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•• I:IStt.l.e\, st.ell.:t.l.ees.

Dr. P. T. Mikluehin, P. E., P. T. Mikluehin anti Assoe., Toronto, Ontario, Canada

Introduction

Latest progress in design and construction af modem masonry slructures and buildings requires, from a designer, an adequate knowledge of long·term effects of applied loads on the general performance of buildings. The evalualion of Ihese long·term effects must be based on the dala derived from general principies of engineering rheology.

Rheology, as applied lo slructures . is a study of problems connected with changes in stress distributions and defae­mations taking place in slruclures and buildings subjected lo the action of long·term loading.

The knowledge of rheological behaviour of slruclures in general 1,2 and masonry struclures in particular 3,4 is very important for the design and construction af buildings where masonry is combined with other materiais possessing different characteristics af creep. as in reinforced concrete ar structural steel frame-works, working together with the masonry. ar where masonry elements are built af units with different physical characteristics.

The phenomenon of differential creep must also be considered in struclures subjected to variations in the in· tensities of externai loads acting on various elements of the buildings.

Tests have shown that long-term deformation of masonry under the action of loads is accompanied by the change in magnitude of modulus of elaslicity 1,3. Changes in defor· malion characteristics pue lo sue h loading affect the rigidity of structures and must be taken into account in the design of struclures subjecled lO dynamic forces and in ali problems requiring the determination of the frequency of vibrations and other dynamic characteristics of the structure. These con­siderations are important in seismic design of masonry buildings.

68

Rheological Behaviour of r.:tasonry

Since masonry is a composite building material , each component influences, in its own way , the overall rheologici behaviour of rnasonry structures.

Masonry Units

As far as masonry units are concerncd, they possess various degrees of creep. Creep characleristics of these unils depend on the malerials used for Iheir fabricalion and the type 01 process utilized for their rnanufacture.

For inslance , creep of hard buml day units , for ali practicà purposes, is negligible. Soft brick units are subjecled lo I

higher degree of creep.

Mortars

Mortars possess a very considerable degree of creep Stabilization of creep deformations usually takes place 4 lo' hours after applicalion of load. After the slabilization period creep defo,mations are proporlional to applied slress provid<d lhe stresses do nol exceed 0.5 - 0.6 Rm.

Main source of creep in masonry is the mortaL Magnitudt of creep is, essentially , a function of the volurnetric content oi cernent paste in mortar joints5.

Creep increases with the increase in water-cement ratio li mortaL

Nonreinforced Masonry

Masonry, being a cornposite material , has definite rhe& logical properties. The most pronounced increase of cre<P deformations in masonry lakes place in 3 to 4 months ,ft~ the initial momenl of loading. Slabilizalion of creep def~ malions begins after 8 to 10 months. After one year, tol deformalions of masonry is approximalely 1.8 to 2.0 li": greater than instantaneous deformations at the moment

first load application.

Rheological Behaviour of Masonry Structures 69

Tests on masonry specimens to study the creep phe­nomena3 have shown that creep deformations take place during the whole process ofloading.

These tests have proven that there exists , for ali practical purposes, proportionality between creep and the applied stress. Such proportionality holds lrue for slress leveIs nol exceeding 0.5 to 0.6 of ultima te strength of masonry (Rm)' In Ihis case, we have a phenomenon of linear crcep.

For stress leveIs between 0.6 to 0.85 Rm proportionality ceases to exist, and the phenomenon of noo-linear creep takes place.

For stress leveis exceeding 0.85 R m crcep does not become smaller with the time but increases, leading to failure of masonry.

Among other factors influencing creep are relative ambient humidity af the surrounding mediuITI and temperature. Creep increases with a decrease in the relative humidity , and with ao increase in temperature.

Creep of masonry also depends on thickness of mortar joints and shape and size of the masonry units.

There is ceriain amount af decrease in crccp with ao increase in thickness of masonry elements.

Age of masonry at the time of loading also affects creep. With the increase of age , creep diminishes.

Recent rests cooducted 00 vibrated masonry3 show pronounced reduction af creep. Creep deformations of non­vibrated masonry are approximately twice as largc as the creep deformations of vibrated masonry.

If a masonry element consists of several masonry materiaIs with different creep characteristics then, in the course of time, stresses from the material with greater creep characteristics will be partially Iransferred to the malerial with lower creep characteristics.

It means that stresses in materiais possessing higher degree of creep will be reduced, whereas stresses in materiais with lower degree of creep will be increased.

Such stress transfer can resuit in overstressing of one of the masonry materiaIs. It is often the case with the walls covered with the exterior c1adding. In such cases special measures shall be undertaken to make sure that the c1adding would not lake part in resisting forces applied to such masonry walls.

