BEARING CAPACITY OF A STRIP FOOTING ON A LAYERED COHESIONLESS SOIL

Transcription

1 BEARING CAPACITY OF A STRIP FOOTING ON A LAYERED COHESIONLESS SOIL Brad Carter ( ) Senior Report CE 5943 November 2005

2 Abstract Two critical aspects of footing design are: bearing capacity and settlement of the underlying soil. Currently, analytical methods that have not been fully verified are used to determine these aspects of design. The purpose of this study is to verify the theories for the specific case of a strip footing over a layered cohesionless soil with the upper layer being dense and the lower layer being loose. Seven tests were conducted using the geotechnical centrifuge at the University of New Brunswick to monitor the settlements of a model strip footing under central load. From these tests, the ultimate bearing capacity was determined. The observed capacities were compared with the theoretical values predicted in past studies. It was found that the results of this study generally agree with the previous studies, but with minor differences. It was predicted that when the layer of dense soil less than 1.5 times the width of the footing, punching would be the governing mode of failure and when the upper dense layer was thicker then 1.5 B bearing capacity or the dense layer would govern. In the present study, punching was found to govern for a dense layer of soil with a depth equal to the base of the footing and not 1.5 times the base of the footing. It was found that Schmertmann s method generally gave conservative values for the amount of predicted settlement in cases where the dense layer of soil was less than two times the width of the footing. More tests will be needed to confirm this difference. ii

3 Acknowledgements I would like to acknowledge Dr. A. J. Valsangkar for his advice and guidance during the completion of this study. I would further like to thank Mr. Dan Wheaton for his advices on assembly of the model, as well as for fabricating the needed components, and operating the centrifuge. iii

4 Table of Contents Abstract... ii Acknowledgements...iii List of Figures... vi List of Tables... vi List of Symbols... vii 1 Introduction Problem Statement Objectives Background Literature Review Ultimate Bearing Capacity Settlement UNB s Geotechnical Centrifuge Theory and Scale Relations Modeling Considerations Methodology Phases of Study Centrifuge Testing Calibration Model Assembly Final Test Preparations Test Data Analysis Results Test Results Theoretical Results Ultimate Bearing Capacity Settlement Discussions Experimental vs. Theoretical iv

5 5.1.1 Ultimate Bearing Capacity Settlement Problems Encountered Conclusions Recommendations References Appendix A Data Collection Run Graphs Appendix B Bearing Capacity Calculations v

6 List of Figures Figure Reference Diagram (Meyerhof, 1974)... 4 Figure 2-2 Values of Ks (Meyerhof, 1978)... 5 Figure Assumed distribution of strain influence factor with depth (Das, 1990)... 6 Figure Filling the Strongbox with Sand Figure Levelling Sand with Vacuum Figure The Jig Placed in the Strongbox Figure Placing the Model Footing using the Jig Figure Footing with Load Cell and LSC's Figure Apparatus Figure Measured Load vs. Displacement Figure Experimental Results and Theoretical Results List of Tables Table Scale Relations (Stewart, 2000)... 8 Table Channels used for electronics Table Theoretical vs. measured ultimate bearing capacity and φ Table Settlements for 50 kpa pressure vi

7 List of Symbols B c 1 c 2 D E z g H h m I z K p K s N N γ, N q P p q lower q n q u Re Rt s α m α p δ γ t γ φ The width of the footing Correction factor for footing depth Correction factor for creep Depth of footing Young s Modulus Acceleration due to gravity The depth of the soil to the loose layer Height of model Strain influence factor Coefficient of passive earth pressure Coefficient of punching shear Scale factor Bearing capacity factors Total passive earth pressure Bearing capacity of lower layer Net foundation pressure Ultimate bearing capacity Effective radius of rotation Radius at top of model Settlement Acceleration of the model Acceleration of the prototype mobilized friction of truncated pyramid Unit weight of upper soil layer Unit weight of soil Soil internal angle of friction vii

