ANALYSIS OF TAPERED HAUNCHED CONNECTIONS

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					FINITE ELEMENT ANALYSIS OF TAPERED-HAUNCHED CONNECTIONS




                        E. S. B. Machaly1, S. S. Safar2 and A.E. Ettaf3




ABSTRACT:

Corner connections are crucial parts for the safety and adequate performance of steel
frames. Extensive studies were made on the performance of square connections
(without haunches) or those equipped with short triangular haunches. However, little
was mentioned in literature or codes of practice about the behavior of tapered-
haunched connections. This was in spite of their extensive use in medium-to-large
span steel frames to develop more economic design and aesthetically pleasing views.
In this paper, the behavior of tapered-haunched connections was investigated using the
finite element method. The general-purpose finite element program, ANSYS was
utilized in the analysis incorporating material nonlinearity and geometric
imperfections to capture the interaction of yielding and buckling on the different
failure modes of such connections. The magnitude of geometric imperfections was
based on the allowable limit stipulated by the Egyptian Code of practice. Flow of
stresses, yielding zones at failure, distribution of forces on stiffeners and carrying load
capacity were determined analytically for a connection proportioned by the
conventional design method. It was noticed that the location of the plastic zone as well
as the connection capacity were mainly affected by web slenderness and stiffeners
configuration. Moreover, it was concluded that the true forces in the stiffeners were
overestimated by the conventional method.



KEYWORDS:

Tapered Connections, Haunched Connections, Beam-to-Column Corner Connections,
Geometric Imperfections, Material Non-linearity, Finite Element Analysis.
_____________________________________________________________________
1.   Professor of Steel Structures and Bridges, Faculty of Engineering, Cairo University.
2.   Associate Professor of Steel Structures and Bridges, Faculty of Engineering, Cairo University.
3.   Teaching Assistant, Structural Engineering Department, Faculty of Engineering, Cairo University.
1. INTRODUCTION:
Tapered haunches are widely used in medium-to-large span frames in order to
minimize the cost of steel in framed structures and to give a more pleasant
view. Figure 1 shows the main dimensions and the geometric configuration of a
typical tapered-haunched connection.




+




    Fig. 1. Geometric configuration of a typical tapered haunched connection

The analysis of tapered knees was carried out previously by several researchers
using both elastic and plastic theories ([1], [2], [3] and [4]). One of the
remarkable attempts in this field was by Fisher [1], who proposed a simplified
analytical model for the tapered connection based on simple plastic theory. The
location of the neutral axis was found to be slightly shifted towards the upper
tension flange in case of identical haunch flanges thicknesses. This was
attributed to the effect of the haunch slope. To account for the impact of axial
force on the plastic moment of the connection, interaction curves were
proposed in which the reduction in the plastic moment capacity was only
required if the axial force value was greater than 15% of the force that causes
section yielding. Moreover the effect of shear-moment interaction on the
plastic capacity was reflected in adopting a reduced yield stress for longitudinal
stresses based on Von-Mises yielding criterion [7].


The location of the critical section was found to be dependent on the slope of
the haunch. If this slope was greater than a certain value the plastic hinge will
be formed at the junction of beam and haunch and if it was smaller than that
value, the plastic hinge will be formed at the corner. Such certain value was
designated as the haunch critical angle at which the yielding will occur
throughout the haunch length, and was estimated to be 12° for the range of
haunch dimensions considered in the study [1]. The critical unsupported length
for the haunch flange was found to be five times the haunch flange width
unless special precautions were made to control strains. Strain control and
enforcing strain hardening at certain points to limit spreading of yielding
throughout the haunch length can be achieved by either increasing the haunch
slope or increasing the haunch flange thickness.


A stiffener was assumed at each point of abrupt change in flange direction.
Stiffeners located at flange tips were used to resist the unbalanced vertical
component in the sloping flange of the haunch assuming no contribution for the
web. On the other hand, the diagonal stiffener was designed to resist the greater
of the unbalanced flange force or the shear force in excess of the web panel
zone capacity. The web contribution was also neglected in resisting the
localized unbalanced component of flange forces.


The main drawbacks of the conventional approach were identified in neglecting
web contribution in resisting the unbalanced flange force components,
prohibiting the use of un-stiffened or semi-stiffened connections and imposing
the use of a flange brace at every point of abrupt change in direction which
may be difficult in some circumstances. Moreover, there was no explicit design
procedure or a certain recommended connection configuration provided by
such conventional methods. In this work, a finite element model was
established for a typical tapered-haunched connection proportioned by the
conventional method [1]. Both material nonlinearity and               geometric
imperfections were included in the analysis to incorporate possible interaction
of yielding and buckling at failure. The purpose of the analysis established
herein was to assess numerically the general behavior of tapered knees and
investigate the strength of un-stiffened and semi-stiffened connections.