Reinforced Masonry

Ali above rncntioned considerations are valid for pIain masonry. In reinforced masonry, lhe study of rheological behaviour indicates that the magnitude cf creep is consider­ably smaller as compared with plain masonry.

In reinforced masonry columns subjected to the action of axial loading, redistribution of normal stresses in longitudinal reinforcement and surrounding masonry takes place due to the phcnorncnon of crccp.

For instance, tests have shown that, after a period of I year, stress in stecl reinforcemcnt can be 1.6 to 1.7 times higher than ai lhe moment of load applicalion , whereas slress In masonry can be reduced to 0.6 - 0.65 of the original stress magnitude3. This fact must be considered during the design of reinforced masonry structures.

Relationship Between Stresses and Deformalions Under Sustained Loads

Case of Linear Creep

In arder to establish the relation between stresses and deformations for long-tenn loading, the following assumptions will be made:

1. masonry can be considered as uniform building material , 2. the relationship between creep deformation and stress is

Bnear, 3. the law cf superposition is valid for creep deformations.

To differentiate between two kinds of deformations taking place in masonry under the action of applied stresses , deformalions at lhe momenl of load application will be called instantaneous elastic deformations, and deformations develop­ing with the time will be called creep deformations.

The magnitude of lhe inslantaneous deformations depends on the age of masonry. Magnitude of creep deformations depends on both lhe age of masonry and the duration of loading.

If the age of masonry is T and the instant for which the deformation is to be found is t, then the tolal unit deformation for masonry element subjected to the action of an axial force at any time can be determined from the equation:

where:

li (t, T) li EL(t, T) + lI CR (t, T)

I li EL (I, T) = E(T)

is the instantaneous elastic unit deformation and,

lICR = F(t , T)

is the creep unit deformation.

(Ia-I)

If at the inslanl t = T, normal mess . a = a(T,) which changes with time then lhe lotaI relative deformation .(1) will be;

t aa(T) '(I)=a(TI)·lI(t,Tl)+ J --' lI(t,T)' dT

TI aT

or after lhe integralion by parts:

• (I) = a(l) E(t) -

t ó J a(T)-ó A(t,T)dT TI T

(10-2)

(I0-3)

70 Designing, Engineering, and Constructing with Masonry Products

From this equalion follows Ihal lolal relalive defonnation consists af instantaneous elastic relative deformation:

_ a(l) fEL - E(I)

and creep relative defonnation:

t 6 - f a(r)8t.(I,r)· dr

rI r

(10-4)

(10-5)

Relalive creep deformalion , Equalion 10-5 , is always positive, if a(r) is grealer Ihan zero.

Ihus:

(10-6)

If lhe normal slress a(r) due lo lhe aclion of long-Ierm loading is conslanl, a(r) = cons!. as il is lhe case for lhe majorily of buildings, Ihen lhe Equalion 10-2 becomes:

(10-7)

Equalion 10-7 can be presenled in lhe following form:

f (I, r) = (10-8)

where:

(10-9)

with:

Preceeding analysis holds true for axial stress conditions. Idenlical relalionships belween elaslic and creep deformations can be eslablished for shear slresses and shear deformations.

The knowledge of normal and shear slresses wilh corre­sponding deformations is importaot for analysis af various stress conditions. In the general case af three-dimensional stress distribution, a definite functional relationship between componenls of deformations and slresses can be established. This relalionship would fully describe slress-slrain condilion in lhe slruclure subjecled lo lhe aclion of exlernal forces, laking inlo accounl lhe influence of creep and change of modulus of instantaneous defonnations.

Case Df Non Linear Creep

The above menlioned considerations of lhe Iheory of cr", are valid for cases when stresses do nol exceed 0.5 R a~ consequenlly when linear relalionship belween creep der~. rnations and corresponding st resses holds true.

If a is grealer Ihan 0.5 R", , but does nol exceed 0.85 R, then nno-linear creep takes place. ]n this case, the relationshir belween slresses and deforrnalions can be described by t~ equation:

(lO-lO,

Here, as before, f (I ,r) is lolal relalive defom13lion of masonn aI lhe inslanl I, subjecled lo slress a(r) applied lo lhe age d rt,E(rl) is modulus of elaslicily, and Fo(l,r) is a functi" determined frorn experiments. This funcHoo expresses noDo linear relationship betwecn stress and creep deformations.

It can be shown Ihal lhe 10laI deformation f (I) can ~ expressed in lhe following form:

f (t) = fEL(I) + 'L.eR(I) [1 + a(I)] (10-111

where fEL (I) is inslanlaneous elaslic deformalion, fL.eR (t) is linear creep deformalion and a(1) is a funclion expressi" the influence af non-1inearity af relationship betwecn stressa and deformalions. When a(t) = O , Ihen Equalion lO-li desenbes the functianal relationship betwecn stresses arlC deformations for the case af linear creep. The relationship Equation 10-]] , is lhe general equalion of lhe Iheory 0/ nao-linear creep.

