DESIGN METHODS OF A TIMBER-CONCRETE T-CROSS-SECTION UDC : (045)=20
|
|
- Mervin Heath
- 6 years ago
- Views:
Transcription
1 FACTA UNIVERSITATIS Seres: Archtecture and Cvl Engneerng Vol., N o 5, 3, pp DESIGN METHODS OF A TIMBER-CONCRETE T-CROSS-SECTION UDC 64.6: (45)= Radovan Cvetkovć, Dragoslav Stojć Faculty of Cvl Engneerng and Archtecture, Unversty of Nš, E-mal: radovancvetkovc@yahoo.com, dragoslavstojc@yahoo.com Abstract. Ths paper deals wth the composte tmber-concrete structures.. By combnng tmber and concrete n a new type of composte materal and usng the best propertes of both materals (the hgh tensle strength of tmber and the hgh compressve strength of concrete) a new type of composte structure s obtaned, whch can have many applcatons, dependng on the dfferent buldng condtons, due to the certan advantages t has over concrete or steel structures. Here, the desgn procedures accordng to the theory of elastcty based on the exact method and approxmate method are gven n order, and partcularly accordng to the regulatons and recommendatons of the modern concept for desgn of tmber structures and concrete structures, gven n Eurocode 5 and based on the lmt states of bearng capacty and usablty of structures.. INTRODUCTION The analyss of composte tmber-concrete beams requres knowledge of a relatonshp between stress and dormatons for all three components, tmber, concrete and shear connectors. The complexty of problems les n determnaton of ths relatonshp and requres an ntroducton of a large number of parameters, whch complcate the calculatons. For practcal calculatons, certan smplfcatons can be made, and certan assumptons to facltate reachng the soluton n a relatvely easy way. The approxmate calculaton method for sem rgd structures s more approprate for use n engneerng because the calculus procedure s smpler. The calculatons of the desgn of the rgd composte structure are gven, consderng there s no relatve slp n the nterface, va the transformed secton method. In ths method, the concrete secton s transformed n a tmber secton, and the neutral lne remans n the same orgnal poston. The wdth of the cross-secton depends on the rate E c /E t. On the other sde the secton wth two materals n sem rgd structures wll show Receved May 5, 4
2 33 R. CVETKOVIĆ, D. STOJIĆ yeld neutral lnes. Dependng on theconnecton stffness, we can ntroduce a reducton of the fectve nerta moment "I" and compare t to the momentum of nerta of the rgdly jonted secton.. THE EXACT METHOD The theoretcal analyss, based on equatons of equlbrum, s done accordng to exact method, [5] whch makes possble the accurate determnaton of the strengths, flows and dsplacements. On the other sde, the approxmated method, whch ncorporates smplfcatons n the problem and facltates both the calculatons and desgn procedures, and shows that the smpler equatons can be appled. The slp between mechancally connected tmber and concrete s taken nto account n structural models by means of the slp modulus. The basc assumptons to composte structures, for example, concrete-wood T-beams, are the followng: Dsplacements owng to bendng are small and, therore, the small dsplacements theory s vald; Dsplacements owng to shear dormatons are neglgble n each element; Bernoull-Navers hypothess about plane sectons reman plane and perpendcular to dormed axs of the secton after dormaton s not vald along the whole cross-secton, but t s ndvdually vald for both tmber cross-secton and concrete cross-secton. Tmber and concrete are sotropc elastc materals and Hooks law s vald; Load-slp relatonshp for the connector can be approxmates to elastc-lnear. Connectors are placed at certan dstance and can be regarded as equvalent contnuous connecton. If a beam s subjected to any transversal loadng wth ntensty "q" that may vary along ts axs, f the boundary condtons of the beam do not have the concrete meanng, n regards to the assumed stress and dormaton of coupled cross secton (see fgure.), usng both the prncples of statc equlbrum and dsplacements compatblty, we can obtan the dfferental equaton that dnes the phenomenon dependng on dsplacement "w" [6]: b E,A, I h u=u -u =w (h /+h /) E,A, I h u w x N V b V +dvn +dn M V v M +dm N +dn a u N M M+dM dx V+dV Fg.. System, cross secton, dormaton, stress, element dx.
3 Desgn Methods of a Tmber-Concrete T-Cross-Secton 33 where: w M M α w = + α (.) ( EI) ( EI) r α = k + + EA E A (.) ( EI) = E A + E A (.3) EAEA ( EI) ( EI) r = + EA + EA (.4) (EI) the bendng stffness of uncoupled cross secton, (EI) the bendng stffness of coupled cross secton, E modulus elastcty of a concrete E modulus elastcty of a tmber, A cross-secton area of a concrete part, A cross-secton area of a tmber part, M, N, V adequate nternal forces of cross secton elements (parts), the others geometrcal sgns are accordng to fgure. If we know the soluton "w" to a specfc set of boundary condtons, the nternal forces for the whole secton and each element of secton are gven by: ( EI ) M = w ( EI) w α α EI q (.5) α V = w EI w q (.6) α M + EI w N = (.7a) r N M + EI w = (.7b) r M M = EIw (.8a) = EIw (.8b) V = M + vr (.9a) V = M + vr (.9b)
