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Dan Abrams + Magenes Course on Masonry

Dan Abrams + Magenes Course on Masonry

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Lesson14 Lesson14 Presentation Transcript

  • Seismic design and assessment of Seismic design and assessment of Masonry Structures Masonry Structures Lesson 14 October 2004 Masonry Structures, lesson 14 slide 1 The reality of most historical centers: what is “the building”? (Carocci et al., 1993) Masonry Structures, lesson 14 slide 2
  • PROGRESSIVE GROWTH STOREYS ADDED TO IN PLAN EXISTING BUILDINGS A: Existing cell B and C: Added cells (Giuffré, 1993) Cells A and B built after C Masonry Structures, lesson 14 slide 3 Damage in most vulnerable buildings: Tyipically, partial overturning mechanisms of façade walls or corner walls Masonry Structures, lesson 14 slide 4
  • Typical distribution of damage in historical centers. Masonry Structures, lesson 14 slide 5 A strong (arguable) statement: Historical buildings can be thought of as quot;…made by an assemblage of partial structures, and each of them can be easily singled out. Walls, floors, roofs, are isostatic structures resting the one on the other and, at the same time, joining the one to the other. It can be asserted that always the damage affects the weakest part of the building, and the analysis has the task of pointing out which partquot;. A. Giuffré, 1989 Masonry Structures, lesson 14 slide 6
  • A weaker (wiser) statement: Although the building as a whole is a redundant (hyperstatic) structure, simpler subsystems can be identified, which make up the structure and which can be treated in many cases as statically determined. Focus is on equilibrium and on the compatibility of external and internal forces with the strength of each subsystem. First step in modelling: understand the response mechanisms of vulnerable subsystems. Masonry Structures, lesson 14 slide 7 Catalog of damage mechanisms due to earthquakes: DAMAGE DUE TO INSUFFICIENT QUALITY OF MASONRY (AS TYPICAL IN DOUBLE-LEAF WALLS DOUBLE- Masonry Structures, lesson 14 slide 8
  • Catalog of damage mechanisms due to earthquakes: OUT-OF-PLANE INSTABILITY OF OUT- OF- DOUBLE-LEAF WALLS DOUBLE- Masonry Structures, lesson 14 slide 9 Catalog of damage mechanisms due to earthquakes: GLOBAL OVERTURNING OF FAÇADES FAÇ Masonry Structures, lesson 14 slide 10
  • Catalog of damage mechanisms due to earthquakes: GLOBAL OVERTURNING OF FAÇADES FAÇ Masonry Structures, lesson 14 slide 11 Catalog of damage mechanisms due to earthquakes: OVERTURNING OF FAÇADES FAÇ CARRYING OVER CORNER “WEDGES” Masonry Structures, lesson 14 slide 12
  • Catalog of damage mechanisms due to earthquakes: PARTIAL OVERTURNING OF FAÇADES FAÇ Masonry Structures, lesson 14 slide 13 Catalog of damage mechanisms due to earthquakes: OVERTURNING OF FAÇADES FAÇ Masonry Structures, lesson 14 slide 14
  • Catalog of damage mechanisms due to earthquakes: PARTIAL OVERTURNING OF FAÇADES: EFFECT OF OPENINGS FAÇ Masonry Structures, lesson 14 slide 15 Catalog of damage mechanisms due to earthquakes: PARTIAL OVERTURNING OF FAÇADES: EFFECT OF OPENINGS FAÇ Masonry Structures, lesson 14 slide 16
  • Catalog of damage mechanisms due to earthquakes: PARTIAL OVERTURNING OF FAÇADES: EFFECT OF OPENINGS FAÇ Masonry Structures, lesson 14 slide 17 Catalog of damage mechanisms due to earthquakes: DAMAGE DUE TO THRUST FROM ROOF STRUCTURE Masonry Structures, lesson 14 slide 18
  • Catalog of damage mechanisms due to earthquakes: DAMAGE DUE TO THRUST FROM ROOF STRUCTURE Masonry Structures, lesson 14 slide 19 Catalog of damage mechanisms due to earthquakes: DAMAGE DUE TO THRUST FROM ROOF STRUCTURE Masonry Structures, lesson 14 slide 20
