73 research outputs found
Thin walled I-beam under complex loads: Optimization according to stress constraint
Razmatrana je optimizacija složeno optereÄenih tankozidih nosaÄa popreÄnih preseka oblika I- profila izloženih savijanju i ograniÄenoj torziji. Iz opÅ”teg sluÄaja, kada momenti savijanja deluju oko obe glavne težiÅ”ne ose istovremeno sa bimomentom, izdvojeni su neki posebni sluÄajevi koji se razmatraju u zavisnosti od sluÄaja optereÄenja. Problem je redukovan na odreÄivanje minimalne mase, t.j. minimalne povrÅ”ine predloženog oblika popreÄnog preseka tankozidog nosaÄa, za data složena optereÄenja, materijal i geometrijske karakteristike. Zbog toga je povrÅ”ina popreÄnog preseka izabrana za funkciju cilja. Pretpostavlja se da odnos debljine i Å”irine pojedinih delova popreÄnog preseka nije konstantan. Uvedeno je naponsko ograniÄenje. Pri formiranju osnovnog matematiÄkog modela poÅ”lo se od pretpostavki teorije tankozidih Å”tapova sa jedne strane i osnovnih pretpostavki problema optimalnog projektovanja sa druge. KoriÅ”Äena je metoda Lagranžovog množitelja. Rezultati analitiÄki dobijenih jednaÄina za matematiÄki model, numeriÄka reÅ”enja, kao i uÅ”teda mase, izraÄunati su za tri sluÄaja optereÄenja.Optimization of a thin-walled open section I-beam loaded in a complex way subjected to the bending and to the constrained torsion, is considered. From the general case when bending moments about both principal axes appear simultaneously with the bimoment, some particular cases can be considered depending on the loading case. The problem is reduced to the determination of minimum mass i.e. minimum cross sectional area of structural thin-walled beam elements of proposed shape, for given complex loads, material and geometrical characteristics. That is why the area of the cross section is taken as the objective function. The ratios of thickness and length of the parts of the cross section are assumed to be non constant. The stress constraint is introduced. The starting points during the formulation of the basic mathematical model are the assumptions of the thin-walled beam theory from one side and the basic assumptions of the optimum design from the other. The Lagrange multiplier method is used. Solutions of analitically obtained expressions for the mathematical model, numerical solutions, as well as the saved mass, are calculated for three loading cases
Thin walled I-beam under complex loads: Optimization according to stress constraint
Razmatrana je optimizacija složeno optereÄenih tankozidih nosaÄa popreÄnih preseka oblika I- profila izloženih savijanju i ograniÄenoj torziji. Iz opÅ”teg sluÄaja, kada momenti savijanja deluju oko obe glavne težiÅ”ne ose istovremeno sa bimomentom, izdvojeni su neki posebni sluÄajevi koji se razmatraju u zavisnosti od sluÄaja optereÄenja. Problem je redukovan na odreÄivanje minimalne mase, t.j. minimalne povrÅ”ine predloženog oblika popreÄnog preseka tankozidog nosaÄa, za data složena optereÄenja, materijal i geometrijske karakteristike. Zbog toga je povrÅ”ina popreÄnog preseka izabrana za funkciju cilja. Pretpostavlja se da odnos debljine i Å”irine pojedinih delova popreÄnog preseka nije konstantan. Uvedeno je naponsko ograniÄenje. Pri formiranju osnovnog matematiÄkog modela poÅ”lo se od pretpostavki teorije tankozidih Å”tapova sa jedne strane i osnovnih pretpostavki problema optimalnog projektovanja sa druge. KoriÅ”Äena je metoda Lagranžovog množitelja. Rezultati analitiÄki dobijenih jednaÄina za matematiÄki model, numeriÄka reÅ”enja, kao i uÅ”teda mase, izraÄunati su za tri sluÄaja optereÄenja.Optimization of a thin-walled open section I-beam loaded in a complex way subjected to the bending and to the constrained torsion, is considered. From the general case when bending moments about both principal axes appear simultaneously with the bimoment, some particular cases can be considered depending on the loading case. The problem is reduced to the determination of minimum mass i.e. minimum cross sectional area of structural thin-walled beam elements of proposed shape, for given complex loads, material and geometrical characteristics. That is why the area of the cross section is taken as the objective function. The ratios of thickness and length of the parts of the cross section are assumed to be non constant. The stress constraint is introduced. The starting points during the formulation of the basic mathematical model are the assumptions of the thin-walled beam theory from one side and the basic assumptions of the optimum design from the other. The Lagrange multiplier method is used. Solutions of analitically obtained expressions for the mathematical model, numerical solutions, as well as the saved mass, are calculated for three loading cases
