Encyclopedia of Earthquake Engineering

Living Edition
| Editors: Michael Beer, Ioannis A. Kougioumtzoglou, Edoardo Patelli, Ivan Siu-Kui Au

Secondary Structures and Attachments Under Seismic Actions: Modeling and Analysis

  • Nicola NisticòEmail author
  • Alessandro Proia
Living reference work entry
DOI: https://doi.org/10.1007/978-3-642-36197-5_150-1

Synonyms

Introduction

Seismic evaluation of a construction has to include, with a given level of accuracy, the interaction phenomena between (1) soil, (2) foundation, (3) structural part of the construction to be intended as the part in elevation, and (4) nonstructural part to be intended as a part of the construction with minor or no task to the structural capacity.

From a conceptual point of view, it could be easy to assert the following principle:

the seismic evaluation of a construction has to be performed based on 1) the definition of Structural Resisting System (SRS) 2) a proper model implementation of the SRS 3) a proper analysis of the SRS.

Given that the previous principle could resolve the problem, the SRS verification remains one of the goals of the analysis process: it can be pursued with a probabilistic approach (ATC 58 2012) (1) defining the required performance (2) based on the predictable loss (direct and indirect) consequent to a given seismic event.

A general approach, in which the strategy can be framed, is the performance-based approach that (1) defines a given number of performance levels (PLs), (2) chooses a seismic level for each PL, and (3) requires a given performance for each PL. Examples of PLs are operational, immediate occupancy, life safety, and collapse prevention.

The performance-based approach delegates the applicator of it the definition or selection of the most appropriate tools to be applied for the (1) identification of the resisting system (RS), (2) structural analysis (modeling included), (3) capacity definition, and (4) verification.

The first step (RS identification) is not an easy task: the RS includes the soil and the construction that on the other hand can be split in structural and nonstructural elements which are not supposed to have any role in the global resistance of the construction with regard to neither of the so-called vertical loads or of the horizontal loads such as those that schematize wind and seismic actions.

The nonstructural elements, having no role in the seismic capacity, are generally considered as additional weight to be included in the mass evaluation, neglecting the structural interaction between them and the structural resisting system.

For many of the nonstructural elements, the absence of interaction can be considered realistic so that they can be considered as attached elements (from which the name attachment derives) that having a proper structure (mass, stiffness, structural capacity) have to be verified with regard to the seismic action transferred to them (from the resisting system).

The attachments (as previously introduced) are objects with their morphology so that they can be schematized with either continuous or discrete models opportunely connected to the structure they are attached to. Examples of attachments are (1) furnitures, (2) technical systems, and (3) art objects of a museum.

If the structure-attachment interaction is negligible as well as the soil-structure interaction, a cascade procedure can be adopted evaluating the seismic action (in terms of time histories or response spectrum) (1) at the base of the structure in elevation and then (2) transferring it at the points to which the nonstructural element is attached.

The cascade procedure, not considering the primary (SRS) and secondary (attachments) systems as a whole entity (PS system), cannot be adopted when the two systems are tuned; that means their periods are similar and the attachment could be acting as a tuned mass damper (TMD) for the primary system.

The seismic analysis of the attachments can be performed by means of different strategies (Villaverde 1997; Chen and Soong 1988) among which linear and nonlinear analyses can be included: the PS system can be analyzed as a global system with an evident computational effort.

The need of efficient and accurate methods to analyze the PS systems inspired methodologies (Igusa and Der Kiureghian 1985a, b) based on (1) modal synthesis, (2) perturbation theory, and (3) random vibrations.

The decoupling of the secondary system (from the P system) allows the evaluation of the seismic action in terms of Floor Response Spectra whose approach is similar to the approach that governs the decoupling between soil and structures: a Response Spectra is defined and applied at the base of the structure, including in it the effect of the propagation of the seismic action in the soil. Similarly, a spectra (FRS) is defined at the base of the attachment, including in it, with a cascade procedure, the effect of the propagation of the action at the soil (at first) and, subsequently, at the elevation structure.

The definition of a Response Spectra at the base of the attachment solves the problem since the attachment can be analyzed with traditional methodologies that are, for example, Seismic Modal Analyses or Time Histories Analyses based on acceleration histories compatible with the given Floor Response Spectrum.

Usually the effect of the propagation of the seismic event (from the soil to the attachment) is performed considering a linear behavior of the primary structure: this is supposed a realistic assumption for new conceived structure when operational and immediate occupancy PLs are considered.

The linear structural behavior of the principal system could be considered as nonrealistic in some cases, where the system’s nonlinearity could produce effects (on the attachment) more severe than those evaluable with a linear behavior assumption (Chaudhri and Villaverde 2008).

The attachments, as discussed so far, are secondary elements that do not give any contribution to seismic resistance of the primary system and, in these terms, can be classified as secondary nonstructural elements (NSEs) to be distinguished from the secondary structural elements (SEs) that have a negligible role in supporting the seismic action but can have a specific role in transferring the vertical load to the foundation system.

Secondary elements, either attachment (secondary NSEs) or structural elements (SEs), are both systems having their structure, opportunely linked to the primary structure: they have to be adequately modeled in order to be analyzed with the strategies common to the seismic branch, such as (1) static analyses (linear and nonlinear), (2) modal response spectrum analyses, and (3) time domain analyses either linear or nonlinear.

