Denne oppgaven beskriver hva som påvirker skjærkraftkapasiteten til rektangulære slakkarmerte betongbjelker. Fire modeller; 45 graders fagverksmodell, fagverksmodell med variabel helning, trykkfeltsteori og modifisert trykkfeltsteori, vil bli diskutert og utledet. Dimensjoneringsrutinene i de fire standardene Eurokode 2, NS 3473, CSA A23.3 og fib Model Code 2010 presenteres, før de sammenlignes ved hjelp av databaser med forsøksresultater for bjelker med og uten skjærarmering. NS 3473 foreskriver å benytte in situ-fastheter i dimensjonering. De samme reduserte fasthetene oppnås i Eurokode 2 og fib Model Code 2010 ved å innføre faktorene α_c_c og α_c_t på henholdsvis trykk- og strekkfastheten. Disse fasthetsreduksjonene ble ikke benyttet da standardene ble sammenlignet med forsøksresultatene. Resultatene fra sammenligningen viste en trend til at kapasitetsuttrykkene i NS 3473 er tilpasset in situ-fastheter, ettersom standarden viste en større tendens til å overestimere kapasiteten enn de andre standardene da det ble benyttet karakteristiske fastheter. Dette gjelder spesielt for bjelker uten skjærarmering, der kapasiteten beregnes ved hjelp av den forenklede metoden. NS 3473 har allikevel vist seg å ha de laveste variasjonskoeffisientene og dermed det mest konsistente sikkerhetsnivået. Standarden ble ikke analysert i dybden med in situ-fastheter og det er derfor ikke trukket flere konklusjoner. Eurokode 2, CSA A23.3 og Model Code 2010 har alle vist tendenser til å underestimere kapasiteten for bjelker uten skjærarmering med høye lengdearmeringsforhold og lave a/d-forhold. Dette antas å ha sammenheng med den økte buekapasiteten som kan oppnås for lave a/d-forhold. Ingen av standardene tar hensyn til buekapasiteten i kapasitetsuttrykkene. Analyser med indre stavmodeller ble ikke gjennomført. Disse tre standardene har også vist tendenser til å underestimere kapasiteten for bjelker med små tilslagsstørrelser. Dette gjelder spesielt for CSA A23.3 og Model Code 2010 som baserer seg på den forenklede modifiserte trykkfeltsteorien der effekten av aggregate interlock er antatt bestemmende. Det antas derfor at kapasitetsreduksjonen på grunn av små tilslagsstørrelser overvurderes. Eurokode 2, CSA A23.3 og Model Code 2010 ble vurdert til å ha tilnærmet samme sikkerhetsnivå uttrykt med γ_m_o_d_,_5_% = 0.7. Ved å inkludere materialfaktorene oppnådde alle standardene γ_m_o_d_,_5_% > 1.0. For bjelker med skjærarmering viste Eurokode 2 en tendens til å underestimere kapasiteten når bjelkene inneholder lite skjærarmering. Dette ser ut til å gjelde spesielt i tilfeller der betongbidraget forventes å være stort, og det antas at underestimeringen skyldes at betongbidraget utelates. Standarden kompenserer for dette ved å tillate lave rissvinkler, som også fører til en tendens til å overestimere kapasiteten for bjelker med mye skjærarmering. CSA A23.3 og Model Code 2010 beregner rissvinkler som stemmer bedre med det som forventes i virkeligheten. Dette kombinert med at betongbidraget inkluderes fører til at standardene underestimerer mindre for lave skjærarmeringsmengder, og ikke har like store tendenser til å overestimere kapasiteten med mye skjærarmering. Totalt ble sikkerhetsnivået i Eurokode 2, CSA A23.3 og Model Code 2010 vurdert til å være γ_m_o_d_,_5_% = 0.7. Ved å inkludere materialfaktorene oppnådde ingen av standardene γ_m_o_d_,_5_% > 1.0. Dette antas å ha sammenheng med at bjelker med høye verdier for ρ_w f_y_k ikke oppnår flytespenning i skjærarmeringen før kollaps og at beregningsmodellene derfor overestimerer kapasiteten. En forenklet analyse av skalaeffekten viste en tendens til at skalaeffekten fortsatt var til stede for bjelker med skjærarmering. Dette ble bekreftet av et økt sikkerhetsnivå for CSA A23.3 da skalaeffekten ble forsøkt innført i kapasitetsberegningen.

