CC BY
155
УДК 539.42
Effect of notch depth and radius on the critical fracture load of bainitic functionally graded steels under mixed mode I + II loading
H. Salavati, Y. Alizadeh, and F. Berto1
Amirkabir University of Technology, Tehran, 15916-34311, Iran 1 University of Padova, Vicenza, 36100, Italy
Functionally graded steels are produced from austenitic stainless steel and carbon steel by controlling the chemical distribution of chromium, nickel and carbon atoms at the remelting stage through electroslag remelting process. In the present paper, the strain-energy density criterion is employed to assess the critical load of rounded V-notched components made of functionally graded bainitic steel. A crack arrester configuration under mixed mode loading is considered. The flow (yield/ultimate) strength and fracture toughness are assumed to vary exponentially along the notch depth direction while the Young's modulus and the Poisson's ratio are assumed to be constant. The control volume, which is a reminiscent of Neuber's elementary structural volume, depends on the ultimate tensile strength out and the fracture toughness KIC in the case of brittle or quasi-brittle materials subjected to static loadings. Since, out and KIC are not constant along the notch depth, the control volume which can be obtained numerically as a function of the variation of these material properties through the specimen width. Different values of the notch root radius (from 0.2 to 2.0 mm) and notch depth (from 5 to 7 mm) are considered. The assessed critical fracture loads are in sound agreement with the experimental results.
Keywords: functionally graded steel, strain energy density, critical load, notch radius, notch depth
1. Introduction
Several local failure criteria have been proposed by different researchers to assess the fracture loads of notched components. Some criteria are directly based on point and averaged stress parameters along the provisional crack propagation line. The theory of critical distances enclose some approaches, all of them using a characteristic material length parameter when performing fracture assessments on any kind of stress risers. The origins and the basis of the theory of critical distances can be found in Refs. [1-5] while the formalization of the approach and the systematic successful application to notched components the reader can refer to Refs. [6-9].
The idea of stress averaging is linked to the fictitious notch rounding approach as first discussed and clearly showed in Ref. [1]. The influence of plane stress and plane strain conditions on the application of the fictitious notch rounding approach and in particular on the calculation of the multiaxiality factor s was highlighted in Refs. [10, 11]. In those references the importance of considering the real notch opening angle for fictitious notch rounding application has been underlined showing some applications to fatigue loading [12-16].
Another worth mentioning approach is based on the cohesive zone model. It was first proposed for concrete and later successfully extended to brittle or quasi-brittle failure of a large bulk of materials [17-22] and in particular polymethylmethacrylate specimens tested at room and low temperature [17, 21]. In those works both sharp and blunt U- and V-notches were considered.
A recent approach, based on the strain energy density, has been proposed and successfully applied for the fracture assessment of notched components [23, 24]. The strain-energy density approach is based on the evaluation of the averaged strain energy density over a control volume. The criterion has been applied to assess the fracture behavior of different materials under mixed mode loading and torsion loading [25-36]. As well shown in Refs. [37-48] the criterion can be applied to assess the fatigue strength of welded joints and notched components.
Due to its nature, the strain-energy density is able to include 3D effects through the plate thickness, different boundary conditions [49-55] and higher order terms [5658]. An important advantage of the approach with respect to the stress based criteria is the possibility to use coarse meshes for the direct evaluation of the strain-energy density in a control volume [59-62].
© Salavati H., Alizadeh Y., Berto F., 2014
A complete review of the volume-based strain energy density approach applied to V-notches and welded structures have been done in [63-65]. The approach is extended here to the fracture assessment of functionally graded materials subjected to mixed mode loading.
Functionally graded materials are advanced materials belonging to the family of engineering composites made of two or more constituent phases in which the composition, structure and/or specific properties vary continuously and smoothly in a preferred direction. This produces combinations of properties that could not be achieved by employing homogeneous materials composed of similar constituents. In this regard, well-known metal-ceramic functionally graded materials are usually used to enhance the properties of thermal-barrier systems. In fact, cracking or delamina-tion, often observed in conventional multilayer systems, can be avoided by employing functionally graded materials due to the smooth transition of the material properties inside the component [66].
