Some explicit solutionsfor flat and depressed masonry arches

Concerning the analysis of the mechanical behaviour

of masonry arches, two main lines of theoreticalinvestigation can be distinguished: the first regards

arches as a system of rigid voussoirs subject tofriction and unilateral constraints and attempts toevaluate the system' s distance from collapseconditions. The second, following instead acontinuum mechanics approach and an «elastic» or«pseudo-elastic» logic, aims to determine theevolving stress and strain fields in the arch. Within

this framework, the greatest difficulty has beenmodelling the behaviour of very complex materials,

in particular, masonry. In fact, masonry, aheterogeneous, anisotropic material made up ofblocks and some binding matrix, is characterised bysufficiently high resistance to compression, but lowresistance to tension. The comprehensive constitutive

laws able to provide detailed mathematicaldescriptions of the behaviour of such materials oftenprove to be extremely complex.

Following the work of Signorini (1925b), whoperformed the first accurate studies of elastic

material s incapable of withstanding significant tensilestresses, so me authors have pro po sed a non-linearelastic constitutive relation (Di Pasquale 1982, DelPiero 1989, Angelillo 1993). According to this view,masonry arches can be studied as one-dimensional,

non-linear curved elastic beams, so that describingtheir mechanical behaviour becomes a matter of

solving non-linear ordinary differential equations. In

D. AitaR. BarsottiS. Bennati

fact, if the structure is isostatic, a simple integration

leads to determination of explicit expressions for thedisplacements and rotation of any given point on theline ofaxis. In this case, the line of thrust and thepossible arch regions of non-linear behaviour areknown. If the structure is statically indeterminate,however, the displacements and rotations must beexpressed as functions of the unknown redundantreactions, and then restraining conditions imposed,which 1eads to a non-linear algebraic system (Bennati

and Barsotti 1999; 2001).In the case of «simple» redundant structures, such

as Ilat arches or depressed arches with c1amped ends,the solution can be found explicitly: it is in factpossible to arrive at explicit expressions for both thedisplacements and the rotations. Moreover, theboundaries of the regions which behave non-linearly

can be determined analytically. For such vaultedmasonry structures, the evolution of the stress and

strain fields under increasing loads can be easilystudied. Collapse is reached when a kinematicmechanism emerges and the residual stiffness of the

arch vanishes.

A BRIEF HISTORICAL LOOK

As early as the 18'h century the mechanical behaviourof masonry arch structures attracted a great de al of

interest on the part of researchers, interest which

Proceedings of the First International Congress on Construction History, Madrid, 20th-24th January 2003, ed. S. Huerta, Madrid: I. Juan de Herrera, SEdHC, ETSAM, A. E. Benvenuto, COAM, F. Dragados, 2003.

172 D. Aita. R. Barsotti, S. Bennati

continues unabated today. A first line of reasoning,whose origins date back to some eighteenth-centuryresearch (La Hire 1712; La Hire 1730; Cou]omb1776), conceives of the arch as a system of rigid

blocks and focuses on the mechanisms of collapseand determination of the ultimate load. A secondapproach instead seeks to investigate the evolution of

the stress and strain fields with increasing appliedloads. The first researcher to de al with arches asdeformable continua was the renowned Frenchengineer Navier. In his 1826 Résumé des Leí'ons,

Navier sets forth a theory of arches based on the limitanalysis of a system of rigid blocks, after which he

adds the following observation:

Dans la réalité les voussoirs n'étant parfaitement dors, on

ne peut admettre que les pressions s'exercent ainsi contre

des aretes. Cela n' empeche pas que l' on ne puissecalculer, avec une exactitud e suffisante, I'équilibre des

vaútes d'apres les regles énancées précédemment : mais

il paraít nécessaire d'avoir égard 11 l'élasticité de la

matiere des voussairs. Cette questian serait un cas

particulier d'une questian plus générale, qui consiste 11

déterminer les effets qui se produisent dans un corps

élastique de figure quelconque, soumis 11 l'actian de

diverses farces (Navier 1826, 164).

