Keywords: Piping inspection robot, Caterpillar locomotion, Force control algorithm, BeagleBone (BB) black

pipelines of unknown diameters, the reconfigurable pins of BB black are employed to apply a voltage in order to achieve the desired clamping force for having contact with pipeline walls. By setting threshold limits, the voltage and current supply to the motors can be cut-off at the end of clamping or retraction operations. With the help of ADC, the output voltage can be displayed to the user on a computer. For smooth functioning of the robot, the parameters of the motor such as the maximum velocity, direction of rotation and nominal current are calibrated prior to the start of experiments. The output voltage and currents of the motors are generated and plotted for a locomotion cycle of the robot inside a 74 mm diameter straight pipeline.

Prototyping a piping inspection robot using a BeagleBone black board

Swaminath Venkateswaran a , Damien Chablat b , Ramakrishnan

Ramachandran c

a. Ecole Centrale de Nantes, Laboratoire des Sciences du Numérique de Nantes (LS2N), UMR CNRS 6004, 1 rue de la Noë, 44321 Nantes, France, swaminath.venkateswaran@ls2n.fr b. CNRS, Laboratoire des Sciences du Numérique de Nantes (LS2N), UMR CNRS 6004, 1 rue de la Noë, 44321 Nantes, France, damien.chablat@cnrs.fr c. Mechatronics Lab, Department of Design & Automation, School of Mechanical Engineering (SMEC) Vellore, Vellore Institute of Technology (VIT), Vellore, India, ramakrishnan.rc@vit.ac.in

Résumé :

Dans le cadre d'un projet avec AREVA, un robot d'inspection de canalisation de 50 à 94 mm de diamètre a été développé. En utilisant des pattes, des moteurs à courant continu, des vis-écrous et des cartes électroniques, le robot imite la locomotion d'une chenille. Deux algorithmes peuvent être utilisés pour le contrôle de ce robot: le contrôle en position ou le contrôle en effort. Le premier est utilisé pour l'estimation de la position d'un robot lorsque l'environnement est connu mais nécessite des codeurs pour identifier la position des pattes. Dans le cas du contrôle en effort, l'algorithme permet au robot de se déplacer dans un environnement inconnu. En utilisant une carte électronique BeagleBone (BB) black, un algorithme de contrôle en effort est développé pour ce robot d'inspection. Comme le robot travaille à l'intérieur de tuyaux de diamètres inconnus, les sorties PWM de la BB black sont utilisées pour générer la tension nécessaire qui permettant le clampage du robot. En fixant ces seuils, la tension et le courant dans les moteurs peuvent être évalués pour détecter la fin des mouvements de clampage ou de repliement. Avec l'aide des ADC, la tension de sortie est affichée à l'utilisateur sur un ordinateur. Pour le bon fonctionnement du robot, les paramètres du moteur tels que la vitesse maximale, le sens de rotation et le courant nominal sont étalonnés au début des expériences. La tension de sortie et les courants consommés sont générés et tracés pour un cycle de locomotion du robot dans un tuyau droit de 74 mm de diamètre.

Abstract :

As a part of a project with AREVA, an inspection robot for 50-94 mm diameter range pipelines have been developed. By using leg mechanisms, DC motors, spindle drive, and control boards, the robot accomplishes the locomotion of a caterpillar. Two algorithms can be classified for the control of this robot via Displacement control algorithm and Force control algorithm. The former deals with the estimation of the position of a robot inside a known environment and it requires sensors for identifying position as well as for initiation procedures. In the case of force control, the algorithm is highly suitable for a robot that moves inside an unknown or closed environment. Using a BeagleBone (BB) black board, the force control algorithm is developed for the inspection robot under study. As the robot works inside

Introduction

Piping inspection robots play a significant role for industries such as nuclear, sewage as they reduce human effort and impose lesser risks to human life. Such inspection robots employ one or more electronic boards for their control and movements when moving through various trajectories of pipelines. Some of the well-known devices are the Raspberry Pi [START_REF] Upton | Raspberry Pi user guide[END_REF], Arudino [START_REF] Smith | Introduction to arudino a piece of cake[END_REF] and BeagleBone (BB) black [START_REF] Coley | Beaglebone black system reference manual[END_REF]. The choice of these boards or micro-computers purely depends on the intended application. Leccese et al. [START_REF] Leccese | A smart city application: A fully controlled street lighting isle based on raspberry-pi card, a zigbee sensor network and wimax[END_REF] studied on the comparison of these three micro-computers. Arduino is usually used for programming with low-level C++ languages and it can also be used to create an interface between sensors. An interesting example is the vibro-impact capsule type inspection robot prototype of Yusupov et al. [START_REF] Yusupov | Development of a self-propelled capsule robot for pipeline inspection[END_REF] where an Arduino board is used to propel the robot inside a Poly-vinyl chloride(PVC) test pipe. Raspberry Pi and BB Black, on the other hand, works efficiently for high-end programming languages such as Python and MATLAB. These micro-computers operate on Linux platform with access to the internet. Raspberry Pi is intended for applications with GUI and cannot be compatible for interfacing with external sensors. However, it has compatibility for camera interfacing which makes it efficient for moving inside dark pipelines. The robot of Abdellatif et al. [START_REF] Abdellatif | Mechatronics design of an autonomous pipe-inspection robot[END_REF] employs a Raspberry Pi for robot control and detection of cracks inside a dark environment. A bio-inspired caterpillar type robot was proposed by Henry et al. and Chablat et al. [START_REF] Henry | Multi-objective design optimization of the leg mechanism for a piping inspection robot[END_REF], [START_REF] Chablat | Mechanical design optimization of a piping inspection robot[END_REF]. This robot is capable of working inside pipelines having diameters between 50-94 mm. A prototype of this robot was developed at LS2N, France and the testing was carried out inside a transparent PVC pipe of 74 mm diameter and 2.5 m length. A BB black is utilized for the control and locomotion of this robot. This paper concentrates on the architecture of this robot and the results of voltage and currents induced on the motors during a locomotion cycle.

In general, two algorithms can be classified for the control of a robot namely Displacement control and Force control. The former is used for a robot that works in an open environment and moves through a known distance. Since the diameter of any instance of a locomotion step is unknown, the force control algorithm is employed where through voltage control, the necessary force is applied to the motors in order to clamp the robot legs with pipeline walls. DC motors and spindle drives provided by Maxon is used in this robot to accomplish the caterpillar locomotion. The outline of the paper is as follows. In the first section, the locomotion procedure and the electronic architecture for the prototype is presented. Followed by that, the force control algorithm is exploited. The results of voltage induced on the motors during a locomotion cycle are then presented. Finally, the paper ends with conclusions and future works.

Assembly of the robot

Locomotion principle and architecture

The robot proposed in [START_REF] Henry | Multi-objective design optimization of the leg mechanism for a piping inspection robot[END_REF], [START_REF] Chablat | Mechanical design optimization of a piping inspection robot[END_REF] mimicks the locomotion of a caterpillar. The general motion of a caterpillar comprises of a front and rear elongation module and a central blocking module. Anthierens el al. [START_REF] Anthierens | Design of an electro pneumatic micro robot for in-pipe inspection[END_REF] developed a similar inchworm type electro-pneumatic robot that uses bellows to accomplish this locomotion. However, inside nuclear power plant pipelines there exists dust particles and deposits which limits the use of bellows and it is complex to apply a force control algorithm for systems using bellows. Electrical actuators are thus used in the robot to accomplish the caterpillar locomotion which is accomplished in six steps [START_REF] Chablat | Mechanical design optimization of a piping inspection robot[END_REF]. The outline of the robot inside a pipeline of radius r is shown in Figure 1. A slot-follower leg mechanism is used in the front and rear modules to have contact with pipeline walls during locomotion. In Figure 1, AP represents the slot-follower leg. O is the reference point for the leg mechanism. AC represents the active prismatic link which has a stroke length of ρ. The leg is hinged at B in order to slide and have contact with pipeline walls. The stroke length of the central module is represented by ρ c . A Maxon EC motor (brushless DC motor) coupled with a Maxon spindle drive is used to accomplish the clamping and elongation processes. The specifications of the motor and spindle drive are provided below in Tables 12.

P(z,y)

Interfacing with BB black and ESCON controllers

For the command and control of the robot, a BeagleBone (BB) black micro-computer is used. The BB black offers 92 pins which comprise of the Pulse-Width modulation (PWM), General-Purpose Input/Output (GPIO), Analog to Digital converters (ADC) and reconfigurable digital ports [START_REF]Beaglebone black[END_REF]. Each Maxon motor is interfaced with the BB black through their ESCON 36/3 servo controller by three pins.

One pin is connected to the PWM which is used to operate a motor on a duty cycle in a particular direction. The other pin is connected to a GPIO which is used to enable or disable the motor functioning in a boolean way. The third pin is connected to the ADC to read the motor voltage. The PWM of the BB provides a maximum voltage of 3.3 V and it is used to operate the motors with control over the speed.

