7. Component Load State and Analysis of Stresses

Transcription

1 SREGH O AERIALS ehhanosüsteemide komponentide õppetool 7. Component Load State and Analsis of Stresses 7. Load State of a Component 7. Stress heor and Stress Analsis Priit Põdra 7. Component Load State and Analsis of Stresses SREGH O AERIALS 7.. Load State of a Component Priit Põdra 7. Component Load State and Analsis of Stresses

2 Structural odel is a Basis for Strength Analsis SRUCURAL ODEL = graphical representation of an ideal mechanical sstem together with relevant dimensions and other data Real Structure Simplified echanical Sstem Bar is deformable Supports are absolutel rigid Joints are absolutel rigid Ideal echanical Sstem Parameters with negligible influence are eliminated (Saint Venant Principle) oo simple structural model large uncertaint of analsis results Real Shaft oo comple structural model voluminous calculation work Shaft Structural odels Loads in principal centroidal planes plane plane Structural odel for orsion L L L L L L L L L Priit Põdra 7. Component Load State and Analsis of Stresses Internal orces Analsis with the ethod of s orce and oment Projections at a Q Centroid (Internal) orce Sstem = interaction of resultant force and resultant moment, that are applied to the section centroid Combination of,, andq depends on particular task Resultant force and resultant moment are represented b the projections about the section principal centroidal aes Priit Põdra 7. Component Load State and Analsis of Stresses 4 Q

3 Steps of Internal orces Analsis. Choosing of the sections to be analsed; Diagrams of Internal orces and Principal Plane. Determining of the sections centroid; A B. Determining of the sections principal centroidal aes; A B 4. Reduction of loads into components about the sections L L L principal centroidal aes; Diagram of Q is not shown 5. Calculation of section resultant force and resultant moment components about the principal centroidal aes together Diagram of Principal Plane with determination of signs (using the method of sections): A - aial force (in direction of ), B - shear force Q (in direction of and ), Diagram of Q is not shown - bending moments (around and ), - torque (around ); Diagram of 6. Distribution of internal force components along the bar ais is described b the inthernal force diagrams. Load state of each section is determined b the Diagram of combination of internal force components acting there Priit Põdra 7. Component Load State and Analsis of Stresses 5 or Load States of a Component () Component LOAD SAE = set of internal forces and respective deformations, acting in the loaded bar According to the number of different internal forces, acting simultaneousl in the component (and/or at some of its sections), the load states are divided as: SIPLE Load States = just one internal force is acting at the component cross-section (, Q, or ) COBIED Load States = a combination of several internal forces (simple load states) is acting at the component cross-section he SIPLE Load States are: ESIO/COPRESSIO = onl the aial force is acting (or is relevant) at the component cross-sections; ORSIO = onl the torque is acting (or is relevant) at the component cross-sections; BEDIG = just one and onl bending moment is acting (or is relevant) at the component cross-sections (planar bending); SHEAR = just one and onl shear force Q is acting (or is relevant) at the component cross-sections Priit Põdra 7. Component Load State and Analsis of Stresses 6

4 Properties of Simple Load States Each Simple Load State is chaeracterised b: direction (relative to section) and tpe (force or moment) of internal force; cpecific deformation and specific failure mechanism in the case of breakage. ESIO & COPRESSIO Pure SHEAR ORSIO Pure BEDIG. he cross-section remain plain (Bernoulli hpothesis). he shape of cross-sections does not change. he area of cross-sections does not change 4. he cross-sections remain perpendicular to the bar ais 5. he cross-sections remain parallel relative to each other and the initial position 5. he cross-sections rotate 6. he ais of (straigth) bar remains straigth 6. he bar ais curves 7. he length of bar changes 7. he length of bar does not change Priit Põdra 7. Component Load State and Analsis of Stresses 7 Component Deformations and ailure ESIO & COPRESSIO Pure SHEAR ORSIO Pure BEDIG ormal Deformation angential Deformation ormal Deformation Compressive Deformation e ensile Deformation Shearing Deformation Short Component Q orsional Deformation Bending Deformation Distance between crosssections changes in the direction of component ais ailure in Compression (shear) Cross-sections move relative to each other in direction perpendicular to component ais Shearing ailure Cross-sections rotate relative to each other around the component ais orsional ailure (shear) Cross-sections rotate relative to each other around the cross-sections principal centroidal ais Bending ailure ailure in ension Priit Põdra 7. Component Load State and Analsis of Stresses 8