Morphological and Phenomenological Aspects Versus Modeling and Analysis

Depending on the structural resisting system typology and the construction usage, the NSE typology can be wide (see Fig. 1, for the building case) and their taxonomy can be found in Taghavi and Miranda (2003) where a comprehensive database of nonstructural components is presented covering different aspects such as, among others, cost information from which is deducible that the structural cost of a building could be not relevant with respect to the global cost: the office buildings structural costs, even if relevant, are only 18 % of the total cost of construction that can be split in cost of (1) structural elements, (2) secondary structural elements, and (3) nonstructural elements such as the contents are. The cost of the nonstructural elements can be estimated to reach the 70 % of total construction costs if the hotel buildings are concerned, while it is lower in office buildings (62 %) and hospitals (48 %) where contents (such as medical equipment) can be estimated to be 44 % of the total cost.
Fig. 1

Typology of building nonstructural elements. Reproduced from FEMA 74 (2005)

Economic loss due to seismic nonstructural damage can be relevant: during the 1994 Northridge earthquake, the nonstructural damage was (Kircher 2003) about 50 % of the global building damage which was estimated to be $18.5 billion.

Most of the NSEs have limited seismic performance because they are not properly attached to the primary structures, so that, depending on their slenderness, they can (see Fig. 2) (a) topple over, (b) slide and topple, and (c) slide. The loss of capacity of the NSEs or their connections can cause damage to other equipment (see Fig. 2) and injury to people, so careful attention has to be paid to the design of the connection (see Fig. 2).
Fig. 2

Principal rigid body mechanism

Unlike the old conceived NSEs, the new-generation elements can have a good seismic performance, thanks to the wide range of connections that can be adopted to link the NSE to the structure. Depending on the case at hand, the design can include (1) seismic joints opportunely designed to accommodate seismic displacements, (2) seismic isolators to reduce the acceleration level, and (3) dissipative device to reduce the level of acceleration, velocity, and displacement.

So the new-generation NSEs cannot be conceived without an adequate strategy for the connection design, an example of which is reported in Fig. 3 where the case of a machinery (for cement production) mounted on a steel structure attached on a reinforced concrete building has been reported: dissipative devices have been introduced at the base (of the steel structure) to reduce the seismic action on both the machinery and building.
Fig. 3

(a) Cement industrial building. Machinery mounted upon an rc structure. (b) Cement Industrial Building Applications: Italy

Referring (see Fig. 3) to the previously introduced example (where the equipment can be considered as an attachment of the primary structure), the following indications can be given for the seismic analysis:
  1. 1.

    If the NSE is rigidly connected to the structure and its mass (MNS) is not negligible with respect to the building mass (MS), a global analysis of the PS system is required. In case of modal spectra time history (TH) analysis, some approximation in the damping definition is needed due to the different damping of the attachment (steel structure) with respect to the reinforced concrete structure. More appropriate step-by-step TH analyses can consider the element damping, properly modeling it by means of dashpots when nonclassical damping is present.

     
  2. 2.

    If MNS is negligible with respect to MS, but the NSE period (TNS) is close to structural period (TS), the so-called tuning happens and a global analysis of the PS system is required, not excluding positive effect.

     
  3. 3.

    If MNS is negligible with respect to MS, and TNS is not close to TS, the decoupling could be considered and the strategy for the analyses could be oriented to the definition of the seismic action at the level of the connection, considering the structural system as a stand-alone element subjected to a seismic action at the base of it.

     
The following strategy can be considered: (a) time history analyses applied either to the whole system or to the stand-alone attachment, considering the acceleration histories recorded at the level of the attachment connection and (b) spectrum-based analyses defining an acceleration spectrum consistent with the time histories recorded at the points where the attachment is connected to the structures: the generated spectrum is usually named Floor Response Spectrum (FRS) even if (as the case reported in Fig. 4) the FRS has been evaluated where the attachment is linked and not at the floor level.
Fig. 4

FRS generation

The FRS definition is an important task for the NSE analysis, being the referred analysis tool to be adopted, due to its recognized simplicity in conjunction with good level of reliability (in those cases where the decoupling can be adopted). It is possible to affirm that the usually adopted acceleration spectrum is to structural analysis as the FRS is to the analysis of nonstructural elements, so that the FRS is a period-dependent function that can be evaluated for different value of the attachment damping given a specific soil and structure characterized by their own periods and dampings.

Differently from the secondary nonstructural elements, the secondary structural elements (SE) require different strategies of modeling and analysis. It is useful to introduce them as reported in CEN (2008): some structural elements (i.e., beams and columns) can be designed as seismic secondary elements, neglecting their contribution to the global seismic resistance so that their stiffness and strength can be neglected. As further specified in CEN (2008), the SE and the joint (that link them with structure) have to be designed considering (1) the vertical gravitational load, (2) displacement consequent to the seismic action, and (3) second-order effects that include the flexural moments evaluated considering the deformed element shape (P-Delta effect).

Clearly the previous definition of the SE supposes that they have a negligible influence in the global structural behavior.

