RELIABILITY-BASED PROBABILITY ANALYSIS FOR PREDICTING FAILURE OF EARTH BRICK WALL IN

COMPRESSION

A. A. Adedeji

Department of Civil Engineering University of Ilorin, PMB 1515

Ilorin, Kwara State, Nigeria. aaadeji@unilorin.edu.ng or amadeji@yahoo.com

ABSTRACT Reliability evaluation, based on compressive strength, of earth wall with respect to its service life was carried out in this paper. At present, reliability-based design studies for a typical earth wall structure may not consider the effects of cement plaster and mortar joints. The wall components of cement plaster, cement joint mortar and its units were taken into account in the analysis. Fault tree analysis, as in a conventional masonry, has given possible failure combinations for the strawbale wall under vertical loads. Using constant failure rate (CFR), for the wall having strength of 2.06N/mm2 at 28 days old, the service age of 149 years gives a reliability of 0.5. A reliability-based analytical approach, in minimising the expected lifetime cost, is emphasized in this work for an earth walled residential building. The results of the analysis has indicated that plastered earth wall, under compression, has failure probability of 0.77 for its optimum design value.

Key words: Reliability model, failure, earth, cement plaster, fault tree analysis, compression

INTRODUCTION

The need to research into the earth (mud and laterite) material is imperative, as its strength in compression, for a low-rise building, compares favourably with other common wall materials (concrete and sandcrete block, earth, burnt and unburnt brick, to mention a few). An urgent demand for standard, durable and cost-effective building materials suggests that there is the need to look inwards for such materials.

Cement plastered earth, even in its form, is a very reliable material in building construction if necessary expertise is followed. Earth walled structure has many advantages: heat/cold insulation performance, low shrinkage, local availability and its Low-Tech construction. A wall may not perform its intended functions due to structural failure, such as cracks, water penetration, and poor finishes. All these may be attributed to poor workmanship (improper jointing and poor plumbing), lack of adherence to specifications and misuse by over-stressing beyond the stated capacity of the brick wall.

Since the validity of any particular design method rests on the extent to which it can perform, this research intends to: determine how reliable this wall is as a load bearing element is with respect to its failure rate, so as to incorporate the reliability values into design for its service life.

The main objective of this work therefore is to evaluate the reliability-based probability of the Earth wall failure rate for a two storey residential building.

To achieve this objective, earth panel were moulded, masonry prisms specimens were produced in laboratory environment. Units (bricks) were subjected to non-destructive tests, while prisms were crushed to failure and their properties were recorded. Reliability-based design approach for the earth wall under compressive load has been proposed based on the results of the properties obtained from the test. The cost of the wall was minimized. RELIABILITY-ENHANCING TECHNIQUES

In research, the term reliability means “repeatability” or “consistency”. A measure is considered reliable if it would give the same result over and over again assuming that what we are measuring isn’t changing.

Reliability evaluation of a product can include a number of different analyses, depending on the phase of the product life cycle (Okpala and Kotingo 2007). David (2001) described the reliability engineering activity as an ongoing process starting from the conceptual phase of a product life cycle.

Trochim (2006) classified reliability estimators into four types. These are: (i) Inter-Rater reliability and it is used to assess the degree to which different raters give consistent estimates of the same phenomenon, (ii) Test-Retest Reliability in which the consistency of a measure from one time to another is assessed, (iii) Parallel-Forms reliability is the assessment of the consistency of the results of two tests constructed in the same way from the same content domain and (iv) the Internal Consistency reliability used to assess the consistency of results across items within a test.

The brick wall design is usually based upon vertical design loads for a reasonable performance (BS 5628, 1985). A wall may be designed to carry a load of 50kN, and can still carry up to 60kN load, but would not be so reliable to this extent. A balance must therefore be achieved between rating, cost and reliability. Partial redundancy is employed to accomplish the required functions so as to reduce the strength of the wall. For instance, spalling or cracking of plaster may allow severe water or frost penetration into the wall fibre thereby reducing, to an extent, the wall strength. In other word, Test-Retest, which is especially feasible in most experimental and quasi-experimental designs without a treatment control group was embarked upon in this work.

