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Snap-through behaviour of cables in flexible structures
Computers and Structures 77 (2000) 361±369
www.elsevier.com/locate/compstruc
Snap-through behaviour of cables in ¯exible structures Malcolm A. Millar a,*, Majid Barghian b a
Department of Civil and Structural Engineering, UMIST, P.O. Box 88, Manchester M60 1QD, UK b Department of Civil and Structural Engineering, Tabriz University, Tabriz 711-51664, Iran Received 7 February 1998; accepted 17 July 1999
Abstract A study of the static and dynamic behaviour of ¯exible structures that exhibit the snap-through phenomenon is presented in this paper. A single cable example is presented to show that dynamic analysis can be an eective method of solving static snap-through problems. This paper presents an in-depth study of the behaviour of a guyed mast structure that has received attention in a number of previous elastic investigations [Peterson C. Abgespannte Maste und Schornsteine, Statik und Dynamik. Berlin: Wilhelm Ernst and Sohn Verlag, 1970; Argyris JH, Dunne PC, Angelopoulos T. Comp Meth Appl Mech Engng 1973;2:203±50; El-Katt MTH. Non-linear dynamic analysis of cable structures. PhD Thesis, UMIST, 1987]. The Williams toggle is an example of a rigid frame snap-through behaviour [Williams FW. Quart J Mech Appl Mech 1964;17:456±69]. Meek and Xue [Meek JL, Xue Q. Comput Meth Appl Mech Engng 1996;136:347±61] present a similar approach to the authors using dynamic analysis to trace the snap-through behaviour of rigid framed structures without having to follow the drooping load path. The very ¯exible guy snap-through response has not been previously reported. Ó 2000 Civil-Comp Ltd. and Elsevier Science Ltd. All rights reserved. Keywords: Non-linear; Cables; Flexible; Structures; Snap-through
1. Introduction An elastic structure with non-linear softening characteristics, in which the structural stiness decreases with increasing load, may ultimately become unstable. At this stage, the stiness of the structure is zero. In the unstable state the stiness the structure will naturally seek another stable equilibrium state if one exists at the instability load. The move to a new stable equilibrium state may involve extremely large changes in deformation. This behaviour is commonly referred to as the Ôsnap-throughÕ phenomenon and is illustrated in Fig. 1. A study of the behaviour of two ¯exible structures that exhibit snap-through is presented. The ®rst structure is a simple suspended cable which is used to
*
Corresponding author.
present the fundamental snap-through behaviour. The second structure is a guyed mast structure that has received attention in a number of previous elastic investigations, but the details of the guy snap-through response have not been previously reported. Due to their lightweight construction and the overall ¯exibility, the guyed masts that support the transmitters often have relatively large displacements when subjected to external forces such as wind, snow and ice. The performance of the transmitters is affected by these deformations; therefore, a close check on the detailed static and dynamic behaviour of the tower is a design requirement. A ®nite element program (FINELE) has been developed by the authors at UMIST to analyse linear, non-linear elastic static and dynamic problems [1] for structures comprising frames, trusses, grid and cable elements. An option to use either time integration or modal superposition techniques is available for solving the equations of motion.
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Fig. 1. Snap-through phenomenon.
2. Examples of snap-through behaviour 2.1. A simple suspended cable The curve of the cable is modelled as a series of straight links with a constant axial stiness. A time step of 0.01 s has been used for each of the three cases of Section 2.1. 2.1.1. Response due to gravity and imposed loading (case 1) A simple cable comprising ®ve elements is considered hanging under its own self-weight as shown in Fig. 2. Gravity loads of )0.01 kN each are applied at nodes 2±5 assuming the Z-direction is positive upwards. The area of the cable, the modulus of elasticity and the initial unstrained lengths are assumed as 500 mm2 , 105 kN/mm2 and 3000 mm, respectively. The vertical displacements of the cable under the gravity loads are as follows: for nodes 2 and 5, )0.2094 mm and for nodes 3 and 4, )0.3147 mm. The horizontal displacements are approximately zero. Additional static loads of )1.0 kN each are applied at nodes 2, 3, 4 and 5. The displacements are presented in Table 1. The same example is considered as a dynamic problem. The displacements under gravity loads are added to the original coordinates and the dynamic loads of
Fig. 2. Cable structure. Table 1
Fig. 3. Dynamic response of nodes 2±5.
