Simulation of elliptical shaped crack propagation in the zone of steel bar welded joint and lay-in panel of the top column node point of the sports facility

1. Task setting

The top column node of a longspan sports facility is a welded joint of a steel bar (main load-bearing element), a lay-in panel, side ribs and a foundation. The column node is produced from constructional steel C345 GOST 27772088, welding chords amount to 16...20mm.

The node is fixed beneath the foundation and loaded with a bending moment of 9.44 t/cm = 926 N/m (non-nominal near-critical loading transmitted to the lay-in panel as a result of the project deficiency).
This reseach analyses the operation of this node and its brittle fracture resistance by microspall in conditions of a non-nominal loading and in the presence of the initial defect.

Fracturing of this type is very dangerous as it occurs abruptly and develops at high speed without any remarkable macroplastic deformation.

However, the disastrous stage of an uncontrolled high-speed main crack propagation in a way of brittle fracture is preceded by a stage of a relatively slow 'viscous' crack propagation starting from the initial defect and ending with the critical point (when the brittle fracture by rupture occurs). Thus, the dangerous situation can be detected and prevented in due time given the corresponding building monitoring. This 'viscous' crack propagation occurs as a result of a cyclic load that is described by a cycle assimilation factor in case small cracks and damages occur in the crack front area by every peak loading and concequently the maximum crack opening.

In order to estimate the crack strength of the node, its model should contain the initial defect - a damage in the form of an external elliptical shaped crack which position and surface orientation depend on the maximum principal tensile stress pattern.

This analysis is aimed to estimate the rate of growth of such a crack, the maximum safe crack depth, the time of the crack growth until distruction (the 'life cycle' of the node with a developint defect) and the spread of these parameters.

The task solution included 3 stages.

1st stage. The finite element model of the node without cracks has been examined with the aim to determine potentially dangerous zones where they are likely to appear and propagate. In order to get a detailed view of the stress-strain state, the model includes the welded joints shape and the contact interation in the zones 'steel bar - lay-in panel' and 'steel bar - side ribs'. The stress-strain state of the model under steady loads transmitted through the welded joints and contact elements has been determined, and the zone with the maximum principal tensile stresses . As expected, they found to be localised in the welded joint area, namely in the welded joint 'steel bar - lay-in panel' (see fig. 1). Subsequently, the model of an elliptical marginal crack has been inserted into this zone in a way that the crack surface is normal to the principal stresses vectors .

2nd stage. The chain of finite element models of the node has been analysed that differ from each other only by the crack width. A series of computations according to the finite elements methods have determined the strain-stress state of model under steady loads, and the maximum stress intensity factor has been computed in its front area according to the method of movement approximation of crack edges.

3rd stage. On the basis of the stress intensity factor in the front area of the crack as well as of the crack strength of the material and loading characteristics, the following factors have been estimated:

• crack growth rate,
• maximum safe crack depth,
• time of crack propagation until distruction,
• spread of the factors mentioned.

The strain-stress state and the stress intensity factor have been computed according to the method of finite elements with the help of the software ANSYS 11.0 SP1. The model of the half part of the top column node has been analysed, and the corresponding boundary conditions have been written on the plane of symmetry. The crack is located in the zone of the welded joint 'steel bar - lay-in panel', has an elliptical form, and its surface is normal to the vectors of the principal tensile stresses. The general view of a geometric model of one of the variants (crack width = 10mm) is shown in the figures 2...3.

Fig. 2. The geometric model of an elliptical shaped crack with crack width = 10mm in the zone of the steel bar welded joint and the lay-in panel.

The finite element grid has generally a regular strusture and contains mainly elements of the first order SOLID45 with a number of contact elements CONTA175 and TARGE170 of the type 'node - surface' (51359 nodes and 44645 elements with the contact, 52660 nodes and 47487 elements without the contact). In the crack area, more accurate elements of the second order SOLID95 have been used. In the front zone of the crack a concentration of the FE grid hase been performed, and singular elements have been build. The SOLID95 elements ringing the crack front have been modified to singular ones by a shift of middle nodes by a quarter of the rib length with the aim of implementation of the form function with the root feature. The final model with contact elements and the inserted crack includes 92365 nodes and 65713 elements. Итоговая модель с контактными элементами и «вписанной» трещиной насчитывает 92365 узлов и 65713 элементов. The general view of the finie element model of one of the variants (crack width = 10mm) is shown in the figures 3...4.

Fig. 4. The finite element model of an elliptical shaped crack with crack width = 10mm in the zone of the steel bar welded joint and the lay-in panel.

Fig. 5. The finite element model of an elliptical shaped crack with crack width = 10mm (closeup picture).

