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Calculation of buckling loads of reinforced panels
00457949(94)oo4o6-4
TECHNICAL CALCULATION
Compurers& SfructuresVol. 55, No. I, pp. 173-176. 1995 Copyright 0 1995 El&r Science Ltd Printedin Great Britain.All rights reserved 0045.7949/95 s9.50 + 0.00
NOTE
OF BUCKLING LOADS OF REINFORCED PANELS Jo& Maria Minguez
Departamento
de Fisica Aplicada
II, Facultad de Ciencias, Universidad 48080 Bilbao, Spain (Received
I November
de1 Pais Vasco,
Aptdo
644,
1993)
Abstract-An easy and fast approach to the calculation of the buckling load of a longitudinally reinforced panel is presented, taking into account the torsional rigidity of the stiffeners, as well as their inertia against bending. Previously, it has been shown how the torsional resistance of thin-walled stringers and, consequently, their behaviour when reinforcing a compressed panel, depends mainly on whether they have a closed or an open cross-section.
INTRODUCITON
of utility in the practice of the design of reinforced thin plates and for the calculation of buckling loads, without adding special difficulties, but accounting for the whole resistance of the stiffeners.
In many engineering structures, it has become increasingly important to save weight, not so much for the cost of the material, but to alleviate the structure itself. This is the case in aircraft and ship construction, where thin panels are being widely used. An easy and useful way to reduce the weight of a panel is to redistribute its material to make better use of it, which can be attained by removing some material from the panel itself and reinforcing it with stiffeners. Thus, in recent years, the use of stiffened panels has become very common and a good deal of the strength of materials literature is being devoted to this topic [14]. In the calculation of buckling loads of reinforced panels, a generally accepted rule consists of considering the inertia of the reinforcing stringers against bending, while taking no account of their torsional rigidity. This is corroborated by the fact that in classic textbooks [7], when studying the stability of reinforced plates, the torsional strength of the ribs is neglected and so ignored. The main reason for this is that as most of the stiffeners used in reinforcing thin plates are thin-walled, their torsional rigidity is effectively small, due to local buckling effects that can appear, as will be shown. However, in the buckling of a reinforced panel, as the panel skin undergoes bending in both directions, along its length and across its width, the longitudinal ribs stand up to bending and twisting, both effects depending on their cross-section. Consequently, if the torsional strength of the reinforcing stiffeners is not negligible, the above-mentioned rule leads to inaccurate calculations of the buckling loads. In fact, the use of closed cross-section stringers, with higher torsional strength, can be an easy way to increase the rigidity of thin plates and so the buckling loads. This was shown analytically by the author in [8], where the buckling modes of reinforced panels were deduced, taking into account the torsional rigidity of the stiffeners, and the buckling loads were rigorously calculated. Later, the same point was demonstrated experimentally [9]. Nevertheless, it is not realistic to believe that such long and laborious calculations as shown in detail in [8] are going to be performed for the design of a stiffened panel. For this reason, the aim of the present work is to summarize, from all the mathematical work, some practical rules that may be
BENDING AND TORSIONAL STRENGTH STRINGERS
OF THIN-WALLED
A panel reinforced with longitudinal thin-walled stiffeners, as represented in Fig. 1, is going to be considered as subjected to a longitudinal compressive load. Detailed and rigorous analysis of the buckling of a reinforced panel must be performed by considering the panel skin and the stringers as a whole, to deduce the overall buckling deflected surface of the panel and hence the critical load. This was done in [8], by developing the energy method. Nevertheless, as a first and faster approach, the panel can be divided into longitudinal strips, each laterally limited by two stiffeners. Then every strip can be studied as an independent panel with the boundary conditions provided by the stiffeners. This is why the strength and the behaviour of the stiffeners must be well-known, for they restrain the panel edges accordingly. Regarding the bending behaviour of the stringers, as they are attached to the panel skin, their only possible bending is that within a plane perpendicular to the panel. Then the critical stress that would produce their buckling is given by R2 n,, = n*E-, a2 R being the radius of giration of the cross-section and a the length of the panel and the stringer. In practice, the stress applied over the panel and the stiffeners is always lower than this value. Therefore, the overall buckling of the stringers does not take place, but they remain straight. However, if the applied load were higher than the critical value, the stiffener-could buckle in such a way that the higher the applied stress, the shorter the semi-wave of its buckling deflected line. With respect to the torsional strength of the stiffeners, the following expression can be assumed, from (71, for the 173
Technical
174
Note
Fig. 2. Bending
of the flanges of a Z stringer twisting.