Functionally graded steels are a new interesting research field which involves many researchers all around the world. These new materials allow more advantageous combinations of strength and ductility when compared to the traditional high strength steels. Functionally graded steels have been recently obtained from austenitic stainless steel and plain carbon steel by controlling the chemical distribution of chromium, nickel and carbon atoms at the remelting stage through the so-called electroslag remelting (ESR) process [67]. Studies on transformation characteristics of functionally graded steels produced via electroslag remelting have shown that by selecting appropriate arrangement and thickness of original electrodes made of ferritic and austenitic steels, composites with graded ferrite, and austenitic regions together with emerged bainitic and/or martensitic layers can be obtained:
(«C Y o) ),
(Yo a Yo) —YMY),
(ao Y oao Y o)-(ao Y o Y oao)"
ESR
ESR
■>(aßYMY), ■>(aßYßa).
In the above transformations ao and Y o are original ferritic and austenitic stainless steels in the primary electrode, a and y are ferritic and austenitic graded regions in the final composite and P and M are the bainitic and martensitic layers in the final composite.
In [68] the tensile behavior of functionally graded steels with different configurations was experimentally investigated and modeled by using the modified rule of mixture. In that work, variation of the yield strength in the graded region has been estimated by means of a linear expression between the yield strength gradient and the Vickers micro-hardness profile of the composite. In particular, Vickers microhardness profile of austenitic graded region of the aPY functionally graded steels has been modeled by using the strain gradient plasticity theory [69].
The same theory was used to model the tensile strength of functionally graded steels in Ref. [70]. The main advantage of this improved model with respect to the previous one [68] is that for determining the mechanical properties of the functionally graded steels, the microhardness of each layer (i.e. Vickers microhardness profile) is not necessary as input parameter.
Other important aspects are related to the characterization of functionally graded steels under hot working conditions. In Ref. [71], the flow stress of functionally graded steels under hot compression loading was assessed by applying constitutive equations in combination with the rule of mixture. In that paper, a theoretical model has been proposed to assess the flow stress of bainitic <xPy and martensitic yMy graded steels under hot compression conditions based on the Reuss model for the overall strains.
In Refs. [72-74] the Charpy impact energy of crack divider specimens was measured experimentally and modelled by means of two different methods. The obtained results showed that the Charpy impact energy of the specimens depended on the type and the volume fraction of constituent phases. Charpy impact energy of the functionally graded steel was considered to be the sum of the Charpy impact energy of constituent layers by means of the rule of mixtures. The first theoretical model correlates the Char-py impact energy of functionally graded steels to the Charpy impact energy of the individual layers through Vickers microhardness of the layers [72]. In the second model, the Charpy impact energy of all layers was related to the area under the stress-strain curve measured from plain tensile tests [73, 74].
In Refs. [75, 76], the effect of the distance between the notch tip and the position of the median phase on the Charpy impact energy has been investigated. The results have shown that when the notch apex is close to the median layer, the impact energy reaches its maximum value due to the incre-
Electrode —i
Initial electrode
Output : water
Copper mould ' Skin of slag
Molten slag V
ç Metal pool
Solidification direction Solidified ingot
Fig. 1. Electroslag remelting device used for specimens manufacturing
Chemical composition of original ferritic and austenitic steels, %
Table 1
C Ni Cr Mo Cu Si Mn S P
316L (y) 0.01 9.58 16.69 1.89 0.43 0.53 1.50 0.04 0.04
AISI 1020 (a ) 0.11 0.07 0.12 0.02 0.29 0.19 0.63 0.08 0.01
ment of the absorbed energy by plastic deformation ahead of the notch tip. On the other hand, when the notch apex is far from the median layer, the impact energy strongly decreases.
The brittle or quasi-brittle static failure of functionally graded steels by considering a crack arrester configuration was studied by Barati et al. [77]. The paper dealt with specimens made of yMy functionally graded steel weakened by U-notches. As remarked in that work, when the notch tip is placed in the transition region between austenite and martensite layers, the fracture load of functionally graded steel is greater than that of the corresponding homogeneous steel. On the other hand, when the notch is placed in the transition region between martensite and austenite layers, the fracture load of the homogeneous steel is higher than that of the functionally graded steel. In that study, the Young's modulus and the Poisson's ratio have been assumed to be constant, while the ultimate tensile strength and the fracture toughness KIC has been varied exponentially through the specimen width.
