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Special facilities of a PC-based structural design software package for the marine industry
Advances in Engineering Software 17 (1993) 87-94
Special facilities of a PC-based structural design software package for the marine industry Ai-Kah Soh & Chee-Kiong Soh Nanyang Technological University, Nanyang Avenue, Singapore 2263 Most of the existing PC-based structural design software packages are not devised for the design and analysis of marine structures. Therefore, users who are not particularly familiar with structural analysis may find them difficult to use. This paper presents the procedure for developing a software package that can be easily employed to design and analyse marine structures. Some of the useful facilities are described in detail with their possible applications highlighted.
Key words: marine structures, stiffened panels, structural design software package, buckling analysis, computer simulation. INTRODUCTION
MODELLING OF STIFFENED PANELS
Most o f the existing structural design software packages operating on personal computers, e.g. SAP90 and MSC/ PAL, 1 are not tailored for the design and analysis of marine structures. This can be seen from the difficulties encountered in using these packages to analyse a barge structure which consists o f a large number of stiffeners at the bulkheads, sideshells and deck plates. It has been a common industrial practice to model a complete barge structure or a significant portion of it using beam and truss members 2 instead of plate and shell elements due to the enormous size of core memory and disk storage required in order to use the latter. However, it is not an easy task to determine the correct truss and beam section properties for simulating unstiffened and stiffened panels, e.g. bulkheads and sideshells. In most cases, the users of these existing packages predict the section properties based on some arbitrary assumptions which are not supported by technical literature. Therefore, a facility which assists the user to determine the correct section properties is particularly useful to marine structure designers. Moreover, code check facilities which are relevant to marine structures are not commonly available in the existing packages. These facilities should be provided to reduce, if not totally eliminate, manual checks. The objective o f this paper is to introduce a low-cost computer-aided structural design system tailored for marine structures. Possible applications of such system to various marine structures will also be discussed.
It has been a common industrial practice to simulate a plate with or without minor stiffeners by a truss member, as shown in Fig. 1, based on the assumption that the said plate behaves like a shear panel. However, the main difficulty encountered in such simulation is the determination o f the correct section properties for the simulated truss member. Many structure designers predict the section properties based on some arbitrary assumptions which are not supported by technical literature. As a result, the simulation may be either too conservative or too optimistic. The former would only give rise to a costly design. But, the latter might lead to catastrophic structural failure. Basically, the contributions of minor stiffeners are not insignificant and, therefore, they should not be ignored in modelling panels that are reinforced with minor stiffeners. In fact, a number of minor stiffeners can be grouped together and simulated by an 'equivalent' beam, as shown in Fig. 2. Note that the diagonal trusses are employed to simulate the shear effects of the panels between the simulated beams. However, the section properties of the 'equivalent' beam should not be derived based on the section properties of the plating and minor stiffeners to be simulated. Moreover, the section properties of the trusses should not be determined based on arbitrary assumptions.
Buckling behaviour of stiffened panels Most of the stiffened panels of a barge structure, e.g. bulkheads and sideshells, are designed to withstand very large in-plane loadings. Therefore, buckling is the governing mode of failure of these stiffened panels.
Advances in Engineering Software 0965-9978/93/$06.00 © 1993 Elsevier Science Publishers Ltd. 87
Ai-Kah Soh, Chee-Kiong Soh
88
st: i f'f'ener"
~X--Minor
sbiP~ener
I
I
I I I
(a) / / / / / / / / / / / ./ / / /
(b)
Fig. 1. Simulation of a stiffened panel (conservative approach). (a) A panel with major and minor stiffeners; (b) the 'equivalent' framed structure; - - ,
'equivalent' beam; - - - , 'equivalent' truss to account for shear effect.
Thus, the section properties of the 'equivalent' beams should be predicted based on the buckling behaviour of panels reinforced with minor stiffeners. Williams and Wittrick 3 and Home and Narayanan4 have done significant amounts of work on panels with multiple stiffeners. However, the data obtained from the parametric studies performed by these researchers were not adequate for formulation of empirical equations to determine the section properties of 'equivalent' beams. The procedure for developing such equations, which can be easily incorporated into any existing software package, will be described in detail. The buckling behaviour of a panel reinforced with a single stiffener, as shown in Fig. 3, was studied prior to the analysis of multiple stiffener panels. The reasons were twofold: (i)
To determine the restraints to be imposed, in addition to the end restraints, in order to ensure that there was no unnecessary local buckling and that the mode of global buckling was correct. The additional restraints required can be ascertained by comparing the buckling load obtained with that predicted by the established Euler's formulae5 for elastic buckling. These additional restraints would then be applied to those models with multiple stiffeners to enable compatible comparisons be made between the buckling loads of models with different numbers of stiffeners.
(ii) To provide a reference buckling load for the study of the effects of spacing between stiffeners and the slenderness ratio on buckling load. Note that the panel width of the single-stiffener model was conservatively assumed to be 32 times the panel thickness, as shown in Fig. 3. This is in accordance with Clause 1.9.1.2 of the AISC6 specifications and codes of practice.