Design of Masonry Elemenls

Engineering design af masonry structures must be based (li!

the simultaneous consideration af strength, stability, a]'l(l defonnabilily crileria.

Design melhods based on elassical working slress analY'" only, do nol lake inlo accounl lhe redislribulion of in ler .. slresses due lo rheological phenomena laking place in struc' lures, caused by creep and shrinkage of masonry.

Sizes of many slruclural elemenls may be diclaled b) consideralion of Iheir deformabilily and nol by lhe magniludt

of slresses oblained from lhe elassical slress analysis. AnalY'" of lhe effecl of long-Ierm Ioads on slress dislribution! structural elements is af paramount importance for propet design of masonry slruclures.

Creep affecls lhe slrenglh and slabilily of ali slrucUlrJ elemenls of a building and a building ilself as a whole. T\US means that the design af modern masonry structures must tale

crcep into aceount in every element of the structure aJ1ll consider the overaU rheologicaI behaviour of lhe building in I~ lolalily.

Rheological Behaviourof Masonry Structures 71

It has been established that massive masonry elements, under the influence of long-term loading, within the limits of linear creep, show certain amount af increase in ultimate strength.

But if the stresses produced by long-term loads exceed the stress levei corresponding to linear creep, then the ultimate compressive strength of masonry will be reduced .

})esign for Strength and Stability

Tests have proven that , for ali stress leveis , slender masonry ,Iements (columns, walls) with slenderness ratios greater than 8, subjected to long-terrn loadiog, show a reductioo of the ultimate strength3 Therefore, aUowable loads on such ele­ments must be reduced.

Nonreioforced Masoory

Case of Coocentric Compressive Load. AUowable load cao be determined on the basis of the followiog formula:

where:

or

A R",

P d . I P=-C-+ PI . I

c

P d 1 -'- +P <;rp' A'" C 1.1 "m

c

total design load,

(lO-12)

(10-13)

(lO-14)

design compressive force due to dead load , coefficieot taking into consideratioo the effect of creep on streogth of ao element, design compressive force due to tive load , stress factor , taking ioto consideration the slender­ness ratio of the element6,7,8 cross'sectional arca af an element, basic allowable compressive stress.

Coefficient Cc depeods 00 the slenderness l /h ratio of an element). lt cao be determined from the following formula:

Cc = 0.021 (55 .7 - l /h)

where :

8 <; Cc <; 20

The above mentioned considerations are valid'r for the uniform distribution af compressive stresses across the cross section af masonry elements.

A series of tests cooducted 00 the eccentrically loaded masonry elements have shown that ir the compressive stresses do not exceed 0.5 Rm the hypotheses of plane sections can be considered as valid, for both instaotaneous and loog-term load applications.

Neutral axis changes its positioo with time but oot considerably; as a result , one can assume for alI practical design calculations that the position of oeutral axis remains unchaoged. Such assumption greatly simplifies design calcu­lations.

Case of ao Eccentric Axial Load. For small eccentricities allowable load can be determined from the formula:

(10-15)

where: '" - coefficient of stress reductioo , taking ioto consider­

ation the magnitude of the excentricity of applied compressive load 6 ,7,8

Other factors have the same meaning as in the previous case.

For larger excentricities some modification of the above mentioned formula shaU be made.

Case of an Eccentric Axial and Transversal Loads. In the case of a beoding forces acting together with the axialloads, desigo bending momeot can be detennined from the followiog formula:

here: Md.l = bending moment due to dead load, Ml.1 = momeot due to live load ,

(lO-16)

Cc ;:: coefficient taking into consideration the influence of creep.

Reinforced Masonry

Design of reinforced masonry is based on the following assumptions:

1. hypothesis of plaoe sectioos is applicable, 2. alI tensile forces are resisted by the reinforcement , 3. modulus of elasticity of reioforcemeot and masoory

remain constant. Theory of design of reinforced masoory elemeots should

take into consideration the influence of elastic and inelastic phenomena which are takiog place in these elements.

For iostaoce, for elemeots subjected to axialloads, allow­able load P can be determined from the following formula:

P <; rp . A(a . b R", + P . R,) (10-17)

72 Designing, Engineering, and Constructing with Masonry Products

where: l/J

A R", Rs p a b

cocfficient taking ioto account the influence of slenderness fatio, cross-sectiona1 arca of the element , ultima te strength of masonry, allowable stress in reinforcement , percentage of reinforcement , facto r of safety , coefficient taking into account the influence of crcep.

