AN ANAL YTICAL APPROACH FOR W ALLS SUBJECTED TO STATIC AND DYNAMIC OUT-OF-PLANE POINT LOADS

Brian Hobbs l , Michael Ting2 and Matthew Gilbert3

I . ABSTRACT

Previous research on the behaviour of masoruy walls under out-of-plane lateral loading has concentrated on distributed pressure loadings, mainly related to wind and sometimes to earth pressures or blast loading. In some circumstances walls may be subjected to significant out-of-plane concentrated loading Perhaps the most common incidence of such loading is that arising from accidental vehicIe impact on bridge parapets or other masoruy walls adjacent to highways Whilst there has been a significant amount of research into vehicIe impact on metal and reinforced concrete barriers, little attention appears to have been given to masoruy walls. This paper describes the development of an analytical approach for out-of-plane concentrated loading. Initially the analytical development is concemed with static loads and a range of possible failure modes are considered. On the basis of a parametric study the likely criticai mode of failure is identified. The analysis is then extended to incIude transient dynarnic loading of the type associated with vehicIe impacts.

2. INTRODUCTION

This paper is concerned with the use of simple analytical tools to analyse masoruy walls

Keywords: Impact; Masoruy; Walls.

ISenior Lecturer, Department of Civil and Structural Engineering, University of Sheffield, PO Box 600, Mappin Street, Sheffield, SI 4DU, UK

2Former postgraduate student, Department of Civil and Structural Engineering, University of Sheffield

3Research Associate, Department of Civil and Structural Engineering, University of Sheffield.

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subjected to out-of-plane static and dynamic point loads. A considerable amount of attention has been given to the behaviour of masonry walls subjected to out-of-plane distributed loads resulting from wind or blast loading [1,2], but point loads seem to have been largely overlooked.

In the U.K there is no approved method of assessing the capability of a masonry wall to resist velúcle impacts. As a consequence unreinforced masonry parapet walls cannot, for example, be constructed on new masonry arch bridges. However, in the case of these latter structures provision of RC, stee\ or reinforced masonry parapets is very expensive, the former also having the disadvantage of being unsightly. Recent suggestions that parapets on existing UK bridge structures should also conform to approved standards has created fears that ali existing masonry parapets will have to be strengthened or replaced unless their ability to sustain impact loadings is proven.

In contrast to the majority of walls in buildings, masonry parapet walls are often subjected to harsh environrnental conditions - from ali sides. For tlús reason, and because many existing walls were built using weak cement or lime mortars, emphasis is given to the assessment of walls containing masonry with poor tensile, flexural and shear-bond characteristics.

Tlús paper initially considers a number of possible failure mechanisms for free-standing cantilever walls subjected to static point loads. The type of failure mechanism likely to be most criticaI in practice is examined in further detail and procedures for carrying out dynamic analyses are developed.

3. INVESTIGATION OF POSSIBLE FAILURE MODES

A number of possible failure modes have been identified. These are overturning, shear failure and horizontal arclúng failure; each mode is shown in fig. 1. In each case the following notation has been used:

I .. = wall length p = density of masonry

( .. = wall tlúckness

h .. = wall height

3. 1 Overturning

J1. = coefficient of friction

c = masonry shear - bond strength

The resistance of a cracked section of wall, loaded by a point load Pg, at height Yp from the base of the wall (the wall rotates about a Iúnge at height Yfb; any cruslúng of the masonry at the Iúnge point will be negligible given the limited height ofmost parapets).

p = pl .. t;(h"-Yfb) (1) g 2(yp - Yfb)

Alternatively, the resistance Pl of a similarly loaded uncracked wall can be written as:

P =1 ( 2 (fio: +p(h .. -Yfb ») (2) I .... 6(y - )

p Yfb

330

~1

lW

L p

YfS tçi}P (a) Overturning failure (b) Shcar failure

rw !w x

~I-FRONT ELEVATION SIDE ELEVATION

Cc) Punching shear failure

---------1; --------~~ _________ I~f ________ . ______ ~v ~on ~ 1

/lk 11{2 l 11{2 J ~ '1 bing' Iino,

,~~i~ ERaNT ELEVATION

(d) Triangular arching failure

TI J II{2 ?r 11/2 _ . J

~t: !~~ ~--7('---7 / // +yf L.

EROM ElEVATION

(e) Rectangular arching failure

Fig. I Failure modes

331

Section) - 1

Secllon 1 - 1

-wherefkx is the flexural strength ofthe masoruy. Whilst an overtuming failure may occur in the case of short walls, unit interlock will be likely to prevent overtuming of a short section of a long wall (although in the case of modem walls built with movement joints overtuming would probably be possible betweenjoints) .

3.2 Shear failure

A global shear failure of a wall will be considered initially, comprising a large section of the wall shearing along a plane at depth Yft from the top of the wall.

