STATIC NONLINEAR ANALYSIS Advanced Earthquake Engineering CIVIL-706 Instructor: Lorenzo DIANA, PhD 1
By the end of today s course You will be able to answer: What are NSA advantages over other structural analysis methods? How to perform an NSA? What are the key elements when performing NSA? 2
Source: FEMA 440 Earthquake Engineering Assessment 3
Earthquake Engineering Assessment Structure Action Static Dynamic Linear Linear static analysis Linear dynamic analysis Non-linear Non linear static analysis Non linear dynamic analysis 4
Linear static analysis Seismic forces are applied as equivalent static forces Regular (plan and vertical) buildings NTC 2008: T 1 =C 1 x H 3/4 T 1 C 1 H period of the structure factor related to material height of the structure 5
Equivalent Force Method F i = F h x z i x W i / z j x W j F h = S d (T 1 ) x W x λ / g W total mass of the structure λ factor related to the number of storyes W i and W j mass of i and j z i and z j height of i and j 6
Equivalent Force Method 7
Linear dynamic analysis Seismic forces are applied as a dynamic action on structures Regular (plan and vertical) buildings NO mix structures M ẍ + C ẋ + K x = - M e x ẍ g M C K e x mass matrix damping matrix stiffness matrix vector of seismic direction 8
Linear dynamic analysis Response spectrum method Overlapping of effects (SRSS: square root of the sum of the square) 9
Linear dynamic analysis 10
Linear dynamic analysis 11
Linear dynamic analysis 12
Non linear time history analysis Advantage: the complexity of the dynamic load is considered + the dynamic behavior of structure Disadvantage: time consuming 13
Non linear static procedure How does it work? Capacity vs. Demand Sa Sd 14
Nonlinear static procedure Necessary elements? Acceleration-displacement response spectrum (ADRS) Capacity Curve V (force) Δ (disp.) Sa Sd 15
Nonlinear static procedure Developing capacity curves: Displacement-based methods 16
Nonlinear static procedure Assumptions Response: maximum displacement Deformation: most often following the first mode Benefits Displacement-based analysis Nonlinear behavior of the structure Drawbacks Simplified approach (static) Damping difficult to represent 17
Content Capacity curve Seismic damage Acceleration-Displacement Response Spectrum (ADRS) Performance Point Large-scale vulnerability assessment 18
Content Capacity curve Seismic damage Acceleration-Displacement Response Spectrum (ADRS) Performance Point Large-scale vulnerability assessment 19
Capacity Curve Capacity curve defines the capacity of the structure independent to any seismic demand Objectives: Estimate the maximum horizontal displacement Estimate the (global) ductility of the structure 20
Capacity Curve Pushover Simplified analytical method (EC8) DBV method 21
Capacity Curve Pushover Load pattern: distribution of forces should represent the dynamic behavior Triangular (if building regular enough) Following the first mode Other simplified distributions depending on the building Base Shear 22
Capacity Curve Pushover Load pattern: importance definition Effect of lateral load can change the capacity curve Base Shear 23
Capacity Curve Pushover Different model type Structural model (FEM, AEM, etc.) 24
V Capacity Curve Force/Displacement to spectral coordinates Equivalent SDOF Sa Δ Sd Γ= mass participation factor m*= equivalent mass Sa = V m Γ Sd = Δ u Γ 25