For the idle stage, 50% duty cycle is set as a reference where the current is 0 A. For the safe running of motors, 80% of the duty cycle of the PWM is only used. At 80% duty cycle the motor performs either clamping or retraction operation with clockwise rotation and at 20% duty cycle the motor performs declamping or elongation operation with anti-clockwise rotation. The interfacing between BB black and one set of EC motor and its controller is provided in Figure 2.

In Figure 2, the J2 cable of each ESCON controller is connected to its EC motor unit. The J1 cable of each controller is connected to a power source of 12 V. The cables J5 and J6 are connected to the PWM, GPIO and ADC pins of BB black. The PWM, GPIO and ADC pin address on BB black for each motor and its servo controller is listed below in Table 3. Before the start of experiments, the DC motors and ESCON controllers are calibrated in a computer through a USB cable through port J7 as per the specifications mentioned in Tables 12. The output voltages from PWM which are generally analog is passed to the ADC to display the outputs to the user. The ADC of BB black, however, supports a maximum voltage of 1.8 V. Thus, the output analog voltage before passing to ADC of BB black is calibrated in the ESCON software in accordance with the nominal current of the motor. The BB black also offers Enhanced Quadrature Encoder Pulse (eQEP) pins which can be used to determine the position/displacement of robot at any instance of locomotion. However, this feature is not included in the current control loop of the robot. The control methodology for the

Control methodology-Force control algorithm

In the real world application, the robot under study will be implemented for inspection inside dark pipelines with no visual information about its current position. A force control algorithm is employed which is used to control the locomotion of robot inside an environment of unknown size. By applying the necessary clamping force from the power source to the motors, the robot can adapt its legs and clamp itself to pipeline walls. Initially, interfacing between BB black and each motor as well as its servo controller is done as per Figure 2. The BB black is connected to a computer and this board operates on Debian OS 7.9. The activation of pins ‡ specified in Table 3 are done using the linux terminal of BB black micro-computer. When a PWM duty cycle is applied to the motor, the spindle drive enables the leg mechanisms to clamp or de-clamp or allow the central motor to elongate or retract. The motors rotate at a constant speed during its duty cycles. A simpler form of the pseudo-code for one forward motion is provided in Algorithm 1 in Appendix. At the end of one loop cycle, the robot remains in an over-constrained posture wherein both the leg mechanisms are clamped with the walls of the pipeline. The prototype of the robot developed at LS2N, France is depicted in Figure 3.

In Figure 3, A represents the working of robot inside a test pipe and in B, the side view of the robot is shown. In C, the ESCON controllers for the robot is represented. The following operations are ‡ Pin activation of BB black in Debian 7.9 available inhttps://github.com/stvt1991/Control-of-piping-inpsection-robot/tree/master/ Pin-activation • Forward motion (twice)

• Reverse motion (twice)

In order to ensure that there is no continuous running of motors after an operation is finished, threshold limits are being set for the clamping/retraction as well as de-clamping/elongation phases. The upper threshold T U is set at 1.5 V which is 83% of the maximum voltage of ADC and the lower threshold is set at 0.42 V. These values are determined based on experimental trials. Also, these values are ideal when operating inside variable sections as the diameter of any locomotion step is unknown. When the limits are reached, the torque reaches its stall point and the PWM is set to idle phase.

Results of experiments for a locomotion cycle

With the interfacing between controllers and BB black being done, the parameters defined below in Table 4 are calibrated for the ESCON controllers before the algorithm is initiated. The experiment is carried out inside a 2.5 m length PVC transparent tube of 74 mm diameter as depicted in Figure 3. The ADC of the BB black generates the output voltage which is extracted and the results can be traced in The total time taken for the completion of the entire algorithm is 179 seconds. At 89.5 seconds, two sets of forward motions are completed and the reverse motion is initiated. At the end of experiments, the robot returns to its initial position to an over-constrained posture. The de-clamping operation on both legs could be performed to take out the robot. In Figure 4, the red and blue lines indicate the left and right leg actuators. The orange lines highlight the central motor. The threshold voltages are indicated by black and gray lines. It could be observed from the results that higher noises are generated in the output voltage of ADC. There are several factors which could contribute to these noises. When one set of leg mechanisms are clamped and the other end is free, a bending phenomenon was observed which could lead to potential numerical errors in the measure. In order to overcome this issue, a numerical filter is applied. Using MATLAB, the frequency of the noise is measured. With the help of Savitzky-Golay filtering technique in MATLAB [START_REF]Savitzky-golay filtering[END_REF], the measured frequency is matched with the desired frequency of BB black (2 kHz) for reducing the noise.

Using the smoothened output voltages, the current induced on the actuators can be estimated by the equation:

I = V t I c V idle -I c (1)
In Eqn. (1), V t is the ADC output voltage at a given time t of the locomotion cycle. I c is the nominal current of motor which is taken from Table 1. V idle is the ADC voltage at 50% duty cycle which is taken as 0.9 from Table 4. However, the current value obtained from Eqn. (1) does not provide the actual current as there exists power losses in the DC motor that is generated due to the resistance of the windings. The actual current measured from the motor is given by:

I a = I -I n (2)
Here, I n is the no-load current of motor and this value is subtracted from the result of Eqn. ( 1) to obtain the actual current induced in the motor. The results of actual current induced on each motor during the locomotion cycle is represented in Figure 5. It could be observed in Figure 5 that smoother results are obtained with the application of filter. Also, higher starting voltages and currents are observed for each motor when a duty cycle is applied. This is because the motor rotates at a constant velocity and the time taken to attain this velocity when a duty cycle is applied is very short. This can be however rectified by following a motion profile such as "bangbang", where the velocity is gradually increased until it reaches a constant value [START_REF] Khalil | Modeling, identification and control of robots[END_REF]. The de-clamping operation of motors M 1 and M 3 is not done completely in order to achieve larger displacements as the output velocity of the spindle drive is very low of around 0.43 mm/s [START_REF] Chablat | Dynamic model of a bio-inspired robot for piping inspection[END_REF]. However, the central actuator always extends/retracts to a maximum and this could be observed in Figures 4 and5.

With the help of the output currents, the torques induced on the motors followed by the forces can be calculated by the below equations.

τ = I a K T G η m η s (3)
F = 2πτ p (4)
The parameters in Eqns. ( 3)-( 4) such as the torque constant (K T ), gear ratio (G) and efficiencies (η m and η s ) can be taken from Tables 1-2. In Eqn. (4), p represents the pitch of the screw in the spindle drive. The output forces and torques can then be studied for comparison with the results of simulation estimated in [START_REF] Chablat | Dynamic model of a bio-inspired robot for piping inspection[END_REF].

Conclusions and future works

The control of a bio-inspired piping inspection robot was studied in this article. Using a BB black micro-computer, the robot was tested inside a transparent PVC pipe of 74 mm diameter for a period of 179 seconds. It was possible to study the voltage induced on the motors during the cycle using the ADC of BB black. By applying numerical filters, the results were smoothened in order to derive the actual currents induced on the motors after power losses. The currents generated can then be used for estimating torques and forces induced on the motors during the locomotion cycle. The force control algorithm allowed us to carry out the experiments "blind" without going through a complex procedure. By experimental trails, the threshold limits were derived wherein the forces necessary to clamp the robot with pipelines walls can be attained. Higher threshold limits cannot be set as they not only damage the motors but also can possibly deform the leg mechanisms after the contact is established.

The force control algorithm allowed us to test the robot inside a pipe of 74 mm diameter. However, with varying pipeline diameters, it cannot be assured that the same threshold limits can hold good. This is one of the major limitations of this algorithm. In order to overcome this issue, in the future, the existing BB black will be replaced by a BB blue micro-computer. With the BB blue, the eQEP can be incorporated to the robot and it will be possible to detect the displacement of legs for an unknown pipeline diameter. The algorithm will be extended with the eQEP that will have a relation between leg displacement and the threshold limits. Another advantage of the BB blue is that it requires comparatively lesser cables for connection between the ESCON controllers over the existing BB black.

Figure 1 :

 1 Figure 1: Outline of the robot inside a pipeline of radius r

Figure 2 :

 2 Figure 2: Layout of the interface between BB black to one EC-motor and its servo controller

Figure 3 :

 3 Figure 3: Prototype of the robot developed at LS2N, France **

Figure 4 :

 4 Figure 4: Output voltages of motors measured from ADC during experiments

Figure 5 :

 5 Figure 5: Actual currents induced on motors during the experiments

Table 1 :

 1 Specifications of the Maxon EC-motor (Part No. 283828)[START_REF] Motors | program 2017/18. high precision drives and systems[END_REF]

	Parameters	Value
	Nominal voltage	12 V
	Nominal current (I c )	0.456 A
	No load speed	13500 rpm
	No load current (I n )	60.2 mA
	Nominal speed	5840 rpm
	Efficiency (η m )	54 %
	Torque constant (K T ) 7.82 mNm/A
	Speed constant (N

v ) 1220 rpm/V For the control of motors, three ESCON 36/3 servo controllers offered by Maxon

[START_REF] Motors | program 2017/18. high precision drives and systems[END_REF]

are used, with one controller for each motor. This controller ensures speed and torque control of the motor units. Using the motor units, spindle drive, coupling screws and standards fasteners such as circlips, studs, nuts and bolts,

Table 2 :

 2 Specifications of the Maxon Spindle drive (Part No. 424749)[START_REF] Motors | program 2017/18. high precision drives and systems[END_REF]

	Parameters	Value
	Spindle screw diameter and pitch (p) φ5 x 2 mm
	Reduction ratio (G)	455:1
	Max. static axial load	500 N
	Max. spindle length	102 mm
	Efficiency (η s )	63 %
	the entire robot can be assembled. A 3D model of the robot is also realized using CATIA software

† .