Buckling analysis of single-stiffener panels Figure 4 shows the typical finite element model for single-stiffener panels. The model was analysed using a well-established finite element software package, called 'PAFEC', 7 which runs on a VAX8800. This package is suitable for the present study because it has the capability to perform buckling ana!ysis. The model employed eight noded semi-loof curved shell elements,8 'PAFEC' element type 43210, which can be used for any thin, generally curved and folded shell problems. The two ends of the model were fully restrained. In addition, some other restraints were required to prevent local buckling. However, special care was taken to ensure that the structure was not over-restrained, which would give rise to unrealistic buckling loads. With reference to Fig. 5, the additional restraints to be imposed on the model were as follows: (i) the rotational degree of freedom, ~bz, was restrained at all nodes lying on the stiffener;
PC-based structural design software package for the marine industry (ii)
(i)
the ratio of the distance between two adjacent stiffeners and the length of the panel, w/l; (ii) the slenderness ratio of the corresponding single-stiffener for out-of-plane buckling of the panel, l/r; (iii) the ratio of panel to stiffener thickness, tl/t2; (iv) the ratio of stiffener width to depth, b2/bl. Figure 7 shows the typical finite-element models for panels with two and three stiffeners. The boundary conditions imposed on the models were similar to those for the single-stiffener model and, thus, the implementation procedure will not be reiterated here. The convergence study carried out for the two- and threestiffener models indicated that the optimum number of elements was 880 and 1360, respectively. Table 1 shows the buckling load of the two-stiffener panel normalised with respect to that of the corresponding single-stiffener, Pc~Per, for various values of w/l and l/r. This table clearly illustrates the influence of the geometric parameters on buckling load. An empirical equation for predicting the buckling load of a two-stiffener panel can be obtained using a geographical method in which the variation of the normalised buckling load with respect to each geometric parameter is assumed to be in the form of the parameter raised to a power. A plot of the normalised buckling load versus the appropriate parameter on a loglog scale will yield a straight line whose gradient may be interpreted as the power to which the parameter is raised.
~bz was restrained at all nodes lying on the two side edges of the panel.
A study was carried out to determine the rate of convergence of the model devised, in terms of computation time and accuracy, by varying the number of elements used for the model. This study was to ascertain the number of elements required for the model in order to obtain a reasonably converged buckling load without using excessive computation time. The optimum number of elements for the model was found to be 400. Comparison between the buckling loads obtained by the authors and the corresponding values calculated from Euler's formulae for various slenderness ratios showed that the maximum discrepancy between the two solutions was less than 5%. Thus, it was obvious that the single-stiffener model devised was acceptable and the modelling procedure can be extended to panels with multiple stiffeners.
Buckling analysis of multiple-stiffener panels Figure 6 shows typical two- and three-stiffener panels. The section properties of the 'equivalent' beams of multiple-stiffener panels were predicted by scaling those for the corresponding single-stiffener panels based on the ratio of the buckling loads obtained for the former and the latter. The geometric parameters which affect the buckling loads of multiple-stiffener panels are as follows:
.,,'X"-"
Panel
I=
minor
wibh bhree
et:ieOeners
Pane I
w i bh
~wo
~I -I-
I: I I I I I l
~1
==
I t I I I I
I I
89
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I I I I I I I I
II II
I
I
I
ii (a)
/' I I I I I I I I I i
\
\
/
\
I
\
I
\
I I
\
I
\
/
\
/
\
/
\
I
\
I
\
I
\
I
\
I
\
I
\
I
\ l
\ I
\ I
\ \
I (b)
Fig. 2. Simulation of a stiffened panel (improved method). (a) A panel with major and minor stiffeners; (b) the equivalent framed structure; - - , 'equivalent' beam; - - - , 'equivalent' truss to account for shear effect.
Ai-Kah Soh, Chee-Kiong Soh
90
Fig. 3. Typical single-stiffener panel. The following procedure can be employed to derive the empirical equation: (i)
(ii) (iii) (iv)
(v)
plot a graph of log (Pc/Pcf) versus log (w/l) to determine the gradient ml, where Pc and Pcf are the buckling loads of the two-stiffener panel and the corresponding single-stiffener panel, respectively; plot a graph of log [(Pc/Pcf)/(w/l)mq versus log (l/r) to determine the gradient m2; plot a graph of long [(P¢/Pcf)/(w/l) m' (l/r) m2] versus log (tilt2) to determine the gradient m3; plot a graph of log [Pe/Pcf)/(w/l)m'(l/r) m~ (q/t2)mq versus log (b2/bl) to determine the gradient m4 and the intercept, c, on the log (Pc/ Pcf) axis. the empirical equation in the form
Pc/Pcf=C(w/l) ml (I/r) m2 (tilt2) m3 (b2/bl) m4 can then be developed.
