This section provides a parameter sensitivity study.
4.8.1 Strain Hardening exponent
Figure 4.34 shows the results of the strain hardening exponent study. The fracture strain used for calibration is 0.17 and the fracture energy is 48 N/mm. The ultimate load for the simulation with strain hardening exponent 0.06 is 70,65kN, for exponent 0.07: 71,82kN, for exponent 0.09: 71,96kN and for exponent 0.15 the ultimate load is 72,38kN. This is deviation of the experiment of respectively 3.4%, 5.1%, 5,3% and 6,0%. There is almost no deviation between the displacement at fracture for the four hardening exponents.
Figure 4.34; Load – Displacement curve, strain hardening exponent study
To see the influence of the strain hardening exponent on element level the stress - strain curves of two elements are plotted. For the inner element (Figure 4.35) the stress – strain graph is plotted in Figure 4.37. For the outer element (Figure 4.36) the stress – strain graph is plotted in Figure 4.38.
0 10 20 30 40 50 60 70
0 5 10 15 20 25 30 35
Load (kN)
Displacement (mm)
Sample 10 Exponent = 0,06 Exponent = 0,07 Exponent = 0,09 Exponent = 0,15
Figure 4.38 shows the stress-strain curve for the outer element. Here the strains at damage initiation for exponent 0.06 and 0.07 are not significantly different, for exponents 0.09 and 0.15 there a decrease in strain at damage initiation for increasing hardening exponent. The strain at final failure shows no significant difference.
Figure 4.37; Stress – Strain graph Inner Element
Figure 4.38; Stress – Strain graph Outer Element 0
There is a difference in ultimate fracture strain (!̅;6Ñ) between the inner element and the outer element. This is having an effect on the slope of the damage evolution. The two elements fail in different ways. The inner element fails due to shear fracture. And the outer element fails due to void formations as discussed in section 2.2. For the different failure modes different stress triaxialities are valid. In Figure 4.41 the stress triaxiality is plotted to the displacement. The outer element has, for the main part of the simulation, a stress triaxiality of 0.75. Whereas the inner element has a stress triaxiality of -0.57. After approximately 22mm the triaxiality of the outer element increases till around 8.5 and then decreases again till a value of -8.5. Because the element is already completely damaged (Figure 4.40) there is no further influence of the element on the ultimate load of the simulation.
Figure 4.39; Triaxiality - displacement graph -10
4.8.2 Fracture Strain
Together with the strain hardening exponent, the fracture strain determines the shape of the stress triaxiality to fracture strain graph. The strain listed in the legend of the figures in this section refers to the fracture strain used to calibrate the fracture locus. Figure 4.41 shows the plot of the global load – displacement curve for the simulations with a fracture energy of 48 N/mm and a hardening exponent of 0.07. By changing the fracture strain the slope of the simulation, the failure load and failure strain change. A higher fracture strain results in a higher ultimate load. The fracture strain listed in section 4.2 is calculated with Eurocode 9. The fracture strain used here is 0.17 and is lower than the calculated fracture strain of 0.22.
Figure 4.41; Load – Displacement curve, fracture strain study
Figure 4.42 depicts the stress – strain graph of the inner element. The damage initiation point is different for the simulations. By increasing the fracture strain used for calibration an increase is measured for the strain at damage initiation.
For the outer element in Figure 4.43 there is almost no distinction between the strain at damage initiation for the simulations. Also, almost no distinction in the ultimate strain.
