NATIONAL TECHNICAL UNIVERSITY OF ATHENS LABORATORY OF EARTHQUAKE ENGINEERING Design of Earthquake-Resistant Structures Basic principles Ioannis N. Psycharis
Basic considerations Design earthquake: small probability of occurrence during the life of the structure. It wouldn t be wise to design the structure to sustain this ground motion without any damage at all. Flexural yielding: the stiffness is temporarily reduced but the structure regains its strength and stiffness when unloading starts. Economical approach: Allow the structure to exceed its elastic limit during the design earthquake. Control the extend of the damage. Exclude unwanted types of damage (e.g. brittle failure). Repair any damage after the earthquake.
Seismic response of SDOF systems x(t) x (t) u(t) g u(t) m m g p(t)=-m x (t) K, C = K, C x (t) g ẍ (t) g Seismic forces (d Alembert): Restoring force: Damping force: p( t) m x ( t) f s f d V i g ( t) C u ( t) K i u Equation of motion: p f s f d mu mu Cu Ku mx g
Seismic response of SDOF systems Equation of motion: u 2ζωu ω u 2 x g Eigenfrequency: ω K m Eigenperiod: T 2π m Κ Damping coefficient: ζ 2 C mk Duhamel s integral: 1 u( t) x ω d t 0 g ( τ) e ω ( t τ dτ ζω ( t τ ) sin d )
μετακίνηση, u (mm) μετακίνηση, u (mm) φασματική μετακίνηση, SD (mm) μετακίνηση, u (mm) επιτάχ. εδάφ. (m/sec 2 ) Response spectrum 1.0 0.5 1.03 Τ=0.1 sec, ζ=5% 2.5 Σεισμός Καλαμάτας, 1986 (Long) 0.0 0.0-0.5-1.0 0 2 4 6 8 10 χρόνος, t (sec) -2.5 0 2 4 6 8 10 χρόνος, t (sec) 6.0 60 3.0 Τ=0.2 sec, ζ=5% 0.0 50-3.0-6.0-5.88 0 2 4 6 8 10 χρόνος, t (sec) 40 30 16.0 8.0 T=0.3 sec, ζ=5% 20 15.38 0.0-8.0-16.0-15.38 0 2 4 6 8 10 χρόνος, t (sec) 10 0 1.03 5.88 0.0 0.1 0.2 0.3 0.4 0.5 0.6 περίοδος, Τ (sec)
SD (cm) SA (g) Response spectrum Kalamata, 1985 earthquake Response spectra for ζ = 0, 2%, 5%, 10%, 20% 18 16 14 12 10 8 6 4 2 3.6 3.2 2.8 2.4 2.0 1.6 1.2 0.8 0.4 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Ðåñßïäïò Ô (sec) 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Ðåñßïäïò Ô (sec)
SD (cm) SA (g) Pseudo-Spectra For small values of damping (ζ 20%) SA ω 2 SD = PSA SV ω SD = PSV Limits: T 0: SD 0 SV 0 SA x g,max T : SD x g,max SV x g,max SA 0 18 16 14 12 10 8 6 4 2 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Ðåñßïäïò Ô (sec) 3.6 3.2 2.8 2.4 2.0 1.6 1.2 0.8 0.4 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Ðåñßïäïò Ô (sec)
PSV (cm/sec) Logarithmic representation 1000 0.1 g PSA 100 SD 100 cm 0.01 g 10 10 cm 0.001 g 1 1 cm 0.1 0.01 0.1 1 10 Ðåñßïäïò Ô (sec)
PSA (m/sec 2 ) ADRS - representation 8.0 7.0 T=0.5 sec T=1.0 sec 6.0 5.0 T 2π SD PSA 4.0 3.0 T=1.5 sec 2.0 1.0 0.0 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 SD (cm)
ΦΑΣΜΑΤΙΚΗ ΤΑΧΥΤΗΤΑ, PSV (cm/sec) Characteristic regions PSA=100g PSA=10g PSA=1g 100.00 C D SD=100cm PSA=0.1g 10.00 B SD=10cm E PSA=0.01g SD=1cm 1.00 A PSA=0.001g SD=0.001cm SD=0.01cm SD=0.1cm 0.10 0.01 0.10 1.00 10.00 ΠΕΡΙΟΔΟΣ, T (sec) PSA=0.0001g
Effect of soil conditions Design spectrum
Ground types according to EC 8 Design spectrum
EC 8 Elastic response spectrum S e (T) = spectral acceleration for period T a g = γ I a gr = design ground acceleration γ Ι = importance factor a gr = reference peak ground acceleration on type A ground S = soil factor η= 0.10 0.05+ζ
S e / a g EC 8 Elastic response spectrum 2.5 S η 2.5 S η T C /T S 2.5 S η T C T D /T 2 0 Τ Β Τ C T D Περίοδος, T (sec) S S S S e e e e T (T) ag S 1 η 2.5 1 TB (T) a (T) a (T) a g g g S η 2.5 TC S η 2.5 T TC T S η 2.5 2 T D for 0 T T B for T B T T C for T C T T D for T D T 4.0 sec