Table 2 shows the buckling load of the three-stiffener panel normalised with respect to that of the corresponding single stiffener panel, Pc/Per, for various values of w/l and l/r. The above procedure can again be employed to establish an empirical equation for predicting the normalised buckling load of a threestiffener panel. The proposed procedure can also be extended to establish empirical equations for panels with more than three stiffeners. The empirical equations obtained can then be used to estimate the section properties of the 'equivalent' beams based on the above-mentioned conservative approach.
Simulation of unstiffened panels using trusses An unstiffened panel can be simulated by a truss member based on the assumption that it behaves like a shear plate. A plate of length a, width b and thickness t is subject to a diagonal force F, as shown in Fig. 8. This plate is to be simulated by a truss member which is represented by a pair of dashed lines in Fig. 8. The crosssectional area of the member is assumed to be A. By considering the shear deformation of the plate, we obtain 6 = v/(aTl)2+(b72) 2
Fig. 4. Finite element model of a typical single-stiffener panel.
where 71 and % are the angular rotations of sides AB and AC, respectively, and 6 is the diagonal elongation of the plate.
PC-based structural design software package for the marine industry
91
,%,gg#i Fig. 5. Additional restraints imposed on a stiffened panel with encastred ends. A s s u m i n g that point D is m o v e d in the direction o f A D to point D' after d e f o r m a t i o n , then aTl b b3'2 - a
F = rat cos 0 + rbt sin 0 = rt~
i.e.
(2)
+ b2
By considering the stress-strain relations of the truss m e m b e r , we obtain
b23"2 3'1 = a2
F E6 A - Jhr-4-y
Since shear strain 3' = 3'1 + 3'2, we obtain
t~ -
stress r in the plate, i.e.
(3)
Substitute eqns (1) and (2) in eqn (3), we obtain
3"ab ( a2 + b2)1/2
(l)
T h e applied force F can be expressed in terms of shear
/
Ca~)
~
- c t ~ A
E~/ab - (a 2 + b2 )
/
(4)
A
(b)
Fig. 6. Typical (a) two- and (b) three-stiffener panels.
Ai-Kah Soh, Chee-Kiong Soh
92
Table 2. Normalised buckling loads of three-stiffener panels
w/ l
l/ r
Pc/ Pcf
0.0671 0.0806 0.0940 0-1074 0.1343 0.1611 0.1880 0.2149 0.0806 0.0806 0.0806 0.0806 0.0806 0.0806
200 200 200 200 200 200 200 200 160 170 180 185 190 195
2.110 2.249 2.378 2-575 2.764 2.960 2-620 2-219 2-051 2.130 2.185 2.212 2.220 2.235
Serial no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14
(a)
q/t2 = 1, b2/bl = 0.75. (v) (vi) (vii) (viii)
BOUNCON SOLVE CODECH GRAPHICS
Fig. 7. Finite element models for typical (a) two- and (b) threestiffener panels.
F U N C T I O N S OF T H E M O D U L E S
Substituting ~- = G'y in eqn (4),
' I N P U T ' is for entering data which are related to the structure to be analysed, and storing them on appropriate data files which can be used in any of the following modules when necessary. ' P R O P E R T Y ' is an important facility which assists the user to determine the properties of the beam or truss members used for simulating stiffened and unstiffened panels, respectively. The proposed procedure for simulation of stiffened and unstiffened panels can be easily implemented in ' P R O P E R T Y ' . ' R E O R D E R ' is for renumbering the node numbers
A - (a2 + b2)l'StG
(5)
abE where E and G are the moduli of elasticity and rigidity of the plate, respectively.
F R A M E W O R K OF T H E S T R U C T U R A L D E S I G N SOFTWARE PACKAGE A structural design software package devised for marine structures should consist o f the following basic modules: (i) (ii) (iii) (iv)
INPUT PROPERTY REORDER STIFF A~
Table 1. Normalised buckling loads of two-~i~ener panels Serial no. 1 2 3 4 5 6 7 8 9 10 11
wit
l/r
Pc/ Pef
0"0806 0"0940 0-1074 0-1343 0"1611 0-0806 0"0806 0"0806 0"0806 0-0806 0"0806
200 200 200 200 200 160 170 180 185 190 195
1"557 1"458 1"358 1-190 1-125 1"344 1"364 1"412 1"462 1'511 1'544
tl/t 2 : 1, b2/b 1 = 0-75.
a
: B
j'/ '/r
J't
Fig. 8. Simulation of a plate by a truss member.