0 10 20 30 40 50 60 70
0 5 10 15 20 25 30 35
Load (kN)
Displacement (mm)
Sample 10 Strain = 0,18 Strain = 0,17 Strain = 0,16
Figure 4.42; Stress – Strain graph Inner Element 0
50 100 150 200 250
0 0,1 0,2 0,3 0,4 0,5
Stress (N/mm2)
Strain
Strain = 0,18 Strain = 0,17 Strain = 0,16
0 50 100 150 200 250
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7
Stress (N/mm2)
Strain
Strain = 0,18 Strain = 0,17 Strain = 0,16
4.8.3 Fracture energy
As discussed in section 4.3.2 the fracture energy determines the failure displacement of the elements. Figure 4.44 shows the fracture energy study. By increasing the fracture energy from 46 till 50 N/mm the ultimate load of the specimen increases as well as the displacement at fracture. For the investigated fracture energies, a linear relationship results between the fracture energy and the ultimate load (or displacement at fracture). The fracture energy also has an influence of the slope of the last part of the simulation. By increasing the fracture energy, the slope of the simulation becomes steeper.
Figure 4.44; Load – Displacement curve, fracture energy study. (Energy in N/mm)
In Figure 4.45 the stress – strain relationship is plotted for the different simulations for the inner element. The strain at damage initiation is equal for all simulations. The damage evolution branch of the graph as well as the ultimate fracture strain are different for the simulations. The final failure strain and the corresponding stress are scattered. This could be caused by the difference in failure mode, shear fracture for the inner element and failure due to void formation for the outer element.
Figure 4.46 depicts the stress – strain graph for the outer element. The strain at damage initiation point are all equal. The ultimate strain at fracture shows a linear relationship between the fracture energy and the ultimate failure strain. By increasing the fracture energy there is an increase in the ultimate fracture strain.
0
Figure 4.45; Stress – Strain graph Inner Element. (Energy in N/mm) 0
50 100 150 200 250
0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45 0,5
Stress (N/mm2)
Strain
Energy = 46 Energy = 48 Energy = 50
0 50 100 150 200 250
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7
Stress (N/mm2)
Strain
Energy = 46 Energy = 48 Energy = 50
4.8.4 HAZ Material
The stress – strain relationship for the HAZ obtained by Eurocode 9 in section 4.2 could be a false representation of the actual stress – strain relationship of the HAZ. An additional study is performed with the 0.2% proof stress and ultimate stress reduced with an additional 5%. The 0.2% proof stress becomes 48% of the parent metal and the ultimate tensile stress 60% of the parent metal (Table 4.4).
Table 4.4; Material Paramaters Additional Simulation
Alloy 6005A Reduction Alloy 6005A HAZ
vw.x 259.46 0/yyQ ê,.Q 0.48 =,.Q 124.54 0/yyQ simulation is 68.78 kN a deviation of the experimental failure load of 0.7%. The displacement at failure has a deviation of 0.5%. The load – displacement curve of the simulation has an almost perfect fit on the experimental result, at the transition from elastic to plastic deformation a little deviation is noticed, between a displacement of 0 and 5mm. After failure a big drop of the load is measured in the simulations and in the experiment. At a load of around 40kN the displacement increases again with an almost constant force. The perfect fit of this simulation can not only be allocated to the change of the strength parameters of the HAZ. As mentioned in the paragraphs above the strain hardening exponent, fracture strain at the calibration point and the fracture energy also have an influence on the simulation. For this simulation the strain hardening exponent had a value of 0.07, the fracture strain at calibration point was 0.22 and the fracture energy is 52 N/mm.
Figure 4.47; Load – Displacement curve 0
With the parameter of the HAZ the dog bone specimens of section 3.3 can be modeled as described in section 4.7.6. Figure 4.48 depicts the stress – strain curve of the dog bone specimens and the simulations. The simulations are made with an HAZ of 10mm and 20mm as mentioned in the legend. Both simulations show an 0.2% proof stress higher than both experimental results. The ultimate stress of the simulations is lower than experiment 12.2 but higher then experiment 12.1. The average values of the experiment and the input for the ultimate stress has a deviation of 1% (Table 4.5).