P =c/w tw + J.l1w t", p(hw - Yft ) (3)

Once again this failure mode is most likeiy to occur in the case of short walls. However, a more localised "punching shear failure" could conceivably occur in a longer wall . The force required to "punch out" a small section (of height ~ width Ir) of a wall will be at least:

(4)

In fact unit interlock will cause the actual resistance to be somewhat greater than this. However, such failures may not be cornrnon in practice; the section ofwall initially being "punched out" may physically jam, causing an arching or other type offailure to occur.

3.3 Horizontal arching failures

The ability of masoruy walls to resist applied lateral loading by horizontal arching action has been studied by a number of workers, inc1uding McDowell et a1.[3), and Anderson[4) However, work has tended to concentrate on the resistance of masoruy wall panels in framed buildings to distributed applied loading, rather than on the resistance of continuous masoruy structures (such as masoruy parapet walls) to point loads.

3.3 .1 Triangular

The triangular horizontal arching failure pattem which will be considered has a similar geometry to typical RC-slab yield line failure pattems. In fact somewhat inappropriately named "yield line" analyses are now routineiy used to assess the lateral resistance of masoruy wall panels under wind loading (e.g. BS5628 PU). However, whilst these analyses rely on the existence of some masoruy tensile bond strength, here it will be assumed that the masoruy possesses zero tensile strength, and that all resistance is attained via horizontal arching action. Initially a simple arching model has been used, similar to that put forward by Hodgkinson[5) . The model effectively assumes that three hinges form at failure, the thrust in the regions of these hinges actually being distributed over an arbitrary distance of f3 x tw , as shown on figo 2. Thus, assuming the existence of a rectangular stress block, the horizontal thrust can be ca1culated provided a value for k, the compressive strength ofthe masoruy, is available. The elasticity ofthe material is

not inc1uded in the simplified model.

Thus, using these assumptions, the resistance of the wall to the application of the point load, Pres ' can be derived to produce the expression given in (5).

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30302 Rectangular

~ -.- --:--:-

jl --.­_o _ -- -'

tw

Figo 2 Horizontal arching action - simplified model

(5)

tw

Alternatively, a simple rectangular horizontal arching failure pattern can be considered. In this case the length of the failure zone will be constant (at 9 down to a depth of YI .

4k t:P(1- fJ)(h.,., - Y I) ~= ~

li A rectangular pattern can only occur once any bonding between the units and mortar near the base of the wall has broken. However even if some bonding is present the strength of the wall resulting from arching action and that required to overcome the initial resistance to shearing are not likely to be additive (given the brittle nature of the unit-mortar bond) . Thus, unless the initial shearing resistance exceeds the arching resistance, the former can justifiably be neglected in assessing the ultimate failure loado

3.4 Parametric study

Figure 3 shows the predicted intluence of the wall failure length for ali modes of failure except that of punching shear failure . The punching shear failure mode is likely to be highly dependent upon the loaded area, a parameter which is not considered here.

Z e.1 ., " c:

'" ;; ';;;

~

o

h. (mm)

'. (mm)

If mm)

P

2 4

1000

330

0.1

-- Flexural --- Shear

---- RecI-arch

6 8 10 12 14 16 18 20 Wall failure length, ~ (m)

E. (kN/nun') 18 t;.,,(N/rrun') 0.35

f,,(N/mm') 20 c(N/mm') 0.15

p{kglm') 2400 I-' 0.64

Y,(mm) 500 Yfmm) o

Fig. 3 Parametric study - wall resistance vs. wall failure length

333

Clearly the likelihood of a given failure mo de occurring in practice will depend to a large extent on the geometry of the wall under consideration, as well as the material properties. Of the arching failure pattems considered it is clear that the rectangular mode is likely to be most criticai, at least when the shearing resistance of the base of the wall is small . Additionally photographs of actual vehicle impacts on masonry walls appear to indicate that (rectangular) horizontal arching failures are common, at least in the case of "Iong" walls. This mo de offailure is therefore studied in more detail.

4. RECTANGULARARCHINGFAll.URE

The resistance of any fiat arch to applied loading is highly dependent upon the elasticity of the constituent material, and, very importantly, upon that of the abutments. In the case of a long masonry wall it is clear that the "abutments" when horizontal arching is taking place are actually sections of wall themselves. Flat arches exert very high horizontal thrusts; therefore the overalllength of the wall will be of some consequence.