Table 3 :

 3 BB black pin address for each motor and its servo controller Motor

PWM (duty cycle) GPIO (Enable/Disable) ADC (Results)

	Motor-1	P9_14 (DigIN 1)	GPIO_14 (DigIN 2)	AIN0 (AnOUT 1)
	Motor-2	P8_13 (DigIN 1)	GPIO_47 (DigIN 2)	AIN1 (AnOUT 1)
	Motor-3	P9_31 (DigIN 1)	GPIO_68 (DigIN 2)	AIN2 (AnOUT 1)

Table 4 :

 4 ESCON controller calibration before initiation of Force-control algorithm PWM

Duty cycle ADC voltage (V) Nominal current (A)

				Motor speed (rpm)
	20%	0	-0.46	10800 (Counter-clockwise)
	50% (idle)	0.9	0	0
	80%	1.8	0.46	10800 (Clockwise)
	MATLAB. The results of output voltage of each motor for two forward and reverse motions is depicted
	in Figure 4.

† 3D model of the robot in CATIA and simulation available inhttps://www.youtube.com/watch?v=7z6-by83mtw

Appendix


A Lapprand
email: aude.lapprand@insa-lyon.fr

F Boisson

F Delolme

F Méchin

J-P Pascault

Reactivity of isocyanates with urethanes: Conditions for allophanate formation

Keywords: Urethane, allophanate, NMR, MALDI-TOF

The reactions between two polyisocyanates, 4,4'-methylenebis(phenyl isocyanate) (MDI) and trimer of isophorone diisocyanate (tIPDI) and a model aryl-alkyl diurethane were carried out at high temperature ( ≥ 170°C) and several NCO/urethane ratios. A combination of 1 H & 13 C NMR and MALDI-TOF spectroscopies was used and allows the identification of reaction products. When MDI or tIPDI is reacted with a diurethane at high temperature and after cooled, allophanates are formed but no isocyanurates. 13 C NMR was used to quantify the different reaction products obtained under different experimental conditions. Only a few allophanates ( ≤ 10%) are obtained after 1h of reaction.

Introduction

The chemistry of polyurethanes rests above all on the reactions between alcohol and isocyanate, which have been widely studied [1]. The formation of urethane groups results from the addition between isocyanates and alcohols, with kinetics depending on the nature of the two reagents and of the solvent. The electronic effects of substituents on the reactivity of isocyanates are of great importance, since it was shown that phenyl isocyanate was fifty times more reactive than ethyl isocyanate with respect to methanol [2]. The effect of solvent [3][4][5] on the rate of the isocyanate-alcohol reaction was also confirmed, the rate of reaction decreasing approximately in order of the increasing dielectric constant of the solvent and the ability of the solvent to hydrogen bond with alcohol.

In addition, a number of side reactions may take place depending upon the reaction conditions such as the temperature, the ratio between alcohol and isocyanate, and the presence of catalyst (Fig 1). These side reactions result in the formation of allophanates and isocyanurates under special conditions, together with ureas and biurets if there are traces of water during the reaction (Fig 2)

The determination of allophanate and biuret has been of great interest. Kogon [6][7][8] followed the disappearance of the absorption band in the near infrared to study the formation of allophanate in the reaction of urethane with aryl isocyanate. Furukawa and Yokoyama [9] used aliphatic primary amine to distinguish the allophanate and the biuret from carbamate and disubstituted urea due to the selective degradation of allophanate and biuret by this amine. Sumi et al. [10] showed that urea, urethane, allophanate and biuret linkages exhibited their own characteristic chemical shifts in 1 H NMR spectra. Okuta [11] dissolved the formed polyurethane networks in dimethylsulfoxide, DMSO and dimethylacetamide, DMAc, by degrading the allophanate and biuret linkages with aliphatic primary amine followed by NMR analysis of the degradation products to yield quantitative information about crosslinks. Spirkova et al. [12] and Schwetlick et al. [3] used HPLC to identify the reaction products with model compounds, in the same way as Sebenik [13] with SEC and 13 C NMR.

According to Spirkova, no secondary reactions [12] can occur at room temperature without catalyst. At 50°C, in acetonitrile, Schwetlick [3] identified 3 different reaction conditions: at equimolar NCO/OH ratios, with or without catalysts like tin carboxylates and common tertiary amines, urethane is the main product when isocyanates react with alcohols, the kinetic rate constants are in the order k a >> k b ≈ k c (see Fig 1 for the reactions). At equimolar NCO/OH ratios, and with catalysts such as aminals, aminoalcohols, amidines and carboxylate, phenolate and alkoxide anions, the order of rate constants is k a < k b ≈ k c . In the same way, at high NCO/OH ratios, isocyanurate is the only final product formed via the urethane and allophanate as detectable intermediates.

At 130°C, and with a NCO/OH ratio of 4, Sebenik et al. [13] found that the kinetic rate constant for the formation of allophanates from isocyanates and polyols lied in the range 4-7.10 -5 L.mol -1 .s -1 . Kogon [8] found that at 100-140°C, the rate constant of ethyl-α,γ-diaryl allophanates increased from 1.0.10 -6 to 6.0.10 -6 L.mol -1 .s -1 . The allophanate formed was an intermediate reaction product in the formation of isocyanurate when the equilibrium of the phenyl isocyanate/ethyl urethane reaction (NCO/urethane = 12.6) was approached.

Allophanate can dissociate, ending to the reformation of urethane and isocyanate. The stability of allophanate depends on the type of structure and on temperature. By using model aliphatic, cycloaliphatic and aromatic compounds, it was shown [14,15] that aromatic allophanates are the least stable compounds and dissociate to a considerable extent, even if heated for a short time only (10 min at 165-210°C) [16]. The rate of allophanate dissociation observed by Kogon [17] increases with temperature, from 0.39.10 -4 to 1.65.10 -4 s -1 at 132 to 150°C. According to Parodi [1], the fact that measurable dissociation occurred at 106°C shows that the allophanate crosslink would be relatively unstable.

The dissociation of allophanates is due to the reversible character at high temperature of the addition reactions of isocyanate with compounds that have active hydrogens. This proceeds by the same mechanism as the dissociation of the urethane linkages. Bayer [START_REF] Bayer | Das Diisocyanate-Polyadditions Verfahren[END_REF] reported the upper stability temperature for different urethane groups shown in Table 1.

Yang et al. [START_REF] Yang | [END_REF], in the same way as Camberlin and Pascault [20], studied the dissociation of urethanes synthesized from 4,4'-diphenyl-methane diisocyanate and 1,4-butanediol. These urethanes were found to dissociate thermally to free isocyanate and alcohol at elevated temperatures. Degradation is possible even in the solid state if annealing time is long enough, i.e. 2 hours at 170°C. One may expect a most probable molar mass distribution after the random scission process during the degradation and the random reformation of urethanes during cooling. In any experiment involving urethanes heated at high temperatures, the possibility of degradation must be taken into account.

In this study, the reactivity of the urethane linkage with polyisocyanates at high temperature has to be known. Lu et al. [21] investigated reactions between urethanes and many functional compounds, except isocyanates. A model urethane compound, dioctyl 4,4'-methylenebis(phenyl carbamate) (OMO) was synthesized by capping 4,4'-methylenebis(phenyl isocyanate) (MDI) with n-octanol, in the same way as Lu et al. (Fig 3). This low molar mass diurethane was reacted with either MDI, or trimer of isophorone diisocyanate (tIPDI) by varying 2 parameters, stoichiometry and temperature.

The products of the reaction between OMO and MDI or t.IPDI have been characterized by SEC, 1 H and 13 C NMR and MALDI-TOF, in order to be able to anticipate the results of a further grafting experiment involving a thermoplastic polyurethane, TPU, and MDI or tIPDI.

The final aim of this work is indeed to perform and to control the chemical modification of TPUs in order to obtain polymers that can be self-crosslinked after processing, for better thermomechanical characteristics by grafting isocyanates on the urethane units of TPU chains and thus introducing new reactive groups able to post-react [22,23].

Experimental

Analytical techniques.

Size exclusion chromatography (SEC) was performed in THF, using a Waters chromatograph equipped with 3 microstyragel columns and a refractive index detector. The column calibration was done referring to polystyrene standards.