8 ω Δz Angular rotation speed Thickness of sand layer viii

9 1 Introduction 1.1 Problem Statement This study examines the bearing capacity of a strip footing on a layered cohesionless soil using physical modeling. The results of the experiment are compared with existing theories. 1.2 Objectives The study had the following objectives: Model a footing on a layered soil using a geotechnical centrifuge. Compare the results with the existing theories of the bearing capacity and settlement of footings on layered soils. 1.3 Background A footing is used to transmit the load from a structure to the soil on a larger area to reduce the pressure. Different types of footings are used for different applications. The footing type used in this study is a strip footing which is largely used to support a linear load such as a load bearing wall or a retaining wall. A strip footing is rectangular in shape but its length is much greater than its width. Analysing a strip footing is a simple case as it can be analysed in two dimensions. When a structure is built, the soil on which the footing is to be founded is generally compacted to provide a firm base. Since the compaction extends only a finite depth into the soil it can create a dense soil over loose soil condition. It is important to know in these situations, how much load a soil can support so that the soil does not fail in bearing capacity under the loads applied by the structure. There are theoretical formulas which can be used to determine bearing capacity for such layered soils. These formulas have been in use for many years and have not been thoroughly verified. The lack of testing is due to the difficulty of the tests. One option of testing is to build a full size footing and test it for determining bearing capacity. While this would provide good results, a test such as this is both expensive and difficult to accomplish. A second option is to test 1-g 1

10 scale models. This is the most common method as it is much less costly and the scale models are easy to handle. However, the results from testing in this manner may not be fully accurate as some factors are lost in the scaling(taylor, 1995). A third approach is to use centrifuge modeling. A scaled model when accelerated higher than that of gravity will behave as a larger model. This allows for ease of handling and more representative results for prototype structures. In order to study bearing capacity of a footing in the specific case of dense sand over loose sand, the thickness of the upper layers in relation to the width of the footing was varied. 2

11 2 Literature Review 2.1 Ultimate Bearing Capacity If the ultimate bearing capacity of the dense sand layer is much greater than that of the underlying loose deposit it can be approximated by considering the failure as an inverted uplift problem. At the ultimate load a sand mass having an approximately truncated pyramidal shape is pushed into the lower sand layer in such a way that the friction angle φ and the bearing capacity of the lower layer are mobilized in the combined failure zones (Meyerhof, 1974). The ultimate bearing capacity of the layered soil should be equal to bearing capacity of the lower layer plus the punching resistance of the upper layer and the contribution due to surcharge. The forces on the failure surface of the sand can be taken as total passive earth pressures inclined at an angle δ acting upwards on a vertical plane through the footings edge. Therefore the ultimate bearing capacity according do Meyerhof (1974), q u, of a strip footing can be taken as 2Ppsin( δ ) q u = q lower + + γ t D... (1) B Where: q lower = bearing capacity of the lower sand layer, P p = total passive earth pressure, δ = mobilized friction on truncated pyramid, B = footing width, and γ t = unit weight of upper sand 3

12 Figure Reference Diagram (Meyerhof, 1974) Pp can be determined as follows: P p 2 2D γ t H (1 + ) K p = 0.5 H... (2) cos( δ ) Where: H = dense sand layer thickness K p = coefficient of passive earth pressure It is convenient to use the coefficient of punching shear resistance through the footings edge so that: Ks tan φ = Kp tan δ... (3) Substituting equations (2) and (3) into (1) gives us q u = q lower γ t H + 2 2D (1 + ) K H B s tan( ϕ) + γ D t... (4) With a maximum of: q u = 0.5γ t BNγ+γ t DN q... (5) 4

13 The maximum occurs when the H/B ratio is high enough such that the upper layer no longer fails through punching and its bearing capacity governs the bearing capacity of the layered soil system. The factor Ks can be determined from the following figure provided by Meyerhof and Hanna s 1978 study on the Ultimate bearing capacity of foundations on layered soils under inclined load. Figure 2-2 Values of Ks (Meyerhof, 1978) 5