X-FEM, den utvidede elementmetoden, er en relativt ny metode for simulering av sprekkvekst. X-FEM bygger på rammeverket til vanlig elementmetode, FEM. I X-FEM berikes nodene i et element for å inkludere diskontinuiteter og singulariteter innenfor elementet. X-FEM kan dermed beskrive arbitrær sprekkvekst uten å endre nettet i området rundt sprekkspissen. LS-DYNA er X-FEM koplet med en kohesiv materialmodell. Den kohesive materialmodellen lar brukeren definere en tri-lineær kohesiv lov for sprekkvekst. Den kohesive loven i denne oppgaven er definert til å være lineær og er dermed kun gitt av traksjonsspenning og maksimal seperasjon før sprekkvekst. Alle numeriske simuleringer er utført med plan-tøyningselementer i LS-DYNA. Oppgaven fokuserer på å evaluere X-FEM i LS-DYNA. X-FEM brukes for å simulere sprekkvekst i fire prøvestykker. To av prøvestykkene er modellert ut i fra forsøksdata fra et eksperiment som vi har utført ved NTNU og de to andre er hentet fra litteraturen. Resultat fra X-FEM sammenlignes med resultat fra en elementerosjonsmetode med skademodell fra SIMLab. X-FEM gav konsekvent bedre resultater enn elementerosjonsmetoden. Det ble observert en nettavhengighet, denne ble undersøkt for en av modellene. Sprekken ble bredere og forgrenet seg da elementstørrelsen ble redusert. X-FEM approksimerte sprekkstien godt i de numeriske modellene.

The subject of this thesis is related to verification of wind induced dynamic response of the Svinesund Bridge in the time domain by the use of autoregressive simulations. This thesis starts by giving a theoretical study in the field of time–series simulation of wind induced dynamic loading, with emphasis on the Schur decomposition by AR model and polynomial approximation. This is a procedure that generates a multivariate wind field velocity vector by decomposing the PSD matrix into the basis of the eigenvectors. The advantages of this method is that very few spectral modes exhibit significant power, meaning that one could truncate the spectral modal matrix only evaluating the m first eigenvectors (m << n). Another advantage is that each component of the eigenvectors are very regular functions, which is true independent of the analytical model assumed for the PSD function. This allows, by the use of standard finite element procedure for approximating the eigenvectors, the generation procedure, via standard AR model, to only require a limited number of samples of univariate coherent processes to describe the multivariate wind field, and this number is independent of the number of components in the process. Another advantage making use of the regularity of eigenvectors is that only a small number of subdivision frequencies are needed to approximate the eigenvectors in a polynomial form by a standard finite element procedure. All these advantages combines makes this a very computational effective procedure for generation of a multivariate wind field velocity vector. As a bonus the physical meaning of eigenvectors and eigenvalues of the PSD matrix provides useful information about the stochastic wind process in view of a structural analysis. This because the eigenvalues could be seen as the power of n independent processes W_{1}(t),...,W_{n}(t), while the eigenvectors are mode shapes, similar to the structural modes, associated with the wind field velocity. After the theory is explained an explanation of how the time domain simulations of wind induced dynamic response was created using the NatHaz On-line Wind Simulator (NOWS) is given. Here a discussion regarding choices made in regard to different inputs to NOWS as well as how the simulated wind field velocities was to be applied onto the FEM. These simulations was then applied to a finite element model of the Svinesund Bridge, created in the computer software Abaqus, using several different strategies. Then results were extracted from the model in the form of accelerations from these simulations. These accelerations along with the simulated wind field velocities was then compared with real measurements obtained when the storm Per passed Svinesund Bridge on January 14, 2007. After comparing many different velocities and accelerations from different simulations to the measured response it was concluded that the "Full in 2 points" simulation with approach z_{1} created the best representation of the measured response. The "Full in 2 points" simulation uses a drag coefficient of C_{D}=0.15 on both the windward and the leeward box girder, while the arch has a drag coefficient of C_{D}=0.8. The z_{1} approach indicate that the z-coordinates should be given as if the surface is raising linearly between the sea and the arch abutments, and between each arch abutment and pier 5 and 8. NOWS seems to assume that the terrain under a structure is plane, and therefore seems to use the z-coordinates for both defining the height of a point and the spatial separation between to points. So when z_{1} was the most correct approach this would indicate that getting the correct height of each point in a simulation is more important than getting a correct spacial representation of the nodes. It was also concluded that exposure category A and B are ill suited for this location, but that it was difficult concluding which was the better of C and D. In the last chapter sources of error and possible improvements were discussed. Here errors such as not having done a thoroughly calculation of the modulus of elasticity for the arch, taking possible cracking of the concrete into account, which could have great impact on the model were noted. The problems of working with a black box such as the NOWS was also discussed. The summation of this discussion being that if one are to do any improvements the first two considered should possibly bee the modulus of elasticity the concrete arch and finding a different simulation procedure where more controll could be obtained in how the simulation is preformed, either by writing one's own or by finding an open source approach somewhere.