S e / a g EC 8 Elastic response spectrum 2.5 S η 2.5 S η T C /T S 2.5 S η T C T D /T 2 0 Τ Β Τ C T D Περίοδος, T (sec) Ground type Τ Β (sec) Τ C (sec) Τ D (sec) S Α 0.15 0.40 2.50 1.00 Β 0.15 0.50 2.50 1.20 C 0.20 0.60 2.50 1.15 D 0.20 0.80 2.50 1.35 E 0.15 0.50 2.50 1.40
Performance levels Define earthquake loadings with various probabilities of occurrence Define various acceptable level of damage Combine each earthquake loading with an acceptable level of damage Performance level Very small damage Damage limitation level Significant damage Collapse prevention Close to collapse Frequency of occurrence of the seismic excitation Large Frequent earthquakes Small Rare earthquakes Very small Very rare earthquakes unacceptable
Performance requirements of seismic codes Most codes are based on two performance levels: Damage limitation level The structure is expected to experience limited structural and non-structural damage during frequent earthquakes. In this limit state: The structural members retain their strength and stiffness. No permanent deformations and drifts occur. No repair is needed. The seismic action is usually termed as the serviceability earthquake. Reasonable probability of exceedance = 10% in 10 years (mean return period = 95 years). Compliance criteria are usually expressed in terms of deformation limits.
Performance requirements of seismic codes Collapse prevention level Ensure prevention of collapse and retention of structural integrity for an earthquake with a small possibility of occurrence during the life of the structure: Significant damage might happen. The structure should be able to bear the vertical loads and retain sufficient lateral strength and stiffness to protect life during aftershocks. The seismic action is referred as the design earthquake. For structures of ordinary importance: 10% probability of exceedance in 50 years (mean return period of 475 years). Compliance criteria are expressed in terms of forces (force-based seismic design).
S a / (S a gr ) Design concept 2.5 The structure is designed to behave elastically up to a level of seismic forces lower than the one required for fully elastic response. 2.0 1.5 1:q The loads corresponding to the design seismic action are derived by dividing the elastic ones by a reduction factor denoted by R (reduction factor) or 1.0 q (behaviour factor). 0.5 1:q How is q defined? 0.0 0 T B T C T D Period, T
Ductility capacity The collapse mechanism of the structural members is related to their deformation and not to the forces induced to them during the seismic action. In order to comply with the non-collapse criterion, an overall ductile behaviour should be ensured. In other words: the structure should have an adequate capacity to deform beyond its elastic limit without substantial reduction in the overall resistance against horizontal and vertical loads. This is achieved through proper dimensioning and detailing of the structural elements. In addition, capacity design concepts are applied, in order to ensure that ductile modes of failure (e.g. flexure) should precede brittle modes of failure (e.g. shear) with sufficient reliability.
Nonlinear response Force-deformation relation Elastoplastic idealization Same area under the two curves
Basic definitions Yield strength behaviour factor: f s q y = f e f y Ductility factor: μ = u m u y f e f y It can easily be proved: u m u e = μ q y u y u e u m u Larger values of μ correspond to larger plastic deformation more damage. For μ close to 1, the response is close to the elastic.
