Today ranked among
Europe's leading
engineering institutions
• Academic staff: 2,003
• Administrative staff: 1,540
• Students: 11,190
• Undergraduates: 6,803
• Doctoral students:1,200

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Sapienza University of RomeSapienza University of Rome
School of EngineeringSchool of Engineering
VANKE PAVILION
RAMBOLL Computational Design

15. April 11 2015April 11 2015
Sapienza University of RomeSapienza University of Rome
School of EngineeringSchool of Engineering
VANKE PAVILION
RAMBOLL Computational Design

16. April 11 2015April 11 2015
Sapienza University of RomeSapienza University of Rome
School of EngineeringSchool of Engineering
VANKE PAVILION
RAMBOLL Computational Design

17. April 11 2015April 11 2015
Sapienza University of RomeSapienza University of Rome
School of EngineeringSchool of Engineering
VANKE PAVILION
RAMBOLL Computational Design

18. April 11 2015April 11 2015
Sapienza University of RomeSapienza University of Rome
School of EngineeringSchool of Engineering
WORKSHOP PROGRAM OUTLINE
09.15-09.45 Introduction: the Pierluigi Nervi inspiration Konstantinos
09.45-11.15 Grasshopper introduction and exercises Salma & Kareem
11.15-11.30 Morning break
11.30-13.00 Karamba introduction and exercises
•Simple beams (cantilever and simply supported), cross
sections, loads, materials, joints / truss elements
•Introduction to 2nd order analysis, buckling modes,
eigenfrequencies
•Mesh and shell elements (importance of good mesh
qualities and exercises in shell elements)
•Shell analysis tool (force-flow, principal stress, principal
bending)
Kristjan & Mariam
13.00-14.00 Lunch break
14.00-17.00 Design exercise
•Analyze existing structure(s)
•Design of new free-form shed covering the open space
over the Sapienza Engineering Faculty Cloister
17.00-18.00 Group presentations

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Sapienza University of RomeSapienza University of Rome
School of EngineeringSchool of Engineering
isostatic lines – Nervi’s inspiration

20. April 11 2015April 11 2015
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School of EngineeringSchool of Engineering
Isostatic lines define the directions of principal stress to visualize the stress
trajectories in beams and other elements. In other words, they indicate by their
direction at any point, the direction of one of the principal stresses.
Beam with isostatic lines (thick
compression lines and thin tension lines)
isostatic lines – a brief recall
Cantilever beam

21. April 11 2015April 11 2015
Sapienza University of RomeSapienza University of Rome
School of EngineeringSchool of Engineering
Lines of stress within a material subject to
compressive loading graphically describing the
isostatic lines around a hole – in the far right case
with a stiffer infill.
"Hole Force Lines" by Kaidor ‘Form and Forces’, by Edward
Allen and Waclaw Zalewski
isostatic lines – “force follows stiffness”

22. April 11 2015April 11 2015
Sapienza University of RomeSapienza University of Rome
School of EngineeringSchool of Engineering
stress trajectory: a line showing the continuous
change in the orientation of a principal stress
throughout a body.
although trajectories may curve, their intersections
with other principal stresses remain perpendicular.
isostatic lines: stress trajectories in surfaces

23. April 11 2015April 11 2015
Sapienza University of RomeSapienza University of Rome
School of EngineeringSchool of Engineering
isostatic lines: stress trajectories in surfaces
D'Aloisio, Righi, Modelli matematici per l’architettura e il calcolo numerico 2013 2014
Dr. Luca Sgambi, Politecnico di Milano

24. April 11 2015April 11 2015
Sapienza University of RomeSapienza University of Rome
School of EngineeringSchool of Engineering
Considering (Arcangeli) that a 2D
continuous body subjected to normal
forces produces two families of
orthogonal curves (isostatics),
tangential to the principal bending
moment trajectories, along which
torsional moments are equal to zero.
If this continuous body is replaced by
ribs oriented along the isostatics, then
the rib structure and the continuous
body would have identical structural
behavior under identical loading and
support conditions.
isostatic lines – Nervi’s inspiration
Isostatic inspiration for the rib patterns of Nervi’s floor systems
Isostatic Ribbed Floor Slab

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School of EngineeringSchool of Engineering
Source: Nervi, P.L., Aesthetics and Technology in Building, Harvard University Press (1965).
Each 5m x 5m slab of the Gatti Wool
Factory is supported by a central column.
All slabs are monolithically joined along the
perimeter edges.
• red lines: primary isostatics, corresponding
to the maximum principal bending moments
• blue lines: represent the secondary
isostatics, corresponding to the minimum
principal bending moments.
Gatti Wool Factory floors
Design: Nervi, Arcangeli, Cesteli Guidi
ProjectNervi:AestheticsandTechnology,Dale
Clifford,CarnegieMellonUniversity

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School of EngineeringSchool of Engineering
C.C. Guidi e P.L.
Nervi
Lanificio Gatti Roma
1951-53 solaio a
nervature isostatiche
Gatti Wool Factory floors
Design: Nervi, Arcangeli, Cesteli Guidi

27. April 11 2015April 11 2015
Sapienza University of RomeSapienza University of Rome
School of EngineeringSchool of Engineering
Gatti Wool Factory floors
Design: Nervi, Arcangeli, Cesteli Guidi

28. April 11 2015April 11 2015
Sapienza University of RomeSapienza University of Rome
School of EngineeringSchool of Engineering
Source: Nervi, P.L., Aesthetics and Technology
in Building, Harvard University Press (1965).
Each 10m x 10m slab is supported by columns at the four corners and the isostatic patterns
follow one-eighth symmetry
Palace of Labor (Palazzo del Lavoro), Turin
Design: Nervi, Arcangeli, Cesteli Guidi
Source: Project Nervi - Aesthetics and Technology,
Dale Clifford, Carnegie Mellon University

29. April 11 2015April 11 2015
Sapienza University of RomeSapienza University of Rome
School of EngineeringSchool of Engineering
Analogy between natural and artificial forms: in nature there is a “natural reinforcement” along the most
stresses zones
A similarity in nature (?)

