Nonlinear Analysis
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MSTower V6 http://www.mstower.com
Non-Linear Analysis Non-Linear Analysis (also called second-order analysis) performs an elastic analysis in which secondorder effects may be considered. The different second-order effects are described below. Non-linear analysis uses a multi-step procedure that commences with a linear elastic analysis. The load residuals, computed for the structure in its displaced position and with the stiffness of members modified, are applied as a new load vector to compute corrections to the initial solution. Further corrections are computed until convergence occurs. There is no single method of iterative non-linear analysis for which convergence is guaranteed. It may therefore be necessary to adjust the analysis control parameters in order to obtain a satisfactory solution. The solution may not converge if the structure is subject to gross deformation or if it is highly nonlinear. This may be the case as the elastic critical load is approached. Note: You should not attempt to use non-linear analysis to determine elastic critical loads. Results of non-linear analysis should be treated with caution whenever the loading is close to the elastic critical load. More: Second-Order Effects Running a Non-Linear Analysis Troubleshooting Non-Linear Analysis
Second-Order Effects The most important second-order effects taken into account in non-linear analysis are the P-Delta effect (P-Ä) and the P-delta effect (P-ä). These are discussed in detail below.
P-Ä AND P-ä EFFECTS You may independently include or exclude these two major effects. Different combinations of the P-Ä and P-ä settings affect the operation of non-linear analysis as set out in the table below. Node
Axial Hoang Van Cuong - HSD Viet Nam...
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Coordinate Force Update Effects NO NO
YES
NO
NO
YES
YES
YES
Analysis Type Linear elastic analysis with tension-only or compression-only members taken into account. This can be achieved for any load case by selecting linear analysis Analysis includes the effects of displacement due to sidesway but not changes in member flexural stiffness due to axial force. These settings will usually yield satisfactory results for pin-jointed structures. Full account is taken of the effects of axial force on member flexural stiffness while the effects of node displacement are approximated by a sidesway correction in the stability function formulation. These settings normally give minimum solution time with second-order effects taken into account. This is the default analysis type, which provides the most rigorous solution for all structure types.
More: Node Coordinate Update – P-Delta Effect Axial Force Effects – P-delta Effect Flexural Shortening Changes in Fixed-End Actions Non-Linear Members
Node Coordinate Update – P-Delta Effect The P-Delta effect (P-Ä) occurs when deflections result in displacement of loads, causing additional bending moments that are not computed in linear analysis. P-Ä is taken into account either by adding displacement components to node coordinates during analysis or by adding sidesway terms to the stability functions used to modify the flexural terms in the member stiffness matrices. Either small displacement theory or finite displacement theory may be used with node coordinate update. As shown Hoang Van Cuong - HSD Viet Nam...
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in the diagram below, finite displacement theory takes into account the rotation of the chord of the displaced member in computing the end rotations and the extension of the member. Only where large displacements occur would the use of finite displacement theory produce results different from those obtained with small displacement theory.
SMALL AND FINITE DISPLACEMENT THEORIES
Axial Force Effects – P-delta Effect The bending stiffness of a member is reduced by axial compression and increased by axial tension. This is called the P-delta effect (P-ä) and is taken into account by adding beam-column stability functions to the flexural terms of the member stiffness matrices. Member stiffness matrices therefore vary with the axial load and are recomputed at every analysis iteration. The stability functions are derived from the "exact" solution of the differential equation describing the behaviour of a beam-column. The additional moments caused by P-ä are approximated in some design codes by the use of moment magnification factors applied to the results of a linear elastic analysis.
Flexural Shortening Flexural shortening, also called bowing, is the reduction in chord length caused by bending. If the ends of the member are completely restrained against axial movement very high tensions may develop with transverse loading. In practice, however, it is difficult to obtain such restraint. In most structures the effect is small but can give rise to considerable difficulty in obtaining convergence of the analysis. Inclusion of the flexural shortening effect is rarely required for a tower or mast.
