Appunti di teoria

Mechanical Behavior of Masonry

Slide 02

• Masonry made of composite materials:
  • Mortar + Bricks/Stones = Masonry
    • Ductile
    • Brittle
    • a mean behavior between mortar and bricks/stones
• The behavior depends on the regularity of the pattern or layout - Anisotropic Behavior
• Different ratio promotes significance (autotension) which positively affect the global strength of masonry in certain conditions.
• Uniaxial Stress: both mortar and brick have a low tensile strength. Whereas they have relatively high compressive strength.
• We can set a masonry failure criterion under multi-axial stresses: Mohr-Coulomb
  • |σ₁| >> σ₄
  • φ = angle of dilatancy or friction
  • c = cohesion
• If φ = 0 → non coherent material → Mohr-Coulomb becomes Tresca Criterion for Material

Principal Stresses

• τ = (σ_c - σ_t)/(σ_c + σ_t)
• MC: |σ₁ - σ₃| ≤ (c + t_f)

The field of admissible stresses is :

• |−σ_c < σ_I σ_II < σ_t

MC matches the experimental data obtained on brittle materials, such as concrete, mortar and brick.

b c t_c

t_min < d c <= R e reality no activation in the soft peak

Attempt a comparison of the along margin

of yield disc & with b v field

Failure in masonry compression

b' _fd f_mk b' _mk m a representative same body of a piece of brick masonry

masonry is a result of mortar bed joints and stretching in one direction.

Performance numel compared to height and width pure stress states

# pure rotation # pure m. rotation state

m = - 1 m = 1 m = 0 (n t)

We can ' on the equations needed to solve the problem for uniaxial compression force σ

eq i l

b' _{x t} ^y _k <0 n and o = 0 eq. 1

k d σ (rel = concave) ‌ n b f ‌

a homogeneous, isotropic, linear, elastic material

conformely substitution is

( b bond x + abc f + b y + d_m f) x = e x = element

that moves b _m = e m = e _m thickness m = ¾ m

Anisotropia

we can set these equations e b' = ε   _{limit lim c t a}

ok you give parameters to lower a negative c j ok a one-off k cross e bv

on`` individual bar f transversal stoic

system   to

t (b v ± abc f) = 1 Σ ( c_m - e m) Σ b' _x f_c m

( ) e _b + b_c^c d e j σ = σ − mv σ

d_d b < b = 0 j

with b x+ − ∫ dm d f σ = b_e −- e m

(− e + t c j)

d_m = ( c_m – - 1 d c

σ ( a + now d b' ^-1 )

b v m e = ±Δ −  

•

b t x+ v

(k x − e ) t_x_e o − (σ = Σ< )

)(b k ^-1 ) − y)

de t x m

. b_m = ^_m −  

We can now eliminate the load-bearing capacity of the masonry.

b) Wolddorf's proposal less terracing

1) m i  _cobby to >c tensile compression

m ⁰ e^b precalculation d ₁

• this composite is expected to disregard failure parallel to the wall middle plane.

FAILURE FOR COMPRESSION

slp eci

This scenario depies where

failure for

axal comp.

1) failure get after the principle stress failure  nueser before their own

2) in any cases failures occur after

following inequality walls

a) failure without b)failure motion

tension occ. e _{e e' a em b f (2, 3 — comuni)}

this pattern is marked because of the approximation of vertical cracks in the brick (e.g mortar joints).

b − be´ error because 2 initial and transshipment lead condition.

We can equate the PVW:

_ordinates 2 free

since W = D:

To check admissibility, now a fine analyze the free body diagram of block 4, 2, 3.

M = N + P - T_o

To = 2H_o L

N₁ d_t bias. Now d more = d more

P to min phases

can N₁ (H + 2z) = _p

N₂ = p_b λ = ₂ L

P_o = 2P_o b_o

P_t di 1.

L_o = _2P χ(...) chi

SOLVI TRIMMING

so that d bound, for the non - dimensional thought of the links on...

then ∑ ++ yildzy₂ ...

2^nd STEP - convertere la matrice elastica ad 0°

red-given rotation: either one or two.

les expressions peuvent être E_{[C]_}, risultant per inner... un REV con coeff de che consiste di altri... distributions of heterogenéité nel contesto de... macro-omogénéité.

• ... due materiali accross the continuum...
• ... des strainers de ceux cité... x'λi, stretching...
• ... rotational inclusions de aquellas... esempi de celui... adjoint [largos?]
• ... & des éléments de continuità de contact & de strainers.
• ... strongly accross successive fractors de mass. [E']^{,1}... tension... by deduction.
• ... two 3R... its tests, macro additive structure,»
• ... filing de gloire de elementaux approximes... [p omgeving],... [shell] &...

Unit of two-step homo: → it definitatasi staggered move...

• ... de deux step → de un'unit: korun (...) & → il by method de [soluti...] di [mestru] atura aceciations.
• Perdivere suits construction yo umetriar &... reinforced, composte & utilitém ici (p maggio... pas [per] il propre) pour advantage de super-celtsi. [3 cont] 90°.

1. Displacement [fek tek, bie nodu] are defined according to demand... de in this plane accordante [k+, g^h], & FVE divisib... & suppointed aqui.
2. Fiber nod quadrates & adynamic. [Hodado prophas] contenue (indeque) alignamenti à il reforce de dimensionality or... il compo.
3. Measure the cortiy of nodules displacement and stiffness accross the...

b.al + cop is fulfilled only in an average sense

1. Determinare macro scope exists content (in downs fosse).
2. Dec nous ano grounde d'or qui quantità de materi et tesili exchange... matricula...
3. ... a method de disposition walls RVE... distribuites sopplantat à satisfait & only de ridacité e... un numero elevate de contamina...

• ... il roi d'homogeneité secondo ideé di Cufani... Tomai de los tieters...
• ... il condizioni di accrosso & segondamente, solo metàte sul RVE, (o divostrie poete abeste),... é il mascopo accrosso...

result satisfying only if motion and tribo are souredly from a mechanical point of view...

If it does not propagare a new... race spill need to be (qua)...

... unite 1... E'1 + E'λ1...