Stress concentration in castellated I-beams under transverse bending

Although castellated beams are already applied about 100 years, the theory of their stress state still is not still finally elaborated. It can be confirmed, for example, with absence of recommendations on designing of such beams in Eurocode 3. This situation can be explained, first of all, with complexity of problem. To choice the optimal dimensions of castellated beam it is need to appreciate the maximum level of stresses in it, because this is one of important parameters in structural norms. Stress distribution in castellated beams was investigated in any works [1-13] mainly using FEM and experiments. Analytical relations were analyzed in works [1415]. However reliable formula for stress level was not obtained. Very suited instrument for compare of stress state of beams with different web-cutting pattern is the stress concentration factor ασ (SCF), representing by itself non-dimension magnitude. It is possible determine SCF as ratio of maximum equivalent stresses in zone of opening to maximum stress in flange of beam with solid web under given external load. In this work the determination of coefficient ασ was performed for case of transverse flexure.


Introduction
Although castellated beams are already applied about 100 years, the theory of their stress state still is not still finally elaborated.It can be confirmed, for example, with absence of recommendations on designing of such beams in Eurocode 3.This situation can be explained, first of all, with complexity of problem.
To choice the optimal dimensions of castellated beam it is need to appreciate the maximum level of stresses in it, because this is one of important parameters in structural norms.Stress distribution in castellated beams was investigated in any works [1][2][3][4][5][6][7][8][9][10][11][12][13] mainly using FEM and experiments.Analytical relations were analyzed in works [14][15].However reliable formula for stress level was not obtained.
Very suited instrument for compare of stress state of beams with different web-cutting pattern is the stress concentration factor ασ (SCF), representing by itself non-dimension magnitude.It is possible determine SCF as ratio of maximum equivalent stresses in zone of opening to maximum stress in flange of beam with solid web under given external load.In this work the determination of coefficient ασ was performed for case of transverse flexure.

Equivalent stresses in beam
For estimation of the stress concentration level under transverse bending it was initially considered simply supported castellated I-beam, performed on unwasted technology from rolled profile #50 (GOST 8239-72), loaded with concentrated force, applied in middle of span.Depth of holes was adopted equal to 0 667 h .H  , as more useful in structural practice.Web-post width was equal to a side of hexagonal hole, i. e. classic scheme of the beam perforation was considered (Fig. 1).In this case dimensions of beams were l-75-1-17-1.52сm-0.667-1.In common case it can be interpreted as , i. e. lengthtotal height of beamweb thicknessflange width -thickness of flangerelative depth of openingrelative width of webpost.Concentrated force P = 112.5 kN was constant in all cases of loading.Accuracy of calculations by FEM is mainly determined with the finite element sizes: the less FE, the more accurate is calculation.But application of all refined elements is not suitable because of the restricted computer memory and essential increasing of computed time.For example, solution of the equation system with 300000 unknowns in computer with 4 Gb RAМ demand more the one minute.Reducing the size of the equation system and respectively time of calculation can be achieved using different approaches: application the super element method; taking into account the structure symmetry and considering only a half of beam; using no uniform mesh of finite elements and others.Last two approaches are simplest and rather effective, but appear question, what size of elements will be sufficient for getting of demanded accuracy of calculation?Theoretically to determine optimum sizes of FE is rather complex, that is why in most cases they are chosen on base of calculations with successive reducing of element sizes, until difference in results will be negligible.