Hoang Van Cuong - HSD Viet Nam...
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MSTower V6 http://www.mstower.com
Changes in Fixed-End Actions Member fixed-end actions may change between successive analysis iterations owing to displacement of the member and variations in its flexural stiffness caused by axial force. MStower automatically recalculates the fixed-end actions at each analysis iteration and updates the load vector accordingly.
Non-Linear Members Analysis of structures containing tension-only, or cable members requires non-linear analysis. At the conclusion of each analysis step, all members nominated as tension-only or compression-only are checked and either removed from or restored to the model for the next analysis step, according to their deformation. If the removal of non-linear members causes the structure to become unstable, no solution is possible.
Running a Non-Linear Analysis Selecting Load Cases for Non-Linear Analysis Non-Linear Analysis Parameters
Selecting Load Cases for Non-Linear Analysis Non-linear analysis lets you specify the load cases to be analysed and the analysis type (linear or nonlinear) to be used for each. For non-linear analysis a load vector is formed for each load case to be solved, whether a primary load case or a combination load case. There is no need to analyse any load cases for which results are not required. On selecting the Analyse > Non-Linear command, the following dialog box is displayed so you may specify the load cases to be analysed and the analysis type. In the Type column, load cases are identified as Primary or Combination. The second character is a code that specifies whether the load case is to be processed with Linear analysis or Non-linear analysis, or is to be ignored (Skipped).
Hoang Van Cuong - HSD Viet Nam...
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MSTower V6 http://www.mstower.com
SELECTING LOAD CASES FOR NON-LINEAR ANALYSIS The ability to use different analysis types is used for obtaining results for both linear and non-linear analysis in a single pass. This may be necessary where the model includes members to be designed to different codes with different analysis requirements. In general, only "realistic" load cases should be selected for non-linear analysis – there is no point in analysing a wind load case because this load will never exist in isolation. This is particularly important for structures containing cable elements where realistic loads including self weight are required to determine the equilibrium position of each cable, and a solution may not be possible for load cases containing only some load components. Note: The settings in this dialog box will be lost if you subsequently perform a linear analysis. In this case, the analysis type flag (S/L/N) will be unconditionally set to Linear. You must reinstate the analysis type flag if you revert to non-linear analysis.
Non-Linear Analysis Parameters The next dialog box determines the type of non-linear analysis that will be performed for load cases selected for non-linear analysis.
NON-LINEAR ANALYSIS PARAMETERS The dialog box contains the following items:
Node coordinate update (P-Ä) This flag is set if node coordinates are to be updated at each analysis step. It is automatically set for structures containing cable elements. The default setting is on.
Hoang Van Cuong - HSD Viet Nam...
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MSTower V6 http://www.mstower.com
Small/finite displacement theory If the node coordinate update flag has been set, either small or finite displacement theory must be selected. Small displacement theory is the default setting.
Axial force effects (P-ä) If this flag is set member stiffnesses are modified at each analysis step. The default setting is on.
Residual / displacement Specifies the criterion to be used for convergence of the solution. Residual uses a function of the maximum out-of-balance force after analysis. When Displacement is selected, convergence is checked by comparing the convergence tolerance against a generalized measure of the change in displacement between successive iterations. For a satisfactory solution there must be acceptably small changes in the displacement and the residual must be of a low value. The default setting is Residual.
Displacement control Increasing the setting of this control will assist convergence in situations where displacements appear to diverge with successive analysis iterations, or for structures that are initially unstable but become stable as they displace under load. You normally leave this control at minimum and only increase the setting if difficulties are encountered in solution.
Convergence tolerance This value determines when the analysis has converged, determined by checking the change in the convergence criterion between successive analysis cycles. Too small a value will prolong the solution time and may even inhibit convergence. The default value is 0.0005. Do not attempt to achieve "convergence" by increasing the tolerance.