This report deals with shape optimization. Curved beams and thin shells are optimized with respect to internal bending moments. The shape optimization focuses solely on the shape of centroidal axes and middle surfaces, i.e. cross sections and shell thicknesses remain constant. The main part of the report is a presentation of three different cases, each with their own objectives. The scenarios are simplified and the criteria for success is internal bending moments alone; however, the methodology and the lessons learned are transferable also to other cases. 1. Curved beams in general. Optimum shapes are derived analytically for a variety of load cases and methods for estimating continuous optimum shapes are exhibited. Software which optimizes structures based on variations in the values of parameters is then applied. A brief study is done to get familiarized with approaches to the use of the software. Some of the shapes found analytically work as benchmarks. 2. An axisymmetric shell, the dome. Classical theories are studied for determining the internal bending moments of different dome shapes. After establishing the bending moments for a selection of shapes, the software is applied to accommodate the calculations and help determine what is the optimum one. An objective is to get familiarized with the assembling of 3 dimensional models within the software interface. 3. A double curvature shell. With the aid of the software, different suggestions are made to a hypothetical architect regarding the shape of the shell. The author derives an effective method for estimating optimum shapes for curved beams. He comes to the perception that analytical approaches might not outweigh the constructive benefits of alternative methods when the complexity of the cases increases. The software proves itself an invaluable tool for form finding; however, it does not offer the same accuracy and possibilities to intervene with the algorithm of the FEA solver as other software. The general understanding of the approaches tested is that independent points and points defined by an interpolation curve are successful and that parameters assigned some level of interdependence is less successful. Some suggestions for further work and possible improvements with regards to the process of structural optimization are presented at the end. These include mainly topology optimization, implementation of constructive measures (discrete) and how to best benefit from a conjunction of methods and types of software.

This thesis is written in relation to the current research at SIMLab in the field of impact engineering. An overall objective is to establish a general numerical model for predicting and simulating the fragmentation process of metallic materials during impact. Previously conducted experiments at SIMLab revealed that blunt nosed projectiles of hardened ARNE tool steel may be fragmented, when struck against thick target plates at certain velocities. Two preceding M.Sc. theses have been written at SIMLab on the same topic, and provided an experimental basis for evaluation of the present numerical results. Simulations were carried out using the nonlinear finite element codes IMPETUS Afea Solver and LS-DYNA. Series of 20 tensile tests were conducted for ARNE tool steel, hardened to nominally Rockwell C values of 40 and 52. A significant variation was found in the failure strain for both materials. Modified Johnson-Cook constitutive models were directly calibrated from the tests, in combination with stochastic failure models based on the Cockcroft-Latham criterion. Numerical simulations of the tensile tests were in good agreement with the experimental observations. Fractography unveiled fracture surfaces with both ductile and brittle characteristics in in the hardened projectiles. A brittle failure criterion was inversely calibrated based on previously conducted bending tests. Previously conducted Taylor tests, which induced fracture and fragmentation, were attempted simulated with the calibrated material models. The simulated plastic deformations in the unhardened projectiles were described in very good agreement with experimental observations. Both element erosion and node splitting have successively been used to model material failure. It became evident that a stochastic variation of the failure parameter can give good results, and seems necessary for capturing the fragmentation process in a brittle metallic material. Node splitting in combination with cubic 64-node elements seems to give good results when the problems are dominated by tension, while element erosion and small linear elements are more suitable for problems dominated by compressional stresses and shear fractures.

