Geometry. Funicular Polygon

1. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
FUNDAMENTALS OF MATHEMATICS I

2. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY

3. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
PREVIOUSLY ON MATHS 1...
I FORGOT TO ASK...

4. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
FUNICULAR POLYGON
EXAMPLE 1. POSITION and FORCES MAGNITUDES
EXAMPLE 2. POSITION and FORCES MAGNITUDES
INTRODUCTION
EXAMPLE 3. POSITION and FORCES MAGNITUDES

5. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
FIRST MINUTE QUESTION
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Password: SAME AS USERNAME

6. FIRST MINUTE QUESTION
Which of these
sentences is true?
Only bars CD, EF, IK and HL are zero
A
B
C
D There are no zero bars in the truss
Only bars AC, CD, EF, IK and HL are zero
Bars BC, CD, CF, FE, FG and IK are zero

7. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
Source: CNN

8. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
Source: CNN

9. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
Source: Gaudí Work

10. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
Source: Gaudí Work. Blog Estructurando

11. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
FUNICULAR POLYGON
The figure formed by a light string hung
between two points from which weights are
suspended at various points.
A force diagram for such a string, in which
the forces (weights and tensions) acting on
points of the string from which weights are
suspended are represented by a series of a
djacent triangles.
McGraw-
Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The Mc
Graw-Hill Companies, Inc.
Source: The Art of Spansh Bridge Designs. Princeton University

12. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
Source: Quque Goberna

13. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
Source: Quque Goberna

14. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
1.- Determine in a geometrical way the coordinates of the intermediate points B, C and D of the funicular polygon
formed by the represented forces. Assume the side joints are A with coordinates [0, 4 m] and E [12 m, 6 m]. Assume
also that the horizontal component of all the forces is 40 kN.
2.- Determine the internal forces in all the cable segments
Seafarers Bridge - Melbourne, Australia
2009 –[Nicholas Grimshaw]

15. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
1.- Determine in a geometrical way the coordinates of the intermediate points B, C and D of the funicular polygon
formed by the represented forces. Assume the side joints are A with coordinates [0, 4 m] and E [12 m, 6 m]. Assume
also that the horizontal component of all the forces is 40 kN.
2.- Determine the internal forces in all the cable segments
4 3 2 3
x
4
2
12
6
A
B
C
D
E
10
15
5
y

16. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY

17. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY

18. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
THERE ARE INFINITEPOSIBLE FORCES DIAGRAMS and FUNICULAR DIAGRAMS. WE NEED SOME REQUIREMENTS
IN THIS EXAMPLE WE HAVE TWO REQUIREMENTS:
- HORIZONTAL FORCE 40 kN
- THE FUNICULAR POLYGON SHOULD START at A AND FINISH at E

19. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
HORIZONTAL FORCE = 40 kN
so the pole is 40 kN away from the vertical forces
FIRST GUESS: NO MATTER ITS VERTICAL POSITION

20. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
HORIZONTAL FORCE = 40 kN
so the pole is 40 kN away from the vertical forces
FIRST GUESS: NO MATTER ITS VERTICAL POSITION

21. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
HORIZONTAL FORCE = 40 kN
so the pole is 40 kN away from the vertical forces
FIRST GUESS: NO MATTER ITS VERTICAL POSITION

22. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
HORIZONTAL FORCE = 40 kN
so the pole is 40 kN away from the vertical forces
SECOND GUESS: THE FUNICULAR POLYGON ENDS at E

23. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
HORIZONTAL FORCE = 40 kN
so the pole is 40 kN away from the vertical forces
SECOND GUESS: THE FUNICULAR POLYGON ENDS at E

24. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
HORIZONTAL FORCE = 40 kN
so the pole is 40 kN away from the vertical forces
SECOND GUESS: THE FUNICULAR POLYGON ENDS at E

25. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
HORIZONTAL FORCE = 40 kN
so the pole is 40 kN away from the vertical forces
SECOND GUESS: THE FUNICULAR POLYGON ENDS at E

26. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
HORIZONTAL FORCE = 40 kN
so the pole is 40 kN away from the vertical forces
SECOND GUESS: THE FUNICULAR POLYGON ENDS at E

27. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
HORIZONTAL FORCE = 40 kN
so the pole is 40 kN away from the vertical forces
SECOND GUESS: THE FUNICULAR POLYGON ENDS at E

28. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
Puente de Bacunayagua
Cuba - 1959 - [Luis Sáenz Duplace]

29. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
1.- Determine in a geometrical way the coordinates of the intermediate points B, C and D of the funicular polygon
formed by the represented forces. Assume also that the horizontal component of all the forces is 500 kN.
2.- Determine the internal forces in all the cable segments

30. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY

31. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY

32. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY

33. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY

34. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
Today’s words?

35. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
Poly Corporation Headquarters. Beijing 2007. SOM

36. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
Poly Corporation Headquarters. Beijing 2007. SOM

37. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY