Chapter 3: Introduction to OpenSees

Chapter 3: Introduction to OpenSees

OpenSees, the Open System for Earthquake Engineering Simulation is an object-orientated, open source framework. It has been developed in Berkeley University. As an open source, it allows users to develop sequential, parallel and grid-enabled finite element computer applications for simulating the response of structural and geotechnical systems subjected to earthquakes and other hazards. The following presentation get ideas from official documentation and wikipedia web pages provided by the OpenSees website [1], [2], [3],[7], [8], [9], [10], [11].

OpenSees is primarily written using C++ programming language and uses several fortran and

C numerical libraries for linear equation solving, and material and element routines.

As an object-orientated program, OpenSees is seen as a collection of objects. Each object is capable of receiving messages, processing data, and sending messages to other objects. Each object is of a particular type or Class. The class defines the data and methods of an object. The conceptual approach for simulation is summarize in the following images.

Fig. 3.1: a) simulation framework ; b) comparison of codes [1]

OpenSees is comprised of a set of modules to perform creation of the finite element model specification of an analysis procedure, selection of quantities to be monitored during the analysis, and the output of results. In finite element analysis, an analysis is used to construct 4 main type of object:

ModelBuilder Domain Analysis

Recorder

3.1 Domain

Domains hold the state of the model at time and and are responsible for storing the objects created by the ModelBuilder object and for providing the Analysis and Recorder objects access to these objects.

Other Classes associated with Elements and provided by OpenSees are:

3.2 ModelBuilder

3.3 Analysis

Analysis move the model from state at time to state at time .

The most important Analysis provided by OpenSees are:  Static Gravity Analysis

 Static Pushover

- Monotonic Pushover: one directional displacement-controlled static lateral loading. - Reversed Cyclic Pushover: one directional displacement-controlled static lateral

loading, with displacement cycles imposed in positive and negative directions.

- Uniform Sine-Wave: sine-wave acceleration input to all the node restrained in specified direction.

- Multiple-Support Sine-Wave: different sine-wave displacements are specified at particular nodes in specified directions.

- Uniform Earthquake: the same earthquake acceleration input (from file) at all the restrained nodes in specified direction.

- Multiple-Support Earthquake: different earthquake displacement input (from file) are specified at particular nodes in specified direction.

- Bidirectional Earthquake: different input are specified for two directions. The same acceleration is input at all nodes restrained in specified direction.

Uniform Sine Multiple-Support Sine

Uniform Earthquake Multiple-Support Earthquake

3.4 Recorder

Recorders monitor user-defined parameters in the model during the analysis.

OpenSees uses Tcl (Tool Command Language) scripting language and its interpreters as wish and tells.

3.5 Uniaxial OpenSees materials used

3.5.1 Elastic Material

A simple elastic material is used as example to compare the results obtained with more sophisticated materials. In fig. 3.2 the stress-strain diagram is shown.

Fig. 3.2: stress-strain diagram of elastic material [1]

The parameter determining the behavior is: -

3.5.2 Elastic-Perfectly Plastic Material

As the simpler elastic material, also the elastic-perfectly plastic material have been considered as a comparator. In fig. 3.3 the stress-strain diagram is shown.

Fig. 3.3: stress-strain diagram of elastic-perfectly plastic material [1]

The parameters determining the behavior are: -

- -

3.5.3 Hardening Material

An hardening material, combining both linear kinematic and isotropic hardening is used. The material includes optional visco-plasticity. The stress-strain diagram is shown below.

Fig. 3.4: stress-strain diagram of hardening material [1]

The parameters determining the behavior are: -

- -

3.5.4 Steel01 Material

Provided by OpenSees, Steel01 is an uniaxial bilinear steel material with kinematic hardening and optional isotropic hardening.

In fig. 3.6 and the stress-strain diagram is shown both in monotonic and cyclic envelopes. In fig. 3.7 the influence of isotropic hardening on material behavior is shown.

(a) (b)

Fig. 3.6: monotonic (a) and cyclic (b) stress-strain diagram of steel01 material [1]

(a) (b)

Fig. 3.7: cyclic hysteretic stress-strain diagram of steel01 material with isotropic hardening in compression (a) and in tension (b) [1]

The parameters determining the behavior are: -

-

-

Isotropic hardening is introduced by the following optional parameters:

- - - -

3.5.5 Steel02 Material

Provided by Giuffré-Menegotto-Pinto, Steel02 is a non linear steel material with isotropic strain hardening. In fig. 3.8 and the stress-strain diagram is shown both in monotonic and cyclic envelopes. In fig. 3.9 the influence of isotropic hardening on material behavior is shown.