)1.0 kN each are added to the initial gravity loads at nodes 2±5. The structure is damped and settles down to the static solution (Fig. 3) using a damping coecient of 15%. 2.1.2. Response due to gravity and reversed imposed loading (case 2) The cable as used in case 1 is subjected to the gravity loads plus an upward force of +1.0 kN. A static analysis fails because the snap through phenomenon occurs. Using the dynamic analysis with a similar damping coecient, the static solution is obtained without the snap-through instability. The ®nal response gives the following displacements as shown in Table 2 with the dynamic response given in Fig. 4. 2.1.3. Response due to a de®ned load-time function (case 3) The same cable is analysed as a dynamic problem with a loading pattern that temporarily reverses the 1.0 kN nodal loads spread over a period of 350 s as shown in Fig. 5. The response presented in Fig. 6 shows that the static solution obtained after approximately 80 s returns after a similar period of time when the applied load has reTable 2
Node
Horizontal displacement (mm)
Vertical displacement (mm)
Node
Horizontal displacement (mm)
Vertical displacement (mm)
2 3 4 5
)0.02 )0.0099 0.0099 0.02
)8.4597 )12.532 )12.532 )8.4597
2 3 4 5
)0.0193 )0.0096 0.0096 0.0193
28.523 42.786 42.786 28.523
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Fig. 4. Dynamic response of nodes 2±5.
Fig. 5. Load±time function curve.
Fig. 7. De¯ected shapes of the cable when modelled with 30 elements at time intervals: (a) 0±50 s, (b) 50±195 s, (c) 195±260 s, and (d) 260±500 s.
Fig. 6. Dynamic response of nodes 2±5.
turned to a value of )1.0 kN. The positive loads produce similar displacements to those obtained in Table 2 and Fig. 4. A damping coecient of 15% has again been used to obtain rapid convergence of the displacements to the static solution when the load level is constant. Fig. 6 shows the behaviour when the cable is modelled using ®ve elements. A more detailed study of the
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behaviour based on a model using 30 elements is given in Fig. 7 which shows the shape of the cable as it changes from a sagging to a hogging pro®le and back to a sagging pro®le as the time dependent loading shown in Fig. 5 is applied. 2.2. A guyed transmitter tower The static and dynamic behaviour of the guyed transmitter mast shown in Fig. 8 is presented. The element properties are given in Table 3. The static behaviour was analysed under gravity loads of )250 kgf ()2.45 kN) applied at each tower point, acting in the positive direction of the Y-axis. Every upper cable node has a gravity load of )150 kgf ()1.47 kN) at the intermediate points and the lower cables have corresponding values of )75 kgf ()0.74 kN). Wind loads of 1000 kgf (9.81 kN) are applied at both the tower and cable nodes. The FINELE result is compared with the NONSAP program. The static load de¯ection curve for the horizontal displacement of the top of the mast (point B) is shown in Fig. 9. The masses given in Table 4 were used at each node in the X, Y and Z-directions. The static displacements obtained from the gravity loads are added to coordinates for the dynamic analysis. The responses of nodes A and B due to the wind load of a half sine wave, of maximum value of 300 kgf (2.94 kN) (Fig. 10) applied at each point are shown in Fig. 11. The individual plots are coincident for NONSAP and FINELE in this ®gure. Two wind conditions are considered: the ®rst with the wind in the Y-direction which produces a predominantly
Fig. 9. Static load±displacements (Y-direction) of node B in the transmitter tower.
Table 4 Masses used for the transmitter tower nodes The mass used at each nodal point (kgf s2 /cm) Tower nodes Lower cables nodes Upper cables nodes
0.25482 0.076453 0.152905
Fig. 10. Load±time curve (wind load).
