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Fractural analysis of pre-cracked simply supported beam
G.M. Kravchenko, D.S. Kostenko, K.I. Dolgopolova Don State Technical University, Rostov-on-Don
Annotation: the article considersbehaviour of a simply supported pre-cracked beam made of elasto-plastic material. The aim of the work is to study propagation of min-span fractures in the beam, defining of its behaviour in the local area and developing of techniques and methods to prevent further growth of a crack. There were applied vary verifymethods to analyse deflections, internal forces and stress intensity factor in the area of crack propagation.Analytical and numerical calculation had been used. For numerical solution ANSYS software is used, based on finite element method. According the solutions key features and conclusions are given. Keywords: fracture, pre-cracked beam, stress intensity factor, load-bearing capacity, elasto-plastic material, finite element method.
The aim of the work is to study the load-bearing capacity of the pre-cracked beam with various lengths and widths of the crack. The beam made of elasto-plastic material and considered in 2-dimensional case.
There are three basic types of fractures. In the first group (it is the type I of the cracks) the fracture is originated from tension, in the second (type II.) from shear, and in the third from twisting (type III.) [1].
Fig. 1. - Basic types of fracture
Opening mode (I): the crack surfaces separate symmetrically with respect to the planes XY and XZ.
Sliding mode (II): the crack surfaces slide relative to each other symmetrically with respect to the plane XY and skew-symmetrically with respect to the plane XZ.
Tearing mode (III): the crack surfaces slide relative to each other skew-symmetrically with respect to both planes XY and XZ.
This study deals with only first group of fractures in a beam which cause by vertical load.
The geometry is made with SolidWorks. It was done for each case of the crack (size parameters as various lengths and widths ware varied) and after exported to ANSYS Workbench. The material properties and some parameters of the crack were defined with ANSYS Engineering data satellite. For each case of the input data stress intensity factor (KI), deflections and stresses ware define for different geometry sizes and forms of the crack by ANSYS and also analytical solution [2,3].
For the analysis a 2D element model was built up with SolidWorks for each geometry case of the crack. The analysis was made in ANSYS.
The general parameters of the model show in table 1.
Table № 1
The general parameters of the beam
Length of the beam (S), mm Height of the beam (W), mm Thickness of the beam (B), mm Depth of the crack (a), mm Distribution load (q), kPa Young's module (E), GPa
6000 800 400 50..150 1 210
The geometry was imported to ANSYS, there the boundary conditionals and load are applied according the Figure 2.
q
■
i
Fig. 2. - Principal scheme of the beam
The next step is mesh generating of the model. In the area of crack we have to use smaller side sizes of the elements [4,5]. We shall define optimal size of finite elements. To solve this problem we start from elements with 40 mm side size and we will compare the values of deflection with the previous one each step.
Table № 2
Maximum vertical displacements in the mid-span of the beam
in depending on elements size
Element size, Deflection, Difference,
mm m %
40 0,0657
16 0,0658 0,24
8 0,0660 0,19
4 0,0660 0,06
2 0,0660 0,02
As can be seen from the previous table, for those sizes of elements there is no significant difference between deflections of the beam.
Also, we have to check influence of the element size for another parameter of the analysis - Stress intensity factor [6]. This relation is shown in the table.
Table № 3
Relation between stress intensity factor and element sizes
Element size , mm Stress intensity factor (Ki), МР?л Difference, %
40 1,5351
16 3,2552 52,84
8 4,9875 34,73
4 6,0267 17,24
2 6,7728 11.02
1 7,2167 6,15
0.5 7,4681 3,37
It can be seen, that the difference of the Stress intensity factor with 1 mm and 0,5 mm elements size less than 5%, which is acceptable. For the current numerical experiment we have to select element with 0,5 mm side size for the crack area.
Fig. 3. - Finite element model of the beam
The following figure demonstrates the beam with mid-span crack (depth of the crack is« = £0 mm), where the value of stress intensity factor is equal to
0,58065.This value is close to the result comes from analytical solution for the same input data. [7].
I (i.vmm» men (l.sfiilsft mln
Fig. 4. - Stress intensity factor (KI)
The shape of the crack is changed after applying the load. It can be seen from the figure 5.
Fig. 5. - Shapes of the crack
a) original shape; b) shape with crack opening displacement
Figure 6 presents distribution of the normal stress at the top of the crack. There are plastic and elastic material behaviors in this area.
Fig. 6. - Normal (horizontal) stress distribution at the top of crack
Numerical calculation results of the displacement of the beam for the different values of the crack length and constant value of its deep show in table 4.
Table № 4
Maximum values of deflection for different length of the mid-span crack
Deep of the crack, mm Length of the crack, mm Vertical displacement, mm
150 3 18,47
150 4 18,48
150 5 18,47
150 6 18,46
150 7 18,46
150 8 18,46
150 10 18,48
150 15 18,47
150 20 18,46
150 25 18,48
The next step of the research is analysis of the KI changing in depends on shape of the crack [8]. The new models have crack with rounded top and diameters of the rounding are 1 mm, 2 mm, 5 mm, and 10 mm. Geometry of the one of them is shown in Figure 7.