EXPERIMENTAL METHODOLOGY

Specimens of cement plastered earth prism were constructed from bricks size: 75mm x 105mm x 205mm. The specimen prisms (having slenderness ratio i.e. height: thickness = 3) were produced from bricks that were joined together, using 1:10 (cement: sand) mix ratio.

Compressive strength tests were carried out at 3, 7, 14, 21 and 28 days, and the rate of strength gained for the tested specimens were obtained. During the crushing tests by the universal testing machines, a portable ultrasonic non-destructive digital apparatus with an indicating tester was used for testing strength at each section loss due to damage conditions of the wall component and results were recorded. Prisms initial average strength values for 7, 14, 21 and 28 days were obtained as: 1.48, 1.66, 1.89 and 2.06 (N/mm2) respectively. It has been established according to results of tests carried out by the Cement and Concrete Association and reported by Roberts et al (1988) of the minimum influence

that mortar joint has on the strength of wall (Figure 1). The mortar effect was considered only in the assumption of the cost factor of the wall. Since the effect of mortar is taken as mere binder between the panels (Robert et al, 1988), the effect of it in the loss of wall section is negligible and was therefore neglected

ANALYSES AND RESULTS Strength Analysis

From the results of the analysis, the total strength considered was 100N/mm2 (Table 1 and 2). If i is the number of the strength by the end of testing day (i), then Ri is the total of the strength still

remained at the end of day (i), and by Leitch (1988),

)(1 iji

iQ (1)

Ri = 100 – Qi (2)

Mortar Strength (N/mm^2) Figure 1 Effects of mortar strength on wall strength Solid blocks 18.5 N/mm2 Cellular blocks 14.0 N/mm2

TABLE 1. AVERAGE STRENGTH RESULTS FOR 10 BRICKS EACH TESTING DAY Days Strength

gained( i) N/mm2

Cumulative strength(Qi) N/mm2

Average strength ( ) N/mm2

Cumulative strength (Q) N/mm2

Remaining strength (Ri) %

Strength Rate (d)

3 0.084 0.82 0 0 99 0.010 7 1.410 1.494 19 20 80 0.192 14 1.770 3.264 23 43 57 0.288 21 2.010 5.274 27 70 30 0.474 28 2.290 7.564 30 100 0 1.000

TABLE 2 AVERAGE STRENGTH RESULTS FOR 5 PRISMS FOR EACH TESTING DAY

Days

Strength gained ( i) N/mm2

Cumulative Strength (Qi)

N/mm2

Average strength ( ) N/mm2

Cumulative strength (Q)

N/mm2

Remaining strength (Ri) %

Strength Rate (d)

3 0.08 0.08 1 1 84 0.190 7 1.48 1.56 33 34 66 0.214 14 1.66 3.22 22 56 44 0.333 21 1.89 5.11 22 78 22 0.500 28 2.06 7.17 32 100 0 1.000

The data presented in Tables 1 and 2 show values of cumulative strength for each day. The last column di (Table 1 and 2) is related to the strength gained and the total strength remained. This is represented by the following equation.

1

I

ii R

d (3)

where di is the strength rate or rate of gain in strength, which is the value of strength in a given day as the remaining strength at the beginning of the day. This is the probability of obtaining particular strength of a brick during day (i), assuming there are still remaining strength during the day. Constant Failure Rate Model

Constant failure rate (CFR) is employed here for easy algebraic manipulation. Here d is assumed constant with time and according to Leitch (1988):

R(t) = e-dt (4)

where R(t) -=constant rate of failure, t = variable time . The failure density is expressed with respect to constant rate of failure as:

F(t) = de-dt (5)

So that the estimator is expressed as recommended by Leitch (1988):

F(t) = 1-e-dt (6)

The mean time to failure, is the average functioning (without a failure) period for an item or average life cycle of a number of items, is expressed as:

MTTF = 1/d (7) So that reliability :

R = e-t/m (8) where t = period of time, d = failure rate, m = expected average number of break downs

