A parametric investigation of the seismic capacity for masonry arches and portals of different shapes

Engineering Failure Analysis 52 (2015) 1–34

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Engineering Failure Analysis journal homepage: www.elsevier.com/locate/engfailanal

A parametric investigation of the seismic capacity for masonry arches and portals of different shapes Rossana Dimitri a, Francesco Tornabene b,⇑ a b

Department of Innovation Engineering, University of Salento, Via per Monteroni, 73100 Lecce, Italy DICAM Department, School of Engineering and Architecture, University of Bologna, Viale del Risorgimento 2, 40136 Bologna, Italy

a r t i c l e

i n f o

Article history: Received 15 December 2014 Received in revised form 4 February 2015 Accepted 28 February 2015 Available online 12 March 2015 Keywords: Arch Conservation Discrete element modelling Limit analysis Masonry

a b s t r a c t Masonry arches are typical components of historic buildings throughout the world, and their damage or collapse is very often caused by earthquakes. The first-order seismic assessment of masonry structures can be represented by the equivalent static analysis method, which does not capture all of the dynamics, but provides a measure of the lateral loading that the structure can withstand before collapse. This study aims to understand the stability of unreinforced masonry arches and portals (i.e. buttressed arches) subjected to constant horizontal ground accelerations, combined with the vertical acceleration due to gravity. An analytical model based on limit analysis is developed to describe the relative stability of pointed and basket-handle arches and portals with respect to circular ones, for varying geometry parameters. The equivalent static analysis determines the value of the constant lateral acceleration needed to cause collapse of the structure, which coincides with the minimum peak ground acceleration needed to transform the vaulted system into a mechanism. Predictions of the analytical model are compared with results of numerical modelling by the Discrete Element Method (DEM). This numerical model considers masonry as an assemblage of rigid blocks with no-tension frictional joints, and is based on a time stepping integration of the equations of motion of the individual blocks. The satisfactory agreement between predictions of the two approaches validates the analytical model and verifies the potentials of the discrete element framework as a method of evaluating the quasi-static behavior of unreinforced masonry structures. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Vaulted masonry structures are a large part of the world architectural heritage. These structures are particularly vulnerable to seismic events, as shown by the recent damage of invaluable monuments worldwide, which makes necessary to analyze their stability and safety to prevent possible collapse in hazardous conditions. For structural assessment purposes, it is necessary to elaborate models of the mechanical behavior of materials, which can vary widely from very accurate to very simplified ones. In this perspective, different aspects make the static or dynamic analysis of historical masonry constructions a complex task: the geometry data is usually scarse or missing; the constitution of the inner core of the structural elements is unknown, a complete mechanical characterization of the materials utilized is hardly possible, the existing damage of the structure is unknown, the sequence of construction is not documented and the building processes vary from one period to another as well as from one site to another. ⇑ Corresponding author. Tel.: +39 051 2093500. E-mail address: [email protected] (F. Tornabene). http://dx.doi.org/10.1016/j.engfailanal.2015.02.021 1350-6307/Ó 2015 Elsevier Ltd. All rights reserved.

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Analysis of masonry arches and vaulted structures in the framework of limit analysis investigates basic aspects of their behavior at collapse and matches modern analysis techniques with geometrical static principles rising from traditional theories. Three main hypotheses (i.e. no tensile strength, infinite compression strength, and no sliding failure) are clearly assumed by Heyman [1–4] on the mechanical behavior of masonry. This leads to simple computations and eliminates the possibility of failure due to material strength, but permits only failure due to instability, because of the formation of a sufficient number of hinges transforming the structure into a mechanism. The application of limit analysis to the study of the collapse of structural elements like arches, vaults, and flying buttresses under vertical and/or horizontal static loadings seems to be very attractive, as demonstrated in the literature for many applications (e.g. [5–18], among others). Limit analysis is adopted in these works to treat simple masonry elements since complex buildings are an assemblage of elementary structural schemes, and their overall capacity can be somehow derived from the ones of the components. In addition to the collapse of masonry structures due to static loads, many works in the literature have explored the resistance of vaulted systems to lateral accelerations, such as those due to an earthquake. Most research conducted so far on arches and portals under horizontal accelerations has followed an equivalent static approach. With this approach the arch or portal is considered subjected to a constant horizontal acceleration and a vertical one due to gravity. The analysis computes the minimum work needed for the formation of a sufficient number of nondissipative hinges transforming the structure into a mechanism. Oppenheim [19] introduces an analytical model to describe a masonry arch as a single degree of freedom three-bar (fourhinge) mechanism under horizontal ground motion. Assuming that the hinges would only occur in some predefined locations, the governing collapse mechanism is derived with its corresponding minimum horizontal ground acceleration necessary to cause collapse through an iteration procedure. The same iterative approach is used by Clemente and Raithel [20] in order to compute the onset accelerations and the governing mechanisms of circular arches with an overhanging back-fill subjected to constant lateral accelerations. In their work, the authors assumed different models to simulate the structureback-fill interaction, where the most suitable loading pattern for seismic actions is checked. The same problem is then studied by Ochsendorf [21] for a circular arch using the principle of virtual work to determine the ground surface tilt which causes collapse and the corresponding horizontal ground acceleration. The analysis is then extended to portals and triumphal arches by Baratta et al. [22], De Luca et al. [23], DeJong and Ochsendorf [24], analyzing the relative effect of each structural element (i.e. the circular arch and the buttress) on the global stability of the structure. For more complex geometries, however, the limitations of analytical modelling have enhanced the usefulness of numerical modelling for the structural analysis of masonry structures [25–28]. Despite the wide use of FEM for structural analyses, this method is basically tailored toward continuous structures which remain relatively connected during elasto-plastic failure under quasi-static or dynamic loading. Masonry, on the other hand, is heterogeneous, and is separated by joints and fractures throughout, making it unreasonable to model as an elastic continuum. At the same time, the deformations in masonry structures are not due to elastic deformations of the masonry material, and cannot be predicted satisfactorily by an elastic analysis. Failure is brittle and individual units (e.g. stones, bricks) are often free to separate, especially during quasi-static or dynamic loading. Method primarily developed to predict strength failure can give an indication of possible collapse mechanisms, but often struggle to predict collapse. Moreover, the exact stress is sometimes incognizable in a masonry structure, due to the unknown loading history, boundary conditions, or material properties. For all these reasons, performing affordable non linear analyses with FEM still require high expertise. An alternative and appealing approach is instead represented by the Discrete Element Method (DEM) where discrete bodies can move freely in space and interact with each other with contact forces, providing an automatic and efficient recognition of all contacts. For rigid bodies, the contact interaction law can be considered as the only constitutive law. Unlike FEM, in the DEM a compatible finite element mesh between the blocks and the joints is not required. Mortar joints are represented as zero thickness interfaces between the blocks. Representation of the contact between blocks is not based on joint elements, as occurs in the continuum FEM. Instead the contact is represented by a set of point contacts with no attempt to obtain a continuous stress distribution through the contact surface. Large displacements and rotations of the blocks are allowed with the sequential contact detection and update of tasks automatically. This differs from FEM where the method is not readily capable of updating the contact size or creating new contacts. The numerical technique based on DEM has different advantages with respect to FEM, e.g. the low storage, simple to code, suitable for parallel processing, same algorithm for statics and dynamics. The DEM was initially developed by Cundall [29] to model blocky-rock systems and sliding along rock mass. The approach was later applied to evaluate the statics [14,15,30–34] or dynamics [24,35–42] of masonry structures including arches and buttresses with failure occurring along mortar joints. These studies demonstrated the suitability of DEM to perform analyses of masonry structures and to describe realistically the ultimate load and failure mechanisms. However, the studies conducted thus far on masonry structures including arches were mainly focused on the structural behavior of semicircular geometries. On the contrary, there is a lack of information about the relative efficiency of different arch shapes, for any possible mechanism of collapse, such as pointed or basket-handle arches. Pointed arches are typical structural elements of the Gothic architecture. They allowed the Gothic cathedrals to reach larger heights than the Romanesque ones, while bearing lower thrusts for given loads and spans and reducing the weight on the lateral walls. Differently, the basket-handle arches generate a lower opening, for a given span. Their principal application was in bridges, though there are also examples of basket-handle arches in portals or ancient masonry buildings. It is well known however, as the ancient master builders were able to use geometrical rules, developed through centuries of trial and error, to build

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complex arches and vaults. They usually scale up the same proportions for new larger elements, without knowing anything else about the material properties or allowable stresses. The ideal geometries and proportions of arches and vaults were typically related to the in-plane dimensions and/or height of the space to be covered, and in many cases dictated by some practical difficulties in the building site. This makes clearly interesting a parametric assessment of masonry structure for varying geometry parameters, as herein performed both analytically and numerically. More specifically, the equivalent static analysis, based on the geometry and independent of the dimensional scale, is herein used to determine the minimum constant lateral acceleration for collapse of masonry arches and portals of different shape. The limit analysis represents the theoretical basis to examine stability from the analytical point of view, while the DEM, is herein adopted to model the discontinuous nature of the structures in the numerical formulation. The remainder of this paper is organized as follows: the limit analysis is first reviewed and detailed for pointed and basket-handle arches and portals in Section 2. A systematic assessment of the sensitivity of the quasi-static response to the geometry parameters is given in Section 3 for masonry arches, and in Section 4 for masonry portals. Section 5 is concerned with the numerical modelling where a brief overview of the DEM is first given, and numerical results are then discussed and compared with the analytical predictions, and results from FEM. Finally, conclusions are drawn in Section 6. 2. Analytical modelling 2.1. Problem statement Under the usual assumptions of limit analysis of infinite compressive strength, no tensile strength, and absence of sliding failure, masonry becomes an assemblage of rigid ashlars which are not able to slide or crush, but can only disconnect forming non-dissipative hinges [3]. At incipient collapse, a general shaped arch subjected to an horizontal constant acceleration will be transformed into an asymmetric mechanism, with formation of four hinges: two hinges at the extrados and two hinges at the intrados. This section considers two types of arches, namely pointed and basket-handle arches, as illustrated in Fig. 1a and b. In order to study the comparative stability of these shapes, their main geometry parameters are here defined in dimensionless form: the t=Rcirc ratio between the arch thickness t and the centerline radius of the reference circular arch Rcirc ; the angle of embrace 2a; for pointed arches, the e=Rcirc ratio between the eccentricity of the arch centre e and Rcirc ; for basket-handle arches, the ratios r=Rcirc and d=Rcirc ; r being the smaller radius and d the distance of the middle centre from the horizontal springline of the reference circular arch (Fig. 1). Collapse of the arch requires four non-dissipative hinges to form, two at the extrados (B and D, Fig. 2), and two at the intrados (A and C, Fig. 2). The location of the four hinges, A at the intrados, B at the extrados, C at the intrados, and D at the extrados, must be found by iterative calculations. The hinge locations are defined by the angles aA ; aB ; aC , and aD measured from the centre of the coordinate system. Each rigid block defined between two consecutive hinges are subjected to the vertical acceleration of gravity, g, and the lateral acceleration defined as a factor k, multiplied by the acceleration g. For all kinematically admissible positions of the hinges, a corresponding value of constant lateral acceleration can be computed applying the equilibrium in the form of the principle of virtual work. Basing on the initial assumption of rigid material, the internal work dissipated by the system is zero; therefore, from the principle of virtual work, the total work done by the acceleration is

I M2

B

g g

M1

C

g

I

g

B

M3

M2

M1

y

y

A

D

t B A

C

t

D

r

A

e

Rcirc (a)

D

C D

B A

x

2

x

2

M3

C

d Rcirc (b)

Fig. 1. Geometry of the pointed (a) and the basket-handle arch (b).