The thesis includes a summary of important theory in the fields of impact engineering and plasticity theory, and a literature study is carried out on aluminium designations, alloying and heat treatment. The true stress-strain curves of aluminium alloy AA6070 in O-, T4-, T6-, and T7-configurations are identified by tensile tests where the behavior is continuously measured to fracture. The aluminium was delivered as 20 mm rolled plates; microstructural images and strain ratios are reported. Material constants are found by direct calibration and inverse modeling. Ballistic tests are done in a laboratory using 7.62 mm APM2 bullets, 20 mm blunt projectiles and 20 mm ogival projectiles (CRH = 3). From these tests the ballistic limit curves and the ballistic limit velocities are found for all temper/projectile combinations. In the material tests it was shown that the O-temper is the most ductile temper and consequently almost no fragmentation takes place in the ballistic tests for this temper. The T6-temper proved to be brittle, and fragmentation was commonly seen in the ballistic tests. The degree of fragmentation is found to be of vital importance for the ballistic performance. The Cockcroft-Latham fracture criterion is implemented by using the plastic work to fracture. Numerical analyses are performed with the IMPETUS Afea Solver and LS-DYNA, with 3D-models and 2D axisymmetric models, respectively. Ballistic limit curves and the ballistic limit velocities are calculated on the basis of the numerical results and then compared to the experimental values. Limited sensitivity studies are conducted on mesh size, heat-expansion dependency, strain-rate dependency, etc. Overall the results from 2D axisymmetric models are found to be consistent with previous studies, and the 3D-analyses carried out with the IMPETUS Afea Solver gave some good results. The IMPETUS Afea Solver proved to be a user-friendly finite element program with some powerful features. In addition to the numerical studies, a thorough derivation of the Cylindrical Cavity Expansion Theory (CCET) is given. Results from CCET are good for 7.62 mm APM2 bullets. A case-study where the ballistic performance is determined without conducting any experiments is also conducted. The results are promising, but less conservative than the original simulations, due to the inability to calibrate a fracture criterion. Some suggestions for further work within the field of impact engineering and the application of the finite element method are provided at the end of the thesis, followed by an appendix that includes graphical representations and photographs of the multiple material tests and program-codes written for use in the thesis.

Fra april 2010 er Eurokode 8 eneste gjeldende standard for jordskjelvdimensjonering i Norge. I standarden deles konstruksjoner inn i klassene lav, medium og høy duktilitet. I Norge er det kun anledning for å benytte klassene for lav og medium duktilitet.I klassen for medium duktilitet (DCM) åpnes det for å redusere lastene på grunn av energiopptaket knyttet til plastisk deformasjon. For å få til dette divideres lastene med konstruksjonsfaktoren q. Denne faktoren er avhengig av hvor duktilt bygget er. Mer duktile bygg kan benytte høyere konstruksjonsfaktor. I duktilitetsklasse medium varierer denne mellom 1,5 og 4,5 for betongkonstruksjoner.For å benytte seg av en konstruksjonsfaktor over 3,0 må bygget designes slik at de plastiske deformasjonene fordeles jevnt. Dermed gir rammesystemer, koplede veggsystemer eller kombinasjoner av disse høyest konstruksjonsfaktor. For disse kan grunnfaktoren på 3,0 økes med en overstyrkefaktor. Denne faktoren finnes ved å dividere last hvor konstruksjonsmessig ustabilitet intreffer med last ved første flyt i konstruksjonen.I Norge er det vanlig å benytte seg av heis- og trappesjakter for å føre horisontale krefter til fundament. Oppgaven undersøker om heis- og trappesjakter med store utsparinger får en slik fordeling av plastiske deformasjoner at de oppnår en overstyrke som tilsvarer den som tillates for ovennevnte bæresystemer, og dermed rettferdiggjør bruk av en høyere q-faktor.Dette blir undersøkt ved å utføre en monoton pålastning til brudd (en såkalt push-over analyse) og resultatene viser at overstyrken øker med størrelsen på utsparingene. Med en prosentvis utsparing på 30% eller høyere per etasje kan man forvente en overstyrke på 1,2-1,4 som tilsvarer en konstruksjonsfaktor på 3,6-4,2. Noe uventet ble det oppdaget at når utsparingene nærmer seg 100% synker overstyrken mot 1,0 igjen. Så store utsparinger er av liten praktisk interesse og påvirker likevel ikke konklusjonen.Konklusjonen blir dermed at betongvegger med store utsparinger kan tilnærmes rammesystemer eller koplede veggsystemer.