2. FINITE ELEMENT MODELING:
The complexities involved in the real behavior of knee connections impose the
use of a sophisticated finite element model. Both material and geometric
nonlinearities must be incorporated in the model to initiate the interaction
between buckling and yielding. For this reason, the general purpose finite
element program ANSYS [5] was used for its powerful capabilities in solving
such complicated problems that include special features such as plasticity and
large deformation effects.


2.1 Geometric Configuration:
The tapered knee geometric configuration was selected as per Fisher [1] and
Machaly [2]. Dimensioning of the tapered knee components was performed in
accordance with the Egyptian Code of Practice for Steel Construction and
Bridges [6] to support the straining actions induced on the beam-to-column
connection of a portal frame with 30 m span and with 8 m height when
subjected to gravity and wind loads. The beam section was selected as an I-
shaped section with web plate 386x7 mm and flanges 280x14 mm, i.e.
W(386x7/280x14)       while    the   column     section    was    designed   as
W(485x7/280x15). The haunch flange dimensions were taken similar to those
of the beam flange, whereas the haunch web thickness was increased to 8 mm.
The slope of the taper was assumed to be equal to 9°.
2.2 Modeling of Connection Plate Elements:
Figure 2 illustrates the finite element model selected for the connection in
which flange and web plates were modeled by an iso-parametric finite strain
shell element in ANSYS element library designated as Shell 181[5]. Such an
element was selected for its higher stability and better modeling for large
deformation problems. Shell 181 is a four nodded shell element with six
degrees of freedom at each node. It has both bending and membrane
capabilities. It is suitable for analyzing thin to moderately thick shell structures.
It is well suited for linear, large rotation and/or large strain non linear
applications. The elements special features include: Plasticity, stress stiffening,
large deflection and initial stress import.




             Fig. 2. Finite element model for tapered haunched connection


2.3 Boundary Conditions and Loading:
The finite element model was extended beyond the haunch tips to a distance
equals to twice the beam depth horizontally and equals to twice the column
depth vertically in order to minimize end conditions effect on results. The far
end of the column was restrained in the three spatial directions x, y and z. The
connection upper flange was restrained in the out-of-plane direction at points
corresponding to purlins locations. At the haunch tips, an out-of-plane restraint
was provided whereas the corner re-entrant point was left un-braced in the out-
of-plane direction. The model was loaded by reporting the bending moment,
normal force and shear force at the location of the beam end from the portal
frame analysis and converting such forces into equivalent nodal forces [9]. The
computed nodal forces were scaled to produce the value of the beam theoretical
plastic moment [1] at the haunch-to-beam junction. Due to the small values of
normal and shear forces, the reduction of the beam plastic moment due to
moment-shear-normal interaction was not considered during load application.


2.4 Material Model:
The idealized stress-strain curve for mild steel based on elastic- perfectly
plastic behavior was employed [8]. The value of the yield stress was taken as
2.4t/cm2 and the Young's modulus was chosen to be 2100 t/cm2. Isotropic
hardening and Von Mises yield criterion were employed throughout the non-
linear analysis.


3. FINITE ELEMENT RESULTS:
In this section the finite element results are presented. Three connection
configurations were studied based on stiffener configuration. At first, the case
of un-stiffened connection was solved with no edge or diagonal stiffeners.
Second, a semi-stiffened connection with diagonal stiffener only was
considered and finally the case of a fully stiffened connection with both
diagonal and edge stiffeners was solved. The analysis of each configuration
was conducted to determine the elastic buckling load, plastic limit load
neglecting geometric imperfections and the inelastic limit load considering
material nonlinearity and geometric imperfections.
3.1 Case of Un-Stiffened Connection:
3.1.1 Elastic Buckling Analysis:
Figure 3 illustrates the contour plot of out-of-plane displacements
corresponding to the fundamental buckling mode of the un-stiffened
connection. It was noticed that the connection lost its stability primarily due to
web buckling at corner re-entrant under the effect of the concentrated
unbalanced flange forces at that point. The buckling load did not exceeded 0.33
times the plastic moment of the adjacent beam section, Mp.