Earth wall failure

Weak panel

Separation of mortar from wall

Weak panel

Plaster failure

Cement not enough

Panel cracks

Bleeding segregation

Excess cement

Insuffici- ent cement

Imposed load exceed capacity + high excentricity

Imposed load exceed capacity + high excentricity

Plaster is stronger that the base wall

Plaster washed ourt by water

Plater cracks to spall

Excess of cement

Excess of cement

Poor mix design

Poor mix design Poor

assessment tender wall

Poor assessment study wall

Excess of cement

Implelegate cement

Poor curing

Poor workmanship

To A

Earth wall failure

Weak panel

Separation of mortar from wall

Weak panel

Plaster failure

Panel rack

Bleeding segregation

Excess cement

Insufficient cement

Imposed load exceeds capacity + high excentricity

Imposed load exceeds capacity + high excentricity

Plaster is stronger that the base wall

Plaster washed out by water

Plater cracks to spall

Excess of clay Excess of

clay

Poor mix design

Poor mix design Poor

assessment slender wall

Poor assessment study wall

Excess of cement

inadequate cement

Poor curing

Poor workmanship

A

Figure 2. Fault-tree analysis for earth wall

Specification of Reliability Fault tree analysis

Based on experience and past works (Adedeji 2004, Adedeji 2002, Adedeji 2001) on the failure of masonry In predicting reliability, all possible combination of failure in failures of wall could occur due to possible combination of faults. This could be due to the panel not properly moulded into required standard, improper curing or mixing. It could also be due to strong or weak mortar joint. Fault tree analysis composed by this work, gives a pictorial view of these possibilities and their relationships to each other as in Figure 2.

The occurrences of the above events (Figure 2) are not mutually exclusive as two, three or more of these conditions can occur at the same time. This may lead to the failure of the wall. Block diagram analysis

It has been necessary to specify not only for the performance characteristics of the Earth wall at early stage but also the reliability and its allowable characteristics. A realistic reliability figure at this stage must be determined for a wall life span. In this paper, a design approach was based on early work of Frangopol and Hendawi (1994) and Lin and Frangopol (1996).

Considering the Reliability Panel Diagram (RPD) as shown in Figure. 3, the reliability of each wall component was indicated. This assumption was used to determine the wall total cost factor (K) in equation (10).

Figure 3. Block diagram analysis for earth wall

Table 3 indicates the reliability of panel, mortar, plaster and wall with time. At this stage a

reliability figure must be determined for wall life span. In this paper, a design approach used is based on the early work of Frangopol and Hendawi (1994) and Lin and Frangopol (1996).

TABLE 3. RELIABILITY OF BRICK, MORTAR, PLASTER AND WALL WITH TIME BbR Time

(year) BmR Time

(year) BpR Time

(year) Rw Time

(year) 0.84 114 0.93 48 0.95 34 0.78 65 0.50 146 0.89 76 0.91 62 0.65 95 0.72 178 0.85 106 0.87 91 0.56 95 0.72 215 0.81 138 0.83 122 0.42 192 0.68 252 0.77 171 0.79 154 0.42 192 0.64 292 0.73 20 0.75 188 0.35 229 0.60 335 0.69 243 0.71 224 0.29 267

BbR = panel reliability, BmR = mortar reliability, BpR = plaster reliability, Rw = wall reliability

Bb

Bm

Bp

Wall

Bb = panel Bm = mortar Bp = plaster

Estimation of Capacity Loss of Sectional Area The compressive strength of loaded Earth wall members depends mainly on the total available

sectional area of the wall. For a uniform loading, the total wall crushing failure area, as a function of time t, is expressed as:

A(t) = LxB for t≤ T (9a) (intact or undamaged )

and A(t) = L [B – dr(t -T), for t > T (9b)

(damaged section) where L, B = length, thickness of wall respectively, T = time of service initiation, dr = rate of damage (disintegration) on panel and plaster.

