5.1. Cross-Section and the Strength of a Bar
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1 TRENGTH OF MTERL Meanosüsteemide komponentide õppetool 5. Properties of ections 5. ross-ection and te trengt of a Bar 5. rea Properties of Plane apes 5. entroid of a ection 5.4 rea Moments of nertia 5.5 Principal Moments of nertia Priit Põdra 5. Properties of ections TRENGTH OF MTERL 5.. ross-ection and te trengt of a Bar Priit Põdra 5. Properties of ections
2 Bar ross-ection and trengt Te probleem of trengt nalsis: re te bar cross-section sape and dimensions optimum? Tick branc is stronger tan tin branc wic tickness is sufficient? Priit Põdra 5. Properties of ections Parameters of ross-ection trengt Tension and ompression Bar strengt depends on ow te load is applied to it ross-ection Torsion ear F D N diagram ross-ection Parameter of trengt rea Dimension; [m ] Wen D times, ten strengt = 4 times ross-ection D diagram Priit Põdra 5. Properties of ections 4 max M max T W ross-ection Parameter of trengt Polar moment of resistance W Dimension; [m ] Wen D times, ten trengt = 8 times F Q ross-ection diagram D ross-ection Parameter of trengt rea Dimension; [m ] Wen D times, ten strengt = 4 times
3 trengt of Bar ross-ection under Bending n te problems of bending: Te cross-section strengt is caracterised b te moments of resistance about te principal centroidal axes of inertia Beam Under Bending ction entroid and Principal xes of ross-ection entroid Te axis of te beam is crooked due to load m Principal xes nalsis of a cross-section of te beam (bar) under bending: Determine te location of cross-section centroid Determine te direction of principal axes alculate te area moments of inertia Priit Põdra 5. Properties of ections 5 TRENGTH OF MTERL 5.. rea Properties of Plane apes Priit Põdra 5. Properties of ections 6
4 General Definition of a Problem ross-ection = planar geometrical sape alculation of cross-section parameters = problem of planar geometr General Geometr Parameters of a Planar ape Rectangular oordinate stem d rea Element rea Outline mportant Parameters of a ape for trengt nalsis: urface rea tatic Moments of rea Location of entroid Direction of entroidal Principal xes of nertia Priit Põdra 5. Properties of ections 7 rea Moments of a Planar ape rea tatical Moments [m ] d statical moment about - axis d statical moment about axis d d rea nertia Moments, [m 4 ] xial nertia Moments d axial inertia moment about axial inertia d area product of axis moment about axis inertia about axes d area polar inertia moment about axis rea moments are alwas calculated about some predetermined axes Priit Põdra 5. Properties of ections 8
5 TRENGTH OF MTERL 5.. entroid of a ection Priit Põdra 5. Properties of ections 9 tatical Moments and entroidal xes Te values of static moments d d depend on te location of respective axes and rea tatical Moments entroidal xes xes, tat satisf = centroidal axes Tere is infinite number of centroidal axes entroid rea entroid = point were all centroidal axes intersect ross-ection rea entroid = point, were te bar axis crosses te cross-section Priit Põdra 5. Properties of ections
6 alculation of rea entroid Location Te dimensions and location of a sape about -axes must be known Determine suc location of axes, tat a Distance between and d b Distance between and a d b d d b Taking: (accordingl), ten: d a b b Priit Põdra 5. Properties of ections oordinates of a entroid Geometr and dimensions of te sape are known Known axis sstem entroid rea centroid coordinates can be calculated onl, if respective axes are predefined oordinates of te entroid rea tatical Moment about xis rea Priit Põdra 5. Properties of ections
7 tatical Moment of a imple ape imple ape = te sape for wic: R ircle R location of te centroid is known surface area can be easil calculated area moments can be easil calculated R b Rectangle d d b b Priit Põdra 5. Properties of ections tatical Moment of a ombined ape ombined ape = te sape ombined ape for wic: location of te centroid is not known surface area is not eas to calculate area moments are not eas to calculate division into simple sapes is eas ombined sape is dividend into parts: ombined sape part can be eiter positiive or negative d d d tatical Moment of a ombined ape () () ()... (accordingl) d... tatical Moment of a ombined ape = total of te parts statical moments Priit Põdra 5. Properties of ections 4 () () () bout te same axis...
