Modellazione Ricerca in Italia sulla tematica della modellazione di strutture murarie Complesso comportamento non lineare del materiale muratura (disomogeneit( disomogeneità,, anisotropia, asimmetria di comportamento compressione-trazione, non linearità del legame sforzi-deformazioni), dell interazione FRP e rottura fragile del FRP interazione muratura- Schemi di calcolo 2D 3D Normativa italiana sulla muratura Normativa o linee guida sull applicazione di FRP
Modellazione della muratura Approccio Micromeccanico Approccio Fenomenologico Omogeneizzazione Modello Discreto a blocchi Modello Continuo Macromeccanico Modellazione multiscala
Approccio Micromeccanico Modello Discreto a blocchi
Approccio Micromeccanico Omogeneizzazione Modello Continuo Macromeccanico
Modello a blocchi Interfaccia con attrito alla Coulomb (Luciano-Sacco, 1990) σ 0 τ µ σ
Masonry structures UC for masonry material Luciano-Sacco, I. J. Solids Struct., 1997 Simple damage model: the cracks occur only in the mortar material which behaves in a perfect elasticbrittle manner, the bricks are indefinitely elastic, the mortar thickness is small, so that the cracks can develop only vertically or horizontally, when a fracture starts to develop, a full failure of a mortar junction is supposed.
Possible damaged states of the masonry Paths S1 S2 S3 S8 S1 S2 S4 S8 S1 S5 S3 S8 S1 S5 S6 S8 S1 S7 S4 S8 S1 S7 S6 S8
Homogenization Solve the UC problem for each possible state: in V Displacement representation form su V Periodicity continuity Solved via finite element method
It is set: it is solved the problem for each unit strain tensor; the averages of the local stresses in the UC are evaluated:
Overall elastic moduli of the undamaged and damaged states
Strength of the masonry Cohesive Coulomb criterion c cohesion µ friction coefficient Yield function at the mortar joint 1 2 3 4 5 6 7 8
Structural computation
Shear test for different values of the friction coefficient for c=100 MPa for different values of the cohesion for µ=1
One-dimensional reinforced masonry elements Out-of-plane bending of masonry wall arc achitrave vaults reinforced by circumferential chain.
Masonry modelling One-dimensional masonry element Unit cell: one-half of the repetitive microstructure Aim: reinforced masonry overall response resultant axial force axial strain bending moment average curvature damage and plasticity of the mortar, damage and plasticity of the blocks, delamination of the composite from the masonry, failure of the reinforcement. L = L b + L m L b and L m L g L u = L + L g Masonry total length of the unit cell block and mortar half lengths length of perfect adhesion unglued length zone A M = b h constant rectangular cross-section Reinforcements A + R and A R top and bottom FRP areas
Plasticity Mortar and block Constitutive equation Traction (no plasticity effects) ε p =0 for σ 0 D damage parameter ε p plastic strain E Young modulus Damage σ =(1 D) E (ε ε p ) Evolution law: 0 for ε e <ε ± c 0 Ḋ = ε ± c ε ± u ε ± u ε ± c ε 2 ε for ε ± c <ε d <ε ± u d 0 for ε ± u <ε d Compression effective stress σ = σ 1 D yield function with hardening f( σ,α) = σ (σ y + Kα) α internal hardening variable K plastic hardening parameter ε + c and ε + u undamaged and completely damaged + traction compression ε d elastic strain
Numerical applications Stress-strain response for the block, the mortar and the wh masonry Material properties: mortar 4 2 E m = 5000MPa σ y,m =3MPa K m =500MPa ε + c,m =1E 4 ε+ u,m =4E 4 ε c,m =10E 4 ε u,m =40E 4 block E b = 15000MPa σ y,b =10MPa K b =1500MPa ε + c,b =1E 4 ε+ u,b =6E 4 ε c,b =15E 4 ε u,b =60E 4 S t r e s s [ M P a ] 0-2 -4-6 -8-1 0 Masonry M a s o n r y BMortar lo c k M o r t a r Block -1 2-0. 0 0 6-0. 0 0 4-0. 0 0 2 0. 0 0 0 0. 0 0 S t r a i n Geometrical parameters: L b =25mm L m =5mm b =130mm h =250mm masonry blocks 50 130 250mm mortar layers of 10mm. strain concentration in the mortar masonry behavior worse than the response of the two components.
Carbon?ber-reinforced plastic composite material E c = 200000MPa f + R = f R = 2500MPa L g =22mm A R = A + R = A R Axial force N tot versus the strain e tot in tension Axial Force [N] 18000 16000 14000 12000 10000 8000 6000 4000 A R = 0 mm 2 A R =10 mm 2 A R =20 mm 2 A R =30 mm 2 A R =40 mm 2 A R =50 mm 2 Bending moment - curvature for the reinforced masonry M o m e n t [ N m m ] 2.4 x 1 0 6 2.2 x 1 0 6 2.0 x 1 0 6 1.8 x 1 0 6 1.6 x 1 0 6 1.4 x 1 0 6 1.2 x 1 0 6 1.0 x 1 0 6 8.0 x 1 0 5 6.0 x 1 0 5 4.0 x 1 0 5 2.0 x 1 0 5 A R = 0 m m 2 A R = 1 0 m m 2 A R = 2 0 m m 2 A R = 3 0 m m 2 A R = 4 0 m m 2 A R = 5 0 m m 2 0. 0 0. 0 5. 0 x 1 0-7 1.0 x 1 0-6 1. 5 x 1 0-6 2. 0 x 1 0-6 C u r v a t u r e 2000 0 0.00000 0.00003 0.00006 0.00009 0.00012 Strain 1. 2 x 1 0 6 Axial force N tot versus the strain e tot in compression Axial Force [N] 0-20000 -40000-60000 A R = 0 mm 2-80000 A R =10 mm 2 A R =20 mm 2 A R =30 mm 2-100000 A R =40 mm 2 A R =50 mm 2 M o m e n t [ N m m ] 1. 0 x 1 0 6 8. 0 x 1 0 5 6. 0 x 1 0 5 4. 0 x 1 0 5 2. 0 x 1 0 5 A R = 2 0 m m 2 0. 0 0.0 6.0 x 1 0-7 1.2 x 1 0-6 1. 8 x 1 0-6 C urvature -120000-0.0012-0.0009-0.0006-0.0003 0.0000 Strain
Bending moment - curvature for the reinforced masonry Delamination effects: axial behavior Moment [Nmm] 14000000 12000000 10000000 8000000 6000000 4000000 2000000 0 0.00000 0.00001 0.00002 0.00003 0.00004 0.00005 0.00006 0.00007 0.00008 Curvature [mm -1 ] Axial Force [N] 0-10000 -20000-30000 -40000-50000 -60000-70000 -80000-90000 -100000-110000 G c =0.004 N/mm G c =0.020 N/mm -120000-0.0015-0.0012-0.0009-0.0006-0.0003 0.0000 Strain Position of the neutral axes Delamination effects: bending behavior Neutral axes Yn [mm] -125-100 - 75-50 - 25 0 25 50 75 100 125 0 100 200 300 400 500 600 y Time Yn g Yn b Yn m Yn tot Moment [Nmm] 4500000 4000000 3500000 3000000 2500000 2000000 1500000 1000000 500000 G c =0.004 N/mm G c =0.020 N/mm 0 0.000000 0.000005 0.000010 0.000015 0.000020 0.000025 Curvature [mm -1 ]