PC-based structural design software package for the marine industry
v
\\\\
Fig. 9. A riser clamped to an offshore jacket. employed by the user in order to minimise the frontwidth which affects the size of memory and computation time required for analysing the structure. The reverse Cuthill McKee method 9 has been proven to be an effective node renumbering technique. 'STIFF' is for forming the stiffness matrices of all members and storing them on files. 'BOUNCON' is for implementing the boundary conditions. This module provides two important facilities for node and member releases. Node releases are for implementing support conditions, whereas member releases are for imposing boundary conditions at those member ends that are not fully restrained. The usefulness of the member-release facility can be illustrated by an example in which a riser is clamped to an offshore jacket, as shown in Fig. 9. This riser can be treated as structural members by releasing all the undesirable restraints at the member ends. Member releases can be accomplished by modifying the local stiffness matrices of the members concerned. The procedures for implementing member releases has been described in detail by Chang and Tay 1° and it will not be reiterated here.
93
'SOLVE' forms the primary stiffness matrix of the structure analysed, imposes boundary conditions at the support nodes of the structure, performs Gaussian elimination and then determines the displacement components at all nodes. The module also determines member forces and support reactions once all the nodal displacements have been obtained. 'CODECH' performs necessary member and connection checks for the whole structure to ensure that the member sizes chosen are adequate and there are no punching shear problems. The AISC6/AP111 codes are commonly used for designing steel structures, and the latter is particularly useful for marine structures. Therefore, it is advisable to include these design code checks in the package. Moreover, automatic re-sizing of members should also be included in this module to save the effort for re-analysing the structure. 'GRAPHICS' makes use of the input and output data of the structural analysis to display the undeformed and/or deformed geometries of the structure analysed. Some of the useful facilities provided by 'GRAPHICS' are listed below: (i) rotation of the model about three principal axes; (ii) zooming in to show the details of the model; (iii) panning across the screen to reveal off-screen portions; (iv) re-drawing of the original model; (v) displaying of node and member numbers in order to facilitate better correlation between the graphics shape and the actual data; (vi) magnifying of the displacements by a factor specified by the user to render easier visual observations; (vii) automatic scaling so that the initial display of the model can be matched with the graphics frame.
APPLICATIONS The structural design package discussed earlier has wide
Fig. 10. Computer model for a section of barge structure.
94
Ai-Kah Soh, Chee-Kiong Soh
offshore jacket which can be easily analysed and designed using the proposed package. CONCLUSIONS The above illustrations clearly show that the existing structural design software packages are not tailored for the design and analysis of marine structures. However, some of the useful facilities described above, e.g. the facility which assists the user to determine the section properties of the members used for simulating unstiffened and stiffened panels, can be easily implemented on any of the existing packages to reduce the time required for devising computer models. Alternatively, a low-cost PC-based structural design system can be developed without much difficulty, based on the procedure outlined above. REFERENCES
Fig. 11. Computer model of a typical offshore jacket. applications. It can be used for the design, transportation and installation of marine structures, and also for the design of onshore steel structures, buildings, etc. Figure 10 shows the computer model for analysing a section of barge structure. The procedure for setting up this model is as follows: (a) (b) (c) (d)
idealise the structure to be analysed; number all the nodes and members; determine the nodal co-ordinates for all nodes; input the dimensions of unstiffened and stiffened panels to enable the software to calculate the properties for each of the simulated members; (e) calculate the properties of other structural members; (f) set boundary conditions; (g) determine all the loading conditions to be studied.
Figure 11 shows the computer model for a typical
1. Falk, H. & Beardsley, C.W. Finite element analysis packages for personal computers, Mech. Engng, 1985, 107 (l), 54-70. 2. Cook, R.D. Elements based on assumed displacement fields, Concepts and Applications of Finite Element Analysis, 2nd edn, Wiley, New York, 1981, pp. 77-112. 3. Williams, F.W. & Wittrick, W.H. Numerical results for the initial buckling of some stiffened panels in compression, Aeronautical Quarterly, 1972, XXIII, 24-40. 4. Home, M.R. & Narayanan, R. An approximation method for the design of stiffened steel compression panels, Proc. Inst. Civil Engineers, September 1975, 2, 501-4. 5. Benham, P.P. & Crawford, R.J. Buckling instability, Mechanics of Engineering Materials, Longman Scientific and Technical, Harlow, Essex, 1987, pp. 278-302. 6. AISC. Manual of Steel Construction, 8th edn, American Institute of Steel Construction, Chicago, USA, 1980. 7. PAFEC Data Preparation, Publication of PAFEC Limited, Strelley Hall, Nottingham, UK, 1989. 8. Irons, B.M. The semiloof shell element, Finite Elements for Thin Shells and Curved Members, Wiley, New York, 1976, pp. 197-222. 9. Cuthill, E.H. Several strategies for reducing the bandwidth of matrices, Sparse Matrices and their Applications, ed. D.J. Rose & R.A. Willoughby, Plenum Press, New York, USA, 1972, pp. 157-66. 10. Chang, K.C. & Tay, S.H. Low cost structural design computer system for offshore structures, B. Engng Project Report, Nanyang Technological Institute, Singapore, February 1986. 11. API, RP2A, Recommended Practice for Planning Designing and Constructing Fixed Offshore Platforms, 17th edn, API, Dallas, USA, 1987.