(a) (b)

Fig. 3.8: monotonic (a) and cyclic (b) stress-strain diagram of steel01 material [1]

(a) (b)

Fig. 3.9: cyclic hysteretic stress-strain diagram of steel02 material with isotropic hardening in compression (a) and in tension (b) [1]

The parameters determining the behavior are: - - - -

Isotropic hardening is introduced by the following optional parameters:

- - - -

3.6 OpenSees sections used

A section defines the stress-deformation response at a cross section of a beam-column or plate element.

3.6.1 Uniaxial Section

This section uses a previously-defined Uniaxial Material object to represent a single section force-deformation response quantity. The force-deformation quantity that can be modeled by this section are , , , , ,

3.6.2 Fiber Section

A fiber section has a general geometric configuration formed by sub-regions of simpler, regular shapes (quadrilateral, circular and triangular regions) called patches. The sub-commands patches are used to define the discretization of the section into fibers. The Quadrilateral Patch

Command is used. The geometry is defined by four vertices, as illustrated in fig. 3.10.

Fig. 3.10: quadrilateral patch [1]

3.6.3 Section Aggregator

This command is used to create a section witch groups previously-defined Uniaxial Material in a single section force-deformation model.

3.7 OpenSees elements used

3.7.1 Elastic Beam-Column Element

With linear elastic behavior. This elements are used to create a simple model for compare the results of more accurate modeling. The input parameters are:

- - - - - -

3.7.2 Elastic Beam-Column Element with zero-Length spring elements

The element are modeled using linear elastic beam-column elements. The non linear behavior is modeled using the concentrated plasticity concept of zero-length element with rotational springs. The springs follow a bilinear hysteretic response based on the “Modified Ibarra Krawinkler Deterioration Model”. A zero-length elements connect two nodes at the same coordinates.

3.7.3 Non linear Beam-Column Element

There are two types of force-based non linear beam-column element:  Non linear BeamColumn Elements (distributed plasticity)  Beam With Hinges (concentrated plasticity with elastic interior)

3.7.4 Non Linear BeamColumn Element

This element is based on the non-iterative force simulation, and considers the spread of plasticity along the element. It is created using a previously-defined section object. The integration along the element is based on Gauss-Lobatto quadrature rule (two integration points at the element ends).

3.7.5 Beam With Hinges Element

This element is based on the non-iterative flexibility formulation, and considers plasticity to be concentrated over specified hinge lengths at the element ends. This type of element is divided in three parts: two hinges at the ends, and a linear elastic region in the middle. The hinges are defined by assigning to each a previously-defined section.

Ela

While the integration of distributed-plasticity force-base elements distributes the gauss points along the entire element length, the beamWithHinges element localizes the integration points in the hinge regions. Two integration points per hinges are used to be able to represent the curvature distribution accurately. Gauss-Radau integration is used.

Fig. 3.13: integration points distribution [1]

The main advantages to this formulation are:

 Non linear behavior is confined to the integration points at elements ends.  The user can specify the length of each hinge.

 It captures largest bending moment at the ends.  It represents linear curvature distributions exactly.

 Characteristic length is equal to when deformations localize. The input parameters are:

- - - - - - - -

The elastic properties are integrated only over the beam interior, which is considered to be linear-elastic. The beamWithHinges element above described has four elastic section and two fiber section at the end.

3.8 OpenSees implementation of the Modified Ibarra-Medina-Krawinkler

deterioration models – common features

The main characteristics of the deterioration model have been discussed in the previous chapter. In this paragraph the attention has been focused on the practical implementation of the models in OpenSees.

The deterioration models previously discussed are assigned to zero-length springs that are typically located at the ends of structural elements (concentrated plasticity concept). These springs are connected to elastic beam-column elements to represent the structural component as part of the building. A three-storey frame model is shown to clarify what said.

