Both monotonic and cyclic displacement-controlled tests have been performed on the frame in order to better comprehend the behavior of the different models

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Chapter 7 Analysis of a plane steel frame

7.1 Abstract

In this chapter a plane steel frame has been extrapolated from a building and analyzed using several combination of material, element and section provided by OpenSees. Both monotonic and cyclic displacement-controlled tests have been performed on the frame in order to better comprehend the behavior of the different models. Finally a simulated earthquake has been applied by imposing an acceleration pattern to the base nodes.

7.2 Problem data

 Structure

University project: OPUS Building number 1

 Material Steel: S235

 Loads

Live load (Q)

Wind load (W)

Snow load (S)

Earthquake action (E) Serviceability limit state

 Loads combinations

Combination number 1: ULS1 Combination number 2: ULS2 Combination number 3: ULS3

Combination number 1: Seismic1 Combination number 1: Seismic2

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 Global geometry

Number of storeys

Resisting system MR Resisting system CB

Span Span

Secondary beam Secondary beam

Storey/height

distribution Storey/height

distribution

Table 7.2.I: Steel structure data

7.3 Summary of the final design

 General

Seismic mass of the building

Behavior factor

Accidental torsion

 Slab

Type

Thickness

 Beams

Type Fully rigid

Discontinuous slab Type Simply supported

Discontinuous slab

Section Section

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 Columns

Section Section

 Bracings

Storey number Dissipative elements Storey number Dissipative elements

1

2 2

3 3

4 4

5 5

 Overstrength factor of the primary seismic elements

Storey number 1 Storey number 1

Table 7.3.I: Final design data

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7.4 Geometrical overview

Fig. 7.1: Building 1 ; X-Y plane

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Fig. 7.3: Building 1 ; Y-Z plane

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Fig. 7.4: Building 1 ; 3D view

7.5 Plane analysis of X-Z plane

7.5.1 OpenSees materials used

To ensure a non linear behavior of the column only non-linear behavior material have been considered. Specifically:

 Uniaxial Material Hardening

 Uniaxial Material Steel01 [ref. par. 3.5]

 Uniaxial Material Steel02 [ref. par. 3.5]

 Modified Ibarra-Medina-Krawinkler deterioration model with bilinear hysteretic response [ref. par. 3.9]

7.5.2 OpenSees sections used

The sections used to model the non linear behavior of the cantilever column have been:

 Fiber section [ref. par. 3.6]

 Zero-length section [ref. par. 3.6]

7.5.3 OpenSees elements used

The element used to model the non linear behavior of the column have been:

 Elastic Element [ref. par. 3.7]

 Nonlinear Beam-Column Elements [ref. par. 3.7]

 Beam With Hinges Elements [ref. par. 3.7]

7.6 OpenSees Models

7.6.1 Model 1

The model 1 of the plane frame has been realized using pre-defined Uniaxial Material Hardening. A modeled Fiber Section of HEB 400 and IPE 400 is applied to Nonlinear Beam-Column element, as shown in Fig 7.5.

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Fig. 7.5: Fiber section discretization of HEB400 and IPE 400 sections

7.6.2 Model 2

The model 2 of the plane frame has been realized following the “concentrated plasticity”

concept, as already described in Par. 5.6.1. The columns have been modeled using Fiber sections. Instead, the beams have been modeled using pre-defined Elastic Beam-Column Element connected to Zero-Length Elements, which serve as rotational springs to represent the structure’s nonlinear behavior. Four rotational springs have been also used to model the fixed base nodes. The springs follow a bilinear hysteretic response based on the Modified Ibarra-Medina-Krawinkler deterioration model with bilinear hysteretic response [ref. Par. 3.9]. The input parameters for the rotational behavior of the plastic hinges have determined using empirical relationship developed by Lignos and Krawinkler in 2010 [10]. Rigid Link elements have been used to model the rigid panel zone in the beam-column connections. EqualDOF function has been used to constrain beam-column joints in the same floor to have the same lateral displacement. An idealized scheme of the model is presented in Figg. 7.6 - 7.9, in which spring’s and rigid links size have been greatly exaggerated for clarity.

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Fig 7.6: Plane frame – Nodes

Fig 7.7: Plane frame – Generic node Where:

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Fig 7.8: Plane frame – Elements

Fig 7.9: Plane frame – Generic node

Where:

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( )

Since a frame member is modeled as an elastic element connected in series with rotational springs at the end, the stiffness of this component must be modified so that the equivalent stiffness of this assembly is equivalent to the stiffness of the actual frame member. Based on the approach described in Appendix B of Ibarra and Krawinkler [8], the rotational springs are made “n=10” times stiffer than the rotational stiffness of the elastic element in order to avoid numerical problems and allow all damping to be assigned to the elastic element. To ensure the equivalent stiffness of the assembly is equal to the stiffness of the actual frame member, the stiffness of the elastic element must be times greater than the stiffness of the actual frame member. In this example, this is accomplished by making the elastic element’s moment of inertia times greater than the actual frame member’s moment of inertia.

In order to make the nonlinear behavior of the assembly match that of the actual frame member, the strain hardening coefficient (the ratio of post-yield stiffness to elastic stiffness) of the plastic hinge must be modified. If the strain hardening coefficient of the actual frame member is denoted and the strain hardening coefficient of the spring (or plastic hinge region) is denoted then:

The theoretical parameters used for the definition of the backbone curve of the zero- length rotational springs have been calculated using the predefined relations provided by Krawinkler and Lignos [9-10] (see Par. 2.5). MathCAD sheets have been used and the results are shown below.

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7.6.2.1 Cross section HEB 400

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The table 7.6.I summarizes all the parameters requested by OpenSees for the definition of the Moment-Rotation backbone curve for the plastic hinges. This give also information about all the theoretical value, obtained by using the above defined relations, as well as the practical values used in all the simulated tests. One can see that for some test the input value of a parameter is much different from the theoretical one. The reason is attributable to the fact that the relations used to define the theoretical value of the parameters derive from cumulative distribution, so that they have mainly a statistical significance.

Cross section: HEB 400

Table 7.6.I: Plane frame – Cross section HEB 400 – IMK model – Requested parameters

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7.6.2.2 Cross section IPE 400

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The table 7.6.II summarizes all the parameters requested by OpenSees for the definition of the Moment-Rotation backbone curve for the plastic hinges. This give also information about all the theoretical value, obtained by using the above defined relations, as well as the practical values used in all the simulated tests. One can see that for some test the input value of a parameter is much different from the theoretical one. The reason is attributable to the fact that the relations used to define the theoretical value of the parameters derive from cumulative distribution, so that they have mainly a statistical significance.

Cross section: IPE 400

Table 7.6.II: Plane frame – Cross section IPE 400 – IMK model – Requested parameters

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7.7 Analysis Performed

7.7.1 Eigen Analysis

At first this simple analysis has been performed to have basic indications about the behavior of the models of the frame. To be sure of the goodness of the results obtained using OpenSees, they have been compared with ones obtained using SAP2000 software.

The seismic masses have been applied to the node and the values are reported in Fig 7.10.

Fig. 7.10: Plane Frame – Seismic Masses Distribution

7.7.2 Monotonic lateral displacement-controlled pushover analysis

A displacement-controlled pushover analysis has been performed on the frame’s models.

At first a vertical load pattern has been applied to the node of the model to simulate the gravity load. The following loads have been considered:

 Self weight of the steel members

 Concrete slab with an inter-axis of 6 m (internal frame)

 Imposed loads

Then time has been reset to zero and a displacement-controlled test has been performed. The maximum displacement of the free node (node 24) has been set equal to 10% of the total height of the buildings, so 1750 mm. In addiction an horizontal load pattern has been applied to every node, by following the equations:

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Where:

Fig. 7.11: Plane frame – Displacement-controlled pushover analysis Horizontal load pattern

7.7.3 Ground Motion simulation analysis

A dynamic analysis has been performed on both Model 1 and Model 2. First a vertical load pattern has been applied to the node of the model to simulate the gravity load. The following loads have been considered:

 Self weight of the steel members

 Concrete slab with an inter-axis of 6 m (internal frame)

 Imposed loads

Then time has been reset to zero and the earthquake simulation test has been performed. All the ground nodes of the plane frame have been subjected to the following acceleration pattern (Akz 1).

