Prediction of the stress level and stress concentration in cellular beams with circular openings

Although stress concentration in vicinity of regularly located circular openings under axial tension is well researched [1], in this work the problem was investigated in aspect of transverse bending of cellular beam, when simultaneous action of shear force and bending moment take place and pure bending. For today the main shapes of perforations are hexagonal and circular openings, which are produced on unwaste technology. Comparison of strength of such beams for choice of optimal design solution is possible in first turn with estimation of their stress state. In spite of numerous works dedicated to estimation of stress level in perforated beams as foreign [2-14], so and Russian [15, 16] authors, there is no analytical expression for evaluation of stresses in zones of concentration. In all works the analysis of stress state is performing on base of calculation by finite element method, sometimes accompanied with experiments. More often in them there are only numerical values and colored fields of stress distribution. To generalize such results or use them for predicting of stress level in designed beam is rather difficult. In the work it was obtained the analytical expression for maximum equivalent stresses on Mises in beams with circular openings. These beams are distinguished with wide spectrum of sizes (Fig. 1), but in this work it was changed only one parameter of perforation – relative width of web-posts. The relative depth of openings was remained unchangeable and equal to ξ=0.667. It was also investigated influence of web thickness on the stress level. Investigation of stress distribution in cellular beams with circular openings was made under two kinds of loading: -under transverse bending when shear force at any section is constant and flexure moment is changing on lineal law; under pure bending, when shear force is absent. Stress concentration was evaluated under pure bending.


Introduction
Although stress concentration in vicinity of regularly located circular openings under axial tension is well researched [1], in this work the problem was investigated in aspect of transverse bending of cellular beam, when simultaneous action of shear force and bending moment take place and pure bending.
For today the main shapes of perforations are hexagonal and circular openings, which are produced on unwaste technology.Comparison of strength of such beams for choice of optimal design solution is possible in first turn with estimation of their stress state.In spite of numerous works dedicated to estimation of stress level in perforated beams as foreign [2][3][4][5][6][7][8][9][10][11][12][13][14], so and Russian [15,16] authors, there is no analytical expression for evaluation of stresses in zones of concentration.In all works the analysis of stress state is performing on base of calculation by finite element method, sometimes accompanied with experiments.More often in them there are only numerical values and colored fields of stress distribution.To generalize such results or use them for predicting of stress level in designed beam is rather difficult.
In the work it was obtained the analytical expression for maximum equivalent stresses on Mises in beams with circular openings.These beams are distinguished with wide spectrum of sizes (Fig. 1), but in this work it was changed only one parameter of perforationrelative width of web-posts.The relative depth of openings was remained unchangeable and equal to ξ=0.667.It was also investigated influence of web thickness on the stress level.
Investigation of stress distribution in cellular beams with circular openings was made under two kinds of loading: -under transverse bending when shear force at any section is constant and flexure moment is changing on lineal law; -under pure bending, when shear force is absent.Stress concentration was evaluated under pure bending.

Theoretical approach
Before to come to estimation of the stress level it was solved task of determination of equivalent stresses in vicinity of opening under transverse bending.It was considered simply supported beam, performed from plate elements and loaded with concentrated force in mid-span.Such loading produce deformation of transverse bending under constant shear force.
, which completely determine their geometry.Incoming magnitudes are interpreted as:  .In common form it can be ex- pressed via stress components as: The value of stress concentration factor   can be calculated on relation: where TT max  is maximum stress in flange of beam with solid web, determined on technical theory of flexure as: where x М is flexure moment caused with external load in section х, where stresses eqv max  are determining; W is modu- lus of beam's cross section with solid web: So far as under transverse flexure on value eqv max  influent the shear force V and bending moment M, magnitude eqv max  can be represented as sum of two terms: where  and М  are numerical coefficients, determined from FEM calculations.
Magnitude of flexure moment Мx for n-th opening can be written as: where s is step of openings; n is ordinal number of opening, in vicinity of which the equivalent stresses eqv max  are determining.
In common case the step of openings can be determined as: Taking into account that Substitution of Eq. ( 4), Eq. ( 6) and Eq. ( 8) in Eq. ( 5) after simple transformation leads to relation So for calculation eqv max  it is need to determine numerical coefficients V  and М  .Initially it was considered simply supported cellular beam loaded with concentrated force Р in mid-span; in second case the beam was loaded with two equal forces Р , applied at equal distances from supports, that produced deformation of pure flexure in the middle part of beam.
For getting reliable results by FEM in vicinity of openings it was adopted refined mesh of finite elements with sizes .Such combination of dimensions assured high accuracy of solution with minimum waste of the calculation's time.For additional reducing the equations system NEQ it was taken in a view the symmetry of structure, which allow considering only half of beam.Once more approach of reducing dimensions of NEQ was application of mesh of FE not along the whole contour of opening but only on half of it where high stresses are expected.Beside of this, refined mesh was applied not at every hole but only near every fifth, so as number of openings along the beam was too big and reached 100-130.Indicated peculiarities allow reducing the number of equations till 150-300 thousands.
Under transverse bending it was investigated influence of web thickness w t and relative width of webposts  under constant relative height of openings 667 0.   on a level of equivalent stresses near openings.
Stress state of all beams was appreciated for simply supported beams under the same shear force , which is equal to reaction of support (Fig. 3).9).In the ended right column of Table 1 the total number of openings in considered beam is indicated.For constant length of beam the value N will be different because of different width of web-posts.
For convenience of analysis the numerical data obtained by FEM near the contours of openings were joined in Table 1, in which values of stresses eqv max  , calcu- lated by Eq. ( 9) are also shown.