Case study in: Lestuzzi P. and Badoux M. (2013) Évaluation parasismique des constructions existantes Unreinforced masonry (URM) Example URM building in Yverdon-les-Bains 26
Unreinforced masonry (URM) Architectural plan Example 27
Unreinforced masonry (URM) Example Simplification for the modeling 28
Unreinforced masonry (URM) Key-parameter: Frame effect (coupling of horizontal/vertical elements) Zero moment flexible slab (w/o frame effect) 2 h h 0 to ta l 3 Infinitely stiff lintels (total frame effect) 1 h h 0 s to r e y 2 29
Unreinforced masonry (URM) Extreme case studies: Example 2 h h 6.7 m 0 to ta l 3 1 h h 1.2 5 m 0 sto rey 2 30
Unreinforced masonry (URM) Example A) Frame effect not considered Wall lw=2.0 m lw=2.4 m lw=4.0 m lw=5.0 m lw=6.0 m TOT [dir Y] Nxd [kn] 180 160 1200 360 1520 - VRd [kn] 21.7 24.6 123.3 113.6 304.1 618 δu [%] 0.8 0.8 0.8 0.8 0.8 - Δy [mm] 10.3 7.1 10 5.6 10.5 7.9 Δu [mm] 27.7 25.3 27.5 24.2 27.9 24.2 k [kn/mm] 2.1 3.5 12.4 20.2 29 77.8 31
Unreinforced masonry (URM) Example A) Frame effect not considered V Rd, R = l w N xd 2 h w 1 1.15 N xd l w t w f xd V Rd, S = 1.5 f vd0 0.85 f xd + 0.4 N xd 0.5 N xd 32
Unreinforced masonry (URM) Example A) Frame effect not considered h p 3 h 0 h p κ y = V Rd H tot + 6 E I eff G A eff y = δ y H tot δ y = d y hp u = y + d u d y δ u = d u hp = 0.8% 33
Lateral strength [kn] Unreinforced masonry (URM) Example A) Frame effect not considered 700 600 500 400 300 200 100 Lw = 2.0 m Lw = 5.0 m Lw = 6.0 m Building (dir Y) 0 0 5 10 15 20 25 30 Top ultimate displacement dir Y [mm] 34
Unreinforced masonry (URM) Example A) Frame effect not considered V Rdy = (4*21.7+2*113.6+1*304) = 618 kn k E = (4*2.1+2*20.2+1*29) = 77.8 kn / mm Δ y = 618 / 77.8 = 7.94 mm Δ u = 24.2 mm 35
Unreinforced masonry (URM) Example A) Frame effect not considered Wall lw=2.0 m lw=2.4 m lw=4.0 m lw=5.0 m lw=6.0 m TOT [dir X] Nxd [kn] 180 160 1200 360 1520 - VRd [kn] 21.7 24.6 123.3 113.6 304.1 837.6 δu [%] 0.8 0.8 0.8 0.8 0.8 - Δy [mm] 10.3 7.1 10 5.6 10.5 8.5 Δu [mm] 27.7 25.3 27.5 24.2 27.9 25.3 k [kn/mm] 2.1 3.5 12.4 20.2 29 98.6 36
Lateral strength [kn] Unreinforced masonry (URM) Example A) Frame effect not considered 900 800 700 600 500 400 300 200 Lw = 2.4 m Lw = 4.0 m Building (dir X) 100 0 0 5 10 15 20 25 30 Top ultimate displacement [mm] 37
Unreinforced masonry (URM) Example A) Frame effect not considered V Rdx = (4*123.3+14*24.6) = 837.6 kn k E = (4*12.4+2*14*3.5) = 98.6 kn / mm Δ y = 837.6 / 98.6 = 8.50 mm Δ u = 25.3 mm 38
Unreinforced masonry (URM) Example B) Frame effect considered Wall lw=2.0 m lw=2.4 m lw=4.0 m lw=5.0 m lw=6.0 m TOT [dir Y] Nxd [kn] 180 160 1200 360 1520 - VRd [kn] 57.8 65.6 219.4 180.3 541.2 1133 δu [%] 0.8 0.8 0.8 0.4 0.8 - Δy [mm] 10.2 7.6 12.0 6.5 14.5 9.8 Δu [mm] 27.6 25.7 29.0 14.9 30.9 14.9 k [kn/mm] 5.7 8.6 18.3 27.6 37.4 115.4 39
Lateral strength[kn] Unreinforced masonry (URM) Example B) Frame effect considered 1200 1000 800 600 400 Lw = 2.0 m Lw = 5.0 m Lw = 6.0 m Building (dir Y) 200 0 0 5 10 15 20 25 30 35 Top ultimate displacement [mm] 40
Unreinforced masonry (URM) Example B) Frame effect considered V Rdy = (4*57.8+2*180.3+1*541.2) = 1133 kn k E = (4*5.7+2*27.6+1*37.4) = 115.4 kn / mm Δ y = 1133 / 115.4 = 9.81 mm Δ u = 14.9 mm 41