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Fig. 7.12: Ground Acceleration pattern [Akz 1]

The obtained acceleration spectrum is represented in Fig. 7.13 where, anticipating the following results of the Eigen Analysis, the first Period of the structure and the coresponding value of acceleration are shown.

Fig. 7.13: Acceleration Spectrum

 DAMPING

This model uses Rayleigh damping which formulates the damping matrix as a linear combination of the mass matrix and stiffness matrix :

-3,0

-2,0 -1,0 0,0 1,0 2,0 3,0

0 2 4 6 8 10 12 14 16 18 20

Acceleration [m/s²]

Time [sec]

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9

0,0 0,5 1,0 1,5 2,0 2,5 3,0

Sa [g]

T [sec]

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Where:

mass proportional damping coefficient stiffness proportional damping coefficient

A damping ratio of 2%, which is a typical value for steel buildings, is assigned to the first three modes of the structure. The OpenSees rayleigh command allows the user to specify whether the initial, current, or last committed stiffness matrix is used in the damping matrix formulation.

As already described [see Par. 7.6.2], the stiffness of the elastic frame elements has been modified. As explained in Ibarra and Krawinkler [9], the stiffness proportional damping coefficient that is used with these elements must also be modified. As the stiffness of the elastic elements was made times greater than the stiffness of the actual frame member, the stiffness proportional damping coefficient of these elements must also be made times greater than the traditional stiffness proportional damping coefficient.

 FREE VIBRATIONS

At the end of the 20 seconds of effective ground motion, 10 seconds have been added to the analysis, so that the amount of the analyzed time has been 30 seconds. This has been done in order to verify the behavior of the structure under free vibrations.

7.7.4 Incremental Dynamic Analysis

As final step an Incremental Dynamic Analysis (IDA) has been performed on both Model 1 and Model 2. By analogy with passing from a Single Static analysis to an Incremental Static Pushover, one arrives at the extention of a Single Time-history analysis into an Incremental one, where the seismic “loading” is scaled [21]. The IDA study is a multi- purpose and widely applicable method and its objectives include:

- Understand the range of response or “demends” versus the range of potential levels of a ground motion record.

- Understand the structural implication of rarer/more severe ground motion levels.

- Understand the changes in the nature of the structural response as the intensity of ground motion increases.

- Produce estimates of the dynamic capacity of the global structure system.

- Give, in the case of multi-records IDA study, the stability (or variability) of all these items from a ground motion record to another.

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In this thesis work a Single-record IDA has been realyzed. Thus, starting from the based acceleration time history (as explained in Par. 7.7.3), the intensity of the ground motion has been scaled by using a Ground Motion Scale Factor, hereinafter GMfactor.

The results of the IDA study have been presented as the IDA Curve, which is a plot of a Damage Measure (DM) versus one or more Intensity Measures (IM), where:

 is a non-negative scalar that characterizes the addictional response of the structural model due to a prescribed seismic loading. In this work the maximum value of the interstorey drift ratio has ben chosen as DM.

 is a non negative scalar that characterizes the intensity of the ground motion. Both the Acceleration Spectrum Value and the GMfacton have been chosen as IM.

The GMfactor has been scaled from 0,5 to 8,0. In the following image the acceleration time history in the cases of GMfactor=1,0 and GMfactor=8,0 have been put together in order to understand the extreme limit reached with the analysis.

Fig. 7.14: Base Ground Acceleration Displacement Pattern [GMfacor=1,0] and Limit Ground Acceleration Displacement Pattern [GMfacor=8,0]

-25,0 -20,0 -15,0 -10,0 -5,0 0,0 5,0 10,0 15,0 20,0 25,0

0 5 10 15 20

Acceleration [m/s2]

Time [sec]

GM factor=1,0 GM factor=8,0

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7.8 Analysis Results

7.8.1 Eigen Analysis

7.8.1.1 SAP2000 Model

The results, in term of the first four periods and deformed shapes of the structures, are in Fig 7.15.

