1. A relation between Structures and Network Flows Through Graph Representation It was found that the same type of representation, a Graph, can be associated with more than one domain, say Network Flow and One-Dimensional Structures Then for each engineering system s i Structure and s i Network we can construct a system so that T(s i ) =s i Network sisi T sisi Structure B A t s 1 4 2 3 5 P 1 4 2 3 5 P B

2. Network Flow and its connection to structures The following slides will demonstrate the close relation between maximum flow in networks and applying a maximal force in a one dimensional structure.

3. One dimensional structures In 1965, William Prager established the relation between network flow and plastic theory for one dimensional structures. Prager used graph theory in order to establish the connection between the two seemingly different areas.

4. One dimensional structures A one-dimensional structure is a solid structure built of rods and discs, where all rods are parallel to each other. Ground Rods Disc P

5. Network Flow and its connection to structures The rules are pretty straight forward – each disc is replaced by a vertex and each rod by an edge. structure network A B t s 1 4 2 3 5 P 1 4 2 3 5 P B A s t

6. One dimensional structures Applying more than the allowable force will in turn transform the structure into a mechanism. Cable – A rod which preserves its length but cannot accept any compression (Recski) Strut – A rod which cannot accept any tension P s A B t Energy preservation law: P s B A t

7. Network Flow - Example 3 5 1 10 5 A B s t

8. Network Flow - Example One can see that the maximum flow is 9. 3 5 1 10 5 A B s t

9. One dimensional structures- and its link to networks Lets assume now that we have the following one-dimensional structure: p s 3 10 A B 1 5 5 t

10. One dimensional structures- and its link to networks Where the numbers refer to the maximal allowable force in each rod. p s 3 10 A B 1 5 5 t

11. One dimensional structures- and its link to networks Now we have to find the maximal force that can be applied on the structure before any of its discs start moving. p s 3 10 A 1 B t 5 5

12. One dimensional structures- and its link to networks The optimal solution here is 9, and one of the paths is marked in the graph: A B s t t 3 0 3 0 52 1 52 1 10 5 4 5 4 5 0 5 0 1 0 1 0 P B A s Now, finding a cutset in the structure in which all rods are saturated means we have a mechanism: Augmenting Path Min Cut Max Flow Isomorphic

13. Network Flow and its connection to structures What we get is the graph representation of the structure which is isomorphic to the network. Now, finding the maximal allowable force is the same objective as finding the maximum allowable flow. structure Network p s 1 2 1 4 A 3 43 B 2 5 5 P t A B s t

14. A relation between Structures and Network Flows Through Graph Representation It was found that the same type of representation, a Graph, can be associated with more than one domain, say Network Flow and One-Dimensional Structures Then for each engineering system s i Structure and s i Network we can construct a system so that T(s i ) =s i Network sisi T sisi Structure B A t s 1 4 2 3 5 P 1 4 2 3 5 P B

15. A relation between Structures and Linear Programming Through Matroid theory It was found that the same type of representation, a Matroid can be associated with more than one domain, say LP (Linear Programming) and Multi- Dimensional Structures Then for each engineering system s i LP we can construct a system s i Structures so that T(s i )=m i =T(s i ) Matroid mimi LP sisi T sisi Structures T Max st Q(M)*F=0 B(M)*D=0

16. Frames Trusses Electronic circuits Dynamical system Static lever system PGR Potential Graph Representation LGR Line Graph Representation RGR Resistance Graph Representation FGR Flow Graph Representation FLGR Flow Line Graph Representation PLGR Potential Line Graph Representation Planetary gear systems Determinate beams Serial robots Stewart platform Pillar system The map of domains Dual RMR Resistance Matroid Representation Plane kinematical linkage LP Linear Programming Operational research Dual Network Flow Max st Q(G)*F=0 B(G)*D=0 F=k*D Q(M)*F=0 B(M)*D=0

17. Matroid Representation The scalar cutset matrix defines the matroid M Q =(S,F) where S is the set of columns of Q(G) and F is a family of all linearly independent subsets of S. P 3 2 1 S={1,2,3,P) F={{1},{2},{3},{P},{1,2},{1,3},{1,P},{2,3},{2,P}, {3,P}}

18. StructuresNetwork FlowsMatroid s t s t s t Choosing determinate structure Choosing augmenting path Choosing a base Choosing a cut Removing these rods makes the structure not rigid Choosing a self stress Choosing a cycle Choosing a set of linearly dependant members Force law Flow law Q(M)*F=0 Allowable force in each rod The weights in the edges The maximum values of the members

19. Choosing a base B(M)*D=0 Q(M)*F=0 Choosing a set of linearly dependant members A group of manufacturing workers A group of manufacturing workers and an administrative/non- manufacturing worker The sum of hours that a worker manufactures the products equals to the sum of his Working hours A units cost equals to the sum of the hours multiplied by the workers wages The workers hour constraint MatroidOperational Research Multi-Dimensional Structures P