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[Mon Oct 04 11:07:13 2004]
Integriranje ćemo provesti duž horizontalne elementarne trake širine dz od točke do točke prema slici 6.3.
We can carry out the integration along the horizontal elementary line of width dz from point to point according to Figure 6.3.
[Mon Oct 04 11:07:13 2004]
Doprinos te trake ukupnoj poprečnoj sili iznosi
Contribution of this line in total section force amounts to
[Mon Oct 04 11:14:36 2004]
duž ruba presjeka je
along the cut edge there is
[Mon Oct 04 11:16:51 2004]
Kako je doprinos svake elementarne trake jednak nuli, to je ukupna poprečna sila
As the contribution of the every elementary line is equal to zero, that is the total section force
[Mon Oct 04 11:29:26 2004]
Budući da je doprinos svake trake širine
Since is the contribution of every line of width
[Mon Oct 04 11:31:25 2004]
bit će sveukupni doprinos prvog člana podintegralne funkcije izraza (6.21) jednak nuli
the total contribution of the first member of a subintegral function of the expression (6.21) will also be equal to zero
[Mon Oct 04 11:34:37 2004]
U tu svrhu razmotrimo doprinos vertikalne trake širine
For that purpose let us consider the contribution of vertical line of width
[Mon Oct 04 11:37:46 2004]
plohu razapetu iznad poprečnog presjeka
a surface stretched above the cross section
[Mon Oct 04 11:39:35 2004]
Ta ploha prolazi kroz rub poprečnog presjeka.
This surface goes through the edge of the cross section.
[Mon Oct 04 11:39:35 2004]
Ako dvostruki integral u
If double integral in
[Mon Oct 04 11:48:37 2004]
pri uvijanju neokruglog štapa što ga polarni moment tromosti
at twisting the uncircular rod which the polar moment of inertia
[Mon Oct 04 11:52:42 2004]
ima pri uvijanju okruglog štapa
has at twisting of round bar
[Mon Oct 04 11:56:11 2004]
odgovara polumjeru
matches the radius
[Mon Oct 04 12:00:28 2004]
omogućavaju nam da potpuno riješimo problem uvijanja štapa neokruglog presjeka pod uvjetom da smo odredili funkciju
enable us to fully solve the problem of the torsion of a rod with noncircular cross-section, providing that we have determined the function
[Mon Oct 04 12:03:47 2004]
Ta se funkcija određuje rješavanjem parcijalne diferencijalne jednadžbe (6.15) uz zadovoljenje rubnog uvjeta.
That function is determined by solving the partial differential equation (6.15) by satisfying the boundary condition.
[Mon Oct 04 12:09:16 2004]
Jednadžba (6.15) može se riješiti i numerički.
15) can be solved numerically, too.
[Mon Oct 04 12:09:41 2004]
Mi ćemo rješavanje problema uvijanja ilustrirati na relativno jednostavnom slučaju eliptičnog poprečnog presjeka.
We will illustrate the solving of the problem of twisting on a relatively simple case of elliptic cross-section.
[Mon Oct 04 12:09:41 2004]
Osim toga navest ćemo gotovo rješenje za pravokutni presjek.
Furthermore, we will quote the completed solution for the rectangular cut.
[Mon Oct 04 12:10:23 2004]
Najveću korist iz poznavanja parcijalne diferencijalne jednadžbe
The biggest benefit from knowing the partial differential equation
[Mon Oct 04 12:14:28 2004]
Uvijanje štapa eliptičnog poprečnog presjeka
Torsion of a rod with elliptic cross-section
[Mon Oct 04 12:16:11 2004]
prikazan je eliptični poprečni presjek štapa koji je opterećen na uvijanje.
shows the elliptic cross-section of a rod which has a load in torsion.
[Mon Oct 04 12:16:11 2004]
Jednadžba ruba presjeka glasi
Equation of cut edge is as follows
[Mon Oct 04 12:16:47 2004]
Pretpostavit ćemo torzijsku funkciju naprezanja u obliku
We will assume the torsional stress function in the form
[Mon Oct 04 12:17:20 2004]
Ovim je automatski ispunjen rubni uvjet
This automatically fulfills the boundary condition
[Mon Oct 04 12:19:41 2004]
Treba još pokazati da funkcija
What needs to be shown is that the function
[Mon Oct 04 12:21:30 2004]
Štap eliptičnog poprečnog presjeka
Rod with elliptical cross-section
[Mon Oct 04 12:26:34 2004]
Da to pokažemo, odredit ćemo potrebne derivacije.
To show this, we will determine necessary derivations.
[Mon Oct 04 12:26:34 2004]
Te derivacije su
Those derivatives are
[Mon Oct 04 12:30:49 2004]
Ako te derivacije uvrstimo u
If we add the derivatives into
[Mon Oct 04 12:34:49 2004]
Najveće posmično naprezanje javlja se u točkama
The biggest shear stress appears at points
[Mon Oct 04 12:36:28 2004]
Vidimo da se maksimalno posmično naprezanje javlja u onim točkama ruba koje su najbliže središtu poprečnog presjeka.
We see that the maximum shear stress appears in those edge points which are closest to the center of cross section.
[Mon Oct 04 12:36:28 2004]
To pravilo vrijedi općenito, a ne samo za eliptični presjek.
This rule applies generally, and not just for the elliptical cut.
[Mon Oct 04 12:36:59 2004]
Uvijanje štapa pravokutnog poprečnog presjeka
Torsion of a rod with rectangular cross section
[Mon Oct 04 12:41:39 2004]
Ne ulazeći u pojedinosti rješavanja problema navest ćemo samo konačne rezultate uvijanja štapa pravokutnog poprečnog presjeka.
Without getting into details of problem solving, we will list only the final results of torsion of a rod with rectangular cross section.
[Mon Oct 04 12:41:39 2004]
Raspored naprezanja po pravokutnom presjeku prikazan je na slici 6.5
The arrangement of stresses along the rectangular cut is shown on Figure 6.5
[Mon Oct 04 12:43:05 2004]
Raspodjela posmičnih naprezanja po pravokutnom poprečnom presjeku
Distribution of shear stresses along the rectangular cross section
[Mon Oct 04 12:45:54 2004]
Kako smo i očekivali posmično naprezanje u uglovima presjeka jednako je nuli.
As we have expected, the shear stress in cut angles is equal to zero.
[Mon Oct 04 12:45:54 2004]
Najveće posmično naprezanje
The biggest shear stress
[Mon Oct 04 12:46:28 2004]
javlja se na sredini dulje stranice.
appears at the centre of a longer side.
[Mon Oct 04 12:49:41 2004]
Na sredini kraće stranice djeluje naprezanje
At the center of the shorter side there is a stress
[Mon Oct 04 12:50:20 2004]
posmična naprezanja i za kut uvijanja glase
shear stresses and for the angle of twist are
[Mon Oct 04 12:51:23 2004]
može se dovoljno točno odrediti pomoću izraza
can be sufficiently correctly determined with the expression
[Mon Oct 04 12:55:00 2004]
L.
L.
[Mon Oct 04 12:59:00 2004]
Prandtl je 1903.
In 1903, L. Prandtl
[Mon Oct 04 12:59:30 2004]
uočio da je diferencijalna jednadžba koju zadovoljava St.
Prandtl noticed that differential equation which is fulfilled by St.
[Mon Oct 04 14:09:51 2004]
Venantova torzijska funkcija naprezanja (6.14) formalno potpuno analogna diferencijalnoj jednadžbi elastične membrane.
