orce Network Aalysis usig Complemetary Eergy Adrew BORGART Assistat Professor Delft Uiversity of Techology Delft, The Netherlads A.Borgart@tudelft.l Yaick LIEM Studet Delft Uiversity of Techology Delft, The Netherlads yaickliem@gmail.com Summary The method preseted solves statically idetermiate force etworks, which are graphical represetatios of forces (force polygos) of a structure, by usig complemetary eergy. Statically idetermiate force etworks, which have odes where four or more members come together, have for each ode multiple possible force polygos that make euilibrium. This holds for force etworks of structures i oe plae, such as trusses, as for geometric three-dimesioal structures, such as shells. I the case of the latter the surface of thrust of the shell is discretized ito a etwork of forces (euivalet to the thrust lie of a arch), with discrete loads at its vertices. Keywords: graphic statics, force etwork aalysis, force desity, statically idetermiate trusses, shell structures ad complemetary eergy.. Itroductio The force path of a applied load o a shell structure is ot easy to determie, because shells are statically idetermiate. This meas that to determie a force etwork for aalysig a shell s structural behaviour several possible load paths have to be cosidered []. The best result is umerous possible states of euilibriums withi the thickess of the shells surface or defied evelope, fig.. By addig to the force etwork aalysis the cocept of duality betwee geometry ad the i-plae iteral forces of etworks, similar to the duality of the force polygo ad form diagram for lie structures (cable ad arch, fig..), it is possible to have a direct graphical relatio betwee the force etwork ad its possible solutios []. However, the exact solutio is ot provided for by this method, because of the highly idetermiate ature of shell structures, which makes it impossible without the fiite elemet method to calculate the correct iteral forces give i a particular situatio. igure.: force polygos ad relatig thrust lies of arch, possible solutios of force etwork for shell structures rom the force etwork for each ode its reciprocal diagram, the force polygo is draw. This force polygo is scaled by keepig the assumed force desity of the ode costat, to fid the force polygo with the lowest complemetary eergy. The result for each ode is assembled to form the force diagram of the etire structure, the reciprocal diagram of the force etwork.
. Complemetary eergy The potetial eergy accumulated i a elastic body is called strai eergy. The area uder the stress-strai curve is the strai eergy (E v ), the area above the curve is the complemetary eergy (E c ) see fig... [3] complemetary eergy is expressed i stresses: σ Ecompl = E the complemetary eergy per uit of bar legth euals: E compl compl σ N = dv = E EA V per bar with leght l: E N = l EA igure.: strai ad complemetary eergy 3. orce etworks for statically idetermiate trusses Trusses are iterally statically determiate if each ode has three members coected to it (threevalet system, fig. 3.). The truss is the etirely composed of triagles. Statically determiate trusses ca be aalysed by usig graphic static, such as Cremoa diagrams. Trusses are iterally statically idetermiate if each ode has four or more members coected to it (four-valet or higher-valet systems, fig. 3.). Statically determiate ad idetermiate trusses ca be aalysed by usig a matrix method such as the direct stiffess method, the flexibility method, the fiite elemet method or by usig complemetary eergy. igure 3.: three- ad four-valet odes ad their respective force polygos ig. 3. shows a example of a statically idetermiate truss. Although it is composed of triagles because i oe ode four forces come together the structure is statically idetermiate. This ca be solved by makig the structure statically determiate by makig the force i a bar redudat (i fig 3.; the ukow statically idetermiateφ ) ad solvig this by miimizig the al complemetary eergy of the structure (fig. 3.3). igure 3.: iterally statically idetermiate truss
N Ecompl = l EA E =, 4, φ+ 3, 6φ E compl, compl. φ 7 = 0 φ= igure 3.3: solvig the statically idetermiateφ from the complemetary eergy of the members or statically idetermiate trusses with umerous bars solvig the therefore eually umerous redudats is umerically somewhat cumbersome. This problem ca also be solved by searchig for a closed force polygo (which is the reciprocal diagram of the topology of the structure, the force etwork) with the lowest al complemetary eergy of the structure. To be able to objectively compare all the possible force polygos with each other, ad thus searchig for solutios with the same magitude of the applied load, the al sum of the force desity of the structure for each solutio has to be kept costat, fig. 3.4. - search for a force polygo with the lowest complemetary eergy: E = l + l + l = miimum compl, 3 = + l l l3 3 3 - by keepig, for each solutio, the al force desity () of the structure costat: + = costat, l 3, l 3, l igure 3.4: miimizig the al complemetary eergy of the structure, the force etwork This procedure ca be doe by usig a computatio tool such as Grasshopper. This was used to solve this example. Because i the force etwork all members have the same properties, whe calculatig the complemetary eergy the stiffess EA is omitted from the euatios. 4. orce etworks for shell structures Shell structures are statically idetermiate, ad thus also their surface of thrust, which is the 3D euivalet of the thrust lie of a arch. A plaar projectio of the surface of thrust gives the force etwork; its reciprocal diagram is the force polygo. We are dealig with a statically idetermiate structure ad the etwork is a four-valet system. irst we will cosider a four-valet force etwork with oly oe ode with a arbitrary load v, fig 4.. l 3 l l 4 l l = l = l 3 = l 4 = 5 igure 4.: oe ode four-valet force etwork