Fig.3. Contour plot of out-of-plane displacements at the fundamental buckling
                     mode of un-stiffened Connection, cm


3.1.2 Inelastic Analysis Neglecting Geometric Imperfections:
In this section, the plastic limit load of the connection was computed by
conducting a nonlinear static analysis for a geometrically perfect connection
configuration. The Von Mises yield criterion with isotropic hardening was
employed. The load was applied on the model incrementally and solution
obtained at each load step iteratively using the full Newton Raphson technique.
The analysis was terminated when the limit load was reached. The contour plot
of the equivalent stresses at the plastic limit load is illustrated in Fig.4.


Figure 4 shows that the yielding zone was concentrated near the corner re-
entrant zone due to the localized effect of the concentrated unbalanced flange
forces. The connection could not develop the beam plastic moment and the
plastic limit load did not exceed 0.87 Mp at which excessive deflection of the
beam took place and the connection lost its stiffness.




 Fig.4. Equivalent stress distribution at plastic limit load neglecting geometric
                                imperfections, t/cm2


3.1.3 Inelastic Buckling Analysis Considering Geometric Imperfections:
To initiate the interaction between yielding and buckling, geometric
imperfection were introduced in the model. This was done by scaling the
amplitude of the first buckling mode (Fig. 3) of the connection to the allowable
limit in the Egyptian Code [6] such that the bowing in web would not exceed
hw/150 and the lateral deformation in the flange would not exceed hw/75. The
scaled buckled shape was used to modify the nodal coordinates of the perfect
configuration. The analysis was conducted using the full Newton-Raphson
technique as illustrated in Sec 3.1.2. The analysis was terminated when the
limit load at which the connection loses its stability was reached. Figure 5
illustrates the equivalent stresses distribution computed at the limit load.


It was noticeable that the yielding zone was still concentrated at the web panel
zone. Nevertheless, it was extended towards the tension flange. The
imperfections had also initiated some yielding in the column flange. However,
it was the deterioration of the web panel zone capacity which limited the
connection capacity. The failure was mainly attributed to inelastic buckling of
the web at the web panel zone. The limit load did not exceed 0.28 Mp.




   Fig. 5. Equivalent stresses distribution at limit load considering geometric
                               imperfections, t/cm2


Based on the above results, only the elastic buckling and the inelastic analysis
will be considered for the remaining connection configurations to be studied
herein since it was believed that the plastic limit load ignoring geometric
imperfections is a bit theoretical and can hardly be achieved in practice.
3.2 Case of Semi-Stiffened Connection:
The effect of adding diagonal stiffeners in the web panel zone was explored.
An overall increase in strength was expected since diagonal stiffeners will
support part of the unbalanced flange force component at the corner re-entrant
and will strengthen the panel zone against shear buckling.


3.2.1 Elastic Buckling Analysis:
Figure 6 shows the contour plot of the out-of-plane displacements of the first
buckling mode. It was noticeable that although the corner re-entrant lacks an
out of plane support (Sec. 2.3), the diagonal stiffener greatly reduced the out-
of-plane deformation at that point. The first buckling mode was mainly
composed of lateral displacement of the flange resembling the out-of-plane
flexural buckling mode of a pinned-pinned strut. The buckling load was
increased to 1.67 Mp.




Fig. 6. Contour plot of out-of-plane displacements at the first buckling mode of
                    a diagonally stiffened tapered knee, cm
3.2.2 Inelastic Buckling Analysis:
The nonlinear static analysis of an imperfect diagonally stiffened tapered knee
showed that at limit load the yielding zone was transferred from the web panel
zone to the haunch tips (see Fig. 7). This was attributed to the additional
strength provided by the diagonal stiffener at the web panel zone and also due
to the effect of the concentrated unbalanced flange forces at the haunch tips. It
is to be noted that the yield zone formed at the haunch tip with the beam was
not symmetrically generated at the haunch tip with the column since the
column cross sectional dimensions provide larger load carrying capacity (Sec.
2.1). The limit load of the connection was increased due to the addition of the
diagonal stiffener from 0.28 Mp to 0.72 Mp.


3.3 Case of Fully Stiffened Connection:
In this case additional stiffeners are introduced at the haunch tips to resist the
unbalanced flange force components at such points (see Fig. 1).


3.3.1 Elastic Buckling Analysis:
Figure 8 shows the contour plot of the out-plane displacement at the first
buckling mode of the connection. The buckling mode was mainly composed of
web local buckling without buckling in the flanges. The buckling load of the
connection was increased to 3Mp. This result reflects the major influence of
stiffeners in controlling the buckling behavior of the connection.