Venant's torsional stress function (6.14) is formally completely analogous to a differential equation of elastic membrane.
[Mon Oct 04 14:09:51 2004]
Ova analogija nam omogućava da na jednostavan i lak način sebi predstavimo izgled funkcije
This analogy enables us to present in a simple and easy way the look of the function
[Mon Oct 04 14:13:40 2004]
kao i samu raspodjelu posmičnih naprezanja.
as a distribution of shear stresses.
[Mon Oct 04 14:13:40 2004]
U tu svrhu izvest ćemo diferencijalnu jednadžbu elementa membrane koja je jako prednapregnuta vlačnim silama
For that purpose we will produce a differential equation of membrane element which is very prestressed by tensile forces
[Mon Oct 04 14:16:15 2004]
Membrana je zanemarivo male težine i u početku ravna.
Membrane has a very small weight and initially it is straight.
[Mon Oct 04 14:17:28 2004]
Membrana je napeta preko otvora koji ima oblik poprečnog presjeka štapa i duž ruba učvršćena.
Membrane is stressed across the opening which has a form of the cross section of the rod and is fastened across the edge.
[Mon Oct 04 14:17:28 2004]
Ispupčenje membrane postižemo malim pretlakom koji djeluje s jedne strane membrane
Protuberance of the membrane is attained with the small prepressure which takes effect on one side of the membrane
[Mon Oct 04 14:18:14 2004]
Kako su sile prednaprezanja
As the prestressing forces
[Mon Oct 04 14:20:18 2004]
Ravnoteža elementa membrane u pravcu osi x glasi
Balance of a membrane element in the direction of axis x is as follows
[Mon Oct 04 14:21:28 2004]
Sile na elementu elastične membrane
Forces on the elastic membrane element
[Mon Oct 04 14:23:33 2004]
Pri postavljanju ove jednadžbe uzeli smo u obzir da je membrana blago ispupčena i da su kutovi nagiba tangenata u smjeru osi
By setting up this equation, we have taken into consideration that the membrane is mildly protuberant and that angles tangent inclination are in the direction of axes
[Mon Oct 04 14:25:06 2004]
U tom slučaju je sinus kuta približno jednak tangens kuta, tj. vrijedi
In that case is the sine is approximately equal tangent, i.e. there applies
[Mon Oct 04 14:26:45 2004]
U tom slučaju je sinus kuta približno jednak tangensu gdje su
In that case is the sine is approximately equal to the tangent where
[Mon Oct 04 14:33:43 2004]
vidjet ćemo da su one formalno analogne i da zadovoljavaju analogne rubne uvjete.
we will see that are they are formally analogous and that they satisfy analogous edge properties.
[Mon Oct 04 14:33:43 2004]
Prema tome i njihova rješenja će biti analogna, tj. oblik membrane
Consequently, their solutions will be analogous, i.e. the membrane shape
[Mon Oct 04 14:34:18 2004]
Nagib tangente u pravcu osi
Tangential slope in the direction of the axis
[Mon Oct 04 14:35:01 2004]
dok je negativan nagib tangente u pravcu osi
while the negative tangential slope in the direction of the axis
[Mon Oct 04 14:35:58 2004]
To slijedi na temelju jednadžbi
This results based on the equations
[Mon Oct 04 14:36:58 2004]
Da bi analogija bila potpuna očito mora vrijediti
In order for the analogy to be complete, the following has to apply
[Mon Oct 04 14:40:07 2004]
obujam kojeg obuhvaća membrana
is the volume covered by the membrane
[Mon Oct 04 14:45:29 2004]
Na temelju membransko-torzijske analogije provodimo misaoni eksperiment koji nam omogućava određivanje traženih veličina:
Based on the membrane-torsion analogy we carry out the contemplative experiment which enables us to determine the required magnitudes:
[Mon Oct 04 14:45:29 2004]
naprezanja
stresses
[Mon Oct 04 14:46:26 2004]
te relativnog kuta uvijanja
and the angle of twist per unit length
[Mon Oct 04 14:55:05 2004]
Međutim, tu analogiju možemo iskoristiti za neposredno i stvarno određivanje traženih veličina pomoću stvarnog eksperimenta.
However, we can use this analogy for direct and actual determining of required magnitudes through the use of the real experiment.
[Mon Oct 04 15:00:32 2004]
U tu svrhu u nekoj ploči napravimo otvor koji svojim oblikom odgovara poprečnom presjeku štapa i jedan okrugli otvor.
For that purpose, we make an opening in a plate which by its shape corresponds the cross section of the rod, and we make another circular opening.
[Mon Oct 04 15:01:15 2004]
Zatim se preko otvora prevuče tanka membrane (npr.
Then we stretch a thin membrane across the opening (e.g.
[Mon Oct 04 15:01:15 2004]
opna od sapunice) koja se ispupči malim pretlakom
soap membrane) which protrudes with small prepressure
[Mon Oct 04 15:08:08 2004]
Pokus s elastičnom membranom za primjenu membransko-torzijske analogije
Experiment with the elastic membrane for the application of membrane-torsion analogy
[Mon Oct 04 15:12:44 2004]
u točki u kojoj želimo odrediti posmična naprezanja
at the point at which we want to determine shear stresses
[Mon Oct 04 15:20:28 2004]
višestruko povezani poprečni presjek
multiply connected cross section
[Mon Oct 04 15:23:09 2004]
Kod šupljih štapova poprečni presjeci su višestruko povezani i imaju više odvojenih rubova
With hollow rods the cross sections are multiply connected and have several separate edges
[Mon Oct 04 15:42:21 2004]
Na pločice su čvrsto privezani klizači koji se nalaze u vertikalnim vodilicama.
Skaters which are in vertical guides are firmly attached on the plates.
[Mon Oct 04 15:43:01 2004]
Tako osiguravamo horizontalni položaj pločica, tj. konstantnu visinu membrane duž unutarnjih rubova.
That way we ensure the horizontal position of the plate, i.e. the constant height of the membrane along the inner edges.
[Mon Oct 04 15:53:59 2004]
Pretpostavljamo da su pločice zanemarive težine.
We assume that the plates have a very small weight.
[Mon Oct 04 15:53:59 2004]
Ako eksperiment provodimo u realnosti, težinu pločica možemo poništiti koristeći se odgovarajućim utegom koji je s pločicama vezan pomoću niti koje su prebačene preko kolotura, kako je prikazano na slici 6.8.
If we carry out the experiment in reality, weight of the plates can be nullified by using an adequate pendulum which is connected to plates with threads that are thrown over a pulley, as shown in Figure 6.8.
[Mon Oct 04 15:54:29 2004]
Uvijanje tankostjenih zatvorenih profila
Twisting of thinwalled closed profiles
[Mon Oct 04 15:57:34 2004]
Analiza naprezanja i deformacija u tankostjenim profilima opterećenim na uvijanje mnogo je jednostavnija od analize punih neokruglih profila.
Stress and deformation analysis in thinwalled profiles loaded to the torsion is much simpler than the analysis of filled nonround profiles.