Starting from the classification of the secondary elements in NSEs (attachments) and SEs, the following can be asserted: (1) given a construction, it includes a principal (P) and a secondary (S) structure; (2) if the S structure has a negligible influence on the P structure, the whole construction (PS) structural behavior can be decoupled; (3) the S structure can be classified in structural (SEs) and nonstructural (NSEs) elements; (4) the SEs have to be designed for vertical gravitational load (transferred from the PS to them, including the self-weight loads) considering the seismic-induced displacement (P-Delta effect included); (5) the NSEs have to be verified with regard to self-weight loads and seismic action transmitted by the P structure; (6) the secondary structure elements and their supports (links) have to be verified in order to avoid that their partial or total failure can induce injury to people or important objects; and (7) if the interaction between P and S systems is not negligible, a global PS analysis is required.

In the following, some of the principal characteristic regarding secondary structural and nonstructural elements will be described.

Secondary Structural Elements

Typical case of SE is internal and external building partitioning system (Glass Systems included): their seismic contribution is usually neglected in the seismic analysis (1) accepting (for severe earthquake) their damage and (2) imposing that out-of-plane collapse (see Fig. 5) has to be prevented.
Fig. 5

Molise (Italy): Seismic event occurred in 2002 (October 31 (M = 5.4) and November 1 (M = 5.3). Example of damaged infilled frame: in-plane and out-of-plane mechanisms

The infilled partitioning system can have a role in transferring the vertical load even if they can have a minor contribution in seismic global capacity. If they have no role in the vertical load as well as in the seismic P structure capacity, their classification as NSE (attachments) is reasonable.

If no flexible joints are considered (in between the P and S systems they represent), the absence of collaboration with the P system is not judicable by means of qualitative considerations, but it can require a structural analysis of the PS system including them as structural elements. In this case, the designer can follow some suggestions such as those included in CEN (2008) that consider a structural system as SE if its global stiffness is lower than 15 % of the P system stiffness. The models concerning the infilled frames systems are well known.

The damage level in the partitioning system is usually controlled imposing a threshold to the interstory drift (see Table 1) as function of the performance level and construction usage (ASCE 2002).
Table 1

Drift control: usually adopted values as function of usage and performance level

Performance level

Damage description and downtime/loss

Drift control

Immediate occupancy: usually required for construction which usage is considered strategic

Negligible structural damage; essential systems operational; minor overall damage. Downtime/loss: 24 h

0.3 % (stiff joint), 0.6 % (deformable joints)

Life safety

Probable structural and nonstructural damage; no collapse; minimal falling hazards; adequate emergency egress. Downtime/loss: possible total loss

0.5 % (stiff joint)

1.0 % (deformable joints)

Collapse prevention

Several structural and nonstructural damage; incipient collapse; probable falling hazards; possible restricted access. Downtime/loss: probable total loss

Not required

Attachments

Typical cases of attachments are parapets, windows, partitioning systems, antennas, electrical power systems, and furnitures. Depending on their components, they can be sensitive to the seismic acceleration or deformation (see Table 2).
Table 2

NSE classification (ATC/BSSC 1997) and element sensitivity with regard to acceleration and deformation

Component

Sensitivity

Component

Sensitivity

A

D

A

D

A. Architectural

B. Mechanical equipment

1. Exterior skin

1. Mechanical equipment

 Adhered veneer

S

P

 Boilers and furnaces

P

 

 Anchored veneer

S

P

 General mfg. and process machinery

P

 

 Glass blocks

S

P

 HVAC equipment, vibration isolated

P

 

 Prefabricated panels

S

P

 HVAC equipment. Nonvibration isolated

P

 

 Glazing systems

S

P

 HVAC equipment, mounted in-line with ductwork

P

 

2. Partitions

2. Storage vessels and water heaters

 Heavy

S

P

 Structurally supported vessels (category 1)

P

 

 Light

S

P

 Flat bottom vessels (category 2)

P

 

3. Interior veneers

3. Pressure piping

P

S

 Stone, including marble

S

P

4. Fire suppression piping

P

S

 Ceramic tile

S

P

5. Fluid piping, not fire suppression

4. Ceilings

 Hazardous materials

P

S

 (a) Directly allied to structure

P

 

 Nonhazardous materials

P

S

 (b) Dropped, furred, gypsum board

P

 

6. Ductwork

P

S

 (c) Suspended lath and plaster

S

P

   

 (d) Suspended integrated ceiling

S

P

   

5. Parapets and appendages

P

    

6. Canopies and marquees

P

    

7. Chimneys and stacks

P

    

8. Stairs

P

S

   

A acceleration sensitive; D deformation sensitivity; P primary response; S secondary response

Modeling and Analysis

NSEs are elements characterized by their mass and stiffness, and independently of the seismic action they are subjected to, they can be modeled and analyzed based on FEM strategies considering either their linear or nonlinear behavior.

In general the NSE is a system composed by subsystems with a structural complexity (see Fig. 6) that can require 3D complex models to be calibrated by means of experimental tests (Fig. 6a) including identification strategies: dynamic tests can be carried out by means of shacking tables (Fig. 6c).
Fig. 6

(a) Morphology of a bushing. (b) Bushing experimental test carried out at UC Berkeley (CA). (c) Cabinet experimental test carried out at UC Berkeley (CA)

The experimental tests in support of modeling and analysis implementation are especially required either when the importance of NSE usage is considered strategic or when the cost of it justifies the experimental activity. In some cases, a qualification procedure can be required, generally ruled by international standard (Gilani et al 1999; IEEE 2005).