Table 4.5; Material parameters HAZ
Experimental Input Alloy 6005A HAZ Simulation
vw.x 145.71 0/yyQ =,.Q 124.54 0/yyQ
vz 163.67 0/yyQ =" 165.38 0/yyQ
{ 68702 0/yyQ Å 67463 0/yyQ
|z 0.09 !" 0.22
} 33 # 17
Figure 4.48; Stress – Strain curve 0
0,00 0,02 0,04 0,06 0,08 0,10 0,12 0,14
Stress (N/mm2)
5 Design equation
The analytical solution for punching shear failure is derived from the punching shear formula used in the Eurocode for steel [3]. Eq. 5.1 is the design equation for steel hollow section joints. Here =É, is the yield stress of steel and I, is the thickness of the chord. The equation assumes that the effective parts are the height of the brace (h1) and a part of the width of the brace (b1). The effective part of the width is denoted as K7,6. K7,6 is depended on the slenderness ratio of the chord YK,a [. The calibration factor (cI, f) for steel is determined experimentally by Wardenier [14] and has a value of 10.
0ë,í~77Ñ=
The yield stress of steel is replaced by the ultimate stress of the HAZ (=",$*ñ). Furthermore, the formula has a calibration factor cf. Replacing this factor for a value for aluminum gives an analytical solution for the punching shear strength of a hollow section joint (Eq. 5.2).
0ë,OÑ"óò8"ó=
The formula takes into account the effective parts of the brace. The height of the brace (h1) is considered entirely effective, and the width (b1) of the brace is partly effective. Factor cf is calibrated using the ultimate load of sample 10. This results in a factor cf equal to 17.85.
To get a suitable value for cf a parameter study is performed. Four types of X-joint are simulated, as shown in Table 5.1. The ratio between the width of the chord and thickness of the chord have values from 20 till 30. For all simulations the calibration factor is calculated and the average factor for cf is 11.90 with a standard deviation of 1.87. This value for cf is used to calculate the forces for ôöõ,vúõùzûü in Table 5.1. On 3 out of 7 cases the formula predicts the ultimate load lower than the ultimate load of the simulation. The maximum deviation between the formula and the simulation is 5.18%
Table 5.1; Parameter study
Brace [mm] Chord [mm]
The material used in this research is Alloy 6005A – T6. The Ramberg Osgood number for the HAZ is 18. To see what the influence is of a higher ultimate tensile stress another set of simulations are performed with the material parameters of Table 5.2. The ultimate tensile stress of the parent metal is 1.5 times higher than for Alloy 6005A-T6. The 0.2% proof stress is not changed. The Ramberg Osgood number for the HAZ is reduced here till 7.
The parameters for the strain hardening exponent, strain at calibration point and the fracture energy are not changed.
Table 5.2; Material Parameters; higher ultimate stress
Parent Metal Reduction HAZ
vw.x 259.46 0/yyQ Çw.x 0.53 vw.x 137.51 0/yyQ
vz 413.45 0/yyQ Çz 0.65 vz 268.74 0/yyQ
{ 67463 0/yyQ { 67463 0/yyQ
|z 0.12 |z 0.22
} 9 } 7
Table 5.3 shows the results of the parameter study with a higher ultimate stress. Looking at the average values of the Cf factor calculated with f0,HAZ the value is 65% higher than mentioned earlier. The difference between the Cf factor calculated with fu,HAZ is 35%. The correlation between the calibration values calculated with the ultimate stress is better than a calibration factor calculated with the 0.2% proof stress. The values for ôöõ,vúõùzûü here are calculated with an average Cf value of 7.74. In 5 out of 7 cases the formula predicts a lower value for the simulation.
Table 5.3; Parameter study; higher ultimate stress Brace [mm] Chord [mm]
6 Conclusions and recommendations
This research investigates the punching shear failure mode of welded joints between rectangular hollow sections.
Both experiments and numerical (finite element) simulations are performed. Special emphasis is devoted to the capture of fracture in the finite element simulations. Finally, a design equation is provided for the punching shear failure mode.
