Francesca Zuttion

Lorena Redondo-Morata

Arin Marchesi

Ignacio Casuso
email: ignacio.casuso@inserm.fr

High resolution & High-Speed

Keywords: High Speed Atomic Force Microscope, HS-AFM, cantilever, Electron Beam Deposited tip, plasma EBD tip sharpening, tip-sample approach, oscillation amplitude, PID, feedback optimization, image treatment

The advent of high-speed atomic force microscopy (HS-AFM) in recent years has opened up new horizons for the study of structure, function and dynamics of biological molecules. HS-AFM is capable of 1000 times faster imaging than conventional AFM. This circumstance uniquely enables the observation of the dynamics of all the molecules present in the imaging area. In the last ten years, the HS-AFM has gone from a prototype-state technology that only a few labs in the world had access to (including ours) to an established commercialized technology that is present in tens of labs around the world.

In this protocol chapter we share with the readers our practical know-how on high resolution HS-AFM imaging.

Introduction

The atomic force microscopy (AFM) [START_REF] Binnig | Atomic Force Microscopy[END_REF] is a powerful tool for direct visualization of biological samples in an aqueous solution with sub-molecular resolution. It allows non-invasive imaging in physiological conditions of unlabeled biological samples, such as nucleic acids [START_REF] Ido | Beyond the helix pitch: Direct visualization of native DNA in aqueous solution[END_REF][START_REF] Ares | High resolution Atomic Force Microscopy of double-stranded RNA[END_REF][START_REF] Hansma | Reproducible Imaging and Dissection of Plasmid DNA Under Liquid with the Atomic Force Microscope[END_REF] and cell membrane proteins [START_REF] Henderson | Actin Filament Dynamics in Living Glial Cells Imaged by Atomic Force Microscopy[END_REF][START_REF] Schabert | Native Escherichia coli OmpF porin surfaces probed by atomic force microscopy[END_REF]. AFM has rapidly emerged as an effective structural analysis tool, complementing atomic structure data acquired by other techniques such as X-ray crystallography, NMR and electron microscopy. In contrast to these latter methods, which heavily rely on ensemble averaging, AFM exhibits a superior signal-to-noise ratio, enabling molecules to be observed individually under aqueous environment. To acquire a topographic image, a sharp tip attached to the free end of a flexible cantilever is brought into contact and scanned over a sample and its profile height is recorded over the selected scan area. The feedback control allows adjusting the force applied to the sample to avoid damaging fragile biological structures. As a result of the imaging process, during which the distance between the sample stage and the AFM probe varied following the sample topography, a 3D profilometric image of the sample surface is obtained. With the advantage of operating in nearly physiological conditions, AFM is an ideal platform to study biological dynamic processes at the molecular scale. Nevertheless, the framerate of one image every few minutes of conventional AFM is not sufficient to visualize most dynamic processes of biological molecules that take place in its majority at sub-second time scales. The introduction of the high-speed atomic force microscopy (HS-AFM) in 2001 opened the door to the visualization of biological molecule dynamics [START_REF] Ando | A High-speed Atomic Force Microscope for Studying Biological Macromolecules[END_REF]. The technical development of the HS-AFM was performed by the laboratory of Toshio Ando in the University of Kanazawa, Japan.

HS-AFM allows imaging at maximum rates of 5-10 frames per second [START_REF] Ando | A High-speed Atomic Force Microscope for Studying Biological Macromolecules in Action[END_REF][START_REF] Ando | High-speed AFM and applications to biomolecular systems[END_REF][START_REF] Casuso | Characterization of the motion of membrane proteins using high-speed atomic force microscopy[END_REF][START_REF] Ando | High-speed atomic force microscopy coming of age[END_REF]. The operating principle of the HS-AFM is based on the miniaturization of the moving components of the AFM (cantilever and scanner) to increase their velocity by 1000 times, in the interest of achieving reaction speeds of tens of microseconds. For the HS-AFM probe, the gain in speed does not suppose a lost in performance. In the case of the scanner, the X,Y,Z displacement range is decreased ~1:10 over conventional AFMs. With this technique, centrosome [START_REF] Gorle | Computational Model and Dynamics of Monomeric Full-Length APOBEC3G[END_REF][START_REF] Pan | Nanoscale Characterization of Interaction of APOBEC3G with RNA[END_REF], DNA origami [START_REF] Rajendran | Programmed Two-Dimensional Self-Assembly of Multiple DNA Origami Jigsaw Pieces[END_REF], micelles [START_REF] Inoue | Direct observation of surfactant aggregate behavior on a mica surface using high-speed atomic force microscopy[END_REF] and many other biological systems have been imaged at high speed and high resolution.

In this book chapter, we describe in detail our HS-AFM imaging procedure for high resolution and high speed imaging, applicable to any version of the Ando-type HS-AFM system "SS-NEX".

Such procedure is based on our experience imaging individual molecules [START_REF] Casuso | Characterization of the motion of membrane proteins using high-speed atomic force microscopy[END_REF][START_REF] Munguira | Glasslike Membrane Protein Diffusion in a Crowded Membrane[END_REF] or big macromolecular assemblies [START_REF] Chiaruttini | Relaxation of Loaded ESCRT-III Spiral Springs Drives Membrane Deformation[END_REF][START_REF] Mierzwa | Dynamic subunit turnover in ESCRT-III assemblies is regulated by Vps4 to mediate membrane remodelling during cytokinesis[END_REF].

Materials

High-Speed AFM

The protocol here described is adapted for any Ando-type HS-AFM setup "SS-NEX", commercialized by the Research Institute of Biomolecule Metrology (RIBM), Japan. A picture of the HS-AFM hardware of our "SS-NEX" HS-AFM is presented in Figure 1, where the cantilever is mounted up-facing in a cantilever holder that comes with a pool of 110 µl while the sample is mounted on top of the scanner facing down. In general, imaging experiments are performed in tapping mode where the tip is oscillated at constant amplitude. For that, the cantilever is excited at its resonance frequency with the help of a miniaturized piezoelectric actuator. The sample can move in the XY-plane and also in the Z vertical direction thanks to three small multilayer piezoelectric actuators. The Z-piezo the one displaced at the highest frequency (nominal resonance frequency ~ 600 kHz and ~ 150 kHz once glued on the scanner). The deflection of the cantilever is monitored by an optical system. A laser beam with a 20x optical microscope objective is focused on the backside of the cantilever and the reflected beam tracked with a 15 MHz bandwidth photodiode which sends the signal to a Lock-In amplifier (Fourier Analyzer). The signal goes to the Proportional Integral Derivative (PID) controller which compares the amplitude readout with the value settled by the operator (set point amplitude, A) to feed the adjustment of the clearance distance between the sample stage and the HS-AFM probe by the feedback loop.

Sample preparation and mounting

1. Glass-rod (small glass cylinder of 1.5 mm in diameter and 2 mm in height purchased from RIBM, Japan or SCHOTT, Switzerland), colorless nail polish (nitrocellulose dissolved in butyl acetate or ethyl acetate), mica sheet, puncher purchased from RIBM, cyanoacrylate glue (Super Glue®) or two components epoxy adhesive (Araldite®), adhesive tape (Scotch®), cellulose wipers (Kimwipes®), humid hood, microfuge tube of 1 ml.

HS-AFM probe

Plasma sharpening of the Electron Beam Deposited tips

1. A plasma system for surface treatment. We use a 40 kHz Diener ® Zepto system.

2. Piece of 2x1x1 cm 3 of glass or Teflon.

3. Double sided carbon conductive tape.

4. Bottles of oxygen and helium gas.

Image treatment

1. Image processing software supplied with your AFM instrument (see Note 1).

MATLAB (MathWorks, USA

). This is a proprietary programming language specifically designed for simple and efficient matrix manipulation, data plotting and algorithm implementation (see Note 2). A 30-day trial version is available for free download (https://www.mathworks.com). All the procedures described in section 3.5 can be carried out in MATLAB software environment. Routines written in this language for batch image processing described in section 3.5 are available at https://sites.google.com/view/fm4b-lab/group-members/dr-arinmarchesi.