[Mon Oct 04 15:57:34 2004]
Zatvoreni tankostjeni profil koji je opterećen na uvijanje prikazan je na slici
Closed thinwalled profile which is loaded to the torsion is presented on Figure
[Mon Oct 04 16:02:45 2004]
Uvijanje štapa zatvorenog tankostjenog presjeka
Torsion of the rod with closed thinwalled cut
[Mon Oct 04 16:03:43 2004]
Nagib membrane je praktički konstantan i iznosi
Membrane inclination is practically constant and amounts to
[Mon Oct 04 16:07:57 2004]
Prema torzijsko-membranskoj analogiji naprezanje je proporcionalno nagibu membrane, tj. vrijedi
According to torsional-tympanic analogy the stress is proportional to membrane inclination, i.e. there applies
[Mon Oct 04 16:08:47 2004]
Ako uvedemo pojam toka posmičnih sila
If we introduce the concept of shear forces flow
[Mon Oct 04 16:09:51 2004]
kao produkt posmičnog naprezanja i širine stijenke t, bit će
as a product of shear stress and wall width t, there will be
[Mon Oct 04 16:12:20 2004]
Do ovog zaključka možemo doći direktno razmatranjem ravnoteže elementa stijenke sa slike 6.9.
We can arrive to this conclusion directly by considering the balance of wall element in Figure 6.9.
[Mon Oct 04 16:15:18 2004]
Element je ponovo prikazan u uvećanom mjerilu na slici 6.11.
Again, the element is presented in enlarged proportion on Figure 6.11.
[Mon Oct 04 16:15:47 2004]
Na temelju simetrije zaključujemo da posmična naprezanja
Based on the symmetry we conclude that shear stresses
[Mon Oct 04 16:18:28 2004]
Sile na elementu tankostjenog presjeka
Force on the element of thinwalled cut
[Mon Oct 04 16:21:30 2004]
površina koju zatvara središnja linija između unutarnjeg i vanjskog ruba poprečnog presjeka.
is the surface closed with the centre line between the internal and external edge of the cross section.
[Mon Oct 04 16:21:44 2004]
Izraz (6.45) možemo odrediti i pomoću analogije.
45) can also be determined with the analogy.
[Mon Oct 04 16:21:44 2004]
Prema (6.37) vrijedi
According to (6.37) there applies
[Mon Oct 04 16:34:27 2004]
znak integrala
symbol of integrals
[Mon Oct 04 16:36:19 2004]
nagib
incline
[Mon Oct 04 16:38:27 2004]
Vidimo da za tok posmičnih sila vrijedi jednadžba kontinuiteta kao i za tok tekućine.
We see that the equation of continuity is the same for the flow of shear forces as is for the flow of liquids.
[Mon Oct 04 16:38:27 2004]
Torzijski moment tromosti odgovara dvostrukom volumenu kojeg zatvara membrana, tj.
Torsion moment of inertia corresponds to double volume which is closed by the membrane, i.e.
[Mon Oct 04 16:43:21 2004]
Štap poprečnog presjeka prema slici 6.13.
Rod with cross section according to Figure 6.13.
[Mon Oct 04 16:43:21 2004]
opterećen je na uvijanje momentom
is loaded on the torsion with a torque
[Mon Oct 04 16:44:03 2004]
Odrediti raspored naprezanja i kut uvijanja
Determine the arrangement of stresses and the angle of torsion
[Mon Oct 04 16:45:34 2004]
Zadatak ćemo riješiti pomoću membransko-torzijske analogije.
We will solve the problem with the membrane-torsion analogy.
[Mon Oct 04 16:45:34 2004]
Analogna membrana prikazana je na slici 6.13.
The analogous membrane is presented on Figure 6.13.
[Mon Oct 04 16:46:55 2004]
iako su zbog jasnoće slike prikazani pretjerano uvećani
although because of clarity they are excessively enlarged
[Tue Oct 05 09:38:15 2004]
Pravokutni tankostjeni presjek s dva otvora i njemu analogna membrana
Rectangular thinwalled cut with two openings and an analogous membrane on it
[Tue Oct 05 09:44:35 2004]
Dvostruki volumen ispod membrane analogan je torzijskom momentu tromosti, tj.
Double volume below the membrane is analogous to the torsion constant, i.e.
[Tue Oct 05 09:45:07 2004]
Sada možemo odrediti relativni kut uvijanja
Now we can determine the angle of twist per unit length
[Tue Oct 05 09:59:08 2004]
Posmične su sile tada dane izrazom
Shear forces are then presented with the expression
[Tue Oct 05 10:01:26 2004]
Uvijanje štapa otvorenog tankostjenog presjeka
Rod torsion of the open thinwalled cut
[Tue Oct 05 10:06:09 2004]
Izrazi za kut uvijanja i naprezanja
Expressions for the angle of twist and angle of stress
[Tue Oct 05 10:07:46 2004]
Štap otvorenog tankostjenog presjeka prikazan je na slici 6.14.
Rod with the open thinwalled cut is presented on Figure 6.14.
[Tue Oct 05 10:32:54 2004]
Da bismo štap mogli smatrati štapom tankostjenog presjeka njegove dimenzije moraju imati sljedeća ograničenja
In order to consider the rod the one with the thinwalled cut its dimension must have the following limitations
[Tue Oct 05 10:37:17 2004]
Nekoliko oblika otvorenog tankostjenog presjeka prikazano je na slici
Several shapes of the open thinwalled cut are presented on the picture
[Tue Oct 05 10:58:04 2004]
Središnjicom presjeka nazivamo crtu koja dijeli debljinu stijenke po sredini.
Center of the section is the line which divides the width of the wall along the middle.
[Tue Oct 05 11:03:11 2004]
Pretpostavit ćemo za početak da svaki dio ima konstantnu debljinu
Let us assume for the start that any part has a constant thickness
[Tue Oct 05 11:06:12 2004]
Prije nego priđemo analizi štapova tankostjenog presjeka razmatrat ćemo uvijanje štapova koji imaju pravokutni presjek velikog omjera
Before we approach the analysis of rods with thinwalled section we will consider the torsion of rods which have a rectangular large-scale section
[Tue Oct 05 11:09:35 2004]
Otvoreni poprečni presjeci tankostjenog štapa
Open cross sections of thinwalled rod
[Tue Oct 05 11:13:18 2004]
Zamislimo sada prema membransko torzijskoj analogiji da smo preko presjeka razapeli membranu.
Let us now imagine, according to the membrane torsional analogy, that we have stretched a membrane across the section.
[Tue Oct 05 11:13:18 2004]
Tada je krutost štapa proporcionalna obujmu
Then the rigidity of the rod is proportional to the volume
[Tue Oct 05 11:15:05 2004]
a naprezanje najvećem nagibu membrane
and the stress of the largest inclination of the membrane
[Tue Oct 05 11:18:59 2004]
Ako presjek prema slici 6.16a zakrivimo prema slici 6.16b, oblik membrane samo će se neznatno promijeniti.
If we twist the section on Figure 6.16a according to Figure 6.16b, the shape of the membrane will only slightly change.
[Tue Oct 05 11:19:10 2004]
Možemo smatrati da su obujam ispod membrane i njen najveći nagib ostali približno isti.
We can consider the volume below the membrane and its largest inclination approximately the same.
