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# Lesson9 2nd Part

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Dan Abrams + Magenes Course on Masonry

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### Lesson9 2nd Part

1. 1. Seismic design and assessment of Seismic design and assessment of Masonry Structures Masonry Structures Lesson 9, continued October 2004 Masonry Structures, lesson 9 part 2 slide 1 Limitations of the storey mechanism approach To perform a separate analysis for each storey, it is necessary to make assumptions on the boundary conditions of the piers, i.e. on their rotational restraints: fixed-fixed, or fixed- free, or other. These assumptions are strongly affected by the strength and stiffness of the coupling horizontal structural elements: plain unreinforced masonry spandrel beams, or r.c. slabs, or r.c. ring beams, which may or may not crack or fail as horizontal loads increase. The state of stress of these elements cannot be determined accurately on the basis of a separate analysis for each storey, but only from a global analysis of the whole multi-storey structure. In principle, only by knowing how much the coupling element are stressed can the engineer judge if cracking or failure can be expected, and, as a consequence, what kind of boundary conditions can be assumed for the piers. A variation in the axial force of the piers may take place under the overturning effect of the horizontal loads, affecting the flexural and shear strength of the individual piers. This effect may not be of relevance in low-rise squat buildings, but it can be in a more general context. Again, an evaluation of this effect can be made only very approximately with a separate storey-by-storey analysis. Masonry Structures, lesson 9 part 2 slide 2
2. 2. Limitations of the storey mechanism approach The storey-mechanism approach must therefore always be applied with a clear understanding of its meaning and limitations, otherwise it can lead, in some cases, to unrealistic and unconservative results. The engineer can improve to some extent the results with a proper choice of boundary conditions (end rotation) for the piers, but still some structural configurations of multi- storey walls or buildings cannot be analysed properly with such method. Masonry Structures, lesson 9 part 2 slide 3 URM MASONRY SPANDREL BEAMS UNDER at first cracking SEISMIC ACTION Crack patterns from an experimental cyclic test on a full- at ultimate scale masonry building prototype (University of Pavia, 1994) Masonry Structures, lesson 9 part 2 slide 4
3. 3. Strength of urm spandrel beams Very little information is available on the behaviour of urm spandrel beams subjected to cyclic shear. A proposal for strength evaluation which could be suitable for applications is as follows. Unreinforced masonry spandrels can be considered as structurally effective only if they are regularly bonded to the adjoining walls and resting on a floor tie beam or on an effective lintel. The verification of unreinforced masonry coupling beams, in presence of a known axial horizontal force, is carried out in analogy of the vertical walls. If the axial load is not known from the model (for instance, when the analysis is carried out with the hypothesis of in-plane infinitely rigid floors), but horizontal elements with tensile strength (such as steel ties or r.c. ring beams) are present in proximity of the masonry beam, the resisting values may be assumed not greater than the following values associated to the shear and flexural failure mechanisms. Masonry Structures, lesson 9 part 2 slide 5 Strength of urm spandrel beams The shear strength Vt of an unreinforced masonry coupling beam, connected to a r.c. ring beam or a lintel and effectively bonded at the ends, may be computed in a simplified way as follows: Vt = h t fv0 where: h is the section height of the masonry beam; t is the width (thickness) of the beam fv0 = is the shear strength in absence of compression. Masonry Structures, lesson 9 part 2 slide 6