Fig. 7.15: Plane frame – SAP2000 Deformed Shapes and Fundamental Periods

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7.8.1.2 OpenSees Model 1

The results of the Eigen analysis on Model 1, in term of the first four periods and deformed shapes of the structures, are shown in Fig 7.16.

Fig. 7.16: Plane frame – OpenSees Model 1 - Deformed Shapes and Fundamental Periods

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The results of the Eigen analysis on Model 2, in term of the first four periods and deformed shapes of the structures, are shown in Fig 7.17.

Fig. 7.17: Plane frame – OpenSees Model 2 - Deformed Shapes and Fundamental Periods The following table summarizes the first four fundamental periods of all the considered models. One can observe the good correspondence between them. This reassure on the goodness of the OpenSees model definition.

Table 7.8.I: Plane frame – Eigen Analysis – Fundamental periods

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7.8.2 Monotonic lateral displacement-controlled pushover analysis

The results of this test are presented as a Capacity curve in two diagrams. The first one (Base shear versus Displacement of the free node curve ) is shown in Fig. 7.16. The second one (Base shear/Seismic weight versus displacement ratio ) is shown in Fig. 7.17.

Fig. 7.18: Plane frame – OpenSees Model 1 – Capacity curve 1

Fig. 7.19: Plane frame – OpenSees Model 1 – Capacity curve 2

One can clearly observe that the behaviour of the pre-defined Uniaxial material Steel01 applied to the fibers of the section determines the absebce of the descendant branch in the diagram. This aspect can be solved by using a more sophisticated model that include deterioration within it. Model 2 is an exemple of it.

0 200 400 600 800 1000 1200 1400 1600

0 500 1000 1500 2000

Base Shear [kN]

Displacement node 24 [mm]

0 0,01 0,02 0,03 0,04 0,05 0,06

0 0,02 0,04 0,06 0,08 0,1 0,12

Base Shear / Weigth

Displacement node 24 / H building

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The results of this test are presented as a Capacity curve (Base shear versus Displacement of the free node curve ). The diagram provided also information about the base shear of all the four fixed node.

Fig. 7.20: Plane frame – OpenSees Model 2 – Capacity curve

In the following image the capacity curves from the pushover analysis of Model 1 and Model 2 have been put in the same diagram in order to underline the different behavior of the two model. One can observe that this is the same within the elastic range of deformation. When yield occurs the two curves strongly separate and the degrading behavior of Model 2 appears.

0 100 200 300 400 500 600 700 800 900 1000

0 500 1000 1500 2000

V base [kN]

V1 V2 V3 V4 Vtot

0 200 400 600 800 1000 1200 1400 1600

0 500 1000 1500 2000

V base [kN]

Model 1 Model 2

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The deformed shape of the monotonic pushover analysis has represented in Fig 7.22. The displacement of the five floors have been added.

Fig. 7.22: Plane frame –OpenSees Model 2 – Deformed shape of pushover analysis 7.8.2.3 Evaluation of the behavior factor

By using a geometrical construction on the capacity curve of Model 2, it’s possible to evaluate the ratio , thus the ratio between the multiplier of the horizontal forces which corresponds to the point where a number of sections, sufficient for the development of overall structure instability, reach their plastic moment resistance ( ) and the multiplier of the horizontal forces which corresponds to the point where most strained cross-section reaches its plastic resistance ( ).

0 200 400 600 800 1000 1200

0 500 1000 1500 2000

V base [kN]

Model 2

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Which is a more sophisticated approximation of the value 1,35 given by the Par. 6.3.2 of Eurocode 8 for structure characterized by moment resisting frame and more than one storey and more than one span.

7.8.3 Ground Motion Simulation Analysis

The first results are presented in Figures 7.23 – 7.28 as Floor Displacement Time History diagrams.