  • Catalog of damage mechanisms due to earthquakes: RIGID DIAPHRAGMS AND R.C. BEAMS SOMETIMES NOT EFFECTIVE IN PREVENTING DAMAGE OF WALLS Masonry Structures, lesson 14 slide 21 Catalog of damage mechanisms due to earthquakes: LOCAL DAMAGE IN WALL –JOISTS CONNECTIONS Masonry Structures, lesson 14 slide 22
  • Catalog of damage mechanisms due to earthquakes: LOCAL DAMAGE DUE TO POUNDING OF ADJACENT BUILDINGS Masonry Structures, lesson 14 slide 23 Mechanical approach to damage mechanisms: limit analysis •Many of the collapse mechanisms are partial, in the sense that they involve specific sub-structures or components. •Collapse is due, most of the times, to loss of equilibrium rather than to the exceedance of some level of stress. •When horizontal acceleration are high enough to trigger a mechanism, it may be assumed that the different parts can be idealized as rigid bodies. •The lateral force capacity of the subsystem can be related to a corresponding acceleration. •Static threshold resistance can be evaluated through limit analysis and the application of the principle of virtual work. Masonry Structures, lesson 14 slide 24
  • Principle of virtual work If a system which is in equilibrium under the action of a set of externally applied forces is subjected to a virtual displacement (velocity), i.e. an infinitesimal displacement (velocity) pattern compatible with the system’s constraints, the total work (power) done by the set of forces will be zero, i.e. the vanishing of the work done during a virtual displacement is equivalent to a statement of equilibrium The principle of virtual work (PVW) is particularly useful when the structural systmem is complex, involving a number of interconnected bodies, in which the direct equilibration of forces may be difficult Masonry Structures, lesson 14 slide 25 Example of limit analysis using the PVW: out-of-plane analysis of a simple wall PVW: Ψ infinitesimal rotation of the lower body λP2 Φ = Ψ× h1/h2 rotation of upper body P2 λ (P1δ1x +P2δ2x )- (P1δ1y +P2δ2y +S δNy)=0 λ (P1δ1x +P2δ2x )= P1δ1y +P2δ2y +S δNy Wactive= Wresisting λP1 P1 λ= (P1δ1y +P2δ 2y +Sδ Ny )/ (P1δ 1x + P2δ2x ) find minimum λ δλ /dx = 0 x λ min Masonry Structures, lesson 14 slide 26
  • By putting: h2 = 1 H ( x − 1) and h1 = H x x B 2 x + ( S / P )( x + 1) x then : λ= H x −1 By requiring that dλ/dx = 0 we obtain: P+S x = 1+ 2 S Masonry Structures, lesson 14 slide 27 In some simple cases it is possible to write directly equilibrium equations and evaluate λ without recurring to PWV: Mres = Mactive(λ) = λMactive(λ=1) λ = Mres /Mactive(λ=1) in this case, for the two mechanisms below, equilibrium about A and about B are easily written, and two different values of λ will be found, λ1 and λ2. The lowest of the two will be the “true” mechanism. Masonry Structures, lesson 14 slide 28
  • Factors influencing static threshold •geometry and restraints of mechanism •amount, spatial distribution and nature of vertical loads •friction forces •forces coming from devices such as tie-rods •compression strength of masonry Masonry Structures, lesson 14 slide 29 Geometry, restraints of mechanism, spatial distribution and nature of vertical loads •Geometry can be assumed on the basis of the knowledge of the seismic behaviour of similar structures or can be identified considering the presence of previous cracks; •moreover, the quality of the connections between walls, the masonry texture (brickwork), the presence of tie-rods, the possible interactions with other parts of the building and with adjacent buildings have to be considered. •The definition of the geometry and restraints is also strictly related to how vertical weigths and associated horizontal inertia forces are transferred to the walls. Masonry Structures, lesson 14 slide 30
