A Semi-analytical Approach for Masonry Arch Dynamics

Abstract

A semi-analytical approach is proposed for modelling the plane dynamics of a masonry arch, treated as a system of rigid elements with friction and unilateral contacts at each joint. By generalising the method proposed in previous research, the analytical approach is firstly applied to the plane dynamics of a rectangular block simply supported on a moving base. In this case, where the contact although sometimes extended is unique, dynamics is formulated as a frictional contact problem, and conditions for onset of motion according to various mechanisms are fully analytically identified; moreover, criteria for evaluating contact reactions during either smooth or non-smooth dynamics are outlined. The method is then extended to the case of the arch, where each element is characterized at most by a double extended contact; criteria for the onset of motion and evaluation for each element of contact reactions during the dynamic evolution are then identified. The approach proposed constitutes a first step for performing dynamic analysis through either an event-driven or a time-stepping numerical procedure.

Appendix: List of Symbols

C _k , C _k,k+1 :

Absolute and relative instantaneous rotation centres in the arch mechanism

g :

Gravity acceleration

G :

Mass centre of the block

G ⁽ⁱ⁾ :

Mass centre of i-th voussoir

${\boldsymbol{H}}_{\mathrm{P},{\mathrm{n}}_{\mathrm{j}}^{(i)}}^{\left(\mathrm{i}\right)}$ :

Generalised direction associated with unit vector n ⁽ⁱ⁾_j at point P ⁽ⁱ⁾ of i-th mega-voussoir

${\boldsymbol{H}}_{{\mathrm{C}}_{\mathrm{k},\mathrm{k}+1}}^{\left(\mathrm{k}+1\right)}$ :

Generalised direction associated with ${\boldsymbol{n}}^{\left(\mathrm{k},\mathrm{k}+1\right)}$ at point C _k,k+1 of (k+1)-th mega-voussoir

i, j :

Counters of arch voussoirs and joints, respectively

I _n, I _t :

Normal and tangential impulsive reactions

k :

Counter of mega-voussoirs in the arch mechanism

k _s(t):

Acceleration of the ground motion in g units

${\boldsymbol{n}}^{\left(\mathrm{k},\mathrm{k}+1\right)}$ :

Unit vector lined with contact and rotation centres of k-th and (k+1)-th mega-voussoirs

N _P :

Gradient operator of the position of point P

N _P,n, N _P,t :

Normal and tangential vectors of the gradient operator N _P

Ṅ _P,n, Ṅ _P,t :

Normal and tangential vectors of derivative of N _P

${{\boldsymbol{N}}_{\mathrm{A}}}^{-}$, ${{\boldsymbol{N}}_{\mathrm{A}}}^{+}$ :

Negative and positive generalised Coulomb’s boundaries for contact at point A

N ⁽ⁱ⁾_P :

Gradient operator of the position of point P ⁽ⁱ⁾ belonging to i-th voussoir of the arch

N ⁽ⁱ⁾_P,n , N ⁽ⁱ⁾_P,t :

Normal and tangential vectors of gradient operator N ⁽ⁱ⁾_P of i-th voussoir

(O, x, y):

Reference system fixed on boundary Γ

P :

Generic point of the block

Q :

Centre of contact for the block

P ⁽ⁱ⁾_j , ${Q}_{\mathrm{j}}^{\left(\mathrm{i}+1\right)}$ :

Antagonist and candidate contact points at j-th joint of the arch

r _P,t :

Tangential position of point P

ṙ _P :

Velocity of point P

ṙ _P,n, ṙ _P,t :

Normal and tangential velocities of point P

${\ddot{r}}_{\mathrm{P},\mathrm{n}}$, ${\ddot{r}}_{\mathrm{P},\mathrm{t}}$ :

Normal and tangential accelerations of point P

${\dot{r}}_{\mathrm{Q},\mathrm{n}}^{+}$, ${\dot{r}}_{\mathrm{Q},\mathrm{t}}^{+}$ :

Post-impact normal and tangential velocities of point Q

ṙ ⁽ⁱ⁾_P , ṙ ⁽ⁱ⁾_Q :

Velocity of points P ⁽ⁱ⁾ and Q ⁽ⁱ⁾ belonging to i-th voussoir of the arch

${\dot{r}}_{\mathrm{P},{\mathrm{n}}_{\mathrm{j}}^{\left(\mathrm{i}\right)}}^{\left(\mathrm{i}\right)}$, ${\dot{r}}_{\mathrm{Q},{\mathrm{n}}_{\mathrm{j}}^{\left(\mathrm{i}\right)}}^{\left(\mathrm{i}+1\right)}$ :