14 2.2 Settlement Schmertmann s method is commonly used for estimating the amount of settlement due to a load. It is based on a simplified version of vertical strain under a footing in the form of a strain influence factor, I z, as seen in Figure 2.3. The settlement is determined with the equation: (Craig 1997) 4B I z s = c1 c2qn Δz... (6) E 0 z Where, c 1 = correction factor for footing depth c 2 = correction factor for creep q n = net foundation pressure B = width of footing I z = strain influence factor E z = Young s Modulus Δz = thickness of the layer Figure Assumed distribution of strain influence factor with depth (Das, 1990) 2.3 UNB s Geotechnical Centrifuge The centrifuge was commissioned in 1990 and is capable of inertial acceleration of up to 200gs at an effective radius of 1.6 meters. Each arm of the centrifuge can hold a load up to 100 kg. The data acquisition is on arm supporting up to 16 channels and transmits to 6

15 the control room in real time through slip rings with one slip ring reserved for the introduction of fluids. These data are then collected and stored by a computer program called GEN Theory and Scale Relations A centrifuge accelerates the model by rotating it at high revolutions per minute (rpm). When the model is accelerated in this manner it experiences forces like earth s gravity but increased proportionally by a factor N. When the model is spun at high g s, it behaves like a prototype structure, N times its size. (Taylor, 1995) The inertial acceleration of the model α m is denoted by: α m = ω 2 R e... (7) Where ω = angular rotational speed R e = the effective centrifuge radius As the radius varies over the depth of a model, an effective radius needs to be determined since the factors influenced by gravity will also vary over the depth of the model. Re = R t + H m /3... (8) Where R t = Radius at the top of the model H m = the height or depth of model Since N is the multiple of the model feels over the field of gravity it can be seen that α m = Nα p... (9) Where α p the acceleration the prototype experiences The acceleration that the prototype experiences is due to gravity therefore it can be seen therefore g can replace α p with g α m = Ng... (10) Combining equations (7) and (10), ω 2 R e = Ng... (11) 7

16 Only forces which are normally affected by gravity are influenced by the centrifuge as since it acts as an increased gravitational force on an object. The scaling laws are presented in the following table. Table Scale Relations (Stewart, 2000) Quantity Full Scale (Prototype) Centrifuge s Linear Dimension 1 1/N Area 1 1/N 2 Volume 1 1/N 3 Velocity 1 1 Acceleration 1 N Mass 1 1/N 3 Force 1 1/N 2 Stress 1 1 Strain 1 1 Density Modeling Considerations There are some factors that affect the results of the centrifuge tests that would not occur in the real situations - such as, the model being confined in a box. The wall boundaries exert sidewall friction on the soil. Sidewall friction is not major factor for this particular experiment as all measurements are being taken from the surface and the model footing is placed in the centre of the box a distance of 4.5B from the walls. Particle size must be accounted for as well; it is logical to assume that the particle size increases by a factor of the force of gravity like the model does, so fine sand could increase in size to represent gravel (Taylor 1995). It is suggested that the effects of particle size can be avoided by keeping the major model dimensions greater than 15 times that of the soil. 8

17 Another consideration is the fact that the factor N varies through the depth of the model. In this experiment the distance from the top to the bottom of the model is 100 mm which is relatively small. An effective radius is used which is located at the top third of the model is used. The variation of N from the top of the model to the bottom of the model is approximately 8%. 9

18 3 Methodology Five tests were performed on a model of a footing at a rate of 30g s. The depth of the dense sand was varied with the loose sand to determine how this affected the results. While a range near 30g s was used for the testing this was not on purpose; instead, it is due to the difficulty of running the centrifuge at an exact g-level. This should not affect the results as it would only change the proportional value N which can be accounted for in the calculations. The factor H will be examined as a ratio to the width B. At 4 times the width of the footing most of the stress from the load on the footing will have been dissipated. For that reason, 4 times the width of the footing will be the lower bound for the thickness of the dense sand with tests being run at 0B, 1B, 2B, 3B, and 4B. 3.1 Phases of Study There were 3 phases of study: 1) Design and assembly of the model and apparatus: in this phase all the materials needed to perform the tests were gathered and any modifications to existing equipment were performed. 2) Testing the model on the centrifuge: during this phase the model was run on the centrifuge 5 times at varying depths of dense and loose sand. 3) The final phase was analysis of the data: During this phase all of the collected data from phase 2 were analysed. 3.2 Centrifuge Testing Calibration The load cell was calibrated first. The load cell allows for the measurement applied load to the footing to be measured. At loads of varying and known intensity voltages were 10

19 measured in the load cell, a linear regression of these data points allowed for the establishment of the calibration curve. The LSC s were calibrated with the GEN2000 computer program using a two point calibration method. The distance LSC was extended was measured and noted in the GEN2000 program. Finally the LSC was fully compressed and this was also noted in the GEN2000 program. The procedure was then repeated for the other LSC Model Footing The model footing was a steel bar with the base dimensions of 194 mm x 25 mm. When accelerated to a rate of 30g s it will simulate a footing of 5.82 m x 0.75 m. This will approximate a strip footing Sand Placement The strongbox used to hold the sand had internal dimensions of 197 mm x 254 mm x 194 mm deep. Since only 100 mm of room in the box was needed for the sand a 76 mm thick wooden block was placed at the bottom. This allowed the soil sample to be closer to the top of the box and the measuring devices. Sand was air pluviated into the Strongbox from a hopper which was suspended over the strongbox bolted on the centrifuge arm. The total depth of sand was 100 mm, which was composed of a layer of dense sand over layer of loose sand. The loose sand was placed in the strongbox first by maintaining a drop height of 0.5 cm. The dense sand was placed in the box using a drop height of 40 cm. The dense layer was placed in approximately 1 cm lifts rather than the constant height at which the loose soil was maintained. Figure 3.1 shows the placement of the dense layer. 11

20 Figure Filling the Strongbox with Sand Once the Box had been filled to slightly above 100 mm, the surface of the sand was vacuumed level with a shop-vac and a leveling guide. This ensured that the top of the sand was level. Vacuuming is demonstrated in Figure 3.2. Figure Levelling Sand with Vacuum Model Assembly Once the sand was placed, the final assembly of the model and the measuring instruments was performed. First the model footing was placed with a light wooden jig that was constructed to allow for correct model placement. The model was carefully placed on the 12

21 sand so as not to compact the sand underneath. This process is shown in Figures 3.3 and 3.4 Figure The Jig Placed in the Strongbox Figure Placing the Model Footing using the Jig Once the footing was placed the jig was removed and the pneumatic ram was bolted on the strongbox. A small ball-bearing was placed on the center of the footing to act as a contact point between the piston and the footing. Once the ball bearing was placed on the footing the piston was carefully extended until it was almost touching the ball-bearing. The LSC s were then ready to be placed into their machined locations on the pneumatic ram. The LSC s were placed in such a manner that they are fully compressed. As the footing is pushed into the soil they extend and correspond to the footing settlement. Finally the lines which carry the nitrogen to operate the pneumatic ram are connected to the hydraulic slip ring of the centrifuge. A photograph of the footing, LSC s, and the load cell is shown in Figure 3.5 and a sketch of the final apparatus can be seen in Figure

22 Figure Footing with Load Cell and LSC's Figure Apparatus Final Test Preparations The centrifuge needs to be properly balanced as the model is mounted on one arm of the centrifuge. The weight of sand is approximated by the sand having an approximated density of 1500kg/m 3 and a known volume. Finally the strongbox, wood block and apparatus all have known masses allowing an equal mass to be placed on the other arm of the centrifuge. 14

23 The final procedure is for all the measuring devices to be connected into the electronic panel on the centrifuge. The devices were connected into the following channels: Table Channels used for electronics Instrument Channel LSC 2 2 LSC 1 3 Load Cell 1 Accelerometer 0 Finally the centrifuge area was cleared of all debris and the room vacated prior to starting the centrifuge Test From the control room the data logger is started followed by the starting of the centrifuge. The data is logged during the acceleration of the centrifuge. Though these data were not used in the actual calculation, they give an indication of how the model is performing under self weight. This was especially important with the test when entirely loose sand was tested as the model nearly failed under its own self weight. The centrifuge is accelerated to as close to 30 g s as possible without going under 30 g s. Once the required speed has been reached, the load is then applied the footing with a tank of nitrogen located in the control room. Due to the valve on the nitrogen tank it was difficult to slowly increase the pressure, so rather then smooth increases the pressure was increased in steps. As load was being increased, the deflection of the footing was monitored up to a displacement of 13 mm. Finally the footing was unloaded and the centrifuge stopped. During the slowing down of the centrifuge data logging was not carried out. 15