5 Combined Load States of a Component COBIED Load State = several internal forces are simultaneousl acting at the componet cross-sections COBIED Load State = combined interaction of SIPLE load states ost common COBIED load states are: Bending + Shear = bending moment and shear force Q are acting at the same principal plane of component cross-section Bending + Bending = bending moments (shear forces Q ma also be present) areacting in both principal planes of component cross-section Eccentric ension or Compression orsion + Bending = aial force and bending moment(s) (at one or both principal planes) are acting at the component cross-section (shear forces Q ma also be present) = torque and bending moment(s) (at one or both principal planes) are acting at the component cross-section (shear forces Q ma also be present) Priit Põdra 7. Component Load State and Analsis of Stresses 9 Internal orces of Combined Load States Principle of orces Superposition Impact of a sstem of loads = sum of single load impacts Analsis of combined load state is based on the analsis of consistuent simple load states Solutions for simple load states give the solution of combined load state Bending + Shear Bending + Bending Eccentric Compression Q Bending and shear at the same principal plane orsion + Bending Bending or Bending + Compression and bending orsion and bending in one Shear in both principal in one or both principal or both principal planes planes planes Priit Põdra 7. Component Load State and Analsis of Stresses 0

6 SREGH O AERIALS 7.. Stress heor and Stress Analsis Priit Põdra 7. Component Load State and Analsis of Stresses Internal orce is a Resultant of Stresses ESIO & COPRESSIO Pure SHEAR ORSIO Pure BEDIG Q diagram diagram diagram diagram Aial force-resultant equation da A Shear force-resultant equation Q da A orque-resultant equation Priit Põdra 7. Component Load State and Analsis of Stresses A da Internal orce-resultant Equation = mathematical function of section internal force and stress SRESS = intensit of internal force at some point of the section Bending moment-resultant equation A A da da

7 Pingete koosmõju sisepinna punktides Igale sisepinna sisejõule vastab oma selle staatilisest seosest tulenev pingelaotus: Pikijõud pikkepinge Paindemoment paindepinge Vääne vildakpaindega epüür epüür ma O eed on normaalpinged mõjuvad sisepinnaga risti O h ma Põikjõud Q lõikepinge Q Väändemoment väändepinge ma O ma eed on nihkepinged mõjuvad piki sisepinda Analüüsida on vaja punktide (erinevatel sisepindadel mõjuvate) pingekomponentide koosmõju. b ma epüür ma epüür O : ma O : O : ma ma ; ; h ; b ma ma ma Priit Põdra 7. Component Load State and Analsis of Stresses Load State Stresses and Stress heor AXIU normal stress What is, for the given load state,:? AXIU shear stress Important problem of stress analsis Strength Condition: aimum Stress Design Stress SIPLE Load States aimum Stresses: act at the known sections, can easil be calculated WE OW: Distributions, values and directions of stresses at the CROSS-SECIOS COBIED Load States aimum Stresses: sections are not known, need more sophisticated analsis WE EED O OW: AXIU values and directions of normal and shear stresses at the component sections SRESS HEORY = Part of the Strength of aterials, that studies the interrelations among stresses, that act at the same point of componet, but on the sections of different angle of inclination Priit Põdra 7. Component Load State and Analsis of Stresses 4

8 Stress State of a Point Stress State of a Component Point = set and interrelation of stresses, that act at sections of given point s of a loaded component in equilibrium aterial point i 0 Component Each section at that point ma have different value of normal stress Each section at that point ma have different value of shear stress I II III 4 5 Sisepind IV Evan at the sections of same inclination, the stress values of different points ma be different HOOGEOUS SRESS SAE = at the sections of same inclination, the stress values of different points are equal Priit Põdra 7. Component Load State and Analsis of Stresses 5 PRICIPAL PLAE at the Point of a Loaded Component Onl normal stress (shear stress is absent) is acting at the crosssection points due to AXIAL load Both normal stress and shear stress are acting at inclined sections due to AXIAL load Stress valuea at the section depend on the inclination of that section Cross- under AXIAL LOADIG = PRICIPAL PLAE (at all points of that section) 0; = 0 PRICIPAL PLAE = section (of the loaded component at the given point) which has no shear stress Important Concept Priit Põdra 7. Component Load State and Analsis of Stresses 6

9 PRICIPAL SRESSES at the Point of a Loaded Component PRICIPAL SRESS = normal stress (tensile or compressive) acting at the principal plane (section, where shear stress = 0) HREE PERPEDICULAR PRICIPAL PLAES where the shear stresses are absent can be found at each point of the loaded componentga, so that: Principal stresses are designated:,, wo principal stresses are alwas the etremal stresses at that point Position of principal planes depend on the load state = maimum tensile stress at that point (+) = maimum compressive stress at that point (-) Stress states are characterised based on the principal stresses: HREE-dimensional stress, if 0; 0; 0 PLAE stress, if 0; =0; 0 OE-dimensional stress, if 0; = 0; = 0 Priit Põdra 7. Component Load State and Analsis of Stresses 7 Stress Analsis of Combined Bending and Shear Combained bending and shear action Point to be studied CROSS-SECIO normal stress at point the I h Cross- Q diagram A diagram Q Cross- Structural odel Q diagram Q diagram CROSS-SECIO shear stress at point the Q 4 Q A h Calculate the maimum stresses at the point!!! Priit Põdra 7. Component Load State and Analsis of Stresses 8

