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A code requirement for the compressive strength of plate elements
Marine Structures 1 (1988) 71-80
A Code Requirement for the Compressive Strength of Plate Elements
C. G u e d e s Soares Naval architecture and Marine Engineering Section, Technical University of Lisbon, Av. Rovisco Pais, IST, 1096 Lisbon, Portugal (Received 8 December 1987; accepted 5 January 1988)
ABSTRACT Two design equations are derived in a simple form suitable for incorporation in design codesfor merchant ships and warships. The equations are derived from a full description of the variables that govern plate strength which are weighted by the probability density function of the plate geometric variables that are applicable to each ship type. Key words." Plate strength, collapse strength, residual stresses, initial distortions, uncertainty modelling, code design, safety factors, code format. 1 INTRODUCTION Design codes represent the accepted rules of practice in the design of the structures that they govern. They provide a balance between safety and economy, avoiding the clearly unsafe and uneconomical designs. The relationship between e c o n o m y and safety depends on the type of structure considered and for a code to be optimised, consideration must be given to its scope. This includes the type of structures and loading, as well as the range of dimensions of the components. In formulating a code requirement, a balance must be achieved between its accuracy and simplicity, both of which are desired features. As discussed in greater detail in Ref. 1, the design variables that are 71 Marine Structures 0951-8339/88/$03.50 © 1988 Elsevier Applied Science Publishers Ltd, England. Printed in Great Britain.
72
C. Guedes Soares
chosen to be included in the code requirement must be the ones that are more relevant to the feature under consideration. By neglecting less important variables, the design method is not expected to reproduce exactly the strength of all components to which it is applicable, but only to do so in an average sense. Any design method is also expected never to give large discrepancies in any individual case to which it is applied. Faulkner 2 proposed a design method for rectangular plate elements subjected to uniaxial compressive loads. The equation accounts explicity for plate slenderness and for the effect of weld-induced residual stresses. The equation was derived from a comparison with experimental results of plates that have a certain degree of initial distortions, although it does not include that variable explicitly. Recently, the author has studied the problem in a systematic way, showing that in the range of common values of plate slenderness, the compressive strength of the plate elements could change by as much as 60%. Furthermore, the strength of nearly perfect plates could be reduced by as much as 20% in the normal range of variation of initial distortions and of residual stresses. This difference in strength is also similar to the one between simply supported and clamped plates. To be consistent, one should account explicitly for all the variables that have the same degree of influence on the strength. This was the view taken in Ref. 1 where Faulkner's method was extended to account explicitly for the effect of the level ofintial distortions. This was achieved by calibrating the design formulas with results from experiments and from calculations with sophisticated numerical methods. In having a design equation that accounts explicitly for plate slenderness, for the level of residual stresses and initial distortions as well as boundary conditions, one has a good tool for design and for strength assessments. The problems involved in using the same equation for both purposes, as well as the requirements for its format, are discussed in Ref. 3. In design one does not always know the values of all relevant variables, as opposed to what happens in the analysis.4 The most significant variables must always be included in the design equation but the uncertainty about the value of less relevant variables advises in many situations that they should not be included explicitly in the design equation. In this case, they must be substituted by their effect, in an average sense. However, their average effect will depend on the scope of the code, i.e. on the family of structures and on the range of dimensions to which it will apply. In the case of plate elements a simple design equation could involve explicitly only the plate slenderness, which is the most important
Requirement for the compressive strength of plate elements
73
variable governing compressive strength. This would represent a simplification of the equations proposed by Faulkner a a n d by the author) In this work such a simple design equation is derived from the recent results) It will be derived for two different code scopes to illustrate the methodology that should be adopted, as well as the different results to which it leads.
