CYPE 3D

Pushover analysis can be used to realistically evaluate the non-linear response of a structure subjected to seismic forces through the incremental application of lateral loads under displacement control.

This procedure can be used to:

• Study the seismic behaviour of the building, identifying the sequence of plastic hinge formation and the potential mechanisms of collapse.
• Identify structural deficiencies and areas of concentrated damage, thereby facilitating the design and assessment of reinforcement measures.
• Determine the structure's capacity curve and identify its performance point.
• Assess new-build structures during the design stage, as well as existing structures for seismic assessment, rehabilitation, or strengthening.

CYPE 3D can be used to perform a pushover analysis on a structure entered into the program, using the options in the "Pushover" group on the "Analysis" tab.

For each load case in the list (1), the following is defined:

• the "Initial load state" (2),
• the "Control node" (3),
• and the "Lateral load patterns" (4).

Resolution process

The section entitled "Solution process" begins by defining the "Iterative linearisation method". The iterative linearisation method controls how iteration is performed within a load step to satisfy the equilibrium relationship R(u) = Fe - Fi(u) = 0. The available methods are:

• Newton-Raphson
  This method is suitable for small or medium-sized models (< 100,000 degrees of freedom (DOFs)). It is the most conservative method. It minimises the residual between iterations because it performs factorisation of the system’s tangent matrix at each iteration, which allows the analysis to progress more accurately when many steps are chained together or the system has a response that is highly sensitive to small perturbations. However, performing the factorisation at each iteration makes it slower than other methods and, at the same time, more sensitive to oscillations.

• Modified Newton-Raphson
  This method is suitable for large models (> 100,000 degrees of freedom (DOFs)). It performs the factorisation of the system’s tangent matrix only at the start of the step and thereafter works with an approximation, which makes it faster than the Newton-Raphson method and stable against oscillations. It has linear convergence.

• Krylov-Newton
  This method is suitable for large models (> 100,000 degrees of freedom (DOFs)). It factors the tangent matrix at the start of the step and uses the previous residuals to correct the direction of the displacement. It is very robust when the tangent matrix is ill-conditioned.

• Newton-Raphson (linear search, step size 0.5)
• Newton-Raphson (linear search, factor 0.8)
  These methods are suitable for small or medium-sized models (< 100,000 degrees of freedom (DOFs)). They work in the same way as the Newton-Raphson method, but in this case, a search is performed in the direction of the displacement increment vector to minimise the residual. This is useful when the Newton-Raphson method oscillates without converging.

For any method, the following must be specified:

• The "Number of consecutive retries when a step fails".
• The "Number of iterations per retry".
• The "Minimum offset increase per retry".

You can also tick the relevant boxes for "Apply other algorithms before subdivision retries" and/or "Apply an exhaustive convergence search strategy during retries".

Convergence test

The convergence test calculates the Euclidean norm of the estimated parameter, taking into account all the degrees of freedom in the model. It is used to determine whether the iterative linearisation method has reached an approximate equilibrium within the specified tolerance range.

In the "Convergence test" section, you can define the "Type" by selecting one of the following:

• Balance of forces
  This test is usually sufficient to ensure the overall accuracy of the solution obtained. If selected, the program checks that the change in displacement between iterations is less than the specified tolerance. This test uses a single tolerance for the degrees of freedom of translation and rotation. The internal units used are metres and radians.
  • When selecting this test, you must choose the "Convergence tolerance" from the available options. The "Attempt to complete the analysis using the displacement equilibrium test" option is also available (for which you must select the "Convergence tolerance (displacements)" and enter the "Step subdivision factor").

• Displacement equilibrium
  If this test is selected, the program checks that the residual force between iterations is less than the specified tolerance. The internal units used are tonnes. This test is recommended in situations where displacement equilibrium is reached before force equilibrium, which is usually reflected in an irregular displacement-shear force curve and significant variation in forces and moments.
  • When selecting this test, you must choose the "Convergence tolerance" from the available options.

• Balance of forces and displacements
  If this test is selected, the program checks that both the balance of forces and the balance of displacements are satisfied. The results are as accurate as possible, but it can be difficult to get the model to achieve the target displacement.
  • When selecting this test, you must select the "Convergence tolerance (forces)" and the "Convergence tolerance (displacements)".