In the present investigation, functionally bainitic graded steels have been produced by means of electroslag remelting method. As expected, the flow (yield/ultimate) strength of each layer as well as the fracture toughness have been found to vary exponentially along the notch depth. From the produced functionally graded material some specimens have been machined with the aim to study the fracture behavior of the notched material under mixed mode I + II loading. Different notch tip radii have been considered as well as different notch depths. The influence of these geometrical parameters on the fracture load has been investigated. Finally the critical load has been assessed by employing the criterion based on the averaged value of the strain-energy density, just mentioned in the first part of the introduction. Due to the variability of the mechanical properties inside the material the control volume for the application have been obtained numerically.
2. Experimental procedures
To fabricate functionally graded steels, a miniature electroslag remelting apparatus has been employed. In typical electroslag remelting process, as illustrated in Fig. 1, the slag is used to heat the solid material and to produce a controlled molten metal bath from the initial electrode, which melts by means of electrical induction occurring between the electrode and the bottom part of the mould.
The slag used was a mixture of 50 % of CaO and 50 % of Al2O3. The functionally graded steels have been obtained from ingots made of a ferritic steel (AISI 1020) and an austenitic steel (AISI 316L). The diameter of the ingots was equal to 45 mm. The chemical composition of the two materials is summarised in Table 1. The mechanical properties of the single phases present in the employed functionally graded steels are summarized in Table 2.
The height of initial ferritic and austenitic slices was 150 and 210 mm, respectively. The two parts were joined by means of spot welding and were prepared for the re-melting stage and successively inserted vertically in the electroslag remelting furnace. The furnace contained a 70x70 mm2 squared copper mould. The remelting process was carried out under a constant power supply of 16 kVA. During the remelting process, the composition of the fer-rite/austenite interface changed. By so doing the amount of alloying elements reached a sufficient level that allowed, after cooling, to obtain the bainitic phase. In fact, the formation of the bainitic layer is usually obtained by diffusion of the alloying elements, such as nickel (Ni) and chromium (Cr) atoms, from the austenite layer to the ferrite layer and, vice versa, by the diffusion of carbon (C) atoms from the ferritic layer to the austenitic layer (Fig. 2, a). Careful studies of the microhardness profile of the gamma/alpha interface indicate the formation of an intermediate bainitic phase between the two pre-existing layers, as shown in Fig. 2, b.
After remelting, the composite ingots were forged at 980 °C and the specimens were cooled in air. The final
Table 2
Mechanical properties of single phase steels present in the considered functionally graded steels [68, 69]
Single phase Yield strength, MPa [68] Ultimate strength, MPa [68] KIC, MPa • m05 [69] Poisson's ratio v Elasticity modulus, GPa
Ferritic 245 425 45.72 0.33 207
Austenitic 200 480 107.77 0.33 207
Bainitic 1025 1125 82.08 0.33 207
Cr, Ni'^X^
Austenitic y \ Ferritic
region / ic region
/ iin •3 —
vh/
C
m
Fig. 2. Initial electrode before remelting (a), ingot after remelting (b)
ingots were 22 mm high. They were also polished and etched. The graded region in the median part of the ingots was 20 mm in height. The etching solution (Kalling) contained 5 g of copper chloride, 100 ml of hydro-chloric acid, 100 ml of water and 100 ml of ethanol. The microstructures of the transition region between ferrite and bainite as well as the new bainitic layer are shown in Fig. 3.
The final height of 18 mm was reached by grinding. Dealing with the bending tests of functionally graded steels, there are two possible configurations called crack divider and crack arrester (Fig. 4). In the crack divider configuration the plane which contains the notch tip is perpendicular to the layers while in the crack arrester configuration the plane which contains the notch tip is parallel to the layers. In the present work, the aPY specimens drawn from the ingots
were characterized by a length of 90 mm, a width of 18 mm and thickness of 9 mm, in agreement with the main standard ASTM E1820 [78] for the crack arrester configuration.
The load was applied normally to the interface layers at a distance b from the notch bisector line in order to obtain a mixed mode loading condition (Fig. 5). In the first part of the experimental activity, the notch radius and the notch opening angle were kept constant and only the notch depth was varied. The load-displacement curves were recorded and used to obtain the critical load. For each geometrical configuration three tests were repeated. In the second part of the experimental activity the notch radius has been varied from 0.2 to 2.0 mm keeping constant the notch depth.