Navier's idea of regarding masonry arches aselastic structures arose in an already fertile scientificcontext. One need only consider his text, Mémoire sur

les lois de l'équilibre el du mouvement des sofidesélasliques (Navier 1827), which offered an importantcontribution to defining the bases of the moderntheory of elasticity. At the same time, Cauchy was

hard at work studying e]astic bodies, resulting in hisvolumes, Exercices de Mathématiques (Cauchy1826), and Poisson on establishing his mathematicaltheory 01' elasticity, proposed in two fundamentalpapers (Poisson 1812; Poisson 1829), which were

shortly to be foIJowed by the ground-breaking works

ofLamé and Clapeyron (1833).After Navier, little by little the interpretative model

of masonry arches as hyperstatic elastic systems carneto be established. Studies based on the principies ofminima, whose approach was primarily abstract andtheoretical, alternated with work grounded onexperimental considerations, which offered rapid

methods for overcoming the static indeterminacyinherent in such problems. In the early 19'hcentury, F.J. Gerstner (1831, 405) introduced for the first time

the concept of the line of thrust, which was widelyutilised by subsequent authors. lt is a well-known fact

that for an arch in equilibrium under assigned loads,one can trace an infinite number 01'admissible lines ofthrust, of which only one is the «Irue» one. However,in his attempt to determine this true line of thrust,Gerstner introduced further hypotheses, which turnedout to be completely arbitrary.

Of the many authors that took up Gerstner' s ideas

and deve]oped them, those most worthy 01' note areMoseley (J 843) and Scheffler (J 857), whose

solutions were based on the principie of leastresistance, and Méry, who in his celebrated work(Méry 1840) put forth a simple method for

determining the value of the horizontal thrust at thekeystone under the hypothesis that tensile stresses are

absent in every section of the arch. We should alsorecall the contribution of Drouets (1865), whoproposed a «principie of minimum action», according

to which the stress state occurring within an arch isthat corresponding to the lowest value of themaximum compressive stresses. as well as those 01'ViIJarceau (J 854), Carvallo (1853), DenfertRochereau (J 859), Saint-Guilhem (J 859) and Dupuit(1870), all 01'whom added to our understanding of the

behaviour of masonry arches. However, in thisframework, one of the most important contributionscarne from Durand-Claye, who introduced themethod of «stability areas» (Durand-Claye 1867,66),by which it is possible to determine the set of

admissible lines of thrust via a graphicalreconstruction, while accounting for both the

structure's overall stability, and the limited resistanceof the masonry material.

A change in perspective ensued from the lines 01'reasoning foIJowed by Crotti, who maintained that theposition which the thrust lines will assume within the

arch can only be identified by considering the

material' s elastic properties, the mortar' s degree 01'stiffness, the varying thickness of the vaults andjoints, in short, all those factors that contribute to

defining the «internal constitution of the vaults»(Sinopoli and Foce 200 1). In fact, Crotti writes: «ogni

altro metodo e inevitabilmente fallace in quantola vera teoria della resistenza delle volte puaessere fondata solo sulla teoria del!' elasticitaopportunamente modificata e semplificata» [aIJ other

methods are inevitably unsound, in that the truetheory of the resistance of vaults can be based only

Some explicit soll1tions for tlat and depressed masonry arches 173

upon the theory of elasticity, suitably modified andsimplified] (Crolti 1875). He was moreover the

first to introduce clear equations for kinematiccompatibility and constitutive equations, thereby

succeeding in furnishing a rational solution to theassociated problem of static indeterminacy. It is notsurprising that Crolti's choice of methodology was

taken up again toward the end of the 19th century byEurope' s most i lIustrious structural engineers,including Italy's Alberto Castigliano (1879).