No. load steps You may apply loads in a stepwise fashion which may assist in obtaining a solution for flexible structures by keeping displacements small at each load increment. This parameter is usually left at its default value of 1.
Iterations per load step The maximum number of analysis iterations for each load step. This parameter is used to stop the analysis if convergence is taking an excessive time. The default value is 50, but larger values are often applicable for very flexible structures or models containing large numbers of cable elements.
Relaxation factor The relaxation factor is applied to incremental displacement corrections during analysis. The optimum value for the relaxation factor depends on the type of the structure. As a general rule, structures which "soften" under load (i.e., displacements increase disproportionately with load) have an optimum relaxation factor between 1.0 and 1.2 while structures which "harden" under load have an optimum relaxation factor as low as 0.85. Caution is recommended in changing the relaxation factor from the default value of 1.0; if the relaxation factor is too far from optimum the analysis may require an excessive number of iterations for convergence or it may not converge at all.
Oscillation control This control facilitates convergence when the solution oscillates owing to the removal and restoration of tension-only or compression-only members. The default setting is off.
As the analysis proceeds, the analysis window displays key information for each selected load case. At each analysis iteration the maximum values of residual and displacement are displayed in correct user units. Note that at this stage the values shown are from the most critical degree of freedom, i.e., residuals Hoang Van Cuong - HSD Viet Nam...
6
MSTower V6 http://www.mstower.com
may be either forces or moments, and displacements may be either translations or rotations.
Troubleshooting Non-Linear Analysis It is possible to perform a successful linear analysis for structures that are incapable of resisting the imposed loads. Non-linear analysis is a more complete simulation of the behaviour of a structure under load and the procedure may fail to provide a solution where a linear analysis succeeds. This may occur, for example, if some compression members are slender and buckle. Where non-linear analysis fails to converge, the following tips may be helpful:
Make sure that a linear analysis can be performed. If not, troubleshoot the linear analysis before continuing with the non-linear analysis.
Is a full non-linear analysis necessary? If the only significant non-linear effect is the presence of tension-only or compression-only members, set the analysis type to L for these load cases. In other cases, a successful analysis may result if either node coordinate update or axial force effects are excluded.
Examine the analysis log file. It contains information about members that have become ineffective because of slenderness or member type.
Perform an elastic critical load analysis to check the frame buckling load. If it is greater than the imposed load non-linear analysis is not possible.
Is the structure too flexible? Remove excessive member end releases (pins). Sometimes, in diagnosing convergence problems, it is helpful to remove ALL releases and reinstate them in stages.
Adjust non-linear analysis parameters.
Instability Instability detected during linear analysis is usually due to modelling problems and some of the common causes of these are discussed elsewhere. Because a non-linear analysis considers the effects of axial force on member stiffness it is able to detect a range of instability that linear analysis cannot. For example, non-linear analysis may detect buckling of individual members or of the whole frame. The manner in which a structure is modelled and the analysis parameters used can have some bearing on the stage of the analysis when instability of individual members is detected and the way in which it is subsequently treated. If an unstable member is detected during the update process at the end of each iteration, it will be deleted from the following iteration in much the same way that a tension-only member would be. The presence of unstable members is reported in the Analysis window and details are written to the static log file. However, if the instability is not in a single member but localized in a small group of members it may not be detected until the completion of the analysis. In this case, the presence of the instability will be reported in the Analysis window and some diagnostic information will be written to the static log file to assist you in correcting the problem. Even though the analysis has failed, results are available and may be used to determine corrective measures, e.g. increase some member sizes or, perhaps, change to tension-only members. The results of Hoang Van Cuong - HSD Viet Nam...
7
MSTower V6 http://www.mstower.com
an analysis in which instability has been reported are useful for diagnosis but should not be used for other purposes. An elastic critical load analysis will often assist in locating the cause of local instabilities.
Hoang Van Cuong - HSD Viet Nam...