The damage model used in this research is progressive damage and failure. This model is implemented in the material behavior of the elements used in the finite element simulation. First the undamaged stress – strain relationship is used until a damage initiation point. The damage initiation criteria is based on the fracture strain – stress triaxiality space. At different stress triaxialities different fracture strains are valid. Various models exist to capture the relationship between stress triaxiality and fracture strain [8]. One of these models is the Maximum Shear (MS) – stress criterion. This model is easy to apply because it uses one calibration point. In addition, Bao and Wierzbicki [8] demonstrate that the model is reasonably accurate. For these reasons, this model is applied in the current research. The calibration of the MS – stress criterion is based on dog bone specimens. The fracture strain used for this calibration needs further research. The fracture strain could be the strain at the ultimate tensile stress (start of necking) or the strain at ultimate fracture of the dog bone specimen. For the calibration of the MS – stress criterion the hardening exponent of the material is required. This can be obtained by the true stress – strain relationship of the HAZ. After damage initiation the stress – strain relationship of the material does no longer accurately represent the behavior. Therefore, a stress – displacement approach is used. The ultimate displacement is determined by the fracture energy. To get a representative value for the fracture energy further research needs to be conducted.
The Punching Shear failure mode occurred on both experimental specimens. The load – displacement graph shows an almost perfect correlation between the two experiments. After the elastic part the specimens show a considerable strengthening behavior with large deflections before failure. At failure there is a clear deviation between the two specimens. Both specimens show crack initiation from the leg of the weld with a snap thru mechanism at the ultimate load. The specimens show a similar strain ratio between the height of the brace and the width of the brace. The width of the brace is activated, after large deflections of the chord face already occurred. Some parts of the welding were insufficient and during testing the weld detached from the chord face.
It is advised to start the welding at the width of the brace, to ensure a good weld along the height of the brace.
After calibration of the strain hardening exponent, fracture strain and fracture energy. The force – displacement curve can be predicted reasonably well with the finite element simulations. In addition, the strains at a local level do also correspond with the experiments. As well as the location and the shape of the crack. With the same calibration values the stress – strain curve of a dog bone specimens of the HAZ can be predicted with finite element analyses. Due to these reasons it is likely that the used approach and calibration parameters are valid.
The design equation presented in this research gives a reasonable estimation of the ultimate load fur punching shear failure. This is supported with a small parameter study with different slenderness ratios of the chord face.
Further research needs to be performed to get a reliable design equation. Different parameters need to be varied as well as the aluminum alloy.
With the method presented for the capture of fracture in the finite element model, a model can be build that can predict all failure modes. Once that model is completed and validated a large parameter study can be conducted into all failure modes resulting in design equations for aluminum hollow section joints.
7 References
[1] CEN, Eurocode 9; NEN-EN-1999-1-1, Brussel, 2011.
[2] A. Association, Aluminum design manual (ADM), 2015.
[3] CEN, Eurocode 3; NEN-EN-1993-1-8; Design of steel structues; Part 1-8; Joints, Brussel, 2011.
[4] I. F. Soetens, "Welded connections in aluminium alloy structures," Heron, vol. 32, no. 1, 1987.
[5] S. d. Jongh, "The reistance of welded T and Y-joints of rectangular hollow sections in aluminium,"
Technische Universiteit Eindhoven, Eindhoven, 2016.
[6] Y. Bao and T. Wierzbicki, "On fracture locus in the equivalent strain and stress triaxiality space,"
International Journal of Mechanical Sciences, vol. 46, pp. 81-98, February 2004.
[7] B. Y, "Prediction of ductile crack formation in uncracked bodies," Cambridge, 2003.
[8] T. Wierzbicki, Y. Bao, Y.-W. Lee and Y. Bai, "Calibration and evaluation of seven fracture models,"
International Journal of Mechanical Sciences, vol. 47, pp. 719-743, 2005.
[9] ISO, Metallic materials - Tensile Testing -, 2009.
[10] W. Hosford and R. Cadell, "Metal Forming," in Mechanics and Metallurgy, Cambridge, Cambridge University Press, 2007, pp. 34-36.
[11] A. Hillerborg, M. Modeer and P. Petersson, "Analysis of Crack Formation and Crack Growth in Concrete by means of Fracture Mechanics and Finite Elements," Cement and Concrete Research, vol. 6, pp. 773-782, 1976.