Fig. 3.14: example of OpenSees implementation of hysteretic models [8]

3.8.1 Stiffness of rotational springs at member ends

As a consequence the structural properties of the member are a combination of the properties of the sub-elements. The rotational stiffness of the member, , can be derived from the

structural properties of the frame. For example for a beam subjected to double curvature bending:

The rotation stiffness of the beam can be related to the stiffness of the spring, , and the stiffness of the beam-column element, , according to the following equation:

Evident choice for sub-element stiffnesses appear to be either: 1) , in which case

2) , in which case

Both options are not desirable in the context of a computer analysis with OpenSees. An infinite beam stiffness would force all deformations into the plastic hinge springs, leading the problem that the elastic spring stiffness, which has to be defined a priori, would be the same regardless

of the moment gradient. In reality the gradient may change during the time history analysis. The second option, instead, considering an infinite spring stiffness, would cause numerical instability problems and would also make it impossible to express strain hardening and post-capping stiffness as fraction of the elastic spring stiffness.

In order to avoid the problem of the second option and to minimize the problems associated with the first, it has been decided to use an elastic spring stiffness that is times larger than the rotational stiffness of the beam-column element.

Where a value on has been considered. The stiffness of the sub-elements can be expressed as a function of the total stiffness of the member and the multiplier .

A large value of accomplishes to minimize the problems of the first option, because changes in moment gradient can be accounted, and the problems of the second option, because it permits the incorporation of all backbone and cyclic deterioration characteristics in the plastic hinge rotation spring. But the properties of these springs need to be modified because the deterioration properties belong to the full element and not to the spring alone.

3.8.2 Parameters for deteriorating springs

In Par. 3.8.2 the elastic stiffness of the member has been related to the elastic stiffness of the rotational springs and the beam-column element. Additional parameter must be adjusted in the non linear range to reproduce the moment-rotation relationship at the end of the elements. The following image presents the hysteretic response of a beam member which is the combination of the individual moment-rotations of the springs and the beam-column element. The beam-column element remains elastic during all the Time History Analysis and the non linear response in entirely due to the springs.

Fig. 3.15: moment-rotation relationship for a member based on the moment-rotation of the plastic hinge springs and elastic beam-column element [10]

3.8.2.1 Strain hardening coefficient

The strain hardening coefficient of the plastic hinge of the spring, must be adjusted to

obtain the strain hardening coefficient for the moment-rotation of the member, . Because

the sub-element are connected in series, the increment in rotation of the total element results:

Where is the increment in strength developed in the inelastic range and is the stiffness

of the spring for the strain hardening branch. By substituting eq. (3.4), (3.5) in (3.6) and by considering

, results:

3.8.2.2 Post-capping stiffness coefficient

This parameter is calculated in the same way that of the strain hardening branch, because it’s also expressed as a fraction of the elastic stiffness of the member.

3.8.2.3 Ductility capacity of the spring

This has to be adjusted to obtain the correct ductility capacity of the moment-rotation relationship at the end of the member. By adding the deformation of the sub-elements in the elastic and inelastic range it is possible to derive the following equation that provides the ductility capacity of the spring:

3.8.2.4 Parameter for cyclic deterioration

To simulate the correct rate of deterioration of the moment-rotation relationship of the member the parameter have to be adjusted. For the spring the expression is:

3.9 OpenSees implementation of the Modified Ibarra-Medina-Krawinkler

deterioration model with bilinear hysteretic response (bilinear material)

The previously defined “Modified Ibarra-Medina-Krawinkler deterioration model with bilinear

hysteretic response” (ref. par. 2.5) has been implemented in the OpenSees analysis software.

Fig. 3.16: Modified Ibarra-Medina-Krawinkler model – Moment-rotation curve [3]

For the definition of the model the following parameters are requested:

Theoretical

definition

OpenSees

definition

Description

integer tag identifying material

elastic stiffness

strain hardening ratio for positive loading direction

strain hardening ratio for negative loading direction

_{effective yield strength for positive loading direction}

effective yield strength for negative loading direction (negative value)

Cyclic deterioration parameter for strength deterioration

_{Cyclic deterioration parameter for post-capping strength deterioration}

Cyclic deterioration parameter for acceleration reloading stiffness deterioration (is not a deterioration mode for a component with Bilinear hysteretic response).