8
Non-Linear Analysis Non-Linear Analysis (also called second-order analysis) performs an elastic analysis in which secondorder effects may be considered. The different second-order effects are described below. Non-linear analysis uses a multi-step procedure that commences with a linear elastic analysis. The load residuals, computed for the structure in its displaced position and with the stiffness of members modified, are applied as a new load vector to compute corrections to the initial solution. Further corrections are computed until convergence occurs. There is no single method of iterative non-linear analysis for which convergence is guaranteed. It may therefore be necessary to adjust the analysis control parameters in order to obtain a satisfactory solution. The solution may not converge if the structure is subject to gross deformation or if it is highly nonlinear. This may be the case as the elastic critical load is approached. Note: You should not attempt to use non-linear analysis to determine elastic critical loads. Results of non-linear analysis should be treated with caution whenever the loading is close to the elastic critical load. More: Second-Order Effects Running a Non-Linear Analysis Troubleshooting Non-Linear Analysis
Second-Order Effects The most important second-order effects taken into account in non-linear analysis are the P-Delta effect (P-Ä) and the P-delta effect (P-ä). These are discussed in detail below.
P-Ä AND P-ä EFFECTS You may independently include or exclude these two major effects. Different combinations of the P-Ä and P-ä settings affect the operation of non-linear analysis as set out in the table below. Node
Axial Hoang Van Cuong - HSD Viet Nam...
1
MSTower V6 http://www.mstower.com
Coordinate Force Update Effects NO NO
YES
NO
NO
YES
YES
YES
Analysis Type Linear elastic analysis with tension-only or compression-only members taken into account. This can be achieved for any load case by selecting linear analysis Analysis includes the effects of displacement due to sidesway but not changes in member flexural stiffness due to axial force. These settings will usually yield satisfactory results for pin-jointed structures. Full account is taken of the effects of axial force on member flexural stiffness while the effects of node displacement are approximated by a sidesway correction in the stability function formulation. These settings normally give minimum solution time with second-order effects taken into account. This is the default analysis type, which provides the most rigorous solution for all structure types.
More: Node Coordinate Update – P-Delta Effect Axial Force Effects – P-delta Effect Flexural Shortening Changes in Fixed-End Actions Non-Linear Members
Node Coordinate Update – P-Delta Effect The P-Delta effect (P-Ä) occurs when deflections result in displacement of loads, causing additional bending moments that are not computed in linear analysis. P-Ä is taken into account either by adding displacement components to node coordinates during analysis or by adding sidesway terms to the stability functions used to modify the flexural terms in the member stiffness matrices. Either small displacement theory or finite displacement theory may be used with node coordinate update. As shown Hoang Van Cuong - HSD Viet Nam...
2
MSTower V6 http://www.mstower.com
in the diagram below, finite displacement theory takes into account the rotation of the chord of the displaced member in computing the end rotations and the extension of the member. Only where large displacements occur would the use of finite displacement theory produce results different from those obtained with small displacement theory.
SMALL AND FINITE DISPLACEMENT THEORIES
Axial Force Effects – P-delta Effect The bending stiffness of a member is reduced by axial compression and increased by axial tension. This is called the P-delta effect (P-ä) and is taken into account by adding beam-column stability functions to the flexural terms of the member stiffness matrices. Member stiffness matrices therefore vary with the axial load and are recomputed at every analysis iteration. The stability functions are derived from the "exact" solution of the differential equation describing the behaviour of a beam-column. The additional moments caused by P-ä are approximated in some design codes by the use of moment magnification factors applied to the results of a linear elastic analysis.
Flexural Shortening Flexural shortening, also called bowing, is the reduction in chord length caused by bending. If the ends of the member are completely restrained against axial movement very high tensions may develop with transverse loading. In practice, however, it is difficult to obtain such restraint. In most structures the effect is small but can give rise to considerable difficulty in obtaining convergence of the analysis. Inclusion of the flexural shortening effect is rarely required for a tower or mast.
