The castellated beams deflections calculated with theory of composed bars

practice. Namely beams with such openings will be considered in the article. For today there are three different methods of calculation of the castellated beams deflections: method based on the theory of composed bars (TCB) [3]; method using the theory of Vierendeel truss [4] and the finite element method (FEM) [5]. First two methods are approximate ones but FEM is an exact method. It’s seems the best of all to use FEM for calculation of deflections, but application of FEM demands of existence of rather expensive program complex for modeling of calculated structure (for example, ANSYS) and beside of this a researcher is to be qualified in using ANSYS, so as even small deviations in adopted boundary conditions or description of model can bring to significant distortions in results. That is why different analytical approximate methods are elaborated for prediction of the castellated beams deflections. In Russian practice the method of the theory of composed bars, elaborated by А.R. Rzanizyn [3] is more popular and in abroad Vierendeel method is widespread, application of which to calculation of castellated beams was even included in one of the previous variants of Eurocode 3 [4]. In the work the task of obtaining reliable relation for prediction of deflections of the castellated beams was put. A criterion of the calculation accuracy is the finite element method results. Calculation of deformations of perforated beams deflections with the theory of composed bars was modified by author with integration of differential equation in Fourier series [6].


Introduction
In Russian structural Norms SN&R [1], as in Eurocode 3 [2], one of the basic demand to castellated beams is securing of necessary rigidity, i.e. restriction of relative deflection f / l .For different structures admissible value of this magnitude is different, but more often for beams it had to satisfy to condition 1 250 f / l /  . In SN&R [1] the relative height of openings in castellated beams is restricted by value 0 667 h / H .
 , as more useful in structural practice.Namely beams with such openings will be considered in the article.
For today there are three different methods of calculation of the castellated beams deflections: method based on the theory of composed bars (TCB) [3]; method using the theory of Vierendeel truss [4] and the finite element method (FEM) [5].First two methods are approximate ones but FEM is an exact method.It's seems the best of all to use FEM for calculation of deflections, but application of FEM demands of existence of rather expensive program complex for modeling of calculated structure (for example, ANSYS) and beside of this a researcher is to be qualified in using ANSYS, so as even small deviations in adopted boundary conditions or description of model can bring to significant distortions in results.That is why different analytical approximate methods are elaborated for prediction of the castellated beams deflections.
In Russian practice the method of the theory of composed bars, elaborated by А.R. Rzanizyn [3] is more popular and in abroad Vierendeel method is widespread, application of which to calculation of castellated beams was even included in one of the previous variants of Eurocode 3 [4].In the work the task of obtaining reliable relation for prediction of deflections of the castellated beams was put.A criterion of the calculation accuracy is the finite element method results.
Calculation of deformations of perforated beams deflections with the theory of composed bars was modified by author with integration of differential equation in Fourier series [6].