[Tue Oct 05 11:19:10 2004]
Prema tome možemo pisati
Therefore, we can write
[Tue Oct 05 11:19:43 2004]
Posmična naprezanja u tankom pravokutnom poprečnom presjeku
Shear stresses in thin rectangular cross section
[Tue Oct 05 11:28:10 2004]
Analizu uvijanja samoga štapa provest ćemo uz pretpostavku da se njegovi poprečni presjeci zakreću kao krute figure, tj. svaki dio poprečnog presjeka zakreće se za isti kut, pa možemo pisati
Analysis of torsion of alone rod have a good time beside the assumption that his cross sections turn as stiff figures, i.e. the any part of cross section turns for the same angle, so can write
[Tue Oct 05 11:30:44 2004]
Analizu uvijanja samoga štapa provest ćemo uz pretpostavku da se njegovi poprečni presjeci zakreću kao krute figure, tj. svaki dio poprečnog presjeka zakreće se za isti kut, pa možemo pisati
Analysis of rod torsion will be carried out with the assumption that its cross sections turn as stiff figures, i.e. each part of cross section is turned at the same angle, so we can write
[Tue Oct 05 11:31:53 2004]
je n broj posebnih dijelova presjeka.
where n is the number of special parts of sections.
[Tue Oct 05 11:31:53 2004]
Svaki dio presjeka nosi jedan dio momenta uvijanja
Each part of the section carries one part of the twisting moment
[Tue Oct 05 11:35:21 2004]
Torzijski moment tromosti
Torsion constant
[Tue Oct 05 11:37:24 2004]
Najveće posmično naprezanje u i-tom dijelu presjeka prema (6.50) iznosi
The biggest shear stress in i part of the section according to (6.50) is:
[Tue Oct 05 11:38:52 2004]
torzijski moment otpora definiran izrazom
the torsional moment of resistance defined with the expression
[Tue Oct 05 11:58:07 2004]
Usporedimo li dva poprečna presjeka jednake površine i oblika od kojih je jedan zatvoren, a drugi rasječen, vidjet ćemo da su nosivost i krutost zatvorenog profila mnogo veće od nosivosti i krutosti otvorenog profila.
If we compare two cross sections of identical surface and shape of which one is closed, and other is cut, we will see that the bearing capacity and rigidity of a closed profile is much greater than the bearing capacity and rigidity of an open profile.
[Tue Oct 05 11:58:27 2004]
Na slici 6.18.
Figure 6.18.
[Tue Oct 05 11:58:41 2004]
prikazana su dva ovakva profila s odgovarajućim membranama.
presents two such profiles with corresponding membranes.
[Tue Oct 05 11:58:41 2004]
Ako je maksimalni nagib membrane
If the maximum membrane inclination is
[Tue Oct 05 12:04:32 2004]
Tanki trapezasti poprečni presjek
Thin trapezoidal cross section
[Tue Oct 05 12:07:56 2004]
Ponekad dio tankostjenog otvorenog presjeka nema jednoliku debljinu , tj. ima oblik tankog trapeza.
Sometimes a part of the thinwalled open section does not have an even thickness, i.e. it has a form of a thin trapezoid.
[Tue Oct 05 12:07:56 2004]
Prema slici 6.17 može se primijeniti za taj dio izraz
According to Figure 6.17 we can use the following for this expression
[Tue Oct 05 12:08:54 2004]
Usporediti maksimalno naprezanje
Compare a maximal stress
[Tue Oct 05 12:12:58 2004]
Zatvoreni tankostjeni presjek ima puno veći
Closed thinwalled section has a much larger
[Tue Oct 05 12:15:19 2004]
Integral u nazivniku gornjeg izraza iznosi
Integral in denominator of the above expression is
[Tue Oct 05 12:17:29 2004]
Maksimalno posmično naprezanje izračunat ćemo pomoću izraza
The maximum shear stress can be calculated using the following expression
[Tue Oct 05 12:22:46 2004]
Omjer kuta uvijanja otvorenog i zatvorenog profila iznosi
Ratio of the angle of twist of the open and the closed profile is
[Tue Oct 05 12:25:16 2004]
Također je omjer posmičnih naprezanja dan izrazom
Also, the proportion of shear stresses is given in the expression
[Tue Oct 05 12:28:50 2004]
Vidimo da je kut uvijanja 112 puta veći, a maksimalno posmično naprezanje 18 puta veće kod otvorenog, nego li kod zatvorenog profila uz isti moment uvijanja
We see that the angle of twist is 112 times larger, and that the maximum shear stress is 18 times greater for the open, than for the closed profile with the same twisting moment
[Tue Oct 05 12:30:27 2004]
To znači da je krutost zatvorenog profila 112 puta veća, dok je nosivost (uz zanemarivanje koncentracije naprezanja) 18 puta veća.
This means that the rigidity of the closed profile is 112 times larger, and that the bearing capacity is (with neglecting the stress concentrations) 18 times larger.
[Tue Oct 05 12:32:40 2004]
Štap presjeka prema slici 6.19 opterećen je momentom uvijanja
Section rod, according to Figure 6.19, is loaded with the twisting moment
[Tue Oct 05 12:40:29 2004]
Kako će se promijeniti tok naprezanja, ako se uklone dva unutrašnja rebra?
How will the stress flow change, if we remove two internal ribs?
[Tue Oct 05 12:41:30 2004]
prikazana je analogna membrana
shows the analogous membrane
[Tue Oct 05 12:43:38 2004]
Tankostjeni presjek s tri otvora i njegova analogna membrana
Thinwalled section with three openings and its analogous membrane
[Tue Oct 05 12:46:14 2004]
Uzmemo li u obzir da je po membransko-torzijskoj analogiji
If we take into consideration that according to membrane-torsion analogy
[Tue Oct 05 12:48:03 2004]
Torzijski moment tromosti jednak je po analogiji dvostrukom obujmu ispod membrane, tj
Torsion constant is equal is according to the analogy to double volume below the membrane, i.e.
[Tue Oct 05 12:49:24 2004]
Relativni kut uvijanja iznosi
Angle of twist per unit length is
[Tue Oct 05 12:50:43 2004]
Da smo umjesto približnih vrijednosti za površine
If instead of approximate values for surfaces
[Tue Oct 05 12:55:59 2004]
Ako uklonimo unutrašnja rebra, imat ćemo po čitavom vanjskom rubu presjeka jedinstven tok posmičnih naprezanja
If we remove internal ribs, then we will get a unique flow of shear stresses along the entire external boundary of the section
[Tue Oct 05 14:17:26 2004]
Na unutrašnjim uglovima otvorenih profila pojavljuje se koncentracija tangencijalnog naprezanja.
On internal angles of open profiles appears the concentration of tangential stresses.
[Tue Oct 05 14:17:26 2004]
Izgled membrane u ovakvom uglu prikazan je na slici 6.20a, a raspored naprezanja na slici 6.20b
Look of the membrane in the such corner is presented on Figure 6.20a, and the arrangement of stress is presented on Figure 6.20b
[Tue Oct 05 14:18:24 2004]
Veličina maksimalnog naprezanja na mjestu koncentracije može se približno izračunati pomoću izraza
The size of maximal stresses on the place of concentration can approximately be calculated with the expression
[Tue Oct 05 14:20:48 2004]
Koncentracija posmičnih naprezanja u uglu otvorenog tankostjenog presjeka
Concentration of shear stresses at the corner of open thinwalled section
[Tue Oct 05 14:22:57 2004]
Ako se u uglu sastaju dva dijela različite debljine u njima će djelovati različita posmična naprezanja.
If in the corner meet two parts of different thickness within themselves will function different shear stresses.
[Tue Oct 05 14:25:15 2004]
Ako se u uglu sastaju dva dijela različite debljine u njima će djelovati različita posmična naprezanja.
If two parts of different thickness meet at the angle, different shear stresses will appear in them.