Hoang Van Cuong - HSD Viet Nam...
3
MSTower V6 http://www.mstower.com
Changes in Fixed-End Actions Member fixed-end actions may change between successive analysis iterations owing to displacement of the member and variations in its flexural stiffness caused by axial force. MStower automatically recalculates the fixed-end actions at each analysis iteration and updates the load vector accordingly.
Non-Linear Members Analysis of structures containing tension-only, or cable members requires non-linear analysis. At the conclusion of each analysis step, all members nominated as tension-only or compression-only are checked and either removed from or restored to the model for the next analysis step, according to their deformation. If the removal of non-linear members causes the structure to become unstable, no solution is possible.
Running a Non-Linear Analysis Selecting Load Cases for Non-Linear Analysis Non-Linear Analysis Parameters
Selecting Load Cases for Non-Linear Analysis Non-linear analysis lets you specify the load cases to be analysed and the analysis type (linear or nonlinear) to be used for each. For non-linear analysis a load vector is formed for each load case to be solved, whether a primary load case or a combination load case. There is no need to analyse any load cases for which results are not required. On selecting the Analyse > Non-Linear command, the following dialog box is displayed so you may specify the load cases to be analysed and the analysis type. In the Type column, load cases are identified as Primary or Combination. The second character is a code that specifies whether the load case is to be processed with Linear analysis or Non-linear analysis, or is to be ignored (Skipped).
Hoang Van Cuong - HSD Viet Nam...
4
MSTower V6 http://www.mstower.com
SELECTING LOAD CASES FOR NON-LINEAR ANALYSIS The ability to use different analysis types is used for obtaining results for both linear and non-linear analysis in a single pass. This may be necessary where the model includes members to be designed to different codes with different analysis requirements. In general, only "realistic" load cases should be selected for non-linear analysis – there is no point in analysing a wind load case because this load will never exist in isolation. This is particularly important for structures containing cable elements where realistic loads including self weight are required to determine the equilibrium position of each cable, and a solution may not be possible for load cases containing only some load components. Note: The settings in this dialog box will be lost if you subsequently perform a linear analysis. In this case, the analysis type flag (S/L/N) will be unconditionally set to Linear. You must reinstate the analysis type flag if you revert to non-linear analysis.
Non-Linear Analysis Parameters The next dialog box determines the type of non-linear analysis that will be performed for load cases selected for non-linear analysis.
NON-LINEAR ANALYSIS PARAMETERS The dialog box contains the following items:
Node coordinate update (P-Ä) This flag is set if node coordinates are to be updated at each analysis step. It is automatically set for structures containing cable elements. The default setting is on.
Hoang Van Cuong - HSD Viet Nam...
5
MSTower V6 http://www.mstower.com
Small/finite displacement theory If the node coordinate update flag has been set, either small or finite displacement theory must be selected. Small displacement theory is the default setting.
Axial force effects (P-ä) If this flag is set member stiffnesses are modified at each analysis step. The default setting is on.
Residual / displacement Specifies the criterion to be used for convergence of the solution. Residual uses a function of the maximum out-of-balance force after analysis. When Displacement is selected, convergence is checked by comparing the convergence tolerance against a generalized measure of the change in displacement between successive iterations. For a satisfactory solution there must be acceptably small changes in the displacement and the residual must be of a low value. The default setting is Residual.
Displacement control Increasing the setting of this control will assist convergence in situations where displacements appear to diverge with successive analysis iterations, or for structures that are initially unstable but become stable as they displace under load. You normally leave this control at minimum and only increase the setting if difficulties are encountered in solution.
Convergence tolerance This value determines when the analysis has converged, determined by checking the change in the convergence criterion between successive analysis cycles. Too small a value will prolong the solution time and may even inhibit convergence. The default value is 0.0005. Do not attempt to achieve "convergence" by increasing the tolerance.