A more detailed but nevertheless sim pie iterative analytical procedure which allows the effects of displacements on the out-of-plane resistance of a wall to be determined is now described. The philosophy behind the method is simply that as a very small outward displacement of the section of wall under consideration would cause the formation of three hinges, these can be assumed to be present initially. Now, suppose a given wall, as shown on figo 4, is displaced incrementally by a small amount dincr.

lo dh, Ir lo

p

Fig. 4 Revised horizontal arching model

After a given increment of displacement has been applied, the total out of plane displacement, dou" can be written as d

OU1 = d

OU1 + dmcr ' The diagonal shortening of I d can

therefore be written as

=::.> d l =-J(t. -douY +(lrf2+dJ +ld

Thus the horizontal thrust F can be calculated iTom:

[d (lf / 2 +dJ +d 1

I (ld - di) h F = E,. A,. ( )

Irf2+dh

(7)

(8)

- where E,. is the modulus of elasticity of the masonry, and Aw is the cross sectional area of the wall (hf x tJ. By resolving forces in the appropriate directions the out-of-plane resistance, Pres , can be calculated - refer to equation (9).

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( I -d ) P =2F e oul

'" ( 11 / 2 + dJ (9)

Now, when the horizontal thrust F calculated at a given displacement exceeds that calculated previously, the crushed depth of masonry, dCro'h , should be increased accordingly:

d =~ cro,h r h lkc f

(lO)

Thus the etfective thickness ofthe wall le becomes le = Iw -dcro'h ' and ld =~t; +(///2)2 .

Now, ifthe "abutments" are not rigid, the length ofwall required to carry the horizontal thrust Fis :

981J..l ( lwhf P) I = Q

F (11)

Unless the wall is located between fixed "abutments" the elasticity of the masonry beyond the arching zone will cause the "abutments" to displace by an amount dh , where dh can be calculated from :

d - ---.!L h - 2 E,.. A,..

(12)

The above procedure should be repeated (using a fairly large number of small increments of displacement) to obtain a resistance vs. out-of-plane displacement response (e.g . P", vs. dou,). The procedure can be repeated until P,,, ~ O .

The fiictional resistance at the base of the sections of wall arching outwards has been ignored - this is a conservative assumption. Conversely, the assumption (a) that the masonry yields in a ductile manner, (b) that ali shortening of the masonry in the wall takes place in a horizontal direction and (c) that the forces in the regions of the hinges are distributed at an angle of 90° are ali non-conservative.

Figures 5(a) and 5(b) show the predicted influence ofwall failure length on the resistance of a given wall when the "abutments" are fixed and allowed to deform respectively. Figures 6(a) and 6(b) show comparable plots indicating the predicted influence of the compressive strength ofthe masonry. Finally, figs . 7(a) and 7(b) show comparable plots indicating the predicted influence ofthe elasticity ofthe masonry (11 taken as 0.64) .

The figures indicate that the "abutment" conditions are of great importance Other factors which are likely to greatly atfect the ability of a masonry wall to resist static out­of-plane loads by arching action include the presence of vertical cracks and/or unfilled (or partially filled) vertical mortar joints; these are common in practice.

5. DYNAMIC ANALYSIS OFWALLS

Emphasis has so far been placed on calculation of the static resistance of masonry walls subjected to out-of-plane loading However, as outlined previously, the dynamic case must be considered in order that the resistance of walls to vehicle impact loading, for example, can be determined. In this case inertial forces are also involved; given the

335

1~'r--------------------------------,

z '" . ~ 1 ~ .~

lI:

hf(mm)

Iw(mm)

I~mm)

50

hf(mm)

Iw(mm)

Vmm)

hf(mm)

Iw(mm)

I~mm)

lf=2m

lf=4m

Lf=6m

U=8m

Lf=10m

250 100 150 250 300 Displacement (mm)

1000 E.{kN/mm2) 18 hf(mm) 1000 E. (I<."I/mm2) 18

330 e,,(N/mm') 20 I.(mm) 330 e,,(N/mm2) 20

p(kglm') 2400 I~mm) p (kglm' ) 2400

(a) Wall WIth fixed "abutments" (b) Wall WIth movable "abutments"

50

\000

330

8000

Fig. 5. Influence ofwaJl failure length

150 Oisplacement (mm)

Ew{kN/mm2)

f,. (N/mm2)

p(kglrn')

18

2400

250

hf(mm) 1000

I.(mm) 330

I~mm) 8000

Oisplacement (mm)

E.{kN/mm') 18

e,,(N/mm2)

p(kglrn') 2400

(a) Wall WIth fixed "abutments" (b) Wall WIth movable "abutments"

Fig. 6. Influence of compressive strength of masonry

150 Displacement (mm)

1000 E.{kN/mm2)

330 e,,(N/mm')

8000 p (kglrn')

250

20

2400

z '" . o c

>l ~

hf(mm)

Iw(mm)

I~mm)

Displacement (mm)

1000 E.{kN/mm2)

330 e,,(N/mm2)

8000 p (kglrn')

20

2400

Ca) Wall WIth fixed "abutments" (b) Wall WIth movable "abutments"

Fig. 7. Influence of masonry elasticity

336

350

3SO

3SO

o

" E ~ " ~ Ci

relatively high mass per unit length of virtually ali masonry walls the inertial effects are likely to be significant when calculating the dynarnic resistance ofmost masonry walls.