3.3.2 Inelastic Buckling Analysis:
The analysis of the fully stiffened connection revealed that failure was
primarily due to the flange yielding at haunch tips rather than web yielding that
was prohibited by the effect of diagonal and edge stiffeners (see Fig 9). The
limit load of the connection was increased to reach 0.98 Mp indicating that the
connection adequate capacity was maintained by the addition of diagonal and
edge stiffeners. Since failure was mainly attributed to yielding, the effect of
geometric imperfections was insignificant and the plastic moment of the beam
section was almost reached.




              Fig.7. Equivalent stresses distribution at limit load
                       for a semi-stiffened knee, t/cm2




Fig. 8. Contour plot of out-of-plane displacements at the fundamental buckling
                     mode of a fully stiffened connection
A summary of the finite element results was illustrated in Table 1. For each
connection configuration studied, the connection capacity together with the
type and location of failure was listed for the purpose of comparison. It was
concluded that the addition of stiffeners not only pronounced significantly the
connection elastic buckling load, but also altered the buckled shape. The elastic
buckling load of the fully stiffened connection was almost ten times as much as
the un-stiffened connection. On the other hand, the addition of stiffeners had a
less significant effect on the plastic limit load since failure took place by
yielding. The addition of diagonal stiffeners increased the plastic limit load by
5 % whereas the addition of edges stiffeners increased the plastic limit load by
10% approximately.




           Fig.9. Contour plot of equivalent stresses at limit load for
                        fully stiffened connection, t/cm2


The limit load of un-stiffened knee revealed that failure was mainly attributed
to buckling of the web at corner re-entrant since it was slightly less than the
elastic buckling load. On the other hand, the plastic limit load greatly exceeded
the elastic buckling load. Therefore, it was noticeable that geometric
imperfections had minor effect on such connections unlike semi-stiffened
knees that were greatly affected by geometric imperfections (the limit load was
reduced by 26% by the introduction of geometric imperfections) since failure at
limit load was mainly attributed to inelastic buckling. On the other hand, fully
stiffened connections were insignificantly affected by geometric imperfections
since failure was caused by flange yielding at haunch tips.


                  Table 1 Summary of Finite Element Results
                Un-stiffened          Semi-stiffened            Fully-Stiffened
                Connection              Connection               Connection
Analysis    Moment      Type &      Moment      Type &        Moment      Type &
  Case      Capacity    location    Capacity    location      Capacity   location
                       of Failure              of Failure                of Failure
                                                                           Web
                       Buckling                Buckling                  buckling
 Elastic    0.33 Mp    of web at    1.67 Mp    of haunch      3.0 Mp       near
Buckling                 corner                  flange                  diagonal
                                                                         stiffener
                        Yielding                 Web                     Yielding
                       of web at                yielding                 of flanges
 Plastic    0.87 Mp      corner     0.91 Mp    at haunch      1.0 Mp     at haunch
                                                  tips                      tips
                       Buckling                  Web                     Yielding
                       of web at                yielding                 of flanges
Inelastic   0.28 Mp      panel      0.67 Mp    at haunch      0.98 Mp    at haunch
Buckling                  zone                    tips                      tips


4. COMPUTATION OF FORCES IN STIFFENERS:
The simple analysis adopted by the conventional method [1, 2] for computing
forces in stiffeners by considering equilibrium of flange forces is inaccurate as
it ignores the interaction between the connections component based on their
relative stiffness. On the other hand, the effect of stiffener thickness on
connection moment capacity was not explicitly demonstrated. In this section,
the effect of stiffener thickness on moment capacity in a fully stiffened
connection was demonstrated. The stiffener thickness was varied from 0.5 to 2
times the thickness stipulated by the conventional approach. The variation in
elastic buckling load and plastic limit load was recorded in each case and
normalized with respect to their respective values when stiffener thickness
computed by the conventional method was utilized. Figure 10 illustrates the
results obtained. It is to be noted that the limit load was excluded in such
comparison since the failure of fully stiffened connections was mainly caused
by yielding rather than by inelastic buckling (Table 1).