0
m
Fig. 3. The microstructure of transition region from original ferrite to median bainite of a|Py functionally graded steel (a) and bainitic region of aPY functionally graded steel (b)
Fig. 4. Two configurations of bending tests of functionally graded steel: crack divider configuration (a) and crack arrester configuration (b)
Fig. 5. Geometry of the specimen under three-point bending in the crack arrester configuration. L = 90 mm, S = 72 mm, W = = 18 mm, 2a = 60°, b = 5 mm
3. Mechanical properties
Figure 6 shows the Vickers microhardness profile of the aPy composite. The thickness of the bainitic layer is approximately equal to 2 mm, as visible from the figure.
In order to model flow (yield/ultimate) strength of the considered aPy composite, the specimen has been divided into three homogeneous regions made of a, y, and P phases and two graded regions made of a and y phases (see Fig. 7). The thickness of each region has been obtained from the Vickers microhardness profile. The number of layers in the a region is ma and the number of layers in the y region is my.
The flow (yield/ultimate) strength of the constituent elements in the a region changes from the original a 0 steel on one side, to the bainitic layer on the other side. Similarly, the flow (yield/ultimate) strength of the constituent elements in the y region varies from the bainitic layer on one side to the original y 0 steel on the other side. Without entering too much in the details, it has been assumed in agreement with Ref. [68] that the flow stress of each layer varies proportional to the Vickers microhardness of that layer:
a) = Mp)-°fK) vH(a,)+ f ' VH(P) - VH(a 0) '
+ at(a o)VH(p)-qf(P)VH(ao) . = 1 VH(P) - VH(ao) ' '
1 ..., ma ,
(1)
Fig. 6. Vickers microhardness profile versus depth in aPy composite
a f( Y i) =
Gf(P)-af( Y o) -VH( Y') +
VH(P) - VH( Y o)
+ af(Yo)VH(P)-af(P)VH(Yo) . m (2)
+--~-, ' — 1,..., mY. (2)
VH(P) - VH(Yo) Y
In Eq. (1) and (2) af(P), af(a0) and af(y0) are the flow (yield/ultimate) stresses of the bainitic layer, of the original ferritic steel and of the original austenitic steel, respectively. VH(a') is the Vickers microhardness of each layer; VH(a0) and VH(y0) are the Vickers microhardness of a0 and y0 steels, respectively; and as said above, ma and mY are the number of elements/layers in which regions a and Y have been divided to describe the variation of the mechanical properties inside the material.
The fracture toughness KIC of the constituent elements in the a region varies from that of the original a 0 steel, on one side, to that of the bainitic layer, on the other side. Similarly, the fracture toughness KIC of the constituent elements in the Y region varies from that of the bainitic layer, on one side, to the original y0 steel, on the other side. Moreover, it is assumed that the fracture toughness of each element within the region of the composite obeys the exponential functions as reported below:
Kic/„)(Xi) — Ki,
-ln[(Kic)p/(Kic )a0 ]
MC( a0)
K IC( y)( xi) = K
"ln[(K ic)y „/( kIC)P ]
IC( P)
(3)
(4)
In Eqs. (3) and (4) xi is the position of each element/layer in the a or y regions, xa, Xp and xY are the positions of the original ferritic layer, of the original bainitic layer and the original austenitic steel, respectively, (KIC) , (Kic)p and (KIC)Y 0 are the fracture toughness corresponding to a 0, P and y0, respectively.
4. Control volume
In the strain-energy density approach applied under plane strain conditions to a homogeneous material the control radius Rc is a function of the fracture toughness KIC, the ultimate tensile strength aut, and Poisson's ratio v. It can be expressed according to the following expression:
R —
(1 + v)(5 -8 v)
4 n
K
IC
(5)
In homogeneous materials, Rc is constant. Accordingly to Refs. [24, 79] the control volume in V-notched specimens
Fig. 7. Dividing the aPY composite to different regions
'E
Crack initiation j
Fig. 8. Control volume for homogenous materials: mode I loading (a); mixed mode loading (b)
under mode I loading conditions is centered in relation to the notch bisector line (Fig. 8, a). Under mixed mode loading the critical volume is no longer centered on the notch tip, but rather on the point where the principal stress reaches its maximum value along the edge of the notch (Fig. 8, b), see Ref. [25].
However, in non-homogeneous materials, Rc varies from point to point due to the variation of the material mechanical properties. In a non-homogeneous medium with a smooth unidirectional variation of the mechanical properties along notch depth direction, the outer boundary of the control volume assumes the shape shown schematically in Fig. 9 (for mode I and mixed mode loading).