ImageJ (NIH)

. This is a free, Java-based image processing platform (https://imagej.nih.gov/ij/). Although not specifically designed for AFM image treatment, a number of user-written plugins exists to efficiently tackle several of the procedures described in section 3.5. A routine written in MATLAB to convert the .asd HS-AFM file format in multipage 32-bit .tiff files supported by ImageJ is available at https://sites.google.com/view/fm4b-lab/group-members/dr-arinmarchesi.

Methods

Sample Preparation

1. Glue a cleaned glass-rod on the Z-piezo by using nail polish and let it dry for at least 20 min. Glass-rods can be cleaned by sonication (f ~ 35 kHz) in acetone and stored in a plastic box until use.

2. Glue a mica disk on the glass-rod with cyanoacrylate glue and let it dry for 10 min (see Note 3).

3. Cleave the mica disk with adhesive tape to obtain a clean, atomically flat surface.

To facilitate the cleavage, mica can be previously cleaned with acetone (see Note 4).

4. Deposit a 1.2 µl-2 µl drop of the sample solution on the mica surface immediately after cleavage. Cover the sample with a humid hood to avoid drying (Figure 2, see Note 5).

5. Fill a cap of a microfuge tube of 1 ml with the working buffer solution (~ 300 µl).

Immerse the sample to wash it, this will remove the excess of molecules that did not adsorbed on the surface. Repeat the procedure at least five times without the need to replace the buffer between rinses.

6. Keep the sample wet with the working buffer solution under the humid hood until it is mounted on the HS-AFM.

Sample Mounting

1. Wash abundantly the cantilever holder with 2% anionic detergent, tap water, acetone, 70% ethanol and ultrapure water. Dry it gently with air or nitrogen flow if possible (see Note 6).

2. Set a small cantilever on the cantilever holder: position the cantilever on the trench of the cantilever holder and fix it with the cantilever holding screws (Figure 3, see Note 7).

3. Mount the cantilever holder on its base by placing the two magnets of the cantilever holder in contact with the micrometer screws of the base.

4. Wash the pool with 110 µl ultrapure water for 10 times.

5. Fill the pool with 110 µl of the working buffer.

6. Switch on the laser controller, select the "I mode" and adjust the intensity taking into account the SUM signal and the cantilever thermal fluctuation. This means that the chosen value should allow reaching a high SUM signal (in the range of 0.5-1.2 V) when the intensity of the reflected laser spot from the cantilever on the photodetector is maximized, while the thermal fluctuation of the cantilever is kept lower (in the range of 10 mV -20 mV) (see Note 8).

7. Focus the cantilever on the camera screen and find the infrared laser spot (𝜆 = 808 nm) (see Note 9). 8. Align the laser on the cantilever and adjust the photodiode parameters: use the micrometer screws of the cantilever holder base to adjust both the lateral and vertical positions of the holder until the intensity of the reflected laser spot from the cantilever on the photodetector is maximized. This corresponds to a maximum value of the SUM parameter on the photodetector controller. Set the DIFF parameter of the photodetector controller as close as possible to zero by using the Y-photodetector micrometer screw. In general, the SUM value should be around 0.5-1.2 V for new cantilevers. 9. Troubleshooting the laser alignment: If the SUM is too low, then check the position of the cantilever and try to adjust it. If the cantilever is not in focus point of the laser beam, modify its vertical position by using the Z-cantilever holder micrometer screw. The SUM will increase as the cantilever reaches the focus point of the laser beam. If the laser spot is not well centered, use the XY-cantilever holder micrometer screws to adjust the lateral position of the cantilever. If the back of the cantilever is dirty or is an already-used cantilever then the SUM signal could be lower, in that case replace the cantilever.

10. Place the scanner holder. The scanner holder presents on the backside one linear and one circular grooves. The grooves have to be placed in correspondence with the support screws of the stage that determine the distance between the cantilever holder and the scanner. Ensure that the scanner holder is well positioned and does not move. Check the position of the screws of the scanner holder and ensure that they are retracted.

11. Carefully mount the sample scanner (Figure 4) on its holder by first accommodating the back (cable) side, putting the magnets in contact with the screws of the scanner holder and then placing the front side of the scanner. During this step, the sample will be plunged in the pool. Pay constant attention to the distance between the scanner and the cantilever while plunging in the sample, always looking at the video monitor. If they are too close to each other, then use the handy controller ("UP") to move the step motors and lift up the scanner (see Note 10).

12. Check the alignment of the glass-rod and the cantilever by looking if the glass-rod is in the same vertical plane of the cantilever.

13. Connect the scanner to the break-out box (see Note 11).

Tip-sample approach in amplitude modulation mode

1. Before approaching the sample surface and the tip, check in the video monitor (video camera or CCD) that there are no bubbles around the surface or the AFM chip (that would cause abnormal cantilever deflection). Likewise, ensure that in the camera display there are no particles or dust diffusing around in the liquid, due to contamination (see Note 12). The edges of both the sample support (mica, glass, etc.) and the cantilever should look clean.

2. The tip is placed relative to the sample stage around a millimeter away in the Zaxis, as shown in Figure 5A. In the camera display, the surface edge must be blurred or unfocused, ensuring that there is enough distance between the substrate surface and the cantilever (Figure 5B). Bring the sample close to the cantilever using the micrometer screws in XY-plane, as indicated by the arrow in Figure 5A.

3. In the Z-axis, bring the sample close to the cantilever performing a coarse approach with the Z-step motor (see Note 13). For approaching, the tip is exactly positioned in the vicinity of the edge of the surface (Figure 5C and 5D, see Note

14).

4. To find the first (or fundamental) resonance frequency of the cantilever, measure the thermal fluctuation of the HS-AFM probe cantilever performing a power spectrum of the photodiode signal (see Note 15). 5. Using the Fourier Analyzer, apply an AC sinusoidal voltage to the excitation piezo located on the cantilever holder, to excite the cantilever at the measured resonance frequency in point 3.4 (see Note 16).

6. Select the cantilever free amplitude between 0.1-0.5 V (see Note 17). This typically corresponds to amplitudes between 1-5 nm (see Note 18). 7. The amplitude set point of the cantilever should be lower than the free amplitude.

In general, for the approach the amplitude set point is set around 20 % -40 % lower than the free amplitude. 8. Ensure that everything is plugged, Z-scanner, isolation table (if used), X-, Y-and Z-drivers and PID are on (see Note 19). 9. Set parameters to start approaching: center the offset for the output voltage of the X-, Y-, and Z-drivers (see Note 20). In the PID controller: the integral gain is set typically at 20 % -40 % of the maximum, the proportional and derivative gains are set to minimum. The tilt for the X-and Y-axes is also centered. If there are two time constants in the controller (slow and fast), switch to slow. 10. Approach automatically, selecting the velocity of the step motor and the percentage of amplitude reduction that will halt the automatic approach (see Note 21).

11. When the tip is very close to the surface, the free oscillation amplitude may vary significantly -usually increasing-due to unspecific interactions. If the oscillation amplitude varies, stop the automatic approach and readjust the set point amplitude 20 % -40 % lower than the free amplitude. Proceed again to approach automatically.

12. If the software detects a tip-sample contact (when the set point amplitude approximates the free amplitude, detected thanks to the read-out of the amplitude and the PID output), the automatic approach stops automatically. In case of a manual approach, stop approaching when the Z-piezo starts to retract.

13. After the automatic approach stopped, offset the voltage of the piezo-driver until the output voltage in the PID is close to 0 V (see Note 22).