High-resolution liquid NMR spectroscopy was carried out with a Bruker DRX 400 spectrometer operating at 400 MHz for 1 H and 100.6 MHz for 13 C. Spectra were obtained with a 5 mm-QNP probe at 298K. Deuterated chloroform, CDCl 3 , or dimethylsulfoxide, DMSO-d6, were used as solvents. Chemical shifts values are given in ppm with reference to internal tetramethylsilane.

All mass spectra were obtained using a MALDI-TOF Voyager-DE STR (Applied Biosystems) equipped with a nitrogen laser emitting at 337 nm. Samples were dissolved in THF or DMSO and mixed with Dithranol as matrix and NaI as cationization agent (10/1/1, v/v/v). An aliquot of 1 µL of the resulting mixture was spotted on the MALDI sample plate and air-dried.

Materials.

Octanol was purchased from Riedel-deHaën and used as received. MDI, provided by Huntsman (Suprasec 1100) contains ∼98% of 4,4' diphenylmethane diisocyanate and ∼2% of 2,4' diphenylmethane diisocyanate with Tf = 38°C. It was kept at 4°C, in order to prevent dimer formation.

OMO was synthesized according to a previously published procedure [21] and characterized by 1 H and 13 C NMR. The melting point of OMO was found by DSC between 117 and 120°C, as reported by Lu et al. [21].

The trimer of IPDI, supplied by Degussa (Vestanat T 1890/100) is an amorphous polyisocyanate, with a Tg = 70°C. According to a recent publication [24], the so-called trimer of IPDI is formed by many oligomers, not only trimer but also pentamer, heptamer or higher molar mass oligomers of IPDI (Fig 4), resulting from condensation of isophorone diisocyanate. (Fig. 5)

OMO modification.

OMO was mixed with MDI or IPDI trimer in a reaction flask at temperatures between 160°C and 210°C, under argon, using varying isocyanate/urethane ratios, i/u. At various reaction times, samples were removed and quenched in an ice-water bath. One part was then dissolved in tetrahydrofuran (THF) with 0.5%w/w secondary amine (diethylamine or di n-butylamine). The secondary amines are expected to react and neutralize the unreacted NCO. Both neutralized and non neutralized samples have been analyzed.

Results and discussions.

Urethane dissociation and OMO stability.

Urethanes are known to dissociate at high temperature, the reversible degradation releases free isocyanate and alcohol (Fig. 1-a). After heating at 180°C, even during 1 hour, the SEC chromatogram of OMO after cooling back to room temperature shows no difference.When heating at 210°C, the SEC curves indicate a decrease in OMO concentration while at this temperature, octanol is partially volatilized. It means that even if all OMO molecules could be dissociated at 180°C, alcohol and isocyanate functions would remain in stoichiometric ratio and during cooling, the NCO+OH reaction would be favored, and OMO reformed. But when heating at 210°C, alcohol and isocyanate are no longer in stoichiometry and the total amount of OMO can not be recovered after cooling. Furthermore, using the same model diurethane Lu et al. [21] showed that only 1% of urethane was converted after 60 min at 200°C.

Method for analysis of the reaction products between OMO and isocyanates.

When OMO is heated with an isocyanate, MDI or tIPDI at high temperature, in the range 170-200°C, the OMO dissociation reaction occurs; but as there is now excess isocyanate groups, during cooling down to room temperature for analysis and according to Figure 1, four main reactions can be expected.

When reacting isocyanates with alcohols, studies [3-5, 25, 26] have shown that side reactions are favored in solution and consequently the choice of the solvent to use for NMR analysis is a real problem. By calculating rate constants for the primary and secondary reactions, the results of Anzuino et al. [25] checked the hypothesis of allophanate formation in dimethylacetamide, without catalyst, at 10-30°C. Authors found that the allophanate/urethane ratio increased with increasing [NCO]/[OH] ratio and temperature. Moreover, an isocyanate compound is not stable in this solvent, since Matsui [26] showed the cyclization of the isocyanate group in DMAc at 40°C. Matsui et al. demonstrated that trimerization, which is known to be catalyzed by bases or metals, could take place slowly even without catalyst. They supposed that DMAc, as a slightly basic solvent, might enhance the reaction, being itself a catalyst.

In the first step of the study, NMR analysis of non-neutralized samples was carried out with DMSO-d6 as solvent.

In the C=O region of the 13 C-NMR spectra for non-neutralized samples, isocyanurate functions were identified by their resonances at 149.03 ppm. According to literature [13,27], the other four peaks have been attributed to the C=O groups from urethane (153.67 ppm), allophanate (154.88 ppm and 151.06 ppm), and aryl-aryl urea ( 152.57 ppm) functions (see Figure 6-b).

A detailed study showed that the samples changed very quickly (less than one hour) in DMSO-d6, at room temperature, without any catalyst. These side reactions lead to the disappearance of allophanate functions, the appearance of isocyanurates while no isocyanurate functions are present in the unchanged sample as shown in Figure 6.

Even for an unreacted sample of OMO + MDI, comparing the 1 H NMR spectrum recorded before and after a 13 C NMR acquisition, it was shown that the time required for the 13 C NMR experiment was sufficient to permit the formation of allophanates, because of the reaction of urethanes with isocyanates, at 25°C, in DMSO-d6. The 13 C NMR spectrum also indicates the presence of isocyanurate groups, which are present neither in the OMO spectrum nor in the MDI spectrum, but result from a reaction between both during the acquisition.

As assumed by Anzuino et al. [25] and Matsui et al. [26], the polar character of the solvent might be responsible for the relatively fast secondary reactions occurring in DMSO-d6. It should be noticed than the time required for the acquisition of the 13 C NMR spectrum is incompatible with the rapid evolution of the non neutralized products in DMSO-d6, contrary to some literature indications [28]. Once again, the high reactivity of isocyanate groups is pointed out as the necessity to neutralize the unreacted functions before analysis.

CDCl 3 , which is less polar, was then used to follow the reaction as the products were shown not to develop in this solvent. On the other hand, with this solvent, the NH proton signals were no more attributable because their resonances were very close to those of aromatic protons. The most suitable method was then found to carry out 1 H NMR analysis in DMSO-d6 and to use CDCl 3 for 13 C NMR analysis. Furthermore, a comparative NMR analysis of both neutralized and non neutralized samples has been achieved using these conditions in order to ensure that neutralization by a secondary amine did not degrade the products of reaction.

Reaction between the diurethane OMO and the aromatic isocyanate, MDI

The reactions were carried out at T = 170-200°C and the samples cooled to room temperature and neutralized or not by a secondary amine for different analyses.

A SEC chromatogram is given in Fig. 7. SEC peaks of both starting materials (OMO and MDI) are formally identified, and decrease during reactions while new peaks corresponding to compounds of higher molar masses appear and increase. Assumptions were made about these peaks, but to clearly identify the reaction products, a combination of 1 H & 13 C NMR and MALDI-TOF spectroscopies was used and appears very useful.

Enlargements of the C=O regions (b) and the NH regions (a) of 13 C and 1 H NMR spectra, respectively are shown in Figure 8 for a OMO+MDI mixture with equimolar NCO/urethane ratio, heated at 192°C for 60 minutes.

As mentioned before, three main peaks are observed in the C=O region of the 13 C-NMR spectra for the non-neutralized sample corresponding to urethane (153.91 ppm) and allophanate (151.67 and 165.10 ppm). These functions are also characterized by their NH proton resonances found in DMSO-d6 at 9.50 ppm for urethane and 10.65 ppm for allophanate.

Thanks to NMR analysis, allophanates have been identified as the main functions formed during the reaction between the low molar mass diurethane OMO and MDI at high temperature. No isocyanurate has been found as reaction product.

A complementary study was achieved by MALDI-TOF since this technique allows the separation and the identification of the reaction products according to their molar mass. However, the same molar mass could correspond to different molecules. When the sample is neutralized by a solution with a secondary amine, an increase in the molar mass is observed in the molecules carrying one or many unreacted isocyanates. That is why comparing the spectrum of non neutralized and neutralized sample allows the clear identification of the reaction product.

The formation of many monourethanes (OM), which are generated by the dissociation of the urethane bond of OMO in presence of excess isocyanate is showed at m/z 403.4 (Fig. 9). Numerous compounds of higher molar masses have been identified as grafted di-or monourethanes. These grafted molecules are the result of reactions between urethane bonds and isocyanates, ending in allophanate linkages. Those molecules can further react, with dissociated alcohol for example, ending to branched molecules. NMR and MALDI-TOF lead to the same conclusions, since only compounds involving allophanate bonds are found, and no molecules carrying isocyanurate linkages. Compared to the literature, it is a surprising result as it was not expected to form allophanate during reaction at such high temperatures. Moreover, many authors have considered this function as an intermediate reaction product in the formation of isocyanurate. In our reaction conditions, i.e. equimolar NCO/urethane ratio and temperature between 170 and 200°C, no isocyanurate is observed even after one hour of reaction.