No. load steps You may apply loads in a stepwise fashion which may assist in obtaining a solution for flexible structures by keeping displacements small at each load increment. This parameter is usually left at its default value of 1.
Iterations per load step The maximum number of analysis iterations for each load step. This parameter is used to stop the analysis if convergence is taking an excessive time. The default value is 50, but larger values are often applicable for very flexible structures or models containing large numbers of cable elements.
Relaxation factor The relaxation factor is applied to incremental displacement corrections during analysis. The optimum value for the relaxation factor depends on the type of the structure. As a general rule, structures which "soften" under load (i.e., displacements increase disproportionately with load) have an optimum relaxation factor between 1.0 and 1.2 while structures which "harden" under load have an optimum relaxation factor as low as 0.85. Caution is recommended in changing the relaxation factor from the default value of 1.0; if the relaxation factor is too far from optimum the analysis may require an excessive number of iterations for convergence or it may not converge at all.
Oscillation control This control facilitates convergence when the solution oscillates owing to the removal and restoration of tension-only or compression-only members. The default setting is off.
As the analysis proceeds, the analysis window displays key information for each selected load case. At each analysis iteration the maximum values of residual and displacement are displayed in correct user units. Note that at this stage the values shown are from the most critical degree of freedom, i.e., residuals Hoang Van Cuong - HSD Viet Nam...
6
MSTower V6 http://www.mstower.com
may be either forces or moments, and displacements may be either translations or rotations.
Troubleshooting Non-Linear Analysis It is possible to perform a successful linear analysis for structures that are incapable of resisting the imposed loads. Non-linear analysis is a more complete simulation of the behaviour of a structure under load and the procedure may fail to provide a solution where a linear analysis succeeds. This may occur, for example, if some compression members are slender and buckle. Where non-linear analysis fails to converge, the following tips may be helpful:
Make sure that a linear analysis can be performed. If not, troubleshoot the linear analysis before continuing with the non-linear analysis.
Is a full non-linear analysis necessary? If the only significant non-linear effect is the presence of tension-only or compression-only members, set the analysis type to L for these load cases. In other cases, a successful analysis may result if either node coordinate update or axial force effects are excluded.
Examine the analysis log file. It contains information about members that have become ineffective because of slenderness or member type.
Perform an elastic critical load analysis to check the frame buckling load. If it is greater than the imposed load non-linear analysis is not possible.
Is the structure too flexible? Remove excessive member end releases (pins). Sometimes, in diagnosing convergence problems, it is helpful to remove ALL releases and reinstate them in stages.
Adjust non-linear analysis parameters.
Instability Instability detected during linear analysis is usually due to modelling problems and some of the common causes of these are discussed elsewhere. Because a non-linear analysis considers the effects of axial force on member stiffness it is able to detect a range of instability that linear analysis cannot. For example, non-linear analysis may detect buckling of individual members or of the whole frame. The manner in which a structure is modelled and the analysis parameters used can have some bearing on the stage of the analysis when instability of individual members is detected and the way in which it is subsequently treated. If an unstable member is detected during the update process at the end of each iteration, it will be deleted from the following iteration in much the same way that a tension-only member would be. The presence of unstable members is reported in the Analysis window and details are written to the static log file. However, if the instability is not in a single member but localized in a small group of members it may not be detected until the completion of the analysis. In this case, the presence of the instability will be reported in the Analysis window and some diagnostic information will be written to the static log file to assist you in correcting the problem. Even though the analysis has failed, results are available and may be used to determine corrective measures, e.g. increase some member sizes or, perhaps, change to tension-only members. The results of Hoang Van Cuong - HSD Viet Nam...
7
MSTower V6 http://www.mstower.com
an analysis in which instability has been reported are useful for diagnosis but should not be used for other purposes. An elastic critical load analysis will often assist in locating the cause of local instabilities.
Hoang Van Cuong - HSD Viet Nam...
8
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