1. Symposium in honour of Gianpietro Del Piero Sperlonga September 30th and October 1st 2011.The static theorem of limit analysis for masonry panels M. Lucchesi, M. Šilhavý, N. Zani
2. Constitutive equationsIn order to characterize a masonry-like material we assumethat the stress must be negative semidefinite and that thestrain is the sum of two parts: the former depends linearlyon the stress, the letter is orthogonal to the stress andpositive semidefinite, X − Sym I œ I/ I0 I 0 − Sym X œ ‚ÒI / Ó I0 † X œ !where I (?) œ " Ðf? f?T Ñ #is the infinitesimal strain and ‚ is symmetric and positivedefinite, i.e., 2
3. E" † ‚ÒE# Ó œ E# † ‚ÒE" Óß E † ‚ÒEÓ ! for each E − Symß E Á !PropositionFor each I − Sym there exists a unique tripletÐX ß I / ß I 0 Ñ that satisfies the constitutive equation.Moreover, a masonry-like material is hyperelastic.We write AÐI Ñ œ " X ÐI Ñ † I #for the stored energy. 3
4. NotationsH reference configuration of the body ` H œ W fß such that W f œ g? displacement field, such that ul W œ != À f Ä ‘$ surface traction, À H Ä ‘$ body forceX À H Ä Sym stress tensor s S b Ω D 4
5. Limit analysisThe limit analysis deals with a family of loads _Ð-Ñ thatdepend linearly on a scalar parameter - − ‘, _ Ð - Ñ œ Ð , - ß =- Ñ œ Ð , ! - , " ß = ! - = " Ñ , ! , =! permanent part of the load , " ß =" variable part of the load - loading multiplier,! and ," are supposed to be square integrable functions onH, =! and =" are supposed to be square integrable functionson f . 5
6. Let Z œ Ö@ − [ "ß# ÐHß ‘$ Ñ À @ œ ! a.e. on W×be the Sobolev space of all ‘$ valued maps such that @ andthe distributional derivative f@ of @ are square integrableon H.For each Ð-ß @Ñ − ‘ ‚ Z we define the potential energyof the body MÐ-ß @Ñ œ ( AÐI Ð@Ñ.Z ( @ † , - .Z ( @ † =- .E H H fwhere ( AÐI Ð@Ñ.Z His the strain energy and _Ð-Ñß @ œ ( @ † , - .Z ( @ † =- .E H fis the work of the loads. 6
7. Moreover, we define the infimum energy M! Ð-Ñ œ inf ÖMÐ-ß @Ñ À @ − Z ×ÞProposition(i) The functional M! À ‘ Ä ‘ Ö _× is concave, M! Ð+- Ð" +Ñ.Ñ +M! Ð-Ñ Ð" +ÑM! Ð.Ñfor every -, . − ‘ and + − Ò!ß "Ó.Therefore, the set A À œ Ö- − ‘ À M! Ð-Ñ _×is an interval. 7
8. (ii) The functional M! is upper semicontinuous, M! Ð-Ñ lim supM! Ð-5 Ñ 5Ä_for every - − ‘ and every sequence -5 Ä -.We interpret the elements of A as loading multipliers forwhich the loads _Ð-Ñ are safe, i.e. the body does notcollapse. 8
9. Each finite endpoint -- of the interval A is called acollapse multiplier with the interpretation that for - œ --or at least for - arbitrarily close to -- outside A the bodycollapses.We say that a stress field X is admissible if X − P# ÐHß Sym Ñand that X equilibrates the loads _Ð-Ñ œ Ð, - ß =- Ñ if ( X † I Ð@Ñ.Z œ ( @ † , .Z ( @ † = .E - - H H ffor every @ − Z .We say that the loads _Ð-Ñ are compatible if there exists astress field X that is admissible and equilibrates _Ð-ÑÞ 9
10. Proposition (Static theorem of the limit analysis)The loads _Ð-Ñ are compatible if and only if M! Ð - Ñ _ .That is the loads _Ð-Ñ are safe (i.e. - − A) if and only ifthere exist a stress field X which(i) is square integrable,(ii) takes its values in Sym ,(iii) equilibrates the loads _Ð-Ñ. 10
