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Seismic design of buildings Analysis and design of earthquake resistant buildings Roberto Tomasi 11.05.2017 Roberto Tomasi Seismic design of buildings 11.05.2017 1 / 22
Seismic design of buildingsAnalysis and design of earthquake resistant buildings
Roberto Tomasi
11.05.2017
Roberto Tomasi Seismic design of buildings 11.05.2017 1 / 22
Overview
1 Elements of dynamics
2 Standards Design Rules
3 Capacity design
4 References
Roberto Tomasi Seismic design of buildings 11.05.2017 2 / 22
Elements of dynamics
Non Linear SDOF System
In the previous lecture an elastic behaviour of the structure was assumed inorder to study its dynamic behaviour under seismic loads. Is thishypothesis realistic? Can we really design earthquake resistant structurewithout damages?An earthquake is a rare natural phenomenon that produces exceptional(very high) loads on the structures. Designing structures that behave in theelastic range might be too expensive.We can accept that some damages occur taking into account the nonlinear behavior of a structure, that in most of cases can be represented byan elasto-plastic model, characterized by:
Fy = Strength
k = Stiffness
µ = uuu0
= Ductility
Roberto Tomasi Seismic design of buildings 11.05.2017 3 / 22
Elements of dynamics
Anelastic SDOF System
The equation of motion has a similar formulation;the only difference is that now the internal force isnot linear dependent by the relative displacement.
Mu(t) + cu(t) + ku(t) = −Mx0(t)
The solution can not be obtained in the same wayof a linear SDOF system. A numerical integrationin time domain (Time history analysis) have tobe done, even if it can be very time consuming incase of many degree of freedom systems.Some past studies have demonstrated that themaximum displacement of a non linear SDOFsystem is very similar to the correspondinglinear system one (Newmark Hypothesis).
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Elements of dynamics
Design Response Spectrum
From Newmark’s hypothesis:
ue,max = ua,max = µ · uy
For the elastic system the maximum force can becalculated as:
Fs,e,max = m · SA,e
From the picture it is easy to realize that:
Fs,e,max
Fs,y=
umax
uy= µ
The maximum force for the anelastic system can be calculated as:
Fs,y =Fs,e,max
µ= m ·
SA,eµ
= SD,e ⇒
The design force can be evaluatedreducing the elastic force by theductility!We can define a reduced responsespectrum defined as design responsespectrum.
Roberto Tomasi Seismic design of buildings 11.05.2017 5 / 22
Elements of dynamics
Design Response Spectrum
Fs,y =Fs,e,max
µ= m ·
SA,eµ
= SD,e
The analysis of a non linearstructure can be performedassuming an elastic behaviourand reducing the forces by thefactor q!!!
The higher the ductility, the lowerthe design force!!!If we design a ductile structure we canreduce the elastic force by a coefficientcalled factor q, that is equal to theductility. This means to reduce theelastic spectrum by the factor q.
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Elements of dynamics
Role of ductility in seismic response
Sd(T ) = Se(T )/q
• The ductility properties of the structure reduces the level of the action.• The q-factor represents the ductility level of the structures.
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Standards Design Rules
Energetic Approach
What’s the physical meaning of q-factor? Why can we reduce theelastic forces?From the integration of the equation of motion it can be obtained:
Ek(t) + Evd(t) + Eh(t) = Ein(t) + Es(t)
Ek = Kinetic Energy; Evd = Energy Dissipated via Viscous Damping; Eh = Hysteretic Energy;
Ein = Input Energy; Es = Recoverable Elastic Energy;
The input energy expressed in the energy formulation is the true total energy input tothe system. If we want to reduce the energy absorbed by the structure, caused by theelastic strain energy we need to increase the hysteretic energy, equal to the amount ofthe dissipated energy. We can reduce the elastic force if the structure can dissipate theinput energy, by means of its hysteretic behaviour. However this implies damages to thestructure.
q ⇒ ductility⇒ dissipated energy
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Standards Design Rules
What’s the value of q-factor?
Standards give the q-factor for a lot of different structural type and fordifferent materials. The designer can choose between a high level ofductility (CDH) or a medium level (CDM). In the first case q-factor ishigher.In order to ensure the selected ductility level, a lot of design rules areexplained according to the capacity design approach. Constructiondetails are becoming increasingly important!!!
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Standards Design Rules
Structural types for timber structures
Structural Type Example
1.Cross Laminated Timber (X-Lam) system, i.e. buildings comprised ofX-Lam shear walls according to XX (reference to the Material Propertiessection) with the specifications given in YY (reference to the CapacityDesign Rules section).
2.Light wood-frame system, i.e. structures in which shear walls are madeof timber frames to which a wood-based panel or other type of sheathingmaterial according to XX (reference to the Material Properties section)are connected according to the specifications given in YY (reference tothe Capacity Design Rules section).
3.Log House building system, i.e. structures in which walls are made bythe superposition of rectangular or round solid or glulam timber elements,prefabricated with carpentry joints at their ends and with upper andlower grooves according to specifications given in YY (reference to theCapacity Design Rules section).
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Standards Design Rules
Q-factors for timber structures [Proposal]
Structural type DCM DCH
X-Lam buildings 2 3Light-Frame buildings 2,5 4Log House buildings 2 -Moment resisting frames 2,5 4Post and beam timber buildings 2 -Mixed structures made of timber framing and masonryinfill resisting to the horizontal forces.
2 -
Large span arches with two or three hinged joints - -Large span trusses with nailed, screwed, doweled andbolted joints
- -
Vertical cantilever systems made with glulam or X-Lamwall elements
2 -
For structures designed in accordance with the concept of low-dissipative structuralbehaviour (DCL) the behaviour factor q should not be taken greater than 1,5.