One view to the optimization of thin-walled open sections subjected to constrained torsion
U ovom radu je razmatrana optimizacija tankozidih konzolnih konstrukcionih elemenata otvorenih popreÄnih preseka izloženih ograniÄenoj torziji. Cilj rada je odreÄivanje minimalne mase, tj, odreÄivanje minimalne povrÅ”ine popreÄnog preseka tankozidih konstrukcionih elemenata oblika I i U-profila za zadata optereÄenja, materijal i geometrijske karakteristike. Zbog toga je za funkciju cilja odabrana povrÅ”ina popreÄnog preseka nosaÄa. Za kriterijum ograniÄenja odabran je kriterijum ograniÄenja deformacija. Pri formiranju osnovnog matematiÄkog modela poÅ”lo se od pretpostavki teorije tankozidih Å”tapova sa jedne strane i osnovnih pretpostavki problema optimalnog projektovanja sa druge. Primenom metode Lagranžovog množitelja izvedene su jednaÄine Äija reÅ”enja predstavljaju optimalne odnose dimenzija popreÄnog preseka izabranog oblika. Dobijeni rezultati su iskoriÅ”Äeni pri numeriÄkom proraÄunu primenom Metode konaÄnih elemenata.One approach to the optimization of thin-walled open section cantilever beams subjected to constrained torsion is considered. The aim of this paper is to determine the minimum mass i.e. minimum cross- sectional area of structural thin-walled I-beam and Channel section beam elements for given loads material and geometrical characteristics. The area of the cross-section is assumed to be the objective function. The displacement constraints are introduced. The starting points during the formulation of the basic mathematical model are the assumptions of the thin-walled beam theory from one side and the basic assumptions of the optimum design from the other. Applying the Lagrange multiplier method, the equations of which the solutions represent the optimal values of the ratios of the parts of the chosen cross sections are derived. The obtained results are used for numerical calculation applying the Finite Element Method
Nonlinear Approach to Thin-Walled Beams with a Symmetrical Open Section
The principle of virtual work is applied to thin-walled beams with a cross-section with the middle line of an arbitrary curvilinear shape and with a continuously varying thickness. Six equilibrium equations and a seventh one related to the constrained torsion are derived taking into account general sectorial coordinates. The obtained relations are applied to structural elements with one longitudinal plane of symmetry with the shape similar to real turbine blades. All geometrical characteristics are calculated for one structural element with a modified cross-section shape. It has been shown that it is not recommendable to neglect the influence of secondary effects
One view to the optimization of thin-walled open sections subjected to constrained torsion
U ovom radu je razmatrana optimizacija tankozidih konzolnih konstrukcionih elemenata otvorenih popreÄnih preseka izloženih ograniÄenoj torziji. Cilj rada je odreÄivanje minimalne mase, tj, odreÄivanje minimalne povrÅ”ine popreÄnog preseka tankozidih konstrukcionih elemenata oblika I i U-profila za zadata optereÄenja, materijal i geometrijske karakteristike. Zbog toga je za funkciju cilja odabrana povrÅ”ina popreÄnog preseka nosaÄa. Za kriterijum ograniÄenja odabran je kriterijum ograniÄenja deformacija. Pri formiranju osnovnog matematiÄkog modela poÅ”lo se od pretpostavki teorije tankozidih Å”tapova sa jedne strane i osnovnih pretpostavki problema optimalnog projektovanja sa druge. Primenom metode Lagranžovog množitelja izvedene su jednaÄine Äija reÅ”enja