2 C O M P R E S S I V E S T R E N G T H OF PLATE E L E M E N T S F a u l k n e r proposed a method of predicting the strength of rectangular plates with initial imperfections, by which the plate strength t~F is given by: OF = O'm/Cr0 = { ~ b - mt~b
(1)
where a m is the plate average compressive stress, cr0 its yield stress, Ob is the strength of a plate with initial distortions but without residual stresses and A~b is the strength reduction due to the presence of residual stresses. These parameters are given by: al
a2
forfl > 1.0
(2)
where/3 = (b/t) (Cro/E) 1/=is the plate slenderness, b and t are the breadth and thickness of the plate, E is the Young's modulus of the material and the constants are: a l = 2.0
and
a2 = 1.0 for simple supports
(3a)
al = 2.5
and
a2 = 1.56 for clamped supports
(3b)
The residual stresses ar are assumed to be uniformly distributed across the central zone of the plate~ being equilibrated by two strips of tensile yield stresses at the edges, each with a breadth ofr/t. This assumption of stress distribution together with equilibrium considerations gives the value of the residual stresses as: err _
ao
277 (b/t) - 2r/
(4)
The strength degradation induced by those stresses is given by: A~b
-- O'r E t
a0 E
(5)
where E t is the tangent modulus of elasticity which can be approximated
C. Guedes Soares
74
by a simple expression given in Ref. 5 or by the Ostenfeld-Bleich parabola: 2 Et_( E
tz3fl 2 ) a4 +pr(1 --pr)
Et - l'0 E
for
0
(6a)
for
fl < l'9/X/~
(6b)
wherepr is the ratio of the material's proportional limit ap (reduced by the residual stresses) and the yield stress, i.e.pr = (Op - ar)/a0. The constants depend on the boundary conditions: a3 = 3.62 a4 = 13.1 for simple supports (7a) a3 = 6.31
a4
= 39.8
for clamped supports
(7b)
The proportional ratio can vary between 0.5 and 0.75, but Faulkner advises the use of 0.5.2 The generalisation of Faulkner's method proposed by the author I to account explicitly for the amplitude (6o) of initial distortions is: OG = (1"08~b){( 1
A~b '~ (1 - 0.0078r/)} 1"08~b]
{ 1 - (0"626- 0"121fl)~} (0"665 + 0"006r/ + 0"36 6° t + 0" 14/3)
(8)
where the first term in brackets predicts the strength of perfect plates, the first and the second indicate the strength of flat plates with residual stresses, the first and third give the strength of plates with initial deflections and no residual stresses, and the four terms should be used for plates that have both initial deflections and residual stresses. The model uncertainty that is associated with this expression can be described by a coefficient of variation (COV) of 0.07. 3 CHARACTERISTICS OF SHIP PLATES Ship plates have proportions which are different from those encountered in other metal structures. Even when making comparisons between different ship types, one will find different trends. To give an idea of two limiting cases, a statistical description of plate dimensions is presented here for the cases of tankers and of warships. The data base that was used for tankers consists of typical plates in the deck and bottom of the midship section of 130 tankers built between 1973
Requirementfor the compressive strength of plate dements
75
P ,40
Frigates
,36.32" .28.24 .20•16.12" .08 .04"
4
7 a
D
b
.40.
Tankers
• 361 .32" .28.24" .20.16. .12. .08' .04-
1
2
I
3
5 a
8
9
b
Fig. 1. Probability density distribution of plate aspect ratio in frigates and tankers. and 1986 with lengths between 66 m and 390 m. In the case of warships, the data is mostly from frigates. Figure 1 shows probability density functions for the distribution of the plate aspect ratio (a -- a/b), while Fig. 2 indicates the distribution of
76
C. Guedes Soares
P 36-
33-
3o-
Frigates
.27-
.24-
.21 -
.18
o
,15-
.12-
,09-
.06-
.03-
h
l
10
20
30
40
50
60
70
80
90
100
,
,
I
I
,
110
120
130
140
150
bit
[--1---I 160
170
180
P~ .36-
.33-
3o-
Tankers
27-
.24 -
.21-
.18-
.15.12-
.o9-
.06-
.03i
10
I
L~)
30
40
50
60
70
I,
8o
bit 90
Fig. 2. Probability density distribution of plate slenderness in frigates and tankers.