I mars 2011 ble Eurokode 2 innført som eneste gjeldene regelverk for dimensjonering av betongkonstruksjoner og det har oppstått en del diskusjon rundt behandlingen av gjennomlokking. Regelverket er tungvint i bruk da beregningsmetodikken avhenger av hvilken søyletype som betraktes. Beregningsmetodene som brukes blir for kant- og hjørnesøyler avanserte, noe som har ført til store forenklinger i regelverket. Ved betraktning av hjørnesøyler tas det for eksempel kun hensyn til geometri i beregningen av tillatt aksiallast.På grunn av de store manglene i regelverket er det utarbeidet en ny Model Code ved navn Model Code 2010 som danner grunnlaget for en ny utgave av Eurokode 2 på område gjennomlokking. Model Code 2010 er basert på en ny mekanisk modell, i motsetning til Eurokode 2 som er empirisk. Denne oppgaven ser nærmere på bruken av Model Code 2010 og det sammenlignes mot Eurokode 2. Først utføres et litteraturstudie for å finne det teoretiske grunnlaget til Model Code 2010. Den mekaniske modellen beskrives i sin helhet og alle forenklinger som er gjort for å komme frem til endelige prosjekteringsregler presenteres. Eksempelberegninger er inkludert for å illustrere bruken av Model Code 2010. Til slutt sammenlignes de to regelverkene i en parameterstudie og kapasitetene som beregnes vurderes opp mot kombinasjoner av aksiallaster og momenter funnet fra en lineær-elastisk analyse av en dekkekonstruksjon.Model Code 2010 viser seg å beregne forholdsvis like kapasiteter som Eurokode 2 for indre søyler. For kant- og hjørnesøyler beregnes kapasiteten etter Model Code 2010 mer eller mindre på samme måte som for indre søyler og resultatene er fornuftige. Forenklingene som er gjort for kant- og hjørnesøyler i Eurokode 2 kommer klart frem og resultatene avviker mer fra Model Code 2010. Det er ikke utført sammenligninger mot utførte forsøk, men i og med at Model Code 2010 tar hensyn til flere parametere og viser mer kontinuitet i beregningsmetodikken er det nye regelverket en klar forbedring. Model Code 2010 gjør også beregningene for konstruksjoner med skjærarmering enklere ettersom rektangulært armeringsgrid tillates.

In the present thesis impact against offshore pipelines was studied. Incipient fracture in the pipes after impact was of special interest. The present work was a continuation of previous theses on the subject, and part of an ongoing research program between SIMLab (NTNU) and Statoil ASA.Component tests and numerical simulations using ABAQUS/Explicit and LS-DYNA were carried out to investigate the problem. Two pipes were impacted at different velocities in the pendulum accelerator at the Department of Structural Engineering, NTNU. The most exposed regions were investigated metallurgically to examine the material behavior at large deformations due to the impact.Pre-compressed notched tension tests at large compressive plastic strains (up to 40 %) were conducted. A material model with combined kinematic and isotropic work-hardening was calibrated based on the material tests. Inverse modeling was carried out with the optimization tool LS-OPT. In addition, the material model consisted of an uncoupled fracture criterion. The combined material model was used in the numerical simulations.The material tests revealed a distinct Bauschinger effect, transient and permanent softening, as well as a work-hardening stagnation during reversed loading. Simulations of the material tests in ABAQUS/Explicit corresponded well with the experiments. However, simulations of the pipe in ABAQUS/Explicit revealed little improvement regarding the global response compared to previous theses. Further, the volume element model in LS-DYNA revealed hydrostatic tension in the critical element during rebound, where incipient fracture was assumed to appear. This was supported by metallurgical investigations revealing a crack through 75 % of the thickness after impact at the highest velocity. It is clear that this decreases the strength of the pipes. Hence, cracks will propagate through the thickness earlier than expected during stretching in the simulations. This indicates that the single-parameter assumption regarding the fracture criterion used in this thesis ceases to be valid for the component test. To recreate the fracture in the impacted pipes an additional formulation must be included.

Polymers are the up and coming engineering material. The number of possibleapplications is increasing fast and so is the need for a good polymer material model. Thescientists at SIMLab at NTNU have developed such a material model. In this thesis is thematerial model calibrated to represent an impact modified polypropylene used by Toyota.The validation started with the calibration. A review of the theoretical background of thematerial model and a detailed explanation of the calibration procedure are given todescribe the derivation of the calibrated material model. The model is then applied in FEsimulationsof the material tests and the two validation tests. The first validation test is aquasi-static tension test using a specimen with a centric hole. This test resembles thematerial tests but the centric hole makes the load response slightly more complex. Theother test is an impact test using a drop tower. This test validates the material modelsability to represent dynamic problems and also tests the representation of the strain ratedependency.The material model proves that it is capable of representing all the tests in this thesis quite accurately.