due to its
stability to the rib and prevents it suffering a deformation in the way in which the Z stringer is deflected from its original shape. Consequently, the two terms of eqn (2) are valuable when dealing with R stiffeners, which means that they have got a considerable torsional strength. All this will be applicable when studying the buckling of reinforced thin plates.
BUCKLING
Fig.
1. Diagrammatic
representation of the longitudinally reinforced panel.
LOAD OF A PLAIN PANEL WITH VARYING EDGE CONDITIONS
Considering that the critical stress of a long flat panel, with no reinforcement at all, undergoing a longitudinal compression is given by
12
twisting moment thin-walled rib:
undergone
by
any
cross-section
o,, = KE--, b2
of a
where @ is the angle turned by the section, X is the ordinate of such a section along the panel, C = GJ is the torsional rigidity of the stiffener, equal to the product of the shearing modulus of elasticity of the material by the torsion constant of the cross-section, and C, = EC,, is the warping rigidity of the rib, equal to the product of the modulus of elasticity of the material by the warping constant of the cross-section. Nevertheless, concerning the particular shape of a stiffener, its torsional characteristics need to be analysed to get a better understanding of its twisting and of some local buckling that can appear with it. For this reason, a 2 stringer and a stiffener with R cross-section are going to be studied as representative of open and closed cross-section ribs respectively. The first term of expression (2) comes from the shearing stresses supported by the cross-section, whereas the second term represents the opposition the flanges of the stringer offer to their bending contained in their own plane, which is produced by the rotation of the section, as shown in Fig. 2 for a Z stringer. If the cross-section of the rib is not closed, the strips of their flanges undergoing the bending compression cannot be stable, for they are very long, and local buckling occurs (Figs 2 and 3). This is why the second and most significant term of the twisting moment in eqn (2) can be neglected. Then, ignoring the whole torsional strength of open crosssection stringers is not a rough approach. Otherwise, in the case of hat-shaped stringers, although they have an open cross-section, this is closed and fixed on the panel skin along the two flanges. This circumstance gives
(3)
where I and b are, respectively, the thickness and the width of the panel and the constant K depends on the boundary conditions, it is possible to find a value of K according to the particular edge conditions of every case. At the same time, those boundary conditions can be accurately reflected in a factor such as
U=P,
(4)
L, being the equivalent length corresponding to pinned edges. If k = 0.3 is the Poisson’s ratio of the material, the two factors, U and K, acquire the values given now in the following definite cases represented in Fig. 4.
Fig. 3. Buckling
of the flanges of a Z stringer compression when twisting.
due to their
Technical
Note
175
(C) One edge free,
-_ -_
I
(a) Two edges free:
u
----------
So K = 1.200 when (I = 0.5.
.
b
(D) Both pinned edges This is the case of reference. Consequently, the factor U is unity, whereas the result reached from [7] for K is 3.615.
Lp
1
the other built in
In this case, it is clear that the equivalent length corresponding to pinned edges is exactly double than the panel width, whereas K can be calculated again from [7]. Now C = 1.328, which gives the result K = 1.200.
1
(b) One edge free. the other articulated: K = 0.412 U = 0.25 SoK=3.615whenCJ=l. (E) One pinned edge, the other fixed The equivalent length corresponding to pinned edges is 1.42 b, as shown in Fig. 4. The value of K in this case is calculated in [S], for a panel with an edge built-in by means of a hat-shaped stiffener, very strong against twisting, and deduced to be 5.133.
