PSI - Issue 11

Michela Monaco et al. / Procedia Structural Integrity 11 (2018) 388–393 Author name / Structural Integrity Procedia 00 (2018) 000–000

390

3

In what follows the structural geometry of the wall in which the examined masonry pier is contained allows a perfectly antisymmetric behavior of the panel with respect to its vertical axis.

a

b

Fig. 2. (a) Masonry strut within the panel; (b) Corresponding cantilever beam.

The geometry of the reacting strut is so that defined by a symmetric shape, forming an angle 2 ϕ with the vertical axis of the panel (see Figure 1.b), not necessarily coincident with the axis 1 ϕ , indicating the slope of the line connecting the medium points of the load distributions. Due to the symmetry of the problem, reference can be made to a cantilever beam with variable cross section as in Figure 2, loaded with the combination of the load R and the transportation moment such as:

cos (

)

1 N R T R M Rd = = = 1

2 ϕ ϕ −

sin (

)

(1)

2 ϕ ϕ ϕ −

cos

1

To solve the problem, two distinct domains have been defined on the beam length where the variation of stiffness function is continuous with its derivative. The problem can be formulated in variational form by introducing the potential energy of the system with boundary conditions defined at the two domains ends and requiring that the solution be the function minimizing the PE that also satisfies the displacement boundary conditions. It is straightforward to show that the strain energy density, denoting with the indexes 1 and 2 the two domains defined on the strut, is:

   

1 2         +

   

L

L

L

L

1 2

1

2

1

2

0 

0 

0 

0 

F v (2)

2 ϕ ( , , ) B C φ v v

PE

2 EI z dz χ 1 1 1 1 ( )

2 EI z dz χ 2 2 2 2 ( )

2 EA z dz ε 1 1 1 1 ( )

2 EA z dz ε 2 2 2 2 ( )

=

=

− ⋅

+

+

C

Where PE is the total potential energy, the first term at the second member is the bending strain energy density, the second term is the axial strain energy density and the third one is the potential energy of the external forces and: