Behavior of Interacting Parallel Strip Footings

by

Mien Jao

Assistant Professor, Department of Civil Engineering
Lamar University - Beaumont, TX 77710
e-mail: jaomu@hal.lamar.edu

Mian C. Wang

Professor, Department of Civil Engineering
The Pennsylvania State University - University Park, PA 16802
e-mail: mcw@email.psu.edu

H. C. Chou

Graduate Assistant, Department of Civil Engineering
Lamar University, Beaumont, TX
77710
e-mail: hungchieh@hotmail.com

C. J. Lin

Assistant Professor, Department of Civil Engineering
Lamar University, Beaumont, TX 77710
e-mail: lincx@hal.lamar.edu

ABSTRACT

The settlement, tilting, and soil plastic yielding behavior of closely spaced parallel strip footings were presented. The results of study were obtained from computer analyses using a finite element computer program. Footings analyzed were both surface and embedded footings with a footing width of 610 mm. The foundation soil was compacted kaolin having cohesion of 158.5 kN/m2 and an internal friction angle of 8.0°. Based on the results of the study, it was concluded that footing interaction may cause unsymmetrical plastic yielding of the foundation soil under each footing, footing tilting, and an increase in footing settlement. Both footing settlement and tilt angle decrease with increasing footing spacing. The effect of footing interaction on footing behavior becomes insignificant when the footing spacing approaches six times footing width.

KEYWORDS: Strip Footings, Interaction, Footing Behavior, Footing Settlement, Footing Tilting.

INTRODUCTION

When two parallel strip footings are closely spaced, they may interact, resulting in different footing behavior than that of isolated strip footings. Footing behaviors of particular interest in foundation design include bearing capacity, settlement, and tilting. Of these, the bearing capacity behavior of closely spaced strip footings has received more studies than the other two. Examples include Stuart (1962), West and Stuart (1965), Das and Larbi-Cherif (1983 a and b), Das et al. (1993), and Wang et al. (1994). Since very little information on footing settlement and tilting behavior is available, this paper presents the results of a study on this subject matter.

The study was conducted using the method of finite element analysis. Footings analyzed were reinforced concrete strip footings supported by a cohesive soil. In the finite element computer program, the concrete footing was treated as a linear elastic material. The foundation soil was characterized as a nonlinear elastic perfectly plastic material. Within the elastic range, the hyperbolic stress-strain law was used, whereas, the Ducker-Prager yield criterion was adopted to model the plastic behavior. The computer program used was the same as that adopted in other studies, such as Hsieh and Wang (1992) and Wang et al. (1994).

COMPUTER ANALYSIS

Both surface and embedded strip footings were analyzed. The depth of embedment (Df) was equal to footing width (B), which was 610 mm (2 ft). Each footing carried a vertical central load. The cohesive foundation soil was compacted kaolin. The soil properties used in the analysis were 158.5 kPa (cohesion), 8.00 (internal friction angle), 15.75 kN/m3 (unit weight), 19,860 kPa (initial modulus in compression), 0.77 (soil constant), and 0.39 (Poisson’s ratio). The concrete footing properties used were 22.8 x 106 kPa (modulus in compression), 24.4 kN/m3 (unit weight), and 0.30 (Poisson’s ratio).

The analysis was made for varying footing spacings. For each footing spacing, the footing pressure was applied incrementally. Under each pressure, footing settlement, soil yielding pattern, and footing tilting data were determined. These data were used to analyze and discuss the effect of footing spacing on footing interaction.

FOOTING SETTLEMENT

Footing settlement varies with footing pressure intensity. For ease in comparison, the footing settlement under the ultimate load was used in the analysis. Also, the settlement data were nondimensionalized by using the settlement ratio between interacting (Si) and non-interacting (So) footings. The settlement ratios (Si/So) were related with footing spacings. The relationships between footing settlement ratio and footing spacing to footing width ratio (D/B) were obtained for the surface and embedded footings, in which D is the center-to-center footing spacing. Such a relationship was also determined from elastic solution for surface footings. The graphical relations between So/Si and D/B are summarized in Figure 1. Also included in the figure is the relationship derived from the test results of Das and Larbi-Cherif (1983a).