11. Rectangular panelsIn the study of the static of masonry panels we verify thatthe problem of finding negative semidefinite stress fieldsthat equilibrate the loads is considerably simplified ifinstead of stress fields represented by square integrablefunctions we allow the presence of curves of concentratedstress. p o x y λ Ω+ γλ h λ Ω− λq b 11
12. ExampleIn the following example, firstly, we will obtain a stressfield which is a measure, whose divergence is a measure,that equilibrate the loads in a weak sense. Then we will seea procedure that, starting from this singular stress field,allow us to determine a stress field which is admissible andequilibrates the loads in the classical sense. 12
13. p x O y γλ λ H œ Ð!ß ,Ñ ‚ Ð!ß 2Ñ § ‘# ß W œ Ð!ß FÑ ‚ Ö2×ß f œ ` H ÏW ß , - œ !ß Ú :4, on Ð!ß ,Ñ ‚ Ö!× =- Ð<Ñ œ Û -3ß on Ö!× ‚ Ð!ß 2Ñ Ü ! elsewherewhere : !ß - ! and < œ ÐBß CÑ. 13
14. p x O y Ωλ + γλ a Ωλ − λThe singularity curve # - can be obtained by imposing theequilibrium of the shaded rectangular region with therespect to the rotation about its left lower corner. Weobtain # œ ÖÐBß CÑ − H À C œ Ê B× - : - 14
15. which divides H into the two regions H- , „ H- œ ÖÐBß CÑ − H À C Ê : + B×, - H- œ ÖÐBß CÑ − H À C Ê B×Þ : -The singularity curve # - has to be wholly contained intothe region H and this implies ! - --with :, # -- œ # Þ 2 15
16. For the regular part of the stress field X<- (defined outside# - ) we take œœ - :4 Œ 4 if < − H+ X<- Ð<Ñ -3 Œ 3 if < − H- Þ The stress field defined on # - can be obtained by imposingthe equilibrium of the shaded rectangular region with therespect to the translation. We obtain X=- Ð<Ñ œ È:- <Œ< k<k Þ < k<kWe note that is the unit tangent vector to # - . 16
17. The stress field T- œ X<- X=-has to be interpreted as a tensor valued measure. That is,a function defined on the system of all Borel subsets of Hwhich takes its values in Sym and is countably additive, T Œ E3 œ ! T- ÐE3 Ñ _ _ - 3œ" 3œ"for each sequence of pairwise disjoint (borelian) sets.T- is the sum of an absolutely continuous part (withrespect to the area measure) with density X<- and asingular, part concentrated on # - , with density X=- . Boththe densities X<- and X=- are regular functions. 17
18. We can prove that the stress measure T- weaklyequilibrates the loads Ð, - ß =- Ñ, that is ( I Ð@Ñ. T œ ( @ † , .Z ( @ † = .Eß - - H H ffor any @ − Z . 18
19. Integration of measuresBy the static theorem of limit analysis, in order to assertthat the loads _Ð-Ñ are compatible we need an equilibratedstress field that is a square integrable function and thenequilibrated stress measures are not enough.Now we describe a procedure that in certain cases allowsus to use the stress measure T- to determine a squareintegrable stress field X - . Crucial to this procedure is thefact that both the loads Ð=- ß , - Ñ and the admissibleequilibrating stress measure T- depend on a parameter -. 19
