Displacement-based methods EDCE: Civil and Environmental Engineering CIVIL Advanced Earthquake Engineering

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1 Displacement-based methods EDCE: Civil and Environmental Engineering CIVIL Advanced Earthquake Engineering EDCE-EPFL-ENAC-SGC

2 Content! Link to force-based methods! Assumptions! Reinforced concrete: chord rotation! Capacity curve! Examples: RC shear walls and frames! Unreinforced masonry! Example: masonry shear walls EDCE-EPFL-ENAC-SGC

3 Link to force-based methods! Application of behavior factor q specified in the codes SIA ! For existing buildings, non ductile behavior (q = 1.5 or q = 2.0 for reinforced concrete)! Displacement-based method: more realistic approach of real seismic behavior! Allows to justify an higher value of q EDCE-EPFL-ENAC-SGC

4 Displacement-based Method (DbM)! Applicable to deformable structures! Brittle failure modes excluded: - shear failure - bending with concrete failure before steel yielding - bending with steel failure without sufficient plastic deformation - failure of overlapping zones or anchorage zones EDCE-EPFL-ENAC-SGC

5 Displacement-based Method (DbM)! More realistic approach of real behavior lateral force large change in deformation level small change in force level Joe s Beer! Food! Joe s Beer! Food! lateral displacement EDCE-EPFL-ENAC-SGC

6 Method included in SIA 2018! Slides prepared by Prof. Alessandro Dazio, IBK- ETH Zürich i b k Erdbebeningenieurwesen und Baudynamik EDCE-EPFL-ENAC-SGC

7 Reinforced concrete: chord rotation shear wall column beam V! V M V r V V M l! M r M M V l EDCE-EPFL-ENAC-SGC

8 Reinforced concrete: chord rotation! Angle formed by the tg to element axis at M max location and the chord joining this point to the zero moment point! Represents the element sollicitation (removal of rigid displacements)! Verification by the comparison (demandcapacity) of chord rotations at the element level EDCE-EPFL-ENAC-SGC

9 Chord rotation!: definitions shear wall column beam V! V M rigid body rotation = chord = tangent to element axis = zero moment point V r V V M l! M r M M V l EDCE-EPFL-ENAC-SGC

10 Yielding chord rotation! y F y moment curvature F y # y! y L v $ y = F y L 3 v 3EI = M y EI " L 2 v 3 = # y " L 3 v " L v =! y L v V y V y M y M y " y M y EDCE-EPFL-ENAC-SGC

11 Ultimate chord rotation! u F u moment curvature L v! u h pl = plastic zone h sp = strain penetration L pl = plastic hinge V u h pl L pl pl M u st M u ( 0.08! L + 0.! f d ) L = a 022! EDCE-EPFL-ENAC-SGC v s h sp bl " p " u " y

12 Ultimate chord rotation! u " p " u " y # y # u # p F u L v! y! p! u L pl L pl % u = $ u L v = $ y L EDCE-EPFL-ENAC-SGC v + (# " # )! L! (L " 0.5! L ) u y pl v pl M u V u

13 Non-linear force-displacement relationship # F F! F u F y L v reality approximation V M # y # u # y = f(" y ), F y = f(m y ), # u = f(" y, " u ), F u = f(m u ), # EDCE-EPFL-ENAC-SGC

14 Bending moment-curvature relationship M M n M y " y = " y M n / M y " y " y " EDCE-EPFL-ENAC-SGC

15 Bending moment-curvature relationship! shear wall N = kn Ø20 36 Ø8 2 Ø20 EDCE-EPFL-ENAC-SGC

16 Bending moment-curvature relationship bending moment [MNm] M n M y Nominal yield " y First yield $ s = $ y $ c = EDCE-EPFL-ENAC-SGC " y Ultimate $ s = $ s,max, $ c = $ c,max Nominal strength $ s = $ c = curvature [10-3 m -1 ] " u

17 Bending moment-curvature relationship bending moment [MNm] M n M y 1 st approximation " y " y " u EDCE-EPFL-ENAC-SGC modified approximation curvature [10-3 m -1 ]

18 Bending moment-curvature relationship! nominal yield curvature " y EDCE-EPFL-ENAC-SGC

19 Bending moment-curvature relationship! Material mechanical properties EDCE-EPFL-ENAC-SGC

20 Material mechanical properties a) concrete BH300, SIA 168/68 f ck (t) 3) f sk b) steel IIIa, SIA 168/68 3) 2) f tk stress [MPa] ) 1) f cd acc. to SIA 262 2) f ck acc. to SIA ) f ck (t) acc. to SIA ) $ cu strain [ ] stress [MPa] ) f sd acc. to SIA 262 2) f sk approx. 3) f sk type 1) $ uk strain [ ] EDCE-EPFL-ENAC-SGC