slendemess (b/t). It is clear that the distributions are different for each ship type. The m e a n and C O V in each case is given by:
a/b = 3.2
V = 0.32
for frigates
(9a)
Requirementfor the compressive strength of plate elements V = 0.16
for tankers
(9b)
b / t = 60
V = 0.42
for frigates
(10a)
b / t = 46
V = 0-25
for tankers
(10b)
a/b
= 4.7
77
The m e a n values of b / t correspond to slendernesses (13) of 2.0 a n d 1.5 respectively for tankers a n d frigates. As well as having different typical dimensions, ship plating in warships will similarly be expected to have different characteristic initial distortions and residual stresses. Faulkner's measurements of initial plate distortions in warships indicated that 6o/t = K f l 2
(11)
where K = 0.12 w h e n fl < 3 and K = 0.15 w h e n fl > 3. Jastrzebski and Kmiecik 6 have been conducting measurements in merchant ships suggesting another relation: 6o/t = O.O094(b/t) - 0.205
(12)
which has even a different functional dependence on plate slenderness. It will not be discussed here which one is more correct, but instead, they will be taken as representative of different ship types. Finally, concerning the level of residual stresses existing in the plates, very little information is available. F a u l k n e r 2 conducted some measurements which were c o m p a r e d with others in Ref. 4 where the width of the weld affected zone (77) was considered to be described by a m e a n and a COV of 5.25 and 0.07 respectively. This statistical description is assumed to be applicable to any type of ship. Thus, in this work, the differences between the characteristics of ship plating in warships a n d m e r c h a n t ships are concerned with plate slenderness and amplitude of initial distortion.
4 D E R I V A T I O N OF T H E D E S I G N E Q U A T I O N S To obtain a design equation that depends only on plate slenderness, it is necessary that the effect of the other parameters is taken at their expected values. Expressing eqn (8) as:
~o(fl, 0,60) = q~b(fl).B(fl,0,60)
(13)
onc aims at determining the bias factor B which will depend on the distribution ofthc governing parameters. Its mean value and variancc is given by weighting thc model error B by its probability of occurrence:
78
C. Guedes Soares
f f fBro,0,/50), fro, no,s0)dfl
(14)
dr/d/50
¢3r2 =
( Jf ,f{B(13, r/,/50)-
~}2 f(/3, r/,/50)d/3 dr/dr0
(15)
where f(/3, 17,/50) is the joint probability density function of the variables /3,17,/50. This function represents the probability that for a given ship the typical plates of the midship section have a specified value of/3, 17and/50. It is assumed that these effects are independent so that: f(/5, 77, i50) = 3q (fl) .f2 (/7) .f3 (/50)
(16)
The probability density function of plate slendernessjq is indicated in Fig. 2. The functions f2 and f3 are assumed to be Normal distributions with the mean values equal to 5.25 and to eqns (11) and (12) respectively. Using eqns (8) and (13)-(16) results in the following values = 0.824
V = 0.016 for merchant ships
(17a)
= 0.814
V = 0.030 for warships
(b/t <
110) (17b)
These values of the bias, combined with eqns (13), (2) and (3), result in two simple equations which can be used to assess the strength of plates in each type of ship. For design one is interested in incorporating some safety against the uncertainty of the governing parameters. Formal approaches to derive the partial safety factors are available 7 but in the present context one can use a characteristic value of bias which is between 2 and 3 standard deviations below the mean: Be = B (1 - kV)
(18)
where k is a constant. F o r k = 2.5, the probability thatB is smaller than Be is of the order of a few percent and the resulting characteristic design equations are:
t~em-
1.6 fl
0.8 f12
(19)
~¢m-
1.5 /3
0.75 ~2
(20)
for merchant ships (m) and warships (w), in the case of simply supported plates. The corresponding equations for clamped plates are obtained by substituting the constants in eqn (3a) by those in eqn (3b). If one were to include plates with b/t greater than 110, which represent 4% of the warship data, the values in eqn (17b) would change toB = 0.884
Requirement for the compressive strength of plate elements
79
and V = 0.468, the latter of which seems too high to be representative of the population, and eqn (18) with k = 2.5 would not be applicable. Equations (19) and (20), which are based on characteristic values, can be used to design plates which are expected to be fabricated with the common values of initial distortion and weld-induced residual stresses in merchant ships and warships. They differ from eqn (2) which does not incorporate implicitly the effect of residual stresses but does incorporate implicitly the effect of the initial distortions that were present in the set of experiments in which it was based. Equation (8), which was derived in Ref. 1, accounts explicitly for all the effects that influence plate strength. Equations (19) and (20) only account explicitly for the most important parameter, representing the effect of the others by their mean values for the two ship types.