In the present work, a numerical subroutine has been employed to obtain the boundary of the control volume case by case, in agreement with the variation of the mechanical properties inside the material.
In the numerical approach, firstly the initial Rc (control radius along the direction of the provisional crack initiation point) has to be evaluated, KIC and aut vary from point to point, the ultimate tensile strength of each layer aut (xf) is obtained by Eq. (2) and the fracture toughness of each layer Kc( x) is obtained by Eq. (4). Therefore, the following equation has to be numerically solved to obtain x0 (see Fig. 10):
x0 - a + P = (Rc(initial) + P) cos 4 (6)
Then, Rc along the direction of the provisional crack propagation can be calculated as:
R
(1 + v)(5 -8 v)
c(initial)
4n
KIC( x0) ° ut(
\2
(7)
In a homogenous steel, the angle 0* (see Fig. 8) can be obtained analytically [20, 51]. In non-homogeneous steels,
Crack initiation angle
Fig. 9. Control volume for functionally graded steel materials: mode I loading (a); mixed mode loading (b)
Fig. 10. Notch depth and x-y Cartesian coordinate systems, x0 is the coordinate of the point where crack initiation occurs
it is not possible to use the same set of equations. Therefore, in the numerical subroutine, the angle 0 has been increased with a discrete step until the boundary of the control volume intersects the notch edge. In the present work, the outer boundary of the control volume has been divided in n1 +1 points where n1 is the integer part of the angle 0* and n2 +1 points where n2 is the integer part of the angle 02 (see Fig. 11). For 9<0< 0**
X' — a-p + [Rc(X') + ro]cos(0 + 9) + (p-r0)cos 9, y — [Rc(X') + r0]sin(9 + i) + (p - Tq) sin 9, for 0 <0<9
X' — a-p + [ Rc( xt) + ro]cos(9-0) + (p-ro)cos ^ yi —[ Rc( Xi) + ro] sin(9 - i) + (p- ro) sin 9,
(8)
(9)
a, MPa
■ 973 1 664 : 355 - 45 1-264
Maximum principal stress
Fig. 12. Principal stress contour lines for the case p = 1 mm, a = 5.5 mm, 2a = 60° and b = 5 mm
for 9<0<02
X — a -p + [ Rc (X') + ro]cos(-9 + 0) + (p- ro)cos 9,
yt — -[ Rc( Xi) + ro]sin i + (p-ro)sin 9, (10)
where Rc( x) is calculated as follows:
Rc( X ) =
(1 + v)(5 -8 v)
4 n
Kic( X )
aut( X)
\2
(11)
Subscript i is related to the ith layer in the graded region.
5. Critical load evaluation
In the present work, the averaged value of the strain-energy density over the control volume is applied to predict the critical load of the V-notched specimen under mixed mode loading as follows:
F W
j app — _app (1?)
Fc _c ' (12)
where Fapp is the applied load in the numerical model, _app is the averaged strain-energy density over the control volume related to Fapp, Fc is the critical load and Wc is the critical value of the strain-energy density. The critical value Wc has been evaluated accordingly to the following expression:
2
Wc = auL. c 2E
(13)
In Eq. (13) a ut is the ultimate tensile strength of the layer of material including the notch tip accordingly to Eq. (2)
= Rc + ^0
Fig. 11. Discretization of the control volume boundary
Fig. 13. Strain-energy density contour lines in the control volume for the case p = 1 mm, a = 5.5 mm, 2a = 60° and b = 5 mm
Fig. 14. Comparison between the finite element method (FEM) and experimental values of the critical load for different notch depths in the constant notch radius
Fig. 15. Comparison between the finite element method and experimental values of the critical load for different notch radii and a constant notch depth
and E is the Young's modulus. We must emphasize that the Young's modulus (E = 207 GPa) and the Poisson's ratio (Y= 0.33) have been assumed to be constant along the specimen width. In this research, finite element method has been utilized to obtain directly the strain-energy density.
The maximum stress occurring along the edge of the notch as well as the averaged strain-energy density was evaluated numerically by using ABAQUS 6.11. For each geometry, two models were created: the first was mainly oriented to the determination of the point where the maximum principal stress was located with respect to the notch bisector line (angle 9 in Fig. 11); in the second model the control volume, where the strain energy density was averaged, was placed along the edge in the position determined in the first model and determining the contour points by using Eqs. (8)-( 11). The analysis was carried out by means of eight-node elements under plane strain and linear-elastic hypotheses. The applied load was constant and equal to 5 kN. Figures 12 and 13 show the maximum principal stress and strain-energy density contour lines inside the control volume, near the notch tip. The strain energy density is approximately symmetric in relation to the ideal line normal to the edge and crossing the point of the maximum principal stress.