When the abutments are rigid the wall can be considered as a single degree of freedom dynarnic system. Thus, using a numerical integration procedure and given that a" á" aI represent the angular displacement, velocity and acceleration respectively at time step f ,

solutions can be obtained as follows : a ,= a,_, +ótá,_, (13)

á , =: á ,_, +ót aI_I (14) Using the geometry of the wall the out-of-plane displacement can be calculated; this value will be used to calculate P"" the static resistance of the wall (using equations 7 -9).Thus:

d =_['1 tan(-(a -tan-'(í)-t ») OUI 2 I 'r/2 e

(15)

Rearranging the equation ofmotion, the angular acceleration can be calculated :

.. ' / (PoPP - ~eJ a = -'--"'-'-----"'--I 2(

(16)

- where Iw is the rotational inertia of the walL Iw will decrease as the depth of crusrung dcrush increases (and the effective wall thickness te decreases correspondingly). l.e.

I =H I t p('} +t; + f:) (17) w 1 1 '" 12 4 12

The procedure described above can be used for any specified applied force-time rustory (Popp vs. time) ; in general verucle impacts are comparatively slow and soft Figure 8 shows a typical displacement-time response for a number of applied force-time histories, the force Popp varying bilinear1y between O~Pmo.x ~O in time O~O . 05s~O . ls .

100

50

o

. 10Q-!::---:~--::-r-::----::-r::--:c::--:---:-~ 0.00 0.05 0. 10 0.15 0.20 0.25 0.30

TIme (5)

hr (mm) 1000 E. (k.N/mm') 18

I.(mm) 330 t;.,(N/mm') 20

I!mm) 8000 p(Jcgim') 2400

Flg. 8 Dynanuc response of masonry wall

337

...•...

O'~4-;C-~-'~O-~'C2 -~'4~-'~6-~'-B~~ Fallure length , If (m)

h,(nun) 1000 E. (JcN/mm') 18

I.(nun) 330 t;.,(N/mm') 20

I! rrun) p(Jcglm' ) 2400

Flg. 9 Influence offatlure length on wall resistance

The resistance of the wall is overcome when the predicted dynamic response fails to retum to a free vibration response after impact In practice rnctional forces will provide damping which wiJI prevent the waJI from ever vibrating freely; rnctional forces can readily be included in an analytical procedure ofthe type described previously.

Figure 9 shows the predicted influence of the waJI failure length on both the dynamic and static resistance of a masonry wall; the bilinear force-time history mentioned previously has been used to obtain the dynamic values. The figure indicates that inertial forces have a beneficiai influence on the resistance of masonry waJIs subjected to impact loadings, at least when horizontal arching failure modes are being considered.

The aforementioned dynamic analysis uses the assumption that the waJI section under consideration is heJd between rigid "abutments", this is unlikely to be the case in practice. A similar procedure which is suitable for obtaining solutions applicable to walls with movable "abutments" can also be devised, in which the waJI can be considered as a two degree of freedom dynamic system. However, inertial effects wiJI ensure that the influence of movable "abutments" on the dynamic resistance of masonry walls will be somewhat less than in the static case.

7. CONCLUSIONS

This paper has described some of the preliminary analytical work which has been undertaken in connection with a study of the behaviour of masonry walls subjected to out-of-plane point loading. The significance of inertial forces in resisting dynamic (impact) loadings ofthe type associated with vehicles has been highlighted.

A programme of tests on full size masonry walls using a drop hammer/rotating quadrant impact rig is about to begin. It is envisaged that the simple analytical tools described in this paper will be used in conjunction with F.E. models to investigate the predicted influence ofthe parameters involved, thereby enabling comparison with the test results.

8. REFERENCES

I . Moore, lF.A et aI. , "The Resistance of Masonry to Lateral Loading (Colloquium)", The Structural Engineer, VoI. 64A, No.11 , 1986, pp319-350.

2. Astbury, N.F , "Gas Explosions in Load Bearing Brickwork Structures", BCRA special publication No.68, Stoke-on-Trent, 1970.

3. McDowell, E.L et aI. , "Arching Action Theory of Masonry Walls" , ASCE, Joumal ofStructural Division, Proceedings paper 915,1 956.

4. Anderson, c., "Arching Action in Transverse Laterally Loaded Masonry Wall Panels", The Structural Engineer, VoI. 62B, No. l , 1984, ppI 2-23 .

5. Hodgkinson, RR. , et aI. , "Preliminary Tests on the Effects of Arching Action in Laterally Loaded WaJIs", Proceedings of the 4th Intemational Brick Masonry Conference, Paper 4.a.5, Brugges, 1976.

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