Special facilities of a PC-based structural design software package for the marine industry Ai-Kah Soh & Chee-Kiong Soh Nanyang Technological University, Nanyang Avenue, Singapore 2263 Most of the existing PC-based structural design software packages are not devised for the design and analysis of marine structures. Therefore, users who are not particularly familiar with structural analysis may find them difficult to use. This paper presents the procedure for developing a software package that can be easily employed to design and analyse marine structures. Some of the useful facilities are described in detail with their possible applications highlighted.
Key words: marine structures, stiffened panels, structural design software package, buckling analysis, computer simulation. INTRODUCTION
MODELLING OF STIFFENED PANELS
Most o f the existing structural design software packages operating on personal computers, e.g. SAP90 and MSC/ PAL, 1 are not tailored for the design and analysis of marine structures. This can be seen from the difficulties encountered in using these packages to analyse a barge structure which consists o f a large number of stiffeners at the bulkheads, sideshells and deck plates. It has been a common industrial practice to model a complete barge structure or a significant portion of it using beam and truss members 2 instead of plate and shell elements due to the enormous size of core memory and disk storage required in order to use the latter. However, it is not an easy task to determine the correct truss and beam section properties for simulating unstiffened and stiffened panels, e.g. bulkheads and sideshells. In most cases, the users of these existing packages predict the section properties based on some arbitrary assumptions which are not supported by technical literature. Therefore, a facility which assists the user to determine the correct section properties is particularly useful to marine structure designers. Moreover, code check facilities which are relevant to marine structures are not commonly available in the existing packages. These facilities should be provided to reduce, if not totally eliminate, manual checks. The objective o f this paper is to introduce a low-cost computer-aided structural design system tailored for marine structures. Possible applications of such system to various marine structures will also be discussed.
It has been a common industrial practice to simulate a plate with or without minor stiffeners by a truss member, as shown in Fig. 1, based on the assumption that the said plate behaves like a shear panel. However, the main difficulty encountered in such simulation is the determination o f the correct section properties for the simulated truss member. Many structure designers predict the section properties based on some arbitrary assumptions which are not supported by technical literature. As a result, the simulation may be either too conservative or too optimistic. The former would only give rise to a costly design. But, the latter might lead to catastrophic structural failure. Basically, the contributions of minor stiffeners are not insignificant and, therefore, they should not be ignored in modelling panels that are reinforced with minor stiffeners. In fact, a number of minor stiffeners can be grouped together and simulated by an 'equivalent' beam, as shown in Fig. 2. Note that the diagonal trusses are employed to simulate the shear effects of the panels between the simulated beams. However, the section properties of the 'equivalent' beam should not be derived based on the section properties of the plating and minor stiffeners to be simulated. Moreover, the section properties of the trusses should not be determined based on arbitrary assumptions.
Buckling behaviour of stiffened panels Most of the stiffened panels of a barge structure, e.g. bulkheads and sideshells, are designed to withstand very large in-plane loadings. Therefore, buckling is the governing mode of failure of these stiffened panels.
Advances in Engineering Software 0965-9978/93/$06.00 © 1993 Elsevier Science Publishers Ltd. 87
Ai-Kah Soh, Chee-Kiong Soh
88
st: i f'f'ener"
~X--Minor
sbiP~ener
I
I
I I I
(a) / / / / / / / / / / / ./ / / /
(b)
Fig. 1. Simulation of a stiffened panel (conservative approach). (a) A panel with major and minor stiffeners; (b) the 'equivalent' framed structure; - - ,
'equivalent' beam; - - - , 'equivalent' truss to account for shear effect.
Thus, the section properties of the 'equivalent' beams should be predicted based on the buckling behaviour of panels reinforced with minor stiffeners. Williams and Wittrick 3 and Home and Narayanan4 have done significant amounts of work on panels with multiple stiffeners. However, the data obtained from the parametric studies performed by these researchers were not adequate for formulation of empirical equations to determine the section properties of 'equivalent' beams. The procedure for developing such equations, which can be easily incorporated into any existing software package, will be described in detail. The buckling behaviour of a panel reinforced with a single stiffener, as shown in Fig. 3, was studied prior to the analysis of multiple stiffener panels. The reasons were twofold: (i)
To determine the restraints to be imposed, in addition to the end restraints, in order to ensure that there was no unnecessary local buckling and that the mode of global buckling was correct. The additional restraints required can be ascertained by comparing the buckling load obtained with that predicted by the established Euler's formulae5 for elastic buckling. These additional restraints would then be applied to those models with multiple stiffeners to enable compatible comparisons be made between the buckling loads of models with different numbers of stiffeners.
(ii) To provide a reference buckling load for the study of the effects of spacing between stiffeners and the slenderness ratio on buckling load. Note that the panel width of the single-stiffener model was conservatively assumed to be 32 times the panel thickness, as shown in Fig. 3. This is in accordance with Clause 1.9.1.2 of the AISC6 specifications and codes of practice.