Fig. 8. Guyed transmitter mast. Table 3 Data input for the transmitter tower members
PGN1 PGN2 PGN3 Lower cables Upper cables
Area (cm2 )
Modulus of elasticity, E (kgf/cm2 )
Pre-load (kgf)
1.0 1.0 1.0 1.0 1.0
1.0816 ´ 108 2.112 ´ 107 8.16 ´ 106 1.7115 ´ 107 2.1273 ´ 107
± ± ± )1.0 ´ 104 )1.2 ´ 104
Fig. 11. Response of nodes A and B.
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365
Fig. 12. De¯ection of rear cables and mast (with no damping) due to the wind load in the Y-direction.
symmetrical response and the second when the wind is in the X-direction causing the structure to twist. Plots of the behaviour of the guys under wind conditions are presented in Figs. 12±14. At the early time step and after the imposed load has been removed, the mast and guy displacements are in-
dependent of the damping coecient, and therefore, the plots are coincident. A time step of 0.005 s has been used to analyse and plot the displacements for the following ®gures. In all the ®gures, dotted lines represent the original shape of the cables and mast leg. Fig. 12 shows the de¯ections of
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Fig. 13. De¯ection of rear cables and mast with dierent damping ratios.
the rear cables and mast due to the wind in the Ydirection. Damping has been ignored. Fig. 13 presents the plots of de¯ections of the rear cables and mast for different damping ratios at various times. Finally, Fig. 14 shows the de¯ections when the wind load is applied in the X-direction. Damping has been ignored. The corre-
sponding plans are presented to highlight the out-ofplane displacements of the cables. In Fig. 12 at the beginning of the response, the eect of the wind load is insigni®cant and the initial displacements obtained from static analysis under gravity loads causes the cables to take a sagging shape. The
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367
Fig. 14. De¯ection of rear cables and mast (with no damping), wind load is in the X direction.
shape of the mast is vertical to the ground. The assumed sinusoidal wind load increases and reaches a maximum value at time 1.5 s. Then, the wind load decreases and after time 3 s, the wind load vanishes. All wind is assumed to act horizontally. The maximum de¯ection of the cables of the mast occurs around time 1:75 s. The
mast and cables de¯ect in the Y-direction (wind load direction). The rear cables pass their original straight line and billow up and out. After time 1:75 s, the direction of de¯ection changes. The cables return to a sagging pro®le again after time 3 s and the mast passes through the vertical after time 4 s. The
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Fig. 14. (continued)
vibration of the mast and cables continues, but the amplitude of the de¯ections reduces. The shape of the cables and the mast tends towards the position at time 0. Since damping has been ignored, the cables and the mast still vibrate slowly at time t 12 s.
3. Applications The application of the method presented using the dynamic response of the structure, including the cables, enables the analyst to trace the behaviour of the struc-
M.A. Millar, M. Barghian / Computers and Structures 77 (2000) 361±369
369
ture through a locally unstable region of the load± displacement path. Many commercial programs treat cables as non-linear elastic single elements and are not, therefore, able to trace the structural behaviour of guyed masts through such unstable regions. Loads corresponding to ice on the structure can also be included in the analysis. Both the gravity forces due to the dead weight of the ice and the additional surface area attracting additional wind forces will aect the overall structural behaviour. Temperature eects may also have a detrimental eect on the structural response. A rise in the temperature will make the guys less sti, and hence increase the displacements, whereas a drop in the temperature may produce additional compression forces in the mast that cause it to buckle. All these eects can be modelled using FINELE.
response of cable structures that have local unstable behaviour due to snap-through. Static non-linear problems that exhibit snap-through behaviour can be eectively analysed using a dynamic approach with damping. When snap-through occurs, the static equilibrium equation KD W becomes singular, K being the tangent stiness. However, if a dynamic analysis is used, the equations of motion include both the inertia and damping forces which are non-zero, and therefore, the equations are solvable.
4. Conclusion
Reference
The UMIST ®nite element program, FINELE has been used eectively to trace the non-linear dynamic
[1] Barghian M. Static and dynamic analysis of bar element structures. PhD Thesis, UMIST, 1996.
