ABSTRACT.

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AN ALTERNATIVE APPROACH TO REDUCTION FACTORS FOR THE DESIGN OF SOLID MASONRY WALLS.

by

J. MORTON. THE BRICK DEVELOPMENT ASSOCIATION, U.K.

This paper considers an alternative approach to either British practice of EC6 for reduction factors. Developed from a quasi Rankine approach, a table of reduction factors is developed for straiqht plate walls. These are interchanqeable with, but not of the same exact value to those qiven in BS 5628 or EC6. Because the value of Younq's Modulus is contained with the expression, the proposed approach has the advantaqe of offerinq desiqners a method of evaluatinq existinq buildinqs an inteqral part of rehabilitation and restoration work and old stone masonry buildinqs too. The approach allows irreqular shaped members, such as hollow piers of differinq shapes, cellular or heavily piered walls etc., to be desiqned when the load is positioned within the kern of the section such that no tension is developed within the masonry. The approach can also be readily extended to cover cavity wall desiqn.

INTRODUCTION Solid masonry walls have been desiqned by a variety of methods for many years. The methods have proved successful in desiqn in as much that there have not been siqnificant collapses over the considerable period durinq which they have been in use.

The current British Standard approach, however, has some drawbacks, if not in practice, then from a theoretical point of view. The first major drawback is that neqative values of the desiqn load resistance of walls are qenerated at slenderness ratios only a few percent above the arbitrary maximum value. A second drawback is the amalqamation of the basic stability curves with the condition of local crushinq at the head of the wall. A further, more fundamental criticism, however, could be considered to be the very nature of the method employed. An approach modified from concrete technoloqy has been used which, althouqh not 'wronq' in concept does not permit the desiqner to 'feel' the similitude to a masonry modelo Similarly, the method enables only solid walls to be desiqned. It is not possible to desiqn a major hollow loadbearinq pier, or a solid pier or wall of irreqular planform, usinq the current British Standard approach. Because of the above and as a result of the need desiqn for desiqn quidance for vertical compression members of non sOlid, qeometric section, a new desiqn approach has been developed. (1) The approach is, here, further developed to propose stability curves for plain solid walls throuqh the full ranqe of eccentricities. The proposed approach can readily be extended to cover the desiqn of cavity walls.

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BASIC APPROACH For a wall of any section where the load is applied within the core of the section it has been shown (1) that a capacity reduction factor can be developed for masonry from the followinq equation:-

1 B· = •••••••••••• 1

n/T + l/fi

where B· is a capacity reduction factor.

where n = bendinq stress multiplier.

e y n = 1 + = 1 + 6em

where em is the eccentricity at the criticaI, mid hieqht of the wall.

T = bendinq stress enhancement factor: a value of 1.1 is proposed.

1f 2 fi = c (---

Le/ r

where c is qiven by E = c fk. c is normally taken as 900 or 1000 for masonry.

The application of B*, a reduction factor to account for both eccentricity and slenderness, follows the basic rule:-

fi x Area of Member x fk Pmax =

F.O.S.

In limit state terms this becomes:-

B A fk qamma f Pmax =

gamma m

= The Desiqn Load Resistance

The above equations form the present desiqn approach in BS5628 (2). fi* as defined in Equation 1 above, althouqh different in value from the reduction factor of the same name in BS5628, is nonetheless interchanqeable with it.

In the particular case of true concentric loadinq the Beta equation simplifies to:-

1 fi* = • • • • • • • • • •• 2

1 + l/fi

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In practical masonry design, true concentric loading is more of a concept than a reality.

The derivation and validation of the Equations 1 , 2 above are fully discussed elsewhere (1). They relate to 'no tension' loading where the load is positioned within the kern of the section.

The method has been section analysis of by establishing the positioned outside that:-

extended for high eccentricities and a cracked solid (and only solid) walls has been developed values of two stability parameters. For loads

the kern of the section it has been shown (4)

1 13* = ••• , ••••••• 3

n Arc/r +l/Eíl

This expression is valid for non uniform struts, such as is shown in Figure 1 and is defined by the following relationship:-

11 2 EI Pcrit =

= E PE where PE is the Euler criticaI load of the uniform uncracked strut.