24 3.3 Data Analysis The results of the data collection are then imported into excel as CVS data and plotted as load vs. deflection. The point of failure is determined and the pressure is determined from that, using the footings area load q f =... (12) A Where: load = load at failure A = area of footing Once q f is determined φ is needed. To determine φ the relationship between Nγ and N q (Craig 1997) is used. q f = 0.5γ t B Nγ + γ t D N q... (13) N q =exp(πtan(φ))tan 2 (45 + φ/2)... (14) N γ = (N q -1)tan(φ)... (15) Equations (13), (14), and (15) are then combined to give us an equation which can be used to solve for φ. q f =0.5γ t B (exp(πtan(φ))tan 2 (45 + φ/2)-1)tan(φ) + γ t D exp(πtan(φ))tan 2 (45 + φ/2)... (16) All quantities are known in the equation except φ. All of the measured data can now be compared with the theoretical equations to determine φ. Once φ has been determined the theoretical bearing capacity will be calculated using the equation q u = q lower +γ t H 2 (1+2D/H)K s tan(φ)/b + γ t D (which was shown in the literature review portion of this report). 16

25 4 Results Seven centrifuge tests were completed. The maximum bounds from which φ and q lower was derived from were completed twice, with the other 3 tests were for the H/B ratios of 0.5, 1, 2. The results are presented in Figure 4.1, and individual data runs are presented in Appendix A. 4.1 Test Results The ultimate bearing capacity was determined at 12 mm deflection or a settlement of approximately 50% of the base width. A structure would not tolerate a deflection of nearly 50% of the base width and as such this value was used to define q f. The results which are presented in Figure 4.1 show a general trend of increasing bearing capacity as the thickness of the upper dense layer increases. This result is as expected; as the upper layer becomes thicker more force will be required to punch through the upper dense layer. Once the upper layer reaches sufficient thickness, punching is no longer the governing mode of failure. This is demonstrated in that H/B of 2 and 4 having similar results. The reading from the load cell was multiplied by g or 9.81m/s 2 to determine the force applied to the footing. This force is then divided by the footing area to obtain a pressure. 17

26 Load vs Displacement Pressure (kpa) H/B 0.5 H/B 1 H/B 2 H/B 4 H/B 4 H/B (run 2) Displacement (mm) Figure Measured Load vs. Displacement 4.2 Theoretical Results The experimental results were compared with the theoretical results based on the equations outlined in the literature review Ultimate Bearing Capacity Using equation (16), φ for dense layer was determined to be 43.6 o and φ for the loose layer was determined to be 28.3 o, though φ for the loose layer is not needed in any calculations. The measured bearing capacity for H/B ratio of 0 is used for the value of q lower. Using φ dense, q lower, and γ t in equation (4), theoretical values can be calculated for the different H/B ratios. See Appendix B for calculations. The results are summarized in the following table: 18

27 Table Theoretical vs. measured ultimate bearing capacity and φ H/B Ratio Theoretical Ultimate Bearing Measured Ultimate Bearing Capacity Percent Difference (%) Capacity (kpa) (kpa) φ dense = 43.6 o φ loose = 28.3 o Settlement The theoretical settlement values were determined using Schmertmann s method as outlined in the literature study. The value for Young s Modulus, E, for dense and loose soils was determined by using the measured pressures at 4 mm settlement and solving for E. These E values were then used to calculate the settlements at a pressure of 50 kpa for the H/B ratios of 0.5, 1, and 2. Calculations are shown in appendix C. Table Settlements for 50 kpa pressure H/B Ratio Theoretical Settlement (mm) Measured Settlement (mm) Percent Difference (%)

28 5 Discussions The results are discussed further in this section. 5.1 Experimental vs. Theoretical Ultimate Bearing Capacity The experimental results of this study were compared with the theoretical values based on Meyerhof s findings for the expected bearing capacity of the footing on a layered soil. As seen in the following figure (Figure 5.1) the theoretical estimates for H/B greater than 1 start to overestimate the actual bearing capacity of the soil. It can also be seen that increasing the thickness of the upper sand layer increases the bearing capacity until an H/B of 2 is reached. However the equation which assumes a punching failure for the upper sand layer overestimates the capacity of the upper soil. As the upper layer becomes thicker it is less likely to fail in punching and more likely to be a standard bearing capacity failure. This is likely what is happening between the H/B ratio of 1 and 2. 20