14. Troubleshooting the approach step: several situations may lead to an anomalous engage or a complete approach failure. It can be observed that the Z-piezo does not hold the voltage value (it moves slowly to fully expanded or contracted position, or back and forth): a) Mica is not tightly attached to the glass-rod or the glass-rod is not firmly attached to the piezo stage. Sometimes mica partially exfoliates allowing some fluid leakage between the different layers. All these situations lead to oscillations of the surface while scanning that may prevent a successful approach.

b) Due to a bad cleavage, or an excess of glue between the mica and the glass, the substrate plane is too inclined respect to the tip, hence the contact is not stable.

c) The cantilever is tip-less (see Note 23). In this case, decreasing the amplitude set point also changes the SUM value in the photodetector (negative deflection values). d) There are big aggregates loosely adsorbed on the surface with strong and unspecific tip-sample interactions that vary the oscillation amplitude. Try to approach with lower amplitude set point, higher free oscillation amplitude or a different area; eventually reconsider the sample preparation.

e) The solution is dusty or there is material adsorbing on the cantilever, which may lead to false engaging in the surface. Exchange the buffer solution in the pool or eventually increase the free oscillation amplitude. Eventually reconsider the sample preparation.

HS-AFM imaging

At this point the clearance distance between the sample stage and the HS-AFM probe is sufficiently small for the Z-piezo range to bring sample and the tip in and out of contact.

1. Out of contact, turn off the cantilever excitation, with the AFM probe, observe in the oscilloscope the width of the thermal vibration of the cantilever AThermal.

2. Turn on the excitation of the cantilever oscillation and set an out of contact oscillation A0 = 10•AThermal.

3. Set a limited pixel number (~ 100 x 100 pixels), a set point amplitude A>A0 so the tip is separated from sample, a small scan size (~ 50 nm x 50 nm), and a scan rate of 1 frame/second (see Note 24). 4. Start scanning, then slowly decrease the A, when it is below A0, the highest features on the sample are visualized. In the case of observing very high objects of hundreds of nanometer height that could damage the probe, move away to another region. If the area is appropriate for the HS-AFM tip to scan, decrease A a little more, so most of the area scanned is under slight tip contact. 6. Localization of 'area of interest': Increase the scan area progressively to several hundred nanometers and increase the number of pixels accordingly. It is recommended to have between 1 and 2 pixels per nanometer. Look for the area of interest where to perform the imaging, displacing on the sample using the sample stage. Use always minimal force (maximal possible set point amplitude A) for imaging to reduce the risk of tip damage. In the case that it is not found an area of sufficient interest in the available scan area, withdraw the sample from the tip, displace macroscopically using the screws and approach on other area (see Note 26).

7. In the case you localize an area of interest, zoom in to frame your experiment, the frame should show clearly all the areas that are important for posterior data analysis.

8. Optimization of oscillation amplitude: Reduce the amplitude excitation signal, the probe will go out of contact, reduce the set point amplitude A to regain tip-sample contact, further reduce the excitation signal, further reduce A… repeat this procedure until you arrive to a cantilever oscillation that is A0 ~5•AThermal. 9. Check again the feedback saturation limit position by increasing the feedback speed of the I gain, next slow down slightly the I gain to avoid feedback resonance, and next increase the feedback speed of the P gain parameter to increase contrast on fine topographic details. 10. Speed increase: progressively increase the imaging speed, evaluate the loss of quality on the image relative to speed increase. Select the speed that provides a fair compromise between image quality and maximum possible speed.

11. If the sample presents height jumps over a few nanometers, the standard PID feedback speed of reaction may be too slow to correctly contour the sample during downhill scanning (so-called parachuting), in this case the HS-AFM dynamic PID should be turned on [START_REF] Kodera | Dynamic proportional-integral-differential controller for high-speed atomic force microscopy[END_REF].

12. Dynamic PID tuning: set all values to 0 (the upper threshold, the lower threshold, and their corresponding gains). Slow down the imaging rate to its lowest speed where parachuting is observed. Next, increase the gain of the upper threshold until the parachuting disappears. Next, increase the imaging speed until parachuting reappears and again increase the gain of the upper threshold until the parachuting disappears. Repeat this action until resonances start appearing in the imaging, then reduce slightly. The gain of the lower threshold is usually set to 0 unless the sample shows very sharp uphill slopes (for example, trenches).

13. The shape of the tip apex is the most important parameter regarding the resolution.

If the resolution obtained is not satisfactory, tip sharpening of the amorphous carbon tip deposited by electron beam deposited (EBD) on the end of the small cantilever is advised, for this use the gas plasma chamber. i) In a piece of clean glass or Teflon, place on the border a strip of double sided carbon tape. Position the HS-AFM probe in a way that only the cantilever chip is in contact with the carbon tape, hence the cantilever and its tip are poked out in the air (see Note 27).

ii) Introduce the HS-AFM cantilever probe with its support in the plasma chamber of the plasma. Select the gas to fill in the chamber (see Note 28). Run the plasma chamber for a few tens of seconds, inert gases require longer times. Typical parameters we use are pressure ~ 1 mbar, power ~ 20 W.

iii) Image the sample with the sharpened tip. If resolution is not good enough further repeat steps i) to iii). Figure 6 shows the image quality improvement on Outer Membrane Porin F (OmpF) following consecutive sharpening cycles (see Note 29).

HS-AFM video processing and displaying

HS-AFM produces large data sets of image sequences that are affected by a number of instrument-specific peculiarities (such as drift of the piezoelectric elements, contrast discontinuities along the fast-and slow-scan axes, feedback parameters, etc.) that need to be accounted for before any analysis is performed.

Hereafter we provide some clues and suggestions in order to fix these instrument-specific hitches in what we believe is an accurate and time effective manner.

1. Open the HS-AFM data file with your favourite AFM software or programming language. Our own routine to read .asd HS-AFM file formats in MATLAB is available at https://sites.google.com/view/fm4b-lab/group-members/dr-arinmarchesi. To normalize the brightness across all the image stack subtract from each frame the frame average intensity (see Note 30).

2. Visually inspect your image stack and remove all the corrupted frames eventually present (changes in imaging conditions might result in insufficient forces or even the tip being for a certain amount of time out-of-contact). You might also consider removing frames skewed or dilated by sudden and fast motions of the scanner or sample.

3. Align all the frames in the stack to a reference image (see Note 31) in order to compensate for X-and Y-axis drift (see Note 32). This can be first tried in an automated way using normalized cross-correlation (or other image registration algorithms) and reiterated throughout all the frames using control flow statements (see Note 33). Visually inspect the aligned image stack and delete -or manually overlay-badly registered frames to the reference image. To attain this, first filter each frame with a 2D Gaussian blur filter to remove high frequency noise. Afterwards, identify three points that sit on the substrate/background of the image. Make sure these very same points are on the substrate only throughout all the frames. Fit each triad with a plane and subtract it from the aligned video.

Perform a horizontal line-by-line levelling (see Note 35). As the background has

been already subtracted, a median (0 order) levelling will suffice (see Note 36). To avoid levelling artefacts make sure to exclude features not belonging to the substrate/background (usually by means of their height) from the computation. Define a region of interest over the substrate and fit the histogram with a Gaussian function to verify that the variance of the background has decreased after the procedure.

6. Fine align all the frames in the stack to a reference image using sub-pixel precision. For this purpose a software package that performs template matching by means of zero mean normalized cross-correlation was developed by our group some time ago [START_REF] Husain | Software for drift compensation, particle tracking and particle analysis of high-speed atomic force microscopy image series[END_REF]. Alternatively, use the ImageJ plugin Template Matching and Slice Alignment (see Note 32).

7. (Optional) Filter frame by frame the image stack with a matrix Gaussian filter to remove unwanted high-frequency noise. Be aware that over-application of lowpass filters or use of large matrices will tend to blur the data.

Notes

1. Although AFM image processing and analysis is in general best done with the software supplied by the instrument manufacturer, software packages provided with HS-AFM instruments might be rather limited, and more powerful third party packages are often needed.

2. Alternatively, a free software compatible with many MATLAB scripts is GNU Octave (https://www.gnu.org/software/octave/). All the procedures described in section 3.5 can be implemented in Octave.

3. Mica disks with a diameter of 1.5 mm are used as sample surfaces. To obtain them, ~ 0.05 mm thick mica sheets are perforated with a puncher. Careful attention must be paid in the selection of the mica disk. The disk should present a regular and flat border since only areas near the edges will be used for imaging.