5 CONCLUSIONS Design equations need to be as simple as possible for repetitive and failfree use but they must include the important variables to be able to predict correctly the strength of the components to be designed. In this work, two simple design equations are obtained by weighting a complete expression by the probability distributions of its less significant variables. These distributions represent the typical cases of merchant ships and warships. The main objective of this work is to draw attention to the idea that simple design equations must account for the probability distributions of typical dimensions, and consequently will be different for each structure type. The results are based on a limited statistical survey and, thus, should be viewed as indicative. However, the method presented here is applicable to other cases and can be used to update the design equations if a better statistical description of the parameters becomes available.
6 ACKNOWLEDGEMENTS The ideas presented in this work have been developed during the author's visit to the Department of Naval Architecture and Ocean Engineering of the University of Glasgow in the summer of 1986. The author is grateful to Professor D. Faulkner for his kind hospitality and for inspiring discussions. The author is also grateful to Dr C. S. Smith from the Admiralty Research Establishment in Dunfermline and to Mr A. C. Viner and Mr A. H. S. Wickham from Lloyd's Register of Shipping
80
C. Guedes Soares
for having provided the statistical data used in this work. This work is part of the Research project 'Structural Reliability' that the author is conducting at C E M U L , the Centre for Mechanics and Materials of the Technical University of Lisbon, which is financially supported by INIC, the National Institute for Scientific Research.
REFERENCES 1. Guedes Soares, C., Design equation for the compressive strength of unstiffened plate elements with initial imperfections. J. Const. Steel Res., 9 (1988), in press. 2. Faulkner, D., A review of effective plating for use in the analysis of stiffened plating in bending and compression. J. Ship Res., 19 (1975) 1-17. 3. Faulkner, D., Guedes Soares, C. & Warwick, D. M., Modelling requirements for structural design and assessment. In Integrity of Offshore Structures, Vol. 3, ed. D. Faulkner, A. Incecik & M. J. Cowling, Elsevier Applied Science, London, 1988, pp. 17-27. 4. Guedes Soares, C., Uncertainty modelling in plate buckling. Struct. Safety, 5 (1988) 17-34. 5. Guedes Soares, C. & Faulkner, D., Probabilistic modelling of the effect of initial imperfections on the compressive strength of rectangular plates. In Proceedings of the 3rd International Symposium on Practical Design of Ships and Mobile Units, Trondheim, Norway, June 1987, Vol. 2, pp. 783-95. 6. Jastrzebski, T. & Kmiecik, M., Statistical investigations of the deformations of ship plates (in French). Bull. Assoc. Tech. Marit. Aeronaut., 86 (1986), 325--45. 7. Lind, N. C., Reliability-based structural codes. Practical calibration. In Safety of Structures under Dynamic Loading, Vol. 1, ed. I. Holland et al., Tapir, Trondheim, Norway, 1978, pp. 149-60.
