## 5 Finite Element Code
### Basic Characteristics of Finite Element Code
## Abbreviations

ANNs

Artificial Neural Networks

BPNNs

Back-Propagation Neural Networks

CDF

Cumulative Distribution Function

DI

Damage Index

DOF

Degrees of Freedom

DS

Damage State

EC2

Eurocode 2

EC8

Eurocode 8

FE

Finite element

FEM

Finite Element Method

HD

Heavy Damage

ID

Insignificant Damage

IM

Intensity Measure Level

logsig

Log-sigmoid transfer function

LN

Lognormal

LS

Limit state

MAFEA

MAsonry Finite Element Analysis

MD

Moderate Damage

ND

No Damage

PGA

Peak Ground Acceleration

PDF

Probability Density Function

purelin

Linear transfer function

RC

Reinforced Concrete

tansig

Hyperbolic Tangent Sigmoid transfer function

URM

UnReinforced Masonry

List of Symbols

_Latin letters (Capital)_

\(A_{fail}\)

Damaged surface area of the structure

\(A_{tot}\)

Total surface area of the structure

\(E\)

Modulus of elasticity

Ex

Modulus of Elasticity across x-axis

Ey

Modulus of Elasticity across y-axis

Gxy

Shear modulus in the xy plane

P(\(\cdot\))

Probability

Sa

Spectral acceleration

Sa(T)

Spectral acceleration at period T

Sd(T)

Spectral displacement at period T

SI

Spectrum Intensity

_Latin letters (Small)_

\(f_{uc}\)

Masonry compressive strength

\(f_{bc}\)

Brick compressive strength

\(f_{mc}\)

Mortar compressive strength

h

Height of masonry wall

t

Thickness of masonry wall

t\({}_{b}\)

Height of brick

t\({}_{m}\)

Thickness of mortar joint

\(v_{u}\)

Relative volume of unit

\(v_{m}\)

Relative volume of mortar

## Appendix A

The analytical form of the anisotropic finite element \(8\times 8\) stiffness matrix is defined by integration over the area of the element by the following, symmetric to the principal diagonal \(\left(K_{ij}=K_{ji}\right)\) coefficients:

### Stiffness Matrix Coefficients \(K_{1j}\) (j = 1)

Figure 37:3

### Stiffness Matrix Coefficients \(K_{2j}\) (j = 1, 2)

Figure 37:4

Figure 37:5

### Stiffness Matrix Coefficients \(K_{3j}\) (j = 1, 2, 3)

Figure 37:6

Figure 37:7

Figure 37:8

### Stiffness Matrix Coefficients \(K_{4j}\) (j = 1, 2, \(\ldots\), 4)

Figure 37:9

Figure 38:0

Figure 38:1

Figure 38:2

### Stiffness Matrix Coefficients \(K_{5j}\) (j = 1, 2, \(\ldots\), 5)

Figure 38:3

Figure 38:4

Figure 38:5Figure 38:6 (A14) Figure 38:7 (A15)

A.6. Stiffness Matrix Coefficients \(K_{6j}\) (j = 1, 2,..., 6)

Figure 38:8 (A16) Figure 38:9 (A17) Figure 40:0 (A18) Figure 40:1 (A19) Figure 40:2 (A20) Figure 40:3 (A21)

A.7. Stiffness Matrix Coefficients \(K_{7j}\) (j = 1, 2,..., 7)

Figure 40:4 (A22) Figure 40:5 (A23) Figure 40:6 (A24) Figure 40:7 (A25) Figure 40:8 (A26) Figure 40:9 (A27) Figure 39:0 (A28)

A.8. Stiffness Matrix Coefficients \(K_{8j}\) (j = 1, 2,..., 8)

Figure 39:1 (A29) Figure 39:2 (A30) Figure 39:3 (A31) Figure 39:4 (A32) Figure 39:5 (A33) Figure 39:6 (A34) Figure 39:7 (A35) Figure 39:8 (A36)

where \(t\) is the thickness of the element, \(\beta=\frac{b}{a}\) (see Figure 6), and \(\lambda=\frac{v_{yx}}{E_{y}G_{xy}}=\frac{v_{xy}}{E_{x}G_{xy}}\) according to Equation (9). ## References

* (1) [PERSON]; [PERSON] Modeling and vulnerability of historical city centers in seismic areas: A case study in Lisbon. _Eng. Struct._**2004**, _26_, 1295-1310. [CrossRef]
* (2) [PERSON]; [PERSON] Simplified indexes for the seismic vulnerability of ancient masonry buildings. _Constr. Build. Mater._**2006**, _20_, 200-208. [CrossRef]
* (3) [PERSON]; [PERSON] The assessment of damage limitation state in the seismic analysis of monumental buildings. _Earthq. Spectra_**2009**, _25_, 323-346. [CrossRef]
* (4) [PERSON]; [PERSON] Automatic fragility curve evaluation of masonry churches accounting for partial collapses by means of 3D FE homogenized limit analysis. _Comput. Struct._**2011**, _89_, 1628-1648. [CrossRef]* _ICOMOS1931_ _ICOMOS1931. The Athens Charter for the Restoration of Historic Monuments. In Proceedings of the First International Congress of Architects and Technicians of Historic Monuments, Athens, Greece, 1931.