21 Material mechanical properties! Characteristic values! Partial factor for deformation capacity % D = 1.3! Ultimate concrete deformation $ cmax = $ cu = (in general)! Peak steel strain $ smax = &$ su - in general: & = if M u <2M cr : & = special cases: & =? EDCE-EPFL-ENAC-SGC

22 Application of DbM! Non-linear element behavior! Equivalent SDOF! Force-displacement relationship of structure! Capacity curve! Seismic displacement demand! Shear strength verification EDCE-EPFL-ENAC-SGC

23 Example: reinforced concrete Main properties!! 5 stories Building end of 60! 1 x direction «wall»! 1 x direction «frame»! Zone 3b, CS C, CO I EDCE-EPFL-ENAC-SGC

24 Example RC: dimensions and structure m 5 =213t C m 4 =380t m 3 =380t B m 2 =380t t A m tot = 1733t EDCE-EPFL-ENAC-SGC

25 Example RC: element dimensions shear wall cross-section beam at mid span beam at connection column type A column type B EDCE-EPFL-ENAC-SGC

26 Example RC: shear walls direction Structure réelle w (MDOF) w m 5 (SDOF) w/ ( m 4 m*=1163t F d F d m 3 m 2 m 1 F d k* h*=11.95m k MDOF = ' k walls EDCE-EPFL-ENAC-SGC

27 Example RC: shear walls direction! Force-displacement relationship global horizontal force F d [knm] F y =744kN F y = 372kN w y / ( =0.039m building shear walls horizontal displacement, w/ ( [m] 0.20 EDCE-EPFL-ENAC-SGC F u =828kN F u =414kN w u / ( =0.119m

28 Example RC: shear walls direction! ADRS spectrum with capacity curve S ad, F d /m* [m/s 2 ] elastic design spectrum (Z3b, CO I, CS C) T=1.55s 1 w R,d / ( S ud, w / ( [m] EDCE-EPFL-ENAC-SGC w d / ( w u / ( capacity curve

29 Example RC: frame direction! Force-displacement rel.:!frames A, B and C C B A! Torsion neglected EDCE-EPFL-ENAC-SGC

30 Example RC: frame direction! Modelisation Elément pilier L pl L!EI g L pl Elément traverse!ei g (var.) EDCE-EPFL-ENAC-SGC L pl L L pl

31 Example RC: frame direction! Column element type A L pl /2 L pl /2 N V N M EDCE-EPFL-ENAC-SGC V M = plastic hinge M/! (rigid-plastic with softening) bending mom. [KNm] bending mom. [KNm] kN -500kN N=-1500kN 0kN curvature [10-3 m -1 ] -1500kN -1000kN -500kN N=0kN plastic rotation [10-3 ]

32 Example RC: frame direction! Beam element Moment [KNm] L pl /2 L pl / V 0 Type 4 Type Type 8 M -200 Type 2 M -300 Type 6 V -400 = Rotule plastique M/! (rigide-plastique avec adoucissement) Rotation plastique [10-3 ] Type 4 Type 8 Type 2 Type 6 Moment [KNm] EDCE-EPFL-ENAC-SGC Courbure [10-3 m -1 ] Type 8

33 Example RC: frame direction! Force-displacement relationship global horizontal force F d [kn] F y =2032kN EDCE-EPFL-ENAC-SGC building = 2 x frame A + 1 x frame B frame B frame A w y =0.098m onset of plastic hinge softening w u =0.182m horizontal displacement W d [m]

34 Example RC: frame direction! Ultimate deformation of frame B w d = Rotule = Rotule (rupture atteinte) F d EDCE-EPFL-ENAC-SGC

35 Example RC: frame direction! ADRS spectrum with capacity curve 5 4 elastic design spectrum (Z3b, CO I, CS C) S ad, F d /m* [m/s 2 ] 3 2 T=1.52s capacity curve 1 EDCE-EPFL-ENAC-SGC w d / ( w R,d / ( S ud, w d /( [m] w u / (

36 Unreinforced masonry Vernier Dörflingen douane Oberriet Yverdon-Les-Bains EDCE-EPFL-ENAC-SGC

37 DbM: unreinforced masonry! Assumptions (FEMA 356) plastic deformations concentrated in some elements (piers) EDCE-EPFL-ENAC-SGC

38 DbM: unreinforced masonry! Limitation: out-of-plane failure excluded EDCE-EPFL-ENAC-SGC

39 Masonry: out-of-plane failure! Control with h/t ratio (FEMA 356) EDCE-EPFL-ENAC-SGC

40 Masonry: deformation capacity! Compression depending (tests EPFZ) W7 W2 lateral force [kn] Schubkraft [kn] W6 W1 W4 V ) Schiefstellung drift EDCE-EPFL-ENAC-SGC