4 33 R. CVETKOVIĆ, D. STOJIĆ + + M EI w V EI w v = = (.) r r When we know the stress from equaton (.) to (.) t s possble to determne normal stress and shear stress. The normal stresses are gven n tmber and concrete, separately observed, by: σ M x N x ( xy, ) = y I + A (.) σ M x N x ( xy, ) = y I + A (.) where y s the dstance between the center of the consdered element and the fber whose stress we want to determne. In order to analyse the shear stress, we can observe the forts n an elementary segment of tmber from a composte tmber-concrete T-beam n fgure : It s possble to wrte Fg.. The forts n an elementary segment of tmber N (, ) * * σ x y da A * = (.3) where: * N resultng normal acton of the stress that acts n the smaller element; * A transversal secton area of the smaller element. Usng equatons (.) and (.3) and by calculatng the equlbrum of the emphaszed elementary segment n fgure, we can fnd:
5 Desgn Methods of a Tmber-Concrete T-Cross-Secton 333 τ * * * VS v A S ( xy, ) = + r bi b A I (.4) The equaton (.4) makes t possble to determne the shear stress n any one pont on the tmber cross-secton. Analogously, the expresson to the evaluaton of the shear stress n the concrete secton can be obtaned by: τ * * * VS v A S ( xy, ) = + r bi b A I (.5) 3. APPROXIMATE METHOD Ths method provdes approxmate analytc solutons[9]. The basc assumptons n ths method are the same as n exact method presented above. The connecton of tmber and concrete can be made by means of many types of metal connectors: nals, steel dowels, rngs, connected perforate metal plates, etc. The glued jonts are regarded as rgd connectons. The slp between mechancally connected tmber and concrete s taken nto account n structural models by means of the slp modulus K = (3.) u where F s s the shear force n the mechancal fastener and u s the slp n the connecton. The shearng flow "v" that appears on nterface of the materals s yelded by: v = s (3.) where "s" s the spacng between connectors. Usng equatons (3.) and (3.), we fnd: where K s the equvalent slp modulus n the jont. Accordng to the elastc prncples of bendng theory: N N M M v F s F s = ku (3.3) k = K / s (3.4) EAu = (3.5a) EAu = (3.5b) = EIw (3.6a) = EIw (3.6b)
6 334 R. CVETKOVIĆ, D. STOJIĆ V V = E I w (3.7a) = E I w (3.7b) v = ku = u u + w a = u u + w( h / + h /) (3.8) From the equlbrum of the two elements (tmber and concrete) n both longtudnal and axal drectons, we obtan: N + v = (3.9a) v N + = (3.9b) / M = V vh (3.a) / M = V vh (3.b) V + V = p = V (3.) where "p" s a generc loadng appled to beam. Addng equatons (3.a) to (3.b), takng nto consderaton the equatons (3.) and consequently dfferentatng t n relaton to x, we fnd: M + M + va+ p = (3.) If usng the elastc prncples changes both nternal forces and moments, the followng system of dfferentable equaton s yelded: " E A u + k u u + w a) (3.3) ( = " E A u + k u u + w a) (3.4) ( = ( E I + EI ) w" k( u u + w a) a = p (3.5) In ths way equatons (3.9a), (3.9b) and (3.) are formulated n functon of the dsplacements u, u and v. The practcal applcaton of a system of dfferentable equaton wll be shown at the example of smply supported beam wth a snusodal dstrbuton loadng as shown n fgure 3, so a smple analytcal soluton can be acheved. Ths s due to dormaton forms n the drectons of the axs whch agrees wth both snusodal and cosne functons [9]. Fg. 3. Snusodal dstrbuton loadng.
7 Desgn Methods of a Tmber-Concrete T-Cross-Secton 335 We can get: p p sn π l x = (3.6) u u cos π l x = u u cos π l x = w w sn π l x = (3.7a,b,c) where: u and u are the horzontal dsplacements at both concrete and tmber centrod, respectvely at the ends of the beam. The maxmum vertcally dsplacement w s at mdpont. These terms, when substtuted n Equatons (3.3), (3.4) and (3.5), produce a system of equatons wth the constants u, u, w, whose soluton allows to determne the stress at both concrete and tmber centers: γema σ = (3.8) where: γema σ = (3.9) γ = (3.) + k a π EA k = (3.) l K γea = a γ E A + E A EA a = a γ E A + E A Stress at the outer fbers of both concrete and tmber: EM σ =, h, h (3.) (3.3) EM σ = (3.4a,b) EM σ =, h EM σ = (3.5a,b), h Although the deductons had been made for mdpont, the expressons for stress calculatons can be extended to others cross-sectons along the length of the coupled beam, beng enough to change M to M(x).
8 336 R. CVETKOVIĆ, D. STOJIĆ The shearng flow along the length of the coupled beam can be calculated by expresson: γveaa v = (3.6) The elastc lne s gven by: M ( x) v ( x) = (3.7) EI Ths means that, consderng a snusodal dstrbuton loadng, we attaned a dfferental equaton to elastc lne. The equaton (3.7), s well-known from bendng theory, wheren we replace the bendng stffness EI of homogeneous beam by the bendng stffness (EI) f of the case of coupled beams. The soluton of ths equaton s much smpler than expresson (3.). 4. DESIGN METHOD ACCORDING TO EUROCODE 5 The desgn of the composte beams s regulated n the appendx B of the Eurocode 5. The stress calculaton for tmber and concrete and the calculaton of the connectors s to be performed n accordance wth the theory of the elastc compound. Accordng to recommendatons from the appendx B of the Eurocode 5, n consstence wth what has been sad, we can calculate the geometrcal propertes, stresses, and characterstcs of connecton of the cross secton shown by fgure 4, accordng to next steps: b h,5h σ σ m, A I,5h a τ max y A I E b h,5h a h z σ m, σ Fg. 4. Geometrcal propertes and stresses The fectve bendng stffness wll be calculated as follows: = n = ( E I + γ E A a ), (4.) where: number of elements consstng composte (complex) cross secton. In case of T- cross secton, that s. E the average value of modulus of elastcty for concrete and tmber, respectvely A = bh, (4.a)