Modeling

Modeling has to take into account all the components that give stiffness and strength contributions, including the connection elements that, if needed, have to be modeled as nonlinear elements.

In many cases, such as the bushing sketched in Fig. 6a, an accurate modeling requires informations about all the subcomponents (coil springs, valves) in terms of mass stiffness and strength. The needed informations are not usually known and the element investigation has to be supported by means of experimental tests devoted either to global information acquisition (frequencies, modal shapes) or to evaluation of the level of performance given a defined seismic action. Experimental tests can include shaking table tests or static tests: this aspect is strictly linked to the qualification process (IEEE 2005).

Seismic Action Modeling and Structural Analyses

Seismic action can be simulated according to the usually adopted strategies that, for the case at hand, include (1) time histories (usually in terms of acceleration) and (2) response spectrum finalized either to modal analyses or to static linear or nonlinear pushover analyses.

Seismic level will depend on the referred performance level that (see Table 1) identifies the required performance associable to a seismic event with a given return period, to be defined based on cost-benefit analysis.

General rules valid for secondary elements are the following:
  1. 1.

    Mass and stiffness uncertainties have to be considered together with spatial distribution of seismic effect in case of extended SE systems.

     
  2. 2.

    Seismic effects on SE have to take into account, in general, both horizontal and vertical components to be evaluated based on a structural model of the principal system.

     
  3. 3.

    If the SE behavior can be decoupled from the principal system, the datum method for the evaluation of the peak acceleration at the SE is based on the Floor Response Spectrum (FRS) that given an SE element, with a defined structural period and damping, attached to a given part of a structure, having its mechanical properties, subjected to a given seismic event (E), allows to define the peak acceleration to which the element will be subjected when the seismic event (E) is transferred at the base of NS element.

     

Based on the knowledge of the FRS, one of the following methods can be adopted: (a) static equivalent forces (including nonlinear pushover analysis), (b) modal analysis, and (c) time history (linear or nonlinear) analyses based on accelerograms compatible with the FRS.

Floor Response Spectra-Based Evaluation

Floor Response Spectra are functions that define the response spectrum of a given response parameter (e.g., acceleration, velocity, displacement) as a function of period and damping of a given structure (attachment) localized at a given point of the construction.

The generally adopted technique for the FRS definition consists in (1) analyzing the P structure (to which the S structure is attached) in the time domain, considering n time histories (e.g., acceleration TH), (2) evaluating (for each TH), at a given point of the structure, the TH of the acceleration and the related response spectrum for a given damping value, and (3) defining one representative spectrum (based on the n available FRSs) having a given overcoming probability (usually a 50 % probability is considered): for low values of n (e.g., minor than 7), an envelope spectrum has to be considered.

Usually the location of the attachment is not known in advance, so that the previous procedure can be applied considering p points obtaining p Response Spectra. For those points that are located at the same level (floor) of the P structure, a single spectrum can be evaluated (enveloping the Response Spectra), naming it Floor Response Spectrum.

If the goal is the evaluation of conservative FRSs, for each floor, a set of points has to be opportunely selected so that both translational and rotational effects are captured: they usually include the floor centroids and one or more corners for each floor.

It is worth mentioning that having defined the P system structural model, it is possible to evaluate a transferring function Hp that (1) knowing the Fourier Transform (F i, input FT) of a given accelerogram (2) allows the definition of the Fourier Transform of the TH acceleration at given point (F o, output FT) so that (3) the inverse Fourier Transform of Fo gives the TH at the considered point that is the required information for the Response Spectrum evaluation.

Alternatively, if the Power Spectrum G i of a given earthquake or of a family of earthquakes is known as well as the previous defined Input Transferring Function (F i), the output Power Spectrum (G o) is evaluable according to Eq. 1.
$$ {G}_o(w)={G}_i(w)\;\left|{\mathrm{H}}_p(w)\right|{}^2 $$
(1)
Knowing G o(W) and the transferring function (HSDOF) of a single-degree-of-freedom system (SDOF), the SDOF Spectra Power Density is evaluable (see Eq. 2) and the related Response Spectrum is the required FRS.
$$ {G}_{\mathrm{SDOF}}(w)={G}_o(w)\;\left|{\mathrm{H}}_{\mathrm{SDOF}}(w)\right|{}^2 $$
(2)
The previously presented approaches are not usually adopted for conventional structures as they are time consuming, so that predictive expressions are proposed in literature or enforced in international recommendations: given the peak ground acceleration, the floor peak acceleration is evaluated by multiplying the PGA by an analytical function, named Sa in the following.

The usually proposed functions (Sa) are based on (1) simplified expressions for the evaluation of the floor acceleration and (2) simplified shape functions representative of the required FRS.