                            1.4

                            1.2

                             1
        Relative strength




                            0.8

                            0.6                                       Elastic Buckling
                            0.4                                       Elastic Plastic

                            0.2

                             0
                                  0   0.5   1                   1.5    2                 2.5
                                                Relative thickness



   Fig. 10 Effect of stiffeners thickness on the connection moment capacity


Figure 10 shows that the elastic buckling load is greatly pronounced when the
stiffener thickness increases from 0.5 to 1.0 times the conventional thickness
since the buckling mode in such region included local buckling in stiffeners.
When stiffener thickness exceeded the conventional thickness, the connection
capacity was increased by 18% due to pronounced stiffener thickness whereas
the stiffener local buckling modes were excluded. The increase in the plastic
limit load was nearly negligible for stiffeners thickness greater than or equal to
0.7 the thickness suggested by the conventional method. This indicated that
providing a smaller thickness than such limit will result in a fast deterioration
of strength. On the other hand, an evaluation of forces developed in stiffeners
showed that the conventional approach is generally conservative. Figures 11
and 12 illustrate the growth of vertical force component in edge and diagonal
stiffeners, respectively, by loading. Vertical component of stiffener forces
computed by the conventional method together with the vertical component of
flange forces were also plotted in Figs. 11 & 12 for comparison. By
investigating the results presented in Figs. 11 & 12, the following can be
concluded:


   1- Forces in stiffeners vary linearly with the applied moment. This was
      because the connection components in the vicinity of the diagonal
      stiffener or the web zone around the edge stiffeners remain elastic over a
      major part of loading.
   2- Force developed in edge stiffener was nearly 65% of corresponding
      force deduced from the conventional method. Moreover, force in
      diagonal stiffener was only 50% of its conventional value. This was
      mainly attributed to two reasons :
          a) The conventional method assumes that the bending moment is
             solely supported by flanges. This overestimates the value of the
             flange forces. In fact a part of the bending moment (about 16%)
             is resisted by the web.
          b) The conventional method totally ignores the web contribution in
             resisting any unbalanced component of flange concentrated force
             although the web has some capacity [9].
   3- Vertical component of stiffener force does not coincide with the vertical
      component of flange forces. Edge stiffener supports a lower load
      compared to flange vertical component whereas the diagonal stiffener
      carried a higher load than expected from applying joint equilibrium
      using the finite element results. This was attributed to the positive
      contribution of the web in case of edge stiffeners. On the other hand, the
      web exerted additional force on diagonal stiffeners due to shear
      deformation of the web panel zone.



                               VERTICAL FORCE IN FLANGE
                      10       Vertical FORCE IN STIFFENER
                       8       Conventional force
           Force(t)




                       6

                       4

                       2

                       0
                           0   0.1     0.2         0.3        0.4   0.5    0.6        0.7   0.8      0.9            1
                                                                                                   Fraction of load
                                              Force growth in the edge stiffener

                      Fig. 11. Growth of force in edge stiffeners by loading


                                     VERTICAL FORCE IN FLANGE
                      40             Vertical FORCE IN STIFFENER
                      35
                                     Conventional force
                      30
                      25
             Force




                      20
                      15
                      10
                       5
                       0
                           0   0.1     0.2          0.3       0.4   0.5    0.6        0.7   0.8      0.9            1
                                                                                              Fraction of loading
                                                 Force growth in diagonal stiffener

 Fig.12. Growth of vertical force component in diagonal stiffeners by loading


5. SUMMARY AND CONCLUSIONS:
In this paper the finite element analysis of a typical tapered knee designed by
the conventional approach was conducted to predict the connection behavior
and evaluate the design assumptions adopted by the conventional design
method. Based on stiffeners used, three connection configurations were studied
designated herein as: un-stiffened, semi-stiffened and fully stiffened
connections. The analysis of each connection configuration was conducted
three times to compute the elastic buckling load, the plastic limit load and the
limit load. Although the elastic buckling load may not govern the connection
capacity, the elastic buckling was established to identify critical regions that are
susceptible to fail by instability and to determine the fundamental buckling
mode that is identified as the worst imperfection shape. The plastic analysis of
the geometrically perfect connection was conducted to obtain the plastic limit
load that is only governed by yielding and to identify critical zones that fail by
yielding. The inelastic buckling analysis was established to determine the limit
load that accounts for the interaction between buckling and yielding. Finally,
forces that develop in stiffeners were evaluated and compared to conventional
design method with emphasis on the effect of stiffener thickness on connection
carrying capacity.