The nonlinear contact static problem has been solved without regard to plastic deformation in the crack front zone. The view of the strain-stress state for the same variant (crack depth = 10mm) is shown in the figures 6...7. The process of opening of the crack is shown in the figure 8.

The stress intensity factor has been calculated under the assumption of the fact that the stress state is near-flat next to the side face of the node.
The mechanical characteristics of the material (construction steel С345 GOST 27772-88): Young's modulus Pa, Poisson ratio , friction factor in contact pairs .

2. Determining the parameteres of the simulation model

In order to simulate the endurance crack growth under the action of a cyclic load on the top columns node, the relation of the crack growth rate in accordance with the Paris' law has been used:
, m/cycl. (1)

l for crack length (depth), N for load cycles number proportional to the effective frequency of the variable loading . for the range of the stress intensity factor, for the cycle assymetry factor, . The stress intensity factor is calculated in accordance with the method of movement approximation of crack edges with the help of the finite element software ANSYS 11.0 SP1.

The values of are observed quantities of the crack strength of the material which are for C435: , the exponent , cyclic crack resistance , crack resistance .

Equation (1) is integrated on the crack length from initial to critical in order to calculate the number of cycles before distruction. The clitical length (depth) of the crack in the moment of total distruction is determined by the moment when the stress intensity factor achieves the cyclic crack resistance. That is, the critical crack depth can be calculated by the following equation:
or. (2)
In integration on the number of cycles is performed from null (because the initial crack is already present) to the number of cycles high enough to the total destruction, when the initial crack width achieves the critical one in accordance with the following equation:

(3)

Results of the mathematical simulation are described in the following articles.

3. Calculating the 'lifetime' of the node
The calculator of the 'lifetime' of the node has been created to estimate the influence of different parameters on the final period of the crack growth to the critical debth in a quick way. The calculator is an MS Excel file including the calculated factors of strass intensities where numerical integration of the correspondence is performed (3).

In order to calculate the influence of such factors as: exponent , parameter , cycle asymmetry factor , effective loading frequency , the control parameters are located on the corresponding sheet as shown in table 1.

A substitution of different values results in an automatic recalculation of the crack growth time and a reconstruction of the graphs as shown in figure 9.

An impact of control parameters and their range:

• effective loading frequency f_eff: a parameter of loading conditions; the smaller - the slower grows the crack;
• Paris's parameter C: material capability; the smaller - the slower grows the crack;
• cycle asymmetry factor R: a parameter of loading conditions; varies through the range 0,92...0,95; the more - the slower grows the crack;
• exponent n: material capability; varies through the range 3,12...3,26; the smaller - the slower grows the crack;
• threshhold value KIN Kth: material capability; varies through the range 5...8 ; the more - the slower grows the crack;
• threshhold value KIN (cyclic crack resistance) Kmax: material capability; taken as 158 ; the more - the slower grows the crack.

Fig. 9. The correspondence of the crack growth rate and its depth and the correspondence of the crack depth and time.

The correspondences of parameter variations are shown in fig. 10...12.

Fig. 11. The correspondence of the crack depth and time in case of a variation of cyclus asymmetry factor

• The operation of the top column node of the sports facility in non-nominal mode (as a result of the project deficiency) and in respect of the initial defect in the welded joint zone has been analyzed.
• The calculation is based on a numerical determination of stresses intensity facor in accordance with the method of movement approximation of crack edges with the help of the finite element software ANSYS.
• Assumed this, the elliptical width in the welded joint zone of the steel bar and the lay-in panel with the with of 1 mm is dangerous as the obtained stress intensity factor exceeds the threshhold one, which will result in an increasing growth of the crack.
• The noncritical crack depth (resulting in a brittle distruction) amounts to 14 mm under such conditions.
• Based on art. 3...4, the estimation of the 'lifetime' of the node corresponds with the growth time of the crack from the depth of 1 mm to the critical depth of 14 mm.
• Assuming the most probable charasteristics of the crack growth resistance (exponent and Paris' factor ) and under the most probable loading conditions (cycle asymmetry factor and effective loading frequency Hz), the 'lifetime' of the node amounts to approximately 35 months. See fig. 9.
• Nondissipative assumptions (exponent , cycle asymmetry factor ) can reduce the 'lifetime' of the node significantly: up to 5 months. See fig. 10...12.
• The greatest impact on the crack growth time have the following factors (in decreasing order):
• loading capability: cycle asymmetry factor ;
• material capability: exponent and Paris' factor ;
• loading capability: effective loading frequency .
• The crack growth rate (minimal 'lifetime' amounts to 5 months) allows to detect the crack by means of monitoring before the moment of destruction (the critical depth of 14 mm) and take measures to prevent the emergency situation.