29 Calculated Results and Measured results qf(kpa) 300 Calculated Results Measured Results H/B Figure Experimental Results and Theoretical Results Settlement Schmertmann s method provided generally good results. In all cases but H/B = 2, the theoretical values were greater than the measured. The value predicted for settlement with H/B = 2 is very close with approximately a 2% difference. This difference could possibly be accounted for with experimental error. Since no tests were completed for H/B ratios between 2 and 4, it is unknown if the values between 2 and 4 are accurately predicted. As can be seen in Figure 5.2, the predicted settlements for the theoretical and the measured values are changing at different rates. This is due to the predicted values following the 2 linear lines in the strain influence diagram, whereas the measured values 21

30 follow a non-linear shape. Though there is a difference in the curves they both have the same general shape, making Schmertmann s method a reasonable prediction. Settlement vs Sand Thickness Theoretical Measured 140 Settlement (mm) H/B ratio Figure Settlement vs Sand Thickness 5.2 Problems Encountered There were few problems encountered in this study. The problems encountered centred on not being able to apply a load in fine enough increments. This created jumps in the data. The solution which was used in this case was to do some of the tests twice to try and pick up missing data points. 22

31 6 Conclusions This study has shown that as the upper dense layer of a soil increases in thickness the ultimate bearing capacity generally increases up until an H/B ratio of 1. The following conclusions can be derived about a footing on dense sand over loose sand Meyerhof s findings for sand over clay can be applied to the dense sand over loose sand with the difference that it only applies until an H/B ratio of 1 is reached. At an H/B ratio of about 1 the failure is less to do with a punching failure from the dense sand but increasingly more of a standard bearing failure. At an H/B ratio of about 2 the dense layer of sand is no longer experiencing punching failure. Schmertmann s method overestimates the settlement which the footing experiences for H/B ratios less than 2, but otherwise it provides a reasonable estimate. 23

32 7 Recommendations From this study 3 recommendations can be made 1) Redo the testing with an apparatus which can apply a load in finer increments so a smoother curve can be derived 2) More testing is needed for H/B ratio of 1 and some testing of for an H/B ratio of 1.5 to get more results around the critical area which the punching failure starts to convert to a bearing failure 3) Test at least one run at a H/B ratio of 3 to determine if the theoretical settlement is still underestimated. 24

33 References Craig, R.F Soil Mechanics, 6 th Ed. Spon Press, London New York Das, B.M, Principles of Foundation Engineering. PWS-Kent, Boston. Meyerhof G.G Ultimate Bearing Capacity of Footings on Sand Layer Overlaying Clay. Canadian Geotechnical Journal, 11(2) pp Myerhof G.G., Hanna A.M, Ultimate bearing capacity of footings on layered soils under an inclined load. Canadian Geotechnical Journal, 15, pp Stewart, Marcie Smooth Strip Footing on a Finite Layer of Sand Over a Rigid Boundary. University of New Brunswick, Fredericton Taylor, RN Geotechnical Centrifuge Technology. Blackie Academic and Professional, Bishopbriggs, Glasgow. 25

34 Appendix A Data Collection Run Graphs 26

35 H/B 0 Data Collection Displacement (mm) Pressure (kpa) 27

36 H/B 0.5 Data Collection Displacement (mm) Pressure (kpa) 28

37 H/B 1.0 Data Collection Displacement (mm) Pressure (kpa) 29

38 H/B 2.0 Data Collection Displacement (mm) Pressure (kpa) 30

39 H/B 4.0 Run 1 Data Collection Displacement (mm) Series1 Pressure (kpa) 31

40 H/B 4.0 Run 2 Data Collection Displacement (mm) Series1 Pressure (kpa) 32

41 Appendix B Bearing Capacity Calculations 33

42 34

43 35

44 Appendix C - Settlement Calculations 36

45 37

46 38