The term Arc refers to the criticaI area ratio which is defined as:-

A Arc =

Ac

where A is the area of the uncracked strut and Ac is the area of the cracked section of the strut at the criticaI section, ie at the position of

cracked section

-L2~ L1--

figure. 1

loadbearing section of wall

maximum deflection. The net effect of these two stability parameters can be seen in Figure. 2. Equation 3 develops a similar shaped stability curve to that given by Equation 1 but gives somewhat lower values of 13*, as would be expected. The case shown is for a structural eccentricity at the head of the wall equal to t/3. A full description of the above analysis for cracked sections is given elsewhere (4).

Figure. 2 compares these two curves with the BS5628 (2) approach and .with test results. It can be seen that at slenderness ratios of up to about 15, the values of 8*, generated by Equation 3, are generally conservative. Above this slenderness ratio, however, the results lie below the dotted line. At high slenderness Equation 3 does not appear to predict safely the behaviour of highly eccentrically loaded walls.

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FINAL FORMULA It is therefore necessary to consider the work already progressed and to propose a method based on the above approach which "is a good prediction equation over the full range of practical slenderness ratios.

Consider two walls, both with high eccentricity, one of low slenderness, say H/t = 1 and one with high slenderness, say H/t = 50. (Figure. 3)

COMPAfUSON OF 886821, ANO CRACKED 8ECTlON REOUCTIOH , ACTORa WITH TEST RESUL T8

.71--f--t-+--t--t--t--+--t-- t--I

.3 • - - - - - - - ",. • ••• r--~+--+-+--+-+-'~~-1-"~'~'---~' ~---~~~

~ ~ 20 H ~ 35 ~O 4fi 60 Slendern ... Ratlo He/ t.

F lqul·C. 2

(3)

p .. p. trl illl!lu l ilr ,_-.:::I .. .. c t ... ng" l lO .. se _s I' Ll oe k '.-, - s tl"CS S "' I oek

(b) (c)

F ilJU1'c . J

KEY

~SECTtOH NeT JI(S($TlNC LO,,"

Iil$SUI1t:b STJI(S$ SlOC>< IH TOP Of NA U

It is realistic to assume that the squat member will behave as predicted by the cracked section analysis described by equation 3. The behaviour of the slender member must be questioned however because of the significance of deflected shape and the considerable instability of the strut system due to its extreme slenderness.

It can be argued that it is more appropriate, as a structural model, to assume, that the cracked section extends throughout the fuII height of the strut. Using this assumption, the question remains, 'What proportion of the member's thickness should remain uncracked?'

Two models commend themselves for further consideration.

1. The load resisting section is based on a rectangular stress distribution. Thus, a member can never give a lower load carrying capacity than that generated by assuming the reduced thickness of the section to be such that the load lies on the centre line of that thickness. (Figure. 3c)

2. The load resisting section is based on a triangular stress distribution. Then the thickness throughout the full height of the member can be taken as the thickness of the uncracked section at the head of the wall. In this case the load lies on the third point of the new (uncracked) thickness of the member. (Figure. 3b)

We can restate the above assumptions in different words. Assuming good workmanship, a wall loaded at high eccentricity should never fail at a load less than it would carry if the wall thickness, t, were assumed to be that shown in Figures. 3b or 3c.

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Examininq both these assumptions, for the particular case of e= t/3, the plots of the effects of the assumptions can be seen. In Fiqure. 4, the difference between the two 'lower bound' curves, can clearly be seen. Assumption 1 can be seen to be more onerous than Assumption 2 as would be expected. It is, however, more certa in

conceptually to be the lowest possible bound than is Assumption 2. Whichever is more appropriate, this approach appears to offer a better solution at hiqh slenderness. The earlier mentioned cracked section analysis is more appropriate for low slenderness.

The question of how to combine a suitable hiqh slenderness solution with the low slenderness approach still remains.