Reaction between the diurethane OMO and the aliphatic isocyanate tIPDI

The study of OMO reacted with tIPDI becomes more complicated, partly because of the various components (trimer, pentamer, heptamer, etc…of IPDI) of the polyisocyanate. Moreover, it should be noticed here that the IPDI molecule as the corresponding IPDI "trimer" exhibits two different isocyanate groups. Consequently, in the 13 C NMR spectrum of tIPDI, two different peaks can be observed, according to the aliphatic or cycloaliphatic nature of the isocyanate groups and also different isocyanurate signals are observed according to the nature of the isocyanate groups involved in these functions [29][30][31][32].

After the reaction of OMO with tIPDI, at T = [170-200°C] and [NCO]/[urethane] = [0.5-1] for 60min, the 13 C NMR spectrum (Fig. 10) reveals the presence of five main signals in the carbonyl region : according to literature [13,24,30,[START_REF] Desilet | IPDI Pentamer of IPDI Heptamer of IPDI[END_REF], the signals at 149.70 and 150.52 ppm are attributed to the C=O from isocyanurates. The signal at 154.08 ppm comes from the C=O of urethane. The C=O resonances from alkyl-alkyl urea are found at 156.66 and 157.40 ppm.

Four minor signals are also detectable at 156.05 ppm, 155.77 ppm, 154.68 ppm and 153.61 ppm.

Depending on the nature of the isocyanate that reacts with the urethane, two different allophanates can be expected (Fig. 11). Each allophanate function contains two carbonyl groups, one coming from the initial urethane function, and the other coming from the isocyanate function. Considering that the chemical shifts of the C=O coming from urethane (C 11b and C 12b in Fig. 11) are expected to be close to one of the allophanate functions from OMO+MDI, meaning close to 156.10 ppm, the peaks at 155.77 and 156.05 ppm were putatively attributed to C 11b and C 12b . It is difficult by NMR to be sure that no isocyanurate functions have been formed during reaction since the corresponding carbonyl signals are expected to be close to those of the starting tIPDI. However a quantitative exploitation of the 13 C NMR spectra (recorded with an inversegate decoupling sequence and a pulse delay of 11.4 s and a pulse angle 70°) shows that the ratio between the integral of isocyanurate signals and the integral of the carbons labeled 8+8' (not involved in the reactions) does not change significantly while heating.

The MALDI-TOF analysis confirms the absence of molecules involving isocyanurate bonds, but indicates the formation of many grafted diurethanes, via allophanate linkages. These molecules can further react with dissociated alcohol. It should be noticed that the different oligomers of IPDI (pentamer, heptamer etc…) have also reacted with the diurethane OMO. As discussed latter, the main difference in OMO + MDI versus OMO + tIPDI reactions observed by MALDI-TOF is the weakness of the peak corresponding to the monourethane (OM).

Comparison between OMO reactions with MDI and tIPDI

From a quantitative analysis of the signals found in the carbonyl region of the 13 C NMR spectra, it was possible to follow the evolution of the amount of urea, urethane and allophanate functions produced or consumed in both reactions versus time (Fig. 12). As expected, isocyanates and urethanes are consumed at the same rate during the reaction. However, only few allophanates are obtained, approximately 10% after 1hour of an OMO + MDI reaction. In the case of a reaction of OMO + tIPDI in the same conditions of stoichiometry and temperature, even less allophanates are formed. However, another experiment at 200°C (figure not represented here) showed that the quantity of allophanate produced by the reaction of OMO + tIPDI increased with temperature.

The main difference between the behavior of OMO + MDI and OMO + tIPDI reactions is the quantity of monourethane evidenced during SEC analysis.

Comparing OM peaks from SEC chromatograms for an OMO + MDI mixture and an OMO + tIPDI mixture (Fig. 13), at equimolar isocyanate over urethane ratios, it is noteworthy that much more monourethanes OM are obtained when MDI is reacted with OMO.

It is well known that aromatic isocyanates are more reactive than aliphatic or cycloaliphatic isocyanates. This could also explain the two different behaviors observed here. Indeed, due to the different reactivities, the diurethane OMO reacted with a cycloaliphatic isocyanate, in our case tIPDI, would produce less allophanates than that reacted with an aromatic isocyanate, here MDI. In the same way, less tIPDI is supposed to react with dissociated alcool, produced by the urethane dissociation at high temperature. This means that the isocyanate and alcohol functions originating in the OMO molecules will remain in a quasi equimolar ratio. Almost all the urethane linkages, dissociated during the reaction at high temperature can then be reformed, so that fewer monourethanes OM will be observed back at room temperature in the case of a reaction OMO + tIPDI.

Conclusion

Thanks to a model study, it was possible to clearly identify allophanates as the main products of reaction between a diurethane and di-or polyisocyanates at high temperature, without any catalyst and when the mixture is cooled to room temperature. Isocyanurates form only in very polar solvents during analysis but do not constitute a reaction product in bulk. When tIPDI is reacted with urethanes, less allophanates and monourethanes are formed, comparing with the reaction of OMO + MDI. Moreover, it has been shown that the combination of NMR and MALDI-TOF analysis can be a very useful characterization method.

These results prove the possibility to create some crosslinks, through allophanate functions withstanding these temperatures, by adding polyisocyanates to a TPU by reactive extrusion. This simple modification has already been used to improve the mechanical and thermal properties of such material [22,23].

Financial support from Novéon (Frontonas-69 F) and Nexans (Lyon-69 F), together with fruitful advice and discussions with Mr Lagneaux, Mr Widawski, Mr Fournier and Mr Pinto, are gratefully acknowledged. Figure 13 : SEC curves of OMO + MDI mixture (---), at i/u=1, 190°C, 60min and OMO + tIPDI mixture (), at i/u=1, 210°C, 120min. -

R 1 NCO R 2 O C H N O R 1 R 2 OH + C N C N C N O O O R 1 R 3 R 4 R 2 OH R 2 O C N O R 1 C NH O R 3
-Figure 1 - - R 1 NCO R 1 NH C O NH R 2 R 2 NH 2 + R 1 NH 2 R 1 N CO CO NH R 2 NH R 3

  The signals at 153.61 and 154.68 ppm were then attributed to either C 11a or C 12a . It is interesting to notice that signals at 155.77 and 153.61 ppm, like signals at 156.10 and 154.68 ppm, have very close integral values so that they should belong to the same allophanate function.

Table 1 :

 1 Thermal stability of different types of urethanes Type of urethane group Approximate upper stability temperature Alkyl-NH-CO-O-Alkyl ~ 250°C Aryl-NH-CO-O-Alkyl ~ 200°C Alkyl-NH-CO-O-Aryl ~ 180°C Aryl-NH-CO-O-Aryl ~ 120°C

Figure 1 :

 1 Figure 1 : Different reactions between NCO and OH groups. The kinetic rate constants of the reactions a, b, c and d are, respectively, k a , k b , k c and k d .

Figure 2 :Figure 3 :

 23 Figure 2 : Reactions of NCO groups in the presence of water

Figure 4 :

 4 Figure 4 : MALDI-TOF spectrum of tIPDI.

Figure 5 :Figure 6 :

 56 Figure 5 : Isocyanurate formation Figure 6 : Expansion of the carbonyl region of 13 C NMR (DMSO-d6) spectrum of OMO+MDI, i/u = 1, 192°C for 60 min, non-neutralized sample a) acquisition time = 28min, b) acquisition time = 3h47min, c) after 5 days.

Figure 7 :

 7 Figure 7: SEC chromatograms of OMO+MDI, isocyanate/urethane=1, reacted at 192°C.

Figure 8 :Figure 9 :

 89 Figure 8: Enlargements of (1) 1 H (NH region) and (2) 13 C (C=O region) NMR spectrum for a OMO+MDI mixture, i/u=1, 192°C for 60 min a) non-neutralized sample b) sample neutralized with diethylamine

Figure 10 :

 10 Figure 10 : 13 C NMR (CDCl 3 ) spectrum of OMO + tIPDI, i/u=1/2, 170°C, for 60 min (sample neutralized with diethylamine). (i) impurity coming from THF.

Figure 11 :Figure 12 :

 1112 Figure 11 : Two different allophanates expected by reacting OMO with tIPDI

Figure 1 :

 1 Figure 1 : Different reactions between NCO and OH groups. The kinetic rate constants of the reactions a, b, c and d are, respectively, k a , k b , k c and k d .

Figure 2 :Figure 3 :-Figure 4 :-Figure 4 -Figure 5 :

 23445 Figure 2 : Reactions of NCO groups in the presence of water

Figure 6 :Figure 7 :Figure 8 :Figure 9 :-Figure 9 -

 67899 Figure 6 : Expansion of the carbonyl region of 13 C NMR (DMSO-d6) spectrum of OMO+MDI, i/u = 1, 192°C for 60 min, non-neutralized sample a) acquisition time = 28min, b) acquisition time = 3h47min, c) after 5 days.

