Jan Bujnak Facult of Civil Engineering, Universit of Zilina, Slovakia Ruzica R. ikolic Facult of Mechanical Engineering, Universit of Kragujevac, Serbia Jelena M. Veljkovic RPP-ZSTV Factor, Kragujevac, Serbia LYSIS OF DIMESIOIG THE ECCETRICLLY LODED COLUMS
Behavior of the real structure is alwas different from the theoretical predictions. The basic causes for this discrepanc: geometric imperfections material flaws loads that do not act at the cross section s center of gravit
Here is considered: the influence of the load eccentricit on behavior of the column loaded in compression. Comparative presentation is given of: the calculation results according to theoretical considerations and results obtained b the rton Perr formula (i.e. according toeurocode 3 standars).
The column shown in Figure is exposed to: compressive force which acts at distance e from the axis of the non-deformed column.
The reference frame is chosen in such a wa that the origin is in the middle between the two ends of the column. The bending moment in the arbitrar cross-section is: M (e ) (1) If the stress does not exceed the proportionalit limit and if the deflection is small, the differential equation of the elastic line (the deformed - deflected position) is: d d EI (e ) dx dx EI EI (e ) ()
The solution of this equation is of the form: sin px Bcospx C (3), B, C = e + and p = (/EI) are the constants, which are being determined from the initial conditions: 0 and B (e ) (4) Substituting this into (3) gives: (e )1 cosx / EI (5)
Thus, the maximal deflection = for x = L/ is: e 1 cos L/ / EI / cos L/ / EI e sec L/ / EI 1 (6) For an value of e, the value of sec L/ / EI lies between - and +, when the argument tends to π/, 3π/, 5π/, and the deflection increases infinitel. If one chooses π/, taking into account that for this value the load is the least, it follows:
L / / EI / EI L (7) from where follows that: cr EI / L (8) what represents the Euler's formula.
The maximal normal stress can be obtained b summing the maximum stress due to the axial load and the stress due to the bending: max M max I c (e i )c where i is the gration (inertia) radius of the cross section. Substituting equation (6) into (9) gives: (9) max / 1 ec / i sec L/ i / EI (10)
1 max ec / i sec L / i / EI (11) This is known as the secant formula. It gives the relationship between the unit load, (needed for dimensioning the column) material characteristics and eccentricit e. The term (L/i) represents the slenderness l, which exists in the Euler's formula for the critical buckling force. The value (ec/i ) is called the eccentricit coefficient.
THE DIMESIOIG FORMULE n example of the eccentricall compressed column.
THE MODIFIED FORMUL When the load is applied in the plane that contains the axis for which the second moment of area is smaller (the z-axis), the column can buckle about both axes, thus both possibilities should be tested: l l v z z 1 k f I Mc l l v 1 k f I Mc (1)
THE MIXED FORMUL The stress in the column can be written as: Mc w (13) I When equation (13) is divided b w, one obtains: / w Mc/ w I 1 (14)
Equation (14) can be rewritten as: / a Mc/ b I 1 (15) (/) is the average unit load of the eccentricall loaded column a is the allowable average unit load (stress) for the centricall loaded column, calculated for the larger value of (L/i) (Mc/I) is the bending moment of the column b is the allowable bending stress
THE YRTO-PERRY FORMUL (Eurocode 3) ( )(f ) cr b b b cr b (15) where: E / l cr b / f is the ield stress is the imperfecton factor
The imperfecton factor can be expressed as: ( l l0 ) (17) where: l f / cr The smallest solution of equation (16) is: 1 ( l l 1 ( l l ) l 4l / l 0 ) l 0 where: b / f α = 0.1 to 0.76
COMPRISO OF VRIOUS FORMULE ILLUSTRTED O THE EXMPLE The data necessar for calculations are: =1810 mm, i z =68.8 mm, i =196 mm, c=30.7 mm, e=15 mm L=6 m effective buckling length.
() CLCULTIOS CCORDIG TO THE MODIFIED FORMUL l z L i z 6000 68.8 87. f k z 1 l z v l 90 1.9 1 87. 116.7 110MPa i =196 mm Mc f l 1 I k l v 0.15 0.307 90 6 / 0.196 1 0.196 1.76 116.7 90.9 MPa 90.9 10 6 181010 6 1.655M
(B) CLCULTIOS CCORDIG TO THE MIXED FORMUL The additional data is the allowable bending stress σ b =190 MPa (according to JUS standards). / a Mc/ b I / a ec /(i b ) 1 1 110 0.15 0.307 / 0.196 190 1 76.7 MPa 76.7 10 6 181010 6 1.397M
(C) CLCULTIOS CCORDIG TO THE YRTO-PERRY FORMUL The additional data is f = 55 MPa 1 ( l l with, α=0.49 and / f 1 0.49(0.1 0.) 0.1 1 ( l l ) l 4l / l 0 ) l 0 1 (0.1 0.) 0.1 0.1 550.4 107.5 MPa 40.1 0.4 107.510 6 181010 6 1.96M
COCLUSIO If one compares the results obtained b the three formulae presented, it can be seen that: all the three results are within 0 % limits, while the value obtained from the rton-perr formula is the most on the safet side. This is et another example of the fact that regulations prescribed b Eurocodes are much more on the safet side, than theoretical and empirical formulae. However, this also means more material consumption. The theoretical results still offer the best compromise between the two contradicting requirements, but the standards must be obeed.