For gluing the mica disk on the glass-rod it can be used whether cyanoacrylate adhesive or a two-components epoxy adhesive. 4. To facilitate the cleaving of the mica disk glued on the glass-rod, you can clean the surface with a cellulose wipe soaked with acetone before performing the cleavage. Acetone can eliminate possible residues of organic material from previous sample preparation and facilitate the attachment of the adhesive tape on the mica sheet that will be removed. The final surface will not be affected by the previous use of acetone.

5. Use a wet cellulose wipe to create a humid environment inside the hood.

Incubation conditions (time, temperature, etc.) will depend on the sample type.

Sample case: Preparation of supported lipid bilayers (SLBs) on mica. POPG and POPE lipid stocks are kept in solution in chloroform. The stock solutions are mixed in a microfuge tube of 2 ml to obtain the desired molar ratio and diluted in chloroform:methanol (3:1, v:v). The organic solvent is evaporated under a gentle nitrogen flow. After the sample is kept in a desiccator under reduced pressure overnight. The "liposome hydration buffer" is heated up at 65 ºC and it is used to rehydrate the sample. Then, the sample is dropped in a thermal bath at 65 ºC for 45 s, following by 45 s vortexing; this heating-vortexing cycle is repeated 5 times.

Finally, the lipid suspension is sonicated for 30 min in an ultrasonic bath (f ~ 35 kHz) to obtain large unilamellar vesicles. 1 µl of the "incubation buffer" and 1 µl liposomes suspension are incubated at room temperature on a freshly cleaved mica under a humid hood for 20 min. The sample is then rinsed 10 times with imaging buffer and then imaged by HS-AFM.

6. Pay attention not to wet the backside of the cantilever holder. 7. To settle the cantilever, completely remove the plate and position the cantilever, then first secure the screw at the bottom of the plate and afterwards the one in the front (see Figure 3). Do not screw to strong, screws can break easily, but enough to ensure the tighten fix of the cantilever and avoid possible vibration during imaging. Alternatively, slide the cantilever into place with the help of a tweezers to rise up the plate and secure the screws as explained before. Pay attention not to damage the cantilever during this operation. The cantilever must be parallel to the holder otherwise it would be easier to break it when mounted on the setup. For imaging experiment, the small cantilever listed in the "Material" section should be used.

8.

Imaging experiments can be performed with the laser diode driver set in constant intensity "I mode" or constant power "P mode", depending on the type of sample and conditions in which to be imaged. For high-resolution imaging, we recommend using the laser diode driver in the "I mode". The "P mode" operates a feedback system that may introduce noise to the image. It can be used to avoid local heating when performing one-day long measurements.

Characteristic values we use are: in the "I mode", set the output between 21.8 and 23 mA, while in the "P mode" set the output between 0.13 and 0.18 mW for the laser diodes used in HS-AFM, although this number may vary with the ageing of the laser diode. 9. To facilitate this task, set the video camera in high light sensitivity ("night shot" mode) so the laser spot is intensified and easily visible on the screen. 10. In the case the wide XY range scanner (scan range: XY = 4 µm x 4 µm, Z = 0.7 µm) is used to perform imaging experiments, the screws of the scanner holder are in the front side of the element. Also in this case, to accommodate the scanner, the backside is first positioned and afterwards the front side is placed by engaging the magnets in contact with the screws. 11. Before connecting the scanner, check the liquid in the pool neither to form a pronounced meniscus nor wet the Z-scanner, because while applying a voltage during imaging may cause an electric arc and consequently damage the Z-scanner.

12. If the fluid in the pool does not look clear or there are particles, dust visible in the video monitor, renew your buffer solution exchanging the fluid several times in the chamber. If the problem persists, consider these situations: i) the adsorption of the sample to the surface is not strong enough, ii) the sample concentration is too high, iii) there is a possible buffer solution-sample interaction (for example, magnesium salts precipitate in phosphate buffer). 13. To prevent the tip crashing on the surface, it is critically to be very attentive to the video camera while performing a coarse approach. Stop the coarse approach when the surface edge starts to define but is still blurred or unfocused. If the tip crashes brutally on the surface it will be most likely damaged.

14. The cantilever position on the surface is important for high-speed imaging. If the cantilever is not placed in the edge but closer to the middle area of the surface, during imaging, the Z-scanner exerts a viscous drag or a hydrodynamic pressure that drags the base or chip of the cantilever, which leads to an increase in the response time and imprecise cantilever deflection [START_REF] Ando | A High-speed Atomic Force Microscope for Studying Biological Macromolecules in Action[END_REF][START_REF] Ando | Filming biomolecular processes by high-speed atomic force microscopy[END_REF][START_REF] Uchihashi | Guide to video recording of structure dynamics and dynamic processes of proteins by high-speed atomic force microscopy[END_REF]. [START_REF] Inoue | Direct observation of surfactant aggregate behavior on a mica surface using high-speed atomic force microscopy[END_REF]. Usually, AFM operating software programs include this function. [START_REF] Munguira | Glasslike Membrane Protein Diffusion in a Crowded Membrane[END_REF]. The fundamental resonant frequency is convoluted when using piezo acoustic excitation of the cantilever with a myriad of peaks due to 'echoes' in the liquid cell pool, the so-called the forest of peaks. HS-AFM imaging is performed selecting a frequency that 'echoes' in the fluid cell in the range of frequencies of the fundamental resonance of the cantilever [START_REF] Carrasco | Cutting down the forest of peaks in acoustic dynamic atomic force microscopy in liquid[END_REF]. 17. Typically, the free oscillation amplitude is set in concordance of the roughness of the sample; the flatter the sample; the lower the amplitude. 18. To convert the amplitude value from V to nm, it should be calibrated once approached using the optical beam detection sensitivity, which requires performing force curves (approach and retract cycles between the tip and the sample). In general, avoid calibrating the sensitivity before imaging -or before the experiment-to prevent tip damage or tip wearing. The calibration can be performed at the end of the experiment, after imaging. At this step -to approach-use an approximate value: the sensitivity does not change significantly for the same type of cantilevers.

19. When the PID controller is active and the amplitude set point is set lower than the free oscillation amplitude, the Z-piezo fully expands (to positive voltage). 20. If the full voltage range of the Z-piezo driver is 50 V, center it to 25 V. If the full voltage range of the X-and Y-piezo drivers are 100 V, center them to 50 V. [START_REF] Ando | Filming biomolecular processes by high-speed atomic force microscopy[END_REF]. A typical percentage of amplitude reduction for halting the automatic approach is ~30 %. Alternatively, manually approach slowly moving the step-motor to bring close the cantilever to the sample surface. This is particularly useful when the surface is rough and/or heterogeneous.

22. Alternatively, slowly up or down with the step-motor. [START_REF] Carrasco | Cutting down the forest of peaks in acoustic dynamic atomic force microscopy in liquid[END_REF]. In general for high-resolution imaging, it is very common to use amorphous carbon or carbon nanofibers tips. These materials are brittle and therefore they break easily. Always scan with very gentle tapping. 24. During the first images the risk of tip damage is high, here it is intended to limit it.

25. If the integral I gain is increased too much, saturation of the feedback loop bandwidth occurs; the Z-displacement cannot be done any faster even if the PID controller is requiring it. Then when I saturation is reached the feedback loop breaks, and error accumulates in the integrator-circuit, an I signal overshoot is created. Next, the circuit will create to compensate an overshoot of the I signal in the other direction. This oscillatory sequence of overshoots creates strips in the topographic image of the HS-AFM. Importantly, the I saturation depends on the topography of the sample under the tip. I saturates on the sample topographies that demand faster motions of the z-displacement, the uphill and downhill motions. 26. To avoid tip damage during the withdraw, first retract the tip from the sample using the Z-piezo displacement; for this set point A>A0 before using the motor. 27. Otherwise, as surfaces get charged with plasma, the cantilever can break due to electrostatic repulsions.

28. The nature of the gas injected in the plasma chamber affects the etching rate of the amorphous carbon EBD tip. For example, oxygen oxidizes the carbon or the EBD tip, hence is more aggressive than inert gases such as helium, whose atoms just physically bombard the carbon. As a general rule, the less is the imaging resolution, the more etching power should be applied on the tip apex.