Figure 14 shows the comparison between the theoretical and experimental values of the critical loads for different values of the notch depth and keeping constant the notch
radius. In all the considered cases the notch tip was placed in the transition region from the original ferritic layer to the median bainitic layer (aP graded region) of the aPY functionally graded steel. As this figure shows, there is a good agreement between the assessed critical loads based on strain-energy density and the experimental results. This agreement is not influenced by the variation of the notch depth.
The main results from the finite element analysis are reported in Table 3 as well as the direct comparison with the results from the tests carried out in the present research.
The comparison between the predicted and experimental critical loads for different notch radii, keeping constant the notch depth, is shown in Fig. 15. In this case, since the notch depth is constant, the ultimate strength and the fracture toughness do not vary as well as the critical value of the strain-energy density. From the figure it is well visible that the critical load slightly increases with increasing the notch radius but the variation is very limited and the sensitivity to the notch radius is found to be almost negligible.
The details of the calculation by varying the notch radius (for a constant value of the notch depth) are also summarized in Table 4.
6. Conclusions
In the present work, the averaged value of the strain-energy density over a control volume has been used to
Table 3
Strain-energy density and critical load in V-notched aPY functionally graded steel for different notch depth in the constant notch radius under mixed mode loading. p = 1 mm, F = 5 kN, E = 207 GPa
aut, MPa Wcr, MJ/m3 WFEM, MJ/m3 Critical load Critical load Percent
a, mm (FEM), kN (experiment), kN of error, %
5.0 515.326 0.643 0.136001104 11.15 11.2 0.44
5.5 627.330 0.950 0.161932145 12.36 12.5 1.32
6.0 762.132 1.403 0.195310321 13.50 13.7 1.48
6.5 925.904 2.070 0.246518605 14.68 14.7 0.13
*FEM — finite element method
Table 4
Strain-energy density and critical load in V-notched a|Py functionally graded steel for different notch radius in the constant notch depth under mixed mode loading. a = 5.5 mm, F = 5 kN, aut = 627.33 MPa, E = 207 GPa
p, mm Wcr, MJ/m3 WpEM, MJ/m3 Critical assessed load (FEM), kN Critical load (experiment), kN Percent of error, %
0.2 0.99 0.16505 12.24 12.25 0.08
0.4 0.99 0.16351 12.29 — —
0.6 0.99 0.16196 12.35 — —
0.8 0.99 0.16040 12.41 12.44 0.24
1.0 0.99 0.15885 12.47 12.54 0.56
1.2 0.99 0.15733 12.53 12.59 0.48
1.4 0.99 0.15588 12.59 12.63 0.32
1.6 0.99 0.15454 12.64 — —
1.8 0.99 0.15336 12.69 — —
2.0 0.99 0.15242 12.73 12.75 0.16
predict the critical loads of V-notched specimens made of functionally graded steels under mixed mode loading. The main findings of the present work can be summarised as follows.
The ultimate stress of each layer of the considered aPy functionally graded steel has been obtained by using the mechanism-based strain gradient plasticity (MSG) theory while the fracture toughness has been considered to vary exponentially along the specimen width.
The approach based on the strain-energy density evaluated in a control volume previously used for homogeneous materials has been successfully extended to functionally graded steels. The outer contour of the volume has been determined numerically.
The average deviation between the theoretical and the experimental values in terms of the critical loads changing the notch depth and keeping constant the notch radius has been found to be very limited (0.8 %).
The average deviation between the theoretical and the experimental values in terms of the critical loads changing the notch radius and keeping constant the notch depth has been also found to be very limited (0.84 %).
Further studies are necessary to investigate more in detail the influence on the critical loads of the geometrical parameters as well as for extending the present method to different functionally graded steels.
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Salavati Hadi, Graduate Student, Amirkabir University of Technology, Tehran, Iran, hadi_salavati@aut.ac.ir Alizadeh Y, Amirkabir University of Technology, Tehran, Iran, alizadeh@aut.ac.ir Berto Filippo, Prof., University of Padova, Italy, berto@gest.unipd.it