A well-known fact about masonry is that althoughit presents excellent resistance to compressive forces(of the order of 10 MPa if made out of bricks, whilc

stone offers much more), it has very limited tensileresistance (only a fraction of a MPa). Its mechanicalbehaviour, radically different under conditions of

tension and compression, renders the results obtainedvia linear elastic analysis highly questionable. Inchapter 6 of his treatise, Castigliano proposessearching for an approximate solution by using an

iterative procedure, during which process only thatpart of the structure that was under compression in

the preceding step is considered to be reactive. In thesecond part of the treatise, Castigliano reportsthe results of his analysis of two masonry archbridges conducted on the basis of the foregoing

considerations: the first is a brick bridge built over theOglio river, while the second bridge, designed by theengineer Mosca, was built of freestone over the Dorariver in Turin. Revelant to this context Perrodil(1872) proposed applying elasticity theory to the

study of masonry vaults after evaluating the results of

tests conducted on an arch of 38~metres' span. Someyears later, Perrodil would utilise the equations oflinear elasticity theory to analysc constant-thicknessmasonry vaults (Perrodil 1876)

It is however to Signorini that the credit goes fordirect-method studies of the mechanical behaviour ofelastic materials unable to bear tensile stresses. Afteraddressing various issues linked to the case of

combined compression and bending of reinforcedconcrete, in 1925 he presented two articles to theNational Academy of the «Lincei». In the first, heproves a theorem for the existence and uniqueness

of the solution for equilibrium problems in thepresence of low tensile-strength material s (Signorini1925a). In it, Signorini examines the case ofcombined comprcssion and bending of a straight

cylinder made of a homogeneous material with

only slight, uncertain resistance to tension and

characterized by non-linear elastic behaviour. In thesecond of the aforementioned articles, he tackles

the same problem assuming a piecewise-linearconstitutivc equation for any longitudinal fibre of thecylinder.

In the 1970's, Signorini's ideas began to be takenup again by other researchers, amongst which

Giovanni and Manfredi Romano. who in their studiesview masonry arches as one-dimensional elements,exhibiting non-linear e\astic behaviour. The authors

moreover propose a numerical procedure for arrivingat the solution, whose convergence they thendemonstrate (Romano 1979).

A NON-LINEAR ELASTlC MODEL

FOR MASONRY ARCHES

Following Signorini's idea, it seems reasonable to

assume non-linear elastic behaviour for any givenlongitlldinal arch fibre. Here, for the sake ofsimplicity, we have chosen a piecewise-linear

constitutive relation, as shown in Figure l.The simple elastic stress-strain constitutive relation

used here (Fig. 1) is a piecewise linear function whichreduces to the basic linear case for stresses comprisedwithin the threshold values CY,and CY¡'denoting the

(J

Ut

EE

t

(Je

Figure 1

The (J-E relation

174 D. Aita, R. Barsotti, S. Bennati

material' s resistance to compression and tension,respecti vely. Outside this range, strain grows at

constant stress. Consequently, denoting by E the

material's Young's moduJus, we have

E=¡~,

E ::; El

E <E<E, ,E2E ,

where eJ,= EE, and a; = EE,. This simple non-linearequation allows accounting for masonry's weaktensile strength and bounded compressive strength.One drawback is that the material is unrealisticanyassumed to be able to transmit low tensile stresses,even in presence of high strains.

Constituti ve equation (1) was adopted by two of

the authors in a previous work (Bennati and Barsotti,2001), and is an improved version of that used in

two other works by the same authors (Bennatiand Barsotti 1999; 2002), in which masonry'scompressive strength was assumed to be unbounded.

In the fonowing, as usuany done in the theory ofthe bending of beams, the cross-section of the arch, ofheight h and unit width, is assumed to remain plane

and normal to the longitudinal fibres after bending.Moreover, the normal stresses eJ, will depend on the

4)

Figure 2

The elastic dornain

(1)

corresponding strains E, according to the samerelation holding under a monoaxial state of stress.Under the foregoing hypotheses, the strain is linearover any given cross-section.