A Code Requirement for the Compressive Strength of Plate Elements
C. G u e d e s Soares Naval architecture and Marine Engineering Section, Technical University of Lisbon, Av. Rovisco Pais, IST, 1096 Lisbon, Portugal (Received 8 December 1987; accepted 5 January 1988)
ABSTRACT Two design equations are derived in a simple form suitable for incorporation in design codesfor merchant ships and warships. The equations are derived from a full description of the variables that govern plate strength which are weighted by the probability density function of the plate geometric variables that are applicable to each ship type. Key words." Plate strength, collapse strength, residual stresses, initial distortions, uncertainty modelling, code design, safety factors, code format. 1 INTRODUCTION Design codes represent the accepted rules of practice in the design of the structures that they govern. They provide a balance between safety and economy, avoiding the clearly unsafe and uneconomical designs. The relationship between e c o n o m y and safety depends on the type of structure considered and for a code to be optimised, consideration must be given to its scope. This includes the type of structures and loading, as well as the range of dimensions of the components. In formulating a code requirement, a balance must be achieved between its accuracy and simplicity, both of which are desired features. As discussed in greater detail in Ref. 1, the design variables that are 71 Marine Structures 0951-8339/88/$03.50 © 1988 Elsevier Applied Science Publishers Ltd, England. Printed in Great Britain.
72
C. Guedes Soares
chosen to be included in the code requirement must be the ones that are more relevant to the feature under consideration. By neglecting less important variables, the design method is not expected to reproduce exactly the strength of all components to which it is applicable, but only to do so in an average sense. Any design method is also expected never to give large discrepancies in any individual case to which it is applied. Faulkner 2 proposed a design method for rectangular plate elements subjected to uniaxial compressive loads. The equation accounts explicity for plate slenderness and for the effect of weld-induced residual stresses. The equation was derived from a comparison with experimental results of plates that have a certain degree of initial distortions, although it does not include that variable explicitly. Recently, the author has studied the problem in a systematic way, showing that in the range of common values of plate slenderness, the compressive strength of the plate elements could change by as much as 60%. Furthermore, the strength of nearly perfect plates could be reduced by as much as 20% in the normal range of variation of initial distortions and of residual stresses. This difference in strength is also similar to the one between simply supported and clamped plates. To be consistent, one should account explicitly for all the variables that have the same degree of influence on the strength. This was the view taken in Ref. 1 where Faulkner's method was extended to account explicitly for the effect of the level ofintial distortions. This was achieved by calibrating the design formulas with results from experiments and from calculations with sophisticated numerical methods. In having a design equation that accounts explicitly for plate slenderness, for the level of residual stresses and initial distortions as well as boundary conditions, one has a good tool for design and for strength assessments. The problems involved in using the same equation for both purposes, as well as the requirements for its format, are discussed in Ref. 3. In design one does not always know the values of all relevant variables, as opposed to what happens in the analysis.4 The most significant variables must always be included in the design equation but the uncertainty about the value of less relevant variables advises in many situations that they should not be included explicitly in the design equation. In this case, they must be substituted by their effect, in an average sense. However, their average effect will depend on the scope of the code, i.e. on the family of structures and on the range of dimensions to which it will apply. In the case of plate elements a simple design equation could involve explicitly only the plate slenderness, which is the most important
Requirement for the compressive strength of plate elements
73
variable governing compressive strength. This would represent a simplification of the equations proposed by Faulkner a a n d by the author) In this work such a simple design equation is derived from the recent results) It will be derived for two different code scopes to illustrate the methodology that should be adopted, as well as the different results to which it leads.