  • Friction forces Masonry Structures, lesson 14 slide 31 Friction forces friction along toothed crack (De Felice and Giannini, 2001) Masonry Structures, lesson 14 slide 32
  • Forces coming from tie rods α0P1 P1 θ1 Masonry Structures, lesson 14 slide 33 Forces coming from tie rods F1 α0P1 P1 θ1 The effect of tie rods can be introduced as an external force whose value depends on displacement Masonry Structures, lesson 14 slide 34
  • Forces coming from tie rods F1 a* strength of anchorage attained a0* (a) α0P1 with tie rod a'0 P1 (b) θ1 without tie rod d *=0.4 d' * u 0 d* d'0* d0* displacement The effect of tie rods can be introduced as an external force whose value depends on displacement Masonry Structures, lesson 14 slide 35 Effect of compressive strength of masonry (a) (b) SHIFTING OF HINGE Wall subject to overturning:: (a) assuming infinite (high) compression strength; (b) with limited (low) compression strength: centre of vertical reaction moves inwards Masonry Structures, lesson 14 slide 36
  • More complex mechanisms Q rs Qr Q rs Qf Ts Q Q fs r T 1 hs 2 φ b Ti h l i α L n (D’Ayala and Speranza, 2002, Restrepo-Velez, 2004) Masonry Structures, lesson 14 slide 37 Qr Qr Qf T 1 φ hs 2 b h l i α L L1 n L2 L1 L2 (Restrepo-Velez, 2004) Masonry Structures, lesson 14 slide 38
  • (D’Ayala and Speranza, 2002) Masonry Structures, lesson 14 slide 39 Use of rigid-body analysis for seismic assessment •Earlier uses of limit rigid-body limit analysis was made essentially on a comparative basis, to evaluate which part of the structure are most vulnerable, and to check the effect of strengthening techniques (e.g. insertion of tie-rods, of rigid diaphragms…) on out-of-plane mechanisms. •The general concept is to have horizontal load multipliers for out-of-plane mechanisms which are higher than the global base shear coefficient of the building, corresponding to the strength associate to in-plane response of walls. •A more recent approach (new Italian seismic code) proposes the use of rigid- body analysis within equivalent static assessment procedures which take into account, in an approximate way, the dynamic nature of the response. Masonry Structures, lesson 14 slide 40
  • Use of rigid-body analysis for seismic assessment 1. definition of a s.d.o.f. mechanism and its kinematics, by idealizing the substructure as a set of rigid bodies which can slide/rotate, separated from each other by fracture lines. 2. evaluation of the static horizontal multiplier of vertical weights α0 that corresponds to the static threshold resistance. To this end, the following forces are applied to the system: -the vertical self weights of the rigid blocks, applied at their centres of mass; -the vertical loads carried by the walls transmitted by floors, roof, etc. ; - a system of horizontal forces, proportional to the vertical weights, and to the loads carried by the walls, if the corresponding inertia forces are expected to be transferred to the walls which are part of the mechanism - if present, external forces (e.g. from tie rods or from friction at boundaries); - if present, internal forces (e.g. due to friction/interlocking among units). Masonry Structures, lesson 14 slide 41 Given a virtual rotation θk to the generic block k, it is possible to establish the corresponding virtual displacements of the points of application of all the forces along the respective directions. The value of α0 can be obtained using the Virtual Work Principle, in terms of displacements: ⎛ n n+m ⎞ n o α 0 ⎜ ∑ Pi δ x,i + ∑ P j δ x, j ⎟ − ∑ Pi δ y,i − ∑ Fh δ h = L fi ⎜ i =1 ⎟ i =1 ⎝ j= n +1 ⎠ h =1 where n is the number of all the dead loads (weights, vertical forces) applied to the different rigid blocks of the mechanism, m is the number of weight forces not directly applied to the blocks, whose inertia forces will be transmitted to the blocks of the mechanism; o is the number of the external forces applied to the blocks but not related to considered masses; Masonry Structures, lesson 14 slide 42