Fig. 1 Calculation scheme of castellated beam
In this work the refined mesh of FE was adopted only in vicinity of one opening, as it gives the least system of equations.Then this refined mesh was displaced in turn to each opening.
It is clear the size of element is to be connected with radius of curvature of hexagonal opening.The smaller fillet radius the lesser elements are to be, otherwise contour of opening takes form not smooth but broken line, and accuracy of calculation will be reduced.Although formally openings in webs performed on unwaste technology are not rounded in reality they have some fillets.In accordance with recommendations of AISC (American Institute of Steel Constructions) in calculations for strength of castellated beams the fillet radius of hexagonal openings is necessary adopt equal to .
Fulfilled analysis show the satisfactory accuracy is reached under sizes of finite elements equal to 0.05r, that is why in calculations sizes of FE near the contour of opening they were equal , and in other parts of beam their dimensions were 20 mm
Under transverse bending the important role in value of ασ play as bending moment M so and shear force V.The first one determines level of normal stress σx, and second is connected with the shear stress τxy in web.In technical theory of flexure the stress state of beam with solid web is considering taking into account only two stress components σx and τxy.In perforated beams near openings the normal stress σy is also achieve big values.That is why in castellated beams under transverse bending takes place complex stress state, the integral parameter of which for evaluation of SCF under joint action of σx, σy and τxy can be equivalent stress von Mises eqv max  .In common form it can be expressed via stress components as: Appreciate value of   with formula: where TT max  is maximum stress in flange of beam with solid web, determined on technical theory of flexure as: where W is modulus of inertia of beam's cross section with solid web: Of course such approach to calculation of SCF on Eq. ( 2) has some peculiarity, because the level of maximum stress eqv max  is measuring in one cross section, and base value  is taken in another, but it has no important sense, because ασ is non-dimension magnitude.Basic advantage here is commodity of calculation TT max  , and correspondence of obtained value ασ to a physical picture of stress state of web in the concentration zone.
With the aim to distinguish influence of V and M on value ασ it was performed calculations under constant value V and practically absent moment М and under joint action of V and M.
For that a simply supported castellated I-beam of arbitrary length, loaded with concentrated force Const P  in middle of span was considered.In this case near ended opening under any length of beam the shear force V will be constant and bending moment (Fig. 2) does not exceed 3%, i. е. this magnitude is rather stable under changing of width of ended web-post.All this allow in further calculations adopt where αV is numerical coefficient, determined from FE analysis.Calculation with FEM shows the coefficient 41 For example, for castellated I-beam with dimensions 1125-75-1-17-1.52cm-0.667-1under action of concentrated force Р = 112.25 kN level of maximum equivalent stress in vicinity of 1-st hole according to Eq. ( 5) will be:


of flexure moment in the same beam with dimensions 1125-75-1-17-1.52сm-0.667-1,but in zone of second and following openings (Fig. 4).As it can be seen, level of maximum stress grows proportionally to moment М.On base of these results the stress eqv max  in vicinity of any opening can be represented as sum of two items: one caused by shear force V in accordance with Eq. ( 5) and second caused by moment М: where αM is numerical coefficient, determined by FEM analysis.Flexure moment М for n-th opening can be approximated as: where s is step of openings; n is ordinal number of opening, in vicinity of which the stress value eqv max  is determining.In common case the step of openings under any perforation can be written as: where Substitution of Eq. ( 4), Eq. ( 7) and Eq. ( 8) in Eq. ( 6) bring to expression: In Eq. ( 9) value (n -2) is written instead of (n -1), because proportional growth of stresses is appearing only beginning from 2nd opening.Taking into account the coefficient is constant for all calculated below beams Eq. ( 9) can be rewritten as: Verify Eq. ( 10) for castellated I-beam 1125-75-1-17-1.52сm-0.667-1loaded with concentrated force obtained analytically by Eq. ( 10) are shown in Table 1.
It is need to note that Eq. ( 10) remains applicable for any length of beam under unchangeable parameters of perforation.So for ratio of beam length (the divergence does not exceed 2.6%).Such result is satisfying to engineering accuracy of calculations.But it is possible to obtain practically full coincidence of results by FEM and by Eq. ( 10) for this beam, if perform correction of coefficient αV, reducing it from 41 to 39.5 and remaining second coefficient αM unchangeable and equal to 6.4.In this case results obtained by FEM and by Eq. ( 10) practically coincide.Values of stresses for other openings of this beam and for beams of other sizes are shown in Table 1.
In Table 1 it were considered castellated beams fabricated on unwasted technology from rolled profiles #45, 50, 55 and 60.They are quite similar in proportions of height and web thickness but main difference is in relations of webarea to flange-area.This difference can be seen in variation of coefficient αV which is changing in very narrow range: from 38.8 to 41.

 
. Results of calculation of beam with dimensions 1125-75-1-17-1.52сm-0.7-1,performed by Eq. ( 10) with value of αV = 46.5 and computed by FEM are shown in Table 2. Stress state of web near the different openings of this beam under concentrated load Р = 112.5 kN is also shown in Fig. 6.
It can be seen from Table 2 and Table 1 the increaseing of the depth of opening leads to growth of coefficient of force αV, i. e. role of shear force in value of equivalent stresses is increasing.Dependence αV on magnitude β is proportional and can be approximated with expression: Similar lineal dependence there is and for beams with other dimensions.2, from which it is the divergence in different calculations does not exceed 2.5%.Results of calculation by FEM are shown in Fig. 7.