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λg B

C

g

λg B

g C

D

A

D

A

(a)

(b)

Fig. 2. Example of hinge formation and line of thrust at collapse in the pointed arch (a), and in the basket-handle arch (b). 3 3 X X Mi g v i þ M i kgui ¼ 0 i¼1

ð1Þ

i¼1

where ui and v i are the horizontal and the vertical displacements of each barycentre of the moving blocks, and M i is the relative mass. Since all displacement components depend on a single arbitrary lagrangian parameter, the above equation results to be a homogeneous equation in the only unknown k. For the analyzed structures, in particular, either w; / or h can be chosen as lagrangian parameters, so that the values of the displacement components can be calculated from the product of the considered rotation by the distance from rotation centre. For the collapse pattern in question, let C 1 ¼ A; C 2 ¼ I; C 3 ¼ D be the absolute rotation centres, and C 12 ¼ B; C 23 ¼ C; C 13 the relative rotation centres. Such points are determined using the condition that when a virtual displacement field is onset, then the absolute centres of any two parts must be aligned with their relative centre, and also any relative centre must be aligned. Rearranging Eq. (1), and solving for k yields

P3 Mi v i k ¼  P3i¼1 M i¼1 i kui

ð2Þ

where k is the multiplier kinematically sufficient to produce the collapse into the structure according to the assumed mechanism. The real hinge locations correspond to the lowest value of lateral acceleration, here indicated as kg (where g is the acceleration of gravity), and this is the constant acceleration leading to collapse of the arch. Hence, k and the hinge locations at collapse must be found by iterative calculations. When the mechanism forms, the line of thrust is contained within the thickness of the arch and is tangent to the boundary in the four hinge points (Fig. 2). The same analysis is then extended to the portal considering the additional parameters B=Rcirc (where B is the width of the buttress) and h=Rcirc (where h is the height of the buttress from the base to the horizontal springline of the reference circular arch). Collapse of a portal also requires four hinges to form, two at the extrados (B and D, Fig. 3), and two at the intrados (A and C, Fig. 3). For the generic portal, three types of mechanism can be activated: a local mechanism, characterized by the local failure of the arch only, without any hinge occurring in the buttresses; a global mechanism, characterized by the presence of two hinges at the base of the buttresses and two hinges in the arch; and a semi-global mechanism, characterized by the presence of one hinge at the base of one buttress and three hinges in the arch [23]. While the global mechanism interests portals under the only ash load actions or low seismic actions and dominant ash load ones, the semi-global or local mechanism interests portals massive or very stocky, respectively, under the only seismic actions or dominant seismic actions plus low ash load actions [22]. In all cases considered herein, the hinge locations corresponding to the lowest value of lateral acceleration are those represented in Fig. 3, i.e. a semi-global mechanism controlled. This type of mechanism is similar to the situation described for the generally shaped arches, except that the extrados hinge D now occurs at the base of the buttress instead of at the arch support. The work calculation necessary to determine the minimum lateral acceleration multiplier k is summarized in the following.  For assumed hinge locations A; B; C, and D, determine the areas (AAB ; ABC ; ACD ) and the centre of mass location of each rigid block (xAB ; yAB ; xBC ; yBC , xCD ; yCD ), whose expression vary with the geometry of the arch considered.  For the assumed hinge location, determine the location of the instantaneous centre, I, for the central portion (xI ; yI ), as follows

mAB xA  mCD xC  yA þ yC mAB  mCD yI ¼ yA þ mAB ðxI  xA Þ

xI ¼

ð3Þ ð4Þ

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R. Dimitri, F. Tornabene / Engineering Failure Analysis 52 (2015) 1–34

(a)

(b)

Fig. 3. Example of hinge locations at collapse in the portal with pointed arch (a) and basket-handle arch (b) (semi-global mechanism).

where mAB and mCD are the angular coefficient of the right lines passing through the hinges A; B, and C; D, respectively, equal to

yB  yA xB  xA y  yC ¼ D xD  xC

mAB ¼

ð5Þ

mCD

ð6Þ

 Apply a small rotation to the portion BC of the arch, here indicated with h, and compute the relative rotations of the two other blocks (AB and CD), here indicated respectively with w and /. These relative rotations are univocally defined by the rotation of one component as the mechanism has one degree of freedom. In other words:

h¼1

ð7Þ

xI  xB w¼h xB  xA xC  xI /¼h xD  xC  Compute the small horizontal and vertical displacements, ui and

ð8Þ ð9Þ

v i of each centre of mass Mi , as follows

uAB ¼ wðyAB  yA Þ

v AB ¼ wðxA  xAB Þ uBC ¼ hðyI  yBC Þ

v BC ¼ hðxBC  xI Þ

ð10Þ

uCD ¼ /ðyCD  yD Þ v CD ¼ /ðxD  xCD Þ  Compute the vertical and horizontal components of work, assuming as positive the horizontal loads in the right direction and the vertical downwards loads. The two components of work are here indicated with

Lv ¼ cgðAAB v AB þ ABC v BC þ ACD v CD Þ

ð11Þ

Lh ¼ kcgðAAB uAB þ ABC uBC þ ACD uCD Þ

ð12Þ

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R. Dimitri, F. Tornabene / Engineering Failure Analysis 52 (2015) 1–34

Table 1 Schematic description of the user-defined routines. Subroutine Define hAmin ; hAmax ; hBmax ; hCmax , hDmin ; hDmax ; dh; M, kinitial If kinitial > 0 Then For i ¼ hAmin to hAmax Step dh hBmin ¼ hAmax þ dh For j ¼ hBmin to hBmax Step dh hCmin ¼ j þ dh For k ¼ hCmin to hCmax Step dh If k > j Then For l ¼ 0 to M Step 1 hD ¼ hDmin  dh  l Calculate Areas/Weights/Centres of position/Virtual displacements/k If k > 0 Then If k 6 kinitial Then aA ¼ i aB ¼ j aC ¼ k a D ¼ hD k ¼ kinitial End If End If Next l l¼0 End If Next k k¼0 Next j j¼0 Next i End if End Subroutine

 Compute the lateral acceleration k by dividing the vertical and the horizontal work as follows

k¼

Lv Lh

ð13Þ

 Repeat the calculation for all kinematically admissible positions of the four hinges.  The critical hinge locations correspond to the lowest value of lateral acceleration kg, which is the constant acceleration leading to collapse of the arch. This process is performed by developing Visual Basic for Applications (VBA) procedures, that is compiling macros in the Excel Visual Basic Editor (VBE), linked to some Excel worksheets. With this approach, the user has to run the macros for each variation of the dimensionless geometry parameters (already defined as t=Rcirc ; 2a; e=Rcirc ; r=Rcirc ; d=Rcirc ). The program output provides the hinge locations A; B; C, and D for the critical mechanism and the minimum lateral acceleration k. A schematic description of the user-defined VBA routines using polar coordinates can be represented, for the generally shaped arch, as reported in Table 1. The previous subroutine is simplified to two or three cycles ‘‘For’’, when studying lateral stability of generally shaped portal frames, depending on whether a semi-global or global mechanism of collapse occur, in such a way as to fix one or two hinges at the base of the buttresses. 2.2. Geometrical definition of collapse mechanisms More details about hinge coordinates, areas and centres of mass of the arch portions between two conservative hinges, are herein provided for pointed and basket-handle arches and portals. 2.2.1. The pointed arches Let us consider first the collapse mechanisms for a pointed arch, and define b the angle enclosed starting from the negative x-axis towards the generic point location at the intrados (P) or the extrados (Q) of the arch (positive if clockwise). hi and he are the angles starting from the negative x0 - or x00 -axis towards the same point locations, whose maximum values (i.e. hmax;i and hmax;e ) are reached by hinges on the axis of symmetry (Fig. 4). The relative distances of P and Q from the centre O are here indicated as li;P and le;Q , respectively, which can be determined as

R. Dimitri, F. Tornabene / Engineering Failure Analysis 52 (2015) 1–34 sin hi li;P ¼ e sinðbh iÞ sin he le;Q ¼ e sinðbh eÞ

for b < 90

;

hi li;P ¼ e sinðsin pbh Þ i

he le;Q ¼ e sinðsin pbhe Þ

;

for b > 90

7

ð14Þ

ð15Þ

Based on these assumptions, the coordinates of the hinges A; B; C; D are determined as

xA ¼ li;A cos bA

yA ¼ li;A sin bA

xB ¼ le;B cos bB

yB ¼ le;B sin bB

xC ¼ li;C cos bC

yC ¼ li;C sin bC

xD ¼ le;D cos bD

yD ¼ le;D sin bD

ð16Þ

The areas AAB ; ABC ; ACD of the arch portions between two consecutive hinges, A and B; B and C; C and D, are given by

 ðhB  hA Þ  2 Re  R2i 2     ðhmax;i  hB Þ  2 ðhmax;e  hmax;i Þ 2 sinðhmax;e  hmax;i Þ ðhmax;i  hC Þ  2 Re  R2i þ 2 Re  Ri Re þ Re  R2i ABC ¼ 2 2 2 2  ðhC  hD Þ  2 Re  R2i ACD ¼ 2