(c) One edge free, the other built in: K = 1.200, U = 0.5
So K = 5.133 for U = 1.42. (F) Both fixed (d) Both pinned edges:
K = 3.615, U = 1 li
(e) One pinned edge, the other fixed: K = 5.133, U = 1.42
So K = 6.327 when U = 2. Now, it is possible to graphically represent K against U for the afore-mentioned cases and to accurately enough draw a continuous curve K = K(U), as in Fig. 5. The values of K for long panels under longitudinal compression and with support conditions in between those calculated above can be interpolated within this curve.
4 Q Both fixed edges: Fig. 4. Values
(A) Two
K = 6.327, U = 2
of K and
(i = b/L,
in six definite cases.
edgesfree
BUCKLING LOAD OF A REINFORCED
In this case, it is evident that the equivalent length corresponding to pinned edges is infinite and, therefore, U is zero. In contrast, it is clear that a long panel with no support along its longitudinal edges will buckle as soon as the slightest compressive load is applied to it, which means that K is null too. So K = 0 when U = 0. (B) One edge free,
oc, = c-,
C is another
constant D=-
The buckling of a longitudinally reinforced plate (see Fig. 1) can be tackled by considering the different zones into which the plate is divided by the ribs, as if they were long flat plates with the longitudinal edges supported by the stiffeners. Whatever support conditions the stiffeners provide for a region of the plate, the factor U = b/L, takes a value from Fig. 5, which determines K and so the buckling load.
I= 6-
790 b%
(5)
equal to 0.456 and Et ’ 12(1 -$)
is the flexural straightforward
PANEL
the other being articulated
Now the equivalent pinned length is four times the panel width. As for the constant K, with these boundary conditions, [7] finds out that the critical stress is given by
where
edges
In this last case, Fig. 4 shows clearly that the equivalent pinned length is half the panel width. As for the numerical factor K, it can be calculated from [7], as in cases (B) and (C). Then, the value of C being 7.0, it is found that K is 6.327.
rigidity of the panel skin. Consequently, a calculation yields for K the value 0.412. So K = 0.412 when Li = 0.25.
0
0.2
I 0.4
I 0.6
I 0.8
, 1.0
, 1.2
, 1.4
, 1.6
, 1.8
, 2.0
U = panel/width equivalent length corresponding to pinned edges Fig. 5. Values of K applicable to long panels with different boundary conditions, against factor U.
176
Technical
A I--
Note bending, that made the applied stress higher than the critical value given by eqn (1).
r
BF
JfB
II
II II
cl-
II
-
II II II II II II II II II II II l-
I
I E
-+c
II
J
(fZ$&i) (&dgel)
4 A
Sec. BB (fmt panel) Sec. BB (second panel) Fig. 6. Buckling modes of two identical panels, the first reinforced with a Z stringer and the second with a stiffener of R cross-section. Usually, the stringers reinforcing panels present enough inertia against bending to prevent their own buckling. Therefore, they are kept straight and the boundary support they offer to a region of the panel corresponds to points beyond point D in Fig. 5. If the torsional strength of the ribs is very low, as it is in the open cross-section stringers, they will act as articulations that allow the rotation of the plate they limit. The point representative of this case would be close to D and correspond to the traditional approach of neglecting the torsional rigidity of the stiffeners. On the contrary, if the ribs are strong against twisting, they will behave as building in the region of the plate they support, in which case the point representative of such boundary conditions will go close to point F in Fig. 5. So most regions of reinforced panels between two stringers will buckle as if they were represented in Fig. 5 between points D and F. Nevertheless, the traditional rule of ignoring the torsional rigidity of the ribs and assuming that they always act as articulations can only be a very inaccurate approach to the problem. In fact, taking K = 3.615 (point D), when it can reach the value K = 6.327 corresponding to point F, in the case of closed cross-section stiffeners, represents an error: t=
6.327 - 3.615 6.327
= 43%,
(6)