29. The final shape of the sharpened apex depends on its initial shape, high resolution apexes are not achievable always by plasma etching, a minimum of sharpness of the initial apex is required. So, in the case that after a few cycles of sharpening the required resolution is not achieved, the HS-AFM probe should be replaced. 30. This operation can be implemented in few simple steps using a for Routines perform in MATLAB this and other described processing steps are freely available at https://sites.google.com/view/fm4b-lab/group-members/dr-arinmarchesi.

31. The performance of image registration also depends upon the choice of the reference image (template). To increase the signal-to-noise ratio you shall build your reference image by averaging over several frames (usually 10-20). For this purpose, we recommend to use the median, which is more robust than the mean.

Cropping is also commonly used to remove unwanted features from the edges of the averaged image.

32. Mechanical and piezo-scanner drifts may result into movies in which the sequential frames are not perfect real-space superposition. Therefore, before any data can be extracted from a HS-AFM image stack, all images must be perfectly aligned with respect to a stable coordinate origin. 36. In this routine, for each row of the image, the row median is subtracted from every element in the row.

  : AC10 from Olympus (fc = 1.5 MHz in air and kc = 0.1 N/m) or USC-F1.5-k0.6 Nanoworld (fc = 1.5 MHz in air and kc = 0.6 N/m). 3. Optical microscope, tweezers, screwdrivers. 4. Rinsing solutions: anionic precision detergent cleaner (Alconox® dilute to 2% in water), ultrapure water (Milli-Q®), tap water, 70% ethanol, acetone. 5. Sample case: Supported Lipid Bilayer (SLB). Lipids: 1-palmitoyl-2-oleoyl-snglycero-3-phospho-(1'-rac-glycerol) (POPG), 1-palmitoyl-2-oleoyl-sn-glycero-3phosphoethanolamine (POPE), chloroform, plastic tube (15 mL Falcon®), microfuge tube of 2 ml, Hamilton syringe, fume hood, vacuum desiccator, vacuum pump, potassium chloride, calcium chloride, working buffers: "liposome hydration buffer" 150 mM KCl, 10 mM Tris (tris(hydroxymethyl)aminomethane), pH 7.4 and "incubation buffer" 10 mM Hepes (4-(2-hydroxyethyl)-1piperazineethanesulfonic acid), 100 mM NaCl, 5 mM CaCl2, pH 7.4.

5 .

 5 Increase of the feedback speed: First, set the proportional (P) and the derivative (D) gains at their slowest possible reaction speed, next localize the integral (I) gain feedback speed saturation, for this increase the I gain slowly until strips of feedback resonance appear in the image (see Note 25), next reduce the I gain to a close to saturation position. If needed, the A can be slightly further decreased below A0 to obtain an optimal surface profile. Typical A values range from 0.90•A0 to 0.5•A0. In the case of having several time constants in the controller it is possible that you may have to switch to the fastest reaction speed one to achieve the I gain saturation. Specific P I D values are not given as they will depend on the gain adjustments of the rest of the feedback loop.

4 .

 4 Remove X-and Y-axis tilting by means of three-point plane fitting (see Note 34).

  33. For this purpose, an excellent option is the ImageJ plugins Template Matching and Slice Alignment (available at https://sites.google.com/site/qingzongtseng/template-matching-ij-plugin). Please note that this routine requires your video to be converted in 16-bit format, which might result in an unwanted loss of information. If this is the case, discard the obtained aligned video and apply the computed X and Y shifts to your original image stack to create a new registered image stack. 34. Nonlinearities of the piezoelectric scanners to the applied voltage can result in curved trajectories of the probe or sample stage (depending on the configuration of your system). If scanner bow artefacts are present, you shall consider fitting your image stack with a second order polynomial surface through five points.35. As the horizontal axis is usually the fast scan axis, any change in imaging conditions over the time of the scan will lead to horizontal discontinuities in the topographs. It will thus be well accounted for by a horizontal line-by-line levelling.

  Figure captions:

Figure 1 :

 1 Figure 1: HS-AFM hardware. Image of the HS-AFM hardware. Relevant parts are signaled: the

Figure 2 :

 2 Figure 2: Sample Incubation. Image of the humid hood used for sample incubation on a sample

Figure 3 :

 3 Figure 3: HS-AFM cantilever holder. Left, top view of the high-speed cantilever holder,

Figure 4 .

 4 Figure 4. HS-AFM sample scanner. Image of the standard HS-AFM sample scanner. Relevant

Figure 5 .

 5 Figure 5. Cantilever-sample positioning for approaching. A) Scheme of the cantilever holder

Figure 6 :

 6 Figure 6: Effect of Helium plasma etching on the EBD tip imaging of OmpF. Left, Image

Acknowledgements

This work was supported by French National Institute of Health and Medical Research (Inserm); the program "Investissements d'Avenir" by the French National Research Agency, ANR-10-LABX-0083 (Labex EFL); Grant ANR-16-CE15-0023 (ANR SalmoTubes); Grant ANR-15-CE11-0020 (ANR MoBaRhE). A.M acknowledges support from the Long Term EMBO Fellowship (ALTF 1427-2014) and the Marie Curie Action (MSCA-IF-2014-EF-655157).


Professor Emilio Bastidas-Arteaga

Professor Mark G Stewart
email: mark.stewart@newcastle.edu.au

Cost-effective adaptation to address climate change impacts

Keywords: Reinforced concrete, adaptation, cost-benefit analysis, corrosion, reliability

Concrete is the predominant construction material for buildings, bridges, wharves, and other infrastructure worldwide. A potentially important factor for asset management is the possible influence of climate change. This may alter the environment to which infrastructure is exposed, and in turn may alter the factors known to affect the corrosion of reinforcing steel, including atmospheric CO 2 concentration, temperature, humidity, ocean acidification, airborne pollutants, etc. This chapter poses the problem of adaptation of deteriorating RC structures. It describes a framework for evaluating the cost-effectiveness of adaptation measures that accounts for deterioration models, probabilistic methods and cost-benefit analysis. The methodology is illustrated with an example focusing on cost-effective adaptation of existing RC structures placed in the coastal French cities of Saint-Nazaire and Marseille.

 22.1. Introduction 22.1.1. BackgroundConcrete is the predominant construction material for buildings, bridges, wharves, and other infrastructure in Europe, Australia and elsewhere. A potentially important factor for asset management is the possible influence of climate change. This may alter the environment to which infrastructure is exposed, and in turn may alter the factors known to affect the corrosion of reinforcing steel, including atmospheric CO 2 concentration, temperature, humidity, ocean acidification, airborne pollutants, etc. A rise in temperature will increase the rate of infiltration of deleterious substances (increased material diffusivity) and increase the corrosion rate of steel. Changes in relative humidity levels may also increase the rate of infiltration of deleterious substances [START_REF] Nguyen | An efficient chloride ingress model for long-term lifetime assessment of reinforced concrete structures under realistic climate and exposure conditions[END_REF][START_REF] Stewart | Climate change impact and risks of concrete infrastructure deterioration[END_REF]. Typically these parameters must be considered as random variables or stochastic processes, and their statistical characteristics will gradually change with time. An appropriate framework for dealing with this, and other climate adaptation problem, is structural reliability and risk-based decision analysis considering updating with inspection data [START_REF] Stewart | Climate Impact Risks and Climate Adaptation Engineering for Built Infrastructure[END_REF]Tran et al. 2016a;b). Several studies have used probabilistic or deterministic methods to assess climate change effects on durability of RC structures placed in several countries [START_REF] Bastidas-Arteaga | Influence of weather and global warming in chloride ingress into concrete: A stochastic approach[END_REF][START_REF] Quinet | L'évaluation socioéconomique des investissements publics. Commissariat Général à la Strategie et à la Prospective[END_REF][START_REF] De Larrard | Effects of climate variations and global warming on the durability of RC structures subjected to carbonation[END_REF][START_REF] Stewart | Climate change impact and risks of concrete infrastructure deterioration[END_REF][START_REF] Talukdar | Carbonation in concrete infrastructure in the context of global climate change: Part 2, Canadian urban simulations[END_REF][START_REF] Wang | Impact of climate change on corrosion and damage to concrete infrastructure in Australia[END_REF]. These researches found that the temporal and spatial effects of a changing climate can increase current chloride ingress or carbonation-induced damage risks and justify the need for comprehensive adaptation. Thus, [START_REF] Stewart | Life-cycle cost assessment of climate change adaptation measures to minimise carbonation-induced corrosion risks[END_REF] used a simplified carbonation model and global IPCC (2007) CO 2 concentration and temperature change data to assess the costeffectiveness of increasing design cover as an adaptation measure. This preliminary analysis found that increasing design cover may not be cost-effective. Nevertheless, the results were based on an oversimplified carbonation model and the authors highlighted that the reported preliminary results cannot be generalised and that further research is needed to better characterise the cost-effectiveness of adaptation strategies. [START_REF] Stewart | Climate change adaptation for corrosion control of concrete infrastructure[END_REF] considered the effect of climate adaptation strategies including increases in cover thickness, improved quality of concrete, and coatings and barriers on damage risks. It was found that increases in design cover ameliorate the RC durability under a changing climate. However, such a study does not include cost-benefit assessment of climate adaptation strategies, and later work in this area proved inconclusive [START_REF] Peng | Damage risks and cost-benefit analysis of climate adaptation strategies of RC structures in Australia[END_REF]. 22.1.2. Aims and Scope Within this context, the main objectives of this chapter are:

• to propose a framework for probabilistic cost-benefit analysis of adaptation measures to ameliorate chloride-induced corrosion in new and existing structures, • to apply the methodology to structures subjected to various environmental conditions and/or located in specific locations in France. These challenges require the consideration of time-dependent damage risks and costs that differ as a function of the construction time and particular geographical conditions. Timedependent damage risks depend mainly on exposure (climate change projections) and technical considerations (design standards at the time of construction, properties of the construction and repair materials, inspection and maintenance strategies, implementation (or not) of standards updating, etc.). The assessment of cost-effectiveness of adaptation strategies is mainly related to economic aspects (discount rate, repair and adaptation costs, etc.) and the level of use of structures (remaining lifetime, assessment time, adaptation times, etc.). This study includes many of the above-mentioned aspects. Section 22.2 poses the problem of adaptation of deteriorating RC structures. Section 22.3 describes the proposed framework for evaluating the cost-effectiveness of adaptation measures. It accounts for deterioration models, probabilistic methods and cost-benefit analysis. Section 22.4 presents the repair and adaptation strategy that will be used in the illustrative example (Section 22.5). The example focuses on adaptation of existing RC structures placed in the coastal French cities of Saint-Nazaire and Marseille.

Adaptation of RC structures

Climate adaptation in terms of the enhancement of adaptive capacity can be done by developing new technologies and materials to counter the impact of increasing corrosion risk under a changing climate. From a practical point of view, three questions about the problem of adaptation of structures emerge:

• which measure could be used to adapt?,

• when is the best time to implement adaptation measures?, and • how to measure the cost and effectiveness of adaptation measures?.

Adaptation Measures

There is a wide range of existing and 'low-tech' options that can enhance the durability of concrete structures and these can be applied to reduce the adverse affects of climate change.

The design options generally include the selection of cover, concrete mix, surface coating barriers, extraction, and cathodic protection. In addition to reducing environmental exposure as much as possible, practical adaptation solutions in a new design may come from increasing cover and strength grade, or any approaches that reduce material diffusion coefficient without compromising the reliability and serviceability of concrete. Adaptation measures for new and existing concrete structures may include:

• Surface treatments,

• Realkanization,

• Extra design cover,

• Increase concrete durability,

• Stainless or galvanised steel reinforcement,

• Corrosion inhibitors,

• Cathodic protection, and • Replace existing cover with new concrete.

Adaptation Time

The time of adaptation is highly variable and dependent on extent and location of corrosion damage. Some adaptation strategies could be applied at time of construction (coatings/surface treatments, reinforcement), and others at time of corrosion initiation (realkanization, chloride extraction). Clearly, it is preferable to use adaptation strategies that are implemented during design and construction rather than in-service (e.g., when corrosion damage occurs) as the latter will be much more costly in terms of direct costs and inconvenience/user delays and other indirect costs [START_REF] Stewart | Climate change adaptation for corrosion control of concrete infrastructure[END_REF].

Cost-Effectiveness of Adaptation Strategies

Adaptation strategies will have varying degrees of effectiveness and cost. Some will require regular maintenance over the life of the structure, such as surface treatments, which will increase their lifecycle cost. Given that there are many millions of new and existing concrete infrastructure in many countries the cost of adaptation can be immense. For this reason, a risk-based approach is needed to assess the optimal level, if any, of adaptation measures. This includes the cost, location, timing and extent of adaptation measures.

An increase in cover thickness can increase the time of carbonation and chloride ingress to reach concrete reinforcement and in turn delay carbonation and chloride-induced corrosion. It is therefore one of the most obvious and simplest adaptation options in the design of concrete infrastructure under a changing climate to maintain structural durability and serviceability.

While the change of cover is considered as the most straightforward design approach to reduce the impact of changing climate, other options may also include the selection or design of concrete materials to reduce the diffusion coefficient of deleterious substances -i.e. slow the ingress of those substances, and hence delay corrosion of concrete reinforcement. In practice, selection of a higher strength grade of concrete is one approach to reduce the diffusion coefficient, in addition to enhancing its mechanical properties. [START_REF] Stewart | Life-cycle cost assessment of climate change adaptation measures to minimise carbonation-induced corrosion risks[END_REF] have conducted a preliminary lifecycle cost assessment to assess the cost-effectiveness of increasing design cover as an adaptation measure to mitigate the effects of carbonation of concrete. The lifecycle cost analysis considered costs associated with extra design cover and expected maintenance/repairs for typical RC structures and elements over the next 100 years considering several IPCC atmospheric CO 2 emission scenarios. The preliminary analysis found that lifecycle costs for the current situation ('do nothing' -use existing covers) are lower than lifecycle costs for proposed increases in design cover. This suggests that although enhanced greenhouse conditions will lead to increased carbonationinduced corrosion of RC structures it may not be cost-effective to increase design covers. However, a more detailed assessment of risks, costs, benefits of adaptation measures and environmental impact may reveal a different conclusion. Further work considered the effects of spatially variable corrosion damage, and the costs and benefits of adaptation measures (Peng andStewart 2014, 2015). What is important is the need to estimate time-dependent changes in damage risks, the effectiveness and cost of one or more adaptation measures, costs of repairing damage, and other criteria needed to assess the optimal level of adaptation measures both now and into the future.

Proposed Adaptation Framework

The main practical problem concerning adaptation of structures lies in evaluating its costs and effectiveness. Figure 22.1 shows the proposed framework for dealing with this problem that combines:

• general methods that could be applied to any structure (deterioration models, stochastic approaches and cost-benefit analysis), with • information specific to each structure (climate change predictions and structural characteristics). The following sections focus on describing the general methods that are employed in the proposed framework. Information specific to each structure will be detailed in the numerical application (Section 22.5). Structural, characteris)cs, 22.3.1. Deterioration Models As mentioned in (Bastidas-Arteaga et al. 2010;[START_REF] De Larrard | Effects of climate variations and global warming on the durability of RC structures subjected to carbonation[END_REF][START_REF] Stewart | Climate Adaptation Engineering and Risk-Based Design and Management of Infrastructure[END_REF], the assessment of climate change consequences on RC durability requires deterioration models that account for the influence of surrounding environmental actions (temperature, relative humidity, rain, etc.). Table 22.1 summarises the characteristics of the models considered in this study. We will focus on modelling deterioration since construction (or repair) until severe concrete cracking. Thus, the whole lifetime will be divided into three stages (Table 22.1) that are modelled based on different principles. For the stage of corrosion immunity (i.e., before corrosion initiation), we use a finite element model that accounts for chloride ingress into concrete by diffusion and convection [START_REF] Bastidas-Arteaga | A comprehensive probabilistic model of chloride ingress in unsaturated concrete[END_REF][START_REF] Saetta | Analysis of chloride diffusion into partially saturated concrete[END_REF]. This model takes into account the influence of temperature and relative humidity variations on the chloride penetration process. The stage of active corrosion is modelled based on electrochemical principles and considers only the effects of temperature [START_REF] Duracrete | Statistical quantification of the variables in the limit state functions, DuraCrete -Probabilistic Performance based Durability Design of Concrete Structures, EU -Brite EuRam III[END_REF]. The stage of corrosion until severe concrete cracking is assessed by two models. The former concerns the time since corrosion initiation until the appearance of a first crack on the concrete surface (El Maaddawy and Soudki 2007). The second represents the time until reaching a threshold crack width w lim [START_REF] Mullard | Corrosion-Induced Cover Cracking: New Test Data and Predictive Models[END_REF]. Both models combine mechanical and electrochemical principles. The corrosion propagation models considered in the second stage that are influenced by temperature variations drive both models. Although corrosion propagation is widely influenced by relative humidity, nowadays there is not a model that accounts for this factor in a comprehensive manner.