AAB ¼

ð17Þ

and the corresponding centres of mass location are computed as

  4 R3e  R3i sinððhB  hA Þ=2Þ hB þ hA cos 3 R2e  R2i ðhB  hA Þ 2   3 3 4 Re  Ri sinððhB  hA Þ=2Þ hB þ hA ¼ sin 3 R2e  R2i ðhB  hA Þ 2

xAB ¼ e  yAB





!    4 R3e  R3i sinððhmax;i  hC Þ=2Þ hmax;i þ hC  e hmax;i  hC xBC ¼ cos 3 R2e  R2i ðhmax;i  hC Þ 2 2ABC    ! R2e  R2i  4 R3e  R3i sinððhmax;i  hB Þ=2Þ hmax;i þ hB hmax;i  hB e cos þ 2ABC 3 R2e  R2i ðhmax;i  hB Þ 2    ! R2e  R2i  4 R3e  R3i sinððhmax;i  hC Þ=2Þ hmax;i þ hC hmax;i  hC sin yBC ¼ 2 2 2ABC 3 Re  Ri ðhmax;i  hC Þ 2   2 2  ! Re  Ri  4 R3e  R3i sinððhmax;i  hB Þ=2Þ hmax;i þ hB hmax;i  hB sin þ 2 2 2ABC 3 Re  Ri ðhmax;i  hB Þ 2   2  R 4 sinððhmax;e  hmax;i Þ=2Þ hmax;i þ hmax;e  þ e hmax;e  hmax;i Re sin ðhmax;e  hmax;i Þ ABC 3 2     Re Ri 2 hmax;i þ hmax;e  sin hmax;e  hmax;i ðRcirc þ eÞ sin ABC 3 2 R2e  R2i

ð18Þ

  4 R3e  R3i sinððhC  hD Þ=2Þ hC þ hD e cos 3 R2e  R2i ðhC  hD Þ 2   4 R3e  R3i sinððhC  hD Þ=2Þ hC þ hD ¼ sin 3 R2e  R2i ðhC  hD Þ 2

ð19Þ

xCD ¼ yCD

ð20Þ

In the equations above, Re ¼ Rcirc þ e þ t=2; Ri ¼ Rcirc þ e  t=2. 2.2.2. The basket-handle arch The basket-handle arch is herein defined by means of three centres of curvature, i.e. two external centres O1 and O3 set on the left and right sides of the axis of symmetry, and one centre O2 on the vertical axis of symmetry. As previously assumed for pointed arches, O is the centre of curvature for the reference circular arch. As also shown in Fig. 5, an absolute coordinate system Oxy and three relative coordinate systems O1 x1 y1 , O2 x2 y2 and O3 x3 y3 are accounted for a complete geometrical description of the arch and coordinates of the hinges. Let h be the angle between the negative x1 - or x3 - axis and the generic

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R. Dimitri, F. Tornabene / Engineering Failure Analysis 52 (2015) 1–34

B

Re

B Re

Q

C

Ri

C

Q

Ri P A

le li

y

e

i,max

Rcirc

i

e,max

Rcirc

O' x'

O x e

A

D

i,max

e,max

y li le

y''

y'

P

D e

i

x'' O''

e

x

O e

e

(a)

(b)

Fig. 4. Pointed arch: geometrical definition of the left (a) and right (b) sides.

Q2 Q1 1 r P1 t

P2 y1

2

2

y

R y

R

max

min

O1 x1 O y2 b

1

3

r

x3

t

x

b

d

O y2

Q3 P3 3

y3 max

O3

min

x

d

Rcirc

Rcirc

O2 x2

O2 x2 (a)

(b)

Fig. 5. Basket-handle arch: geometrical definition of the left (a) and right (b) sides.

hinge location at the intrados (P) or extrados (Q) for the first and last portion of the arch (i.e. portions 1 and 3). This angle can vary within a minimum and maximum value (hmin and hmax respectively), which are here expressed through the main geometry parameters (d; r; t, and 2a) as follows

hmin ¼ p=2  a hmax ¼ p=2  c

ð21Þ

where

b ¼ Rcirc  r

c ¼ arctan ðb cos hmin =ðb sin hmin þ dÞÞ

ð22Þ

The generic position of an hinge set at the intrados or extrados of portion 2 is arbitrarily defined by the angle (hmax þ d), positive if clockwise, starting from the negative x2 - axis towards the generic point location (Fig. 5). The global coordinates of hinges in each portion of the arch are detailed as follows  Portion 1 (P1 ; Q 1 )

xP1 ¼ ½r i cos h þ b cos hmin  yP1 ¼ ½ri sin h þ b sin hmin 

ð23Þ

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R. Dimitri, F. Tornabene / Engineering Failure Analysis 52 (2015) 1–34

xQ 1 ¼ ½r e cos h þ b cos hmin 

ð24Þ

yQ 1 ¼ ½re sin h þ b sin hmin 

 Portion 2 (P2 ; Q 2 )

xP2 ¼ Ri cosðhmax þ dÞ

ð25Þ

yP2 ¼ Ri sinðhmax þ dÞ  d xQ 2 ¼ Re cosðhmax þ dÞ

ð26Þ

yQ 2 ¼ Re sinðhmax þ dÞ  d

 Portion 3 (P3 ; Q 3 )

xP3 ¼ r i cos h þ b cos hmin

ð27Þ

yP3 ¼ r i sin h þ b sin hmin xQ 3 ¼ r e cos h þ b cos hmin yQ 3 ¼ r e sin h þ b sin hmin

ð28Þ

In the equations above, R ¼ ðr sin hmax þ b sin hmin þ dÞ= sin hmax , Re ¼ R þ t=2; Ri ¼ R  t=2; r e ¼ r þ t=2; ri ¼ r  t=2. The lateral stability of the arch is then computed for fixed values of the dimensionless geometry parameters, considering all the possible collapse patterns. In other words, six cases of mechanism shape are here analyzed as follows (Fig. 6).

2

B

C

1 A

2

B

3

D

C 3

1 A

(a)

B

(b)

B

2 C

1 A

3

D

2 C 3

1 A

(c)

D

(d)

B

B

2

2 1

D

C

A

(e)

3

D

1

A

C

(f)

Fig. 6. Possible collapse mechanisms of a basket-handle arch.

3

D

10

     

R. Dimitri, F. Tornabene / Engineering Failure Analysis 52 (2015) 1–34

Case Case Case Case Case Case

1: 2: 3: 4: 5: 6:

A ðportion 1Þ  B ðportion 1Þ  C ðportion 2Þ  D ðportion 3Þ. A ðportion 1Þ  B ðportion 1Þ  C ðportion 3Þ  D ðportion 3Þ. A ðportion 1Þ  B ðportion 2Þ  C ðportion 2Þ  D ðportion 3Þ. A ðportion 1Þ  B ðportion 2Þ  C ðportion 3Þ  D ðportion 3Þ. A ðportion 2Þ  B ðportion 2Þ  C ðportion 2Þ  D ðportion 3Þ. A ðportion 2Þ  B ðportion 2Þ  C ðportion 3Þ  D ðportion 3Þ.

Therefore, for each collapse pattern, the areas AAB ; ABC ; ACD of the single portions are defined as follows. Case 1 ð1Þ

 ðhB  hA Þ  2 2 be  bi 2  d   ðhmax  hB Þ  2 2 C ¼ be  bi þ R2e  R2i 2 2  ðh  h Þ   ð2c  dC Þ  2 max D 2 2 2 ¼ Re  Ri þ be  bi 2 2

ð29Þ

 ðhB  hA Þ  2 2 be  bi 2    ðh  h Þ   ðhmax  hB Þ  2 max C 2 2 2 ¼ be  bi þ c R2e  R2i þ be  bi 2 2  ðhC  hD Þ  2 2 ¼ be  bi 2

ð30Þ

 d   ðhmax  hA Þ  2 2 B be  bi þ R2e  R2i 2 2  ðdC  dB Þ  2 ¼ Re  R2i 2  ðh  h Þ   ð2c  dC Þ  2 max D 2 2 ¼ Re  R2i þ be  bi 2 2

ð31Þ

 d   ðhmax  hA Þ  2 2 B be  bi þ R2e  R2i 2 2  ðh  h Þ   ð2c  dB Þ  2 max C 2 2 ¼ Re  R2i þ be  bi 2 2  ðhC  hD Þ  2 2 ¼ be  bi 2

ð32Þ

 ðdB  dA Þ  2 Re  R2i 2  ðdC  dB Þ  2 ¼ Re  R2i 2  ðh  h Þ   ð2c  dC Þ  2 max D 2 2 ¼ Re  R2i þ be  bi 2 2

ð33Þ

 ðdB  dA Þ  2 Re  R2i 2  ðh  h Þ   ð2c  dC Þ  2 max C 2 2 ¼ Re  R2i þ be  bi 2 2  ðhC  hD Þ  2 2 ¼ be  bi 2

ð34Þ

AAB ¼ ð1Þ

ABC

ð1Þ

ACD Case 2

ð2Þ

AAB ¼ ð2Þ

ABC

ð2Þ

ACD Case 3

ð3Þ

AAB ¼ ð3Þ

ABC

ð3Þ

ACD Case 4

ð4Þ

AAB ¼ ð4Þ

ABC

ð4Þ

ACD Case 5

ð5Þ

AAB ¼ ð5Þ

ABC

ð5Þ

ACD Case 6

ð6Þ

AAB ¼ ð6Þ

ABC

ð6Þ

ACD

For the sake of brevity, the analytical definition of the centre of mass for each portion of the basket-handle arch are given in Appendix A.

11

R. Dimitri, F. Tornabene / Engineering Failure Analysis 52 (2015) 1–34

2.2.3. The pointed portal The other problem addressed in this work is the quasi-static behavior of pointed portals. As for pointed arches, when a local collapse mechanism occurs in pointed portals, Eqs. (14)–(20) have to be used. Conversely, a semi-global mechanism requires some changes in the previously mentioned equations since one hinge is now set at the base of the buttress (point D in Figs. 7 and 8) instead of the arch support (point X in Figs. 7 and 8). Based on the geometrical schemes of Figs. 7 and 8, the global hinge positions are determined as

Re

B

B Re

Q

C

Ri

C

Q

Ri P A

le li

y

y'

e

X

e,max

Rcirc

x

O e

e

e i

x'' O''

O' x'

O x e

X

i,max

Rcirc

i

y li le

y'' A

i,max

e,max

P

e

h

h

D

D

(a)

(b)

Fig. 7. Pointed portal: geometrical definition of the left (a) and right (b) sides.

B

y

C

A

O x

D Fig. 8. Alternative mechanism of collapse of a pointed portal.