Buckling analysis of single-stiffener panels Figure 4 shows the typical finite element model for single-stiffener panels. The model was analysed using a well-established finite element software package, called 'PAFEC', 7 which runs on a VAX8800. This package is suitable for the present study because it has the capability to perform buckling ana!ysis. The model employed eight noded semi-loof curved shell elements,8 'PAFEC' element type 43210, which can be used for any thin, generally curved and folded shell problems. The two ends of the model were fully restrained. In addition, some other restraints were required to prevent local buckling. However, special care was taken to ensure that the structure was not over-restrained, which would give rise to unrealistic buckling loads. With reference to Fig. 5, the additional restraints to be imposed on the model were as follows: (i) the rotational degree of freedom, ~bz, was restrained at all nodes lying on the stiffener;
PC-based structural design software package for the marine industry (ii)
(i)
the ratio of the distance between two adjacent stiffeners and the length of the panel, w/l; (ii) the slenderness ratio of the corresponding single-stiffener for out-of-plane buckling of the panel, l/r; (iii) the ratio of panel to stiffener thickness, tl/t2; (iv) the ratio of stiffener width to depth, b2/bl. Figure 7 shows the typical finite-element models for panels with two and three stiffeners. The boundary conditions imposed on the models were similar to those for the single-stiffener model and, thus, the implementation procedure will not be reiterated here. The convergence study carried out for the two- and threestiffener models indicated that the optimum number of elements was 880 and 1360, respectively. Table 1 shows the buckling load of the two-stiffener panel normalised with respect to that of the corresponding single-stiffener, Pc~Per, for various values of w/l and l/r. This table clearly illustrates the influence of the geometric parameters on buckling load. An empirical equation for predicting the buckling load of a two-stiffener panel can be obtained using a geographical method in which the variation of the normalised buckling load with respect to each geometric parameter is assumed to be in the form of the parameter raised to a power. A plot of the normalised buckling load versus the appropriate parameter on a loglog scale will yield a straight line whose gradient may be interpreted as the power to which the parameter is raised.
~bz was restrained at all nodes lying on the two side edges of the panel.
A study was carried out to determine the rate of convergence of the model devised, in terms of computation time and accuracy, by varying the number of elements used for the model. This study was to ascertain the number of elements required for the model in order to obtain a reasonably converged buckling load without using excessive computation time. The optimum number of elements for the model was found to be 400. Comparison between the buckling loads obtained by the authors and the corresponding values calculated from Euler's formulae for various slenderness ratios showed that the maximum discrepancy between the two solutions was less than 5%. Thus, it was obvious that the single-stiffener model devised was acceptable and the modelling procedure can be extended to panels with multiple stiffeners.
Buckling analysis of multiple-stiffener panels Figure 6 shows typical two- and three-stiffener panels. The section properties of the 'equivalent' beams of multiple-stiffener panels were predicted by scaling those for the corresponding single-stiffener panels based on the ratio of the buckling loads obtained for the former and the latter. The geometric parameters which affect the buckling loads of multiple-stiffener panels are as follows:
.,,'X"-"
Panel
I=
minor
wibh bhree
et:ieOeners
Pane I
w i bh
~wo
~I -I-
I: I I I I I l
~1
==
I t I I I I
I I
89
r"
I I I I I I I I
II II
I
I
I
ii (a)
/' I I I I I I I I I i
\
\
/
\
I
\
I
\
I I
\
I
\
/
\
/
\
/
\
I
\
I
\
I
\
I
\
I
\
I
\
I
\ l
\ I
\ I
\ \
I (b)
Fig. 2. Simulation of a stiffened panel (improved method). (a) A panel with major and minor stiffeners; (b) the equivalent framed structure; - - , 'equivalent' beam; - - - , 'equivalent' truss to account for shear effect.
Ai-Kah Soh, Chee-Kiong Soh
90
Fig. 3. Typical single-stiffener panel. The following procedure can be employed to derive the empirical equation: (i)
(ii) (iii) (iv)
(v)
plot a graph of log (Pc/Pcf) versus log (w/l) to determine the gradient ml, where Pc and Pcf are the buckling loads of the two-stiffener panel and the corresponding single-stiffener panel, respectively; plot a graph of log [(Pc/Pcf)/(w/l)mq versus log (l/r) to determine the gradient m2; plot a graph of long [(P¢/Pcf)/(w/l) m' (l/r) m2] versus log (tilt2) to determine the gradient m3; plot a graph of log [Pe/Pcf)/(w/l)m'(l/r) m~ (q/t2)mq versus log (b2/bl) to determine the gradient m4 and the intercept, c, on the log (Pc/ Pcf) axis. the empirical equation in the form
Pc/Pcf=C(w/l) ml (I/r) m2 (tilt2) m3 (b2/bl) m4 can then be developed.