8 oordinates of a ombined ape entroid oordinates i are not sown tatic Moments of onstituent apes st = Rectangle: nd = Rectangle: rd = Rectangle: oordinate of te st consistuent sape centroid urface area of te st consistuent sape tatic moment of te combined sape urface area of te combined sape rea of ombined ape tatic Moments of ombined ape Priit Põdra 5. Properties of ections 5 oordinates of entroid TRENGTH OF MTERL 5.4. rea Moments of nertia Priit Põdra 5. Properties of ections 6
9 nertia Moments of imple apes () rea properties of simple sapes are given in Engineering Handbooks Formulae for areas, centroids, and inertia moments calculation ccording to definition Rectangle d = bd b d b b d b b d xial Moments of nertia Product of nertia ntegration according to te geometr and dimensions d Pat of derivation is not sown Priit Põdra 5. Properties of ections 7 nertia Moments of imple apes () Transformation for integration purposes b Triangle b dsd d s / b b d Pat of derivation is not sown b 6 4 b b b b d d b b b xial Moments of nertia b Product of nertia d Wen: b b b 7 b b, ten: b 48 sosceles triangle Priit Põdra 5. Properties of ections 8
10 nertia Moments of imple apes () Transformation for integration purposes ircle d = d Polar coordinates Polar Moments of nertia d D 4 d or D D 4 D d d d xial Moments of nertia Product of nertia D 64 4 d Pat of derivation is not sown Priit Põdra 5. Properties of ections 9 rea Moments of nertia about paralleel axes d entroidal xes rea moments of inertia are known about some predefined axes Or vice versa Tese axes sstems are parallel a rea moments of inertia need to be calculated about centroidal axes b are entroidal xes d a b a xial Moments of nertia d (accordingl) d b d b d b b b a Priit Põdra 5. Properties of ections
11 Parallel xis Teorems xial Moments of nertia about Parallel xes rea entroid rea Moment of nertia about some xis entroidal xes e e e urface rea of te ape Distance between te xis and entroidal xis rea Moment of nertia about centroidal xis e Polar Moments of nertia about Parallel xes e e e e Distance between and Non-entroidal xes e rea Products of nertia about Parallel xes e e Priit Põdra 5. Properties of ections nertia Moments of ombined ape () rea moments of inertia of eac constituent sape are known about teir own centroidal axes rea moments of inertia of combined sape need to be calculated about te centroidal axes of combined sape ombined sape is dividend into parts: rea Moments of nertia of ombined ape d d... d (accordingl) d... rea Moments of nertia of onstituent Parts... rea Moment of nertia of a combined sape equals te sum of area moments of inertia of constituent sapes ll of tese are calculated about te same axis Priit Põdra 5. Properties of ections
12 nertia Moments of ombined ape () e e e e e e rea Moments of nertia of onstituent apes st = Rectangle: nd = Rectangle: rd = Rectangle: Priit Põdra 5. Properties of ections e e e e e e rea Moments Moments of nertia of ombined ape Wen te centroidal axes of consistuent sapes are parallel to te centroidal axes of combined sape i i i i i i bout combined sape centroidal axes e e i i i bout consistuent sape centroidal axes i nertia Moments of omplex ape omplex sape is dividend into stripes of equal widt Widt of stripes in direction perpendicular to te axis omplex ape i b i e i omplex ape = ombined sape consisting of man stripes omplex ape ombined ape (wit parts aving parallel centroidal axes) rea Moment of nertia of omplex ape n i bi ei bi or n i e i bi (wen << ) Wen te number of stripes is large and te widt of stripes is small Te moments of inertia of stripes can be neglected Priit Põdra 5. Properties of ections 4
13 nertia Moments about Rotated xes () rea moments of inertia are known about some predefined axes x F d rea moments of inertia need to be calculated about rotated axes D E Transformations OD O E cos sin DF EF ED cos sin ngle of xes Rotation Priit Põdra 5. Properties of ections 5 xial Moments of nertia d d cos sin d cos sin d cos sin cos sin d Product of nertia d nertia Moments about Rotated xes () ssuming te previous transformations Rotated xes xial Moments of nertia cos d sin d sin cos d cos d sin d sin cos d = = = xial Moments of nertia sin cos Priit Põdra 5. Properties of ections 6 cos sin Product of nertia sin cos sin sin
14 TRENGTH OF MTERL 5.5. Principal Moments of nertia Priit Põdra 5. Properties of ections 7 Principal entroidal xes of ape Principal xes = axes, about wic te product of inertia: entoidal xes = axes, about wic te statical moments: Principal entroidal xes = principal axes tat cross te centroid min max Non-entroidal xes Principal entroidal xes Priit Põdra 5. Properties of ections 8 entroidal xes xes intersect at te centroid Te number of centroidal axes = infinit Te number of principal axes = infinit Te number of principal centroidal axes = alwas at least one, sometimes more
15 Principal entroidal Moments of nertia Principal entroidal Moments of nertia = moments of inertia about principal centroidal axes n practice te are often called just Principal Moments of neria Because oter principal axes are not of interest in trengt of Materials Principal entroidal xes min max Te values of principal centroidal moments of inertia are alwas extremal max min xial Moments of nertia Or vice versa = max about suc principal axis, about wic in general te surface area elements distance is bigger Priit Põdra 5. Properties of ections 9 Principal entroidal xes of mmetrical apes For n mmetrical ape: axis of smmetr = principal centroidal axis oter principal centroidal axis is perpendicular to tat and goes troug te centroid (ma also be an axis of smmetr) quare ection ircular ection xis of mmetr Rigt ngle U-ection entroid Principal entroidal xes nfinite number of axes of smmetr Four axes of smmetr For sapes wit more tan two smmetr axes: all axes of smmetr are principal centroidal axes moments of inertia values about tese axes are all equal Priit Põdra 5. Properties of ections
16 Principal entroidal Moments of nertia of n ape. Location of entroid. Rotation ngle of Principal xes arctan Derived from te formula of inertia products about rotated axes. nertia Moments about Freel osen entroidal xes 4. Principal Moments of nertia Y Z cos sin sin cos sin sin Priit Põdra 5. Properties of ections TRENGTH OF MTERL Meanosüsteemide komponentide õppetool THNK YOU! Questions, please? Priit Põdra 5. Properties of ections
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