12

R. Dimitri, F. Tornabene / Engineering Failure Analysis 52 (2015) 1–34

xA ¼ li;A cos bA

yA ¼ li;A sin bA

xB ¼ le;B cos bB

yB ¼ le;B sin bB

xC ¼ li;C cos bC

yC ¼ li;C sin bC

xD ¼ le;X cos bX þ B

ð35Þ

yD ¼ h

The areas of portions AB and BC and the corresponding centre of mass locations are expressed, once more, by Eqs. (17)–(19), whereas the area and centre of mass location of portion CD take the form

ACD ¼

 1   ðhC  hX Þ  2 Re  R2i þ ðxD  le;X cos bX þ BÞ le;X  li;X sin bX þ B h þ li;X sin bX 2 2

ð36Þ

"  #  ðhC  hX Þ  2 4 R3e  R3i sinððhC  hX Þ=2Þ hC þ hX Re  R2i e þ cos 2ACD 3 R2e  R2i ðhC  hX Þ 2  2   2 B le;X  li;X cos bX sin bX  B  2le;X þ li;X þ h þ li;X sin bX þ li;X cos bX þ ACD 2 6ACD

   

  B þ li;X  le;X cos bX le;X  li;X sin bX B þ li;X þ le;X cos bX þ 2ACD "  # 3   ðhC  hX Þ 2 4 Re  R3i sinððhC  hX Þ=2Þ hC þ hX 2 yCD ¼ Re  Ri sin 2ACD 3 R2e  R2i ðhC  hX Þ 2  3 2  le;X  li;X cos bX sin bX B 2 2 2 þ þ li;X sin bX  h 2ACD 6ACD 

  2 2 2 B þ li;X  le;X cos bX le;X  li;X sin bX þ 2ACD xCD ¼

ð37Þ

2.2.4. The basket-handle portal Once again, when local mechanisms of collapse activate in basket-handle portals, the same equations considered thus far for basket-handle arches must be applied. Similarly, for semi-global mechanisms as in Fig. 9, hinge positions at the intrados

Q2

2

Q1 1 r P1 t

min

P2 y1

2

y R

3 X

1

max

O1 x1 O y2 b

R r

x3

t

x b

d

Rcirc

Q3

y

O y2

max

O3

min

x

d

Rcirc

O2 x2

O2 x2

h

h

D

(a)

P3 3 X

y3

D

(b)

Fig. 9. Basket-handle portal: geometrical definition of the left (a) and right (b) sides.

R. Dimitri, F. Tornabene / Engineering Failure Analysis 52 (2015) 1–34

13

Fig. 10. Possible collapse mechanisms of a basket-handle arch.

or extrados of the arch are always defined by Eqs. (23)–(28), while hinge D, set at the base of the buttress, is now defined by coordinates

xD ¼ ½ðr þ bÞ cos hmin þ B ð38Þ yD ¼ h

Six semi-global collapse mechanisms are therefore considered for fixed dimensionless geometry parameters, as shown in Fig. 10, in order to compute correctly the lowest load multiplier, in such a way that k ¼ min ðk1 ; k2 ; k3 ; k4 ; k5 ; k6 Þ. As in ðiÞ

basket-handle arches, for a generic collapse configuration ‘‘i’’, the areas and centres of mass of the rigid blocks AB (AAB ,

14 ðiÞ

R. Dimitri, F. Tornabene / Engineering Failure Analysis 52 (2015) 1–34 ðiÞ

ðiÞ

ðiÞ

ðiÞ

xAB ; yAB ) and BC (ABC ; xBC ; yBC ), are defined by Eqs. (29)–(44), (46), (47), (49), (50), (52), (53), (55), (56), (58), (59). Differently, the area of the block between the last two hinges C and D is computed as

ð iÞ

ð iÞ

ACD ¼ ACX þ

    1 ðhD  hmin Þ  2 t t 2 cos hmin sin hmin þ Bðh þ ðr þ bÞ sin hmin Þ be  bi þ xD þ B  r þ b þ 2 2 2 2

AiCX being the area of portion between hinge C and the arch support X.

t/Rcirc=0.14 t/Rcirc=0.18

1

circ

0.8 0.6 0.4 0.2 2 =120

0 0

0.2

0.4

t/Rcirc=0.12 t/Rcirc=0.16 t/Rcirc=0.20

(b) Horizontal acceleration at collapse, *g

t/Rcirc=0.12 t/Rcirc=0.16 t/Rcirc=0.20

(a)

0.6

0.8

14 12 10 8 6 4 2

2 =120

0

1

0

0.2

e/Rcirc t/Rcirc=0.14 t/Rcirc=0.18

circ

0.8 0.6 0.4 0.2 0

2 =150 0

0.2

0.4

0.6

circ

2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

0.8

1

0.2

0.4

t/Rcirc=0.14 t/Rcirc=0.18

0.6

e/Rcirc

t/Rcirc=0.14 t/Rcirc=0.18

5 4 3 2 1

2 =150 0

0.2

0.4

0.6

0.8

1

e/Rcirc

2 =180 0

1

6

0

0.8

1

t/Rcirc=0.12 t/Rcirc=0.16 t/Rcirc=0.20

(f) Horizontal acceleration at collapse, *g

t/Rcirc=0.12 t/Rcirc=0.16 t/Rcirc=0.20

0.8

7

e/Rcirc

(e)

0.6

t/Rcirc=0.12 t/Rcirc=0.16 t/Rcirc=0.20

(d) Horizontal acceleration at collapse, *g

1

0.4

e/Rcirc

t/Rcirc=0.12 t/Rcirc=0.16 t/Rcirc=0.20

(c)

t/Rcirc=0.14 t/Rcirc=0.18

t/Rcirc=0.14 t/Rcirc=0.18

3 2.5 2 1.5 1 0.5 0

2 =180 0

0.2

0.4

0.6

0.8

1

e/Rcirc

Fig. 11. Lateral stability of pointed arches with angles of embrace 2a equal to 120 (a and b), 150 (c and d) and 180 (e and f).

ð39Þ

R. Dimitri, F. Tornabene / Engineering Failure Analysis 52 (2015) 1–34

Moreover, the centre of mass of portion CD has coordinates   ðiÞ ðiÞ x A ð2B  t=2 cos hmin Þ t t ðiÞ sin hmin cos hmin xCD ¼ CXðiÞCX þ r þ b þ ðiÞ 2 3 ACD 2ACD     ðB  t=2 cos hmin Þ t t cos h sin hmin þ B þ 2 r þ b þ min ðiÞ 2 4 2ACD   B B þ ðiÞ ½h þ ðr þ bÞ sin hmin  þ ðr þ bÞ cos hmin 2 ACD ðiÞ

ðiÞ

ð2B  t=2 cos hmin Þ t 2 2 ¼ þ sin hmin ðiÞ ðiÞ 6 ACD 2ACD   ðB  t=2 cos hmin Þ t t 2 sin hmin þ r þ b þ ðiÞ 2 4 A

ðiÞ yCD

yCX ACX

CD

þ

B ðiÞ

2ACD

½h þ ðr þ bÞ sin hmin ½ðr þ bÞ sin hmin  h ðiÞ

ðiÞ

where the coordinates xCX and yCX refer to the centre of mass for portion CX within hinge C and arch support X.

Fig. 12. Lateral stability of pointed arches with e=Rcirc ¼ 0:5 versus the half angle of embrace.

Fig. 13. Critical hinge locations at collapse for a pointed arch with e=Rcirc ¼ 0:5 and t=Rcirc ¼ 0:12.

15

ð40Þ

16

R. Dimitri, F. Tornabene / Engineering Failure Analysis 52 (2015) 1–34

3. Analytical results for masonry arches 3.1. Introduction The analysis is carried out on arches with an angle of embrace 2a equal to 120 ; 150 and 180 , varying the ratios e=Rcirc and t=Rcirc for pointed arches, and d=Rcirc and t=Rcirc (with r=Rcirc fixed at 0.5) for basket-handle arches. As follows, the lateral stability of the arches is expressed both in terms of kg, and in the dimensionless form k=kcirc , where kg is the horizontal acceleration at collapse of the arch under consideration, and kcirc g is that of the reference circular arch.

t/Rcirc=0.14 t/Rcirc=0.18

2.4 2 =120

2.2 2 circ

t/Rcirc=0.12 t/Rcirc=0.16 t/Rcirc=0.20

(b) Horizontal acceleration at collapse, *g

t/Rcirc=0.12 t/Rcirc=0.16 t/Rcirc=0.20

(a)

1.8 1.6 1.4 1.2

30 2 =120

25 20 15 10 5

1

0 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

d/Rcirc t/Rcirc=0.12 t/Rcirc=0.16 t/Rcirc=0.20

(c)

t/Rcirc=0.14 t/Rcirc=0.18

t/Rcirc=0.12 t/Rcirc=0.16 t/Rcirc=0.20

(d)

0.8

1

1.8 1.6 1.4 1.2

t/Rcirc=0.14 t/Rcirc=0.18

14

Horizontal acceleration at collapse, *g

2 =150

circ

0.6

d/Rcirc

2

2 =150

12 10 8 6 4 2 0

1 0

0.2

0.4

0.6

0.8

0

1

0.2

0.4

t/Rcirc=0.12 t/Rcirc=0.16 t/Rcirc=0.20

(e)

t/Rcirc=0.14 t/Rcirc=0.18

Horizontal acceleration at collapse, *g

2 =180 4 3 2 1 0 0.2

0.4

0.6

d/Rcirc

0.8

t/Rcirc=0.12 t/Rcirc=0.16 t/Rcirc=0.20

(f)

5

0

0.6

0.8

1

d/Rcirc

d/Rcirc

circ

t/Rcirc=0.14 t/Rcirc=0.18

1

t/Rcirc=0.14 t/Rcirc=0.18

6 2 =180

5 4 3 2 1 0 0

0.2

0.4

0.6

0.8

1

d/Rcirc

Fig. 14. Lateral stability of basket-handle arches with angles of embrace 2a equal to 120 (a and b), 150 (c and d) and 180 (e and f). r=Rcirc ¼ 0.