30. April 11 2015April 11 2015
Sapienza University of RomeSapienza University of Rome
School of EngineeringSchool of Engineering
A similarity in nature (?)
left: Wolff’s Law traces bone growth along principle stress trajectories
right: cross section of a human femur
Wolff’s Law developed the theory that bone grows in response to stress or put
another way, the internal patterning of bone is transformable and responsive to
external loading from the environment. The converse is also true, that bone will
degenerate if not subject to loading.
‘OngrowthandForm,’D.Thompson,Dover
reprintof19422nded.(1sted.,1917)1992

31. April 11 2015April 11 2015
Sapienza University of RomeSapienza University of Rome
School of EngineeringSchool of Engineering
Palazzetto dello Sport di Roma
A. Vitellozzi e
P.L. Nervi
Palazzetto dello Sport
Roma 1956-57
sezioni schematiche

32. April 11 2015April 11 2015
Sapienza University of RomeSapienza University of Rome
School of EngineeringSchool of Engineering
A. Vitellozzi e P.L. Nervi
Palazzetto dello Sport
Roma 1956-57
particolare del pilastro a forcella
Palazzetto dello Sport di Roma

33. April 11 2015April 11 2015
Sapienza University of RomeSapienza University of Rome
School of EngineeringSchool of Engineering
Palazzetto dello Sport di Roma

34. April 11 2015April 11 2015
Sapienza University of RomeSapienza University of Rome
School of EngineeringSchool of Engineering
//// Sapienza and DTU workshopSapienza and DTU workshop ////
Sapienza University of RomeSapienza University of Rome,, School of EngineeringSchool of Engineering,, AprilApril 11 201511 2015
Structural optimization in parametricsStructural optimization in parametrics
Optimal lines on free-form surfacesOptimal lines on free-form surfaces
IntroductionIntroduction
Konstantinos GkoumasKonstantinos Gkoumas
StroNGER srl co-founder and partnerStroNGER srl co-founder and partner

35. April 11 2015April 11 2015
Sapienza University of RomeSapienza University of Rome
School of EngineeringSchool of Engineering
Extra slides

36. April 11 2015April 11 2015
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School of EngineeringSchool of Engineering
curved surfaces – tangent plane
The tangent plane to a point on a given surface, is
the plane that goes through that point and that it is
tangent to the surface at that point.
Let F(x,y,z) define a surface that is differentiable at a
point (x0,y0,z0), then the tangent plane to F(x,y,z) at
(x0 ,y0 ,z0) is the plane with normal vector: Grad
F(x0,y0,z0) that passes through the point (x0,y0,z0).
In Particular the equation of the tangent plane is:
Grad F(x0,y0,z0) .
( x - x0 , y - y0 , z - z0) = 0
All tangent lines to a point p on a surface will fall on
the tangent plane to the surface at that point.
Any curve embedded in the surface that passes
through that point will have a tangent at that point
which falls in this tangent plane.

37. April 11 2015April 11 2015
Sapienza University of RomeSapienza University of Rome
School of EngineeringSchool of Engineering
curved surfaces – normal plane, principal curvatures
A normal plane at point p is one that
contains the normal vector, and will
therefore also contain a unique direction
tangent to the surface and cut the surface
in a plane curve, called normal section.
This curve will in general have different
curvatures for different normal planes at p.
The principal curvatures at p, denoted k1
and k2, are the maximum and minimum
values of this curvature, and they measure
the maximum and minimum “bending” of
the surface at that point. They are the
eigenvalues of the Hessian
The lines of curvature are curves which are
always tangent to a principal direction (they
are integral curves for the principal
direction fields).
Eigenvalues of H: k1 k2

38. April 11 2015April 11 2015
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School of EngineeringSchool of Engineering
curved surfaces – mean and Gaussian curvature

39. April 11 2015April 11 2015
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School of EngineeringSchool of Engineering
Surfaces with zero mean curvature are called
minimal surfaces.
Minimal surfaces tend to be saddle-like since
principal curvatures have equal magnitude but
opposite sign.
The saddle is also a good example of a surface with
negative Gaussian curvature.
curved surfaces – specific cases
Surfaces with zero Gaussian curvature are called
developable surfaces because they can be
“developed” or flattened out into the plane without
any stretching or tearing.
For instance, any piece of a cylinder is developable
since one of the principal curvatures is zero.
The hemisphere is one example of a
surface with positive Gaussian curvature

40. April 11 2015April 11 2015
Sapienza University of RomeSapienza University of Rome
School of EngineeringSchool of Engineering
curved surfaces – Gaussian curvature example surfaces
From left to right:
•a surface of negative
Gaussian curvature
(hyperboloid);
•a surface of zero
Gaussian curvature
(cylinder), and;
•a surface of positive
Gaussian curvature
(sphere).
"Gaussian curvature" by Jhausauer - English wikipedia,. Licensed under Public Domain via Wikimedia Commons -
http://commons.wikimedia.org/wiki/File:Gaussian_curvature.PNG#/media/File:Gaussian_curvature.PNG