which is never justified. The region of the curve of Fig. 5 between the points A and D corresponds to panels whose edges are not strongly enough restrained to keep straight. Therefore, the strips of a longitudinally reinforced panel will be represented within this region only if the stiffeners had a very low inertia against
EXPERIMENTAL
EVIDENCE
Experimental evidence of the behaviour of a Z stringer, with open cross-section, as an articulation for the reinforced panel, and of a hat shaped (a) stiffener, with a cross-section closed on the panel skin, as fixing the panel, was provided in [9]. Two identical panels, the first reinforced with a Z stringer along its middle longitudinal line and the second with an R stiffener, buckled into the different modes shown in Fig. 6, when subjected to compression. CONCLUSIONS It has been proved
that, to have a fast and good approach to the buckling load of a reinforced panel, it is necessary (1) to appreciate the bending strength of the stringers which limit every region of the panel; (2) to appreciate their torsional strength and (3) according to them, to choose an approximate value for the factor U, and so a value for the constant K, with which the critical stress of the panel can be calculated through expression (3). Then it is possible to more accurately find out the ultimate strength capacity of the stiffened panel, which depends on the type of in-plane forces and boundary conditions, for, according to the European Recommendations for Steel Construction [lo], in point C.l.2 concerned with the ultimate strength and critical load, the ultimate load of instability must be obtained by multiplying the buckling critical load by a suitable factor c.
REFERENCES 1. G. Sun, R. Mao, Optimization of stiffened laminatedcomposite circular-cylindrical shells for buckling. Compos. Struct. 23, 53-60 (1992). 2. S. Sridharan and M. Zeggane, Postbuckling response of stiffened composite cylindrical shells. AIAA J. 30, 2897-2905 (1992). 3. D. A. Danielson, A. S. Cricelli, C. L. Frenzen and N. Vasudevan, Buckling of stiffened plates under axial compression and lateral pressure. Inc. J. Solidr Struct. 30, 545-551 (1993). 4. Y. L. Guo, Analysis of elastic-plastic interaction buckling of stiffened panels by spline finite strip method. Comput. Struct. 46, 429-436 (1993). 5. B. Morady and I. D. Parsons, A comparison of techniques for computing the buckling loads of stiffened shells. Comput. Struct. 46, 505-514 (1993). 6. H. S. Shen, P. Zhou and T. Y. Chen, Postbuckling analysis of stiffened cylindrical shells under combined external pressure and axial compression, Thin-walled Strut. 15, 43-63 (1993). 7. S. P. Timoshenko and J. M. Gere, Theory of Elastic Stability, 2nd Edn. McGraw-Hill, New York (1961). 8. J. M. Minguez, Torsional rigidity of ribs in the buckling of reinforced panels-analytical study. Comput. Struct. 28, 47-51 (1988). 9. J. M. Minguez, An experimental investigation of the influence of stiffener torsional rigidity on buckling of compression panels. Exp. Mech. 28, 336-339 (1988). 10. European Convention for Constructional Steel Work, European recommendations for steel construction (1978).
TECHNICAL CALCULATION
Compurers& SfructuresVol. 55, No. I, pp. 173-176. 1995 Copyright 0 1995 El&r Science Ltd Printedin Great Britain.All rights reserved 0045.7949/95 s9.50 + 0.00
NOTE
OF BUCKLING LOADS OF REINFORCED PANELS Jo& Maria Minguez
Departamento
de Fisica Aplicada
II, Facultad de Ciencias, Universidad 48080 Bilbao, Spain (Received
I November
de1 Pais Vasco,
Aptdo
644,
1993)
Abstract-An easy and fast approach to the calculation of the buckling load of a longitudinally reinforced panel is presented, taking into account the torsional rigidity of the stiffeners, as well as their inertia against bending. Previously, it has been shown how the torsional resistance of thin-walled stringers and, consequently, their behaviour when reinforcing a compressed panel, depends mainly on whether they have a closed or an open cross-section.
INTRODUCITON
of utility in the practice of the design of reinforced thin plates and for the calculation of buckling loads, without adding special difficulties, but accounting for the whole resistance of the stiffeners.