Acknowledgements Manchester Computing for the use of the NONSAP program is acknowledged.
www.elsevier.com/locate/compstruc
Snap-through behaviour of cables in ¯exible structures Malcolm A. Millar a,*, Majid Barghian b a
Department of Civil and Structural Engineering, UMIST, P.O. Box 88, Manchester M60 1QD, UK b Department of Civil and Structural Engineering, Tabriz University, Tabriz 711-51664, Iran Received 7 February 1998; accepted 17 July 1999
Abstract A study of the static and dynamic behaviour of ¯exible structures that exhibit the snap-through phenomenon is presented in this paper. A single cable example is presented to show that dynamic analysis can be an eective method of solving static snap-through problems. This paper presents an in-depth study of the behaviour of a guyed mast structure that has received attention in a number of previous elastic investigations [Peterson C. Abgespannte Maste und Schornsteine, Statik und Dynamik. Berlin: Wilhelm Ernst and Sohn Verlag, 1970; Argyris JH, Dunne PC, Angelopoulos T. Comp Meth Appl Mech Engng 1973;2:203±50; El-Katt MTH. Non-linear dynamic analysis of cable structures. PhD Thesis, UMIST, 1987]. The Williams toggle is an example of a rigid frame snap-through behaviour [Williams FW. Quart J Mech Appl Mech 1964;17:456±69]. Meek and Xue [Meek JL, Xue Q. Comput Meth Appl Mech Engng 1996;136:347±61] present a similar approach to the authors using dynamic analysis to trace the snap-through behaviour of rigid framed structures without having to follow the drooping load path. The very ¯exible guy snap-through response has not been previously reported. Ó 2000 Civil-Comp Ltd. and Elsevier Science Ltd. All rights reserved. Keywords: Non-linear; Cables; Flexible; Structures; Snap-through
1. Introduction An elastic structure with non-linear softening characteristics, in which the structural stiness decreases with increasing load, may ultimately become unstable. At this stage, the stiness of the structure is zero. In the unstable state the stiness the structure will naturally seek another stable equilibrium state if one exists at the instability load. The move to a new stable equilibrium state may involve extremely large changes in deformation. This behaviour is commonly referred to as the Ôsnap-throughÕ phenomenon and is illustrated in Fig. 1. A study of the behaviour of two ¯exible structures that exhibit snap-through is presented. The ®rst structure is a simple suspended cable which is used to
*
Corresponding author.
present the fundamental snap-through behaviour. The second structure is a guyed mast structure that has received attention in a number of previous elastic investigations, but the details of the guy snap-through response have not been previously reported. Due to their lightweight construction and the overall ¯exibility, the guyed masts that support the transmitters often have relatively large displacements when subjected to external forces such as wind, snow and ice. The performance of the transmitters is affected by these deformations; therefore, a close check on the detailed static and dynamic behaviour of the tower is a design requirement. A ®nite element program (FINELE) has been developed by the authors at UMIST to analyse linear, non-linear elastic static and dynamic problems [1] for structures comprising frames, trusses, grid and cable elements. An option to use either time integration or modal superposition techniques is available for solving the equations of motion.
0045-7949/00/$ - see front matter Ó 2000 Civil-Comp Ltd. and Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 5 - 7 9 4 9 ( 0 0 ) 0 0 0 2 7 - 4
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Fig. 1. Snap-through phenomenon.
2. Examples of snap-through behaviour 2.1. A simple suspended cable The curve of the cable is modelled as a series of straight links with a constant axial stiness. A time step of 0.01 s has been used for each of the three cases of Section 2.1. 2.1.1. Response due to gravity and imposed loading (case 1) A simple cable comprising ®ve elements is considered hanging under its own self-weight as shown in Fig. 2. Gravity loads of )0.01 kN each are applied at nodes 2±5 assuming the Z-direction is positive upwards. The area of the cable, the modulus of elasticity and the initial unstrained lengths are assumed as 500 mm2 , 105 kN/mm2 and 3000 mm, respectively. The vertical displacements of the cable under the gravity loads are as follows: for nodes 2 and 5, )0.2094 mm and for nodes 3 and 4, )0.3147 mm. The horizontal displacements are approximately zero. Additional static loads of )1.0 kN each are applied at nodes 2, 3, 4 and 5. The displacements are presented in Table 1. The same example is considered as a dynamic problem. The displacements under gravity loads are added to the original coordinates and the dynamic loads of
Fig. 2. Cable structure. Table 1
Fig. 3. Dynamic response of nodes 2±5.