9 Desgn Methods of a Tmber-Concrete T-Cross-Secton 337 γ I = bh 3 /, (4.b) γ =, (4.c) ( K ) = + π EAs / a l for = and = 3, (4.d) ( + ) γ ( + ) γ E A h h E A h h = 3 γ EA = For T-cross sectons, h 3 =. The normal stresses are gven by equatons ( EI). (4.e) σ = γ E am/ (4.3a) ( EI) σ, =,5 E hm/ (4.3b) m The shear stress has the maxmum magntude at the pont where normal stresses are equal to zero. Maxmum shear stress at a certan pont along the heght of the tmber element of cross secton should be calculated accordng to the expresson: τ = γ E A a +,5E b h ) V /( b ( EI) ). (4.4),max ( The load of the fastener should be calculated accordng to expresson F = γ EAasV/ EI (4.5) wth = and 3, where s = s (x) dstance between fasteners determned n B.3 and V = V(x). REFERENCES. A. Ceccott, Tmber-concrete composte structures, Structural Tmber Engneerng Proceedngs (STEP ), lecture E3, 995;. H. J. Blaß und M. Schlager, Trag-und Verformungsverhalten von Holz-Beton-Verbundkonstruktonen- Tel, Bauen mt Holz 5/96, pages: , 996; 3. H. J. Blaß, M. van der Lnden und M. Schlager, Trag- und Verformungsverhalten von Holz-Beton- Verbundkonstruktonen-Tel, Bauen mt Holz 6/96, pages: , 996; 4. ENV Eurocode 5: Desgn of tmber structures, Part.: General rules and rules for buldng. European Commttee for Standardsaton. 993; 5. ENV 99--.Eurocode : Desgn of concrete structures, Part.: General rules and rules for buldng, European Commttee for Standardsaton 993; 6. R. Cvetkovć, Behavour of Composte Tmber-Concrete Structures wth Bendng Actons Masters thess, Department of Renforced Concrete and Prestressed Concrete Structures, Ruhr Unversty Bochum, Germany,. 7. B. Stevanovc, Analyss of Composte Tmber-Concrete Structures, Doctoral thess, Faculty of Cvl Engneerng, Belgrade, 3; 8. Demarzo Mauro, Tactano Marcelo:"Semrgd composte wood-concrete T-beams" Proceedngs of World Conference on Tmber Engneerng (WCTE ), P47, Brtsh Columba, Canada, 3. Jul-3. August. 9. Grhammar U. A. and Gopu K. A Composte beam-columns wth nterlayer slp - exact analyss. Journal of Structural Engneerng, New York, v.9, n.4, p. 65-8, Apr.
10 338 R. CVETKOVIĆ, D. STOJIĆ METODE PRORAČUNA SPREGNUTIH KONSTRUKCIJA OD DRVETA I BETONA T PRESEKA Radovan Cvetkovć, Dragoslav Stojć U radu je data analza spregnuth konstrukcja tpa drvo-beton. Povezvanjem drveta betona koršćenjem najboljh svojstava jednog drugog materjala dobja se nov spregnut tp konstrukcje za koj se, zavsno od razlčth uslova građenja, može nac mnogo razloga za prmenu s obzrom na određene prednost u odnosu na beton čelk. Ovde su, redom, date proračunske procedure prema teorj elastčnost zasnovane na tačnom prblžnom metodu posebno, prema pravlma preporukama modernog concepta za proračun drvenh betonskh konstrukcja dath u Evrokodu 5 zasnovanh na grančnm stanjma nosvost u upotrebljvost.
COMPOSITE BEAM WITH WEAK SHEAR CONNECTION SUBJECTED TO THERMAL LOAD
COMPOSITE BEAM WITH WEAK SHEAR CONNECTION SUBJECTED TO THERMAL LOAD Ákos Jósef Lengyel, István Ecsed Assstant Lecturer, Professor of Mechancs, Insttute of Appled Mechancs, Unversty of Mskolc, Mskolc-Egyetemváros,
More informationIndeterminate pin-jointed frames (trusses)
Indetermnate pn-jonted frames (trusses) Calculaton of member forces usng force method I. Statcal determnacy. The degree of freedom of any truss can be derved as: w= k d a =, where k s the number of all
More informationAPPENDIX F A DISPLACEMENT-BASED BEAM ELEMENT WITH SHEAR DEFORMATIONS. Never use a Cubic Function Approximation for a Non-Prismatic Beam
APPENDIX F A DISPACEMENT-BASED BEAM EEMENT WITH SHEAR DEFORMATIONS Never use a Cubc Functon Approxmaton for a Non-Prsmatc Beam F. INTRODUCTION { XE "Shearng Deformatons" }In ths appendx a unque development
More informationDUE: WEDS FEB 21ST 2018
HOMEWORK # 1: FINITE DIFFERENCES IN ONE DIMENSION DUE: WEDS FEB 21ST 2018 1. Theory Beam bendng s a classcal engneerng analyss. The tradtonal soluton technque makes smplfyng assumptons such as a constant
More informationSTATIC ANALYSIS OF TWO-LAYERED PIEZOELECTRIC BEAMS WITH IMPERFECT SHEAR CONNECTION
STATIC ANALYSIS OF TWO-LERED PIEZOELECTRIC BEAMS WITH IMPERFECT SHEAR CONNECTION Ákos József Lengyel István Ecsed Assstant Lecturer Emertus Professor Insttute of Appled Mechancs Unversty of Mskolc Mskolc-Egyetemváros
More informationPlease review the following statement: I certify that I have not given unauthorized aid nor have I received aid in the completion of this exam.
Please revew the followng statement: I certfy that I have not gven unauthorzed ad nor have I receved ad n the completon of ths exam. Sgnature: Instructor s Name and Secton: (Crcle Your Secton) Sectons:
More informationFrame element resists external loads or disturbances by developing internal axial forces, shear forces, and bending moments.
CE7 Structural Analyss II PAAR FRAE EEET y 5 x E, A, I, Each node can translate and rotate n plane. The fnal dsplaced shape has ndependent generalzed dsplacements (.e. translatons and rotatons) noled.