R. Dimitri, F. Tornabene / Engineering Failure Analysis 52 (2015) 1–34

17

3.2. The pointed arch Pointed arches with 2a ¼ 120 are always less stable than the reference circular ones, and their stability decreases as the e=Rcirc ratio increases (Fig. 11a). The relative stability k=kcirc is very weakly influenced by the thickness of the arch. Similar results are found for 2a ¼ 150 (Fig. 11c), but the decrease of stability with e=Rcirc is less pronounced and more influenced by t=Rcirc . Thinner arches are more stable (in relative terms to their circular counterparts) than thicker arches. For 2a ¼ 180 (Fig. 11e), thin pointed arches may be more stable than their reference circular arches. In this case, for each given t=Rcirc ratio, the collapse acceleration reaches the maximum value for a certain e=Rcirc , which can be defined as the optimal ratio (Fig. 11e). The maximum value of collapse acceleration decreases as the thickness ratio increases. The optimal e=Rcirc decreases as the t=Rcirc ratio increases, until it reaches zero (circular arch) for the largest thickness ratio analyzed (0.20). Analyzing the same results in the dimensional form kg, pointed arches reveal, in all cases, a great lateral stability. For a given e=Rcirc parameter, in particular, pointed arches with smaller angles of embrace and larger thickness ratios are found to be more resistant to lateral accelerations (Fig. 11b, d and f). The same results are confirmed also by Fig. 12 where the thickness ratios range from 0.12 to 0.20 and the total angles of embrace vary between 80 and 180° (i.e. a ¼ 40  90 ). For angles of embrace under a critical value, the collapse mechanism is characterized by the formation of two hinges at the supports, and other two hinges within the span. For angles of embrace above this critical value, three hinges form within the span, and only one hinge forms at the support. For small angles of embrace, at which the pointed arch is very stable, the hinge A always forms at the right abutment. Thus, the angle at which hinge A forms is simply 90  a. The central hinge B is typically close to

Fig. 15. Lateral stability of basket-handle arches with d=Rcirc ¼ 0:5 and r=Rcirc ¼ 0:5 versus the half angle of embrace.

180 Hinge D

160 140

Hinge C

120 100 80

Hinge B

60 40

Hinge A

20 0 70

75

80

85

90

Fig. 16. Critical hinge locations at collapse for a basket-handle arch with d=Rcirc ¼ 0:5; r=Rcirc ¼ 0:5 and t=Rcirc ¼ 0:12.

18

R. Dimitri, F. Tornabene / Engineering Failure Analysis 52 (2015) 1–34

A for very stable configurations and tends to remain at about the same location (70 ) as the angle of embrace increases. Similarly, the angle at which hinge C forms remain constant (90 ), and increases rapidly only when a tends to the maximum value amax ¼ 90 . At the same times the hinge at point D results to form always at the extrados of the right support as already demonstrated in the literature by Clemente and Raithel [43] for circular arches. This behavior is illustrated in Fig. 13 for a pointed arch with a thickness ratio of 0.12, in which the angle of each hinge location is measured from the horizontal line below the left abutment. Thus, two different mechanisms can induce collapse into the pointed arches. The first one is a typical four-bar chain, of the type investigated by Oppenheim [19], in which two hinges form at the supports (A and D). The second collapse mechanism corresponds to part of a five-hinge one in which four hinges are necessary for failure, and the angle A, formerly at the left support of the arch, moves into the span. The change in the type of governing mechanism is illustrated by the zig-zag pattern in the location of hinge A (Fig. 13).

t/Rcirc=0.12 t/Rcirc=0.16 t/Rcirc=0.20

1.2 1.18 1.16 1.14 1.12 1.1 1.08 1.06 1.04 1.02 1

(b)

t/Rcirc=0.14 t/Rcirc=0.18

Horizontal acceleration at collapse, *g

circ

(a)

B/Rcirc=0,8 h/Rcirc=3 2 =120

Semi-global mechanism

0

0.2

0.4

0.6

0.8

t/Rcirc=0.12 t/Rcirc=0.16 t/Rcirc=0.20

2 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1

Semi-global mechanism

B/Rcirc=0,8 h/Rcirc=3 2 =120

1

0

0.2

0.4

e/Rcirc

1.1

circ

t/Rcirc=0.14 t/Rcirc=0.18

1.06 1.04 Semi-global mechanism

1.02

B/Rcirc=0,8 h/Rcirc=3 2 =150

1 0

0.2

0.4

0.6

t/Rcirc=0.12 t/Rcirc=0.16 t/Rcirc=0.20

(d)

Semi-global mechanism Local mechanism Semi-global mechanism Local mechanism

1.08

2

2

0.8

1.8 1.7 1.6 B/Rcirc=0,8 h/Rcirc=3 2 =150

1.5 0

0.2

circ

1.2 1 0.8

B/Rcirc=0,8 h/Rcirc=3 2 =180

Semi- global mechanism Local mechanism Semi- global mechanism

Local mechanism Local mechanism

0.2

0.4

0.6

e/Rcirc

0.6

0.8

t/Rcirc=0.12 t/Rcirc=0.16 t/Rcirc=0.20

(f)

Local mechanism

0

0.4

0.8

1

e/Rcirc

1.8

1.4

t/Rcirc=0.14 t/Rcirc=0.18

Semi-global mechanism

1

t/Rcirc=0.14 t/Rcirc=0.18

1.6

1

1.4

1

Horizontal acceleration at collapse, g

t/Rcirc=0.12 t/Rcirc=0.16 t/Rcirc=0.20

0.8

1.9

e/Rcirc

(e)

0.6

e/Rcirc

Horizontal acceleration at collapse, *g

t/Rcirc=0.12 t/Rcirc=0.16 t/Rcirc=0.20

(c)

t/Rcirc=0.14 t/Rcirc=0.18

2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4

t/Rcirc=0.14 t/Rcirc=0.18

Semi- global mechanism Local mechanism Semi- global mechanism

Local mechanism Local mechanism Local mechanism

0

0.2

0.4

0.6

B/Rcirc=0,8 h/Rcirc=3 2 =180

0.8

1

e/Rcirc

Fig. 17. Lateral stability of pointed portals with angles of embrace 2a equal to 120 (a and b), 150 (c and d) and 180 (e and f).

R. Dimitri, F. Tornabene / Engineering Failure Analysis 52 (2015) 1–34

19

3.3. The basket-handle arch Basket-handle arches with 2a ¼ 120 are always more stable than the reference circular ones, and their stability increases as the d=Rcirc ratio increases (Fig. 14a). Thinner arches are less stable (in relative terms to their circular counterparts) than thicker arches. Similar results are found for 2a ¼ 150 (Fig. 14c), but the curves corresponding to the smallest thickness ratios start displaying a maximum point. For 2a ¼ 180 (Fig. 14e), basket-handle arches with the smallest analyzed thickness ratio (0.12) have a large k=kcirc for moderate d=Rcirc values, but become less stable than their reference circular arches for large d=Rcirc values. For each given t=Rcirc ratio, the collapse acceleration reaches the maximum for a certain d=Rcirc ratio (Fig. 14e). The maximum value of collapse acceleration decreases as the thickness ratio increases. The optimal d=Rcirc increases as t=Rcirc increases. Regarding the same results expressed in terms of kg, basket-handle arches with 2a ¼ 120 and 150 are very stable, as the horizontal acceleration at collapse is always much larger than 1g (Fig. 14b and d). Differently, basket-handle arches with 2a ¼ 180 have a large kg for t=Rcirc ratios between 0.14 and 0.20, becoming not so much resistant when t=Rcirc ¼ 0:12 and d=Rcirc ratios approach 1, as kg approaches zero (Fig. 14f). For a given d=Rcirc parameter, basket-handle arches with smaller angles of embrace and larger thickness ratios are more resistant to lateral accelerations (Fig. 14b, d and f). The same results are confirmed also by Fig. 15 for thickness ratios ranging from 0.12 to 0.20 and total angles of embrace varying between 140 and 180 (i.e. a ¼ 70  90 ). As already said for pointed arches, the collapse mechanism is characterized by the formation of two hinges at the supports (A and D), and other two hinges within the span (B and C), for angles of embrace under a critical value. Differently, three hinges form within the span, and only one hinge forms at the support (Fig. 16), for angles of embrace above this critical value. These results all agree with previous results on the circular arch.

4. Analytical results for masonry portals The analyses previously illustrated for the stand-alone arches is repeated for portals having arches of pointed and baskethandle shapes, in order to study the influence of the e=Rcirc and d=Rcirc ratios on the collapse accelerations. Portals with 2a equal to 120 ; 150 and 180 , and with h=Rcirc ¼ 3 and B=Rcirc ¼ 0:8 are analyzed. Results are expressed both in the dimensional form kg, and in the dimensionless form k=kcirc . 4.1. The pointed portal Based on the dimensionless results, it can be deduced as portals with pointed arches having 2a ¼ 120 and 150 are always more stable than their circular counterparts, although k=kcirc is rather small (Fig. 17a and c). For each t=Rcirc , the collapse acceleration reaches a maximum value for a certain e=Rcirc ratio, which can be considered as the optimal ratio in terms of stability. This value increases as the t=Rcirc ratio increases. In general, thinner arches are less stable (in relative terms to their circular counterparts) than thicker arches.

Fig. 18. Lateral stability of a pointed portal with B=Rcirc ¼ 0:8, e=Rcirc ¼ 0:5 and h=Rcirc ¼ 3 versus the angles of embrace.

20

R. Dimitri, F. Tornabene / Engineering Failure Analysis 52 (2015) 1–34

For 2a ¼ 120 , the semi-global mechanism always controls. As 2a increases, the local mechanism controls for a wider range of e=Rcirc values, and especially so for small thickness ratios (Fig. 17c and e), until it becomes the only mechanism for 2a ¼ 180 and the three smallest thickness ratios analyzed. When 2a ¼ 180 and failure is controlled by a local mechanism, thinner arches may become more stable (in relative terms to their circular counterparts) than thicker arches, as previously observed for stand-alone arches, although this does not hold for all thickness ratios. The values of k=kcirc become much larger than one for thickness ratios between 0.12 and 0.14 and ‘‘moderate’’ e=Rcirc values. As opposed to stand-alone pointed arches, portals with pointed arches are in most cases more resistant to lateral accelerations than those with circular arches. This holds both when semi-global and local collapse mechanisms control (Fig. 17e) except for some combinations of thickness ratio and e=Rcirc . The same results are expressed also in terms of kg, as presented in Fig. 17b, d, and f, for variable e=Rcirc parameters, and in Fig. 18 for variable angles of embrace. In general, when the portal collapses with a semi-global mechanism, the stability

t/Rcirc=0.14 t/Rcirc=0.18

1

circ

0.9 0.8 0.7

B/Rcirc=0,8 h/Rcirc=3 2 =120

0.6 0

0.2

0.4

0.6

0.8

t/Rcirc=0.12 t/Rcirc=0.16 t/Rcirc=0.20

(b) Horizontal acceleraion at collapse, *g

t/Rcirc=0.12 t/Rcirc=0.16 t/Rcirc=0.20

(a)

2 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1

Semi-global mechanism

B/Rcirc=0,8 h/Rcirc=3 2 =120

0

1

0.2

t/Rcirc=0.14 t/Rcirc=0.18

1

circ

0.9 0.8 B/Rcirc=0,8 h/Rcirc=3 2 =150

0.7 0.6 0

0.2

0.4

0.6

0.8

2 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1

(e)

t/Rcirc=0.14 t/Rcirc=0.18

0.2

Horizontal acceleration at collapse, g

circ

Semi-global mech. Local mech. Semi-global mech.