[Tue Oct 05 14:25:15 2004]
U izraz (6.64) treba uvrstiti ono koje je veće
The larger one has to be put into the expression (6.64)
[Tue Oct 05 14:32:07 2004]
SAVIJANJE TANKIH KRUŽNIH PLOČA
BENDING OF THIN CYCLE PLATES
[Tue Oct 05 14:33:01 2004]
osnovni pojmovi teorije ploha
basic terms ijn the surface theory
[Tue Oct 05 14:42:18 2004]
Ploča je plosnato tijelo koje je ograničeno dvjema plohama čija je međusobna udaljenost, tj. debljina ploče mala u usporedbi s ostalim dimenzijama.
Plate is a flat body which is limited with two surfaces whose mutual distance, i.e. the plate thickness, is small in comparison with other dimensions.
[Tue Oct 05 14:42:18 2004]
Srednja ploha jest ploha koja je jednako udaljena od vanjskih površi
Middle surface is that surface which is equally distant from the external surfaces
[Tue Oct 05 14:46:11 2004]
Po tome se ploče razlikuju od ljuski kod kojih je srednja ploha zakrivljena
That is what makes the plates different from shells, whose middle surface is twisted
[Tue Oct 05 15:03:18 2004]
Presiječemo li plohu
If we cut through the surface
[Tue Oct 05 15:14:17 2004]
Pomaci i kutovi zakreta tangente pravokutnog elementa ploče
Displacements and angles of tangent twist of a rectangular plate element
[Tue Oct 05 15:14:58 2004]
Budući da su po pretpostavci kutovi
Since it is assumed that the angles
[Tue Oct 05 15:20:12 2004]
Uvijenost ili torzija elementa
Involution or torsion of element
[Tue Oct 05 15:21:52 2004]
Recipročne vrijednosti zakrivljenosti nazivaju se polumjeri zakrivljenosti i označavaju sa
Reciprocal values of curvature are called radiuses of curvature and are marked with
[Tue Oct 05 15:31:13 2004]
Veza između komponenata vektora u novom i starom koordinatnom sustavu dana je izrazom (1.17).
Relation between vector components in new and old coordinate system is given in the expression (1.17).
[Tue Oct 05 15:31:13 2004]
To vrijedi i za komponente vektora položaja, tj. za koordinate
This also applies to vector components of position, i.e. for coordinates
[Tue Oct 05 15:34:28 2004]
Primjenom lančanog pravila o deriviranju funkcija više varijabli možemo dobiti
By applying the chain rule on derivation of functions with more variables, we can get
[Tue Oct 05 15:36:50 2004]
Primijenimo li operatore
If we decide to use the operators
[Tue Oct 05 15:49:36 2004]
komponente tenzora zakrivljenosti
components of curvature tensor
[Tue Oct 05 16:11:17 2004]
Srednjom zakrivljenošću
Mean curvature
[Tue Oct 05 16:12:35 2004]
Gaussova zakrivljenost
Gaussian curvature
[Tue Oct 05 16:19:14 2004]
Fizikalno značenje izraza za transformaciju komponenata tenzora zakrivljenosti objasnit ćemo pomoću slike 8.5.
The physical meaning of expression for component transformation of curvature tensors will be explained on Figure 8.5.
[Tue Oct 05 16:27:15 2004]
Fizikalno značenje izraza za transformaciju komponenata tenzora zakrivljenosti objasnit ćemo pomoću slike 8.5. Slika 8.5a prikazuje kvadratnu ploču s ucrtanom pravokutnom mrežom.
Physical meaning of expressions for component transformation of curvature tensors explain will be explained by Figure 8.5. Figure 8.5a represents a square plate with drawn rectangular net.
[Tue Oct 05 16:27:15 2004]
Slika 8.5b prikazuje istu ploču koja je deformirana djelovanjem četiriju jednakih sila koje djeluju u uglovima ploče
Figure 8.5b represents the same plate which was deformed by the impact of four identical forces which exist in angles of the plate
[Tue Oct 05 16:29:22 2004]
Mohrova kružnica zakrivljenosti
Mohr's circle of curvature
[Tue Oct 05 16:31:00 2004]
Ako je Gaussova zakrivljenost u svim točkama plohe jednaka nuli, ploha je razmotljiva
If Gaussian curvature is equal to zero in all points of plane, than the plane is unfolding
[Tue Oct 05 16:35:38 2004]
Pravokutni element ploče opterećen na uvijanje
The rectangular plate element loaded to torsion
[Tue Oct 05 16:52:18 2004]
predstavlja pogled od brida
represents the view from the rim
[Tue Oct 05 16:54:27 2004]
Taj pogled je prema prethodnom zakrenut za 45
This view is turned towards the previous one for 45
[Wed Oct 06 10:31:21 2004]
Analiza pomaka, deformacija i naprezanja
Analysis of displacements, deformations and stresses
[Wed Oct 06 11:01:33 2004]
Kružna ploča kao element konstrukcije
Cycle plate as a construction element
[Wed Oct 06 11:04:31 2004]
cilindričnim koordinatnim sustavom
cylinder coordinate system
[Wed Oct 06 11:05:17 2004]
Cilindrični koordinatni sustav na kružnoj ploči
Cylinder coordinate system on a cycle plate
[Wed Oct 06 11:10:00 2004]
Ograničenja oblika, pomaka i opterećenja
Limitations of forms, displacements and loads
[Wed Oct 06 11:19:13 2004]
Formule koje ćemo izvesti u ovom poglavlju vrijedit će uz sljedeća ograničenja:
Formulas which derived in this chapter will apply the following limitations:
[Wed Oct 06 11:22:52 2004]
Najveća debljina ploče je određena izrazom
The biggest plate thickness is determined with the expression
[Wed Oct 06 11:23:54 2004]
Progibi ploče su mali
Plate deflections plates are small
[Wed Oct 06 11:24:54 2004]
osnosimetrična, osnosimetrično opterećena i učvršćena
axisymmetric, axisymmetrically loaded and fastened
[Wed Oct 06 11:29:55 2004]
Posljedica toga je da sve geometrijske i statičke veličine ovise o koordinati r, poneke ovise o koordinati z, a nijedna ne ovisi o koordinati
Consequence of this is that all geometric and static sizes depend on the coordinate r, some depend on the coordinate Z, and none depends on the coordinate
[Wed Oct 06 11:31:33 2004]
Geometrijske veličine su pomaci i deformacije, a statičke veličine su naprezanja i komponente unutarnjih sila
Geometric sizes are displacements and deformations, and static sizes are stresses and stress resultants components
[Wed Oct 06 11:34:05 2004]
Pretpostavke o deformiranju i raspodjeli naprezanja
Assumptions about deformation and stress distribution
[Wed Oct 06 11:35:47 2004]
Analizu naprezanja i deformacija provest ćemo uz ove pretpostavke
Stress and deformation analiysis will be carried out with these assumptions
[Wed Oct 06 11:46:43 2004]
Pri deformiranju normale na srednju ravninu ostaju ravne i okomite na elastičnu plohu.
In deforming, the verticals on the middle plane remain straight and vertical to the elastic plane.
[Wed Oct 06 11:46:43 2004]
Elastična ploha jest deformirana srednja ravnina
Elastic plane is a deformed middle plane
[Wed Oct 06 11:48:15 2004]
Elastična ploha ujedno je i neutralna ploha, tj. u njoj su naprezanja
Elastic plane is also a neutral plane, i.e. it contains stresses
[Wed Oct 06 11:52:13 2004]
U ploči vlada približno ravninsko naprezanje.
In the plate there is an approximate plane stress.
[Wed Oct 06 11:52:13 2004]
Naprezanje
Stress
[Wed Oct 06 11:55:16 2004]
zanemarivo je malo u usporedbi s naprezanjima
is negligibly small compared to the stresses
[Wed Oct 06 14:19:54 2004]
Prva je pretpostavka ekvivalentna pretpostavci o ravnim presjecima koju smo uveli u nauci o čvrstoći kad smo razmatrali savijanje štapova.
The first assumption is equivalent to the assumption about straight sections which we have introduced in the strength science when were considering bending of rods.
[Wed Oct 06 14:19:54 2004]
Ona je u cijelosti ispunjena pri čistom savijanju.
It is totally filled during pure bending.
[Wed Oct 06 14:19:54 2004]
Ako osim momenta savijanja djeluje i poprečna sila, ta je pretpostavka samo približno ispunjena, i to tim bolje čim je debljina ploče manja.
If, besides the bending moment there is also a section force, that assumption is only partially fulfilled, and it is better better if the plate thickness is smaller.
[Wed Oct 06 14:19:54 2004]
Kako mi razmatramo samo tanke ploče, odstupanje od te pretpostavke neznatno utječe na rezultate.
Since we are examining only thin plates, departure from this assumption influences the results only slightly.
[Wed Oct 06 14:19:54 2004]
Druga pretpostavka ne može biti u cijelosti ispunjena jer nije moguće ravninu pretvoriti u dvostruko zakrivljenu plohu bez pojave naprezanja.
The second assumption cannot be totally has fulfilled because not it is not possible to convert the plane into double twisted plane without the appearance of stress.
[Wed Oct 06 14:19:54 2004]
Međutim, ako je ploča tanka i ako su progibi mali, odstupanja od te pretpostavke su zanemariva.
However, if the plate is thin and if deflections are small, departures from this assumption are negligible.
[Wed Oct 06 14:19:54 2004]
Treća pretpostavka također nije u potpunosti ispunjena.
The third assumption is also not completely fulfilled.
[Wed Oct 06 14:19:54 2004]
Naime, ploče služe najčešće kao poklopci na koje djeluje tlak
That is, the plates most often serve as lids affected by pressure
[Wed Oct 06 14:25:45 2004]
U tankostjenim posudama tlak rijetko prelazi 1 ili 2 MPa, a često je i manji.
In thinwalled containers pressure rarely crosses 1 or 2 MPa, and it is often smaller.
[Wed Oct 06 14:25:45 2004]
S druge strane, konstrukcije se dimenzioniraju tako da su maksimalna naprezanja blizu dopuštenih.
On the other hand, constructions are dimensioned so that the maximum stresses are near the allowable ones.
[Wed Oct 06 14:25:45 2004]
Dopuštena naprezanja su u čeličnim posudama oko 150 MPa ili veća, ovisno o materijalu i karakteru opterećenja.
Allowable stresses in steel containers are around 150 MPa or more, depending on the material and character of the load.
[Wed Oct 06 14:25:45 2004]
Na temelju toga možemo zaključiti da naprezanje
Based on this, we can conclude that the stress
[Wed Oct 06 14:30:24 2004]
pa njegovo zanemarivanje vrlo malo utječe na točnost rezultata
so neglecting it has a very small influence on the accuracy of results
[Wed Oct 06 14:36:08 2004]
Srednja ploha ploče pod opterećenjem prelazi u slabo zakrivljenu osnosimetričnu elastičnu plohu, zapravo rotacijski paraboloid zadan izrazom
Middle surface of the plate under the load becomes slightly twisted axisymmetric elastic plane, in fact, a rotational paraboloid specified with the expression
[Wed Oct 06 14:41:10 2004]
U toku deformiranja normale na srednju ravninu ostaju ravne i okomite na elastičnu plohu
During deformation, verticals on the middle plane remain straight and vertical to the elastic plane
[Wed Oct 06 14:45:28 2004]
Pozitivni smisao progiba
The positive sense of deflection
[Wed Oct 06 14:52:58 2004]
ortogonalni element duljine
orthogonal element in length
[Wed Oct 06 14:55:29 2004]
aksonometrijski
axonometrically
[Wed Oct 06 14:56:17 2004]
Razmotrimo deformaciju diferencijalnog radijalnog elementa
Let us consider the deformation of differential radial element
[Wed Oct 06 14:57:34 2004]
cirkularnog elementa
circular element
[Wed Oct 06 14:58:35 2004]
Ti se elementi nalaze na udaljenosti
Those elements are at a distance
[Wed Oct 06 15:01:00 2004]
Točke A i D spuštaju se paralelno osi z za iznos w, a točka B za iznos
Points A and D descend parallelly to axis z for the amount of w, and point B for the amount of
[Wed Oct 06 15:04:27 2004]
ali se one zbog zakreta normale još pomiču radijalno za iznos
but because of the turn of vertical they move also radially for the amount
[Wed Oct 06 15:08:35 2004]
Geometrijska analiza deformiranja elementa kružne ploče
Geometric analysis of deformation of cycle plate element
[Wed Oct 06 15:12:05 2004]
Primjena Hookeova zakona
Application of Hooke's law
[Wed Oct 06 15:14:51 2004]
Hookeov zakon za ravninsko stanje naprezanja prema
Hooke's law for plain stress according to
[Wed Oct 06 15:23:43 2004]
pojavit će se i posmično naprezanje
there will also appear shear stress
[Wed Oct 06 15:24:21 2004]
u presjecima okomitim na polumjer
in sections perpendicular to the radius
[Wed Oct 06 15:34:05 2004]
Posmična naprezanja su raspoređena po zakonu parabole, slično kao i kod savijanja ravnih štapova.
Shear stresses are arranged according to the law of parabola, similar as in bendings of straight rods.
[Wed Oct 06 15:34:05 2004]
Raspored normalnih i posmičnih naprezanja prikazan je na slici 8.12. Pri proračunu čvrstoće ploče možemo zanemariti posmična naprezanja jer su redovno manja od normalnih naprezanja i jednaka nuli na mjestu gdje su normalna naprezanja najveća.
Arrangement of normal and shear stresses is shown on Figure 8.12. In calculation of plate strength we can neglect shear stresses because they are regularly smaller than normal stresses and equal to zero on the place where normal stresses are the largest.
[Wed Oct 06 15:34:05 2004]
Posmične komponente naprezanja
Shear stress components
[Wed Oct 06 15:34:50 2004]
jer su radijalni presjeci simetrije
because radial cross-sections are symmetrical
[Wed Oct 06 15:35:40 2004]
jedino preostaje posmično naprezanje
what remains is the shear stress
[Wed Oct 06 15:38:28 2004]
Raspodjela komponenata naprezanja
Distributions of stress components
[Wed Oct 06 15:39:04 2004]
na elementu kružne ploče
on cycle plate element
[Wed Oct 06 15:40:26 2004]
Primjena uvjeta ravnoteže, momenti savijanja i poprečna sila
Application of equilibrium conditions, bending moments and section force
[Wed Oct 06 15:43:51 2004]
Uvjete ravnoteže nećemo postaviti pomoću naprezanja, nego pomoću komponenata unutarnjih sila.
We will not set the equilibrium conditions by means of stress, but by means of components of stress resultants.