More informationModule 11 Design of Joints for Special Loading. Version 2 ME, IIT Kharagpur
Module 11 Desgn o Jonts or Specal Loadng Verson ME, IIT Kharagpur Lesson 1 Desgn o Eccentrcally Loaded Bolted/Rveted Jonts Verson ME, IIT Kharagpur Instructonal Objectves: At the end o ths lesson, the
More informationTHE EFFECT OF TORSIONAL RIGIDITY BETWEEN ELEMENTS ON FREE VIBRATIONS OF A TELESCOPIC HYDRAULIC CYLINDER SUBJECTED TO EULER S LOAD
Journal of Appled Mathematcs and Computatonal Mechancs 7, 6(3), 7- www.amcm.pcz.pl p-issn 99-9965 DOI:.75/jamcm.7.3. e-issn 353-588 THE EFFECT OF TORSIONAL RIGIDITY BETWEEN ELEMENTS ON FREE VIBRATIONS
More informationChapter Eight. Review and Summary. Two methods in solid mechanics ---- vectorial methods and energy methods or variational methods
Chapter Eght Energy Method 8. Introducton 8. Stran energy expressons 8.3 Prncpal of statonary potental energy; several degrees of freedom ------ Castglano s frst theorem ---- Examples 8.4 Prncpal of statonary
More informationFUZZY FINITE ELEMENT METHOD
FUZZY FINITE ELEMENT METHOD RELIABILITY TRUCTURE ANALYI UING PROBABILITY 3.. Maxmum Normal tress Internal force s the shear force, V has a magntude equal to the load P and bendng moment, M. Bendng moments
More informationOne Dimensional Axial Deformations
One Dmensonal al Deformatons In ths secton, a specfc smple geometr s consdered, that of a long and thn straght component loaded n such a wa that t deforms n the aal drecton onl. The -as s taken as the
More informationIn this section is given an overview of the common elasticity models.
Secton 4.1 4.1 Elastc Solds In ths secton s gven an overvew of the common elastcty models. 4.1.1 The Lnear Elastc Sold The classcal Lnear Elastc model, or Hooean model, has the followng lnear relatonshp
More informationPlease review the following statement: I certify that I have not given unauthorized aid nor have I received aid in the completion of this exam.
ME 270 Summer 2014 Fnal Exam NAME (Last, Frst): Please revew the followng statement: I certfy that I have not gven unauthorzed ad nor have I receved ad n the completon of ths exam. Sgnature: INSTRUCTIONS
More informationPlease review the following statement: I certify that I have not given unauthorized aid nor have I received aid in the completion of this exam.
ME 270 Fall 2013 Fnal Exam NME (Last, Frst): Please revew the followng statement: I certfy that I have not gven unauthorzed ad nor have I receved ad n the completon of ths exam. Sgnature: INSTRUCTIONS
More informationOFF-AXIS MECHANICAL PROPERTIES OF FRP COMPOSITES
ICAMS 204 5 th Internatonal Conference on Advanced Materals and Systems OFF-AXIS MECHANICAL PROPERTIES OF FRP COMPOSITES VLAD LUPĂŞTEANU, NICOLAE ŢĂRANU, RALUCA HOHAN, PAUL CIOBANU Gh. Asach Techncal Unversty
More informationThermal expansion of wood and timber-concrete composite members under ISO-fire exposure
Thermal expanson of wood and tmber-concrete composte members under ISO-fre exposure ANDREA FRANGI and MARIO FONTANA Insttute of Structural Engneerng, ETH Zurch, Swtzerland Summary Ths paper dscusses the
More informationMECHANICS OF MATERIALS
Fourth Edton CHTER MECHNICS OF MTERIS Ferdnand. Beer E. Russell Johnston, Jr. John T. DeWolf ecture Notes: J. Walt Oler Texas Tech Unversty Stress and Stran xal oadng Contents Stress & Stran: xal oadng
More informationPlan: Fuselages can. multideck
Lecture 22(18). TRENGTH ANALY OF FUELAGE Plan: 1. tructurally - power fuselage schemes. 2. trength analyss of fuselages cross-sectons. 3. emmonocoque fuselage cross-secton calculaton. Calculaton from external
More informationFinite Element Modelling of truss/cable structures
Pet Schreurs Endhoven Unversty of echnology Department of Mechancal Engneerng Materals echnology November 3, 214 Fnte Element Modellng of truss/cable structures 1 Fnte Element Analyss of prestressed structures
More informationSecond Order Analysis
Second Order Analyss In the prevous classes we looked at a method that determnes the load correspondng to a state of bfurcaton equlbrum of a perfect frame by egenvalye analyss The system was assumed to
More informationI have not received unauthorized aid in the completion of this exam.
ME 270 Sprng 2013 Fnal Examnaton Please read and respond to the followng statement, I have not receved unauthorzed ad n the completon of ths exam. Agree Dsagree Sgnature INSTRUCTIONS Begn each problem
More informationA Mechanics-Based Approach for Determining Deflections of Stacked Multi-Storey Wood-Based Shear Walls
A Mechancs-Based Approach for Determnng Deflectons of Stacked Mult-Storey Wood-Based Shear Walls FPINNOVATIONS Acknowledgements Ths publcaton was developed by FPInnovatons and the Canadan Wood Councl based
More informationTHE EFFECT OF BEAM TO COLUMN CONNECTION IN ARC PORTAL FRAME
THE EFFECT OF BEAM TO COLUMN CONNECTON N ARC PORTAL FRAME Asko Keronen Rakenteden Mekankka, Vol. 26 No 2 1993, ss. 35-5 SUMMARY A full scale rc (renforced concrete) portal frame has been bult n order to
More information829. An adaptive method for inertia force identification in cantilever under moving mass
89. An adaptve method for nerta force dentfcaton n cantlever under movng mass Qang Chen 1, Mnzhuo Wang, Hao Yan 3, Haonan Ye 4, Guola Yang 5 1,, 3, 4 Department of Control and System Engneerng, Nanng Unversty,
More informationORIGIN 1. PTC_CE_BSD_3.2_us_mp.mcdx. Mathcad Enabled Content 2011 Knovel Corp.
Clck to Vew Mathcad Document 2011 Knovel Corp. Buldng Structural Desgn. homas P. Magner, P.E. 2011 Parametrc echnology Corp. Chapter 3: Renforced Concrete Slabs and Beams 3.2 Renforced Concrete Beams -
More informationPlease review the following statement: I certify that I have not given unauthorized aid nor have I received aid in the completion of this exam.