Fig. 7.24: Ground Motion Simulation Analysis – Floor Number

-150 -100 -50 0 50 100 150

0 5 10 15 20 25 30

Displacement Floor 5 [mm]

Ground Motion duration [sec]

Free vibrations zone

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Fig. 7.26: Displacement Time History – Floor Number 4

Fig. 7.27 Displacement Time History – Floor Number 3 -150

-100 -50 0 50 100 150

0 5 10 15 20 25 30

Ground Motion Duration [sec]

-150 -100 -50 0 50 100 150

0 5 10 15 20 25 30

-150 -100 -50 0 50 100 150

0 5 10 15 20 25 30

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The first results are presented in Figures 7.29 – 7.34 as Floor Displacement Time History diagrams.

Fig. 7.30: Ground Motion Simulation Analysis – Floor Number -150

-100 -50 0 50 100 150

0 5 10 15 20 25 30

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Fig. 7.32: Displacement Time History – Floor Number 4 -150

-100 -50 0 50 100 150 200

0 5 10 15 20 25 30

-150 -100 -50 0 50 100 150 200

0 5 10 15 20 25 30

-150 -100 -50 0 50 100 150

0 5 10 15 20 25 30

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The following table summarizes the maximum absolute displacement obtained in the previous analysis, in order to understand the differences between the two OpenSees Models.

-150 -100 -50 0 50 100 150

0 5 10 15 20 25 30

-150 -100 -50 0 50 100 150

0 5 10 15 20 25 30

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Table 7.8.II: Plane frame – Time history – Max displacements and corresponding times

7.8.4 Incremental Dynamic Analysis

7.8.4.1 IDA Curves

The first results of the Incremental Dynamic Analysis have been proposed with the IDA Curves (Fig. 7.35 and 7.36). As previously explained (see Par. 7.7.4) in these curves the maximum value of the interstorey drift ratio has been chosen as Damage Measure.

Instead both the GMfactor and the Correponding Spectrum Acceleration have been used as Intensity Measure. GMfactor has been grown up to 8,0 where 1,0 is the basic accelerogram and also the one in which the plastic behavior of the structure appears.

Fig. 7.36: IDA Curve – Max Interstorey Drift Ratio vs GMfactor 0

1 2 3 4 5 6 7 8 9

0 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09

GM factor

Max Interstorey Drift Ratio

Model 1 Model 2

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Fig. 7.37: IDA Curve – Max Interstorey Drift Ratio vs Peak Applied Acceleration The IDA study is accelerogram and structural model specific. When subjected to differet ground motions, a model often produce quite dissimilar responses that are difficult to predict a priori. However several consideration can derive from the analysis of the previous diagrams:

 All curves exhibit a distinct elastic linear region that ends around so . This elastic branch ends when the first nonlinearity appears.

 The IDA Curve of Model 1 appears to be more regular and characterized by lower displacements for the majority of the GMfactor’s excursion. Instead the IDA Curve of Model 2 is characterized by a weaving behavior. The twisting pattern that Model 1 displays can be seen as a succession of “softening” and “hardening” branchs. It could mean that at times the structure experieces acceleration of the rate of DM accumulation and at others times a deceleration occurs that can be powerful enough to momentarily stop the DM accumulation or even reverse it, thus locally pulling the IDA Curve to relatively lower DM and making it a non-monotonic function of the IM.

 A final softening segments occurs in the IDA Curve of Model 2, when the structure accumulates DM at increasingly higher rates signaling, probably, the beginning of a branch of dynamic instability, thus the point where deformations increase in an unlimited manner for small increments of the IM (GMfactor > 5,5). In fact the OpenSees Model 2 of the plane frame is not able to bear ground motion time history acceleration with a GMfactor major than 5,5. Collapse occurs in the

0 5 10 15 20 25

0 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09 Corresponding Spectra Acceleration [m/s2]

Model 1 Model 2

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report all the collapse times in the analysis with GMfactor from 6,0 up to 8,0. As expected the time of collapse is decreasing with the increase of the GMfactor.

 The computational reasons of the numerical collapse have to be searched in the definition of the IMK deterioration model [see Par. 2.5]. In fact the rates of cyclic deterioration are controlled by theoretical rules, based on the hysteretic energy dissipated when the component is subjected to cyclic loading. Every component is characterized by a reference hysteretic energy capacity, which is a function of the yield moment, the pre-capping rotation and a deterioration parameter. After any excursion a rate of energy is dissipated, until the moment in which the cumulative dissipated energy reached the dissipation capacity.