Ф10
Fig. 7. - Geometry of the rounded crack with diameter 10 mm
These models of the beam were compared to the same one, but with sharp angle in the top of the crack (diameter of the rounding D = 0 ww-a). The results are shown in the following table.
Table № 5
Results for different geometry parameters of the crack
Load, tZf/fi = 0 гкт £>" = 1 гш B=Zmm В = 3 nm £>- = 10 mm
Def., mm Ki, МРа—гии Def., mm Ki, MFtt-vheh Def., mm Ki, NFI-yIKM Def., mm Ki, MFt-v'tHta Def., mm Ki, MPft-rthkj
8 22,06 2.787 22,35 0.237 22,37 0.070 22,39 0.088 22,47 0.058
16 44,12 5.573 44,71 0.475 44,75 0.139 44,77 0.178 44,95 0.116
24 66,18 8.360 67,06 0.712 67,12 0.209 67,16 0.266 67,42 0.175
7 к
d 6 <
1 * *
ro
I 4
d з
2 1
0
i
о
II
ft 1 1 u i I-
4 6 8
Diameter of the crack rounding, mm
10
■ Kl, 8 kN/m ♦ Kl, 16 kN/m Kl, 24 kN/m
12
Fig. 8. - Plot of stress intensity factor values (KI), depends on diameter of the
crack rounding and different values of the load The plot (Fig. 8) shows the relation between stress intensity factor (Ki), diameter of the crack rounding and different values of the load.
This chart describes the relation between deflection of the beam, diameter of the crack rounding and different values of the load.
Fig. 9. - Relation between deflection of the beam, diameter of the crack rounding
and different values of the load
As can be seen from the previous plots, the rounding in the top of crack may significantly decrease value of stresses,whereas the vertical displacements have almost the same values.
To verify numerical solution results we have to compare them with analytical solution results. Following formulas are used as analytical solution.
Let x = et/W. Where a - depth of the crack, w - height of the beam.
Stress Intensity Factor (Ki) can be obtain as K{ = &\>nclY [2], where
- 3C(1 - r) (2,15 - 3,93x+ 2,7**'
To calculate deflection of the beam we use the formula
= — [- -95 - - 93 v.- - -1-1731;;. - -1-151;;. 7*9'i; -v.-:'].
L 'B Or
In case of the crack depth a = 80 mm,stress intensity factor Ks = 0,580S.
Table № 6
Analytical results for the deflection of the beam
Depth of the crack, mm KI, MFaymm Deflection, mm
50 0,3489 19,85
60 0,4392 20,06
70 0,5165 20,29
80 0,5831 20,55
90 0,6443 20,83
100 0,6986 21,14
Depth of the crack, mm Ki, MFaVmm Deflection, mm
110 0,7485 21,48
120 0,7948 21,86
130 0,8384 22,26
140 0,8799 22,70
150 0,9198 23,18
Table № 7
Analytical results of stress intensity factor for different depth of the crack
Depth of the crack, mm Analytical solution ANSYS Difference in Ki, % Difference in deflection, %
Ki, MFav' nun Deflection, mm Ki, M Play mm Deflection, mm
50 0.3489 1.985 0.3510 1.989 0.6 0.2
80 0.5841 2.055 0.5806 2.058 0.6 0.2
120 0.7948 2.186 0.8002 2.191 0.7 0.2
180 1.0340 2.483 1.0409 2.491 0.7 0.3
The table 7 is performed comparing analytical solution results and ANSYS results for stress intensity factor and deflection of the beam. It is clear from the data, that differences between two ways of analysis no more than 0,7%.
J
The following plots illustrate relations between stress Intensity factor (Figure 11) and deflections (Figure 12) of the beam and a/w ratio.
Fig. 11. - Variation of Stress Intensity Factor (KI) with depth of the crack
Fig. 12. - Variation of deflection with depth of the crack
The results of the analysis of the pre-cracked beam under different values of load are listed in Figure 13.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
a/w ratio
Fig. 13. - Plot of stress intensity factor vs. crack depth of the beam
According to the results, the relationship between the stress intensity factor at the crack tip and the crack height is plotted under different loading levels (5 kPa, 10 kPa, 20 kPa), which indicates that the stress intensity factor of the crack tip in the plain concrete beam increase sharply along with the rising of load and crack height (Figure 13).
According the results it can be obtained that the most important role for the load-bearing capacity of the beam plays deep of the crack and shape of its tip. In the case of rounded tip the value of the stresses may be significantly decreased. However, the width of the crack does not cause changing of stress strain state of the beam. It means that the drilling a hole in the top of the crack with diameters 1-5 mm can decrease stresses in the area of crack and prevent its further growth.
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