So, starting with relation (1), we can build the set of

corresponding non-linear constitutive equationsholding at the cross-sectional level between thekinematics parameters, axial strain Eaand curvature X,

and the resultant axial force N and bending moment M.As the stress-strain relation (1) maps an the strains

over a limited range of stress values, it is to beexpected that, at any given cross-section, admissiblevalues of N and M will belong to a bounded region ofthe (N, M) planeo Such region, i.e., the elastic domain,has been plotted in Figure 2.

The three Jinear parts which make up the stress-strain relation naturany induce a partitioning of theelastic domain in the (N, M) plane into six regions:when the point of co-ordinates (N, M), representingthe set of the internal forces at a given section,belongs to region E, then both the strain and the stress

are linear. If the point fans instead into regions B(or D), then the stress distribution is non-linear over

the compression (or tension) side. When the pointbeJongs to region C, the stress distribution is non-

linear over both the compression and tension sides.

M

h2(Ot-0,)/8

h2(Ot- °,)/12

F N

hOt

(point A){

N =ha,(3.1)

M=O

(region B) {N = ha, 1I -E(2E, - 2Eo + xW /8Xh~] (3.2)

M = E(2E,- 2Eo + Xh)2(E, - Eo- Xh)/24X2

(region C) {N = -h(a, + ~)/2+ (~- a)(2Eo - E, - E,)/2X (3.3)M = h2(a - a)/8 - (a - a)(E2 + EE + ~-3(E + E)E + 3~)/6X2{(' t e e el {

e 10 O

(region D){

N = ha, [1 + E(2E,- 2Eo+ xh?/8xha,] (3.4)M =E(2E, - 2Eo - Xh)2(Eo - E, - Xh)/24X2

(region E) {N = EhEo (3.5)M=-EXh'/12

(point f){

N =ha,(3.6)

M=ü

Some explicit solutions for /lat and depresscd masonry arches 175

Finally, when the point coincides with point A (orpoint F), then the stress corresponds to the limit case

where it is constant throughout the cross sectionand egual to a, (or a). As already observed, pointssituated outside the external boundary of region Ccannot be reached. The expressions for the curvesseparating the different regions of the elastic domain,

for M > °, are:

(curve a)6M + Nh - h2a,=°(curve/3)6M + Nh

-h2a =°,

(curve y)(N - h~)(h~ + 3h~ - 4N) + 6M(~ - ~) =°(curve O)(N - ha,)(4N - 3ha, - ha) - 6M(a, - a) = O

(curve qJ)(N - ha)(N - ha) - 2M(a, - a) = °

In each of the six regions, the strain of thebarycentric fibre Eo and curvature X are non-linearfunctions of the axial force and bending moment.After some algebraic manipulations we obtain, in thecase of M > °, the following constitutive relationsbetween the cross-sectional internal forces N and M,and the kinematics parameters Eoand X:

(2.1 )

Analogous relations, omitted here for the sake ofbrevity, hold in the case of M < O.

By using the one-dimensional non-linear eJastic

model described in the foregoing, the elasticeguiJibrium problem of the arch can be written in

terms of simple ordinary differential eguations.To this aim, let us indicate by s the curvilinear

abscissa along the line of the axis, and let u and vstand for the displacements of points along the axisline in the tangential and radial directions,respectively. Simple caJculations, omitted here for the

sake of brevity, show that under the foregoinghypotheses, and in the case the arch is circular with

radius R, the radial displacement v(e) is a solution tothe differential eguation

(2.2)

(2.3)v" + v= -R2X- RE (4)

(2.4) where e = siR, and the prime denotes differentiationwith respect to e. The integral of egn. (4) is

(2.5)v(e) =A coste) + B sin(e) +

+ fBsin(e- úJ)(-R2X(úJ) - RE(úJ»)dúJ

B,"

(5)

where A and B are constants. In turn,

u = -v + RqJ qJ= e - R f X(úJ)dúJ (6)and

where e is a constant (Belluzzi, 1934).