The analysis conducted herein indicated that the connection failure mechanism
depends mainly on stiffener configuration. Un-stiffened connection almost
failed by elastic buckling of the web panel zone and achieved 30% of the beam
plastic moment capacity approximately. Since failure of un-stiffened
connections was governed by elastic buckling, the capacity of such connections
was insensitive to geometric imperfections. In case of semi-stiffened
connections, failure occurred by inelastic buckling of the web at haunch tips
due to unbalanced flange forces. Effect of geometric imperfections was
significant in this connection configuration that achieved almost 67% of the
beam plastic moment. When stiffeners were used at all points of abrupt change
in the flange direction, the connection failed by yielding of flanges at haunch
tips provided that stiffeners width-to-thickness ratio was selected to prohibit
stiffener local buckling. It was noticed that fully stiffened connections achieved
the beam plastic moment and were insensitive to geometric imperfections.
Evaluation of stiffener forces revealed that the conventional method generally
overestimates the forces that develop in stiffeners by at least 50% due to
ignoring the resistance of web to concentrated forces. Provided that local
buckling requirements of the stiffeners were satisfied, the stiffener thickness
stipulated by the conventional method could be reduced by 30% without
reduction in connection capacity.


Finally, the analysis conducted herein provided some light on the complex
behavior of such connections. Due to lack of experimental research work on
such connection type, a comprehensive analytical work is recommended to
evaluate the effect of connection geometric configuration on behavior and to
help develop the conventional design equations to have a better reflection on
the real connection behavior.

REFERENCES :

     1. Fisher, J.W.,Lee, G.C., Yura, J.A. and Driscoll, J.r., "Plastic Analysis
        and Tests of Haunched Corner Connections" WRC Bulletin No.
        91,NewYork, 1969.
     2. Machaly, E.B., "Behavior, Analysis and Design of Steel Work
        Connections", 4th Edition, 2000.
     3. Osgood, W.R., "A theory of flexure for beams with non parallel
        extreme fibres", Journal of Applied Mechanics, ASME, 1939.
     4. Bleich, F., "Design of Rigid Framed Knees", American Institute of
        Steel Construction, Chicago, IL, 1943.
     5. Desalvo, G.J., and Gorman, R.W., "ANSYS User's Manual",
        Swanson Analysis Systems, Houston, PA, 1989.
     6. Egyptian Code of Practice for Steel Construction and Bridges
        (Allowable Stress Design), Code no. (205), 1st Edition 2001.
     7. Bakhoum, M., “Structural Mechanics”, first Edition, 1992.
     8. Salmon, C.G. and Johnson, J.E.," Steel Structures: Behavior and
        Design", Harper and Row Publishers, 4th Edition, 1996.
     9. El-Banna, A.A., " Analysis and Design of Tapered and Curved Beam-
        to-Column Connections", M.Sc Thesis, Cairo University, 2005.
               ‫جحليل سلوك الوصالث النسلوبت باسجخدام طريقت العناصر النحددة‬

                                                                                     ‫النلخص‬
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‫اهيرضٓ هحوم اهيٌشآج . ّ هلد أةريج اهعديد يً ااةحاخ عوٓ اهّصالج اهرنٌيث اهيسحلييث أّ حوىم‬
‫اهحٓ حزّد فيِا اهنيراج ةاةرةث يدوديث ، ةيٌيا هى يذنر اهندير عً اهّصالج اهيسوّةث عوٓ اهرغى يً‬
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‫يً ّفر فٓ نيياج اهحديد اهيسحخديث ّ هيا حعطيَ يً شنل ةياهٓ يةِر.ّ هِذا فولد حى اهحرنيز فىٓ‬
‫ُذا اهةحخ عوٓ دراسث سوّم اهّصالج اهيسوّةث ، ّ ذهم ةاسحخداى ةرٌايج طريلث اهعٌاصر اهيحددت‬
‫‪ ّ ANSYS‬ةحضييً اهالخطيث فٓ نل يً سوّم يادت اهحديد ّ اهشنل اهٌِدسٓ هوّصوث . ّهلد احفق‬
‫عوٓ أً حؤخذ أكصٓ كييث الٌحراف أْ عٌصر يً عٌاصر اهّصوث ةاهلييث اهيسيّح ةِا فٓ اهنىّد‬
‫اهيصرْ 1220.ّ ةاسحخداى ُذٍ اهفوسفث حى اةرا دراسث ييحدت هدراسث اهحّزيعاج اهيححيوث اعصاب‬
‫اهحلّيث. ّهلد ّةد أً ييناٌينيث االٌِيار يعحيد ةصّرت أساسيث عوٓ نيفيث حّزيع أعصاب اهحلّيث .نيا‬
                  ‫ّةد أً اهلّْ اهحليليث فٓ ااعصاب حلل نديرا عً حوم اهيلدرت ةاهطرق اهحلويديث.‬

				
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