[Wed Oct 06 15:43:51 2004]
Naprezanje Uvjete ravnoteže nećemo postaviti pomoću naprezanja, nego pomoću komponenata unutarnjih sila.
Stress
[Wed Oct 06 15:47:34 2004]
U teoriji ploča i ljuski komponente unutarnjih sila uvijek se odnose na jedinicu duljine.
In the theory of planes and shells, stress resultants components always refer to a length unit.
[Wed Oct 06 15:47:34 2004]
Prema tome imamo
Accordingly we have
[Wed Oct 06 15:52:57 2004]
Veza komponenata naprezanja i unutarnjih sila
Link between the components of stress and stress resultants
[Wed Oct 06 15:54:49 2004]
predstavlja moment tromosti jedinične širine poprečnog presjeka.
presents a moment of inertia in the unit width of cross section.
[Wed Oct 06 15:54:49 2004]
Uvedemo li oznaku
If we introduce the mark
[Wed Oct 06 15:56:34 2004]
za fleksijsku krutost ploče, izrazi (8.28) prelaze u
for flectional rigidity of the plate, expression (8.28) become
[Wed Oct 06 15:57:54 2004]
unutarnjih sila i vanjskim opterećenjem
stress resultants and exterior load
[Wed Oct 06 15:59:02 2004]
Uvjeti ravnoteže tog elementa glase
Equilibrium conditions of this element are
[Wed Oct 06 16:07:56 2004]
Opterećenje i unutarnje sile na elementu ploče
Load and stress resultants on the plate element
[Wed Oct 06 16:08:05 2004]
Opterećenje i unutarnje sile na elementu ploče
Load and stress resultants on the plate element
[Wed Oct 06 16:12:42 2004]
Opterećenje i unutarnje sile na elementu ploče
Load and stress resultants on the plate element
[Wed Oct 06 16:14:05 2004]
Diferencijalna jednadžba savijanja kružne ploče
Differential equation of cycle plate bending
[Wed Oct 06 16:21:40 2004]
Taj se sustav jednadžbi dade lako svesti na jednu diferencijalnu jednadžbu s jednom nepoznanicom.
This equation system can easily be reduced to one differential equation with one unknown.
[Wed Oct 06 16:21:40 2004]
Da to postignemo uvrstit ćemo (8.30) u drugu jednadžbu (8.32), pa ćemo dobiti
To achieve this, we will put (8.30) into the other equation (8.32), so will get
[Wed Oct 06 16:29:55 2004]
možemo odrediti poznavajući opterećenje ploče pomoću izraza (8.31) ili još jednostavnije razmatranjem ravnoteže elementa ploče polumjera
can be determined by knowing the plate load with the expression (8.31) or even more simply, with consideration of balance of plate element of radius
[Wed Oct 06 16:31:31 2004]
Opće rješenje diferencijalne jednadžbe (8.33a) glasi
General solution of differential equation (8.33a) is
[Wed Oct 06 16:32:00 2004]
Rubni uvjeti
Boundary conditions
[Wed Oct 06 16:34:31 2004]
Konstante integracije određujemo na temelju rubnih uvjeta, tj. načina učvršćenja ploče.
Integration constants are determined based on boundary conditions, i.e. method of fastening the plate.
[Wed Oct 06 16:34:31 2004]
Ako je ploča puna, onda je u sredini ploče
If the plate is full, then in the middle of the plate
[Wed Oct 06 16:35:50 2004]
određujemo na temelju uvjeta na vanjskom rubu
is determined based on conditions on the outer border
[Wed Oct 06 16:37:51 2004]
Kod prstenastih ploča imamo dva ruba, pa na temelju dva rubna uvjeta možemo odrediti dvije konstante integracije
Ringlike plates have two edges, so based on two marginal conditions we can determine two integration constants
[Wed Oct 06 16:39:28 2004]
Rubovi ploče mogu biti ukliješteni, zglobno oslonjeni ili slobodni.
Plate edges can be wedged in, hingedly supported or free.
[Wed Oct 06 16:39:28 2004]
Posljednja dva mogu biti opterećena rubnim momentom
The last two can be loaded with marginal moment
[Wed Oct 06 16:41:52 2004]
Ako je rub učvršćen i opterećen, prema slici 8.15c i d, rubni uvjet je
If the edge is fastened and loaded, according to Figure 8.15c and d, boundary condition is
[Wed Oct 06 16:42:33 2004]
Za rub ukliješten prema slici 8.15e, rubni uvjet glasi
For the edge wedged in according to Figure 8.15e, boundary condition isutters a sound
[Wed Oct 06 16:43:55 2004]
Kad pomoću rubnih uvjeta odredimo konstante integracije, možemo dobiti konačni izraz
When, by using boundary conditions, we determine the integration constants, we can acquire the final expression
[Wed Oct 06 16:46:01 2004]
Rubni uvjeti za kružnu ploču
Boundary conditions for cycle plate
[Wed Oct 06 17:51:47 2004]
Kako si?
How are you?
[Wed Oct 06 17:51:53 2004]
Kako si?
How are you?
[Wed Oct 06 17:52:00 2004]
Kako si?
How are you?
[Wed Oct 06 17:52:15 2004]
Kako si?
How are you?
[Wed Oct 06 17:52:15 2004]
Kako si?
How are you?
[Wed Oct 06 17:52:15 2004]
Kako si?
How are you?
[Wed Oct 06 17:52:15 2004]
Kako si?
How are you?
[Wed Oct 06 17:52:24 2004]
Kako si?
How are you?
[Wed Oct 06 17:52:24 2004]
Kako si?
How are you?
[Wed Oct 06 17:52:24 2004]
Kako si?
How are you?
[Wed Oct 06 17:52:24 2004]
Kako si?
How are you?
[Wed Oct 06 17:56:46 2004]
Kako si?
How are you?
[Wed Oct 06 17:56:46 2004]
Kako si?
How are you?
[Wed Oct 06 17:56:46 2004]
Kako si?
How are you?
[Wed Oct 06 17:56:46 2004]
Kako si?
How are you?
[Wed Oct 06 18:08:06 2004]
danas je lijep dan.
today is the beautiful day.
[Wed Oct 06 18:09:10 2004]
danas je lijep dani.
today is beautiful days.
[Wed Oct 06 18:09:28 2004]
danas je lijep dani.
today is beautiful days.
[Wed Oct 06 18:09:28 2004]
danas je lijep dani.
today is beautiful days.
[Wed Oct 06 18:09:28 2004]
danas je lijep dani.
today is beautiful days.
[Wed Oct 06 18:09:28 2004]
danas je lijep dani.
today is beautiful days.
[Wed Oct 06 18:09:28 2004]
danas je lijep dani.
today is beautiful days.
[Wed Oct 06 18:09:28 2004]
danas je lijep dani.
today is beautiful days.
[Wed Oct 06 18:09:28 2004]
danas je lijep dani.
today is beautiful days.
[Wed Oct 06 18:09:28 2004]
danas je lijep dani.
today is beautiful days.
[Thu Oct 07 09:20:22 2004]
Da bismo odredili konačne izraze za momente savijanja
In order to determine the final expressions for bending moments
[Thu Oct 07 09:23:41 2004]
Za određivanje progiba
In order to determine the deflection
[Thu Oct 07 09:27:39 2004]
možemo također odrediti iz rubnih uvjeta.
can also be determined from boundary conditions.