2 C O M P R E S S I V E S T R E N G T H OF PLATE E L E M E N T S F a u l k n e r proposed a method of predicting the strength of rectangular plates with initial imperfections, by which the plate strength t~F is given by: OF = O'm/Cr0 = { ~ b - mt~b
(1)
where a m is the plate average compressive stress, cr0 its yield stress, Ob is the strength of a plate with initial distortions but without residual stresses and A~b is the strength reduction due to the presence of residual stresses. These parameters are given by: al
a2
forfl > 1.0
(2)
where/3 = (b/t) (Cro/E) 1/=is the plate slenderness, b and t are the breadth and thickness of the plate, E is the Young's modulus of the material and the constants are: a l = 2.0
and
a2 = 1.0 for simple supports
(3a)
al = 2.5
and
a2 = 1.56 for clamped supports
(3b)
The residual stresses ar are assumed to be uniformly distributed across the central zone of the plate~ being equilibrated by two strips of tensile yield stresses at the edges, each with a breadth ofr/t. This assumption of stress distribution together with equilibrium considerations gives the value of the residual stresses as: err _
ao
277 (b/t) - 2r/
(4)
The strength degradation induced by those stresses is given by: A~b
-- O'r E t
a0 E
(5)
where E t is the tangent modulus of elasticity which can be approximated
C. Guedes Soares
74
by a simple expression given in Ref. 5 or by the Ostenfeld-Bleich parabola: 2 Et_( E
tz3fl 2 ) a4 +pr(1 --pr)
Et - l'0 E
for
0
(6a)
for
fl < l'9/X/~
(6b)
wherepr is the ratio of the material's proportional limit ap (reduced by the residual stresses) and the yield stress, i.e.pr = (Op - ar)/a0. The constants depend on the boundary conditions: a3 = 3.62 a4 = 13.1 for simple supports (7a) a3 = 6.31
a4
= 39.8
for clamped supports
(7b)
The proportional ratio can vary between 0.5 and 0.75, but Faulkner advises the use of 0.5.2 The generalisation of Faulkner's method proposed by the author I to account explicitly for the amplitude (6o) of initial distortions is: OG = (1"08~b){( 1
A~b '~ (1 - 0.0078r/)} 1"08~b]
{ 1 - (0"626- 0"121fl)~} (0"665 + 0"006r/ + 0"36 6° t + 0" 14/3)
(8)
where the first term in brackets predicts the strength of perfect plates, the first and the second indicate the strength of flat plates with residual stresses, the first and third give the strength of plates with initial deflections and no residual stresses, and the four terms should be used for plates that have both initial deflections and residual stresses. The model uncertainty that is associated with this expression can be described by a coefficient of variation (COV) of 0.07. 3 CHARACTERISTICS OF SHIP PLATES Ship plates have proportions which are different from those encountered in other metal structures. Even when making comparisons between different ship types, one will find different trends. To give an idea of two limiting cases, a statistical description of plate dimensions is presented here for the cases of tankers and of warships. The data base that was used for tankers consists of typical plates in the deck and bottom of the midship section of 130 tankers built between 1973
Requirementfor the compressive strength of plate dements
75
P ,40
Frigates
,36.32" .28.24 .20•16.12" .08 .04"
4
7 a
D
b
.40.
Tankers
• 361 .32" .28.24" .20.16. .12. .08' .04-
1
2
I
3
5 a
8
9
b
Fig. 1. Probability density distribution of plate aspect ratio in frigates and tankers. and 1986 with lengths between 66 m and 390 m. In the case of warships, the data is mostly from frigates. Figure 1 shows probability density functions for the distribution of the plate aspect ratio (a -- a/b), while Fig. 2 indicates the distribution of
76
C. Guedes Soares
P 36-
33-
3o-
Frigates
.27-
.24-
.21 -
.18
o
,15-
.12-
,09-
.06-
.03-
h
l
10
20
30
40
50
60
70
80
90
100
,
,
I
I
,
110
120
130
140
150
bit
[--1---I 160
170
180
P~ .36-
.33-
3o-
Tankers
27-
.24 -
.21-
.18-
.15.12-
.o9-
.06-
.03i
10
I
L~)
30
40
50
60
70
I,
8o
bit 90
Fig. 2. Probability density distribution of plate slenderness in frigates and tankers.