)1.0 kN each are added to the initial gravity loads at nodes 2±5. The structure is damped and settles down to the static solution (Fig. 3) using a damping coecient of 15%. 2.1.2. Response due to gravity and reversed imposed loading (case 2) The cable as used in case 1 is subjected to the gravity loads plus an upward force of +1.0 kN. A static analysis fails because the snap through phenomenon occurs. Using the dynamic analysis with a similar damping coecient, the static solution is obtained without the snap-through instability. The ®nal response gives the following displacements as shown in Table 2 with the dynamic response given in Fig. 4. 2.1.3. Response due to a de®ned load-time function (case 3) The same cable is analysed as a dynamic problem with a loading pattern that temporarily reverses the 1.0 kN nodal loads spread over a period of 350 s as shown in Fig. 5. The response presented in Fig. 6 shows that the static solution obtained after approximately 80 s returns after a similar period of time when the applied load has reTable 2
Node
Horizontal displacement (mm)
Vertical displacement (mm)
Node
Horizontal displacement (mm)
Vertical displacement (mm)
2 3 4 5
)0.02 )0.0099 0.0099 0.02
)8.4597 )12.532 )12.532 )8.4597
2 3 4 5
)0.0193 )0.0096 0.0096 0.0193
28.523 42.786 42.786 28.523
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363
Fig. 4. Dynamic response of nodes 2±5.
Fig. 5. Load±time function curve.
Fig. 7. De¯ected shapes of the cable when modelled with 30 elements at time intervals: (a) 0±50 s, (b) 50±195 s, (c) 195±260 s, and (d) 260±500 s.
Fig. 6. Dynamic response of nodes 2±5.
turned to a value of )1.0 kN. The positive loads produce similar displacements to those obtained in Table 2 and Fig. 4. A damping coecient of 15% has again been used to obtain rapid convergence of the displacements to the static solution when the load level is constant. Fig. 6 shows the behaviour when the cable is modelled using ®ve elements. A more detailed study of the
364
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behaviour based on a model using 30 elements is given in Fig. 7 which shows the shape of the cable as it changes from a sagging to a hogging pro®le and back to a sagging pro®le as the time dependent loading shown in Fig. 5 is applied. 2.2. A guyed transmitter tower The static and dynamic behaviour of the guyed transmitter mast shown in Fig. 8 is presented. The element properties are given in Table 3. The static behaviour was analysed under gravity loads of )250 kgf ()2.45 kN) applied at each tower point, acting in the positive direction of the Y-axis. Every upper cable node has a gravity load of )150 kgf ()1.47 kN) at the intermediate points and the lower cables have corresponding values of )75 kgf ()0.74 kN). Wind loads of 1000 kgf (9.81 kN) are applied at both the tower and cable nodes. The FINELE result is compared with the NONSAP program. The static load de¯ection curve for the horizontal displacement of the top of the mast (point B) is shown in Fig. 9. The masses given in Table 4 were used at each node in the X, Y and Z-directions. The static displacements obtained from the gravity loads are added to coordinates for the dynamic analysis. The responses of nodes A and B due to the wind load of a half sine wave, of maximum value of 300 kgf (2.94 kN) (Fig. 10) applied at each point are shown in Fig. 11. The individual plots are coincident for NONSAP and FINELE in this ®gure. Two wind conditions are considered: the ®rst with the wind in the Y-direction which produces a predominantly
Fig. 9. Static load±displacements (Y-direction) of node B in the transmitter tower.
Table 4 Masses used for the transmitter tower nodes The mass used at each nodal point (kgf s2 /cm) Tower nodes Lower cables nodes Upper cables nodes
0.25482 0.076453 0.152905
Fig. 10. Load±time curve (wind load).