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Capacity design
Traditional Design Approach
As seen previously the non linear behavior of a structure can be reppresented by anelasto-plastic model, characterized by strength, stiffness and ductility. Which one is themost important?It depends on the intensity of the expected ground motion. For low earthquakes thestructure should be strength and stiff in order to avoid damages. For high earthquakesthe structure should be ductile to dissipate energy and to avoid the collapse. A verystrength and ductile structure would be best but in most of cases it would be tooexpensive.
How can we design a ductile structure?
If to increase the strength of a structure may be easy (even if expensive), to increase theductility the failure mode must be selected. In fact we have to avoid a brittle failuremode in order to assure a ductile one. In other words it is decided which elements of astructural system will be permitted to yield (ductile components) and which one are toremain elastic (brittle components). This strategy is called:
Capacity Design
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Capacity design
Capacity Design
To explain the capacity design approach we can consider a chain made ofglass rings and hence brittle, and one ring is made of steel and henceductile. Suppose the chain is tauted by a force P.
If the strength of the steel ring is lower than the glass ring one, thebehaviour of chain will be ductile. In fact the steel ring is able to stretch alot before breaking. If the strength of the steel ring is higher than the glassring one, the behaviour of chain will be brittle. In fact the glass ring breaksimmediately after reaching its strength force.
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Capacity design
Capacity Design
In order to get a ductile chain the glass ring needs to be moreresistant than the steel one. Hence the design force for the steel ringwill be equal to P, but for the glass ring, that has to be in the elastic range,the design force will be equal to the resistance of the steel ring, amplifiedby an opportune safety factor: the overstrength factor γRd .
Rd ,steel = P
Rd ,glass = γRd · Rd ,steel γRd > 1
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Capacity design
Capacity Design
Also a structure can be viewed as a chain where some elements are characterized by abrittle failure model some others by a ductile model one. Hence we have to avoid thatbrittle failure happens before yielding of ductile elements.
The designer must choose the right structure failure mode.
Ductile elements will be designed for the analysis internal forces (bending moment,shear,...). The design force for brittle elements are obtained by the equilibrium ofinternal forces after yielding of ductile elements.
Shear Mechanism: ductile behaviourLarger spacing between nails
Strong Hold-downRd,HD ≥ γRd · Rd,nails
Rocking Mechanism: brittle behaviourSmaller spacing between nails
Weak Hold-downRd,HD ≤ γRd · Rd,nails
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Capacity design
The overstrength factor
The overstrength factor is used to ensure that the resistance of the brittleelement is always greater than the ductile one, in order to achieve a globalductile failure of the structure.
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Capacity design
Regularity
Another important topic to ensure a good seismic behavior of structure istheir regularity. This should be take into account in the early stages of theconceptual design of a building.The guiding principles which should be satisfied are:• Structural simplicity, characterized by the existence of clear anddirect path for the transmission of the seismic forces, so that themodeling, studying and designing are subject to much less uncertaintyand the structure seismic behavior is much more reliable.
• Uniformity, characterized by an even distribution of the structuralelements which allow short and direct transmission of the inertia forcescreated in the distributed masses of the building. If necessary thebuilding needs to be subdivided by seismic joint into dynamicallyindependent units.
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Capacity design
Regularity
• Bi-directional resistance and stiffness: Horizontal seismic motion isa bi-directional phenomenon and thus the building structure shall beable to resist horizontal actions in any directions.
• Torsional resistance and stiffness: The structure should possesadequate torsional resistance and stiffness in order to limit thetorsional motions which tend to stress the different structural elementsin a non-uniform way.
• Diaphragmatic behavior at storey level: Floors should act ashorizontal diaphragms that collect and transmit the inertia forces tothe vertical structural systems and ensure that those systems acttogether in resisting the horizontal seismic action.
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Capacity design
Regularity
In relation to the previous principles building structures are categorised intobeing regular and non regular structures.The second ones should be avoid and standards increase the design seismicaction for this type of structures.A structure can be regular or not in elevation or in plan.
Criteria for regularity in elevation Criteria for regularity in plane
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Capacity design
Seismic Analysis of Buildings
Another reason to ensure the regularity in elevation of building is the possibility toreplace the modal analysis of the structure with a simplified method of analysis, calledlateral force method. In fact for regular in elevation buildings the dynamic behaviour is
well represented by just the first mode shape, neglecting the others. The structure canbe modelled just as a SDOF system, which period T1 can be evaluated by a simplifiedequation, function of building height H.
The seismic base shear force Fb can be determined as the product of the total mass ofthe building and the ordinate of the design spectrum at period T1.
T1 = Ct · H3/4 Fb = Sd(T1) ·m · λ λ: correctional factor
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Capacity design
Seismic Analysis of Buildings
For most of regular in elevation building the first mode shape may beapproximated by horizontal displacement increasing linearly along theheight of the building.
The horizontal force acting on the storey i can be calculated as:
Fi = Fb ·zi ·mi∑zi ·mi
zi ; zj are the heights of the masses mi ;mj above the level of application of the seismic action
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References
References
• Villaverde R. , Fundamental Concepts of earthquake engineering, CRCPress 2009
• Di Sarno L. – Elnashai A.S., Fundamentals of earthquake engineering,WILEY 2008
• Chopra A.K. – Dynamics of structures, PRENTICE HALL2007• Penelis G.G. , Kappos A.J. – Earthquake -resistant concrete structure,TAYLOR AND FRANCIS, 1997
• Christopoulos C., Filiatrault A. – Principles of passive supplementaldamping and seismic isolation, IUSS Press 2006
• EN 1998-1 (Eurocode 8-1): Design of structures for earthquakeresistance- General rules, seismic actions and rules for building
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