  • ⎛ n n+m ⎞ n o α 0 ⎜ ∑ Pi δ x,i + ∑ P j δ x, j ⎟ − ∑ Pi δ y,i − ∑ Fh δ h = L fi ⎜ i =1 ⎟ i =1 ⎝ j= n +1 ⎠ h =1 Pi is the generic weight force (block dead load, applied at its centroid, or other weight); Pj is the generic weight force, not directly applied to the blocks, whose mass generates seismic horizontal forces on the elements of the kinematical chain, because not effectively transferred to other parts of the building; δx,i is the horizontal virtual displacement of the point of application of Pi, assuming as positive the positive direction of the considered seismic action; δx,j is the horizontal virtual displacement of the point of application of Pj, assuming as positive the positive direction of the considered seismic action; δy,i is the vertical virtual displacement of the point of application of Pi, assuming as positive if upwards; Fh is the generic external force (absolute value), applied to the block; δh is the virtual displacement of the point of application of Fh , positive if opposite; Lfi is the work of internal forces. Masonry Structures, lesson 14 slide 43 3. Definition of an equivalent s.d.o.f. system with the following characteristics: 2 ⎛ n +m ⎞ n+m ⎜ ∑ Pi δ x,i ⎟ ⎜ ⎟ α0 ∑ Pi α 0g n +m M* = ⎝ n+m ⎠ ∑ Pi i =1 i =1 a* = 0 * = e* = gM* / M e* g ∑ Pi δ 2 x,i i =1 i =1 effective threshold effective mass effective mass acceleration ratio 4’. Simplified “linear” static safety check (ultimate limit state): a gS ⎛ Z⎞ a* ≥ 0 ⎜1 + 1.5 ⎟ with q = 2.0 q ⎝ H⎠ Masonry Structures, lesson 14 slide 44
  • Non linear static methodology 4’’. Non linear static safety check (alternative to linear) 4’’a. Evaluate the evolution of the horizontal multiplier α by progressively increasing the displacement dk of a control point, chosen as suitable by the designer, until the horizontal multiplier reaches the value of zero; Note: when vertical forces are constant and horizontal forces are only proportional to vertical weights, the relationship between α and dk is approximately linear and can be expressed as α = α 0 (1 − d k / d k ,0 ) where dk,o is the displacement corresponding to zero horizontal force. Masonry Structures, lesson 14 slide 45 4’’b. Evaluate the displacement of the equivalent s.d.o.f. system as: a* (a) with variable external forces n+m ∑ Pi δ x,i a 0* (b) linear * d = dk i =1 (a) n +m a'0 δ x,k ∑ Pi i =1 (b) du*=0.4 d'0* d* d'0* d0* an plot the a* – d* curve. Masonry Structures, lesson 14 slide 46
  • 4’’b. The ultimate displacement du* is evaluated conventionally as the lesser of: - 40% of the displacement at zero force - the displacement limit corresponding to locally incompatible conditions (e.g. unseating of floor joists…) a* (a) with variable external forces a0* (a) (b) linear a'0 (b) du*=0.4 d'0* d* d'0* d0* Masonry Structures, lesson 14 slide 47 4’’c. Define effective secant period at 0.4 du* on capacity curve as: d s* Ts* = 2π * as Masonry Structures, lesson 14 slide 48
  • Use of rigid-body analysis for seismic assessment 4’’c. Calculate displacement demand ∆d with the following elastic response spectrum, and compare with displacement capacity: Ts2 ⎛ 3 (1 + Z H ) ⎞ Ts < 1.5T1 ∆ d (Ts ) = a g S ⎜ − 0.5 ⎟ 4π ⎜ 1 + (1 − Ts T1 ) 2 2 ⎟ ⎝ ⎠ 1.5T1Ts ⎛ Z⎞ 1.5T1 ≤ Ts < TD ∆ d (Ts ) =a g S ⎜ 1.9 + 2.4 ⎟ 4π 2 ⎝ H⎠ 1.5T1TD ⎛ Z⎞ TD ≤ Ts ∆ d (Ts ) =a g S ⎜ 1.9 + 2.4 ⎟ 4π ⎝ 2 H⎠ where Z is the height, with respect to ground, of the centroid of all inertial masses involved in the mechanism H is the total height of the building all other parameters (ag, S, TD) are as specified in design acceleration spectra. Masonry Structures, lesson 14 slide 49