It is worth mentioning that given a structure, knowing of it (1) the prevalent modal shape (Φ), (2) the prevalent period (Ts), and (3) the modal participation factor (Γ) if a seismic action is considered, the absolute structural acceleration (üi) associated to the single modal coordinate (Φi) can be evaluated according to Eq. 3 where RS(TS) is the value of the normalized acceleration spectrum, for a given value of the structural damping (ζS) at the prevalent period of the structure.
$$ {\ddot{u}}_i=\mathrm{P}\mathrm{G}\mathrm{A}\cdot \Gamma \cdot \left\{1+\left[{\mathrm{R}}_{\mathrm{S}}\left({\mathrm{T}}_{\mathrm{S}}\right)-1\right]\cdot {\Phi}_{\mathrm{i}}\right\} $$
(3)
Assuming a given analytical function (RFRS), being it dependent on the period (TNS) and the damping (ζNS) of the nonstructural element, the required FRS, associable to the Φi modal coordinate, is equal to
$$ {\mathrm{FRS}}_i=\mathrm{P}\mathrm{G}\mathrm{A}\cdot \Gamma \cdot \left\{1+\left[{\mathrm{R}}_{\mathrm{S}}\left({\mathrm{T}}_{\mathrm{S}}\right)-1\right]\cdot {\Phi}_{\mathrm{i}}\right\}\cdot {\mathrm{R}}_{\mathrm{FRS}}\cdot \left({\mathrm{T}}_{\mathrm{NS}}\right) $$
(4)
Usually, Eq. 4 is simplified adopting (1) a constant value for RS(Ts) evaluated at the plateau of the acceleration spectrum (assumed in the range of 2–3), (2) a simplified expression for the evaluation of modal displacement Φi of a given floor supposed to be equal to z/H where z is the level of the considered floor and H is the total construction height, and (3) a value of Γ between 1 and 1.5.
Based on the previous assumptions, Eq. 4 can be simplified as follows, having assumed RS = 3, Γ = 1:
$$ {\mathrm{FRS}}_i=\mathrm{P}\mathrm{G}\mathrm{A}\cdot \Gamma \cdot \left[1+2\cdot \left(z/\mathrm{H}\right)\right]\cdot {\mathrm{R}}_{\mathrm{FRS}}\cdot \left({\mathrm{T}}_{\mathrm{NS}}\right) $$
(5)
In the following sections, some literature expressions (CEN 2008; FEMA 369 2001; AFPS 2007; KTA 2012) will be given, expressing them in terms of the normalized FRS (Sa) that corresponds to FRSi evaluated for PGA = 1.

It has to be specified that in order to show the trend of the Sa functions, they will be plotted, contextualizing it to the simple structure described in the Case Study section.

CEN (2008)

The following Eq. 6, plotted in Fig. 7, is proposed, supposing ζ S = ζ NS = 5 %.
Fig. 7

Amplification factor Sa for different values of z/H

$$ {\mathrm{S}}_{\mathrm{a}}=3\cdot \frac{1+\frac{\mathrm{z}}{\mathrm{H}}}{1+{\left(1-\frac{{\mathrm{T}}_{\mathrm{NS}}}{{\mathrm{T}}_{\mathrm{S}}}\right)}^2}-0.5\ge 1 $$
(6)
Equation 6 can be evaluated assuming TNS = 0, obtaining (1) the value of the expression adopted for the evaluation of the normalized floor acceleration (Eq. 7a) and (2) the value of the adopted expression for the evaluation of RFRS (Eq. 7b). So that Eq. 6 can be rearranged as reported in Eq. 7c.
It is possible to recognize that (1) a value of 2.5 has been assumed for the evaluation of Rs and (2) the expression adopted for the evaluation of RFRS is supposed to be dependent on the normalized floor height.
$$ \frac{{\ddot{u}}_i}{\mathrm{PGA}}=\left\{1+\left[{\mathrm{R}}_{\mathrm{S}}\left({\mathrm{T}}_{\mathrm{S}}\right)-1\right]\cdot \frac{\mathrm{z}}{\mathrm{H}}\right\}=1+1.5\cdot \frac{\mathrm{z}}{\mathrm{H}} $$
(7a)
$$ {\mathrm{R}}_{\mathrm{FRS}}=\frac{1+\frac{\mathrm{z}}{\mathrm{H}}}{1+1.5\cdot \frac{\mathrm{z}}{\mathrm{H}}}\cdot \frac{3}{1+{\left(1-\frac{{\mathrm{T}}_{\mathrm{NS}}}{{\mathrm{T}}_{\mathrm{S}}}\right)}^2}-\frac{0.5}{1+1.5\cdot \frac{\mathrm{z}}{\mathrm{H}}} $$
(7b)
$$ {\mathrm{S}}_{\mathrm{a}}=\left(1+1.5\cdot \frac{\mathrm{z}}{\mathrm{H}}\;\right)\cdot \left\{\frac{1+\frac{\mathrm{z}}{\mathrm{H}}}{1+1.5\cdot \frac{\mathrm{z}}{\mathrm{H}}}\cdot \frac{3}{1+{\left(1-\frac{{\mathrm{T}}_{\mathrm{NS}}}{{\mathrm{T}}_S}\right)}^2}-\frac{0.5}{1+1.5\cdot \frac{\mathrm{z}}{\mathrm{H}}}\right\}\ge 1 $$
(7c)

FEMA 369 (2001)