Unreinforced masonry (URM) Example B) Frame effect considered Wall lw=2.0 m lw=2.4 m lw=4.0 m lw=5.0 m lw=6.0 m TOT [dir X] Nxd [kn] 180 160 1200 360 1520 - VRd [kn] 57.8 65.6 219.4 180.3 541.2 1796 δu [%] 0.8 0.8 0.8 0.4 0.8 - Δy [mm] 10.2 7.6 12.0 6.5 14.5 9.3 Δu [mm] 27.6 25.7 29.0 14.9 30.9 25.7 k [kn/mm] 5.7 8.6 18.3 27.6 37.4 193.6 42
Lateral Strength [kn Unreinforced masonry (URM) Example B) Frame effect considered 2000 1800 1600 1400 1200 1000 800 600 Lw = 2.4 m Lw = 4.0 m Building (dir X) 400 200 0 0 5 10 15 20 25 30 35 Top ultimate displacement [mm] 43
Unreinforced masonry (URM) Example B) Frame effect considered V Rdy = (4*219.4+14*65.6) = 1796 kn k E = (4*18.3+14*8.6) = 193.6 kn / mm Δ y = 1796 / 193.6 = 9.28 mm Δ u = 25.7 mm 44
Unreinforced masonry (URM) Frame not considered Example Frame considered 45
Unreinforced masonry (URM) Example Seismic safety factor obtained Passing to ADSR format m = m i Φ i (equivalent mass) (1.0+0.75+0.50+0.25+0.10)*250 = 625 t = m* Γ = m iφ i m i Φ = m 2 i m i Φ 2 i (1.0+0.75+0.50+0.25+0.10)*(1.0 2 +0.75 2 +0.50 2 +0.25 2 +0.10 2 )= 2.5/1.875 = 1.33 = Γ 46
Unreinforced masonry (URM) Example Seismic safety factor obtained Frame not considered Frame considered dir V Rd [KN] S ay [m/s2]= V Rd / m* Γ Δ u /Γ [mm] d* [mm] Y 618 0.74 18.2 14.4 1.26 X 838 1.01 19 12 1.58 Y 1133 1.36 11.2 9.9 1.13 X 1796 2.16 19.3 5.6 3.45 α 47
Pictures, formulas and graphs in: Luchini C (2016) PhD Thesis. Development of Displacement-Based Methods for Seismic Risk Assessment of the Existing Building Stock Displacement based vulnerability method (DBV) DBV method = development of capacity curve through few geometrical and mechanical parameters. Method developed for masonry and reinforced concrete existing buildings. MASONRY BUILDINGS 48
Displacement based vulnerability method (DBV) The equivalence between the multi degree of freedom (MDOF) and the single substitute one (SDOF) is established using the procedure proposed by Fajfar (2000) and assumed as reference also in Eurocode 8- Part 1 (2005), thus by introducing the Γ coefficient and the equivalent mass m*. The capacity curve assessment is associated with a certain analysis direction (dir = X,Y). 49
Displacement based vulnerability method (DBV) a y,dir = F dir m Γ Yield acceleration T y,dir = 2π m k dir Fundamental period d 4 = ε d u,s.s. dir + 1 ε d u,unif. dir Ultimate displacement capacity 50
Displacement based vulnerability method (DBV) a y,dir = F dir m Γ Yield acceleration F dir m Γ Total base shear capacity Equivalent mass of SDOF Coefficient representing the modal participation factor and it requires of assuming a modal shape Φ Γ = m iφ i m i Φ i 2 = m m i Φ i 2 51
Displacement based vulnerability method (DBV) a y,dir = F dir m Γ F dir Yield acceleration Total base shear capacity Directly related to the shear strength offered by resistant walls area at the first floor F dir = A 1,dir τ u,dir ξ ζ res τ u,dir ξ ζ res Ultimate shear strength of masonry Coefficient aimed to penalize the strength as a function of the main prevailing failure mode expected at scale of masonry piers, it is assumed equal to 1 in the case of prevalence of shear failure mechanisms and 0.8 in the case of compression-bending failure mechanisms Corrective factor aimed to consider peculiarities of existing buildings 52
Displacement based vulnerability method (DBV) T y,dir = 2π m k dir Fundamental period k dir Stiffness of the SDOF system k dir = ζ rig G H 2 ζ rig N i=1 A i,dir h i Corrective factor taking into account the coupling effect due to spandrels and the flexural contribution 53