176 D. Aita, R. Barsotti, S. Bennati

In the entire elastic domain (with the exception ofthe ]imit points A and F) each of the relations (3) canbe inverted. So in regions B, e, o and E we canobtain explicit expressions for the kinematicsparameters Eoand X as functions of the internal axial

force N and bending moment M. When the arch isstatically determinate, the distribution for N(e) andM(e) are known, and the end conditions in terms of

disp]acement allow for determining constants A, B

and C. In the more genera] case of a staticallyindeterminate arch, instead, it is not known a priori inwhat zones the behaviour is Iinear and where it isnon-linear, and the set of non-linear governingequations (3), (5) and (6) must be so]ved numericallythrough an iterative procedure (Bennati and Barsotti,200]).

However, for simple geometric and Ioadingconditions, such as flat arches or depressed arches

with clamped s]iding ends, the so]ution can be foundexplicit]y: in fact, it is possib]e to write exp]icit

expressions for the disp]acements and rotations.Moreover, the boundaries of the regions whichbehave non-linearly can be determined analytically.

The case of a masonry Dat arch with cIampedsliding ends

Let us consider a masonry t1at arch AB, with clampeds]iding ends, loaded by a transverse load q. uniformly

distributed over its length l. and by an endcompressive force P, as shown in Figure 3.

J~A

-~

Jq

B~ P-~

J.p

e

Figure 3The flat arch

Let us keep the axial 10ad P fixed, and let the

distributed ]oad q increase from zero. For low valuesof q, the behaviour of the flat arch is linear. At anygiven section, the point corresponding to the axial

force (constant and equal to - P) and the bending

moment in the (N, M) plane belong to region E of thee]astic domain (Fig. 2).

If the abscissa x defines the distance of any givencross section from the end A, for increasing load q,

the distribution of stresses will at first be non-linear inthe two portions of the beam near the clamped ends A

and B, where O :S x :S x, and x1 :S x :S I (Fig. 4). In

these two parts, the axial force and the (negative)bending moment fa]] into region D or B of the elasticdomain, depending upon the va]ue of the axial force.

For sma]] axial loads, the non-linearity concerns thetensi]e stresses (region D), whi]e for high axialloads

it will concern the compressive ones (region B).

-M'''(N-M"(N'

-M '(N)

o x, 1

~!'(N)

M(x) M"(;\~M"'(,v

Figure 4

Bending moment diagram (the non-linearity concerns the

two portions of the beam near the clamped ends)

In the lateral part of the t1at arch, where the stressdistribution is non-linear, the expressions for therotation can be written as:

qy;,(x)= q>~(x) + o/¡' (O:Sx:Sx, and X1:SX:Sl) (7)

or

q>~(x)= q>h(X) + o/~ (O:sx:Sx, and x1:Sx:Sl) (8)

where o/;, and o/~ are constants determined byimposing the constraint conditions, while

q>1~(0)= O q>~(0) = O

q>¡,(x,) = q>/i,,(x,) or q>;;(x,) = q>/in(x,) (10)

q>/in (+) = O (~)=Oq>/in 2

Some explicit solutions for t1at and depressed masonry arches 177

1- 2x+

(2MA + Ph + CY,h2 + qlx - qx2)(8MA + 4Ph + 4CY,h2 + qf2)

4yq. (-8MA - 4Ph - 4CYJ¡2 -

qf2)'/

8 (P + CYh)'qr(x) = -. '.d 9 E

(9)

yq(l- 2x). arctan

V -8M" - 4Ph - 4CY,h2- qf2

1-2x+

(2M" - Ph + CY,h2 + qlx - qx2)(8MA - 4Ph - 4CY,h2 + qf2)

4yq. (-8MA + 4Ph + 4CYJ12- qf2)'!'

yq(l- 2x)

V -8MA + 4Ph + 4o;h2 - qf2

and q>/~(x) has an analogous expression, omitted herefor brevity. Of course, the choice of express ion (7) or(8) depends on the value of the normal force N (i.e.,

whether the point (N,M) of the elastic domain belongsrespectively to region D or region B).