ME 270 Sprng 2014 Fnal Exam NME (Last, Frst): Please revew the followng statement: I certfy that I have not gven unauthorzed ad nor have I receved ad n the completon of ths exam. Sgnature: INSTRUCTIONS
More informationI certify that I have not given unauthorized aid nor have I received aid in the completion of this exam.
ME 270 Fall 2012 Fnal Exam Please revew the followng statement: I certfy that I have not gven unauthorzed ad nor have I receved ad n the completon of ths exam. Sgnature: INSTRUCTIONS Begn each problem
More informationEVALUATION OF THE VISCO-ELASTIC PROPERTIES IN ASPHALT RUBBER AND CONVENTIONAL MIXES
EVALUATION OF THE VISCO-ELASTIC PROPERTIES IN ASPHALT RUBBER AND CONVENTIONAL MIXES Manuel J. C. Mnhoto Polytechnc Insttute of Bragança, Bragança, Portugal E-mal: mnhoto@pb.pt Paulo A. A. Perera and Jorge
More informationPlease review the following statement: I certify that I have not given unauthorized aid nor have I received aid in the completion of this exam.
NME (Last, Frst): Please revew the followng statement: I certfy that I have not gven unauthorzed ad nor have I receved ad n the completon of ths exam. Sgnature: INSTRUCTIONS Begn each problem n the space
More informationWeek3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity
Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1-s tme nterval. The velocty of the partcle
More informationDescription of the Force Method Procedure. Indeterminate Analysis Force Method 1. Force Method con t. Force Method con t
Indeternate Analyss Force Method The force (flexblty) ethod expresses the relatonshps between dsplaceents and forces that exst n a structure. Prary objectve of the force ethod s to deterne the chosen set
More informationCHAPTER 9 CONCLUSIONS
78 CHAPTER 9 CONCLUSIONS uctlty and structural ntegrty are essentally requred for structures subjected to suddenly appled dynamc loads such as shock loads. Renforced Concrete (RC), the most wdely used
More informationGEOSYNTHETICS ENGINEERING: IN THEORY AND PRACTICE
GEOSYNTHETICS ENGINEERING: IN THEORY AND PRACTICE Prof. J. N. Mandal Department of cvl engneerng, IIT Bombay, Powa, Mumba 400076, Inda. Tel.022-25767328 emal: cejnm@cvl.tb.ac.n Module - 9 LECTURE - 48
More informationNUMERICAL RESULTS QUALITY IN DEPENDENCE ON ABAQUS PLANE STRESS ELEMENTS TYPE IN BIG DISPLACEMENTS COMPRESSION TEST
Appled Computer Scence, vol. 13, no. 4, pp. 56 64 do: 10.23743/acs-2017-29 Submtted: 2017-10-30 Revsed: 2017-11-15 Accepted: 2017-12-06 Abaqus Fnte Elements, Plane Stress, Orthotropc Materal Bartosz KAWECKI
More informationDESIGN OPTIMIZATION OF CFRP RECTANGULAR BOX SUBJECTED TO ARBITRARY LOADINGS
Munch, Germany, 26-30 th June 2016 1 DESIGN OPTIMIZATION OF CFRP RECTANGULAR BOX SUBJECTED TO ARBITRARY LOADINGS Q.T. Guo 1*, Z.Y. L 1, T. Ohor 1 and J. Takahash 1 1 Department of Systems Innovaton, School
More informationNON LINEAR ANALYSIS OF STRUCTURES ACCORDING TO NEW EUROPEAN DESIGN CODE
October 1-17, 008, Bejng, Chna NON LINEAR ANALYSIS OF SRUCURES ACCORDING O NEW EUROPEAN DESIGN CODE D. Mestrovc 1, D. Czmar and M. Pende 3 1 Professor, Dept. of Structural Engneerng, Faculty of Cvl Engneerng,
More informationAPPROXIMATE ANALYSIS OF RIGID PLATE LOADING ON ELASTIC MULTI-LAYERED SYSTEMS
6th ICPT, Sapporo, Japan, July 008 APPROXIMATE ANALYSIS OF RIGID PLATE LOADING ON ELASTIC MULTI-LAYERED SYSTEMS James MAINA Prncpal Researcher, Transport and Infrastructure Engneerng, CSIR Bult Envronment
More informationFormulas for the Determinant
page 224 224 CHAPTER 3 Determnants e t te t e 2t 38 A = e t 2te t e 2t e t te t 2e 2t 39 If 123 A = 345, 456 compute the matrx product A adj(a) What can you conclude about det(a)? For Problems 40 43, use
More informationELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM
ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM An elastc wave s a deformaton of the body that travels throughout the body n all drectons. We can examne the deformaton over a perod of tme by fxng our look
More informationNumerical Heat and Mass Transfer
Master degree n Mechancal Engneerng Numercal Heat and Mass Transfer 06-Fnte-Dfference Method (One-dmensonal, steady state heat conducton) Fausto Arpno f.arpno@uncas.t Introducton Why we use models and
More informationIrregular vibrations in multi-mass discrete-continuous systems torsionally deformed
(2) 4 48 Irregular vbratons n mult-mass dscrete-contnuous systems torsonally deformed Abstract In the paper rregular vbratons of dscrete-contnuous systems consstng of an arbtrary number rgd bodes connected
More informationChapter 3. Estimation of Earthquake Load Effects
Chapter 3. Estmaton of Earthquake Load Effects 3.1 Introducton Sesmc acton on chmneys forms an addtonal source of natural loads on the chmney. Sesmc acton or the earthquake s a short and strong upheaval
More informationIf the solution does not follow a logical thought process, it will be assumed in error.