Assuminq for the present that

o õ :

COMPARISON OF 80l1D ANO CRACKED SEcnON REOUCTlON FACTOR8

Eco."tr lOlty - 113

1 KEY ~ J- SOU) SECTION Equ. 1 - - CRACKED aEenON Equ. S .8 •• • CRACKED SECTION 1 .. 0.61 -ooCRACKED SECTION t .. t/ 3

.,T'--,----,- --,-- ,--,-----r---.--,---;,---'i

.1t--I--+-+-+--I--+- + -+----jI----1

.8+----1---+--1--+--1--+--+-+---11----1

.6 -:--

.4 _.

.3 • •

figure. 4

the load carryinq capacity of slender members can be accurately predicted by Assumption 2 above, there should realistically be a curve joininq this lower bound assumption, at hiqh slenderness, to the cracked section analysis at low slenderness.

This can be achieved by an iterative method, which takes the form of a difference table where the difference is a linear function of slenderness ratio. An arbitrary decision is required concerninq the value of slenderness ratio at which the interaction curve joins the lower bound solution.

A more convenient and alternative approach is qiven by the equation:-

1 fi* = • • • • • • • • • •• 4

Equation 4, which provides the basic iteration in a simple and convenient formula, is the qeneral equation for stability for solid walls where the applied load acts at an eccentricity qreater than t/6. An additional check on local material failure at the head of the wall still, of course, also requires to be made.

Fiqure. 5 shows equation 4 assuminq trianqular stress distribution: the load therefore lies at the third point of the new thickness.

Fiqure. 6 shows the interaction curve for both assumptions l' 2 above, toqether with wall results. It can be seen that assuminq a thickness of wall where the load is applied at the third point of the new thickness of the section thus ensurinq that all the

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material in the section remains in compression - provides the more appropriate correlation to available test evidence. It is proposed on this basis that assumption 2 - triangular distribution be adopted in designo Figure. 6 also offers comparison with the current BS Beta values, which also incorporate local crushinq.A cut off leveI similar to that in BS5628 for local crushing would also apply to Equation 4 - although it is proper that this be left as a separate check.

COMPARISON DF SOLIO ANO CRACKEO SECTION F1EDUCTION FACTORS

Eccentrlclty - t/ 3

KEY • 9 ~ SOLIO SECnON Equ. 1 - .. CRACKEO SECT10N EQu. 3

• • CRACKED SECTION P o O.!ln - CRACKED SECTION P O. tn/3 . 1 - CRACKED SECTION Equ. "

.11---j--f---j---j--+--+--+---j--f---j

.• '}_--t---I--+--+---j--f--f--+--I---j

.• ,F;:;J:;::::"""'...d:,---t--i--t--i--t--i-...j ~"'" --.. - .. .....

.3 o , ,

35 40 46 60

figure. 5

~ li ~

~

I i u

COMPARISON DF 80lID ANO CRACKEO SECTION REDUCTK>N F ACTORS

Eccenlrlclty - tl3

KEY

+ --t--f---t-,::-. :-: .. ~~~~o SEcnON Squar. 8lock

f--t--f---t-,~-.. ~~~ ~g~~= Ttl.ng. Block

..

.• 9 T .. ,. pnlpln

.1

..

..

.. .3

.2 '. ..\ ....... . '- '- ', o

o 10 " 20 25 SO 36 40 45 60

figure. 6

The correlation is shown in greater detail in Figure. 7 where the test results have been shown for their respective end conditions. Pin/pin end conditions assume he = h: pin/fix conditions assume he = O.75h. Figure. 7 also compares the proposed method with the BS5628 values and with the proposed EC6 approach.

DESIGN PROPOSALS On the above basis it is proposed that:-

1 13* =

nArc/T + 1/L12n ••••••••••• 4

This general equation applies to highly eccentrically loaded walls where the load lies outside the core of the section. It applies to solid wall sections only. It can be further sub-divided for specific cases as follows:-1. For truly concentric loading

1 13* = ••••••••••• 2

1 + 1/n

2. For the eccentricities such that the load remains within the kern of the section, L1 = Arc = 1. and

1 13* = . • . • • . • • • •• 1

(1+3et)/1.1 + 1/n

where et is the eccentricity at the top of the wall.

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3. For e > t/6 AND e ~ 0.3t, Are = 1

1 s* = • • • • • • • • • •• 5

where ee is the eeeentrieity at the position of maximum defleetion.