Since the bending moment M(x) is a knownfunction of x, the abscissas x, and x2 (whichcorrespond to the limit negative value of the bendingmoment, denoted in figure 4 by -M', whose intensity

is given by (2.1) or (2.2)) can easily be expressed interms of both the applied loads and the redundant endrestraining couple MA.

As the structure is symmetric, the boundary andcontinuity conditions are:

depending on the value of the axial force N, whereq>/)x) is the expression for the rotation for

x, :<:: x :<::X2' calculated with the bending moment and

axial force belonging to the region E of the elasticdomain. The three equations (10) form a non-linearalgebraic system, whose solution allows determining

the constants qJ/in and qJ; (or qJ;;)and the redundantend restraining couple M

A'The non-linear equilibrium solution determined by

system (10) holds until the bending moment in the

middle of the beam reaches the border of region E ofthe elastic domain.

If we increase the load q, the non-lineardistribution of stresses will also involve the centralportion of the beam, that is for x2 :<::x :<::x3 (Fig. 5), in

which the normal force N and the positive bendingmoment M will once again belong to the regions D orB, depending on the value of the constant axial

force N.

The respective explicit expressions for the rotationare then:

q>¡;(x) = q>,7(x) + qJ; (Xl :<:: x :<:: x,) (11 )

or

q>;(x) = q>¡;(x) + qJ;; (X2 :<:: x :<:: X3) (12)

where qJ¡~and qJ; are unknown constants, while

+8 (P+CYh)'

q> (x) = -. '.d 9 E

. arctan

(13)

178 D. Aita, R. Barsotti, S. Bennati

and qJl~(X) has an analogous expression, omitted herefor brevity.

x, x,o

114(x)

Figure 5.

Bending moment diagram (the non-linearity concerns the

central partion and the two partions of the beam near the

clamped ends)

If we increase the load q, while maintaining P

constant, the non-Jinearity of the stress distribution

(-(U- U»)'�c

[

cp;(x) = -c , . arctan

2E V3q2

-M"'(}

-Af"(N

wilJ concern both the tensile stresses and thecompressive ones first in the two lateral portions andsubsequently also in the central part of the beam. In

these beam portions , that is for O :'SX :'SXI' x, :'SX :'S land X4 :'S x :'S Xs (Fig. 6), M wiJI belong to region e of

the elastic domain; it will therefore be bounded byvalues M"(N) :'S IMI :'S M"(N) (where M"(N) is givenby (2.3) or (2.4), depending on the value of the axial

force N, whereas M!!T(N) is given by (2.5». Therotations in these two regions are respectively:

-M '(N

AI'(N;

M"(NM"'(,\

cp¿(X) = cp,-(X) + if5¿ (O :'S X :'S XI and Xg:'S x :'S l)(14)

cp~(X) = cp;(x) + if5~ (X4 :'S X :'S xs) ( 15)

and

where if5c and if5Z are once again constantsdetermined by imposing the constraint conditions,while

V -3q(u, - u,)(l- 2x)

~

-6 (MA +ql

x- .!ix2 )(u-

u)- 3(P + uh)(P + uh)

2 2 e', !

and

(-(u - U»)'hrp+(x)= - '

,Loge

2E V3q

In al! the cases previously described, the solutioncan be found by imposing the restraining conditions(for example, that the rotation be nil at the clamped

ends when the intensity of the bending moment at Aand B is less than M!!T(N» and the continuity condition

at the contact points between the different zones

(16)

- V -3q( Uc- u,)(l - 2x) +

+2(17)( ql q

2)6 M +-x--x (u -u)+A 2 2

!,

-3(P + (J,h)(P + uch)

along the fIat arch where the constitutive behaviour is

different.Finally, it is also possible to determine the vertical

displacements by imposing that they be ni! at theclamped ends. For the sake of brevity, their expJicitexpressions have been omitted here.