[Thu Oct 07 09:27:39 2004]
Raspored naprezanja
Arrangement of stresses
[Thu Oct 07 09:30:07 2004]
Maksimalna normalna naprezanja javljaju se na gornjoj, odnosno na donjoj površini, tj. za
Maximum normal stresses appear on upper, or on the lower plane, i.e. for
[Thu Oct 07 09:31:03 2004]
Kružna ploča polumjera
Cycle plate of a radius
[Thu Oct 07 09:32:42 2004]
oslonjena je zglobno duž ruba i opterećena momentom
is supported along articularly down the edge and loaded moment
[Thu Oct 07 09:38:39 2004]
oslonjena je zglobno duž ruba i opterećena momentom
is supported along hingledly along the edge and loaded with the moment
[Thu Oct 07 09:43:55 2004]
maksimalno cirkularno i radijalno naprezanje
maximum circular and radial stress
[Thu Oct 07 09:45:47 2004]
Kružna ploča opterećena po rubu momentom
Cycle plate loaded along the edge with moment
[Thu Oct 07 09:47:05 2004]
Budući da na ploču ne djeluju koncentrirane sile niti kontinuirano opterećenje, bit će poprečna sila
Since the plate is not affected with concentrated forces nor with continuous load, the shear force will be
[Thu Oct 07 09:49:48 2004]
jednadžba savijanja prema
equations of bending towards
[Thu Oct 07 10:02:37 2004]
Uzmemo li u obzir da je
If we take into consideration that
[Thu Oct 07 10:06:08 2004]
Momenti savijanja prema
Bending moments according to
[Thu Oct 07 10:07:57 2004]
Cirkularni i radijalni momenti savijanja konstantni su po cijeloj ploči.
Circular and radial moments of bending are constant along the entire plate.
[Thu Oct 07 10:07:57 2004]
Maksimalna naprezanja prema (8.43) iznose
Maximum stresses according to (8.43) are
[Thu Oct 07 10:15:41 2004]
pa izraz za progib
so the expression for deflection
[Thu Oct 07 10:16:34 2004]
Maksimalni progib nastaje u sredini ploče, tj.
The maximum deflection appears in the center of the plate, i.e.
[Thu Oct 07 10:17:35 2004]
Maksimalni nagib ploče javlja se na rubu tj
The maximum inclination of the plate appears at the edge i.e.
[Thu Oct 07 10:20:23 2004]
Kružna ploča polumjera R oslonjena je zglobno duž ruba i opterećena jednoliko kontinuirano opterećenjem
Cycle plate with radius R is supported hingedly along the edge and loaded with uniformly continuous with the load
[Thu Oct 07 10:21:54 2004]
maksimalni progib i maksimalni nagib elastične plohe ploče
maximum deflection and maximum inclination of elastic planes
[Thu Oct 07 10:36:52 2004]
Kružna ploča slobodno oslonjena po rubu i opterećena jednoliko kontinuirano
Cycle plate freely supported along the edge and loaded uniformly continuously
[Thu Oct 07 10:38:50 2004]
iz uvjeta ravnoteže elementa ploče polumjera
from the equilibrium condition of plate element with radius
[Thu Oct 07 10:40:25 2004]
Prema tome, diferencijalna jednadžba savijanja u ovom slučaju glasi
Consequently, differential equation for bending in this case is
[Thu Oct 07 10:41:43 2004]
Konstantu integracije
Integration constant
[Thu Oct 07 10:44:26 2004]
dobit ćemo konačan izraz za nagib
we will get the final expression for inclination
[Thu Oct 07 10:45:20 2004]
Konstantu integracije odredit ćemo iz rubnog uvjeta
Integration constant will be determined from the boundary condition
[Thu Oct 07 10:47:21 2004]
pa konačno izraz za progib glasi
so finally, the expression for deflection is as follows
[Thu Oct 07 10:48:16 2004]
Maksimalni progib pojavit će se na mjestu
Maximum deflection appears on the place of
[Thu Oct 07 10:49:04 2004]
Maximalni nagib javlja
Maximum inclination appears
[Thu Oct 07 10:54:45 2004]
Oba momenta savijanja imaju maksimalnu vrijednost u sredini ploče
Both bending moments have the maximum rate in the center of the plate
[Thu Oct 07 10:59:47 2004]
Kružna ploča ukliještena je po rubu i opterećena jednoliko kontinuiranim opterećenjem
Cycle plate is hinged along the edge and loaded with uniformly continuous load
[Thu Oct 07 11:00:27 2004]
maksimalni progib
maximum deflection
[Thu Oct 07 11:03:19 2004]
Kružna ploča ukliještena po vanjskom rubu i opterećena jednoliko kontinuirano
Cycle plate is wedged in along the external edge and loaded uniformly continuously
[Thu Oct 07 11:06:03 2004]
U ovom je primjeru opterećenje isto kao i u prethodnom, pa će i izrazi za poprečnu silu biti isti, a time i diferencijalna jednadžba savijanja.
In this example load is the same as and in previous, so the expressions for shear force will be the same, and with that also a differential equation of bending.
[Thu Oct 07 11:06:03 2004]
Opće rješenje i u ovom slučaju glasi
General solution in this case is:
[Thu Oct 07 11:19:21 2004]
pa konačan izraz za kut
so the final expression for the angle
[Thu Oct 07 11:24:22 2004]
Maksimalni progib nastaje u sredini ploče i iznos
The maximum deflection emerges in the center of the plate and is
[Thu Oct 07 11:25:58 2004]
U sredini ploče gubi se razlika između cirkularnog i radijalnog momenta savijanja, pa je
In the center of the plate there is no difference between circular and radial moment of bending, so that
[Thu Oct 07 11:26:36 2004]
Na rubu ploče je
At the plate edge, there is
[Thu Oct 07 11:32:57 2004]
Rješenje primjera 8.3 mogli smo dobiti superponiranjem rješenja iz primjera 8.1 i primjera 8.2. Zamislimo da je prema slici 8.19 ploča opterećena jednolikim opterećenjem q i nepoznatim momentom
Solution of the example 8.3 could have been obtained by superimposing the solution from the examples 8.1 and 8.2. Let us imagine that according to Figure 8.19 the plate is loaded with uniform load q and unknown moment
[Thu Oct 07 11:34:34 2004]
Veličinu ovog momenta odredit ćemo iz uvjeta da je nagib na rubu jednak nuli
Magnitude of this moment will be determined from the condition according to which the inclination on edge is equal to zero
[Thu Oct 07 11:35:26 2004]
nagib na rubu ploče iznosi
inclination at the plate edge is
[Thu Oct 07 11:36:05 2004]
U prvom je članu predznak minus jer
In the first member there is a minus sign because
[Thu Oct 07 11:38:44 2004]
djeluje suprotno nego u primjeru 8.1. Sada možemo iz (a) lako dobiti
functions contrarily to the example 8.1. Now we can from (a) easily obtain
[Thu Oct 07 11:42:13 2004]
superponiranjem i ostale veličine.
superimposing the other magnitudes as well.
[Thu Oct 07 11:43:45 2004]
Ploča opterećena istovremeno jednoliko kontinuirano i momentom
Plate loaded simultaneously uniformly continuously with the moment
[Thu Oct 07 11:45:07 2004]
Kružna ploča konstantne debljine
Cycle plate with constant thickness
[Thu Oct 07 11:46:31 2004]
a na krajevima ukliještena prema slici 8.20
and on ends wedged in according to Figurte 8.20