4. For e > 0.3t 1 + 3ee = 2 at point of maximum defleetion

1 s* = • • • • • • • • • •• 6

DESIGN EFFECTIVE LENGTHS The evidenee of test results suggests that eonventional praetieal values for effeetive lengths should be retained, namely:-

pin/pin fix/fix

he = h he = 0.75h

pin/fix

COMPARISON OF THE PROPOSED METHOD to BS5628 AND EC6

he = 0.75h

It ' is appropriate to eonsider the eomparison between the above proposal and the design approaeh in BS 5628 (2) and the proposal eontained in the draft Euroeode 6 (3), as published for national eomment. These are eompared in Figures 7, 8 and 9 for eeeentrieity ratios of 0.33t, 0.20t and 0.45t respeetively.

1 "

COMPAAISON OF SOLID ANO CRACKED BECTlON REDUCTION FACTORS WITH ECI5, as !i1528 lo TESl RESULTa.

Ecc."trlclty - t/3

1 .. TNI~:n~~':·:I=d~. .. J- as !ill2a K:~ CRACKED SECTlON" Equ. .-••• E C e 111 TESr RESIA. T - phlflx

.1 ~ TESl RESUL T - phlpln

.7t--t----;f__t--j--+--t--t-~f__t-__j

. 6+---"h--jf--t--t---t---t--t---jf--t---I

.51-___ :l--jf__tI.llL '"-j--+--t--t--jf__t-__j

.4i==I=~~V::j:::::;j-T-t----t-T-t-----j

.3+---t---jf--'-f ..... -4I .... ::?--:-I---t--t---jf--t---I ~~-+--+--+--+:~~"~'~,~~+--+--+--4

'I"" ~ " -

Figure. 7

li ~

! i !

1

COMPARISON DF SOLID ANO CRACKED 8ECTION REDUCTION FACTORS WITH ECIS. as 15828 , TESl RESUL T8.

Eccenttlclty - Q.201

.• ~:-BS .'" • • • E C 8 A

KEY - .. CRACKEO SEcnQN Equ. •

.7 ..

.. . 4

.3 '\ " "

"" " ~

\

1. 20 25 3D 315 4. 4' SlencMtnl •• R.tlo Ht;'tI

Figure. B

5.

It ean be seen that EC6 is over optimistie in the slenderness ratio range 10/15 to 25/30. It is eonservative to both BS 5628 and the proposed approaeh at values of slenderness of less than 10/15. EC6 is more generous than BS 5628 at slenderness ratios greater than 20/30, but both EC6 and BS 5628 still produee negative values for the eapaeity reduetion factor for walls for whieh positive wall load test results existo Figure. 10 shows the Beta values whieh are proposed from this work for the full range of eeeentrieities.

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A comparison with wall results has already been given for the case e = t/3 (Figure. 7).

Figure. 11 repeats this comparison for e = t/6. Effective lengths of L and 0.75 x L have been used for pin/pin and pin fixed end conditions respectively. The correlation between theory and practice is good.

. 7.3--+-+-I,--+-+--+-+-j-+-I

.8.3--+-+-If-+-+--t-+-j-+-I J~--~-+--~--+---~-f---r--t---r--l

.4.3---I--+--+--I--+--+--j--+--+---I

.3-h--+-+-If-+-+--+-+-j-+-I ~}-~~4-~f--t-+--t-i-~--r-l

REOUCTION F AClOAS BA8ED OH f,AORTON SOlUTIOH DF

THE RANKINE EQUATION • Tl'IIct ....... _d .. "'.

1 • "'-n ,.tr ... ~

:S~ . ECCENTRICITIES .9 - .-0.015 Eqn. 1 - .-0.10 Eqn. 1 - .-0.15 Eqn. 1

.... • -0.20 Eqn. 6 ..... -0..25 Eqn. 15 ..... -0.30 Eqn. 5 .S •• • • -0.315 EQn. 8' ••• • ·(UQ Eqn. e· .. .• -0.45 Eqn. 8'

;l--r- ........