slendemess (b/t). It is clear that the distributions are different for each ship type. The m e a n and C O V in each case is given by:
a/b = 3.2
V = 0.32
for frigates
(9a)
Requirementfor the compressive strength of plate elements V = 0.16
for tankers
(9b)
b / t = 60
V = 0.42
for frigates
(10a)
b / t = 46
V = 0-25
for tankers
(10b)
a/b
= 4.7
77
The m e a n values of b / t correspond to slendernesses (13) of 2.0 a n d 1.5 respectively for tankers a n d frigates. As well as having different typical dimensions, ship plating in warships will similarly be expected to have different characteristic initial distortions and residual stresses. Faulkner's measurements of initial plate distortions in warships indicated that 6o/t = K f l 2
(11)
where K = 0.12 w h e n fl < 3 and K = 0.15 w h e n fl > 3. Jastrzebski and Kmiecik 6 have been conducting measurements in merchant ships suggesting another relation: 6o/t = O.O094(b/t) - 0.205
(12)
which has even a different functional dependence on plate slenderness. It will not be discussed here which one is more correct, but instead, they will be taken as representative of different ship types. Finally, concerning the level of residual stresses existing in the plates, very little information is available. F a u l k n e r 2 conducted some measurements which were c o m p a r e d with others in Ref. 4 where the width of the weld affected zone (77) was considered to be described by a m e a n and a COV of 5.25 and 0.07 respectively. This statistical description is assumed to be applicable to any type of ship. Thus, in this work, the differences between the characteristics of ship plating in warships a n d m e r c h a n t ships are concerned with plate slenderness and amplitude of initial distortion.
4 D E R I V A T I O N OF T H E D E S I G N E Q U A T I O N S To obtain a design equation that depends only on plate slenderness, it is necessary that the effect of the other parameters is taken at their expected values. Expressing eqn (8) as:
~o(fl, 0,60) = q~b(fl).B(fl,0,60)
(13)
onc aims at determining the bias factor B which will depend on the distribution ofthc governing parameters. Its mean value and variancc is given by weighting thc model error B by its probability of occurrence:
78
C. Guedes Soares
f f fBro,0,/50), fro, no,s0)dfl
(14)
dr/d/50
¢3r2 =
( Jf ,f{B(13, r/,/50)-
~}2 f(/3, r/,/50)d/3 dr/dr0
(15)
where f(/3, 17,/50) is the joint probability density function of the variables /3,17,/50. This function represents the probability that for a given ship the typical plates of the midship section have a specified value of/3, 17and/50. It is assumed that these effects are independent so that: f(/5, 77, i50) = 3q (fl) .f2 (/7) .f3 (/50)
(16)
The probability density function of plate slendernessjq is indicated in Fig. 2. The functions f2 and f3 are assumed to be Normal distributions with the mean values equal to 5.25 and to eqns (11) and (12) respectively. Using eqns (8) and (13)-(16) results in the following values = 0.824
V = 0.016 for merchant ships
(17a)
= 0.814
V = 0.030 for warships
(b/t <
110) (17b)
These values of the bias, combined with eqns (13), (2) and (3), result in two simple equations which can be used to assess the strength of plates in each type of ship. For design one is interested in incorporating some safety against the uncertainty of the governing parameters. Formal approaches to derive the partial safety factors are available 7 but in the present context one can use a characteristic value of bias which is between 2 and 3 standard deviations below the mean: Be = B (1 - kV)
(18)
where k is a constant. F o r k = 2.5, the probability thatB is smaller than Be is of the order of a few percent and the resulting characteristic design equations are:
t~em-
1.6 fl
0.8 f12
(19)
~¢m-
1.5 /3
0.75 ~2
(20)
for merchant ships (m) and warships (w), in the case of simply supported plates. The corresponding equations for clamped plates are obtained by substituting the constants in eqn (3a) by those in eqn (3b). If one were to include plates with b/t greater than 110, which represent 4% of the warship data, the values in eqn (17b) would change toB = 0.884
Requirement for the compressive strength of plate elements
79
and V = 0.468, the latter of which seems too high to be representative of the population, and eqn (18) with k = 2.5 would not be applicable. Equations (19) and (20), which are based on characteristic values, can be used to design plates which are expected to be fabricated with the common values of initial distortion and weld-induced residual stresses in merchant ships and warships. They differ from eqn (2) which does not incorporate implicitly the effect of residual stresses but does incorporate implicitly the effect of the initial distortions that were present in the set of experiments in which it was based. Equation (8), which was derived in Ref. 1, accounts explicitly for all the effects that influence plate strength. Equations (19) and (20) only account explicitly for the most important parameter, representing the effect of the others by their mean values for the two ship types.