predstavljaju optimalne odnose dimenzija popreÄnog preseka izabranog oblika. Dobijeni rezultati su iskoriÅ”Äeni pri numeriÄkom proraÄunu primenom Metode konaÄnih elemenata.One approach to the optimization of thin-walled open section cantilever beams subjected to constrained torsion is considered. The aim of this paper is to determine the minimum mass i.e. minimum cross- sectional area of structural thin-walled I-beam and Channel section beam elements for given loads material and geometrical characteristics. The area of the cross-section is assumed to be the objective function. The displacement constraints are introduced. The starting points during the formulation of the basic mathematical model are the assumptions of the thin-walled beam theory from one side and the basic assumptions of the optimum design from the other. Applying the Lagrange multiplier method, the equations of which the solutions represent the optimal values of the ratios of the parts of the chosen cross sections are derived. The obtained results are used for numerical calculation applying the Finite Element Method
Torsional analysis of open section thin-walled beams
Osnovni cilj ovog rada je da prikaže jedan pristup optimizaciji tankozidnih I, Z i U konzolnih konstrukcionih elemenata otvorenih popreÄnih preseka izloženih ograniÄenoj torziji. Za kriterijum ograniÄenja odabran je kriterijum ograniÄenja deformacija: dozvoljeni ugao uvijanja i dozvoljene ugao uvijanja po jedinici dužine. Za funkciju cilja odabrana je povrÅ”ina popreÄnog preseka nosaÄa. Primenom metode Lagranžovog množitelja izvedene su jednaÄine Äija reÅ”enja predstavljaju optimalne odnose dimenzija popreÄnog preseka izabranog oblika.The main purpose of this paper is to present one approach to the optimization of thin-walled I, Z and channel-section beams subjected to constrained torsion. The displacement constraints are introduced: allowable angle of twist and allowable angle of twist per unit length. The area of the cross-section is assumed to be the objective function. Applying the Lagrange multiplier method, the equations whose solutions represent the optimal values of the ratios of the parts of the chosen cross-sections are derived
An approach to the optimization of thin-walled cantilever open section beams
U ovom radu je razmatran jedan pristup optimizaciji tankozidih konzolnih konstrukcionih elemenata otvorenih popreÄnih preseka izloženih savijanju i ograniÄenoj torziji. Problem je redukovan na odreÄivanje minimalne mase t.j. minimalne povrÅ”ine konstrukcionih tankozidih elemenata oblika I i U-profila za zadata optereÄenja, materijal i geometrijske karakteristike. PovrÅ”ina popreÄnog preseka nosaca je izabrana za funkciju cilja. Uvedeno je naponsko ograniÄenje. Primenom metode Lagranžovog množitelja formirane su jednaÄine Äija reÅ”enja predstavljaju optimalne odnose dimenzija popreÄnog preseka izabranog oblika. Dobijeni rezultati su iskoriÅ”Äeni pri numeriÄkom proraÄunu.An approach to the optimization of the thin-walled cantilever open section beams subjected to the bending and to the constrained torsion is considered. The problem is reduced to the determination of minimum mass, i.e. minimum cross-sectional area of structural thin-walled I-beam and channel-section beam elements for given loads, material and geometrical characteristics. The area of the cross-section is assumed to be the objective function. The stress constraints are introduced. Applying the Lagrange multiplier method the equations, whose solutions represent the optimal values of the ratios of the parts of the chosen cross-section, are formed. The obtained results are used for numerical calculation
Consideration of the horizontal inertial effects at cantilever beams with nonuniform open sections