4. 4. Strength of urm spandrel beams The maximum resisting moment, associated to the flexural mechanism, always in presence of horizontal elements resisting to tension actions in order to balance the horizontal compression in masonry beams, may be evaluated as follows: [ M u = H p h / 2 1 − H p /(0.85 f hu ht ) ] where: Hp is the minimum between the tension strength of the element in tension placed horizontally and the value 0.4fhuht fhu= is the compression strength of masonry in the horizontal direction (in the plane of the wall). The shear strength, associated to this mechanism, may be computed as: V p = 2M u / l where l is the clear span of the masonry beam. The value of shear strength for the unreinforced masonry beam element shall be assumed as the minimum between Vt and Vp. Masonry Structures, lesson 9 part 2 slide 7 Non linear static modelling: beyond the storey mechanism approach “Storey mechanism” Refined finite element Ok up to 2 (3?) storeys Gambarotta & Lagomarsino, Papa Macro-element modelling & Nappi., Lourenço,… Tomaževič, Braga & Dolce fascia maschio nodo MAS3D (Braga, PEFV (D’Asdia & SAM (Magenes, Della TREMURI (Lagomarsino, Liberatore, Spera) Viskovic) Fontana, Bolognini) Penna & Galasco) Masonry Structures, lesson 9 part 2 slide 8
5. 5. Requirements for non linear models • Low or moderate computational burden to allow the modeling of whole buildings: • discretization of the structure with macro-elements: the elements have dimensions comparable to the inter-storey height or with the size of openings (doors, windows), to reduce the number of degrees of freedom of the model. • Reliability of results: • all the fundamental failure mechanisms should be accounted for with suitable failure criteria; • the model should give a good estimate of the overall deformational behaviour under horizontal loads. Masonry Structures, lesson 9 part 2 slide 9 Overview of some macroelement models for urm EQUIVALENT TRUSS APPROACH (Pagano et al., 1984-1990) Masonry Structures, lesson 9 part 2 slide 10
6. 6. Overview of some macroelement models for urm MULTI-FAN MODEL, MAS3D (Braga, Liberatore, Spera, 1990-2000) No-tension stress field simulated as a set of “radial” stress fields for which an analytical formulation in closed form exists. Masonry Structures, lesson 9 part 2 slide 11 Overview of some macroelement models for urm Pier or spandrel elem. “Joint” element PEFV (D’Asdia & Viskovic 1990-today) Linear elastic finite elements with variable (adaptive) geometry. Masonry Structures, lesson 9 part 2 slide 12
7. 7. Overview of some macroelement models for urm TREMURI (Lagomarsino, Penna, Galasco 1997- today) Beam-columns-type elements with internal degrees of freedom and coupling of rotation/axial displacement to simulate rocking. Allows dynamic analysis also. Masonry Structures, lesson 9 part 2 slide 13 Overview of some macroelement models for urm SAM (Magenes, Della Fontana, Bolognini 1998- today) Equivalent 3–d frame model •Simplified strength criteria for all elements, including r.c. ring beams, easily adaptable to code-like formulations. •Simplified multi-linear constitutive rules are used (extension of concepts already present in early storey-mechanism formulations) •Flexural (“rocking”) failure:a plastic hinge is introduced at the end of the effective length where Mu is attained •Shear failure: plastic shear deformation γ occurs when Vu is attained •Suitable for both urm and reinforced masonry. •Crude idealization but effective results especially for prediction of behaviour at ultimate Masonry Structures, lesson 9 part 2 slide 14
8. 8. Nonlinear equivalent frame rigid i offset H1 θ = chord rotation i' ϕ = flexural deform. effective γ = shear deformation length Heff j' rigid H2 offset j V V Spandrel Shear force- Pier element element V shear V u u deformation behaviour in the case of αV u shear failure mechanism γ γ γ γ 1 2 γ = θu− ϕ Masonry Structures, lesson 9 part 2 slide 15 Nonlinear equivalent frame 80 70 F.E.M. SAM (w. brittle spandrels) Total base shear (kN) 60 50 40 30 20 10 0 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 Total displacement at 3rd floor (m) URM wall with weak spandrels: Damage pattern predicted by No storey mechanism refined nonlinear f.e.m. analysis Masonry Structures, lesson 9 part 2 slide 16