41 Masonry: deformation capacity! Influence of compression large compression 600 lateral force [kn] Schubkraft [kn] W7 W2 W6 W1 low compression W Schiefstellung drift EDCE-EPFL-ENAC-SGC

42 Masonry: deformation capacity! In function of normal force (K. Lang, 2002) based on test results 1.0 interstorey drift [% ] Stockwerksschiefstellung [%] maximum interstorey drift " = 0.8 $ 0. 25# u! n Normalspannung [MPa] compression stress [MPa] EDCE-EPFL-ENAC-SGC

43 Strength of unreinforced masonry! In function of the inclination of * 2 EDCE-EPFL-ENAC-SGC

44 Lateral strength of URM elements! Determination with stress fields Principle: superposition of a vertical field and an inclined field EDCE-EPFL-ENAC-SGC

45 Lateral strength of URM elements! Determination with stress fields EDCE-EPFL-ENAC-SGC

46 Lateral strength of URM elements! Determination with stress fields équilibre:!!n v + N n =!!!!!N v e 1v + N n e 1n = M 1!!!!N v e 2v + N n e 2n = M 2 = M 1 + V h limitation des sollicitations: f! = N v l 2v t (cos!) 2! f my N f v = n! (f mx " f my ) l 2n t avec " "l 2v = l w " 2 e 2v " "l 2n = l w " 2 e 2n tan!! tan# résistance:!!v = N v tan! EDCE-EPFL-ENAC-SGC

47 Failure modes! Rocking EDCE-EPFL-ENAC-SGC

48 Failure modes! Shear EDCE-EPFL-ENAC-SGC

49 Lateral strength of URM elements! Simplified formulae (FEMA, EC8) rocking (EC8): V Rd,R = l " N $ w xd N " 1#1,15" xd ' & ) 2 " h 0 % l w " t w " f xd ( shear (EC8): V Rd,S = f vd t l c with f vd = f vd0 + 0,4 N xd / (l c t) f vd! 0,065 f b EDCE-EPFL-ENAC-SGC

50 Masonry: deformation capacity! According to EC8 rocking: " u = h0 0.8! l w [%] shear:! u = 0.4 [%] EDCE-EPFL-ENAC-SGC

51 Unreinforced masonry buildings! Swiss typical building (Yverdon) EDCE-EPFL-ENAC-SGC

52 Example: unreinforced masonry! Simplified typology EDCE-EPFL-ENAC-SGC

53 Example: unreinforced masonry! Lateral strength in transversal direction N d [kn] without coupling effect V Rd [kn] with total coupling effect V Rd [kn] l w = 2.0 m l w = 5.0 m l w = 6.0 m total EDCE-EPFL-ENAC-SGC

54 Example: unreinforced masonry! Displacement determination (Lang 2002) EDCE-EPFL-ENAC-SGC

55 Example: unreinforced masonry! Capacity curve in transversal direction 700 capacity curve in y direction 600 horizontal strength [kn] w R top building displacement [mm] wall 2m wall 5m wall 6m building y EDCE-EPFL-ENAC-SGC

56 Unreinforced masonry buildings! Swiss typical building ( in line building) RC rigid floors, diaphragm effect Very regular, no torsion 3 shear walls, long. direction Zone Z1, a gd = 0.6 m/s 2 Soil E, S = 1.40 H storey = 2.85 m M storey = 470 t V M,base = 880 kn Top, w y = 20 mm Top, w u = 35 mm EDCE-EPFL-ENAC-SGC

57 Unreinforced masonry buildings! Swiss typical building ( in line building) EDCE-EPFL-ENAC-SGC

58 Unreinforced masonry buildings! Swiss typical building ( in line building) "!!! *!! )!! ;,7<74/3015=5./5>/7159?@: (!! '!! &!! %!! $!! %"# #!! %"! "!!!! & "! "& #! #& $! $& %! +,-./ / : )1+,2/-/3,./-4 % 0 $"# $"!!"# EDCE-EPFL-ENAC-SGC !"!! $! %! &! '! #! (! ) * +,*-Γ.//0

59 Masonry: lot of uncertainties! Real element deformation capacity?! Effective coupling effect?! Effective element stiffness (crack pattern)?! Research efforts needed EDCE-EPFL-ENAC-SGC

60 Real behavior?! Coupling effect (frame effect) F 2 flexible floors (without coupling effect): F 1 H tot h 0 h 0 " 2 3 # h tot h st h st l o F 2 l o very stiff lintels (total coupling effect): F 1 H tot h 0 " 1 2 # h st h st h 0 h st l o EDCE-EPFL-ENAC-SGC l o

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