Cyclic deterioration parameter for unloading stiffness deterioration

rate of strength deterioration. The default value is 1.0.

rate of post-capping strength deterioration. The default value is 1.0.

rate of accelerated reloading deterioration. The default value is 1.0.

rate of unloading stiffness deterioration. The default value is 1.0.

pre-capping rotation for positive loading direction (often noted as

plastic rotation capacity)

pre-capping rotation for negative loading direction (often noted as

plastic rotation capacity) (positive value)

post-capping rotation for negative loading direction (positive value)

residual strength ratio for positive loading direction

residual strength ratio for negative loading direction (positive value)

ultimate rotation capacity for positive loading direction

ultimate rotation capacity for negative loading direction (positive

value)

rate of cyclic deterioration in the positive loading direction (this parameter is used to create asymmetric hysteretic behavior for the case of a composite beam). For symmetric hysteretic response use 1.0.

rate of cyclic deterioration in the negative loading direction (this parameter is used to create asymmetric hysteretic behavior for the case of a composite beam). For symmetric hysteretic response use 1.0

3.10 OpenSees implementation of the Modified Ibarra-Medina-Krawinkler

deterioration model with peak-oriented hysteretic response (ModIMKPeakOriented

Material)

3.10.1 Abstract

The previously defined “Modified Ibarra-Medina-Krawinkler deterioration model with peak-oriented hysteretic response” (ref. par. 2.6) has been implemented in the OpenSees analysis software.

Fig. 3.17: Modified Ibarra-Medina-Krawinkler model with peak-oriented hysteretic response Moment-rotation curve [3]

For the definition of the model the requested parameter are the same as the case of the bilinear IMK model, below remembered:

Theoretical

definition

OpenSees

definition

Description

integer tag identifying material

elastic stiffness

strain hardening ratio for positive loading direction

strain hardening ratio for negative loading direction

effective yield strength for positive loading direction

_{effective yield strength for negative loading direction (negative value)} Cyclic deterioration parameter for strength deterioration

Cyclic deterioration parameter for post-capping strength

deterioration

Cyclic deterioration parameter for acceleration reloading stiffness deterioration (is not a deterioration mode for a component with Bilinear hysteretic response).

_{Cyclic deterioration parameter for unloading stiffness deterioration rate of strength deterioration. The default value is 1.0.}

_{rate of post-capping strength deterioration. The default value is 1.0. rate of accelerated reloading deterioration. The default value is 1.0. rate of unloading stiffness deterioration. The default value is 1.0. pre-capping rotation for positive loading direction (often noted as}

plastic rotation capacity)

_{pre-capping rotation for negative loading direction (often noted as}

plastic rotation capacity) (positive value)

post-capping rotation for positive loading direction

_{post-capping rotation for negative loading direction (positive value)}

_{residual strength ratio for positive loading direction}

_{residual strength ratio for negative loading direction (positive value) ultimate rotation capacity for positive loading direction}

_{ultimate rotation capacity for negative loading direction (positive}

value)

rate of cyclic deterioration in the positive loading direction (this parameter is used to create asymmetric hysteretic behavior for the case of a composite beam). For symmetric hysteretic response use 1.0.

rate of cyclic deterioration in the negative loading direction (this parameter is used to create asymmetric hysteretic behavior for the case of a composite beam). For symmetric hysteretic response use 1.0

3.10.2 Backbone curve

In order to define this model the first step is the definition of the backbone curve, as shown in the following image.

Fig. 3.18: Modified IMK model with Peak-Oriented hysteretic response backbone curve [11]

The following parameter are required: -

- -

-

- The advantage of this model is that it can deteriorate in:

-

- - -

Using four parameter that control the energy dissipation capacity of the component.

3.10.3 Effect of parameter variation in Modified IMK model with Peak-Oriented hysteretic

response

Base Case

A fully symmetric hysteretic response has been chosen as a base case. A quantitative representation of the Moment-rotation curve have been presented in the following image.