9. 9. Nonlinear equivalent frame Comparison with experiments: full scale, two-storey, brick masonry building, subjected to quasi static cyclic loading (University of Pavia, 1994-95) 150 160 Wall D - Door wall 100 140 120 Base shear (kN) Base shear (kN) 50 100 0 80 Exp. 1st cycle envelope -50 60 Exp. 2nd cycle envelope Exp. 3rd cycle envelope 40 SAM pushover analysis -100 20 -150 0 -25 -20 -15 -10 -5 0 5 10 15 20 25 0 5 10 15 20 25 Equivalent displacement δeq (mm) Equivalent displacement δeq (mm) Masonry Structures, lesson 9 part 2 slide 17 5-storey urm wall with r.c. ring beams 1.22 2.25 1.45 2.25 1.45 19.12 2.25 1.45 2.25 1.63 2.25 0.64 3.70 1.05 1.74 1.05 2.73 1.05 2.03 2.56 2.03 1.05 2.73 1.05 1.74 1.05 3.70 29.26 Masonry Structures, lesson 9 part 2 slide 18
10. 10. 5-storey urm wall with r.c. ring beams: equivalent frame model Masonry Structures, lesson 9 part 2 slide 19 5-storey urm wall: nonliner equivalent frame pushover analysis Global angular deformation (%) 0.000 0.078 0.156 0.234 0.312 0.390 0.468 1400 0.42 Pushover analysis with Analysis A first-mode (linear) force 1200 0.36 distribution. Base shear coefficient Total base shear (kN) 1000 0.30 Analysis B R.c. beams: elasto-plastic 800 0.24 beam elements (w. flexural Analysis C hinging). 600 0.18 Analysis G The analyses from A to G 400 No r.c. ring 0.12 show the effect of beams 200 0.06 decreasing strength and stiffness of the r.c. beams 0 0.00 on the response of the wall. 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Roof displacement (m) Masonry Structures, lesson 9 part 2 slide 20
11. 11. 5-storey urm wall: nonliner equivalent frame pushover analysis 20 5th FLOOR Coupling elements (masonry soft storey spandrels and r.c. beams) can 16 4th FLOOR affect not only the strength, global overturning but also the overall deformed of cantilever walls shape and collapse mechanism 12 3rd FLOOR Height (m) 2nd FLOOR 8 1st FLOOR 4 Analysis A Analysis C soft storey Analysis G 0 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 Horizontal displacement (m) Masonry Structures, lesson 9 part 2 slide 21 Nonlinear equivalent frame Comparison of a 3-d storey – 1000 Forza alla base-Spostamento mechanism analysis and a 3-d 900 storey mechanism nonlinear frame analysis: two POR SAM 800 storey urm building with rigid 700 floor diaphragms and r.c. ring 600 beams. SAM Forza [KN] 500 The flexural and shear strength 400 criteria of masonry walls are 300 kept the same for both methods 200 100 0 0 0.01 0.02 0.03 Spostame nto [m] Masonry Structures, lesson 9 part 2 slide 22
12. 12. Use of nonlinear static analysis in seismic design/assessment The non linear static analysis is based on the application of gravity loads and of a horizontal force system that, keeping constant the relative ratio between the acting horizontal forces, is scaled in order to monotonically increase the horizontal displacement of a control point on the structure (for example, the centre of the mass of the roof), up to the achievement of the ultimate conditions. A suitable distribution of lateral loads should be applied to the building. At least two different distributions must be applied: -a “modal” pattern, based on lateral forces that are proportional to mass multiplied by the displacement associated to the first mode shape - a “uniform” pattern, based on lateral forces that are proportional to mass regardless of elevation (uniform response acceleration). Lateral loads shall be applied at the location of the masses in the model, taking into account accidental eccentricity. Masonry Structures, lesson 9 part 2 slide 23 Use of nonlinear static analysis in seismic design/assessment The relation between base shear force and the control displacement (the “capacity curve”) should be determined by pushover analysis for values of the control displacement ranging between zero and a sufficiently large value, which must exceed by a suitable margin the displacement demand which will be estimated under the design earthquake (target displacement) . The target displacement is calculated as the seismic demand derived from the design response spectrum by converting the capacity curve into an idealized force-displacement curve of an equivalent single-degree-of-freedom system. For the evaluation of the displacement demand of the equivalent s.d.o.f. system, different procedures can be followed, depending on: • how the seismic input is represented (acceleration spectra, displacement spectra, composite A-D spectra); • how the inelastic and hysteretic behaviour of the structure is accounted for (equivalent viscous damping, ductility demand, energy dissipation demand). Masonry Structures, lesson 9 part 2 slide 24