Figure 11 :Figure 12 :

 1112 Figure 11 : Two different allophanates expected by reacting OMO with tIPDI

Figure 13 :

 13 Figure 13 : SEC curves of OMO + MDI mixture (---), at i/u=1, 190°C, 60min and OMO + tIPDI mixture (), at i/u=1, 210°C, 120min.

  Voyager Spec #1 MC=> BC=>NR(2.00)[BP = 533.6, 32546]

	-
							CO urethane
											13 urethane
				533.6
	CO allophanate
				Diurethane OMO CO allophanate
	CO urea			OMO grafted MDI	CO isocyanurate
				(allophanate) : OM(-gM)O i
							783.9	OMO grafted OM	13+13' allophanate
	Monourethane				914.2	(allophanate) : OM(-gMO)O
	OM
	403.4 413.5	156	154	152	150	ppm		67	66	ppm
	399.1		473.3	535.6 549.6 537.5	628.6 630.6	683.6	916.2 879.9 14 urethane 785.9 761.9	1008.2 1064.1	1164.4 1138.4	1232.3	1295.7	1388.7	1545.0	1770.9
						14+14' allophanate
			42	41	ppm		8+8'
	55	50		45		40	35	30		25	20	15	ppm

-Figure

10

:

13

C NMR (CDCl 3 ) spectrum of OMO + tIPDI, i/u=1/2, 170°C, for 60 min (sample neutralized with diethylamine). (i) impurity coming from THF.

-Figure

10

-


Jean Salençon
email: salencon@lms.polytechnique.fr

Bearing capacity of strip footings with horizontal confinement

Keywords: Soils, Bearing capacity, Confinement, Extrusion, Foundation, Yield design Sols, Calcul à la rupture, Extrusion, Fondation, Force portante

For a strip footing under axial loading, the bearing capacity is influenced by the presence of rigid walls confining the foundation soil. This problem is investigated within the framework of the theory of yield design, considering both a perfectly rough and a frictionless contact condition at the interfaces with the walls in the case of a purely cohesive soil. Upper bounds for the correction factor to be applied to the classical value of the bearing capacity are determined, as functions of the nondimensional geometric parameter of the problem, through the kinematic approach, implementing virtual velocity fields inspired from the solution to the problem of inverted extrusion. In the perfectly rough case, it appears that the new upper bound is a significant improvement of those already available. A very simple relationship is established, which derives the upper bound for the frictionless walls from the upper bound for the rough walls. A general conclusion of the analysis is that, for the values of the geometric parameter that are likely to be encountered in practice, the increase in the bearing capacity due to the presence of the rigid walls remains very small.

Résumé

La capacité portante des fondations superficielles en présence de parois rigides. La présence de parois rigides au voisinage d'une fondation superficielle influe sur la capacité portante de cette fondation. On étudie ce problème dans le cas d'un sol cohérent sans frottement interne en supposant que le contact avec les parois est soit parfaitement rugueux soit sans frottement. En mettant en oeuvre l'approche cinématique du calcul à la rupture avec des champs de vitesse virtuels inspirés de l'étude de l'extrusion inverse, on détermine des bornes supérieures pour le facteur de correction à appliquer au coefficient classique de capacité portante, en fonction du paramètre géométrique adimensionnel du problème. Dans le cas des parois rugueuses, la nouvelle borne supérieure se révèle significativement meilleure que celles disponibles jusqu'alors. Une relation simple est établie qui permet d'obtenir la borne supérieure pour les parois lisses à partir de celle valable pour les parois rugueuses. En conclusion de l'analyse, il apparaît que, pour les valeurs du paramètre géométrique réalistes du point de vue de la pratique, l'accroissement de capacité portante dû à la présence des parois rigides demeure faible.

The problem under investigation

In [START_REF] Salençon | The influence of confinement on the bearing capacity of strip footings[END_REF] we presented the results of a theoretical determination of the bearing capacity of strip footings submitted to axial loading, when the soil foundation, with infinite thickness, is limited in the horizontal directions by rigid vertical walls. The soil foundation was either purely cohesive or frictional assuming a perfectly rough contact condition at the interfaces with the footing and with the rigid walls. The major part of the analysis was devoted to obtaining upper bound estimates within the framework of the yield design theory [START_REF] Salençon | An introduction to the yield design theory and its applications to soil mechanics[END_REF] by revisiting the solution to the problem of the bearing capacity of a surface footing on a soil layer with limited thickness. The virtual velocity fields, which were implemented in the upper bound approach, took place in a soil layer whose thickness was strictly proportional to the gap between the edge of the foundation and the adjacent wall. The same problem will be investigated again, in the case of a purely cohesive soil, in both cases of a perfectly rough or a frictionless (smooth) contact condition (with no tensile strength) at the rigid walls, through a kinematic approach using virtual velocity fields inspired from the analysis of inverted extrusion [START_REF] Hill | The Mathematical Theory of Plasticity[END_REF]. The notations are kept as in [START_REF] Salençon | The influence of confinement on the bearing capacity of strip footings[END_REF]. The width of the footing is denoted B = 2b. The distance between the edge of the footing and the corresponding rigid containing wall is denoted L. The cohesion of the foundation soil is c u with the Tresca strength criterion.

The axial force acting on the footing is denoted F (more precisely, F is the axial force per unit transversal length acting on the footing). The problem is treated as a 2-dimensional one. The theoretical bearing capacity is defined as p ult = F/B when F reaches its maximum value from the yield design point of view.

In consideration of the parameters of the problem, the non-dimensional expression of p ult can be written as:

p ult = c u K c L b N c with N c = π + 2andK c (∞) = 1( 1 )
where K c (L/b) is the correction factor which takes the confining effect into account. In order to distinguish the "rough" and the "smooth" case a superscript "r" or "s" will be introduced in Eq. ( 1), namely: K r c (L/b) and K s c (L/b).

Perfectly rough contact condition with the rigid walls

L/b 2

Referring to Prandtl solution and to the extension of its stress-field by Shield as in [START_REF] Salençon | The influence of confinement on the bearing capacity of strip footings[END_REF], it comes out that:

∀ L b 2,K r c L b = 1 (2)

0 <L/b<2

Investigating the range 0 <L/b<2, we now consider the virtual velocity field constructed, through the use of Geiringer's equations [START_REF] Geiringer | Some recent results in the theory of an ideal plastic body[END_REF] on the symmetrical slip-line field presented in Fig. 1, which is inspired from [START_REF] Hill | The Mathematical Theory of Plasticity[END_REF]. Fig. 1 also shows the corresponding hodograph [START_REF] Green | On the use of hodographs in problems of plane plastic strain[END_REF] when the footing is given a vertical virtual translation motion with velocity U : zone A ′ M ′ SMA moves vertically as a rigid block with velocity U ; across

A ′ M ′ SG and G ′ R ′ Q ′ SMA the velocity field is discontinuous, with a tangential jump equal to U √ 2/2; beneath G ′ R ′ Q ′ SG the soil is motionless; G ′ A ′ C ′
has a translation rigid body motion; since the hodograph and the slip-line field happen to be geometrically similar and orthogonal to each other, the velocity Oa ′ of G ′ A ′ C ′ is normal to the straight line AS with UL/b as vertical component (symmetrically for GAC); the velocity field in the rest of the soil layer is described in the hodograph.

Applying the upper bound approach of the theory of Yield design would require the explicit computation of the maximum resisting rate of work in that virtual velocity field, which should then be compared with the work of the external forces. As a matter of fact a short cut can be taken since Fig. 1 makes it possible to exhibit a so-called "incomplete solution" as introduced by Bishop [START_REF] Bishop | On the complete solution to problems of deformation of a plastic rigid material[END_REF]. Starting from the stress-free surface boundary condition along A ′ G ′ and AG and assuming the vertical direction to be the direction of maximum pressure there, a stress-field is constructed, which is in equilibrium with zero body forces and which saturates the Tresca strength criterion. The network of α and β lines in Fig. 1 is the network of characteristics of this stress-field and the Hencky equations [START_REF] Hill | The Mathematical Theory of Plasticity[END_REF], which are satisfied along these lines, determine the stress-field in the region under concern. Point S,wherethetwo symmetric parts of the network meet each other, is situated on the symmetry axis of the footing and is determined from the condition that the vertical and horizontal directions be the principal axes for the stress-field: hence arcs C ′ R ′ and M ′ N ′ are equal to each other. Only a partial stress-field being constructed does not make it possible to refer to the static approach of the theory of yield design. But, in consideration of Ewing and Hill's theorem [START_REF] Ewing | The plastic constraint of V. notched tension bars[END_REF], since the orientations of the α, β network and of its image in the hodograph are interchanged, the corresponding stressand velocity fields can be combined to produce an incomplete solution [START_REF] Bishop | On the complete solution to problems of deformation of a plastic rigid material[END_REF] with the benefit that the bearing capacity computed by means of the partial stress-field is just the upper bound estimate given by the virtual velocity field. The network in Fig. 1 remains valid until R ′ meets the rigid wall, which corresponds to M ′ meeting the footing. With reference to the geometric parameters of the problem this amounts to: L/b 0.29. For lower values of L/b the network is extended, as shown in Fig. 2, with β lines tangent to the wall and α lines tangent to the footing on the left-hand side and symmetrically on the right-hand side. The entire reasoning remains valid.