Group # Please revew the followng statement: I certfy that I have not gven unauthorzed ad nor have I receved ad n the completon of ths exam. Sgnature: INSTRUCTIONS Begn each problem n the space provded
More informationTorsion Stiffness of Thin-walled Steel Beams with Web Holes
Torson Stffness of Thn-walled Steel Beams wth Web Holes MARTN HORÁČEK, JNDŘCH MELCHER Department of Metal and Tmber Structures Brno Unversty of Technology, Faculty of Cvl Engneerng Veveří 331/95, 62 Brno
More informationMoments of Inertia. and reminds us of the analogous equation for linear momentum p= mv, which is of the form. The kinetic energy of the body is.
Moments of Inerta Suppose a body s movng on a crcular path wth constant speed Let s consder two quanttes: the body s angular momentum L about the center of the crcle, and ts knetc energy T How are these
More informationApplication to Plane (rigid) frame structure
Advanced Computatonal echancs 18 Chapter 4 Applcaton to Plane rgd frame structure 1. Dscusson on degrees of freedom In case of truss structures, t was enough that the element force equaton provdes onl
More informationχ x B E (c) Figure 2.1.1: (a) a material particle in a body, (b) a place in space, (c) a configuration of the body
Secton.. Moton.. The Materal Body and Moton hyscal materals n the real world are modeled usng an abstract mathematcal entty called a body. Ths body conssts of an nfnte number of materal partcles. Shown
More informationPhysics 53. Rotational Motion 3. Sir, I have found you an argument, but I am not obliged to find you an understanding.
Physcs 53 Rotatonal Moton 3 Sr, I have found you an argument, but I am not oblged to fnd you an understandng. Samuel Johnson Angular momentum Wth respect to rotatonal moton of a body, moment of nerta plays
More informationPrinciple of virtual work
Ths prncple s the most general prncple n mechancs 2.9.217 Prncple of vrtual work There s Equvalence between the Prncple of Vrtual Work and the Equlbrum Equaton You must know ths from statc course and dynamcs
More information9.2 Seismic Loads Using ASCE Standard 7-93
CHAPER 9: Wnd and Sesmc Loads on Buldngs 9.2 Sesmc Loads Usng ASCE Standard 7-93 Descrpton A major porton of the Unted States s beleved to be subject to sesmc actvty suffcent to cause sgnfcant structural
More informationPlease initial the statement below to show that you have read it
EN0: Structural nalyss Exam I Wednesday, March 2, 2005 Dvson of Engneerng rown Unversty NME: General Instructons No collaboraton of any nd s permtted on ths examnaton. You may consult your own wrtten lecture
More informationAERODYNAMICS I LECTURE 6 AERODYNAMICS OF A WING FUNDAMENTALS OF THE LIFTING-LINE THEORY
LECTURE 6 AERODYNAMICS OF A WING FUNDAMENTALS OF THE LIFTING-LINE THEORY The Bot-Savart Law The velocty nduced by the sngular vortex lne wth the crculaton can be determned by means of the Bot- Savart formula
More informationSection 8.3 Polar Form of Complex Numbers
80 Chapter 8 Secton 8 Polar Form of Complex Numbers From prevous classes, you may have encountered magnary numbers the square roots of negatve numbers and, more generally, complex numbers whch are the
More informationLAB 4: Modulus of elasticity
LAB 4: Modulus of elastcty 1. Preparaton: modulus of elastcty (chapter15, p.79) Hook s law graphcal determnaton of modulus of elastcty (p.8) determnaton of modulus of elastcty n tenson and flexural stress
More informationINDETERMINATE STRUCTURES METHOD OF CONSISTENT DEFORMATIONS (FORCE METHOD)
INETNTE STUTUES ETHO OF ONSISTENT EFOTIONS (FOE ETHO) If all the support reactons and nternal forces (, Q, and N) can not be determned by usng equlbrum equatons only, the structure wll be referred as STTIY
More informationPreliminary Design of Moment-Resisting Frames
Prelmnary Desgn of Moment-Resstng Frames Preprnt Aamer Haque Abstract A smple method s developed for prelmnary desgn of moment-resstng frames. Preprnt submtted to Elsever August 27, 2017 1. Introducton
More informationStructural Dynamics and Earthquake Engineering
Structural Dynamcs and Earthuake Engneerng Course 9 Sesmc-resstant desgn of structures (1) Sesmc acton Methods of elastc analyss Course notes are avalable for download at http://www.ct.upt.ro/users/aurelstratan/
More informationME 307 Machine Design I. Chapter 8: Screws, Fasteners and the Design of Nonpermanent Joints
Dr.. zz Bazoune Chapter 8: Screws, Fasteners and the Desgn of Nonpermanent Jonts Dr.. zz Bazoune Chapter 8: Screws, Fasteners and the Desgn of Nonpermanent Jonts CH-8 LEC 35 Slde 2 Dr.. zz Bazoune Chapter
More informationChapter 11 Angular Momentum
Chapter 11 Angular Momentum Analyss Model: Nonsolated System (Angular Momentum) Angular Momentum of a Rotatng Rgd Object Analyss Model: Isolated System (Angular Momentum) Angular Momentum of a Partcle
More informationApproximate Method For Probabilistic Presentation Of The Cross-Sectional Properties Of Shipbuilding Structural Profiles And Hull Girder
Summary ABS TECHNICAL PAPERS 2007 10th Internatonal Symposum on Practcal Desgn of Shps and Other Floatng Structures Houston, Texas, Unted States of Amerca 2007 Amercan Bureau of Shppng Approxmate Method
More informationFINITE DIFFERENCE ANALYSIS OF CURVED DEEP BEAMS ON WINKLER FOUNDATION
VOL. 6, NO. 3, MARCH 0 ISSN 89-6608 006-0 Asan Research Publshng Network (ARPN). All rghts reserved. FINITE DIFFERENCE ANALYSIS OF CURVED DEEP BEAMS ON WINKLER FOUNDATION Adel A. Al-Azzaw and Al S. Shaker
More informationAssessment of Site Amplification Effect from Input Energy Spectra of Strong Ground Motion
Assessment of Ste Amplfcaton Effect from Input Energy Spectra of Strong Ground Moton M.S. Gong & L.L Xe Key Laboratory of Earthquake Engneerng and Engneerng Vbraton,Insttute of Engneerng Mechancs, CEA,
More informationANALYSIS OF TIMOSHENKO BEAM RESTING ON NONLINEAR COMPRESSIONAL AND FRICTIONAL WINKLER FOUNDATION
VOL. 6, NO., NOVEMBER ISSN 89-668 6- Asan Research Publshng Network (ARPN). All rghts reserved. ANALYSIS OF TIMOSHENKO BEAM RESTING ON NONLINEAR COMPRESSIONAL AND FRICTIONAL WINKLER FOUNDATION Adel A.