While the plastic deformation are focused in the hinges of the beams, the solver of the program is still able to find a soluction in the equations. Increasing value of the GMfactor of the analysis determines the plastification of the base-column hinges. For that reason the horizontal displacements increase, as well as the second order effects and the stresses. When the base-column hinges reach the dissipation capacity the solver is not able to find a numerical solution to the system of equations and consequently it’s forced to break the analysis, as actually happens for GMfactor bigger than .

The increasing of the GMfactor from to , thus of the intensity of the ground motion energy, determines that the achievement of the aforementioned energy equivalent appears gradually before.

 The IDA Curve of Model 1 is not characterized by a dynamic instability branch because of the nature of the modeled material, which doesn’t consider deterioration or a limit deformation in the stress-strain material law.

7.8.4.2 Maximum required moment and rotation

Other interesting results of the Incremental Dynamic Analysis are recorded in Figg. 7.37 – 7.40. In particular the maximum required Moment of beam-hinges, the maximum required Rotation of beam-hinges, the maximum required rotation of ground-hinges and the residual displacement of the roof storey have been recorded and put into graphs in dipendence with the GMfactor, in order to understand the overall behavior of the structure and the distribution of deformation with the increase of the intensity of the

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 The beam-hinges reach the limit of the elastic branch for a GMfactor around 0,6.

 As consequence, for the rotation of the beam-hinges is major than zero.

 The hinges on the ground node reach the limit of elastic branch as well, but not negligeable residual rotation can be recorded for .

 Once reached the yield limit, the maximum moment of both beams and ground node hinges increases within the hardening branch and then remains quite constant until the appearing of the dynamic instability.

 Regarding the maximum required rotation, both beam-hinges and ground hinges are characterized by a waving but quite regular behavior until . Then, once numerical instability occurs, rotations become unsustainable by the members.

Fig. 7.38: IDA Test – Model 2 - Maximum Recorded Moment for beam-hinges

Fig. 7.39: IDA Test – Model 2 - Maximum Required Rotation for beam-hinges 0

1 2 3 4 5 6 7

0 100 200 300 400

GMfactor

Max Beam-hinges Moment [kNm]

Numerical instability in the analysis

0 1 2 3 4 5 6

-6,25E-17 0,005 0,01 0,015 0,02 0,025 0,03 0,035

GMfactor

Max Beam-hinges Rotation [Rad]

Numerical instability in the analysis

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Fig. 7.40: IDA Test – Model 2 - Maximum Recorded Moment for beam-hinges

Fig. 7.40: IDA Test – Model 2 - Maximum Required Rotation for ground column-hinges

In the following graphic, Fig. 7.41 the residual displacement of both Model 1 and Model 2 is represented. One can clearly observe that the behavior of the two model is similar for . When the intesity of the ground motion increase the two curves diverge and the deteriorating behavior of Model 1 appears. As a consequence the residual displacement of Model 2 increase in a considerable way. As already explained no records of residual displacement are available for because, as dynamic instability occurs, the simulated analysis of Model 2 end before the theorical ground motion duration.

0 1 2 3 4 5 6 7

0 200 400 600 800 1000

GMfactor

Max Ground-hinges Moment [kNm]

Numerical istability in the analysis

0 1 2 3 4 5 6 7

0 0,005 0,01 0,015 0,02 0,025 0,03 0,035

GMfactor

Max Ground-hinges Rotation [Rad]

Numerical istability in the analysis

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Fig. 7.41: IDA Test – Residual displacement of roof storey

7.8.4.3 Evaluation of the Behavior Factor

The behavior factor can be evaluated graphically from the IDA Curve. Called:

One can evaluate the as:

0

1 2 3 4 5 6 7 8

0 50 100 150 200 250 300 350 400 450

GMfactor

Residual displacement of Storey 5 [mm]

Model 1 Model 2

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Fig. 7.42: IDA Test – Evaluation of the behavior factor

This value seems to be reasonable, if compared with the one obtained in the analysis performed with the software RSTAB within the Institute for steel construction of RWTH (q=3,98 – see Par. 7.2)

However the Table 6.1 and Table 6.2 of the Eurocode 8.1-Earthquake general [16]

imposes an upper limit value for the behavior factor of moment resisting frame, characterized by a medium Ductility Class, of 4.0. Therefore the value (corresponding to a ) has been chosen as the behavior factor of the frame.