10 Plane Stress at the Component Points Stresses at the point due to combined bending and shear Internal forces and are absent at that section All internal forces are absent at that section = 0 0 = 0 = 0 Shear stress (with respective internal force) due to the law of complementar shear stresses Both normal and shear stress are acting at the cross-section Principal Stress = maimum tensile stress at the point Principal Stress = maimum compressive stress at the point Cross-section Longitudinal section Principal planes are inclined about the crosssection Priit Põdra 7. Component Load State and Analsis of Stresses 9 Principal stresses at the point Calculate the principal stresses at the point!!! Principal Stresses of a Plane Stress State () General Case of a Plane Stress da Cross- da Longitudinal aterial element with side surface areas of da and da is located at the point Surfaces da and da are small enough in order to consider the stress distributions uniform Q Internal forces of surfaces da and da da da Q da da Equilibrium of the aterial Element Inclined da n Q da Cross- da Longitudinal Q Q t aterial element is sectioned b the surface, that is inclined about the cross-section Equilibrium Equations of the ed aterial Element n t 0 0 Q cosφ sinφ sinφ Q cosφ Q sinφ Q cosφ Q cosφ 0 sinφ 0 Priit Põdra 7. Component Load State and Analsis of Stresses 0

11 Principal Stresses of a Plane Stress State () Surface of inclined Internal orces of section da is small Surface da enough in order to consider the stress distributions uniform da Q da da da dacos dasin ormal and Shear Stress at the Inclined of Point cos sin cos Cross- ormal Stress Equilibrium Equations of the aterial Element using Stresses da dacos dasin dasincos da dasincos dasincos dacos sin Longitudinal ormal Stress sin Angle of Inclination Cross- Shear Stress ind the angle, when = 0 and = ma of that angle is the principal plane at the point Priit Põdra 7. Component Load State and Analsis of Stresses Principal Stresses of a Plane Stress State () Principal Stresses of point Principal Planes Cross- d d unction etremal values are, where its derivative equals 0 d = ma where 0 d sin cos 0 Principal Plane Angle (section, where = ma and = 0) tan d d d = ma where 0 d cos sin 0, where = ma cot Principal Stresses at the Point aimum ensile Stress aimum Compressive Stress ma min aimum Shear Stress at the Point ma Priit Põdra 7. Component Load State and Analsis of Stresses

12 aimum Stresses of the Plane Stress State Principal Stresses of a Plane Stress State Principal Planes aimum Shear Stresses of a Plane Stress State 45 Principal Planes a a Cross- Cross- a a tan cot tan cot 45 of maimum shear stress is inclined b 45 relative to the principal planes Priit Põdra 7. Component Load State and Analsis of Stresses PLAE Stress Applications aimum Stresses for Combined Bending and Shear aimum Stresses of Pure orsion Structural odel Cross- diagram aimum ormal Stresses Q Q Stresses of Cross- 0 ma Q aimum Shear Stress Q diagram Stresses of Cross- 0 0 aimum ormal Stresses aimum Shear Stress ma Priit Põdra 7. Component Load State and Analsis of Stresses 4

13 OE-DIESIOAL Stress Applications aimum Stresses for ension/compression Structural odel Cross- Stresses of diagram diagram Cross- 0 aimum ormal Stresses 0 0 aimum Shear Stress ma OE-DIESIOAL Stress = just one principal stress is acting at the given point (values of other principal stresses equal to 0) maimum Stresses for Pure BEDIG Structural odel Stresses of Cross- aimum ormal Stresses 0 Cross- Priit Põdra 7. Component Load State and Analsis of Stresses 5 diagram 0 diagram aimum Shear Stress ma Principal Plane of One- Dimensional Stress = Component Cross- 0 HREE-DIESIOAL Stress State and Hooke Law in general D Stress State Principal Direction hree-dimensioanal Stress state is to be analsed as a combination of three plane stress states hree Plane Stress States of a hree-dimensional Stress 45 Principal Direction Hooke law for One- Dimensional Stress E Principal Direction Hooke law in General for the Principal Directions E E E = Strain E = aterial odulus of Elasticit, Pa = aterial Poisson Ratio Priit Põdra 7. Component Load State and Analsis of Stresses 6

14 SREGH O AERIALS ehhanosüsteemide komponentide õppetool HA YOU! Questions, please? Priit Põdra 7. Component Load State and Analsis of Stresses 7