In many engineering structures, it has become increasingly important to save weight, not so much for the cost of the material, but to alleviate the structure itself. This is the case in aircraft and ship construction, where thin panels are being widely used. An easy and useful way to reduce the weight of a panel is to redistribute its material to make better use of it, which can be attained by removing some material from the panel itself and reinforcing it with stiffeners. Thus, in recent years, the use of stiffened panels has become very common and a good deal of the strength of materials literature is being devoted to this topic [14]. In the calculation of buckling loads of reinforced panels, a generally accepted rule consists of considering the inertia of the reinforcing stringers against bending, while taking no account of their torsional rigidity. This is corroborated by the fact that in classic textbooks [7], when studying the stability of reinforced plates, the torsional strength of the ribs is neglected and so ignored. The main reason for this is that as most of the stiffeners used in reinforcing thin plates are thin-walled, their torsional rigidity is effectively small, due to local buckling effects that can appear, as will be shown. However, in the buckling of a reinforced panel, as the panel skin undergoes bending in both directions, along its length and across its width, the longitudinal ribs stand up to bending and twisting, both effects depending on their cross-section. Consequently, if the torsional strength of the reinforcing stiffeners is not negligible, the above-mentioned rule leads to inaccurate calculations of the buckling loads. In fact, the use of closed cross-section stringers, with higher torsional strength, can be an easy way to increase the rigidity of thin plates and so the buckling loads. This was shown analytically by the author in [8], where the buckling modes of reinforced panels were deduced, taking into account the torsional rigidity of the stiffeners, and the buckling loads were rigorously calculated. Later, the same point was demonstrated experimentally [9]. Nevertheless, it is not realistic to believe that such long and laborious calculations as shown in detail in [8] are going to be performed for the design of a stiffened panel. For this reason, the aim of the present work is to summarize, from all the mathematical work, some practical rules that may be
BENDING AND TORSIONAL STRENGTH STRINGERS
OF THIN-WALLED
A panel reinforced with longitudinal thin-walled stiffeners, as represented in Fig. 1, is going to be considered as subjected to a longitudinal compressive load. Detailed and rigorous analysis of the buckling of a reinforced panel must be performed by considering the panel skin and the stringers as a whole, to deduce the overall buckling deflected surface of the panel and hence the critical load. This was done in [8], by developing the energy method. Nevertheless, as a first and faster approach, the panel can be divided into longitudinal strips, each laterally limited by two stiffeners. Then every strip can be studied as an independent panel with the boundary conditions provided by the stiffeners. This is why the strength and the behaviour of the stiffeners must be well-known, for they restrain the panel edges accordingly. Regarding the bending behaviour of the stringers, as they are attached to the panel skin, their only possible bending is that within a plane perpendicular to the panel. Then the critical stress that would produce their buckling is given by R2 n,, = n*E-, a2 R being the radius of giration of the cross-section and a the length of the panel and the stringer. In practice, the stress applied over the panel and the stiffeners is always lower than this value. Therefore, the overall buckling of the stringers does not take place, but they remain straight. However, if the applied load were higher than the critical value, the stiffener-could buckle in such a way that the higher the applied stress, the shorter the semi-wave of its buckling deflected line. With respect to the torsional strength of the stiffeners, the following expression can be assumed, from (71, for the 173
Technical
174
Note
Fig. 2. Bending
of the flanges of a Z stringer twisting.
due to its
stability to the rib and prevents it suffering a deformation in the way in which the Z stringer is deflected from its original shape. Consequently, the two terms of eqn (2) are valuable when dealing with R stiffeners, which means that they have got a considerable torsional strength. All this will be applicable when studying the buckling of reinforced thin plates.
BUCKLING
Fig.
1. Diagrammatic
representation of the longitudinally reinforced panel.