The loss of material due to deterioration under constant load is time-dependent. If we consider a residential building of earth walled (RR100), the reliability-based structural optimum design of this wall requires a definition of the 8 design variables and 6 parameters as shown in Table 4. The objective of the optimization process is to minimize the total cost of the wall with the factor:

KC = bm

P

CCC

(10)

where Cp, Cm, Cb = cost of plaster, mortar, brick respectively, the value of Kc = 0.10. TABLE 4. DESIGN VARIABLES (V) AND DESIGN PARAMETERS (P) FOR RR100 BUILDING

The residual capacity factor, due to loss of wall sectional area is expressed as,

ACR =act

damaged

AA

int (11a)

And the sectional area loss

Notation Variables Units V1 Brick height,Hb 150mm V2 Brick thickness, TH 130mm V3 Brick width, B 250mm V4 Wall thickness,Tw 250mm V5 Wall height, Hw 300mm V6 Wall length, Lw 4200mm V7 Mortar thickness, tm 15mm V8 Plaster thickness, tp 15mm P1 Brick strength, b 2.63 N/mm2 P2 Wall strength, w 2.06 N/mm2 P3 Mortar strength, m 2.1 N/mm2 P4 Plaster strength, p 2.1 N/mm2 P5 Live load on wall 103.78 kN/m P6 Dead load on wall 9.54kN/m

Aloss = Ainstact - Adamaged (11b)

where Adamaged is the sectional area reduced or crushed due to spalling of plaster on the masonry prism and Aintact is the area of prism plaster, while the residential capacity factor due to loss in wall strength:

act

damgedCR

int

(12)

where intact is equivalent brick failure. damaged = crushed or damaged on the prism. And the strength loss is expressed as,

)13(int damageactloss

DISCUSSION OF RESULTS Wall Strength and Sectional Loss

In Table 5, the results of the sectional and strength of plastered earth wall were shown, while Figures 5 and 6 show relationship between the residual capacity factor and loss. As the sectional area increases the residual capacity factor remains at maximum value of 0.98, while the residual capacity factor is nonlinear as the loss in strength increases TABLE 5. EARTH WALL SECTIONAL AND STRENGTH LOSS Days intact Aintact damaged Adamaged 3 0.08 27500 0.07 27460 7 1.45 27510 1.40 27470 14 1.60 27500 1.51 27487 21 1.90 27500 1.88 27487 28 2.11 27500 2.01 27489

Figure 4. Residual capacity factor and area loss.

0.00

0.20

0.30

0.50

0.80

1.00

0.0 5.0 10.0 15.0 20.0 25.0 30.0

Res

dual

Cap

acity

Fac

tor

Sectional Area Loss

Figure 5. Residual capacity factor vs strength loss. Optimum design based on service life cost

Minimum material cost may not be the most essential characteristics the utility of a structure If failure consequence is taken into account, more general criteria for the design of wall produces the minimum total expected cost were considered here: (a) initial expected cost Ci (t) and (b) expected failure during life-time (cost due to loss of structure and life Structural optimization design was obtained within the framework of consequences of structural failure. This is necessary to introduce lifetime failure probability Plife in the objective function which is minimized as,

Clife.min = Ci (t) + Ccf= Ci (t) + Cf Plife (15) In which: Plife = 1 - (R(t)) (15a) And Cf = 1000Cb (16) where Cb = cost of a brick. Earth wall cost versus lifetime reliability is shown in Figure 6. At the optimum point, the Earth wall life cost to 30.3% of the cost at the reliability of 0.77.

Loss of Strength

0.00

0.20

0.50

0.80

1.00

0.0 0.20 0.40 0.60 0.80 1.00 1.20

Figure 6. Wall life-time reliability and cost CONCLUSION

A reliability based design approach of an earth (laterite) wall under compressive load has been proposed. The formulation includes partial redundancy and residual (damaged) reliability constraints. Unlike convention wall design, which is based on satisfying code constraints only, the proposed reliability-based approach uses material components or the expected lifetime service of the components. The wall components (cost) are the objective that was minimized. When a wall of a residential low-rise building is under the appropriate vertical load, the reliability which is below 0.5% occurred at the wall age above 149 years.

From the analyses, the optimum design occurs at wall reliability of 0.45 at a cost of N15,000/m2 (in the year 2008), while the maximum cost, in the wall life time, is N 21000 at a reliability of 0.25. The wall under compression has failure probability of 0.77 at its optimum design. ACKNOWLEDGEMENT

I appreciate the help rendered by Kehinde, O.A. for his assistance in producing few data used in this work. My appreciation also extends to the Civil Engineering Department for providing equipment and laboratory for this research. REFERENCES ADEDEJI, A. A. (2004). Deterioration of earth walls and design improvement, Journal of Applied

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