Influence of relative width of web-post at stress level
As it is known, developing of perforated beams is directed on lightening of web with different ways: increasing the depth of openings; increasing the length of opening, by performing them with elongated form such as oval, rectangular or sinusoidal; reducing the width of web-posts.Author proposed technology of fabrication of castellated beams with regular hexagonal openings under any width of webposts [16].That is why influence of relative width of webpost at stress eqv max  is considering below.Results of calculation by FEM on program ANSYS of I-beam 1350-90-1.2-19-1.78сm-0.667-0.5 with relative width of web-post β = 0.5 with sequent displacement of small mesh are shown in Fig. 9. Due to reducing of width of web-posts the number of openings at half of the beam length increased to 7. Analytical calculation of equivalent stress allows confirm, that according to Eq. ( 10) reducing of β lead to less level of eqv max  .In Eq. (10), as in previous variants with β = 1 factor of influence of moment remains the same αM = 6.4,but factor of shear force takes value αV = 37.6.Results of calculation of indicated beam by Eq. ( 10) are shown in Table 4.
Reduce the relative web-post width till β = 0.

Experimental investigation
In order to verify Eq. ( 10) it was put an experiment on steel model in form of double cantilevered I-beam with dimensions 410-38-0.6-12-1cm-0.667-1,loaded by two concentrated forces V = 10 kN applied at the ended sections via dynamometers DR-20 (Fig. 11).Material of beam was steel S345.Installation had two rigid posts located at distance 1 m from each other.Length of each cantilever was 1.5 m.During loading the level of stresses in vicinity of openings was measured by strain gauges with base 1mm located on web in form of strain rosettes.Readings of gauges were registered with Data Acquisition Controller of English firm Schlumberger.Gauges were glue near the fillet corner openings in places, determined with calculation by FEM.Beam was simply supported and loaded symmetrically at ends.
Obtained results indicate the stresses calculated by Eq. ( 12), by FEM and registered in experiment under transverse bending are in good correlation.As it can be seen coefficient αM = 6.4 is constant for all dimensions of beams and different perforation.

Stress concentration factor
Evaluate now stress concentration appearing in web under action of flexure moment M and transverse force V.For this purpose it will be using Eq. ( 2) in which we substitute value of maximum equivalent stress in arbitrary section Eq. ( 10 Substituting Eq. ( 10) and Eq. ( 13) in Eq. ( 2), the stress concentration coefficient ασ is determined as follows where ω * = 6bf tf / Htw + 1 and η = l / H.As it can be seen from Eq. ( 14) SCF does not depend on load factors but is determined only with geometry of beam, relative length η = l / H and parameters of perforation in non-dimension form ξ and β.
It is need to remember the obtained results are applicable to castellated beams with fillet radius of opening r = 0.04h.

Conclusions
1. Analytical expression for SCF for case of transverse bending is obtained as sum of two components reflecting influence of shear force V and flexure moment М respectively.
2. Obtained relations for ασ and for equivalent stresses eqv

.
Now perform with help of calculations by FEM analysis the influence on magnitude eqv max

Fig. 11
Fig. 11 Test set-up with castellated I-beam model 410-38-0.6-12-1cm-0.667-1Results of tests show the measured maximum equivalent stress eqv max  near contour of 2-nd opening was equal to 181 MPa and in vicinity of 3-rd opening it was 190 MPa.Calculation of beam with finite element method (Fig. 12, b and 11, c) indicate values of eqv max  in the same lo- cations equal to 184 MPa and 195 MPa respectively.Difference in values reach 2.5%.Determination of equivalent stress eqv max  appearing in vicinity of third opening in accordance with Eq. (10) for tested beam gives:

3 .
Factor of influence of moment αM = 6.4 does not depend on the relative values ξ and β.4.Factor of influence of shear force αV grows with increasing of depth openings and is almost proportional to value β.5.Stress concentration factor near hexagonal openings under transverse bending can reach value ασ = 4.