LOAD OF A PLAIN PANEL WITH VARYING EDGE CONDITIONS
Considering that the critical stress of a long flat panel, with no reinforcement at all, undergoing a longitudinal compression is given by
12
twisting moment thin-walled rib:
undergone
by
any
cross-section
o,, = KE--, b2
of a
where @ is the angle turned by the section, X is the ordinate of such a section along the panel, C = GJ is the torsional rigidity of the stiffener, equal to the product of the shearing modulus of elasticity of the material by the torsion constant of the cross-section, and C, = EC,, is the warping rigidity of the rib, equal to the product of the modulus of elasticity of the material by the warping constant of the cross-section. Nevertheless, concerning the particular shape of a stiffener, its torsional characteristics need to be analysed to get a better understanding of its twisting and of some local buckling that can appear with it. For this reason, a 2 stringer and a stiffener with R cross-section are going to be studied as representative of open and closed cross-section ribs respectively. The first term of expression (2) comes from the shearing stresses supported by the cross-section, whereas the second term represents the opposition the flanges of the stringer offer to their bending contained in their own plane, which is produced by the rotation of the section, as shown in Fig. 2 for a Z stringer. If the cross-section of the rib is not closed, the strips of their flanges undergoing the bending compression cannot be stable, for they are very long, and local buckling occurs (Figs 2 and 3). This is why the second and most significant term of the twisting moment in eqn (2) can be neglected. Then, ignoring the whole torsional strength of open crosssection stringers is not a rough approach. Otherwise, in the case of hat-shaped stringers, although they have an open cross-section, this is closed and fixed on the panel skin along the two flanges. This circumstance gives
(3)
where I and b are, respectively, the thickness and the width of the panel and the constant K depends on the boundary conditions, it is possible to find a value of K according to the particular edge conditions of every case. At the same time, those boundary conditions can be accurately reflected in a factor such as
U=P,
(4)
L, being the equivalent length corresponding to pinned edges. If k = 0.3 is the Poisson’s ratio of the material, the two factors, U and K, acquire the values given now in the following definite cases represented in Fig. 4.
Fig. 3. Buckling
of the flanges of a Z stringer compression when twisting.
due to their
Technical
Note
175
(C) One edge free,
-_ -_
I
(a) Two edges free:
u
----------
So K = 1.200 when (I = 0.5.
.
b
(D) Both pinned edges This is the case of reference. Consequently, the factor U is unity, whereas the result reached from [7] for K is 3.615.
Lp
1
the other built in
In this case, it is clear that the equivalent length corresponding to pinned edges is exactly double than the panel width, whereas K can be calculated again from [7]. Now C = 1.328, which gives the result K = 1.200.
1
(b) One edge free. the other articulated: K = 0.412 U = 0.25 SoK=3.615whenCJ=l. (E) One pinned edge, the other fixed The equivalent length corresponding to pinned edges is 1.42 b, as shown in Fig. 4. The value of K in this case is calculated in [S], for a panel with an edge built-in by means of a hat-shaped stiffener, very strong against twisting, and deduced to be 5.133.
(c) One edge free, the other built in: K = 1.200, U = 0.5
So K = 5.133 for U = 1.42. (F) Both fixed (d) Both pinned edges:
K = 3.615, U = 1 li
(e) One pinned edge, the other fixed: K = 5.133, U = 1.42
So K = 6.327 when U = 2. Now, it is possible to graphically represent K against U for the afore-mentioned cases and to accurately enough draw a continuous curve K = K(U), as in Fig. 5. The values of K for long panels under longitudinal compression and with support conditions in between those calculated above can be interpolated within this curve.
4 Q Both fixed edges: Fig. 4. Values
(A) Two
K = 6.327, U = 2
of K and
(i = b/L,
in six definite cases.
edgesfree
BUCKLING LOAD OF A REINFORCED
In this case, it is evident that the equivalent length corresponding to pinned edges is infinite and, therefore, U is zero. In contrast, it is clear that a long panel with no support along its longitudinal edges will buckle as soon as the slightest compressive load is applied to it, which means that K is null too. So K = 0 when U = 0. (B) One edge free,
oc, = c-,
C is another
constant D=-
The buckling of a longitudinally reinforced plate (see Fig. 1) can be tackled by considering the different zones into which the plate is divided by the ribs, as if they were long flat plates with the longitudinal edges supported by the stiffeners. Whatever support conditions the stiffeners provide for a region of the plate, the factor U = b/L, takes a value from Fig. 5, which determines K and so the buckling load.