As oted before to objectively compare the possible solutios with the same load, the al force desity of the ode for each solutio has to be kept costat, fig. 4.. 4 4 4 4 = + + + = 3, 5 5 5 5 8 4 8 = + + + = 4 5 5 5 5 4 4 4 4 = + + + = 3, 5 5 5 5, 6 6, 4, 6 6, 4 = + + + = 3, 5 5 5 5 igure 4.: oe ode four-valet force etwork ad possible force polygos
Now for each possible solutio the complemetary eergy is calculated util oe solutio has the miimum complemetary eergy. This ca be doe by scalig the possible force polygos while keepig the force desity for each polygo the same. The scalig of the force polygos ca be easily doe with the followig euatios, by varyig the costat c, fig. 4.3. a ; a b a a = c ( * ) ( ) e= + c c cos α ; e = + c c cosα c δ =δ arcsi si α ; * β =β arcsi siα * e e siδ siβ c = e; d = e siγ siγ b β β β α γ e d δ δ δ c igure 4.3: oe ode four-valet force polygo To summarize the procedure for a oe ode, four-valet or higher-valet, force etwork. - search for the force polygo with the lowest complemetary eergy by keepig, for each solutio, the al force desity costat: compl, i i E = l = miimum i = = l i costat We will ow look at force etworks with multiple odes, such as surfaces of thrust of shell structures. or a force etwork to be i euilibrium with its reactio forces its reciprocal diagram, the force polygo has to have a circumferece which forms a cotiuous lie. If this is ot the case the case the topology of the force etwork does ot represet a force etwork that is physically possible, see broke red lie i fig 4.4. igure 4.4: multiple ode four-valet force etwork ad its reciprocal diagram the force polygo
Whe searchig for the force polygo for a force etwork with multiple odes (the reciprocal diagram) with the least al complemetary eergy of the whole etwork, oly the al sum of the force desities of the boudary (support reactio) members of the etwork has to remai costat for a give load; the referece force desity. The force desities of the itermediate members have o bearig o the solutio, fig 4.5. i,sup ; l i,sup compl,alsys i i E = l = miimum,ref i,sup = = costat l i,sup igure 4.5: multiple ode four-valet force etwork ad boudary braches ad its solutio procedure The procedure to solve a multiple ode, four-valet or higher-valet, force etwork: - geerate a plaar projectio of a proposed discretized surface of thrust; the force etwork or primal grid (the applied loads are discretized as forces actig o the vertices of the discretized surface of thrust), fig. 4.6 - extract relevat iformatio from the primal grid, e.g. legth of member, agle betwee members, member coectivity, to the geerate a reciprocal grid; the force polygos (also see []) with the least al complemetary eergy by keepig the al referece force desity costat (this ca be doe for example by usig a Geeralized Reduced Gradiet (GRG) algorithm), fig 4.7 * search for the force polygo with the lowest complemetary eergy by keepig, for each solutio, the al referece force desity costat: i,sup Ecompl,alsys = i li = miimum,ref = = cos tat l - whe the problem has bee solved the actual force desities ca be determied i all the members, ad thus the correct forces i the members ad the correct reactio forces - geerate the correct discretized 3D surface of thrust by calculatig the z-coordiates of primal grid, usig the (actual) force desities i the members, fig. 4.7 i,sup
igure 4.6: primal grid z ode - euilibrium of iteral forces ad load, calculcatio of z-coordiate to geerate correct discretized surface of thrust (example for ode ): N N N N z z z z z z z z ( ) ( ) ( ) ( ) a m t l 0 + + 4 + z, = 0 la lm lt ll igure 4.7: possible force polygos (reciprocal grids), correct discretized 3D surface of thrust
5. Coclusios By usig complemetary eergy statically idetermiate structures such as trusses ad shell structures ca be easily solved. urther research eeds to be doe for a rigorous mathematical derivatio. Also further physical explaatios eed to be provided for various aspects such as force desity. Never the less the i this paper described method is yet a further step i uderstadig the structural behaviour of shells. Refereces [] O DWYER D., uicular aalysis of masory vaults, Computers & Structures, Vol. 73, No. -5, 999, pp. 87-97 [] BLOCK P., Thrust Network Aalysis, Explorig Three-dimesioal Euilibrium, PhD, MIT, 009. [3] BLAAUWENDRAAD J., Theory of Elasticity, Eergy Priciples ad Variatioal Methods, DUT, 00