1 B/Rcirc=0,8 h/Rcirc=3 Semi- global mech. Semi- global mech. 2 =180

0.5 0 0

0.2

0.4

0.6

d/Rcirc

0.4

0.6

0.8

t/Rcirc=0.12 t/Rcirc=0.16 t/Rcirc=0.20

(f)

Local mech.

Local mech.

1.5

t/Rcirc=0.14 t/Rcirc=0.18

0.8

1

d/Rcirc

Local mech.

2

1

B/Rcirc=0,8 h/Rcirc=3 2 =150

0

1

3 2.5

0.8

Semi-global mechanism

d/Rcirc t/Rcirc=0.12 t/Rcirc=0.16 t/Rcirc=0.20

0.6

t/Rcirc=0.12 t/Rcirc=0.16 t/Rcirc=0.20

(d) Horizontal acceleration at collapse, *g

(c)

0.4

d/Rcirc

d/Rcirc t/Rcirc=0.12 t/Rcirc=0.16 t/Rcirc=0.20

t/Rcirc=0.14 t/Rcirc=0.18

1

2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

t/Rcirc=0.14 t/Rcirc=0.18

Semi- global mech. Semi- global mech.

Local mech. Local mech. Semi-global mech. Local mech. Semi-global mech.

B/Rcirc=0,8 h/Rcirc=3 2 =180

0

0.2

Local mech.

0.4

0.6

0.8

1

d/Rcirc

Fig. 19. Lateral stability of basket-handle portals with angles of embrace 2a equal to 120 (a and b), 150 (c and d) and 180 (e and f).

R. Dimitri, F. Tornabene / Engineering Failure Analysis 52 (2015) 1–34

21

Fig. 20. Lateral stability of a basket-handle portal with B=Rcirc ¼ 0:8, e=Rcirc ¼ 0:5 and h=Rcirc ¼ 3 versus the angle of embrace.

Fig. 21. Computation cycle in UDEC [40].

increases for increasing angle of embrace, increasing e=Rcirc and decreasing thickness. Differently, when the portal collapses with local mechanism, the stability decreases drastically for increasing angles of embrace, increasing e=Rcirc ratios, and decreasing thickness in the arch. 4.2. The basket-handle portal Portals with basket-handle arches having 2a ¼ 120 and 150 are always less stable than their circular counterparts, and the stability decreases as the d=Rcirc ratio increases (Fig. 19a and c). Thinner arches are more stable (in relative terms to their circular counterparts) than thicker arches. For 2a ¼ 120 and 150 , the semi-global mechanism always controls. For 2a ¼ 180 , the local mechanism controls in some ranges of d=Rcirc values (Fig. 19e). The values of k=kcirc become much larger than one for thickness ratios between 0.12 and 0.14 and ‘‘moderate’’ d=Rcirc values. The same results are expressed also in terms of kg, as presented in Fig. 19b,d, and f for variable d=Rcirc parameters, and in Fig. 20 for variable angles of embrace. The stability of portals with basket-handle arches versus the angle of embrace generally shows a minimum, and increases as the thickness ratio decreases. When a local mechanism controls, the collapse acceleration becomes the same of the stand-alone arch and hence displays the trend already illustrated for basket-handle arches. 5. Numerical modelling with DEM 5.1. Introduction Predictions of the analytical model are compared with numerical results obtained from the DEM using the commercial program UDEC [44]. DEM is used to analyze discontinuous materials such as rocks and masonry assemblages. It allows finite displacements and rotations of discrete bodies, including complete detachment, and recognizes new contacts during progression of the analysis. Within discrete element programs, the used code can be classified as a distinct element program, as it uses deformable contacts and an explicit, time-domain solution of the original equation of motion [44,45]. The solution scheme is identical to that used by the explicit finite-difference method for continuum analysis. The time step restriction applies to both contacts and blocks. For rigid blocks, the block mass and interface stiffness between blocks define the time

22

R. Dimitri, F. Tornabene / Engineering Failure Analysis 52 (2015) 1–34

Fig. 22. Mechanisms of collapse of circular arches with Rcirc ¼ 1, and t=Rcirc ¼ 0:2: the analytical (a–c–e) and numerical (b–d–f) configurations.

Table 2 Analytical and numerical results for pointed (e=Rcirc ¼ 0:5, t=Rcirc ¼ 0:2) and basket-handle (d=Rcirc ¼ 0:5; r=Rcirc ¼ 0:5, t=Rcirc ¼ 0:2) arches. Angle of embrace (°)

Analytical results (k)

Numerical results (k)

CIRCULAR ARCH 120 150 180

1.27 0.62 0.29

1.28 0.64 0.31

POINTED ARCH 120 150 180

0.53 0.38 0.24

0.53 0.39 0.25

BASKET-HANDLE ARCH 120 150 180

2.35 1.06 0.54

2.38 1.10 0.56

step limitation. For deformable blocks, the zone size is used, and the stiffness of the system includes contributions from both the intact rock modulus and the stiffness at the contacts (penalty parameters). At any simulation step, when two bodies come in contact, equivalent springs are generated, in the normal and tangential directions, to estimate the magnitude of contact forces that are applied to the bodies pushing them apart. In normal direction, the mechanical behavior of joints is governed by the following equation

MF N ¼ kN  MuN Ac

ð41Þ

R. Dimitri, F. Tornabene / Engineering Failure Analysis 52 (2015) 1–34

23

where kN is the normal stiffness of the contact, MF N is the change in normal force, MuN is the change in normal displacement, and Ac is the contact area. Similarly, the mechanical behavior of joints in tangential direction is controlled by a constant tangential stiffness kT , defined as

MF T ¼ kT  MuT Ac

ð42Þ

where MF T is the change in tangential force, and MuT is the corresponding change in tangential displacement. The Coulomb friction law is used to limit the tangential contact force F T below a certain magnitude taking into account the magnitude of the normal contact force F N and the friction coefficient l at the contact interface. The contact forces are taken into account, together with the gravity forces, in the formulation of the equations of motion. The motion of a discrete body at a certain time step is determined from the equilibrium equations at the previous time step which are integrated using the central difference method to compute the displacements at the current time step. After computing the displacements and rotations of all bodies, for each time step, their corresponding positions are determined. Then, a new cycle of contact detection, contact resolution, application of forces and solution of the equations of motion follows, based on the updated positions of the bodies. This iterative procedure continues until the end of the simulation. Fig. 21 shows the schematic calculation cycle taking place in UDEC analyses.

Fig. 23. Mechanisms of collapse of pointed arches with Rcirc ¼ 1, e=Rcirc ¼ 0:5, and t=Rcirc ¼ 0:2: the analytical (a–c–e) and numerical (b–d–f) configurations.

24

R. Dimitri, F. Tornabene / Engineering Failure Analysis 52 (2015) 1–34

Fig. 24. Mechanisms of collapse of basket-handle arches with Rcirc ¼ 1; d=Rcirc ¼ 0:5; r=Rcirc ¼ 0:5 and t=Rcirc ¼ 0:2: the analytical (a–c–e) and numerical (b– d–f) configurations.

The masonry arches and portals in this work are considered as an assemblage of rigid blocks with frictional joints. The properties of these joints required for the analysis are elastic shear and normal stiffnesses, cohesion and friction angle, tensile strength and dilatancy angle. For consistency with the assumptions of the analytical model, the tensile strength and the cohesion of the joints as well as the dilatancy angle are all taken as zero. The values of the elastic normal and shear stiffnesses are arbitrarily set as 5  103 GPa/m, however it is verified that these values do not influence results. The friction coefficient is set to 1.2 to ensure overturning failure. 5.2. Numerical results: the arches The analyses consider first circular arches, pointed arches with e=Rcirc ¼ 0:5 and basket-handle arches with d=Rcirc ¼ r=Rcirc ¼ 0:5. All of them are also characterized by a thickness ratio t=Rcirc ¼ 0:2 and a density of 2000 kg=m3 . Three angles of embrace are taken into account (2a ¼ 120 ; 150 ; 180 ), as already done for the analytical study. Results are summarized in Table 2 for circular, pointed and basket-handle arches respectively. In general, the numerical results are in perfect agreement with the analytical predictions both in terms of the onset accelerations (kg) and mechanisms of collapse (Figs. 22–24), confirming as the DEM can be a useful tool for the equivalent static analysis of a generally shaped arch under lateral accelerations. 5.3. Numerical results: the portals The same analyses are carried out on circular, pointed and basket-handle portals with the same geometry parameters as considered for the only arches, with the additional parameters B=Rcirc ¼ 0:8 and h=Rcirc ¼ 3 related to the dimensions of

R. Dimitri, F. Tornabene / Engineering Failure Analysis 52 (2015) 1–34

25

Fig. 25. Mechanisms of collapse of circular portals with Rcirc ¼ 1, H=Rcirc ¼ 3; B=Rcirc ¼ 0:8, and t=Rcirc ¼ 0:2: the analytical (a–c–e) and numerical (b–d–f) configurations.

buttresses. Also in these cases the material has a density of 2000 kg=m3 , and the same angles of embrace are taken into account (2a ¼ 120 ; 150 ; 180 ). Results are summarized in Table 3, and the mechanisms of collapse are shown in Figs. 25–27, for circular, pointed and basket-handle portals respectively, revealing once again, a perfect agreement between the analytical and the numerical approaches. In all cases the semi-global mechanism of collapse controls with three hinges forming into the arch and one hinge fixed at the extrados side of the support.