I= 6-
790 b%
(5)
equal to 0.456 and Et ’ 12(1 -$)
is the flexural straightforward
PANEL
the other being articulated
Now the equivalent pinned length is four times the panel width. As for the constant K, with these boundary conditions, [7] finds out that the critical stress is given by
where
edges
In this last case, Fig. 4 shows clearly that the equivalent pinned length is half the panel width. As for the numerical factor K, it can be calculated from [7], as in cases (B) and (C). Then, the value of C being 7.0, it is found that K is 6.327.
rigidity of the panel skin. Consequently, a calculation yields for K the value 0.412. So K = 0.412 when Li = 0.25.
0
0.2
I 0.4
I 0.6
I 0.8
, 1.0
, 1.2
, 1.4
, 1.6
, 1.8
, 2.0
U = panel/width equivalent length corresponding to pinned edges Fig. 5. Values of K applicable to long panels with different boundary conditions, against factor U.
176
Technical
A I--
Note bending, that made the applied stress higher than the critical value given by eqn (1).
r
BF
JfB
II
II II
cl-
II
-
II II II II II II II II II II II l-
I
I E
-+c
II
J
(fZ$&i) (&dgel)
4 A
Sec. BB (fmt panel) Sec. BB (second panel) Fig. 6. Buckling modes of two identical panels, the first reinforced with a Z stringer and the second with a stiffener of R cross-section. Usually, the stringers reinforcing panels present enough inertia against bending to prevent their own buckling. Therefore, they are kept straight and the boundary support they offer to a region of the panel corresponds to points beyond point D in Fig. 5. If the torsional strength of the ribs is very low, as it is in the open cross-section stringers, they will act as articulations that allow the rotation of the plate they limit. The point representative of this case would be close to D and correspond to the traditional approach of neglecting the torsional rigidity of the stiffeners. On the contrary, if the ribs are strong against twisting, they will behave as building in the region of the plate they support, in which case the point representative of such boundary conditions will go close to point F in Fig. 5. So most regions of reinforced panels between two stringers will buckle as if they were represented in Fig. 5 between points D and F. Nevertheless, the traditional rule of ignoring the torsional rigidity of the ribs and assuming that they always act as articulations can only be a very inaccurate approach to the problem. In fact, taking K = 3.615 (point D), when it can reach the value K = 6.327 corresponding to point F, in the case of closed cross-section stiffeners, represents an error: t=
6.327 - 3.615 6.327
= 43%,
(6)
which is never justified. The region of the curve of Fig. 5 between the points A and D corresponds to panels whose edges are not strongly enough restrained to keep straight. Therefore, the strips of a longitudinally reinforced panel will be represented within this region only if the stiffeners had a very low inertia against
EXPERIMENTAL
EVIDENCE
Experimental evidence of the behaviour of a Z stringer, with open cross-section, as an articulation for the reinforced panel, and of a hat shaped (a) stiffener, with a cross-section closed on the panel skin, as fixing the panel, was provided in [9]. Two identical panels, the first reinforced with a Z stringer along its middle longitudinal line and the second with an R stiffener, buckled into the different modes shown in Fig. 6, when subjected to compression. CONCLUSIONS It has been proved
that, to have a fast and good approach to the buckling load of a reinforced panel, it is necessary (1) to appreciate the bending strength of the stringers which limit every region of the panel; (2) to appreciate their torsional strength and (3) according to them, to choose an approximate value for the factor U, and so a value for the constant K, with which the critical stress of the panel can be calculated through expression (3). Then it is possible to more accurately find out the ultimate strength capacity of the stiffened panel, which depends on the type of in-plane forces and boundary conditions, for, according to the European Recommendations for Steel Construction [lo], in point C.l.2 concerned with the ultimate strength and critical load, the ultimate load of instability must be obtained by multiplying the buckling critical load by a suitable factor c.
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