p s (0,t) = Pr t ≥ T dam ⎡ ⎣ ⎤ ⎦ [22.
1] where depending on the asset maintenance policy the owner can specify the limit criteria for repair: threshold chloride content, limit crack width, etc. However, after repair, the timedependent damage risks of the repaired structure will not be the same as the original one p s (0, t) due to changed temperature and RH at the time of repair. Hence, the damage risk for repaired structures exposed to the environment for the first time at time of repair, t rep = iΔt, will change depending on the new climatic conditions and time of repairs (Bastidas-Arteaga and Stewart 2013Stewart , 2015;;[START_REF] Nguyen | An efficient chloride ingress model for long-term lifetime assessment of reinforced concrete structures under realistic climate and exposure conditions[END_REF][START_REF] Stewart | Climate Adaptation Engineering and Risk-Based Design and Management of Infrastructure[END_REF]:

p s,i (iΔt,t) = Pr t ≥ T dam,i ⎡ ⎣ ⎤ ⎦ [22.
2] where T dam,i is the time to damage when the repaired structure is exposed to the environment for the first time after repair and Δt is the inspection length. Figure 22.2 presents a conceptual description of the time-dependent probability of damage for a structure subjected to chloride-induced deterioration under various climate change scenarios. The climate change scenarios are simply represented as relative changes with respect to current climate conditions of a given location after 100 years. The overall trend indicates that the deterioration processes increase probability of damage with time. Figure 22.2 clearly shows that the rate of damage risk is highly dependent on climate change effects. If there is no change in climate (∆RH=0% and ∆T=0ºC), the probability of damage increases with time and remains constant irrespective of time of repair. However, if climate change reduces the environmental relative humidity, i.e. ∆RH= -10% in 100 years, the chloride ingress mechanism slows down, and consequently, the probability of severe cracking decreases. In this case, climate change has a 'positive effect' on RC durability by reducing corrosion damage risk. An opposite behaviour is observed when climate change increases the temperature and relative humidity (∆RH=20% and ∆T=6ºC). In this case, the probability of damage increases if the structure is new or repaired at time t rep . Therefore, the timedependency of damage risks should be considered for a comprehensive cost-benefit analysis of adaptation measures. Costs and benefits may occur at different times so in order to obtain consistent results it is necessary for all costs and benefits to be discounted to a present value. The proposed costbenefit framework could be applied to both new and existing structures. Costs and benefits are measured from the time of the cost benefit assessment, t assess . For example, for an existing structure, if a decision-maker is making a decision in t assess = 2013 about predicted costs and benefits of adaptation measures for a structure built in t construct = 1970, then damage costs incurred prior to t assess =2013 are not considered in the benefit-to-cost ratio as the decision maker is only concerned with costs and benefits that arise after 2013. For a new structure t assess = t construct . If a structure is built in the calendar year t construct , and if it is assumed that corrosion damage is always detected when the structure is inspected, then the expected damage cost measured from year of assessment t assess to end of service life, is E damage (T t ) and is the product of probability of corrosion damage and damage costs, i.e., [22.3] where T t is the service life (typically T t =100 years), Δt is the time between inspections, n is the number of damage incidents, i is the number of inspections, p s,n (t) the probability of the n th damage incidence before time t, r is the discount rate and C damage is the cost of damage including maintenance and repair costs, user delay and disruption costs, and other direct or indirect losses arising from damage to infrastructure. For example, an asset owner should be able to quantify the unit repair cost (€/m 2 ), and if the area of damage is known then repair cost can be estimated.

E damage T t ( ) = p s,n iΔt ( ) -p s,n iΔt -Δt ( ) ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ C damage 1+ r ( ) t construct +iΔt-t assess i=n T t /Δt ∑ n=1 T t /Δt ∑ with iΔt > t assess -t construct , t construct ≤ t assess
Eq. [22.3] can be generalised for costs arising from multiple limit states, such as flexural failure, shear failure, etc. Corrosion damage (severe corrosion-induced cracking) is considered herein as the most influential mode of failure for the estimation of benefits. Eq. [22.3] can be re-expressed as:

E damage T t ( ) = ΔP s,i C damage 1+ r ( ) t construct +iΔt-t assess i=1
T t /Δt ∑ with iΔt > t assess -t construct , t construct < t assess [22.4] where ΔP s,i is the probability of damage incident between the (i -1) th and i th inspections which is a function of time since last repair which is turn is affected by damage risks for original and repaired concrete p s (0,t) and p si (iΔt,t), respectively. The repaired concrete may have the same durability design specifications as the original concrete, or may be repaired to a higher standard (e.g., increased concrete cover). The risk reduction caused by an adaptation measure is thus:

ΔR T t ( ) = E damage-BAU T t ( ) -E damage-adaptation T t ( ) E damage-BAU T t
( ) [22.5] where E damage-BAU (T t ) and E damage-adaptation (T t ) are the cumulative expected damage cost (economic risk) for no adaptation measures (business as usual BAU or existing practice) and adaptation measures, respectively. If an adaptation measure is cost-effective then E damage- adaptation (T t ) will be significantly lower than E damage-BAU (T t ) resulting in high risk reduction ΔR(T t ). In other words, ΔR(T t ) represents the proportional reduction in expected repair costs due to an adaptation measure.

The cost of adaptation, in this case, additional repair costs associated with increased cover, will occur at the same time as the damage (repair) costs are incurred. It follows that the expected cost of adaptation is directly proportional to damage costs

E adapt (T t ) = ΔP s,i C adapt (1+ r) t construct +iΔt-t assess i=1 T t /Δt

∑

with iΔt > t adapt -t construct , t construct < t assess [22.6] where C adapt is the cost of adaptation measures that reduces risk by ΔR and t adapt is the adaptation year. The 'benefit' of an adaptation measure is the reduction in damages associated with the adaptation strategy, and the 'cost' is the cost of the adaptation strategy. The benefit-to-cost ratio BCR(T t ) is:

BCR T t ( ) = E damage-BAU T t ( ) ΔR T t ( ) E adapt (T t ) [22.7]
Clearly, an adaptation measure that results in a benefit-to-cost ratio exceeding unity is a costeffective adaptation measure. Since costs and benefits are time-dependent then it follows that the benefit-to-cost ratio is time-dependent. Thus, an adaptation measure may not be costeffective in the short-term, due to high adaptation cost for example, but the benefits may accrue over time resulting in improved cost-effectiveness in the longer-term. Note that the additional cost of repair (such as increasing cover on repaired concrete) is treated herein as an adaptation cost, and not as a reduced benefit. Since the proposed adaptation framework considers uncertainties (Section 22.3.2), the output of the analysis (BCR) is also variable. This allows the mean BCR and the probability that an adaptation measure is cost-effective Pr(BCR>1) to be calculated. These criteria are used to evaluate the cost-effectiveness of adaptation strategies. Monte-Carlo simulation analysis will be used as the computational tool to propagate uncertainties through the cost-benefit analysis, although analytical methods could also be used -e.g., [START_REF] Stewart | Probabilistic Risk Assessment of Engineering Systems[END_REF].