The following Eq. 8, plotted in Fig. 8, is proposed supposing ζ S = ζ NS = 5 %, where the values of RFRS are reported in Table 3:
Fig. 8

Amplification factor (Sa) shape versus TNS/TN

Table 3

Values of RFRS as function of TNS/TN

TNS/TN

RFRS

TNS /TN < 0.5 and TNS /Tn > 2.0

1.0

0.7 ≤ TNS /TN < 1.4

2.5

0.5 ≤ TNS /TN < 0.7

(7.5 × TNS /TN ) −2.75

1.4 ≤ TNS /TN < 2.0

6 – (2.5 × TNS /TN )

$$ {\mathrm{S}}_{\mathrm{a}}=\left(1+2\cdot \frac{z}{\mathrm{H}}\;\right)\cdot {\mathrm{R}}_{\mathrm{FRS}} $$
(8)
It is possible to recognize that (1) a value of 3.0 has been assumed for the evaluation of Rs and (2) the expression adopted for the evaluation of RFRS is supposed to be independent on the normalized floor height.

AFPS (2007)

The following expression is proposed:
$$ {\mathrm{S}}_{\mathrm{a}}=\sqrt{1+{\varGamma}_S^2\cdot {\mathrm{R}}_{\mathrm{S}}^2\cdot \left(\frac{\mathrm{z}}{\mathrm{H}}\right)2\alpha}\cdot {\mathrm{R}}_{\mathrm{FRS}} $$
(9)
where (1) α is a parameter to be calibrated in order to minimize the difference between the effective modal displacements (Φi) and the proposed simplified expression (z/H), (2) Rs is the value of the normalized structural acceleration evaluated at the fundamental structural period (TS) for a considered value of the structural damping, (3) the participation factor (Γ S ), evaluable according to Eq. 10a, assumes a maximum value of 1.6, if α = 1.5 is imposed, and (4) RFRS values are reported in Table 3 as a function of parameter A (see Eq. 10b) that takes into account the damping (ζ NS) of the nonstructural element.
$$ {\Gamma}_S=\frac{2\alpha +1}{\alpha +1} $$
(10a)
$$ A=\frac{35}{2+{\xi}_{NS}} $$
(10b)
It is worth mentioning that Eq. 9 derives from Eq. 4, with the following assumptions: (1) a unitary participation factor is considered for the ground acceleration, while the principal mode participation factor is considered according to Eq. 10a. (2) The spectral acceleration Rs(Ts) is considered for the evaluation of the floor relative acceleration instead of the spectral relative normalized acceleration (Rs(Ts) − 1). (3) The ground acceleration and the relative structural acceleration are combined through the SRSS (Square Root of the Sum of the Squares) combination rule (Table 4 and Fig. 9).
Table 4

Values of RFRS as function of TNS/TN

TNS/TN

RFRS

TNS /TN < 0.5 and TNS /TN > 2.0

1.0

2/3 ≤ TN /TN < 3/2

A

0.5 ≤ TNS /TN < 1.5

\( A-\left(A-1\right)\times \frac{\left( \log \frac{3}{2}\frac{{\mathrm{T}}_{\mathrm{NS}}}{{\mathrm{T}}_S}\right)}{\left( \log \frac{3}{4}\right)} \)

2 < TNS /TN

\( {K}_T=A-\left(A-1\right)\times \frac{\left( \log \frac{2}{3}\frac{{\mathrm{T}}_{\mathrm{NS}}}{{\mathrm{T}}_S}\right)}{\left( \log \frac{4}{3}\right)} \)

Fig. 9

Amplification factor (Sa) shape versus TNS/TN (α = 1.5, ΓS = 1.6, RS = 2.5)

KTA (2012)

The proposed expression does not give any information to evaluate the floor acceleration (ag), but it only defines the amplification shape (RFRS) reported in Fig. 10 (left), where the f-axis is the component frequency axis (in logarithmic scale) and f 1, f n, and f limit, respectively, are (1) lowest decisive eigenfrequency of the principal system at the lower limit value in the variation range of the system parameters, however, not lower than the rightmost corner frequency of the highest plateau of the associated response spectrum; (2) highest decisive eigenfrequency of the principal system for the upper limit value in the variation range of the component parameters, however, not lower than the rightmost corner frequency of the highest plateau of the associated response spectrum; and (3) upper limit frequency of the associated response spectrum.
Fig. 10

Amplification factor shape (left) and maximum amplification factor (right)

The maximum value of the amplification factor is reported in Fig. 10 (right), where D1 and D2 are respectively the damping ratios (in percentage of critical damping) of structural and nonstructural elements whose suggested values are reported in Table 5.
Table 5

Suggested damping values (in percent of critical damping). Column A: to be adopted for verifying the load-carrying capacity and integrity and for determining the spectra. Column B: in the case of mechanically active components for which the functional capability is verified by a deformation analysis

Components

Damping ratios

A

B

Pipes

4

2

Steel with welded connections and welded components (e.g., vessels, valves, pumps, motors, ventilators)a

4

2

Steel with SL or SLP bolt connections (SLstructural bolt connection with a borehole tolerance ≤ 2 mm; SLPfitted bolt connection with a borehole tolerance ≤ 0.3 mm)