More informationTransactions of the VŠB Technical University of Ostrava, Mechanical Series. article No. 1907
Transactons of the VŠB Techncal Unversty of Ostrava, Mechancal Seres No., 0, vol. LVIII artcle No. 907 Marek NIKODÝM *, Karel FYDÝŠEK ** FINITE DIFFEENCE METHOD USED FO THE BEAMS ON ELASTIC FOUNDATION
More informationRotational Dynamics. Physics 1425 Lecture 19. Michael Fowler, UVa
Rotatonal Dynamcs Physcs 1425 Lecture 19 Mchael Fowler, UVa Rotatonal Dynamcs Newton s Frst Law: a rotatng body wll contnue to rotate at constant angular velocty as long as there s no torque actng on t.
More information8.1 Arc Length. What is the length of a curve? How can we approximate it? We could do it following the pattern we ve used before
.1 Arc Length hat s the length of a curve? How can we approxmate t? e could do t followng the pattern we ve used before Use a sequence of ncreasngly short segments to approxmate the curve: As the segments
More informationDifference Equations
Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1
More informationPhysics 5153 Classical Mechanics. Principle of Virtual Work-1
P. Guterrez 1 Introducton Physcs 5153 Classcal Mechancs Prncple of Vrtual Work The frst varatonal prncple we encounter n mechancs s the prncple of vrtual work. It establshes the equlbrum condton of a mechancal
More informationCapacity of Open Rectangular Shear Walls
ISSN: 9-5967 ISO 900:008 Certfed Volume, Issue, January 0 Capacty of Open Rectangular Shear Walls Essam M. Awdy, Hlal A. M. Hassan Assst Professor, Structural Engneerng, Zagazg Unversty, Egypt Abstract
More informationMaximum Stress Estimation Model for Multi-Span Waler Beams with Deflections at the Supports Using Average Strains
Sensors 05, 5, 778-774; do:0.3390/s5040778 Artcle OPEN ACCESS sensors ISSN 44-80 www.mdp.com/journal/sensors Maxmum Stress Estmaton Model for Mult-Span Waler Beams wth Deflectons at the Supports Usng Average
More informationLab 2e Thermal System Response and Effective Heat Transfer Coefficient
58:080 Expermental Engneerng 1 OBJECTIVE Lab 2e Thermal System Response and Effectve Heat Transfer Coeffcent Warnng: though the experment has educatonal objectves (to learn about bolng heat transfer, etc.),
More informationActual behaviour of composite externally CFRP-reinforced timber beams stress analysis
Actual behavour of composte externally CFRP-renforced tmber beams stress analyss Marcela Karmazínová Abstract The paper s focused on the problems of composte CFRP-renforced tmber beams. CFRP renforcement
More informationANALYSIS OF PILE EQUIVALENT ANCHORAGE LENGTH FOR ELEVATED PILE CAPS UNDER LATERAL LOAD
The th World Conference on Earthquake Engneerng October -7, 008, Bejng, Chna ANALYSIS OF PILE EQUIVALENT ANCHORAGE LENGTH FOR ELEVATED PILE CAPS UNDER LATERAL LOAD M ZHOU Wancheng YUAN and Yue ZHANG Ph.D
More informationChapter 12. Ordinary Differential Equation Boundary Value (BV) Problems
Chapter. Ordnar Dfferental Equaton Boundar Value (BV) Problems In ths chapter we wll learn how to solve ODE boundar value problem. BV ODE s usuall gven wth x beng the ndependent space varable. p( x) q(
More informationPlease review the following statement: I certify that I have not given unauthorized aid nor have I received aid in the completion of this exam.
ME 270 Sprng 2017 Exam 1 NAME (Last, Frst): Please revew the followng statement: I certfy that I have not gven unauthorzed ad nor have I receved ad n the completon of ths exam. Sgnature: Instructor s Name
More informationProfessor Terje Haukaas University of British Columbia, Vancouver The Q4 Element
Professor Terje Haukaas Unversty of Brtsh Columba, ancouver www.nrsk.ubc.ca The Q Element Ths document consders fnte elements that carry load only n ther plane. These elements are sometmes referred to
More information3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X
Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number
More informationA Tale of Friction Basic Rollercoaster Physics. Fahrenheit Rollercoaster, Hershey, PA max height = 121 ft max speed = 58 mph
A Tale o Frcton Basc Rollercoaster Physcs Fahrenhet Rollercoaster, Hershey, PA max heght = 11 t max speed = 58 mph PLAY PLAY PLAY PLAY Rotatonal Movement Knematcs Smlar to how lnear velocty s dened, angular
More informationMAE140 - Linear Circuits - Winter 16 Final, March 16, 2016
ME140 - Lnear rcuts - Wnter 16 Fnal, March 16, 2016 Instructons () The exam s open book. You may use your class notes and textbook. You may use a hand calculator wth no communcaton capabltes. () You have
More informationPhysics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian-1
P. Guterrez Physcs 5153 Classcal Mechancs D Alembert s Prncple and The Lagrangan 1 Introducton The prncple of vrtual work provdes a method of solvng problems of statc equlbrum wthout havng to consder the
More informationFoundations of Arithmetic
Foundatons of Arthmetc Notaton We shall denote the sum and product of numbers n the usual notaton as a 2 + a 2 + a 3 + + a = a, a 1 a 2 a 3 a = a The notaton a b means a dvdes b,.e. ac = b where c s an
More informationLecture Note 3. Eshelby s Inclusion II
ME340B Elastcty of Mcroscopc Structures Stanford Unversty Wnter 004 Lecture Note 3. Eshelby s Incluson II Chrs Wenberger and We Ca c All rghts reserved January 6, 004 Contents 1 Incluson energy n an nfnte
More informationSystem in Weibull Distribution
Internatonal Matheatcal Foru 4 9 no. 9 94-95 Relablty Equvalence Factors of a Seres-Parallel Syste n Webull Dstrbuton M. A. El-Dacese Matheatcs Departent Faculty of Scence Tanta Unversty Tanta Egypt eldacese@yahoo.co
More information( ) = ( ) + ( 0) ) ( )
EETOMAGNETI OMPATIBIITY HANDBOOK 1 hapter 9: Transent Behavor n the Tme Doman 9.1 Desgn a crcut usng reasonable values for the components that s capable of provdng a tme delay of 100 ms to a dgtal sgnal.