Eccentrically loaded reinforced masonry elements, dependo ing 00 lhe magnHude af eccentricity , can be designed according to formulas analogous to formulas for reinforced concrele elemenls. The same holds Irue for lhe design of reinforced masonry subjected to bending forces.

Shearing slresses and diagonal lension shall be checked as welL

Design of Masonry Slruclures on lhe Basis of Deformability Criteria

In addition to the design for the required degree of slrength and stabilily, masonry slructures musl be designed for the required degree of rigidity.

For ali elemenls of masonry structures and the structures themselves, deformations , due to applied loads, should not exceed certain limits. For instance, hjgh masonry wal1s , working logether wilh lhe slruclural frame of the building, should not be subjected to deformations which would resul! in cracking of Ihese walls .

If the deformalions exceed lhe allowable, eilher the rigidily of lhe building frame musl be increased or the slrength of masonry made grealer. The lalter can be achieved by using stronger masonry. by increasing its thickness , ar by intro­ducing proper reinforcement.

Deformations must be reduced to a minimum in reinforced masonry walls , beams, lintels and similar structural elements. Limited deformations are very important for such structures as reinforced masonry water tanks, reservoirs, water ftltration plants and lhe like.

Masonry waUs, carrying exterior cladding must also be checked for deformalions.

]n general , the evaluation of the linear and non-linear creep can be oblained fram Equations 10·8 and 10·11.

Without going into any delails , il can be shown that unil deformations € A' due lo axial forces should salisfy the following simple requirement:

P €A = - -- :s.;;; fALL.

Em' A (10·18)

Unit deformalion due lo lhe action of bending momenls should not exceed lhe allowable:

(10·19)

here: p

Em A M I h Y

is an axialload, modulus of elasticity for masonry I cross-sectionaJ arca of a masonry elemento bending moment, moment of inertja , thickness of masonry , distance from lhe cenler of gravity to the paim under consideration, allowable unit deformation.

In case of eccentrically applied load , unil deformation wi be:

(IQ.2Q)

where: :EM - sum ofmoments due to axial and transversal forces e - eccentricity of ao axialload, other factors are similar to the previous cases.

If lhe deformations due to Ihese forces exceed lbt allowable, an opening of the horizonlal joinls in the tensioo zone of a wall will lake place.

11 means that such a wall must be made thicker. reinforced in lhe zone where opening of the joinls could 1.\, place.

The design of reinforced masonry against excessive defOf· mations must be based on the considerations ana]ogous lo relevant design methods uscd in reinforced concrete structurtS

In preslressed masonry elements, alI deformalions musl ai>! be checked.

The study of deformability of masonry slructures, seen • the lighl of rheological behaviour of masonry , musl ~ intensified to solve new problems arising from the moderr developments in masonry construction.

Conclusion

lt is impossible, within lhe confines of lhis paper. I' present ali material research and theoretical investigatiOl devolcd to lhe rheological behaviour of masonry buildings aO: stroctures.

An altempl was made lo show the imporlance of co< sidering creep phenomena for the adequa te design and construction of modem masonry buildings and to indi"" certain quantitative and qualitalive appraaches lo the solul'" of prablems arising fram analysis of rheological behaviour d such structures.

Presen! slale of pragress in lhe area of engineering design; masonry warrants the inclusion of a series of recoOUOen tions, concerning lhe rheological aspects of masonry CO" slruclion, inlo Building Codes.

In conclusion, we would like to reiterate once more necessity of continuing research in the arca of behaviour of plain and reinforced masonry building

Rheologieal Behaviour of Masonry Slruetures 73

Such research will provide creative architects and engineers with ali necessary data for the design of new modern buildings ",d structures.

References

I. "Symposium on Creep of Concrete", ACI, Publication SP-9, 1964.

2. Arutyunyan, N., "Some Problems in the Theory ofCreep", Pergamon Press, New York, 1966.

3. Polyakov, S. V., and Falevich, B. N., "Masonry Structures", Moscow,1960.

4. "Methodes de Caleul des Murs", Etude Bibliographique, Cahiers du Centre Scientifique et Technique du Batiment, #68, Paris, 1964.

5. NevOle, A. M., "Role of Cement in the Creep of Mortar", ACI lournal, March, 1959.

6. Haller, P. "The Techno1ogical Properties of Brick Masonry in High Buildings", Trans1ation from Schweiz. Bauz. # 76, 1958, National Research Council of Canada, Ottawa, 1959.

7. "Recommended Building Code Requirements for Engi· neered Brick Masonry", SCPI, Washington, May, 1966.

8 . National Building Code, National Research Council, Ottawa, Canada, 1965.