After constructing the network numerically through classical procedures, the stress-field is determined explicitly at each node of the mesh and integration along A ′ M ′ S (or, more conveniently, along A ′ N ′ S), in order to obtain the corresponding upper bound for the bearing capacity, is straightforward. Referring to Eq. (1), we write:

p ult = c u K r c L b N c c u K r c L b ′ N c (3)
where [K r c (L/b)] ′ is the upper bound for the correction factor K r c (L/b).

The results are presented in Fig. 3 together with the upper bound proposed previously [START_REF] Salençon | The influence of confinement on the bearing capacity of strip footings[END_REF] and with the closedform upper bound recently obtained by Puzrin and Randolph [START_REF] Puzrin | Generalized framework for the three-dimensional upper bound limit analysis in a Tresca material[END_REF] through an innovative method. This one becomes rapidly greater than the others when L/b decreases and, in a comparison with [START_REF] Salençon | The influence of confinement on the bearing capacity of strip footings[END_REF], we observe that a significant reduction of the upper bound for the correction factor has been gained from the approach presented here: for the same value of L/b, the increase in the bearing capacity due to horizontal confinement is now reduced by some 50%. This strengthens the conclusion that, for practically relevant values of L/b, the increase in the bearing capacity due to confinement remains very small: smaller than 10% for L/b > 0.72 and smaller than 20% for L/b > 0.45.

It may also be recalled that in the solution proposed in [START_REF] Salençon | The influence of confinement on the bearing capacity of strip footings[END_REF], the thickness of the layer where the virtual velocity field took place was kept equal to d = L √ 2/2 for any value of L/b 2, so that d/b decreased to 0 in proportion with L/b. In the present case it is observed in Fig. 3 that the decrement of d/b with L/b due to the presence of the rigid walls is limited.

Frictionless contact condition with the rigid walls

Since introducing a frictionless contact condition instead of the perfectly rough one results in reducing the resistance of the corresponding interfaces, it is clear from the definition of the bearing capacity that:

∀ L b > 0,K s c L b K r c L b (4)
Therefore [K r c (L/b)] ′ remains an upper bound for the correction factor in the case of the frictionless contact condition. But we can also anticipate that new virtual velocity fields involving slipping of the soil along the walls may now be considered, which would lead to better upper bound estimates for K s c (L/b).

L/b 1

Such a virtual velocity field can be produced from Prandtl classical solution, as shown in Fig. 4, for values of L/b within the range 1 L/b 2. The footing being given a rigid body vertical translation motion with the velocity U , the classical virtual velocity field is maintained in A ′ BA,i nA ′ BC ′ and in ABC; in A ′ C ′ E ′ F ′ (resp. ACEF ) we still encounter a virtual rigid body translation motion parallel to C ′ E ′ (resp. CE) with velocity U √ 2/2; E ′ F ′ (resp. EF) is a velocity discontinuity line such that E ′ F ′ G ′ (resp. EFG) receive the vertical velocity U . This virtual velocity field can still be combined with the stress-field of Prandtl solution restricted to G ′ A ′ AGEBE ′ in order to produce an incomplete solution. It follows that the corresponding upper bound for the correction factor is just:

1 L b 2,K s c L b = 1( 5 ) 3.2. 0 <L/b 1
For 0 <L/b 1, Fig. 5 presents the virtual velocity field derived from Fig. 1 by truncating the slip-line field with the frictionless rigid walls passing through points C and C ′ . The hodograph gives the complete description of the virtual mechanism: in A ′ P ′ R ′ SRCAA ′ the velocity field is the same as in Fig. 1; it is discontinuous across R ′ P ′ A ′ (and symmetric) with a jump equal to U √ 2/2; finally A ′ C ′ G ′ and ACG move vertically with velocity Ub/L. Looking for an incomplete solution with this virtual velocity field, it is clear that restricting the stress field constructed in the "rough" case to the zone limited by the rigid walls in the "smooth" case complies with the frictionless contact condition. Comparison of the orientations of the α, β networks in the hodograph and in the slip-line field completes the proof of an incomplete solution.

The slip-line field in Fig. 5 is valid until A ′ M ′ S becomes tangent to the footing, which corresponds to 0.14 L/b 1. For lower values of L/b the slip-line field is extended, as shown in Fig. 6, with α lines tangent to the footing on the left-hand side (β lines on the right-hand side): the entire reasoning remains valid.

Producing an incomplete solution in both cases makes it easier to apply the upper bound approach. The upper bound estimate for the correction factor being denoted [K s c (L/b)] ′ ,wehave:

p ult = c u K s c L b N c c u K s c L b ′ N c (6)
For smaller values of L/b the correspondence is no more valid but may be retained, if necessary, as a good estimate.

The corresponding results are presented in Fig. 3. The increase of the bearing capacity is not significant for practically relevant values of L/b: it remains smaller than 10% for L/b 0.36 and smaller than 20% for L/b 0.22. This comes from the fact that, in a comparison with the unlimited soil foundation, the overall increase of the resistance of the constituent materials of the system introduced through the existence of the rigid walls is counterbalanced by the frictionless contact condition.

Fig. 1 .

 1 Fig. 1. Slip-line field and hodograph for perfectly rough walls when L/b 0.29.

Fig. 2 .

 2 Fig. 2. Slip-line field for perfectly rough walls when 0 <L/b<0.29.

Fig. 3 .

 3 Fig. 3. Influence of confinement on the bearing capacity: perfectly rough walls [K r c (L/b)] ′ , frictionless walls [K s c (L/b)] ′ , previous upper bound (dashed line), Puzrin and Randolph [8] upper bound (dotted line).

Fig. 4 .

 4 Fig. 4. Slip-line field for frictionless walls when 1 L/b 2.

Fig. 5 .

 5 Fig. 5. Slip-line field and hodograph for frictionless walls when L/b 0.14.

Fig. 6 .

 6 Fig. 6. Slip-line field for frictionless walls when 0 <L/b<0.14.


Confining effect on the bearing capacity of circular footings on a purely cohesive soil

Confining effect on the bearing capacity of circular footings on a purely cohesive soil

Jean Salençon

École polytechnique, Laboratoire de mécanique des solides, 91128 Palaiseau cedex, France ¡¢£¤¥¦£ The bearing capacity of axially symmetrical footings acting on a purely cohesive soil foundation contained by a rigid wall at a finite distance is investigated within the framework of the Yield design theory. Following the same tracks as in a preceding paper devoted to strip footings, the analysis is performed by referring to already existing results concerning the bearing capacity of a circular footing on a soil layer with limited thickness. It comes out that the bearing capacity factor determined by Eason and Shield for a rough circular footing on an unlimited soil foundation is increased by a correction factor that increases when the diameter of the container decreases. Comparison with the results obtained for strip footings acting on a purely cohesive soil in the same conditions shows that the confining effect is significantly lower for a circular footing than for a strip footing.

soils / bearing capacity / confinement / circular footing / yield design

Effet du confinement sur la capacité portante d'une fondation circulaire sur un sol purement cohérent §¨¢©¨On étudie la force portante d'une fondation superficielle circulaire chargée axialement sur un demi-espace limité horizontalement par une enceinte circulaire rigide. On se place dans le cadre de la théorie du calcul à la rupture. Comme dans un article précédent, l'analyse est menée à partir des résultats déjà existants pour la portance d'une fondation circulaire sur une couche de sol d'épaisseur limitée. On montre ainsi que le facteur de capacité portante déterminé par Eason et Shield pour une fondation circulaire rugueuse sur un sol illimité doit, dans le cas présent, être multiplié par un facteur qui croit quand le diamètre de l'enceinte rigide diminue. La comparaison de ces nouveaux résultats avec ceux obtenus pour les semelles filantes dans les mêmes conditions montre que l'effet de confinement est beaucoup moins sensible pour une fondation circulaire. sols / calcul à la rupture / confinement / fondation circulaire / force portante

The problem under investigation

The paper is concerned with the theoretical evaluation of the bearing capacity of a rigid axially symmetrical footing under axial loading, acting on a homogeneous soil foundation with infinite thickness. The diameter of the footing being denoted B = 2b, the soil foundation is assumed to be limited in all horizontal directions at a distance L from the edge of the footing by rigid walls. The contact between the footing and the soil on the one side and between the soil and the rigid walls on the other is assumed to be perfectly rough. As a matter of fact a similar problem has been studied in a preceding paper [START_REF] Salençon | The influence of confinement on the bearing capacity of strip footings[END_REF] where the analysis was restricted to strip footings while the soil foundation could be either purely cohesive or cohesive and frictional. The expected confining effect due to the limited width of the soil foundation was assessed in the form of a correction factor to be applied to the classical bearing capacity factors in Terzaghi's formula, a function of L/b.