More informationGeometrically exact multi-layer beams with a rigid interconnection
Geometrcally exact mult-layer beams wth a rgd nterconnecton Leo Škec, Gordan Jelenć To cte ths verson: Leo Škec, Gordan Jelenć. Geometrcally exact mult-layer beams wth a rgd nterconnecton. 2nd ECCOMAS
More informationIncrease Decrease Remain the Same (Circle one) (2 pts)
ME 270 Sample Fnal Eam PROBLEM 1 (25 ponts) Prob. 1 questons are all or nothng. PROBLEM 1A. (5 ponts) FIND: A 2000 N crate (D) s suspended usng ropes AB and AC and s n statc equlbrum. If θ = 53.13, determne
More information2. PROBLEM STATEMENT AND SOLUTION STRATEGIES. L q. Suppose that we have a structure with known geometry (b, h, and L) and material properties (EA).
. PROBEM STATEMENT AND SOUTION STRATEGIES Problem statement P, Q h ρ ρ o EA, N b b Suppose that we have a structure wth known geometry (b, h, and ) and materal propertes (EA). Gven load (P), determne the
More informationEffect of anisotropy on laminated composite plates containing circular holes
Indan Journal of ngneerng & Materals Scences Vol. 1, June 005, pp. 07-13 ffect of ansotropy on lamnated composte plates contanng crcular holes H Murat Arslan Cukurova Unversty, Cvl ngneerng Department,
More informationTHE SMOOTH INDENTATION OF A CYLINDRICAL INDENTOR AND ANGLE-PLY LAMINATES
THE SMOOTH INDENTATION OF A CYLINDRICAL INDENTOR AND ANGLE-PLY LAMINATES W. C. Lao Department of Cvl Engneerng, Feng Cha Unverst 00 Wen Hwa Rd, Tachung, Tawan SUMMARY: The ndentaton etween clndrcal ndentor
More informationΔ x. u(x,t) Fig. Schematic view of elastic bar undergoing axial motions
ME67 - Handout 4 Vbratons of Contnuous Systems Axal vbratons of elastc bars The fgure shows a unform elastc bar of length and cross secton A. The bar materal propertes are ts densty ρ and elastc modulus
More informationEN40: Dynamics and Vibrations. Homework 4: Work, Energy and Linear Momentum Due Friday March 1 st
EN40: Dynamcs and bratons Homework 4: Work, Energy and Lnear Momentum Due Frday March 1 st School of Engneerng Brown Unversty 1. The fgure (from ths publcaton) shows the energy per unt area requred to
More informationLifetime prediction of EP and NBR rubber seal by thermos-viscoelastic model
ECCMR, Prague, Czech Republc; September 3 th, 2015 Lfetme predcton of EP and NBR rubber seal by thermos-vscoelastc model Kotaro KOBAYASHI, Takahro ISOZAKI, Akhro MATSUDA Unversty of Tsukuba, Japan Yoshnobu
More informationBuckling analysis of single-layered FG nanoplates on elastic substrate with uneven porosities and various boundary conditions
IOSR Journal of Mechancal and Cvl Engneerng (IOSR-JMCE) e-issn: 78-1684,p-ISSN: 30-334X, Volume 15, Issue 5 Ver. IV (Sep. - Oct. 018), PP 41-46 www.osrjournals.org Bucklng analyss of sngle-layered FG nanoplates
More informationCOMPLEX NUMBERS AND QUADRATIC EQUATIONS
COMPLEX NUMBERS AND QUADRATIC EQUATIONS INTRODUCTION We know that x 0 for all x R e the square of a real number (whether postve, negatve or ero) s non-negatve Hence the equatons x, x, x + 7 0 etc are not
More informationModule 2. Random Processes. Version 2 ECE IIT, Kharagpur
Module Random Processes Lesson 6 Functons of Random Varables After readng ths lesson, ou wll learn about cdf of functon of a random varable. Formula for determnng the pdf of a random varable. Let, X be
More informationLecture 12: Discrete Laplacian
Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly
More informationThe Two-scale Finite Element Errors Analysis for One Class of Thermoelastic Problem in Periodic Composites
7 Asa-Pacfc Engneerng Technology Conference (APETC 7) ISBN: 978--6595-443- The Two-scale Fnte Element Errors Analyss for One Class of Thermoelastc Problem n Perodc Compostes Xaoun Deng Mngxang Deng ABSTRACT
More informationENGI 1313 Mechanics I
ENGI 11 Mechancs I Lecture 40: Center of Gravty, Center of Mass and Geometrc Centrod Shan Kenny, Ph.D., P.Eng. ssstant Professor Faculty of Engneerng and ppled Scence Memoral Unversty of Nefoundland spkenny@engr.mun.ca
More informationNUMERICAL DIFFERENTIATION
NUMERICAL DIFFERENTIATION 1 Introducton Dfferentaton s a method to compute the rate at whch a dependent output y changes wth respect to the change n the ndependent nput x. Ths rate of change s called the
More information