Eπιτάχυνση Yield Acceleration διαρροής, (m/sec a y (m/sec 2 ) 2 ) Inelastic response spectra 8.0 7.0 6.0 Inelastic spectra for constant ductility ΑΝΕΛΑΣΤΙΚΑ INELASTIC SPECTRA ΦΑΣΜΑΤΑ μ=1 μ=1.5 μ=2 μ=3 μ=5 μ=8 5.0 4.0 3.0 2.0 1.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Ιδιοπερίοδος, Period, T Τ (sec) (sec)
Ductility factor The damage that will be induced to the structure is directly related to the ductility factor, μ. For the non-collapse performance criterion, certain values can be assigned to the allowable maximum value of μ, depending on: The material (ductile or brittle). The structural system (the more isostatic is the structure the less is the allowable value of μ). The structural irregularities in plan or in elevation and the torsional sensitivity (reduce the allowable value of μ). The connections and the bracing types (steel structures).
S a / (S a gr ) Relations q y - μ For T>T C the equal displacement assumption is made: 2.5 2.0 q y = μ 1.5 For T B <T<T C the equal energy assumption is made: 1.0 0.5 q y = 2μ 1 0.0 0 q y T B T C T D Period, T For T close to zero (very stiff structures) the response is elastic: q y q y μ 2μ 1 q y = 1 1 0 T B T C T C T
S a / (S a gr ) Design value of q Design value of the behaviour factor: 2.5 2.0 q = γ Rd q y (γ Rd = overstrength) 1.5 1.0 1:q Usually, rigid structures possess larger overstrength than flexible ones we usually assume constant value of q for T>T B. q y q μ 0.5 0.0 0 1:q T B T C T D Period, T γ γ Rd Rd q q y q y 2μ 1 1 T B T C T C T
Ductility classes (EC8) Ductility Class High (DCH) Strict detailing criteria should be fulfilled. Provides higher safety margins against local or global collapse under seismic actions stronger than the design earthquake. Ductility Class Medium (DCM) Compared to DCH, certain detailing rules are relaxed. The design leads to slightly easier to construct structures. Provides good performance during moderate earthquakes. Ductility Class Low (DCL) For low seismicity areas. The structure is designed according to EC2 without special seismic considerations. Large values of q are allowed.
Proper detailing Aims to: provide the structure with an adequate capacity to deform beyond its elastic limit without substantial reduction of the overall resistance against horizontal and vertical loads. Example for concrete structures: Special rules are applied for the confinement reinforcement (stirrups) at column-to-beam joints and at critical regions of columns and beams.
Capacity design Aims to: ensure that ductile modes of failure (e.g. flexure) should precede brittle modes of failure (e.g. shear) with sufficient reliability prevent the formation of a soft-story mechanism ensure that certain parts of the structure will remain elastic if it is so desired (e.g. foundation, bridge deck, etc.) Example for concrete structures: At column-to-beam joints, the sum of the design values of the moments of resistance of the columns should be larger than 1.3 the sum of the design values of the moments of resistance of the beams: M Rc 1.3 M Rb
Design procedure Define the seismic loads for: The appropriate seismicity, the soil conditions at the site and the importance of the structure. The appropriate value of the behaviour factor, q Material Structural system Irregularites Ductility class Perform a structural analysis of the structure for the seismic and non-seismic loads, assuming elastic response. Combine the individual load cases according to the code provisions to get the envelop of the member loads.
Design procedure (cont d) Perform the dimensioning of the beams in flexure. Check the beams in shear using the capacity design approach (based on the flexural strength of the beams). Perform the dimensioning of the columns in flexure using the capacity design approach (based on the flexural strength of the beams framing with the columns at the joints). Check columns in shear using the capacity design approach (based on the flexural strength of the columns). Perform a detailed dimensioning of the joints in order to assure their integrity during the design earthquake. Perform the dimensioning of the foundation using the capacity design approach (based on the flexural strength of the columns). Design displacements: d = q d E, d E = from seismic analysis.