U ovom radu se analizira uticaj horizontalnih inercijalnih sila na konzolne nosaÄe sa promenljivim popreÄnim presekom oblika I-profila. Razmatra se varijacija preseka koja ukljuÄuje promenu visine preseka, kao najpraktiÄnije varijante pri projektovanju nosaÄa dizalica. Zbog složenosti postavljenog problema koji ukljuÄuje problem ograniÄenog uvijanja, statiÄki odzivi su dobijeni numeriÄkim metodama. Modeli nosaÄa su usvojeni prema poznatim konstrukcionom preporukama za varijaciju nosaÄa, sa ciljem praktiÄne primene kod konstrukcija konzolnih dizalica. PoreÄenje rezultata je izvrÅ”eno sa poznatim vrednostima efekata uvojnog savijanja kod nosaÄa uniformnog popreÄnog preseka. Sa aspekta postavljenog problema i izabranih modela nosaÄa, prikazane su prednosti koriÅ”Äenja nosaÄa sa promenljivim popreÄnim presekom.The problem of torsion due to horizontal inertial effects is considered at cantilever beam with variable I-section. Linear variation of height is concerned as most practical one for the design of cantilever beams. The solution for adopted cases of beams is obtained numerically, according to the complexity of the given ordinary differential equation which deals with pure torsion along with warping torsion. The models are based on known tailor-made beams with possibility for application in design of jib cranes, as practical aspect of this work. The comparison of results is done with uniform cantilever beam models which can be used as one way for verification of stress state of variable cantilever beams subjected to bending with torsion. The obtained results show corresponding advantages of usage of variable sections
Stress constraints applied to the optimization of a thin-walled Z-beam
Razmatran je jedan pristup optimizaciji tankozidih otvorenih popreÄnih preseka oblika Z - profila, izloženih savijanju i ograniÄenoj torziji. Za data optereÄenja, materijal i geometrijske karakteristike, problem se svodi na odreÄivanje minimalne mase, odnosno minimalne povrÅ”ine popreÄnog preseka konstruktivnih tankozidih popreÄnih preseka izabranog oblika. PovrÅ”ina popreÄnog preseka je izabrana za funkciju cilja. Uvedena su naponska ograniÄenja. Primenjuje se Metoda Lagranžovog množitelja. Rezultati analitiÄki dobijenih jednaÄina za matematiÄki model, numeriÄka reÅ”enja, kao i uÅ”teda u masi, izraÄunati su za tri sluÄaja optereÄenja. Neki rezultati su provereni primenom programa COSMOS.One approach to the optimization of a thin-walled open section Z-beam subjected to the bending and to the constrained torsion is considered. For given loads, material and geometrical characteristics, the problem is reduced to the determination of minimum mass i.e. minimum crosssectional area of structural thin-walled beam of the chosen shape. The area of the cross-section is assumed to be the objective function. The stress constraints are introduced. The Lagrange multiplier method is applied. Solutions of analitically obtained expressions for the mathematical model, numerical solutions, as well as the saved mass, are calculated for three loading cases
Stress constraints applied to the optimization of a thin-walled Z-beam
Razmatran je jedan pristup optimizaciji tankozidih otvorenih popreÄnih preseka oblika Z - profila, izloženih savijanju i ograniÄenoj torziji. Za data optereÄenja, materijal i geometrijske karakteristike, problem se svodi na odreÄivanje minimalne mase, odnosno minimalne povrÅ”ine popreÄnog preseka konstruktivnih tankozidih popreÄnih preseka izabranog oblika. PovrÅ”ina popreÄnog preseka je izabrana za funkciju cilja. Uvedena su naponska ograniÄenja. Primenjuje se Metoda Lagranžovog množitelja. Rezultati analitiÄki dobijenih jednaÄina za matematiÄki model, numeriÄka reÅ”enja, kao i uÅ”teda u masi, izraÄunati su za tri sluÄaja optereÄenja. Neki rezultati su provereni primenom programa COSMOS.One approach to the optimization of a thin-walled open section Z-beam subjected to the bending and to the constrained torsion is considered. For given loads, material and geometrical characteristics, the problem is reduced to the determination of minimum mass i.e. minimum crosssectional area of structural thin-walled beam of the chosen shape. The area of the cross-section is assumed to be the objective function. The stress constraints are introduced. The Lagrange multiplier method is applied. Solutions of analitically obtained expressions for the mathematical model, numerical solutions, as well as the saved mass, are calculated for three loading cases
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