Testing procedures
Performing of openings in webs of castellated beams lead to reducing of their shear rigidity and hence to increasing of its deflections.For simply supported beams its growth can reach 60%, and for a clamped beams even 2-2.5 times compare to deflections of the same dimensions beams with solid web.
In work it was put problem of comparison of different approaches for calculations of deflections of simply supported castellated beam, loaded with distributed load (Fig. 1) under different relative length l/H.Fig. 2 Geometry parameters and calculation scheme of castellated beam according to TCB Below it is considering variant of calculation of deflections with TCB based on integration of differential equation in Fourier series.In work [5] differential equation of the flexure axis of composed beam was obtained in form: here E is Young modulus of material; f is cross area of every T-belt; c K is coefficient of rigidity of elastic layer, formed with web-posts; I is moment of inertia of beam calculated for weakened by opening section; i is proper moment of inertia of T-belt located above opening; M is flexure moment.
From Eq. ( 1) as private cases it can be obtained equation of flexure of packet formed with two bars 2 , not jointed between themselves by shear ties ( 0 c K  ), or equation of flexure of monolith beam EIw M   under c K .For obtaining a simple form of solution Eq. ( 1) was solved in Fourier series for case of action of uniformly distributed load q.Then by decomposition of function of flexure moment M in series by sinus: solution of Eq. ( 1) can be obtained in form of series: Analysis of results of calculations by FEM and TCB of castellated beams show that for simply supported beam (Fig. 1), loaded with uniformly distributed load of intensity q, good results can be obtained remaining only one term of series in decomposition Eq. ( 3), changing coefficient 5 4 /  at 5/384.Then instead of Eq. ( 3) we get: First multiple in Eq. ( 5) represents the deflection ТТ w of simply supported beam, calculated on technical theory of flexure: Moment of inertia I calculated for section with opening have a view: The inertia moment sol I of beam with solid web can be calculated approximately as: Expression ( 8) is approximate so as it is not taken into account proper moment inertia of shelves, influence of which for I-beams usually not exceed 1%.
If take into account that in many cases 100 c K  it is possible to neglect with unity in denominator of Eq. ( 5) and write it in more compact form, convenient for practical calculations: Second additive in brackets in Eq. ( 9) is reflecting the shear component of web deflection of perforated beam.Although the expression for deflection Eq. ( 9) has a compact form it can be simplified more if substitute in it Eq.( 4).Then relation Eq. ( 9) can be written as: For getting of reliable result on theory of composed bars it is need correctly determine coefficient K c , which is function of height of opening h, relative width of web-post с / а , thickness w t , material of beam E and form of web-post.Incorrect finding of value K c can lead to essential loss of accuracy in calculations of the beam deflections.
Unknown coefficient K c can be determined from experiment on static testing of perforated beam or using results of numerical calculation of beam by FEM.In the work it was used second approach.Coefficient of rigidity K c can be calculated as: where Today in structural practice cellular beams with narrow web-posts (Fig. 3, a) are applied widely.Technology of the castellated beams performing suggested by author [7] allow to get any width of web-post independently of side of opening (Fig. 3, b).That is why coefficient    was chosen as function of  .
Using of relation Eq. ( 12) lead to good results for any width of web-post in range 0 3 1 .  .Substitution of Eq. (11) into Eq.(10) lead to:  13), if it is known load, parameters of perforation and dimensions of beam.All calculations of deflections on TCB are convenient to perform with help of electronic tables Excel.Area f of T-belt can be calculated as: Preliminary calculations by FEM showed that better results can be obtained if instead of I in relation Eq. ( 6) use the inertia moment m I determined as average arithme- tic of the inertia moments calculated in two cross sections: in cross section without opening sol I Eq. ( 8) and in cross section with opening I Eq. ( 7): Namely relation Eq. (13) will be used below for calculation of deformations of simply supported beams and only in expression for ТТ w Eq. ( 6) it is need to substitute the inertia moment m I .

Numerical calculation of deflections of castellated beams
Estimation of accuracy of obtained above analytical relation is possible by comparison of numerical results with calculations performed by FEM, which are most reliable.For that it was considered simply supported castellated beams with different values of the web-post slenderness

 
, because with increasing of the web slenderness the influence of perforation at deflections under same relative height of openings is grow.It can be concluded from the second term in Eq. (13), which is proportional to value w h / t .As it is known the perforated beams with hexagonal openings are being extensively used in lightly loaded and long-span composite floor constructions, for example, in multy-level parking garages.Castellated beams are being pushed to span great lengths which can approach 16-28 m height of beam Then calculation of deflection Eq. ( 13) gives:    and web-posts 05 .
  is less of 4% for beams with relative length 15 l / H  (see Table 2).
If compare deflections of perforated beam and beam with solid web ( 900cm l  ) it can be seen (Fig. 5, c) the openings reduce rigidity of beam approximately at 23% for classic perforation with 1   and at 31% for perfora- tion with 03 .
3. For relative length 30 l / H  it is not need to use relation (13) because good results give calculation of deflection with the technical theory of flexure Eq. ( 6) using an average inertia moment m I .Divergence with FEM re- sults is less 1%.
4. The obtained relation Eq. ( 13) can be recommended for including in Structural Norms & Rules for prediction of the castellated beams deflections.

Fig. 1
Fig. 1 Scheme of loading of simply supported beam with uniformly distributed load In the theory of composed bars perforated beam is considering as two bearing T-bars, located above and be-

For s imply supported
beam with height of openings h 0 = 0.667Н coefficient    have a view:

Fig. 3
Fig. 3 Perforated beams with narrow web-posts: a -circular; b -hexagonal openings So, deflection of perforated beam according to theory of composed bars can be easy calculated on relation Eq. (13), if it is known load, parameters of perforation and dimensions of beam.All calculations of deflections on TCB are convenient to perform with help of electronic tables Excel.Area f of T-belt can be calculated as:

1 .
Fig. 5 Deflections of simply supported beams 900 60  0 86 18 1 35сm 0 667 . . .      with narrow webposts: a -03 .  ; b -05 . ; c -with solid web As it can be seen from Table1, accuracy of calculation on TCB is rather high; divergence with FEM does not exceed 2.5% even for shot beams.As it is known, biggest effect of shear on deflections is registered for shot and high beams, i.e. beams with small ratio l / H . Calculation of deflections with FEM was performed with program complex ANSYS using quadrangular elements Shell63 with 6 degrees of freedom in every node., then real height of model will be equal full height of beam H. Due to symmetry it is possible consider only half a beam that sufficiently reduce time of calculations.
f Ht 

Table 2
Deflections (mm) of the simply supported castellated beams with narrow web-posts, load q = 10 kN/m