13. 13. Use of nonlinear static analysis in seismic design/assessment An example of procedure (e.g. as adopted by EC8 and Italian code): Forza alla base-Spostamento 800 T ET T O 705 Step 1: carry out 700 the pushover 600 DLS analysis with the 564 ULS Base shear (kN) chosen force 500 Forza [KN] distribution. Plot 400 capacity curve 300 and determine the performance 200 limit states of 100 interest 0 0 0.01 0.0146 0.02 Spostamento [m] Roof displacement (m) Masonry Structures, lesson 9 part 2 slide 25 Use of nonlinear static analysis in seismic design/assessment Γ = ∑m iΦ 2i Φvibration of the structure,mass displacementdirection, normalized array that represents the in the considered in the first mode of mΦ to the unit value of the relative component of the control point. ∑ i i 2000 Fb Fb 1800 F* = 1600 Γ Base shear [kN] 1400 1200 Step 2: determine an 1000 equivalent bilinear 800 600 s.d.o.f. system 400 200 dc 0 dc 0 5 10 15 20 25 30 d* = Roof displacement [cm] Γ N m* m* = ∑ mi Φ i T * = 2π i =1 k* Masonry Structures, lesson 9 part 2 slide 26
14. 14. Use of nonlinear static analysis in seismic design/assessment Forza alla base-Spostamento 900 800 Capacity F*max curve 700 F*y 600 0.8F*max 0.7F*max 500 Sistema equivalente SDOF TETTO Forza [KN] Equivalent Base shear (kN) 400 Bilineare bilinear SDOF 300 200 100 0 0 d*y 0.01 d*max 0.02 Displacement Spostamento [m] (m) Masonry Structures, lesson 9 part 2 slide 27 Use of nonlinear static analysis in seismic design/assessment Elastic displacement spectrum Step 3: using the elastic response spectrum, calculate ∗ the displacement demand on if T*≥TC d max = d e , max = S De ( T *) the sdof system if T*<TC d e , max ⎡ TC ⎤ ⎢1 + (q * − 1 ) T * ⎥ ≥ d e , max ∗ d max = q* ⎣ ⎦ elastic acceleration m* S e (T * ) spectrum q = * d* = dc Fy* ∗ d max Γ N m* m* = ∑ mi Φ i T * = 2π i =1 k* Masonry Structures, lesson 9 part 2 slide 28
15. 15. Use of nonlinear static analysis in seismic design/assessment Step 4: convert the displacement demand on the equivalent sdof into the control displacement and find target point on capacity curve and Γd max = d c ,max * compare with displacement capacity. 2000 Stato Limite DS 1800 1600 Taglio alla base [kN] 1400 1200 1000 800 600 400 ∗ 200 d max d c , max 0 0 5 10 15 20 25 30 Spostamento copertura [cm] Masonry Structures, lesson 9 part 2 slide 29 Use of nonlinear static analysis in seismic design/assessment Available on ftp site: Relevant chapters of new Italian seismic code (English translation available! Thanks Paolo) Relevant chapters of FEMA 356 Eurocode 8 (see Annex B) Masonry Structures, lesson 9 part 2 slide 30
16. 16. When and how to use storey-mechanism method Eurocode 8: “For low-rise masonry buildings, in which structural wall behaviour is dominated by shear, each storey may be analyzed independently. Such requirements are deemed to be satisfied if the number of storey is 3 or less and if the average aspect ratio of structural walls is less than 1.0. …. New Italian seismic code: “For buildings with number of storeys greater than two, the structural model should take into account the effects due to the variation of the vertical forces due to the seismic action and should guarantee the local and global equilibrium. “ Masonry Structures, lesson 9 part 2 slide 31 Earlier use of storey-mechanism method (Tomaževič) du Φu µu = = Ultimate ductility de Φe q2 +1 q behaviour factor µu ≥ 2 (force reduction factor), specified by code (e.g. 1.5-2.0 for urm) Φ = d/h storey drift a S ⋅ β0 H du , j ≥ Vdesign , j = υ j ⋅ Wtot ⋅ S d (T ; q ) = k j ⋅ Wtot ⋅ g q Masonry Structures, lesson 9 part 2 slide 32
17. 17. Use of storey-mechanism method with present EC8 procedure •Evaluate elastic period of building T1 , e.g. using approximate formulae. •Estimate elastic base shear from elastic acceleration spectrum: Fel,base = Se(T1) Wtot /g = Se(T1) Mtot •Evaluate ratio between interstorey shear Vj of the storey j being considered and the total base shear: N υ j = V j / Fbase Vj = ∑F i= j i where Fi is the seismic force at the i-th floor. •The equivalent sdof is defined by putting F* = Vj and d*= interstorey displacement •Evaluate q* = υj Fel,base /F*y •Calculate d*max= d*y [1+(q*-1)Tc/T1] (not greater than q d*y ) and check d*max≤ du Masonry Structures, lesson 9 part 2 slide 33