26

R. Dimitri, F. Tornabene / Engineering Failure Analysis 52 (2015) 1–34 Table 3 Analytical and numerical results for pointed (e=Rcirc ¼ 0:5, t=Rcirc ¼ 0:2) and basket-handle (d=Rcirc ¼ 0:5; r=Rcirc ¼ 0:5, t=Rcirc ¼ 0:2) portals. B=Rcirc ¼ 0:8; h=Rcirc ¼ 3 in both cases. Angle of embrace (°)

Analytical results (k)

Numerical results (k)

CIRCULAR PORTAL 120 150 180

0.16 0.18 0.19

0.17 0.18 0.21

POINTED PORTAL 120 150 180

0.17 0.19 0.20

0.18 0.20 0.21

BASKET-HANDLE PORTAL 120 0.14 150 0.15 180 0.16

0.15 0.16 0.17

6. Comparison between DEM, FEM and experimental results The suitability of the DEM approach to analyse the quasi-static behavior of masonry vaulted structures is finally evaluated through a comparison with FEM outcomes and experimental results. For this purpose, a classical circular arch is herein chosen as benchmark, as considered by DeJong et al. [40], where the scaled geometry is defined by a thickness ratio t=Rcirc ¼ 0:15, an angle of embrace 2a ¼ 152 , and a radius Rcirc ¼ 20 cm . The discrete arch consists of 13 voussoirs, and is subjected to a constant ground acceleration. The discrete approach is therefore compared to a classical procedure based on the non-linear FEM, as implemented in the computer software STRAND7. It is well known from the literature as different modelling approaches with FEM ca be possibly applied to analyse masonry structures, including equivalent frame [46–48], equivalent material approach [49,50] and micromodelling [51,52]. An equivalent frame approach, also known as ‘‘macro-element approach’’ is herein used to study the in-plane behavior of the arch under the horizontal base motion. Based on this approach, the masonry is modelled as a homogeneous material achieving equivalent mechanical properties using homogenization techniques. The FEM arch is discretized by 60  5 plane-stress 9-node isoparametric elements in the x  y plane (see Fig. 28a). This level of refinement gives a sufficiently accurate approximation of the bending behavior of the sample during the quasi-static process. An elasto-plastic ‘‘tuff masonry model’’ is assigned to the body, with a density q ¼ 2000 kg=m3 , an elastic modulus E ¼ 1:1 GPa, a Poisson’s ratio m ¼ 0:2, a compressive strength f c ¼ 3:3 MPa, and a tensile strength f t ¼ 0:33 MPa (see De Luca et al. [23]). The model is subjected to vertical loads deriving from the self weight and to an horizontal load of increasing intensity, constantly distributed along the height of the arch. Fig. 28a shows the deformed configuration at the last load increment of the analysis, along with the horizontal stress contour superimposed. The stress state resulting from FEM analysis is adequately representative of the collapse mechanism, since it correctly identifies the zones of the arch where the hinges are most likely to occur. The maximum load capacity obtained via FEM is equal to 0.32, and results to be lower than the analytical or DEM-based numerical predictions (i.e. k ¼ 0:37 and k ¼ 0:38, respectively). For this reason, FEM results represent an upper bound limit for the strength capacity, in agreement with the findings of De Luca et al. [23]. A vector representation of the displacements is provided in Fig. 28b, which is clearly representative of a 4-hinge mechanism at collapse, as predicted with limit analysis and DEM (compare Fig. 28a and b with Fig. 28c and d). Despite the fact that the two approaches (i.e. FEM and DEM) are based on very different assumptions, the good agreement between numerical results provides useful indications about the seismic capacity of these importants structural elements. Numerical and analytical quasi-static predictions are finally compared with some experimental results available from the literature, see DeJong et al. [40]. In this experimental campaign some seismic shake table tests were performed by the authors on scaled prototypes of autoclaved aerated concrete, subjected to different earthquakes with varying frequency content and maximum amplitude (i.e. earthquakes occured in Parkfield 1966, El Centro 1940, Golden Gate 1957, Northridge 1994, and Helena 1935). As expected, only a qualitative comparison can be applied between quasi-static predictions and dynamic experimental outcomes because of the different impulse motion applied at the base of the arch. While a quasi-static analysis gives informations about the only minimum acceleration required to transform an arch into a mechanism, the actual dynamic strength of an arch is expected to be higher due to the sliding and/or rocking motions of the blocks. For the given geometry, indeed, the analytical or numerical collapse multiplier (i.e. k ¼ 0:37 or k ¼ 0:38) represents only a lower bound for the dynamic strength of the arch, since its value lies always below the experimental failure domain computed in [40] for the arch named as ‘‘Arch2’’, with magnitudes varying between k ¼ 0:4 and k ¼ 1:3. Regardless, a good matching between theoretical and experimental results can be once again obtained in terms of

R. Dimitri, F. Tornabene / Engineering Failure Analysis 52 (2015) 1–34

27

Fig. 26. Mechanisms of collapse of pointed portals with Rcirc ¼ 1, H=Rcirc ¼ 3; B=Rcirc ¼ 0:8; e=Rcirc ¼ 0:5, and t=Rcirc ¼ 0:2: the analytical (a–c–e) and numerical (b–d–f) configurations.

mechanisms of collapse where 2 hinges form at the support and the other ones throughout the span of the arch, as visible from a comparison between Fig. 28c–e. Based on the above considerations, it is quite evident as the possibility of comparing results of sophisticated numerical analyses or complex experimental investigations with simplified theoretical schemes as given by limit analysis, turns out to be very useful from a practical point of view.

28

R. Dimitri, F. Tornabene / Engineering Failure Analysis 52 (2015) 1–34

(a) 2α = 120°

(c) 2α = 150°

(e) 2α = 180°

(b) 2α = 120°

(d) 2α = 150°

(f) 2α = 180°

Fig. 27. Mechanisms of collapse of basket-handle portals with Rcirc ¼ 1; H=Rcirc ¼ 3; B=Rcirc ¼ 0:8; d=Rcirc ¼ 0:5, r=Rcirc ¼ 0:5, and t=Rcirc ¼ 0:2: the analytical (a–c–e) and numerical (b–d–f) configurations.

7. Conclusions This paper is devoted to the first-order seismic assessment of masonry arches and portals of different shapes, by applying the equivalent static analysis method. Based on this approach, a constant lateral acceleration is applied on the base of pointed and basket-handle arches and portals, and the critical collapse mechanism and the minimum lateral acceleration multiplier are analytically evaluated. For this reason some algorithms are developed for circular, pointed and basket-handle arches and portals, and iterative procedures, based on the kinematic approach of limit analysis, are performed. The predictions of the analytical model are compared with results of numerical modelling with the DEM. Results of the analysis

R. Dimitri, F. Tornabene / Engineering Failure Analysis 52 (2015) 1–34

B

29

C

D A

(a) FEM-Stress contour with possible hinge positions.

(c) Limit analysis.

(b) FEM-displacements.

(d) DEM.

(e) Experimental results [40]. Fig. 28. Quasi-static response for a circular arch: FEM (a and b) vs. limit analysis (c), DEM (d), and experimental results (e).

indicate that the geometry parameters have a pronounced influence on the stability of an arch or a portal. In particular the following main conclusions can be drawn:  Pointed arches with angle of embrace equal to 120° and 150° are always less stable than the circular ones, and their stability decreases as the e=Rcirc ratio (i.e. the degree of ‘‘pointedness’’) increases.  For an angle of embrace of 180°, thin pointed arches may be much more stable than their reference circular arches. For each thickness ratio, the collapse acceleration is maximum for a certain e=Rcirc (optimal ratio).  Basket-handle arches with angle of embrace equal to 120° and 150° are always more stable than the circular ones. Their stability in most cases increases as the d=Rcirc ratio increases.  For 2a ¼ 180 , thin basket-handle arches may be much more stable than their circular counterparts for moderate d=Rcirc values, but become less stable for large d=Rcirc values. For each given thickness ratio, the collapse acceleration reaches the maximum for a certain d=Rcirc ratio (optimal ratio).  The collapse lateral acceleration of a generally shaped arch increases as the thickness increases and the angle of embrace decreases.  At collapse, a generally shaped masonry arch requires two hinges to form at the intrados and two at the extrados, with one hinge forming always at the extrados side of the support.  Portals with pointed arches are in most cases more resistant to lateral accelerations than those with circular arches. This holds both when semi-global and local collapse mechanisms control.  As the angle of embrace increases, the local mechanism controls for a wider range of e=Rcirc values, especially for small thickness ratios.

30

R. Dimitri, F. Tornabene / Engineering Failure Analysis 52 (2015) 1–34

 Portals with basket-handle arches having 2a ¼ 120 and 150 are always less stable than their circular counterparts, and the stability decreases as the d=Rcirc ratio increases. The semi-global mechanism always controls.  For 2a ¼ 180 , the stability may become much larger than that of the portal with circular arch for thickness ratios between 0.12 and 0.14 and ‘‘moderate’’ d=Rcirc values.  Numerical results are in perfect agreement with predictions of the analytical model, and validate the efficiency of the DEM as numerical tool to evaluate the quasi-static behavior of unreinforced vaulted structures.

Appendix A The centre of mass for each portion of the basket-handle arch is computed as in the following. Case 1

ð1 Þ xAB

ð1Þ yAB

ð1 Þ xBC

ð1Þ yBC

ð1 Þ xCD

ð1Þ yCD

" #   3 3 4 be  bi sinððhB  hA Þ=2Þ hB þ hA þ b cos hmin ¼ cos 3 b2e  b2i ðhB  hA Þ 2 " #   3 3 4 be  bi sinððhB  hA Þ=2Þ hB þ hA þ b sin hmin ¼ sin 3 b2e  b2i ðhB  hA Þ 2

ð43Þ

" #    4 b3  b3 sinððh ðhmax  hB Þ  2 hmax þ hB 2 max  hB Þ=2Þ e i þ b cos hmin ¼ be  bi cos 2ABC 3 b2e  b2i ðhmax  hB Þ 2    4 R3  R3 sin ðd =2Þ dC  2 2hmax þ dC C e i  Re  R2i cos 3 R2e  R2i dC 2ABC 2 " #   3 3   ðhmax  hB Þ 2 4 be  bi sinððhmax  hB Þ=2Þ hmax þ hB 2 þ b sin hmin ¼ be  bi sin 2ABC 3 b2e  b2i ðhmax  hB Þ 2 " #    4 R3  R3 sin ðd =2Þ dC  2 2hmax þ dC C e i d Re  R2i sin þ 3 R2e  R2i dC 2ABC 2

ð44Þ

"  #  4 R3  R3 sinðð2c  d Þ=2Þ ð2c  dC Þ  2 2c þ dC þ 2hmax C 2 e i ¼ Re  Ri cos 2ACD 3 R2e  R2i ð2c  dC Þ 2 " #   3 3  4 b  b sinððh ðhmax  hD Þ  2 hmax þ hD 2 max  hD Þ=2Þ e i þ b cos hmin be  bi cos þ 2ACD 3 b2e  b2i ðhmax  hD Þ 2 " #    4 R3  R3 sinðð2c  d Þ=2Þ ð2c  dC Þ  2 2c þ dC þ 2hmax C 2 e i d ¼ Re  Ri sin 2ACD 3 R2e  R2i ð2c  dC Þ 2 " #    4 b3  b3 sinððh ðhmax  hD Þ  2 hmax þ hD 2 max  hD Þ=2Þ e i þ b sin h be  bi sin þ min 2ACD 3 b2e  b2i ðhmax  hD Þ 2