Table 2 shows the buckling load of the three-stiffener panel normalised with respect to that of the corresponding single stiffener panel, Pc/Per, for various values of w/l and l/r. The above procedure can again be employed to establish an empirical equation for predicting the normalised buckling load of a threestiffener panel. The proposed procedure can also be extended to establish empirical equations for panels with more than three stiffeners. The empirical equations obtained can then be used to estimate the section properties of the 'equivalent' beams based on the above-mentioned conservative approach.
Simulation of unstiffened panels using trusses An unstiffened panel can be simulated by a truss member based on the assumption that it behaves like a shear plate. A plate of length a, width b and thickness t is subject to a diagonal force F, as shown in Fig. 8. This plate is to be simulated by a truss member which is represented by a pair of dashed lines in Fig. 8. The crosssectional area of the member is assumed to be A. By considering the shear deformation of the plate, we obtain 6 = v/(aTl)2+(b72) 2
Fig. 4. Finite element model of a typical single-stiffener panel.
where 71 and % are the angular rotations of sides AB and AC, respectively, and 6 is the diagonal elongation of the plate.
PC-based structural design software package for the marine industry
91
,%,gg#i Fig. 5. Additional restraints imposed on a stiffened panel with encastred ends. A s s u m i n g that point D is m o v e d in the direction o f A D to point D' after d e f o r m a t i o n , then aTl b b3'2 - a
F = rat cos 0 + rbt sin 0 = rt~
i.e.
(2)
+ b2
By considering the stress-strain relations of the truss m e m b e r , we obtain
b23"2 3'1 = a2
F E6 A - Jhr-4-y
Since shear strain 3' = 3'1 + 3'2, we obtain
t~ -
stress r in the plate, i.e.
(3)
Substitute eqns (1) and (2) in eqn (3), we obtain
3"ab ( a2 + b2)1/2
(l)
T h e applied force F can be expressed in terms of shear
/
Ca~)
~
- c t ~ A
E~/ab - (a 2 + b2 )
/
(4)
A
(b)
Fig. 6. Typical (a) two- and (b) three-stiffener panels.
Ai-Kah Soh, Chee-Kiong Soh
92
Table 2. Normalised buckling loads of three-stiffener panels
w/ l
l/ r
Pc/ Pcf
0.0671 0.0806 0.0940 0-1074 0.1343 0.1611 0.1880 0.2149 0.0806 0.0806 0.0806 0.0806 0.0806 0.0806
200 200 200 200 200 200 200 200 160 170 180 185 190 195
2.110 2.249 2.378 2-575 2.764 2.960 2-620 2-219 2-051 2.130 2.185 2.212 2.220 2.235
Serial no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14
(a)
q/t2 = 1, b2/bl = 0.75. (v) (vi) (vii) (viii)
BOUNCON SOLVE CODECH GRAPHICS
Fig. 7. Finite element models for typical (a) two- and (b) threestiffener panels.
F U N C T I O N S OF T H E M O D U L E S
Substituting ~- = G'y in eqn (4),
' I N P U T ' is for entering data which are related to the structure to be analysed, and storing them on appropriate data files which can be used in any of the following modules when necessary. ' P R O P E R T Y ' is an important facility which assists the user to determine the properties of the beam or truss members used for simulating stiffened and unstiffened panels, respectively. The proposed procedure for simulation of stiffened and unstiffened panels can be easily implemented in ' P R O P E R T Y ' . ' R E O R D E R ' is for renumbering the node numbers
A - (a2 + b2)l'StG
(5)
abE where E and G are the moduli of elasticity and rigidity of the plate, respectively.
F R A M E W O R K OF T H E S T R U C T U R A L D E S I G N SOFTWARE PACKAGE A structural design software package devised for marine structures should consist o f the following basic modules: (i) (ii) (iii) (iv)
INPUT PROPERTY REORDER STIFF A~
Table 1. Normalised buckling loads of two-~i~ener panels Serial no. 1 2 3 4 5 6 7 8 9 10 11
wit
l/r
Pc/ Pef
0"0806 0"0940 0-1074 0-1343 0"1611 0-0806 0"0806 0"0806 0"0806 0-0806 0"0806
200 200 200 200 200 160 170 180 185 190 195
1"557 1"458 1"358 1-190 1-125 1"344 1"364 1"412 1"462 1'511 1'544
tl/t 2 : 1, b2/b 1 = 0-75.
a
: B
j'/ '/r
J't
Fig. 8. Simulation of a plate by a truss member.