5 CONCLUSIONS Design equations need to be as simple as possible for repetitive and failfree use but they must include the important variables to be able to predict correctly the strength of the components to be designed. In this work, two simple design equations are obtained by weighting a complete expression by the probability distributions of its less significant variables. These distributions represent the typical cases of merchant ships and warships. The main objective of this work is to draw attention to the idea that simple design equations must account for the probability distributions of typical dimensions, and consequently will be different for each structure type. The results are based on a limited statistical survey and, thus, should be viewed as indicative. However, the method presented here is applicable to other cases and can be used to update the design equations if a better statistical description of the parameters becomes available.
6 ACKNOWLEDGEMENTS The ideas presented in this work have been developed during the author's visit to the Department of Naval Architecture and Ocean Engineering of the University of Glasgow in the summer of 1986. The author is grateful to Professor D. Faulkner for his kind hospitality and for inspiring discussions. The author is also grateful to Dr C. S. Smith from the Admiralty Research Establishment in Dunfermline and to Mr A. C. Viner and Mr A. H. S. Wickham from Lloyd's Register of Shipping
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C. Guedes Soares
for having provided the statistical data used in this work. This work is part of the Research project 'Structural Reliability' that the author is conducting at C E M U L , the Centre for Mechanics and Materials of the Technical University of Lisbon, which is financially supported by INIC, the National Institute for Scientific Research.
REFERENCES 1. Guedes Soares, C., Design equation for the compressive strength of unstiffened plate elements with initial imperfections. J. Const. Steel Res., 9 (1988), in press. 2. Faulkner, D., A review of effective plating for use in the analysis of stiffened plating in bending and compression. J. Ship Res., 19 (1975) 1-17. 3. Faulkner, D., Guedes Soares, C. & Warwick, D. M., Modelling requirements for structural design and assessment. In Integrity of Offshore Structures, Vol. 3, ed. D. Faulkner, A. Incecik & M. J. Cowling, Elsevier Applied Science, London, 1988, pp. 17-27. 4. Guedes Soares, C., Uncertainty modelling in plate buckling. Struct. Safety, 5 (1988) 17-34. 5. Guedes Soares, C. & Faulkner, D., Probabilistic modelling of the effect of initial imperfections on the compressive strength of rectangular plates. In Proceedings of the 3rd International Symposium on Practical Design of Ships and Mobile Units, Trondheim, Norway, June 1987, Vol. 2, pp. 783-95. 6. Jastrzebski, T. & Kmiecik, M., Statistical investigations of the deformations of ship plates (in French). Bull. Assoc. Tech. Marit. Aeronaut., 86 (1986), 325--45. 7. Lind, N. C., Reliability-based structural codes. Practical calibration. In Safety of Structures under Dynamic Loading, Vol. 1, ed. I. Holland et al., Tapir, Trondheim, Norway, 1978, pp. 149-60.