Fig. 3.19: Modified IMK model with Peak-Oriented hysteretic response – base case [11]

Modified cases

The effect of the variation of the parameters on the hysteretic response of the model has been shown in the following images. Four Moment-Rotation curves are represented:

a) The

b) The on the hysteretic response of the structural component is shown, when zero residual strength is assumed. One can see that the component fully deteriorates all the way down to zero strength.

c) The is shown, when the deteriorating parameter are selected to be 0,8 instead of 1,0 of the base case. One can see the faster cyclic deterioration in strength and stiffness.

d) The is shown. One can observe the smooth transition to post capping range, compared to the base case, obtained by setting the post capping rotation capacity to 0,1 radians.

c) d)

Fig. 3.20: Parameter variation on Modified IMK model with Peak-Oriented hysteretic response – a) Base case; b) Effect of residual strength; c) Effect of cyclic deterioration parameters; d)

Effect of post-capping rotation [11]

3.11 OpenSees implementation of the Modified Ibarra-Medina-Krawinkler

deterioration model with pinched hysteretic response (ModIMKPinching Material)

3.11.1 Abstract

The previously defined “Modified Ibarra-Medina-Krawinkler deterioration model with peak-oriented hysteretic response” (ref. par. 2.7) has been implemented in the OpenSees analysis software.

Fig. 3.21: Modified Ibarra-Medina-Krawinkler model with pinched responseMoment-rotation curve [11]

For the definition of the model the requested parameter are the same as the case of the peak-oriented model, plus three more parameter to define the pinching behavior:

Theoretical

definition

OpenSees

definition

Description

integer tag identifying material

elastic stiffness

strain hardening ratio for positive loading direction

strain hardening ratio for negative loading direction

effective yield strength for positive loading direction

_{effective yield strength for negative loading direction (negative value)}

Ratio of the force at which reloading begins to force corresponding to the maximum historic deformation demand (positive loading direction)

Ratio of the force at which reloading begins to force corresponding to the absolute maximum historic deformation demand (negative loading direction)

_{Ratio of reloading stiffness}

Cyclic deterioration parameter for strength deterioration

Cyclic deterioration parameter for post-capping strength

deterioration

Cyclic deterioration parameter for acceleration reloading stiffness deterioration (is not a deterioration mode for a component with Bilinear hysteretic response).

Cyclic deterioration parameter for unloading stiffness deterioration

rate of strength deterioration. The default value is 1.0.

_{rate of post-capping strength deterioration. The default value is 1.0.} rate of accelerated reloading deterioration. The default value is 1.0.

rate of unloading stiffness deterioration. The default value is 1.0.

pre-capping rotation for positive loading direction (often noted as

plastic rotation capacity)

pre-capping rotation for negative loading direction (often noted as

plastic rotation capacity) (positive value)

post-capping rotation for positive loading direction

post-capping rotation for negative loading direction (positive value)

residual strength ratio for positive loading direction

residual strength ratio for negative loading direction (positive value)

_{ultimate rotation capacity for positive loading direction}

ultimate rotation capacity for negative loading direction (positive

value)

rate of cyclic deterioration in the positive loading direction (this parameter is used to create asymmetric hysteretic behavior for the case of a composite beam). For symmetric hysteretic response use 1.0.

rate of cyclic deterioration in the negative loading direction (this parameter is used to create asymmetric hysteretic behavior for the case of a composite beam). For symmetric hysteretic response use 1.0

3.11.2 Backbone curve

This model is fully defined in the same way as the previous one. So also in this case the first step is the definition of the backbone curve, as shown below.

Fig. 3.22: Modified IMK model with Pinched hysteretic response – backbone curve [11]

The following parameter are required: - - - - - -

Three additional parameter are requested in order to control pinching:

-

-

-

3.11.3 Effect of parameter variation in Modified IMK model with Pinching hysteretic response

A fully symmetric hysteretic response has been chosen as a base case. A quantitative representation of the Moment-rotation curve have been represented below.

Fig. 3.23: Modified IMK model with Pinched hysteretic response – base case [11]

Modified cases

The effect of the variation of individual parameters on the hysteretic response of the model has been shown in the following images. Four Moment-Rotation curves are represented:

a) The

b) The on the hysteretic response of the structural component is shown, when zero residual strength is assumed. One can see that the component fully deteriorates all the way down to zero strength.

c) The is shown, when pinching parameter are selected to be 0,5 and 0,8 instead of 0,2 of the base case. One can see the great difference in the behavior of the model obtained by putting emphasis on pinching d) The is shown. One can observe the smooth transition to

post capping range, compared to the base case, obtained by setting the post capping rotation capacity to 0,1 radians instead of 0,02 of the base case.

c) d)

Fig. 3.24: Parameter variation on Modified IMK model with Pinched hysteretic response – a) Base case; b) Effect of residual strength; c) Effect of pinching parameters; d) Effect of