The data in Figure 1 show that Si/So decreases with increasing D/B at a decreasing rate and approaches unity around D/B = 6, indicating that footing interaction results in an increase in footing settlement and that the degree of interaction becomes insignificant when footing spacing approaches 6 times footing width. The increased footing settlement due to interaction is primarily resulted from the overlapping of a portion of the stress/strain fields under each footing. The degree of overlapping increases with decreasing footing spacing. When footing spacing reaches about 6 times footing width, the overlapped stress/strain fields do not significantly increase footing settlement.

It is also seen from Figure 1 that at a given D/B ratio, the settlement ratio Si/So is greater for embedded than for surface footings. Due primarily to the effect of overburden pressure, the ultimate load (or ultimate bearing capacity) is greater for embedded than for surface footings. The greater footing load for the embedded footings would cause a larger footing settlement than the surface footings as demonstrated by the figure. However, the difference between embedded and surface footings becomes negligibly small when D/B >4.

It is interesting to note that the data of Das and Larbri-Cherif (1983a) coincide fairly well with the curves of surface and embedded footings except at the left end, where it is bracketted between the two curves. Their test results showed slightly higher settlements for embedded than surface footings. However, the data points were highly scattered without a clear trend of difference in variation with footing spacing between the two different types of footings. As a result, they stated that there was no difference in Si/So vs. D/B relation between embedded and surface footings. It should be pointed out that the soil investigated by Das and Larbri-Cherif (1983a) was a sand and the test footing width equaled 50.8 mm. The close agreement between their test data and the results of the computer analysis suggested that the graphical relation in Figure 1 is valid for a range of soil type and footing width.

D / B

Figure 1. Variation of Settlement Ratio with Footing Spacing

The curve obtained from the elastic solution in Figure 1 is considerably below the other three curves when D/B is small. This indicates that the elastic settlements of interacted footings are smaller than the footing settlements that involve plastic yielding of the foundation soil. Also, the curve is flatter and eventually merges with the other three curves.

The data shown in Figure 1 were analyzed using a statistical regression software package to obtain correlation equations.  The obtained correlation equations between Si/So and D/B are as follows:

For embedded footings under ultimate loading,

Si/So = 1.390 e B/D – 0.471  with R2 = 94.4%

(1)

For surface footings under ultimate loading,

Si/So = 0.972 e B/D - 0.022 with R2 = 91.8 %

(2)

For elastic settlement of surface footings,

Si/So = 0.296 e B/D+ 0.756 with R2 = 93.9 %

(3)

With these correlation equations, it would be possible to estimate the settlement of interacting parallel strip footings within the range of footing spacings analyzed.

PLASTIC YIELDING

The interaction between two adjacent strip footings can be reflected in the plastic yielding pattern of the foundation soil. For an isolated strip footing, the progress of plastic yielding under different footing pressures is illustrated in Figure 2(a), which shows only one-half of the footing width. It is seen that the overall yield pattern is symmetrical and contains an unyielding zone directly underneath the footing when the footing pressure reaches 1,676 kN/m2 (35,000 lb/ft2). The existence of the unyielding zone can be attributed to the combined effects of high confining stress and low shearing stress, because under the footing center, the confining stress is very high and shearing stress is very low.

        718 kPa        1,197 kPa    1,676 kpa

       (15000psf)      (25000psf)     (35000psf)

Figure 2(a). Growth of Yield Zone with Increasing Footing Pressures for a Single Footing

For closely spaced strip footings, the overall yield pattern under the entire system of two footings is symmetrical about the centerline of two-footing system. However, under one footing, the yield pattern is unsymmetrical about the footing center as demonstrated in Figure 2(b). Shown in Figure 2(b) is only one-half of the two-footing system spaced at D/B =4. As seen, up to a footing pressure of 1,676 kN/m2 (35,000 lb/ft2), there still exists an unyielded zone directly underneath the footing center. Plastic yielding progresses downward toward right more than toward left.  As the foundation soil yields, shear failure takes place, resulting in an inward tilting of the footing. An enlarged illustration of a tilted footing with a deformed finite element mesh is presented in Figure 3. Note that Figure 3 shows only one–half of two closely spaced strip footing with D/B = 1.5 and a footing pressure of 2,394 kN/m2 (50,000 lb/ft2).