ð45Þ

Case 2

ð2Þ xAB

ð2 Þ

yAB

ð2Þ xBC

" 3 #   3 4 be  bi sinððhB  hA Þ=2Þ hB þ hA þ b cos hmin ¼ cos 3 b2  b2 ðhB  hA Þ 2 " 3e 3i #   4 be  bi sinððhB  hA Þ=2Þ hB þ hA þ b sin hmin ¼ sin 3 b2e  b2i ðhB  hA Þ 2 " #    4 b3  b3 sinððh ðhmax  hB Þ  2 hmax þ hB 2 max  hB Þ=2Þ e i þ b cos hmin ¼ be  bi cos 2ABC 3 b2  b2i ðhmax  hB Þ 2 " e3 #    4 b  b3 sinððh ðhmax  hC Þ  2 hmax þ hC 2 max  hC Þ=2Þ e i þ b cos hmin be  bi cos þ 2ABC 3 b2e  b2i ðhmax  hC Þ 2

ð46Þ

R. Dimitri, F. Tornabene / Engineering Failure Analysis 52 (2015) 1–34

ð2 Þ

yBC ¼

" #    4 b3  b3 sinððh ðhmax  hB Þ  2 hmax þ hB 2 max  hB Þ=2Þ e i þ b sin h be  bi sin min 2ABC 3 b2e  b2i ðhmax  hB Þ 2 " #    4 b3  b3 sinððh ðhmax  hC Þ  2 hmax þ hC 2 max  hC Þ=2Þ e i þ b sin hmin be  bi cos þ 2ABC 3 b2e  b2i ðhmax  hC Þ 2 " #   4 R3  R3 sinðcÞ e i sin ðhmax þ cÞ  d þ c R2e  R2i 3 R2e  R2i 2c

31

ð47Þ

"

xCD yCD

#   3 3 4 be  bi sinððhC  hD Þ=2Þ hC þ hD þ b cos hmin ¼ cos 3 b2e  b2i ðhC  hD Þ 2 " #   3 3 4 be  bi sinððhC  hD Þ=2Þ hC þ hD þ b sin h ¼ sin min 3 b2e  b2i ðhC  hD Þ 2

ð48Þ

" #    4 b3  b3 sinððh ðhmax  hA Þ  2 hmax þ hA max  hA Þ=2Þ 2 e i þ b cos hmin ¼ be  bi cos 2AAB 3 b2e  b2i ðhmax  hA Þ 2    4 R3  R3 sinðd =2Þ dB  2 2hmax þ dB B e i Re  R2i cos  3 R2e  R2i dB 2AAB 2 " 3 #   3   ðhmax  hA Þ 2 4 be  bi sinððhmax  hA Þ=2Þ hmax þ hA 2 þ b sin hmin ¼ be  bi sin 2AAB 3 b2e  b2i ðhmax  hA Þ 2 " #    4 R3  R3 sin d dB  2 2hmax þ dB B e i d Re  R2i sin þ 3 R2e  R2i dB 2AAB 2

ð49Þ

  4 R3e  R3i sinððdC  dB Þ=2Þ 2hmax þ dB þ dC cos 3 R2e  R2i ðdC  dB Þ 2   3 3 4 Re  Ri sinððdC  dB Þ=2Þ 2hmax þ dB þ dC d ¼ sin 2 2 3 Re  Ri ðdC  dB Þ 2

ð50Þ

"  #  4 R3  R3 sinðð2c  d Þ=2Þ ð2c  dC Þ  2 2c þ dC þ 2hmax C e i Re  R2i cos 2ACD 3 R2e  R2i ð2c  dC Þ 2 " 3 #   3   ðhmax  hD Þ 2 4 be  bi sinððhmax  hD Þ=2Þ hmax þ hD 2 þ b cos h be  bi cos þ min 2ACD 3 b2e  b2i ðhmax  hD Þ 2 " #    4 R3  R3 sinðð2c  d Þ=2Þ ð2c  dC Þ  2 2c þ dC þ 2hmax C e i d ¼ Re  R2i sin 2ACD 3 R2e  R2i ð2c  dC Þ 2 " #    4 b3  b3 sinððh ðhmax  hD Þ  2 hmax þ hD 2 max  hD Þ=2Þ e i þ b sin hmin be  bi sin þ 2ACD 3 b2e  b2i ðhmax  hD Þ 2

ð51Þ

" #    4 b3  b3 sinððh ðhmax  hA Þ  2 hmax þ hA max  hA Þ=2Þ 2 e i þ b cos hmin ¼ be  bi cos 2AAB 3 b2e  b2i ðhmax  hA Þ 2    4 R3  R3 sinðd =2Þ dB  2 2hmax þ dB B e i Re  R2i cos  3 R2e  R2i dB 2AAB 2 " 3 #   3   ðhmax  hA Þ 2 4 be  bi sinððhmax  hA Þ=2Þ hmax þ hA 2 þ b sin hmin ¼ be  bi sin 2AAB 3 b2e  b2i ðhmax  hA Þ 2 " #    4 R3  R3 sin d dB  2 2hmax þ dB B e i d Re  R2i sin þ 3 R2e  R2i dB 2AAB 2

ð52Þ

Case 3 ð3 Þ xAB

ð3Þ yAB

ð3 Þ

xBC ¼  ð3Þ

yBC

ð3 Þ

xCD ¼ 

ð3Þ

yCD

Case 4 ð4 Þ xAB

ð4Þ yAB

32

R. Dimitri, F. Tornabene / Engineering Failure Analysis 52 (2015) 1–34

ð4 Þ

" #    4 b3  b3 sinððh ðhmax  hB Þ  2 hmax þ hB 2 max  hB Þ=2Þ e i þ b cos h be  bi cos min 2ABC 3 b2  b2i ðhmax  hB Þ 2 " e #    4 b3  b3 sinððh ðhmax  hC Þ  2 hmax þ hC 2 max  hC Þ=2Þ e i þ b cos h be  bi cos þ min 2ABC 3 b2e  b2i ðhmax  hC Þ 2 " #    4 b3  b3 sinððh ðhmax  hB Þ  2 hmax þ hB max  hB Þ=2Þ 2 e i þ b sin hmin ¼ be  bi sin 2ABC 3 b2e  b2i ðhmax  hB Þ 2 " #    4 b3  b3 sinððh ðhmax  hC Þ  2 hmax þ hC 2 max  hC Þ=2Þ e i þ b sin hmin be  bi cos þ 2ABC 3 b2e  b2i ðhmax  hC Þ 2 " #   4 R3  R3 sinðcÞ e i sin ðhmax þ cÞ  d þ c R2e  R2i 3 R2e  R2i 2c

ð53Þ

" #   3 3 4 be  bi sinððhC  hD Þ=2Þ hC þ hD þ b cos h cos min 3 b2  b2i ðhC  hD Þ 2 " e3 #   3 4 be  bi sinððhC  hD Þ=2Þ hC þ hD þ b sin hmin ¼ sin 3 b2e  b2i ðhC  hD Þ 2

ð54Þ

"  # 4 R3e  R3i sinððdB  dA Þ=2Þ dB þ dA þ 2hmax ¼ cos 3 R2e  R2i ðdB  dA Þ 2 " #   4 R3e  R3i sinððdB  dA Þ=2Þ dB þ dA þ 2hmax d ¼ sin 3 R2e  R2i ðdB  dA Þ 2

ð55Þ

  4 R3e  R3i sinððdC  dB Þ=2Þ 2hmax þ dB þ dC cos 2 2 3 Re  Ri ðdC  dB Þ 2   3 3 4 Re  Ri sinððdC  dB Þ=2Þ 2hmax þ dB þ dC d ¼ sin 3 R2e  R2i ðdC  dB Þ 2

ð56Þ

"  #  4 R3  R3 sinðð2c  d Þ=2Þ ð2c  dC Þ  2 2c þ dC þ 2hmax C e i Re  R2i cos 2ACD 3 R2e  R2i ð2c  dC Þ 2 " 3 #   3   ðhmax  hD Þ 2 4 be  bi sinððhmax  hD Þ=2Þ hmax þ hD 2 þ b cos hmin be  bi cos þ 2ACD 3 b2e  b2i ðhmax  hD Þ 2 " #    4 R3  R3 sinðð2c  d Þ=2Þ ð2c  dC Þ  2 2c þ dC þ 2hmax C 2 e i d ¼ Re  Ri sin 2ACD 3 R2e  R2i ð2c  dC Þ 2 " #    4 b3  b3 sinððh ðhmax  hD Þ  2 hmax þ hD 2 max  hD Þ=2Þ e i þ b sin hmin þ be  bi sin 2ACD 3 b2e  b2i ðhmax  hD Þ 2

ð57Þ

"  # 4 R3e  R3i sinððdB  dA Þ=2Þ dB þ dA þ 2hmax ¼ cos 3 R2e  R2i ðdB  dA Þ 2 " #   3 3 4 Re  Ri sinððdB  dA Þ=2Þ dB þ dA þ 2hmax d ¼ sin 3 R2e  R2i ðdB  dA Þ 2

ð58Þ

  4 R3e  R3i sinððdC  dB Þ=2Þ 2hmax þ dB þ dC cos 3 R2e  R2i ðdC  dB Þ 2   3 3 4 Re  Ri sinððdC  dB Þ=2Þ 2hmax þ dB þ dC d ¼ sin 2 2 3 Re  Ri ðdC  dB Þ 2

ð59Þ

xBC ¼ 

ð4Þ

yBC

ð4 Þ

xCD ¼ ð4Þ yCD

Case 5 ð5 Þ xAB

ð5Þ yAB

ð5 Þ

xBC ¼  ð5Þ

yBC

ð5 Þ

xCD ¼ 

ð5Þ yCD

Case 6 ð6 Þ xAB

ð6Þ yAB

ð6 Þ

xBC ¼  ð6Þ

yBC

R. Dimitri, F. Tornabene / Engineering Failure Analysis 52 (2015) 1–34

"

ð6 Þ

#   3 3 4 be  bi sinððhC  hD Þ=2Þ hC þ hD þ b cos h cos min 3 b2  b2i ðhC  hD Þ 2 " e #   3 3 4 be  bi sinððhC  hD Þ=2Þ hC þ hD þ b sin hmin ¼ sin 3 b2e  b2i ðhC  hD Þ 2

33

xCD ¼ ð6Þ yCD

ð60Þ

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