7

4

Steel with SLV(P) or GV(P) bolt connections (SLV(P) – preloaded fitted bolt connection; GV(P) – fitted friction-grip bolt connection)

4

2

Cable support structures

10b

7

Fluid media

0.5

0.5

aIf, on account of the design, deformations are possible only in small regions of the structure (low structural damping), the values as listed shall be halved (special cases)

bIn well-substantiated cases, the damping ratio may be increased up to 15 %

In order to compare the obtainable FRS with those previously discussed, the amplification factor (Sa) is plotted in Fig. 11, having assumed (1) a 5 % damping for both structural and nonstructural elements, (2) the FEMA expression (see Eq. 11) for the evaluation of the normalized floor acceleration, (3) f1 = 6.66 Hz that is the rightmost corner frequency of the acceleration plateau of CEN (2008) type 1 Spectrum (A soil), and (4) fn = 11.1 Hz that is the highest decisive eigenfrequency of the principal system described in the Case Study section.
Fig. 11

Normalized floor acceleration based on Eq. 11

$$ {\mathrm{a}}_{\mathrm{g}}=1+2\cdot \frac{\mathrm{z}}{\mathrm{H}} $$
(11)

Verification

As stated in KTA (2012), the verification process has to regard (1) the load-carrying capacity in terms of strength, stability, and secure positioning (e.g., their protection against falling over, dropping down, impermissible slipping); (2) the integrity, that is, the capability of a component above and beyond its load-carrying capacity to meet the respective requirements regarding leak tightness and deformation restrictions; and (3) the functional capability, that is, the capacity of a system or component above and beyond its load carrying capacity to fulfill the designated tasks by way of its respective mechanical or electrical function.

Depending on the importance of the element to verify and the material (conventional or nonconventional material), the verification process could include experimental tests either for the validation of the numerical model or for the qualification of the element itself. The verification procedure can include a) analysis, b) physical experiments, and c) analogies and plausibility considerations.

Based on the introduced classification that distinguishes secondary element in structural and nonstructural, the following criteria can be defined:
  1. 1.

    Secondary structural elements have to be verified with regard to the vertical loads transmitted from the P structure, opportunely combined with the other action considered to be contemporary to the seismic action. The connections have to be verified with regard to the seismic-induced action, including second-order effects such as those induced by the axial load in the deformed configuration (P-Δ effects).

     
  2. 2.

    Secondary nonstructural elements have to be verified with regard to the self-weight loads opportunely combined with the other actions considered to be contemporary to the seismic action.

     
  3. 3.

    For both types of elements (S and NS elements), the action supposed to act contemporary with the seismic action can be consequent to different events such as those pertaining to collisions, explosions, and fires.

     

The verifications have to consider potential damage induced to other elements, which loss of capacity could induce either human or economic loss.

The verification is performed checking that the element capacity will be greater than the demand, defined in terms of different mechanical properties (stresses, forces, displacement) depending on the adopted materials.

In order to define the design forces, the considered floor response acceleration spectra can be reduced to take into account the nonstructural element ductility. If the FRSs have been numerically evaluated, they have to be modified (see Fig. 12) to take into account the structural stiffness uncertainties: (1) an adequate plateau has to be imposed in correspondence of the structural period, (2) the linear envelope has to be properly introduced, and (3) the ductility of the nonstructural element can be considered, properly reducing the FRS (see Fig. 12b, c)
Fig. 12

(a) Design spectra definition. Shape modification: alternative solution. (b) Design spectra definition. Design spectra based on ABCDEF spectrum. (c) Design spectra definition. Design spectra based on ABB’CDEF spectrum

Case Study

The previously described procedures to determine FRSs will be applied to a steel frame system hosting a set of equipments, whose characteristics and localization are reported in Table 6 and Fig. 13 (left), reproduced from KTA (2012).
Table 6

Equipment mass (Ton) and period (sec)

Equipment

Mass

Floor

Period

1

Open image in new window

20

1

0.051

2

Open image in new window

10

1

0.093

3,4,5

Open image in new window

10

2

0.070

6

Open image in new window

10

3

0.034

7

Open image in new window

20

3

0.060

8

Open image in new window

30

4

0.033

9,10

Open image in new window

15

5

0.036

Fig. 13

Case study: geometry (left) and models (center, right)

The maximum acceleration of each equipment can be evaluated by means of (1) time histories considering the interaction between the principal structure and the equipment or (2) FRSs evaluated based on the previously described cascade procedure or on the predictive expressions already presented.

Time History-Based Evaluation of Equipment Accelerations

A detailed model of the PS system could include the secondary system modeled as reported in Fig. 13 (right): PS principal modal shapes are those reported in Fig. 14a. Alternatively equipments can be modeled by means of mass lumped at pertinent position of the floor as reported in Fig. 13 (center): PS principal modal shapes are those reported in Fig. 14b.
Fig. 14

(a) Modal shape: direct modeling of the equipment. Periods (sec): 0.48, 0.158, 0.121. (b) Modal shape: equipment modeled as lumped masses. Periods (sec): 0.49, 0.159, 0.09

Both models have been analyzed by means of time histories, carried out (1) generating 7 accelerograms compatible with the acceleration spectra (PGA = 0.15 g) suggested in CEN (2008) for B soil and low-magnitude events (M < 5.5) (see Fig. 15), (2) considering a constant damping value of 2 % for the structural model and for the equipments, (3) performing a dynamic modal TH analysis evaluating, for each accelerogram, the maximum absolute value of a given quantity (acceleration), and (4) averaging the maximum values obtained (at step 3) for each analysis.
Fig. 15

Target acceleration spectrum (PGA = 0.15 g) and spectra of the generated accelerograms

If the detailed model with interaction (WI) is considered, the evaluated quantities are the mass accelerations of the single equipment.