• Balance of forces or displacements
  If this test is selected, the programme checks that the balance of forces or the balance of displacements is satisfied. It should only be used as a test to analyse model convergence issues.
  • When selecting this test, you must select the "Convergence tolerance (forces)" and the "Convergence tolerance (displacements)".

• Energy balance
  This test should be used as a last resort. If this test is selected, the program checks that the change in energy between iterations is less than the specified tolerance. The tolerance is expressed in units of energy (tonnes × metres in this case). In some deformation modes, this test may return a false equilibrium (when the displacement increment vector is orthogonal to the residual vector).
  • When this test is selected, the "Convergence tolerance" is selected; the option "Attempt to complete the analysis using the displacement equilibrium test" is offered (for which the "Convergence tolerance (displacements)" is selected and the "Step subdivision factor" is entered).

In addition, the "Maximum number of iterations" is entered for each test.

Disturbances

Finally, in the "Disturbances" section, you can tick the relevant box to "Slightly modify the modal load pattern to improve convergence".

Control node

This allows you to select a node in the structure and assign it as the control node for all load cases in the pushover analysis.

The selected node can be viewed under the "Load cases" option, in the "Control node" section. From here, you can also allow the program to select a control node automatically or specify a control node for each load case.

Analyse

This option can be used to configure and run the pushover analysis.

The "Number of vibration modes included in the analysis" can be:

• "Automatic, until a required percentage of the mass has been displaced", which must be entered.
• "User-defined", in which case the number of modes is entered.

You can then enable "Second-order elastic analysis".

"Result storage" can be carried out:

• For "All loads".
• For"One load in every…", enter a specific number in the box that appears on the right.

Among the options for achieving the "Convergence of the analysis", the program offers the following options:

• Apply perturbations to hinge parameters (optional)
  This option can be enabled if the model exhibits convergence issues that prevent the target displacement from being reached. If enabled, the program randomly modifies the yield moment of the hinges by a factor of around 1 in 1000 to try to prevent several hinges from yielding simultaneously, which would trigger a mechanism.
• Assign a parallel spring to each ball joint with a stiffness proportional to the ball joint’s initial stiffness (optional)
  If the analysis stops because the hinges have collapsed prematurely or the minimum desired displacement has not been reached, this option can be activated to enter springs in parallel with the hinges with very low stiffness in an attempt to ensure that, when they collapse, the system matrix remains invertible.

Finally, you can "Consider the finite dimension of the nodes" by ticking the relevant box.

Capacity curve

The capacity curve represents the structural response of the building to increasing lateral loads. If "Capacity curve" is selected under "Result type", the capacity curve is displayed in the blue graph on the right using the relationship between the "Control node displacement" on the x-axis and the "Total shear force" on the y-axis.

Selection of the load case

The capacity curve shown is that obtained for the load case and lateral load pattern selected from the drop-down menus in the "Load case" panel on the left.

On the label of the load case and/or load pattern, the fundamental mode considered is shown in brackets, whilst the percentage value indicates that the results shown have been obtained by applying that percentage of the target displacement.

Furthermore, in this panel, you can also select a specific load from the drop-down menu or navigate between loads using the various buttons provided. When a load is selected, the line on the capacity curve that defines the pair of displacement and shear values corresponding to that load is highlighted in magenta.

This panel also displays a warning if convergence is not achieved for the selected load pattern.

If the target displacement is reached, the "Limit displacement" value is displayed, the "Monitored displacement" (Dx or Dy) is indicated, and the "Control node" reference is shown.

Checking the condition of the plastic hinges

In the "Plastic hinges" panel on the left, you can define the "Classification" by selecting one of the following options:

• "In accordance with the acceptance criteria" ("Immediate Occupancy (IO)", "Life Safety (LS)", "Collapse Prevention (CP)"),
• Or "In accordance with the performance zones" of the hinges ("Zone AB", "Zone BC", "Zone CD", "Zone DE", "Zone EF").

Consequently, the pie chart in the "Condition of hinge" section, in the bottom right-hand corner, shows the percentage of hinges in each condition for the selected load, pattern and load step.