Some exp1icit solutions for tlat and depressed masonry arches 179

M(x)M"(N)

M"'(Aj

-.'.4"'(.'1)-M"(Aj

-M '(N)

X4 Xs

M'(N)

Figure 6

Flat arch: bending moment diagram (the non-linearity ofthe

stress distribution concerns both the tensile and compressive

stresses in the two lateral portions and the centra1 par! of the

tlat arch)

The case of a depressed circular arch withclamped sliding ends

Let us now consider a depressed circular arch ACBwith clamped sliding ends A and B, whose span,radius and centre angle are, respectively, 1, R and 2a,

with smalll/R. We assume that this arch is subjectedto a vertical load q, uniformly distributed over itsspan 1, and a compressive concentrated load P at itsends, as shown in Figure 7.

q ! ! !p-- e

Figure 7Structural scheme of a depressed circular arch with clampedsliding ends

We choose the value of the bending moment Me atpoint C as the unknown redundant reaction. The

formal expression for the bending moment can be

written as a function of the unknown moment Me:

qR2M(8) = Me + PR(l - eos e) - - sin2 e.

2(18)

In order to find this unknown reaction, it isnecessary to impose the constraint conditions. Weimpose that the rotation qJ (8) be nil? at the two

sections A and C, respectively identified by e = a ande= O.

Since the values of the angle 8 are sufficientlysmall, the axial force can be considered constant andthe expression for the bending moment can besimpIified by developing (18) according to a Taylorseries up to the second order:

N(e) =-Pe2

M(e) =Me + (PR -qR2) -.

2(19)

When the axial force and bending moment belong

to the region E of the elastic domain, the solution islinear elastic and the bending moment at the keystoneIS

a - sin a qR2 a-

sin a cos aM =-PR +-e a 4 a

(20)

If we increase the load q, while leaving the axialforce -P constant, the distribution of stresses will benon-linear at first in the two portions of the arch nearthe clamped ends A and B, where the non-lineardistribution will concern the tensile stresses or thecompressive stresses, respectively depending onwhether N and M be long to the region D or B. Then,the central portion of the arch will also startexhibiting non-linear behaviour (with N and Mbelonging to the regions D or B). If the load q still

increases, while P is always constant, the non-linearity of the stress distribution will concern both

the tensile and compressive stresses in the two lateralportions and subsequently extend to the central part ofthe arch as weII. In these portions of the arch, thebending moment M and axial force N will beIong tothe region C of the elastic domain (Fig. 8).

With reference to Figure 8, the explicit expressions

for the rotation qJare:

qJ(~( e)=

qJ;( e) + if5~

for e4 <S e <S es (21)

180 D. Aita, R. Barsotti, S. Bennati

q>/~(()

= q>,~(() + (¡5~«() or q>Z(()

= q>Z(() + (¡5; (e)

for e, :S e :S e2 and, e7 :S e:s es (22)

o, o, o,

for e3 :S e:s e4 and, es :S e :S el> (22)

~~=~~+~~ill~~=~~+~~

depending on the value of the axial force, andq>¿(e) = q>,~(e) + (¡5c

for -a :S e:s e, and es :S e :S a (24) M(A)

In (21-24), (¡5/~' (¡5}¡, (¡5;, (¡5;, (¡5c' and (¡5Z areconstants to be determined by imposing the

restraining conditions, while, for example,

Figure 8Depressed arch: bending moment diagram near the callapseeonditian

e+

2(2Mc - Ph - ~h2)(2M, - Ph + PRfF- - qR2fP - (5,h2)