[12] J. Liu, Y. Bai and C. Xu, "Evaluation of Ductile Fracture Models in Finite Element Simulation of Metal Cutting Processes," Journal of Manufacturing Science and Engineering, vol. 136, 2014.
[13] M. M. R. B. V. Aleo, Development of an equation for aluminum alloys to determine the width of the zone affected by welding.
[14] J. Wardenier, "Hollow Section Joints," Delft University Press, Delft, 1982.
[15] J. Jiang, Artist, Schematic of the pure-mode bilinear traction–separation law used to model the cohesive zone interface. [Art]. 2016.
[16] Y. Bao, "Prediction of ductile crack formation in uncracked bodies," Cambridge, 2003.
A. Experimental Data
During this research experiments are performed on three different failure modes in four test series. The failure mode weld failure is only tested with a chord and a brace plate. Failure mode weld failure is only tested on a full x-joint specimen. Punching shear failure will be tested on a chord with a brace plate and on a full x-joint specimen. The test data for the remaining three specimens is discussed below.
A.1 Weld failure brace plate
Four specimens are available for testing. Two specimens are tested for validation of the finite element model.
If, for validation, more parameters are needed as measured with the first two specimens, this can be done with the remaining two specimens.
A.1.1 Geometry
The specimens are checked for dimensional tolerances. Figure A.1 and Figure A.2 show the dimensions as measured before testing of the specimens.
Figure A.1; Sample 01 [mm] Figure A.2; Sample 01 [mm]
A.1.2 Loading
The loading of the specimens is applied at the top of the brace plate. The top and bottom are clamped in the testing machine. The test is deformation controlled and has a speed of 0.6 mm/min.
A.1.3 Measurement
Measurements are taken in two different ways. The first is for the global force-displacement graph. The load is measured by a load cell of 250kN at the top of the testing machine. This load cell is calibrated regularly in the laboratory. The global displacement is measured on the brace plates and is 400mm apart (Figure A.3). To measure this displacement a special rectangular device is used that bypasses the chord. The device is fixed at the top of the brace and a LVDT measures the displacement at the bottom (Figure A.4).
The second measurement is taken by a DIC (Digital Image Correlation) camera. The DIC camera is placed horizontal in front of the specimen (Figure A.5) looking at the bottom weld of the specimen at the front side.
With the video footage of the DIC camera strains can be measured after the experiment.
Figure A.3; Measurement device [mm] Figure A.4; Measurement device in practice
A.1.4 Setup
Figure A.6 and Figure A.7 show the test setup. In front of the specimen is the DIC camera and a normal camera taking pictures of the top of the specimen. At the back side of the test setup the camera is filming the test. Two devices with LVDT’s are attached measuring the displacement at the front and at the back side of the
specimen.
Figure A.5; DIC camera
A.1.5 Results Sample 01
Figure A.8 depicts the load – displacement graph for sample 01. The maximum load of the specimen is 164,24 kN at a displacement of 2,08 mm. After the initial crack, the crack develops over the width of the weld until final failure at a displacement of 6.20mm.
Figure A.9 shows footage shot with the DIC camera at a displacement of 2 mm. Here the local strains are measured. The red parts of the image are the parts with large positive strains and the purple parts with the larges negative strains. At the top of the image the weld is clearly visible.
With the footage of the DIC software different measurements are taken to see what the strains are at the sides of the brace plate and at the center. In order to keep data compatible with strain gauges which can be used with later studies the measurements here are also taken with virtual strain gauges. Figure A.10 shows the placement of the strain gauges on the specimen, the gauges are placed a bit below the weld. Here traditional strain gauges can be placed in future research.
Figure A.8; Load – Displacement graph
Figure A.9; DIC camera Footage 0
Figure A.11 shows the load strain graph of the strain gauges. The strain gauges at the side of the weld both
Figure A.11 shows the load strain graph of the strain gauges. The strain gauges at the side of the weld both