Displacement based vulnerability method (DBV) 54
Displacement based vulnerability method (DBV) ζ res = ζ 1 ζ 2 ζ 3 ζ 1 ζ 2 ζ 3 It takes into account the non homogeneous size of masonry piers It takes into account the influence of geometric and shape irregularities in the plan configuration It takes into account the influence of spandrels stiffness that directly affects the global collapse mechanism 55
Displacement based vulnerability method (DBV) ζ res = ζ 1 ζ 2 ζ 3 56
Displacement based vulnerability method (DBV) ζ rig = ζ 4 ζ 5 ζ 4 = 1 1+ 1 1.2 G E hp bp 2 coefficient aimed to take into account the influence of the flexural component on the stiffness h p and b p height and width of masonry piers ζ 5 coefficient related to the characteristics of spandrels 57
Displacement based vulnerability method (DBV) ζ res = ζ 1 ζ 2 ζ 3 58
Displacement based vulnerability method (DBV) d 4 = ε d u,s.s. dir + 1 ε d u,unif. dir Ultimate displacement capacity ultimate displacement of masonry piers according to a prevailing failure mode δ u,dir d y,dir ε ultimate drift of masonry piers according to a prevailing failure mode in pier for the examined direction (if shear or flexural one) yield displacement computed starting from a y,dir and T y,dir Fraction assigned to the soft story global failure mode 59
Displacement based vulnerability method (DBV) d 1 = 0.7d y d 2 = ρ 2 d y ρ 2 coefficient varying as a function of the prevailing global failure mode. It is proposed to assume a value equal to 1.5 in case of soft story failure mode and 2 in case of the uniform one 60
Displacement based vulnerability method (DBV) The reliability of the new vulnerability models has been verified through a comparison between the capacity curves evaluated by using the mechanical models and the pushover obtained by non-linear static analyses. 61
Displacement based vulnerability method (DBV) 62
Displacement based vulnerability method (DBV) 63
Displacement based vulnerability method (DBV) 64
Displacement based vulnerability method (DBV) 65
Displacement based vulnerability method (DBV) 66
Displacement based vulnerability method (DBV) 67
Displacement based vulnerability method (DBV) 68
Displacement based vulnerability method (DBV) 69
Content Capacity curve Seismic damage Acceleration-Displacement Response Spectrum (ADRS) Performance Point Large-scale vulnerability assessment 70
Seismic Damage 71
Seismic Damage Damage-based design/assessment: Related to social and economical costs of damage/mitigating measures 72
EMS98 damage grades 73
Content Capacity curve Seismic damage Acceleration-Displacement Response Spectrum (ADRS) Performance Point Large-scale vulnerability assessment 74
Response Spectrum ADSR spectrum Sa Sd 75
Response Spectrum 76
Response Spectrum S d S a ω n 2 ω n k m 77
Acceleration-Displacement Response Spectrum (ADRS) 78
Acceleration-Displacement Response Spectrum (ADRS) 79
SIA norm ADSR Sa Sd T B < T < T C constant acceleration = a gd * 3 factor (2.5; S; η) T < T B increasing acceleration (tending for T=0 to a gd * S) T < T C decreasing acceleration T < T D constant displacment 80
SIA norm ADSR 81
SIA norm ADSR S a = a gd S 1 + S a = 2.5 a gd S η S a = 2.5 a gd S η T C T 2.5η 1 T T B 0 T T B T B T T B T C T T D S a = 2.5 a gd S η T C T D T 2 T D T 82