(

e V qR2 - PR)

arctanV -2Mc + Ph + (5¡h2

2 V qR2 - PR(-2Mc + Ph + ~h2)'

(25)8R(-P- (5h)'m+(e)= ''l'd 9E

+

e+

2(2M, + Ph - ~h2)(2Mc + Ph + PRfP - qR2fF- + ~h2)

(

e V qR2 - PRarctanh

V 2M, + Ph + (5/12

2 V qR2 - PR(-2M, + Ph + (5,h2)

(26)8R(-P- (5h)3m-( e) -

,'l'd - 9E

+

1

VR(cr -

(5)3qy+(e) = - - - '

,Log (2eV -3R(qR - P)( (5 -

(5) + 2VA(M(e),N(e»),2E 2(qR - P)

, , (27)

and

1

VR(cr (5)3

m-(e) = -- -,- ,

arctan (-2eV-3R(qR -P)«(5 -(5) / (2\1 B(M(e), N(e» ))

'1', 2E 3(qR - P) 1 , (28)

with

A(M(e), N(e») = 6M(e)«(5, -(5)

- 3(N(e)-

(5,h)(N(e) - (5,h)

and

B(M(e), N(e») = -6M«()«(5, -~) - 3(N(8) - (5,h)(N(8) - ~h)

Some explieit solutions for t1at and depressed masonry arehes 181

The other rotations have analogous expressions,omitted here for brevity.

The solution can be found by imposing conformityto the restraining conditions and the continuitycondition between different zones, each characterisedby a different constitutive relation. Once the value of

the redundant reaction Me is known, it is also possibleto determine the displacements u(B) and v(B) byimposing the proper end conditions. For the sake ofbrevity, their explicit expressions have been omitted

here.lt is worth noting that, in order to make it clear how

the tensile strength affects the solution, al! theexpressions for the rotations listed above have beenobtained in the general case of al different from zero.

For given values of the axial force P and uniformlydistributed load q, the equilibrium problem leads to anon-linear atgebraic system of equations in theintegration constants which can be solved in a semi-explicit way. By way of example, figures 9, 10, tIand 12 show some results obtained by setting al = 0,3

MPa; ac = 20 MPa; E = 7,000 Mpa; h = 100 cm;1= 1,000 cm; IX =0.1. 7.000

1.000

FLAT ARCHNon-linear elastic so/ution

P=

9850 [daN]

p--=-5000 rdaNl and P

'"14700_1da

-----

~~---, ,q [daN/cr

Figure 9

Flat arch: redundant reaetion MA VS uniformly distributed

load q for different values of the axial force N = -P

With reference to figure 9, it is interesting toobserve that the flat arch exhibits its maximum loadbearing capacity for P = -h(ac + a)/2 = 9850 daN.

As one can see, the rotations at the end sections A

and B and the vertical displacement at the middle

FLA T and DEPRESSEO ARel-

Non-línear eJastic soJutíon

~-'-'~-

-~ '._-

--------...--.-.---

Flatard

' ~

q[daN/a

Figure 10

A t1at and depressed areh with the same span and height:

redundant reactionM"

YSuniformly distributed load q

FLAT ARCH

Rotations

q [daN/cm] 1,2

p

-1000 [daN]

0,8

0,4

oo 0,02 0,06

~-A[rad

Figure 11

Flat arch: uniformly distributed load q YSrotations f{JAat the

ends

FLAT and DEPRESSED ARCHVerticaldisplacement$

q[daN/cm)

Flat arc~

Depressedarch

2V,[cmJ

Figure 12

A t1at and depressed arch wi th the same span and height:uniformly distributed load q vs vertical displacements ve atthe midpoint e

182 D. Aita, R. Barsotti, S. Bennati

section e diverge as load q approaches from below athreshold value, which can be determined by the limitanalysis. This corresponds to the development of acoJlapse mechanism characterised by the formation ofhinges at both ends and the midpoint. Such resultssuggest the advisability of defining a conventional

value of the collapse load as that in correspondence towhich the residual stiffness of the arch becomes lessthan some pre-set fraction of the initial value.

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