Report

Finally, the program allows you to display on screen or print the "List of selected combinations" by clicking the relevant button in the "Capacity curve" window.

This report lists the data for the selected standard, the analysis parameters, the capacity curve, the step-by-step behaviour of the hinges, and the values associated with the performance point for the selected load case and standard.

Plastic hinges

Options in the "Properties-Selection" panel

Clicking on the "Plastic hinges" option opens the "Properties-Selection" window. Here, you must first select one of the following options:

• Bars
  Displays the results for the plastic hinges defined in the selected bar.
• Plastic hinges
  Displays the individual results for the selected plastic hinge.

Load case

The results for the selected load case and lateral load pattern from the drop-down menus in the "Load case" panel on the left will be displayed.

On the label of the load case and/or load pattern, the fundamental mode considered is shown in brackets, whilst the percentage value indicates that the results shown have been obtained by applying that percentage of the target displacement.

In addition, in this panel you can select a specific level from the drop-down menu or navigate between levels using the various buttons provided.

This panel also displays a warning if convergence is not achieved for the selected load pattern.

If the target displacement is reached, the "Limit displacement" value is displayed, the "Monitored displacement" (Dx or Dy) is indicated, and the "Control node" reference is shown.

Plastic hinges

In the "Plastic hinges" panel on the left, you can define the "Classification" by selecting one of the following options:

• "In accordance with the acceptance criteria" ("Immediate occupancy (IO)", "Life safety (LS)", "Collapse prevention (CP)"); additionally, the hinge can be found under "Elastic behaviour (shown in grey);
• "In accordance with the performance zones" of the hinges ("Zone AB", "Zone BC", "Zone CD", "Zone DE", "Zone EF").

This way, the hinges will be represented as coloured spheres in the "Work area", scaled according to their state based on the defined classification and the selected load case, load pattern, and load step, thereby enabling the structure's behaviour to be analysed and potential failure mechanisms to be identified.

Checking the moment-rotation diagram for plastic hinges

When you click on a bar or a plastic hinge (depending on whether the "Bars" or "Plastic hinges" option has been selected in the "Properties-Selection" panel), the program will open a window allowing you to view the moment-rotation diagram (backbone curve) for the hinges assigned to the selected beam or the selected plastic hinge.

Here, you first select the load case, lateral pattern and step from the drop-down menus in the "Load case" panel on the left, in the same way as when viewing the other results of the pushover analysis.

In the "Plastic hinges" panel on the left, you can define the "Classification" as "According to acceptance criteria" or "According to the performance stages of the hinges". This will mark and label the different stages on the diagram using a dashed line.

The "Hinges" panel at the bottom lists the hinges that can be viewed in the window. For each entry in the list, the hinge number, its "Relative position" with respect to the bar, its "Degree of freedom", and its "Status" at the selected step and according to the previously selected classification are indicated. The bottom section displays the "Type" of each hinge and its "Interaction" ("Independent moments" or "Axil-My-Mz" interaction).

Moment-rotation diagram for plastic hinges with independent moments

The graph shows the "Theoretical behaviour" of the selected ball-and-socket joint (at the selected degree of freedom) in orange, and the actual behaviour of the joint obtained from the pushover analysis performed by the software in blue.

A circle (coloured according to the condition of the hinge, using the same criteria as in the workspace)marks the exact point on the actual performance curve where the hinge is located at the selected step. Thus, by navigating between different steps, you can observe the movement of the circle along the actual performance curve.

Moment-rotation diagram for plastic hinges with axial-moment-moment interaction

In the "Axil-My-Mz" interaction hinges, the "Theoretical behaviour" curve is different for each step. However, the hinge’s operating point (represented by the circle shown) always lies on both the actual performance curve (in blue) and the theoretical performance curve for the selected stage (in orange). This curve is scaled by the value of the axil.

The remaining theoretical performance curves associated with the unselected steps are shown in light grey.

Report

Finally, the program allows you to display on screen or print the "List of selected combinations" by clicking on the relevant button in the plastic hinge query window.

This report lists the data for the selected standard, the analysis parameters, and the moment-rotation diagram for each hinge and selected degree of freedom, including the "Hinge properties" and a numerical table showing the "Results per step".