20. p x o y γ μ+ε γμ γ μ−ε μThe idea is to take the average of the stress measure overany set Ð. %ß . %Ñ, where % ! is sufficiently small.Averaging gives the measure Tœ ( " .% - T . -Þ #% .% 20
21. It may happen that this measure, in contrast to T. , isabsolutely continuous with respect to the area measure witha square integrable density.For the previous example we obtain the following result.If ! - -- , then the loads _Ð.Ñ are compatible. In factif Ð. %ß . %Ñ § Ð!ß -- Ñ then the measure Tœ ( " .% - T .- #% .%is absolutely continuous with respect to the area(Lebesgue) measure with density X .X is a bounded admissible stress field on H thatequilibrates the loads _Ð.Ñ. We have X œ X< X = Þ 21
22. The densities X< and X= can be explicitly calculated. Infact, we have X< Ð < Ñ œ ( " .% - X Ð<Ñ. - œ #% .% < Ú :4 Œ 4 if < − H- ÏE Û .3 Œ 3 + Ü Ð#%Ñ" Ð!Ð<Ñ3 Œ 3 " Ð<Ñ4 Œ 4 Ñ if < − H- ÏE if < − Ewhere E œ ÖÐBß CÑ − H À . % :B# ÎC # . %×is the shaded area in the figure, and " # % % !Ð<Ñ œ Ð: B ÎC Ð. %Ñ# Ñß # " Ð<Ñ œ :Ð. % :# B% ÎC % ÑÞ 22
23. Moreover, Ú : # B# X= Ð<Ñ œ Û <Œ< if < − E Ü! %C% otherwiseIt is easily to verify that the stress field X œ X< X =is admissible, i.e. X a square integrable function whichtakes its values in Sym , and that it equilibrates the loads_Ð.Ñ, X 8 œ =. on f , div X œ ! in H. 23
24. Panels with opening a p c x o F1 y γ2h2 Ω1 γ1 Ω3 e Ω2 B γ3h1 d C A q b2 b1 b2 24
25. p a ch2 eh1 qc b2 b1 b2 h2 e B h1 A C b2 b1 b2 25
26. p qc ah2h1 b2 b1 b2 a h2 B Ι h1 C A b2 b1 b2 26
27. Panel under gravity x O λ y b γThe panel is subjected to a side loads and its own gravity on ˜!× ‚ Ð!ß LÑ, =- Ð < Ñ œ œ -3 ! elsewhere. 27
28. The singularity curve is # - œ šÐBß CÑ − H À C œ -,B# Î-›, - œ "Î# È$ÎÞand -- œ -,F # ÎLÞ Ú ,C 4 Œ 4 in H- ßX<- Ð<Ñ œ Û Ü -3 Œ 3 ,B3 4 - 4 Œ 4 , # B# - in H ßwith 3 4 œ 3 Œ 4 4 Œ 3Þ 28
30. By integration we obtain Ú ,C 4 Œ 4 -X Ð<Ñ œ Û .3 Œ 3 ,B3 4 , # B# Î-4 Œ 4 in H ÏE ß Ü W Ð< Ñ in H- ÏEß in Eßwhere E œ Ö< œ ÐBß CÑ À ,-B# ÎC − Ð. &ß . &Ñ×and W Ð<Ñ œ Ð#&Ñ š’ Ð . % Ñ # “3 Œ 3 , # B% " " "#C# # ’ ,BÐ. &Ñ“3 4 , # B$ $C È$ ’Š ‹, B ,CÐ. %Ñ“4 Œ 4 ›Þ & CÐ. &Ñ # # 68 # ,-B# 30
31. References[1] G. Del Piero Limit analysis and no-tension materials Int. J. Plasticity 14, 259-271 (1998)[2] M. Lucchesi, N. Zani Some explicit solutions to equilibrium problem for masonry-like bodies Struc. Eng. Mech. 16, 295-316, (2003)[3] M. Lucchesi, M. Silhavy, N. Zani A new class of equilibrated stress fields for no-tension bodies J. Mech. Mat. Struct. 1, 503-539, (2006)[4] M. Lucchesi, M. Silhavy, N. Zani Integration of measures and admissible stress fields for masonry bodies J. Mech. Mat. Struct. 3, 675-696, (2008)[5] M. Lucchesi, C. Padovani, M. Silhavy An energetic view of limit analysis for normal bodies Quart. Appl. Math 68, 713-746 (2010)[6] M. Lucchesi, M. Silhavy, N. Zani Integration of parametric measures and the statics of masonry panels Ann. Solid Struct Mech. 2, 33-44, (2011) 31
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