.5 - -"::"~

,. ~ 20 H SO 35 40 415 50 Slandern ... Rat lo Halle

.. ~

.. Ü

.7 : j .. I J .

.4

;;

figure. 9

COMPAAISON OF aouo REOUCTlON FAClORS WlTH EC8, 88 8021 , TESl RE8Ul n .

Eccanllldt)' .. 1/0 • ... -'lO ..... 0.71. H • 'c.IIIgn"~'t:IoM.

1 " . ..

..

.7

.. . 5

Figure. 10

COMPARISON OF aouo REDUCTlON FACTORS WITH Eca, BS 6821 & TEST RESUL TS.

Eccenulclty - 0.05t • Ece .. Dr.1I •

1.10 TON : -110O:VI.- t1

i"-.. I , .. "~

r'\" I

'\ , i " . \. .. I

\. , , I ~

.3 KEY" " --- ",," _ 1

.4

.3

~

, ' .. ) " ~

- BS 6828 .... SOU) SECTlON EQu. I F i oI ••• i C e EB TESl RESUl.T· plnlflx •• ~. I o Q TEsr RE8UL T - pln/pln

o w ~ ~ 25 SO 35 40 45 60 8Iendarn ... AaUo He l l.

Fiqure. 11

• •

-B86e28 - - E C 11

,. KEY

F ~ ••• SQlI) BECTIOH Equ. t

2. .. •• 3. 4 • .. 5 •

Figure. 12

Figure. 12 shows the correlation with EC6 which can be achieved by modifying the constants in the proposed approach. The value of Youngs Modulus has been increased in the proposed approach from 900 sk to 1100 fk. Accepting that the range of practical application in design stops at a slenderness ratio of about 30 and accepting that EC6 has a cut off at low slenderness to account for local crushing of the head of the wall, the similitude verges on the exact.

At higher eccentricities, correlation is less exact: the length of the flat portion of the EC6 curve requires to be shortened and the slope requires adjustment to make it shallower. This comment is applicable to many of the graphs and is a conclusion of this work.

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BENDING STRESS MULTIPLIER The approach assumes a value of T = 1.1: this is an important aspecto A conclusion of this work is that resource should be applied to comparing the criticaI strains associated with masonry bending failures to those applying to direct compression failures.

CONCLUSIONS A new method is proposed for the design of normal solid masonry walls which takes account of masonry's propensity to fail by material overstressing at low slenderness, by instability at high slenderness and by an interaction of these two modes at intermediate values of slenderness. The method offers a lower bound approach and is based on a no-tension modele At high eccentricity, it takes account of cracked sections for both material overstressing and instability. The method is seen to be validated by well respected wall test data. The approach has the advantage that, by using one table of Beta values, any section - solid or hollow - whose geometry is defined can be designed if the load transfer at the top of the member is detailed in such a way as to ensure the load lies within the kern of the section, thus avoiding the development of tensile stresses anywhere within the masonry. The method can also adequately deal with the concept of cavity walls.

The comparison between the proposed values given in EC6 is encourag~ng. The proposed method is simple and easy to apply both in desk top work and by computer. It also facilitates the design of different forms of masonry since a no tension model is used and the value of Young's modulus is an explicit parameter which can be altered in the basic equation.

Some consideration will require to be given to redefining the relationship between the characteristic compressive strength of masonry and the strength of the structural unit for the different common mortar designations.

ACKNOWLEDGEMENTS The development of the basic approach on which this paper is based was supported by the Building Research Establishment. The author acknowledges this contribution particularly the kind help and encouragement of Dr R C de Vekey. The paper is published by kind permission of the Director General, The Brick Development Association.

REFERENCES 1. Morton J. The Design of Masonry compression Members

Proc. of 4th Internation Masonry Conference. London October 1989.

2. BS5628: Part 1: 1978 - Use of Masonry: Unreinforced Masonry. The British Standards Institution, London 1978

3. Eurocode No 6. Common Unified Rules for Masonry Structures. Report EUR 9888 EN. Commission of The European Communities. Brussels 1988.

4. Morton J and Horne M R. The Development of stability Parameters for the Cracked section Analysis of Solid Walls at High Eccentricity. Masonry International. (in print)