Fig. 8. Guyed transmitter mast. Table 3 Data input for the transmitter tower members
PGN1 PGN2 PGN3 Lower cables Upper cables
Area (cm2 )
Modulus of elasticity, E (kgf/cm2 )
Pre-load (kgf)
1.0 1.0 1.0 1.0 1.0
1.0816 ´ 108 2.112 ´ 107 8.16 ´ 106 1.7115 ´ 107 2.1273 ´ 107
± ± ± )1.0 ´ 104 )1.2 ´ 104
Fig. 11. Response of nodes A and B.
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365
Fig. 12. De¯ection of rear cables and mast (with no damping) due to the wind load in the Y-direction.
symmetrical response and the second when the wind is in the X-direction causing the structure to twist. Plots of the behaviour of the guys under wind conditions are presented in Figs. 12±14. At the early time step and after the imposed load has been removed, the mast and guy displacements are in-
dependent of the damping coecient, and therefore, the plots are coincident. A time step of 0.005 s has been used to analyse and plot the displacements for the following ®gures. In all the ®gures, dotted lines represent the original shape of the cables and mast leg. Fig. 12 shows the de¯ections of
366
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Fig. 13. De¯ection of rear cables and mast with dierent damping ratios.
the rear cables and mast due to the wind in the Ydirection. Damping has been ignored. Fig. 13 presents the plots of de¯ections of the rear cables and mast for different damping ratios at various times. Finally, Fig. 14 shows the de¯ections when the wind load is applied in the X-direction. Damping has been ignored. The corre-
sponding plans are presented to highlight the out-ofplane displacements of the cables. In Fig. 12 at the beginning of the response, the eect of the wind load is insigni®cant and the initial displacements obtained from static analysis under gravity loads causes the cables to take a sagging shape. The
M.A. Millar, M. Barghian / Computers and Structures 77 (2000) 361±369
367
Fig. 14. De¯ection of rear cables and mast (with no damping), wind load is in the X direction.
shape of the mast is vertical to the ground. The assumed sinusoidal wind load increases and reaches a maximum value at time 1.5 s. Then, the wind load decreases and after time 3 s, the wind load vanishes. All wind is assumed to act horizontally. The maximum de¯ection of the cables of the mast occurs around time 1:75 s. The
mast and cables de¯ect in the Y-direction (wind load direction). The rear cables pass their original straight line and billow up and out. After time 1:75 s, the direction of de¯ection changes. The cables return to a sagging pro®le again after time 3 s and the mast passes through the vertical after time 4 s. The
368
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Fig. 14. (continued)
vibration of the mast and cables continues, but the amplitude of the de¯ections reduces. The shape of the cables and the mast tends towards the position at time 0. Since damping has been ignored, the cables and the mast still vibrate slowly at time t 12 s.
3. Applications The application of the method presented using the dynamic response of the structure, including the cables, enables the analyst to trace the behaviour of the struc-
M.A. Millar, M. Barghian / Computers and Structures 77 (2000) 361±369
369
ture through a locally unstable region of the load± displacement path. Many commercial programs treat cables as non-linear elastic single elements and are not, therefore, able to trace the structural behaviour of guyed masts through such unstable regions. Loads corresponding to ice on the structure can also be included in the analysis. Both the gravity forces due to the dead weight of the ice and the additional surface area attracting additional wind forces will aect the overall structural behaviour. Temperature eects may also have a detrimental eect on the structural response. A rise in the temperature will make the guys less sti, and hence increase the displacements, whereas a drop in the temperature may produce additional compression forces in the mast that cause it to buckle. All these eects can be modelled using FINELE.
response of cable structures that have local unstable behaviour due to snap-through. Static non-linear problems that exhibit snap-through behaviour can be eectively analysed using a dynamic approach with damping. When snap-through occurs, the static equilibrium equation KD W becomes singular, K being the tangent stiness. However, if a dynamic analysis is used, the equations of motion include both the inertia and damping forces which are non-zero, and therefore, the equations are solvable.
4. Conclusion
Reference
The UMIST ®nite element program, FINELE has been used eectively to trace the non-linear dynamic
[1] Barghian M. Static and dynamic analysis of bar element structures. PhD Thesis, UMIST, 1996.
Acknowledgements Manchester Computing for the use of the NONSAP program is acknowledged.