Regarding to the lumped mass system, (1) for each considered accelerogram, an FRS has been generated (Fig. 16a): the FRS is relative to the acceleration of the top left floor corner (no sensible variations in FRSs have been observed if other floor points are considered); (2) having generated (for each floor) seven FRSs, the averaged FRS has been evaluated (Fig. 16b); and (3) for each equipment, depending on the floor it is attached to and its period (TNS), the acceleration has been evaluated through the resulting FRS.
Fig. 16

(a) Floor Response Spectra generations: scheme. (b) Generated Floor Response Spectra. (c) Generated Floor Response Spectra: 1° Floor; Period range 0.07–0.1 s

The results of the performed evaluation are reported in Table 7 where the acceleration of each floor and the acceleration of each equipment are reported, calculated with or without interaction.
Table 7

Floor acceleration (FA, m/sec2) and equipment acceleration evaluated without (F_w/o) and with (F_w) interaction

N E

N F

Floor acceleration

FRS W/0

FRS W

1

2.21

2.7

2.1

2

2.21

6.5

3.2

3

3.26

4.20

4.0

4

3.26

4.20

4.0

5

3.26

4.20

4.0

6

3.65

3.70

3.8

7

3.65

5.10

3.8

8

4.28

4.30

4.3

9

5.25

5.3

5.3

10

5°

5.25

5.3

3.3

Comparing the maximum acceleration evaluated considering the PS system with those evaluated through the cascade procedure, a significant difference (if the Equipment 2 is concerned) between the two approaches can be noticed: the difference is aspectable since the equipment period (0.093 s) is close to the period of the third modal shape (0.090 s) so that (see Fig. 16c) in a very small period range (in between 0.09 and 0.1 s), the acceleration ranges between 7.0 and 4.5 m/s – the already-mentioned tuning effect causes the equipment acceleration reduction if the complete PS system is analyzed in order to include the P-S interaction.

Analytical FRS-Based Evaluation of Equipment Accelerations

It has been already outlined that CEN proposal (CEN 2008) and FEM proposal (FEMA 369 2001) are based on a fixed value (5 %) of structural and equipment damping. So that only the proposal reported in AFPS (2007) and KTA (2012) will be considered in the following.

It has to be specified that (1) concerning the KTA (2012) proposal, the floor acceleration evaluated by means of TH analyses (see Table 7) has been considered and (2) concerning the AFPS (2007) proposal, the FRSs have been evaluated by means of Eq. 9, evaluating the spectral acceleration corresponding to the first structural period (0.49 s) and a participation factor (ΓS) equal to 1.6 that correspond to α = 1.5 (see Eq. 10a).

It is clear (see Fig. 17a, b) that considering equipment periods close to the lowest structural periods, the KTA proposed expressions are more conservative while the AFPS expressions are less conservative (Tables 8 and 9).
Fig. 17

(a) Generated FRSs. Acceleration versus period: (1) numerical simulation and (2) AFPS predictive equations evaluated based on Eq. 9 (α = 1.5, ΓS = 1.6, RS = 2.4). (b) Generated FRSs. Acceleration versus frequency: numerical simulation and KTA predictive equations

Table 8

Normalized floor displacement

Floor

MD

z/H

0.21

0.24

2°

0.45

0.42

3°

0.68

0.62

0.87

0.81

5°

1.00

1.00

Table 9

Equipment (Eq) acceleration

N E

NF

AFPS

KTA

1

1°

2.2

22.0

2

1°

2.2

34.0

3,4,5

2°

3.3

42.0

6

3.7

22.0

7

3.7

42.0

8

4.3

25.0

9,10

5.3

35.0

Summary

The chapter deals with the methodologies focused on the seismic analyses of the so-called secondary (sometimes attachments) elements that are part of a construction whose seismic resistance is delegated to a primary resistant structure.

Although secondary elements can be decontextualized from the primary resistant structures, they will be subjected to seismic action as well and, having their own structures, need to be modeled and analyzed by means of methods included in the general methodologies proper of seismic branch.

Among the methodologies usually adopted for secondary element analyses, the Floor Response Spectra (FRS)-based analyses become popular due to their recognized simplicity.

FRSs provide acceleration (consequently velocity and displacement) to which the secondary element (with a given period and damping) will be subjected to when attached (from which the alternative name attachments derives) to a given part of the structures such as a building floor (from which the name Floor Response Spectra derives).

Given that FRS generation could require onerous numerical analyses, simplified expressions are proposed in literature and discussed in the following together with the general methodologies tailored to secondary element modeling and seismic analyses.

Cross-References

References

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Dipartimento di Ingegneria Strutturale e GeotecnicaUniversità La SapienzaRomeItaly