Content Pushover technique capacity curve Seismic damage Acceleration-Displacement Response Spectrum (ADRS) Performance Point Large-scale vulnerability assessment 83
Determination of the performance point Two of the main static non linear methods to determine the performance point (examples covered here) Equivalent linearization (FEMA 440) Improved Spectrum Method of ATC40 N2 method (equal displacement rule, EC8 approach) 84
Equivalent Linearization Method A little bit of background First introduced in 1970 in a pilot project as a rapid evaluation tool (Freeman et al. 1975) Basis of the simplified analysis methodology in ATC-40 (1996) Improved later in FEMA 440 document (2005) 85
Equivalent Linearization Method Based on equivalent linearization. The displacement demand of a non-linear SDOF system is estimated from the displacement demand of a linear-elastic SDOF system. The elastic SDOF system, referred to as an equivalent system, has a period and a damping ratio larger than those of the initial non-linear system (ATC, 2005). 86
Equivalent Linearization Basic equations 87
Source: FEMA 440 Equivalent Linearization- Performance Point S d = S d T eq ; ζ eq = S d T eq ; ζ =5% η = S a T eq ; ζ =5% T eq 2 4π 2 η 88
Equivalent Linearization- Performance Point S d = S d T eq ; ζ eq = S d T eq ; ζ =5% η = S a T eq ; ζ =5% T eq 2 4π 2 η S d T eq ; ζ eq Spectral displacement of the equivalent system T eq Equivalent period of vibration ζ eq Equivalent viscous damping ratio S d T eq ; ζ =5% Displacement demand of the linear system with 5%-damping elastic ratio η Reduction factor depending from the damping modification factor η = 1 0.5+10ζ eq 89
Equivalent Linearization- Performance Point S d = S d T eq ; ζ eq = S d T eq ; ζ =5% η = S a T eq ; ζ =5% T eq 2 4π 2 η The equivalent period and the equivalent damping ratio are functions of the strength reduction factor of the non-linear SDOF system and, respectively, of the initial period of vibration and of the damping ratio. The various equivalent linear methods differ from each other mainly for functions used to compute T eq and ζ eq. In their work (2008), Lin & Miranda give the equivalent period and the equivalent damping ratio as follows: T eq = 1 + m 1 T 2 R μ 1.8 1 T ζ eq = ζ =5% + n 1 T n 2 R μ 1 90
Equivalent Linearization- Performance Point S d = S d T eq ; ζ eq = S d T eq ; ζ =5% η = S a T eq ; ζ =5% T eq 2 4π 2 η T eq = 1 + m 1 T 2 R μ 1.8 1 T ζ eq = ζ =5% + n 1 T n 2 R μ 1 91
Equivalent Linearization Advantages: Linear computation Use of pushover analysis Drawbacks: Value of damping Not always conservative 92
N2 Method A little bit of background Started in the mid 1980 s (Fajfar and Fischinger 1987, 1989) A variant of the Capacity Spectrum Method (ATC-40) Based on inelastic spectra rather than elastic spectra 93
N2 Method Procedure 94
N2 Method Procedure S S ae, S S R R a d d e Ductility factor Reduction factor Vidic et al. 1992 T R ( 1) 1 T T C T R T T C C Inelastic response spectrum (ADRS) 95
96
N2 Method Procedure 97
N2 Method Procedure (3)- Performance Point if T T C : Basic equations 98
N2 Method Procedure (3)- Performance Point S d S d e 1 ( R 1) R T T C * 99
N2 Method Procedure (3)- Performance Point if T T C : Basic equations 100
N2 Method Procedure (3)- Performance Point S d S d e 101
N2 Method Procedure 102
Content Capacity curve Seismic damage Acceleration-Displacement Response Spectrum (ADRS) Performance Point Large-scale vulnerability assessment 103