PC-based structural design software package for the marine industry
v
\\\\
Fig. 9. A riser clamped to an offshore jacket. employed by the user in order to minimise the frontwidth which affects the size of memory and computation time required for analysing the structure. The reverse Cuthill McKee method 9 has been proven to be an effective node renumbering technique. 'STIFF' is for forming the stiffness matrices of all members and storing them on files. 'BOUNCON' is for implementing the boundary conditions. This module provides two important facilities for node and member releases. Node releases are for implementing support conditions, whereas member releases are for imposing boundary conditions at those member ends that are not fully restrained. The usefulness of the member-release facility can be illustrated by an example in which a riser is clamped to an offshore jacket, as shown in Fig. 9. This riser can be treated as structural members by releasing all the undesirable restraints at the member ends. Member releases can be accomplished by modifying the local stiffness matrices of the members concerned. The procedures for implementing member releases has been described in detail by Chang and Tay 1° and it will not be reiterated here.
93
'SOLVE' forms the primary stiffness matrix of the structure analysed, imposes boundary conditions at the support nodes of the structure, performs Gaussian elimination and then determines the displacement components at all nodes. The module also determines member forces and support reactions once all the nodal displacements have been obtained. 'CODECH' performs necessary member and connection checks for the whole structure to ensure that the member sizes chosen are adequate and there are no punching shear problems. The AISC6/AP111 codes are commonly used for designing steel structures, and the latter is particularly useful for marine structures. Therefore, it is advisable to include these design code checks in the package. Moreover, automatic re-sizing of members should also be included in this module to save the effort for re-analysing the structure. 'GRAPHICS' makes use of the input and output data of the structural analysis to display the undeformed and/or deformed geometries of the structure analysed. Some of the useful facilities provided by 'GRAPHICS' are listed below: (i) rotation of the model about three principal axes; (ii) zooming in to show the details of the model; (iii) panning across the screen to reveal off-screen portions; (iv) re-drawing of the original model; (v) displaying of node and member numbers in order to facilitate better correlation between the graphics shape and the actual data; (vi) magnifying of the displacements by a factor specified by the user to render easier visual observations; (vii) automatic scaling so that the initial display of the model can be matched with the graphics frame.
APPLICATIONS The structural design package discussed earlier has wide
Fig. 10. Computer model for a section of barge structure.
94
Ai-Kah Soh, Chee-Kiong Soh
offshore jacket which can be easily analysed and designed using the proposed package. CONCLUSIONS The above illustrations clearly show that the existing structural design software packages are not tailored for the design and analysis of marine structures. However, some of the useful facilities described above, e.g. the facility which assists the user to determine the section properties of the members used for simulating unstiffened and stiffened panels, can be easily implemented on any of the existing packages to reduce the time required for devising computer models. Alternatively, a low-cost PC-based structural design system can be developed without much difficulty, based on the procedure outlined above. REFERENCES
Fig. 11. Computer model of a typical offshore jacket. applications. It can be used for the design, transportation and installation of marine structures, and also for the design of onshore steel structures, buildings, etc. Figure 10 shows the computer model for analysing a section of barge structure. The procedure for setting up this model is as follows: (a) (b) (c) (d)
idealise the structure to be analysed; number all the nodes and members; determine the nodal co-ordinates for all nodes; input the dimensions of unstiffened and stiffened panels to enable the software to calculate the properties for each of the simulated members; (e) calculate the properties of other structural members; (f) set boundary conditions; (g) determine all the loading conditions to be studied.
Figure 11 shows the computer model for a typical
1. Falk, H. & Beardsley, C.W. Finite element analysis packages for personal computers, Mech. Engng, 1985, 107 (l), 54-70. 2. Cook, R.D. Elements based on assumed displacement fields, Concepts and Applications of Finite Element Analysis, 2nd edn, Wiley, New York, 1981, pp. 77-112. 3. Williams, F.W. & Wittrick, W.H. Numerical results for the initial buckling of some stiffened panels in compression, Aeronautical Quarterly, 1972, XXIII, 24-40. 4. Home, M.R. & Narayanan, R. An approximation method for the design of stiffened steel compression panels, Proc. Inst. Civil Engineers, September 1975, 2, 501-4. 5. Benham, P.P. & Crawford, R.J. Buckling instability, Mechanics of Engineering Materials, Longman Scientific and Technical, Harlow, Essex, 1987, pp. 278-302. 6. AISC. Manual of Steel Construction, 8th edn, American Institute of Steel Construction, Chicago, USA, 1980. 7. PAFEC Data Preparation, Publication of PAFEC Limited, Strelley Hall, Nottingham, UK, 1989. 8. Irons, B.M. The semiloof shell element, Finite Elements for Thin Shells and Curved Members, Wiley, New York, 1976, pp. 197-222. 9. Cuthill, E.H. Several strategies for reducing the bandwidth of matrices, Sparse Matrices and their Applications, ed. D.J. Rose & R.A. Willoughby, Plenum Press, New York, USA, 1972, pp. 157-66. 10. Chang, K.C. & Tay, S.H. Low cost structural design computer system for offshore structures, B. Engng Project Report, Nanyang Technological Institute, Singapore, February 1986. 11. API, RP2A, Recommended Practice for Planning Designing and Constructing Fixed Offshore Platforms, 17th edn, API, Dallas, USA, 1987.