    718 kPa        1,197 kPa      1,676 kPa

   (15000psf)      (25000psf)     (35000psf) 

Figure 2(b). Growth of Yield Zone with Increasing Footing Pressures
for a Pair of Footings Spaced at D / B = 4

 

Figure 3. Deformed Mesh for a Pair of Surface Footings Spaced at D / B = 1.5

 

FOOTING TILTING

The degree of footing tilt angle depends on the extent of soil yielding, which in turn is dependent on the level of footing pressure. Thus, as footing pressure increases, the tilt angle also increases. Figure 4 illustrates the tilt angle vs. footing pressure relation for 610 mm (2 ft) wide surface footings spaced at D/B = 1.5. As shown, the shape of the relation somewhat resembles that of pressure vs. settlement relation. The increase in tilt angle with increasing footing pressure intensifies when the footing pressure approaches the ultimate bearing capacity value. For this footing and spacing, the ultimate bearing capacity is approximately equal to 1,435 kN/m2 (30,000 lb/ft2), and the corresponding tilt angle equals about 2.7°.

(1 psf = 0.04788 kPa)

Figure 4. Variation of Footing Pressure with Rotation Angle

As for the footing settlement vs. spacing relation, it can be expected that the footing tilt angle induced by footing interaction will decrease with increasing footing spacing. Figure 5 illustrates such a relation for 610 mm (2 ft) wide strip surface footings each subjected to 1,435 kN/m2 (30,000 lb/ft2) footing pressure. It is seen that the footing tilt angle (q) decreases with increasing D/B sharply from about 2.7° at D/B=1.5 to about 0.3° at D/B = 4. The correlations equation between q and D/B obtained from a statistical regression analysis is

q = 3.31 e B/D - 3.85 with R2 = 98.5 %

(4)

This correlation provides a rough estimate of footing tilting angle for closely spaced parallel strip surface footings supported by compacted kaolin and subjected to the ultimate loading.

D / B

Figure 5. Variation of Rotation Angle with Footing Spacing

 

SUMMARY AND CONCLUSIONS

The results of a study on the behavior of interacting parallel strip footing were presented. The behavior investigated involved footing settlement, footing tilting, and plastic yielding of foundation soil. The study was conducted by means of computer analyses using a finite element computer program. Both surface and embedded footings with a footing width of 610 mm were analyzed. The foundation soil was compacted kaolin having cohesion of 158.5 kN/m2 and an internal friction angle of 8.0°. From the results of analysis, the effect of footing spacing on footing settlement and footing tilt angle was presented.

Based on the results of this study, it is concluded that footing interaction can cause unsymmetrical plastic yielding of foundation soil under each footing, footing tilting, and an increase in footing settlement. Both footing settlement and tilting angle decrease with increasing footing spacing. The effect of interaction becomes insignificant when the footing spacing reaches about six times footing width.

 

REFERENCES

  1. Das, B. M., Puri, V.K., and Neo, B.K. (1993), “Interference Effects Between Two Surface Footings on Layered soil”. Transportation Research Record No. 1406, Transportation Research Board, Washington, D.C., pp.34-40.
  2. Das, B. M. and Larbi-Cherif, S. (1983a).”Ultimate Bearing Capacity of Closely Spaced Strip Foundations”. Transportation Research Record 945, TRB: 37-39.
  3. Das, B. M. and Larbi-Cherif, S. (1983b). “Bearing Capacity of Two Closely Spaced Shallow Foundations on Sand”. Soils and Foundations, Japanese Society of Soil Mechanics and Foundation Engineering, 23(1): 1-7.
  4. Wang, M. C., Jao, M., and Hsieh, C. W. (1994), “Effect of Underground Cavity on Footing Interaction”. Proceedings, XIII International Conference on Soil Mechanics & Foundation Engineering, New Delhi, India, Vol.2, pp. 575-578.
  5. Hsieh, C. W. and Wang, M. C. (1992), “Bearing Capacity Determination Method for Strip Surface Footings Underlain by Void”. Transportation Research Record No. 1336, Transportation Research Board, Washington, D. C. pp 90-95.
  6. Stuart, J. G. (1962), “ Interference Between Foundation with Special Reference to Surface Footings in Sand”. Geotechnique 12(1): 15-22.
  7. West, J. M. and Stuart, J. G. (1965), “Oblique Loading Resulting from Interference Between Surface Footings on Sand”. Proceedings of 6th International Conference on Soil Mechanics and Foundation Engineering (2): 214-217.

 

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