Table 1
Stresses (МPа) in simply supported beam 4000 -40 -0.3 -10 -0.3 сm -0.667 -ξ under action of shear force V=1 kN Analysis of obtained by FEM data show that for beam with ξ=0.5 magnitudes of eqv max  can be approximated with Eq. ( 9) under values of coefficient of force Divergence in values eqv max  , calculated by Eq. ( 10) and by FEM (Fig. 4, a), is 0.2%, and for 29 th opening it will be on Eq. ( 9 In this case divergence is 2.7%.For some other variants error of approximation with Eq. ( 9) can be bigger, but in average it describes quite well the stress state of cellular beams with circular openings.Now it will be considered diapason of application of obtained Eq. ( 9).In Fig. 5 it is shown results of calculation of stresses of the same beam, that in Fig. 4, a but for openings with numbers n=1-8.Divergence in results obtained by FEM and by Eq. ( 9), near the 6th opening is 2.1%, near the 7th is 0.9%, and near the 8 th is only 0.4%.But for the 4th opening the difference is exceeding already 10.9% (Table 1) and for openings 1-2-3 the divergence is still bigger.
Such big difference in values of stresses is connected with increasing role of shear force near the support because near the first opening influence of shear force is dominant due to small value of flexure moment which is near zero.From here it can be concluded (Fig. 6) the obtained Eq. ( 9   , according to FEM and Eq. ( 9) It is need to note the influence of shear force on value of equivalent stresses is not big, so far as maximum shear stresses from V appear near the neutral axis of beam, and maximum normal stresses from moment M take place in low part of openings.Only in vicinity of ended openings where flexure moment is small equivalent stresses eqv max  reach essential value because of the shear force action (Fig. 5).
Similar calculations performed for beams of the same dimensions and parameters of perforation that in previous case but with thickness of web 0.2 cm w t  show the Eq. ( 9) gives rather good results in this case also.In  9) are shown in Table 2.As it can be seen from Table 2, adopted earlier values of coefficients V  and M  give good results with Eq. ( 9) in this case also.In this distance divergence between data calculated by Eq. ( 9) and by FEM does not exceed 5.3%.Comparison of stress level for different values  allows to note tendency of reducing equivalent stresses on Mises with reducing of web-posts width.Table 2 Stresses (МPа) in simply supported beam 4000 -40 -0.2 -10 -0.3сm -0.667 -ξ under action of concentrated force Р=2 kN, applied in mid-span

Stress concentration factor under pure bending
Consider now level of stresses under pure bending.Deformation of pure bending in central part of beam can be produced by loading of simply supported beam with two symmetrically applied concentrated forces.Research now influence on stress distribution of parameters of the web thickness and width of web-posts.Then for beams 4000-40-tw-10-0.3сm-0.667-0.5 with different thickness of web it is possible to get pictures of stress distribution, represented in Fig. 8, from which it can be seen that under pure bending all stresses remain constant as in shelves so and near the contour of openings.
Level of stresses with reducing of the web thickness is growing because of lesser modulus of section.From here it can be concluded that in cellular beams with circular openings under pure bending SCF is almost proportional to relative width of web-posts.

Conclusions
1.It was obtained empirical Eq. ( 9) for evaluation of equivalent stresses eqv max  on Mises in beams with circular openings under joint action of shear force V=Const and bending moment М=Vx, changing on lineal law.
2. Eq. ( 9) was verified for variants with different web thickness an different relative width of web-posts in range

a b Fig. 1
Cellular beams with different width of web-posts: а) Web perforation is determining with three parameters: diameter of opening d, width of web-post с, representing by itself minimum distance between edges of two adjacent openings, and width of ended web-post 0 с(Fig.2).For denomination of the beam dimensions in work it was used next abbreviated form of writing: relative width of web-post.Dimensions of beam are indicated in centimeters.Program of calculation included investigation of influence on value eqv max  relative width of web-posts  in range of width of web-posts embraces practically whole spectrum of perforation of such beams.

Fig. 2 
Fig. 2 Parameters of cellular beam Derivation of relation for equivalent stresses is realized on analysis of calculation of cellular beams with FEM using program complex ANSYS.Under transverse bending definite role in value of stress concentration factor (SCF)   play as flexure moment M so and shear force V. Flexure moment M determine level of normal stresses x  , and force V is connected

. 4 ,
а the stresses calculated by FEM are equal to 41.9 МPa, and obtained on Eq. (9) have value:


MPa and by FEM 85.1МPа (Fig.4, a).Divergence is not changed.For beam with ξ=0.25 values of coefficients V  and M  will be another ones.The magnitude of coeffi- .4, b) and by Eq. (9) result is represented below:

8 (
Equivalent stresses in beam 4000 -40w t -10 -0.3сm -0.667 -0.5 with different thickness of web: Fig. 8, c) value of SCF became equal to 38 results allow making conclusion that under pure bending the SCF is proportional to the web thickness.Evaluate now influence of the web-posts width on the stress concentration under pure bending.Determine a b Fig. 9 Stresses eqv max  in beam 4000 -40 -0.2 -10 -0.3 сm -0.667 -ξ under pure bending: a) ξ=0.25; b) ξ=0.15 for beam with β=0.667.