The goal of the paper is now, restricting the analysis to a purely cohesive soil foundation, to assess the correction factor in the case of a circular footing and to compare it with the results for a strip footing. Applications of this analysis are expected in the design of soil structures and in the interpretation of reduced scale experiments.

Theoretical framework

The theoretical framework is given by the theory of Yield Design [START_REF] Salençon | An introduction to the yield design theory and its applications to soil mechanics[END_REF] or Limit Analysis. The foundation soil is assumed isotropic and purely cohesive with cohesion c u and it is characterised by a Tresca strength criterion.

The implementation of the theory will mainly follow the same tracks as for the plane problems examined in [START_REF] Salençon | The influence of confinement on the bearing capacity of strip footings[END_REF], with the additional assumption of the Haar-Karman hypothesis [START_REF] Haar | Zur Theorie der Spannungszustände in plastischen und Sandartigen Medien[END_REF]. The kinematic approach with virtual velocity fields being obtained through the method of characteristics will provide upper bound estimates for the correction factor.

One important result obtained in the plane case was to show that upper bound estimates for the assessment of the confining effect due to the limited horizontal width of the soil foundation could be conveniently obtained by revisiting classical solutions to the problem of the bearing capacity of a surface footing on a soil layer of limited thickness. The present analysis is organised in just the same way.

Circular footing on a purely cohesive soil

The considered circular footing is with axis Ox. The axial force acting on the footing is denoted by F and the bearing capacity is: p ult = F/πb 2 . From dimensional analysis it comes out that

p ult = c u K ′ c (L/b)N ′ c , (1)
where N ′ c = 6.05 is the bearing capacity factor for a rough circular footing on an unlimited soil foundation.

Unlimited soil foundation

The theoretical determination of N ′ c within the framework of Limit Analysis was given by Eason and Shield [START_REF] Eason | The plastic indentation of a semi-infinite solid by a perfectly rough circular punch[END_REF] using the equations of axially symmetric plastic flow with the hypothesis of Haar and von Karman. The corresponding mesh of characteristic lines is sketched in Fig. 1 adapted from the original paper. With w denoting the length of the uplifted wave AD, the geometric ratios of this mesh are w/b = C ′ c = 0.88 and B/d = 2.34. For the use of this mesh as a static approach, the stress field was extended in the whole bulk of soil using a method similar to the method given by Bishop [START_REF] Bishop | On the complete solution to problems of deformation of a plastic rigid material[END_REF] in the plane strain case (Fig. 1). For the kinematic approach, an axially symmetric velocity field was constructed: the footing is given a vertical virtual velocity and the volume defined by ABA ′ moves as a rigid body with the same velocity; the region defined by ABCD is deformed with no volume change and the rest of the soil foundation beneath DCBC ′ D ′ is motionless.

Limited soil foundation

Static approach

Obviously a statically admissible stress-field respecting the Tresca yield criterion is obtained in the bulk of soil within the container, whatever L/b 0, through considering the restriction of the stress-field in Fig. 1 within the domain r b + L. Regarding the friction condition at the interface between the wall and the soil (r = b + L), it must also be checked that the normal stress acting on the rough wall is compressive everywhere. Such a verification is not as easy as in the plane case [START_REF] Salençon | The influence of confinement on the bearing capacity of strip footings[END_REF] where the extension of Prandtl's stress-field by Shield's method [START_REF] Shield | Plastic potential theory and Prandtl bearing capacity solution[END_REF] could be used, thus providing an explicit stress field. In the present case we assume that, having checked the condition at the interface, we may state that:

∀L/b 0,K ′ c (L/b) 1. (2)
The practical significance of Eq. ( 2) is that the effect of confinement always results in an increase of the bearing capacity of the footing.

Kinematic approach

• L/b C ′ c
This kinematic approach is obviously still valid when the soil foundation is limited by a rough rigid wall at a distance L from the edge of the footing greater than the width of the uplifted wave: L/b C ′ c = 0.88. Taking Eq. ( 2) into account it follows, that:

∀L/b C ′ c = 0.88,K ′ c (L/b) = 1. (3)
• 0 L/b < C ′ c = 0.88 The investigation of the range 0 L/b < C ′ c = 0.88 follows the same path of reasoning as for the strip footing [START_REF] Salençon | The influence of confinement on the bearing capacity of strip footings[END_REF]. It refers to the solution to the bearing capacity problem for an axially symmetric footing acting on a soil layer of limited thickness h where the bearing capacity appears as a function of B/h and is written in the form

p ult = c u F ′ c (B/ h)N ′ c ( 4
)
with F ′ c (B/ h) a non-dimensional correction factor. Salençon and Matar [START_REF] Salençon | Capacité portante des fondations superficielles circulaires (Bearing capacity of axially symmetrical shallow foundations)[END_REF] made the analysis of that problem using the equations of axially symmetric plastic flow with the Haar-Karman hypothesis in order to determine F ′ c (B/ h). They constructed the network of characteristic lines as presented in Fig. 2, starting from the data on the stress free surface and taking into account the perfectly rough contact condition at the interfaces between the soil layer and the footing on the one side and between the soil layer and the bedrock on the other. The width of the uplifted wave is again denoted by w. This network starts with the same fan of characteristic lines at point A as in Fig. 1. Due to the form of the equations along the characteristic lines in this case (e.g. [START_REF] Shield | Plastic potential theory and Prandtl bearing capacity solution[END_REF]) where the distance to the Ox axis appears explicitly, this fan is not self-similar with respect to A as it used to be in the plane strain case and therefore the ratio w/h = λ ′ c is now dependent on B/h when B/h > 2.34. Although they were not published in the paper we are referring to, the values of the geometrical ratios B/h > 2.34 and w/b corresponding to each other in these networks had obviously been computed and are still available: they are presented in Fig. 3. From this chart we derived λ ′ c as a function of B/h to be used later on. The corresponding value of p ult is obtained as a function of B/h > 2.34 and yields F ′ c (B/ h) in Eq. ( 4). For the interpretation of this solution as a static approach the extension of the stress field in the volumes defined by T ′ ST , R ′ SR and outside ICD is made in the same way as in the strip footing case. In order to make it a kinematic approach, an axially symmetric velocity field was numerically constructed for different values of B/h > 2.34, following the same tracks as in the infinite case by Eason and Shield, applying the equations for the velocity along the characteristic lines. The volume defined by T ′ ST moves as a rigid body with the same vertical velocity as the footing; the volume defined by R ′ SR is motionless as well as the volume outside ICD; the rest of the soil layer is deformed with no volume change. The chart in Fig. 3 now makes it possible to deal with the problem we are presently investigating. Given L/b < C ′ c = 0.88, the corresponding value of B/h is determined from Fig. 3, for which the mesh of characteristic lines in Fig. 2 fits exactly the horizontal width of the soil foundation (Fig. 4). For a kinematic approach, the velocity field determined in the corresponding case for the soil layer is considered and extended assuming the soil mass to be motionless for x h. It follows that an upper bound estimate is obtained for K ′ c (L/b) in the form:

K ′ c (L/b) F ′ c (B/ h) = F ′ c 2λ ′ c (L/b) -1 , (5)
where L/b and B/h are linked to each other according to the chart in Fig. 3. This upper bound is presented in Fig. 5. It appears that the increase of the bearing capacity due to this confining effect remains very small for realistic values of L/b in the case of circular footings.

Comparing axially symmetrical footings with strip footings

The upper bound for K c (L/b) obtained in the case of the strip footing is also presented in Fig. 5. Comparison shows immediately that the confining effect due to the rigid walls is, by far, less important for the axially symmetrical footing than for the strip footing for the same value of the non-dimensional geometric ratio L/b.

From a physical viewpoint this is no big surprise. As a matter of fact one may expect the confining effect to be roughly correlated to the ratio of the volumes of soil that are concerned with the collapse phenomenon under the footing and up to the walls, respectively. Following this track a crude estimate for K ′ c (L/b) can be derived from K c (L/b) in the form:

K ′′ c (L/b) = K c (1 + L/b) 2 -1 . (6)
This heuristic estimate for K ′ c (L/b) is drawn in Fig. 5: the comparison with the upper bound for K ′ c (L/b) in Eq. ( 5) is quite interesting.

The analysis described in the preceding sections confirms that for axially symmetrical footings the assessment of the confining effect due to the limited horizontal width of the soil foundation can be conveniently made by revisiting classical solutions to the problem of the bearing capacity on a soil layer of limited thickness in the same way as for strip footings.

Figure 1 .

 1 Figure 1. Eason and Shield kinematical and static solution [4].

Figure 2 .

 2 Figure 2. Salençon and Matar solution [6].

Figure 3 .

 3 Figure 3. Geometric ratios in Salençon and Matar solution.

Figure 4 .

 4 Figure 4. Proposed solution for assessing the confining effect.

Figure 5 .

 5 Figure 5. Correction factors for the confining effect.