MDOF systems Equation of motion M u + C u + K u = M r x g(t) where: M = mass matrix C = damping matrix K = stiffness matrix r = earthquake direction vector
Natural modes Eigenfrequencies They are derived from the solution of the characteristic equation: K ω 2 M = 0 Eigenmodes (eigenvectors, eigenshapes) They are derived from the solution of the system of equations: where: K ω 2 M φ i = 0 φ i = i th eigenmode φ ji = j th component of i th eigenmode
Eigenmodes node node node node node node 1 st mode 2 nd mode 3 rd mode 4 th mode
Properties Orthogonality φ Τ i M φ j = 0 for i j φ Τ i K φ j = 0 for i j Generalized mass m i = φ Τ i M φ i Generalized stiffness k i = φ Τ i K φ i It can be shown that k i = m i ω 2
Free vibrations For arbitrary initial displacements
Free vibrations For initial displacements according to 1 st mode
Free vibrations For initial displacements according to 2 nd mode
Free vibrations For initial displacements according to 3 rd mode
Modal analysis Displacement at the j th degree of freedom: u j (t) = N n=1 u jn (t) where u jn is the displacement of the j th degree of freedom that corresponds to the n th mode. Response of n th mode: u jn (t) = Y n t φ jn Yn + 2ζ n ω n Y n + ω n 2 Y n = Γ n x g where Γ n is the participation factor of the n th mode: Γ n = φ i Τ M r φ i Τ M φ i
Modal analysis x g(t) u jn (t) = Y n t φ jn Yn + 2ζ n ω n Y n + ω n 2 Y n = Γ n x g
Use of response spectra Maximum displacement of the n th mode at the j th degree of freedom: max u jn = Γ n S d T n, ζ n φ jn T n, ζ n is the spectral displacement that where S d corresponds to period T n and damping ζ n. Maximum seismic force of the n th mode at the j th degree of freedom: max F jn = Γ n S a,d T n, ζ n m j φ jn T n, ζ n is the design spectral acceleration where S a,d that corresponds to period T n and damping ζ n.
Combination of modal responses Significant modes k < N k n=1 M n 0.90 m tot where M n is the effective mass of the n th mode: M n = Γ n φ n Τ M r The effective mass of each mode depends on the direction of the seismic action.
Combination of modal responses Let A n, A m be the maximum value of a quantity A (internal force or displacement) of the n th and the m th mode respectively. SRSS k maxa = ± A n 2 CQC n=1 k k maxa = ± ε nm A n A m n=1 m=1 ε nm = 8 ζ 2 1+r r 3/2 1 r 2 2 +4 ζ 2 r 1+r 2 with r = T n T m
Simplified formulas For planar motion in the plane of the seismic action and for the n th mode: Γ n = N j=1 N j=1 m j φ jn 2 m j φ jn M n = Γ n N j=1 m j φ jn
Design procedure Define structural properties Compute mass and stiffness matrices M and K Estimate modal damping coefficients ζ n Solve the eigen-problem to determine the k lower natural frequencies ω n and modes φ n, 1 n k Compute the corresponding natural periods T n = 2π ω n For a given direction of the seismic action: Compute the participation factors Γ n Compute effective modal masses M n and check that their sum is larger than 90% of the total mass. If not, increase the value of k and repeat the procedure
Design procedure For a given direction of the seismic action, compute the maximum response for each mode n by repeating the following steps: On the design response spectrum read the spectral acceleration S a,d that corresponds to period T n and damping ζ n Compute the seismic force F j,n at each degree of freedom j Perform static analysis of the structure subjected to forces F j,n and determine the modal internal forces and displacements Combine the modal responses using SRSS or CQC for each direction of the seismic action
Spatial combination Let A x, A y, A z be the estimated maximum values of a quantity A that correspond to two horizontal orthogonal directions x, y and the vertical direction z of the seismic action. The maximum value of A for simultaneous action of the earthquake in all directions x, y, z can be estimated as: 1 st Method A = ± A x 2 + A y 2 + A z 2 2 nd Method A = ±A x ± 0.3A y ± 0.3A z A = ±0.3A x ± A y ± 0.3A z or or A = ±0.3A x ± 0.3A y ± A z
Remarks For typical buildings, the vertical component of the seismic action can be neglected. Displacements If d E are the displacements from the above analysis, the actual displacements are calculated as d = q d E where q is the value of the behavior factor used in the design response spectrum. The elastic response of a structure to a specific earthquake can also be computed using the above procedure by substituting the design response spectrum with the elastic spectrum of the ground motion.