Normal velocity of antagonist P ⁽ⁱ⁾ and candidate ${Q}^{\left(\mathrm{i}+1\right)}$ points in the system (t ⁽ⁱ⁾_j , n ⁽ⁱ⁾_j )

R _n, R _t :

Normal and tangential reactions at contact point

${\boldsymbol{R}}^{\left(\mathrm{i},\mathrm{i}+1\right)}$ :

Reaction transmitted by i-th to (i+1)-th voussoir in (t _o, n _o)

${R}_{\mathrm{n}}^{\left(\mathrm{i},\mathrm{i}+1\right)}$, ${R}_{\mathrm{t}}^{\left(\mathrm{i},\mathrm{i}+1\right)}$ :

Normal and tangential reactions transmitted by i-th to (i+1)-th voussoir in (t _o, n _o)

${R}_{{\mathrm{n}}_{\mathrm{j}}^{\left(\mathrm{i}\right)}}^{\left(\mathrm{i},\mathrm{i}+1\right)}$, ${R}_{{\mathrm{t}}_{\mathrm{j}}^{\left(\mathrm{i}\right)}}^{\left(\mathrm{i},\mathrm{i}+1\right)}$ :

Normal and tangential reactions transmitted by i-th to (i+1)-th voussoir in ( t ⁽ⁱ⁾_j , n ⁽ⁱ⁾_j )

S :

Generalised force active on the block

S ^π :

Generalised active force in plane π

S ⁽ⁱ⁾ :

Generalised force active on the i-th voussoir

t :

Time instant

(t, n), (t _o, n _o):

Unit vectors associated with system (O, x, y) for the block and arch, respectively

(t ⁽ⁱ⁾_j , n ⁽ⁱ⁾_j ):

Local unit vectors system associated with i-th voussoir at j-th joint

$\overline{\dot{\boldsymbol{u}}}$ :

Generalised admissible velocity of the block

$\dot{\boldsymbol{u}}$, ü :

Generalised velocity and acceleration of the block

$\overline{{\dot{\boldsymbol{u}}}^{+}}$ :

Generalised admissible post-impact velocity

${\dot{\boldsymbol{u}}}^{-}$, ${\dot{\boldsymbol{u}}}^{+}$ :

Generalised pre-impact and post-impact velocities

${\dot{\boldsymbol{u}}}_{\mathrm{A}}$ :

Mechanism with contact at point A

ü _A :

Generalised acceleration with contact at point A

ü _A,n, ü _A,t :

Normal and tangential generalised accelerations in plane π for contact at point A

$\overline{{\dot{\boldsymbol{u}}}^{\left(\mathrm{i}\right)}}$ :

Generalised admissible velocity of i-th voussoir

${\dot{\boldsymbol{u}}}^{\left(\mathrm{i}\right)}$, ü ⁽ⁱ⁾ :

Generalised velocity and acceleration of i-th voussoir

${\ddot{x}}_{\mathrm{O}}$ :

Acceleration of ground motion

$\varDelta \dot{\boldsymbol{u}}$ :

Generalised velocity variation

Γ :

Boundary of the rigid ground

μ :

Friction coefficient

π :

Plane to which N _A,n and N _A,t belong for ṙ _A,t equal to zero

π *:

Plane to which N _A,n and N _A,t belong for ṙ _A,t different from zero

π ⁽²⁾ :

Plane orthogonal to mechanism ${\dot{u}}^{(2)}$ of the second mega-voussoir

${\boldsymbol{\varPsi}}_{{\mathrm{C}}_{1,2}}^{\left(1,2\right)}$ :

Generalised reaction transmitted by first to second mega-voussoir at C _1,2

${\boldsymbol{\varPhi}}_{\mathrm{P},{\mathrm{n}}_{\mathrm{j}}^{\left(\mathrm{i}\right)}}^{\left(\mathrm{i}\right)}$, ${\boldsymbol{\varPhi}}_{\mathrm{P},{\mathrm{t}}_{\mathrm{j}}^{\left(\mathrm{i}\right)}}^{\left(\mathrm{i}\right)}$ :

Local generalised normal and tangential reactions at point P ⁽ⁱ⁾ of i-th voussoir

${\boldsymbol{\varPsi}}_{\mathrm{P}}^{\left(\mathrm{i},\mathrm{i}+1\right)}$ :

Generalised reaction transmitted at point P ⁽ⁱ⁾ by i-th to (i+1)-th voussoir

Ψ _Q,n, Ψ _Q,t :

Generalised normal and tangential reactions at point Q

Ψ ^*_Q,t :

Generic generalised reaction belonging to Coulomb’s cone

Ξ _Q,n, Ξ _Q,t :

Generalised normal and tangential impulses