By analizing the residual rotation of base-column hinges versus GMfactor curve (Fig.

7.43) it’s clear that push the ground motion intensity over the GMfactor of 4,0 determines the lost of linearity with development of not neglectable rotations of the ground hinges of the columns. Displacements become unacceptables, with the strong increase of second order effects. This may lead to the collapse of the structure by generation of a mechanism.

0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5 5,5 6

- 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08

GM factor

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Fig. 7.43: Residual rotation of base-column hinges

In the following images (Figg. 7.43 – 7.59) all the results obtained with the Dynamic Analisis with GMfactor equal to 4,0 are reported. In particular the following results have been plotted:

 Floor Displacement Time History

 Hinges Moment Time History

 Hinges Rotation Time History

The same results for all the other Ground Motion Factors are stored in the cd annexed to the paper.

0 1 2 3 4 5 6

0 0,005 0,01 0,015 0,02 0,025

GMfactor

Residual rotation of base-column hinges [Rad]

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7.8.4.4 GMfactor=4,0 - Floor Displacement Time History

Fig. 7.44: IDA Test – GMfactor=4,0 – Floor 5 – Displacement Time History

Fig. 7.45: IDA Test – GMfactor=4,0 – Floor 4 – Displacement Time History -500

-400 -300 -200 -100 0 100 200 300 400

0 5 10 15 20 25 30

-500 -400 -300 -200 -100 0 100 200 300 400

0 5 10 15 20 25 30

-500 -400 -300 -200 -100 0 100 200 300 400

0 5 10 15 20 25 30

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Fig. 7.47: IDA Test – GMfactor=4,0 – Floor 2 – Displacement Time History

Fig. 7.48: IDA Test – GMfactor=4,0 – Floor 1 – Displacement Time History -500

-400 -300 -200 -100 0 100 200 300 400

0 5 10 15 20 25 30

-500 -400 -300 -200 -100 0 100 200 300 400

0 5 10 15 20 25 30

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7.8.4.4 GMfactor=4,0 - Hinges Moment Time History

Fig. 7.49: IDA Test – GMfactor=4,0 – Ground Nodes – Moment Time History

Fig. 7.50: IDA Test – GMfactor=4,0 – Nodes Storey 1 – Moment Time History -1000

-800 -600 -400 -200 0 200 400 600 800 1000

0 5 10 15 20 25 30

Moment [kNm]

Ground Nodes

Node 101 Node 201 Node 301 Node 401

-400 -300 -200 -100 0 100 200 300 400

0 5 10 15 20 25 30

Moment [kNm]

Nodes Storey 1

Node 502 Node 601 Node 602 Node 701 Node 702 Node 801

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Fig. 7.51: IDA Test – GMfactor=4,0 – Nodes Storey 2 – Moment Time History

Fig. 7.52: IDA Test – GMfactor=4,0 – Nodes Storey 3 – Moment Time History -400

-300 -200 -100 0 100 200 300 400

0 5 10 15 20 25 30

Moment [kNm]

Nodes Storey 2

-400 -300 -200 -100 0 100 200 300 400

0 5 10 15 20 25 30

Moment [kNm]

Nodes Storey 3

-300 -200 -100 0 100 200 300 400

0 5 10 15 20 25 30

Moment [kNm]

Nodes Storey 4

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7.8.4.4 GMfactor=4,0 - Hinges Rotation Time History

Fig. 7.55: IDA Test – GMfactor=4,0 – Ground Nodes – Rotation Time History -400

-300 -200 -100 0 100 200 300 400

0 5 10 15 20 25 30

Moment [kNm]

Nodes Storey 5

-0,025 -0,02 -0,015 -0,01 -0,005 0 0,005 0,01 0,015 0,02

0 5 10 15 20 25 30

Rotation [Rad]

Ground Nodes

Node 101 Node 201 Node 301 Node 401