For all adaptation options, construction and repair cost data are needed, and such cost data is country, site and structure specific and so it is difficult to make generalisations about these costs. In this chapter, costs are expressed in 2013 euros. Note however, that Eqns. ([22.3] to [22.7]) show that BCR is not dependent on the monetary units, but it is a function of the ratio of damage that is related to adaptation costs. It is assumed that design and inspection costs are similar for different adaptation measures and so are not needed for this comparative analysis. Hence, adaptation strategies will only affect the expected damage costs. As we are concerned about outdoor exposures then the external RC structural elements of interest are slabs, beams and columns. Corrosion damage is assumed to occur on one (exposed) face of a slab and beam, and all faces of a column.

Discount Rates

There is some uncertainty about the level of discount rate, particularly for climate change economic assessments (e.g. [START_REF] Dasgupta | Discounting Climate Change[END_REF] [START_REF] Harrison | Valuing the Future: the social discount rate in cost-benefit analysis[END_REF]. Infrastructure Australia recommends discount rates of 4%, 7% and 10% for infrastructure projects [START_REF] Ia | Outline of Infrastructure Australia's Prioritisation Methodology[END_REF]). Other discount rates vary from 3% (Germany) to over 10% (World Bank) [START_REF] Harrison | Valuing the Future: the social discount rate in cost-benefit analysis[END_REF]. Discount rates are generally assumed constant with time. However, this may not be appropriate when considering intergenerational effects often associated with climate change policy decisions (e.g. [START_REF] Boardman | Cost-Benefit Analysis: Concepts and Practice[END_REF])). Projects with significant effects beyond 30-50 years are considered intergenerational, and so for example, the British Treasury recommends the following time-declining discount rates (HM Treasury 2003): 3.5% (0-30 years), 3.0% (31-75 years), 2.5% (76-125 years), 2.0% (126-200 years), 1.5% (201-300 years), and 1.0% (300+ years). However, there is some controversy about time-declining discount rates (e.g. [START_REF] Viscusi | Rational Discounting for Regulatory Analysis[END_REF])), and the Australian Office of Best Practice and Regulation (OBPR) states that 'there is no consensus about how to value impacts on future generations' and 'Rather than use an arbitrarily lower discount rate, the OBPR suggests that the effects on future generations be considered explicitly' (OBPR 2010). Nonetheless, the 2006 U.K. Stern Review adopted a discount rate of 1.4% [START_REF] Stern | The Economics of Climate Change: The Stern Review[END_REF], and the Australian Garnaut Review adopted discount rates of 1.35% and 2.65% [START_REF] Garnaut | The Garnaut Climate Change Review: Final Report[END_REF]). These relatively low discount rates were selected to not underestimate climate impacts on future generations. However, others suggest higher discount rates when assessing economic impacts of climate change (e.g, [START_REF] Nordhaus | A Review of the Stern Review on the Economics of Climate Change[END_REF])).

The above quantification of discount rates relates mainly to public-sector investments in infrastructure. Private investments in infrastructure, such as the owners of a port, power station, airport, etc., tend to include a risk premium which leads to a higher discount rate [START_REF] Btre | Report 110: Risk in cost-benefit analysis[END_REF]. The marginal rate of return of private investments is suggested as one method to derive discount rates for private investments (e.g. [START_REF] Boardman | Cost-Benefit Analysis: Concepts and Practice[END_REF]). According to [START_REF] Boardman | Cost-Benefit Analysis: Concepts and Practice[END_REF], the best 'proxy' for marginal rate of return of private investments is the before-tax rate of return on corporate bonds -or approximately 4.5%.

In this chapter, discount rates of 2%, 4% and 8% are considered. These discount rates represent the range of discount rates in several countries, and the lower (2%) discount rate is also representative of values used to consider intergenerational and climate change effects.

22.4. Considered Repair and Adaptation Measures 22.4.1. Definition For illustrative purposes, this study considers patch repair as the most common technique to repair corrosion damage in RC structures -e.g., [START_REF] Bre | Residual Life Models for Concrete Repair -Assessment of the Concrete Repair Process[END_REF][START_REF] Canisius | Concrete Patch Repairs Under Propped and Unpropped Implementation[END_REF]. For a patch repair, the concrete cover is typically removed to approximately 25 mm past the steel bars (which are then cleaned of corrosion products) and a repair material is installed. The maintenance strategy assumes that [START_REF] Stewart | Spalling Risks, Durability and Life-Cycle Costs for RC Buildings[END_REF]):

• concrete is inspected at time intervals of Δt;

• patch repair is carried out immediately after corrosion damage has been discovered at time of i th inspection at time iΔt; • damage limit state exceedance results in entire RC surface being repaired;

• repair provides no improvement in durability performance of the repaired structure (i.e., it is repaired with the same cover and concrete quality as the original design specification); • damage may re-occur during the remaining service life of the structure, i.e., multiple repairs may be needed. The maximum possible number of damage incidents is n max = T t /Δt, where T t is the total lifetime. We consider increase in design cover as the climate adaptation measure aimed at reducing the impact of chloride-induced corrosion damage. Increasing the strength grade of concrete, which reduces the diffusion rate, is another possible climate adaptation measure. For more details of this adaptation measure see Bastidas-Arteaga and Stewart (2015[START_REF] Bastidas-Arteaga | Economic Assessment of Climate Adaptation Strategies for Existing RC Structures Subjected to Chloride-Induced Corrosion[END_REF] The cost of repair or replacement and associated user losses, etc. are considerable and for some structures user losses are often much greater than direct repair, replacement and maintenance costs. [START_REF] Val | Life-cycle analysis of reinforced concrete structures in marine environments[END_REF] assumed that the cost of RC bridge deck replacement is double the construction cost based on cost data for removal and replacement costs. However, this is likely to over-estimate the repair costs for most corrosion damage. The estimated cost for concrete patch repair using ordinary Portland cement is 286€/m 2 [START_REF] Bre | Residual Life Models for Concrete Repair -Assessment of the Concrete Repair Process[END_REF][START_REF] Mullard | Life-Cycle Cost Assessment of Maintenance Strategies for RC Structures in Chloride Environments[END_REF][START_REF] Yunovivh | Corrosion Costs and Preventive Strategies in the United States[END_REF]. User losses and other user disruption costs are site and structure specific, but for many RC structures such costs will be minimised if the RC element to be repaired is an external structural member such as walls, columns or facade panels. However, for bridges closure of one lane for a four lane bridge can cause user delay costs of 39,650€ per day [START_REF] Yunovivh | Corrosion Costs and Preventive Strategies in the United States[END_REF]. To allow for a minor user disruption cost (Bastidas-Arteaga and Stewart 2013Stewart , 2015) ) assumed that the total failure cost was C damage =325€/m 2 (approximately $450-500/m 2 in United States dollars). On a more detailed cost analysis, (Bastidas-Arteaga and Stewart 2016) considered costs estimated for the repair of RC slabs and beams of the Agri-foodstuffs terminal of the Nantes Saint-Nazaire Port [START_REF] Srifi | Gestion, surveillance et pertinence des méthodes de réparation des ouvrages maritimes[END_REF]). Table 22.2 summarises the costs of damage that include costs of damage reports, preparation of the building site, repair (removal and reconstruction of the cover) and operating losses. These costs were computed taking into account recent maintenance operations for three repair alternatives. It is noted from Table 22.2 that most of the costs are related to concrete removal and repair operations, followed by the costs of studies and preparation of the building site and operation losses. For the alternative 3, the replacement of corroded steel requires a longer repair time and so increasing operational losses.

Cost for the Adaptation Strategy

The baseline case for construction cost per unit volume (C cv ) including forms, concrete, reinforcement, finishing and labour is approximately 490-850€/m 3 , 910-1010€/m 3 and 780-1560€/m 3 for RC slabs (4.6-7.6 m span), RC beams (3.0-7.6 m span) and RC columns (300 mm × 300 mm to 900 mm × 900 mm), respectively (RSMeans 2012). These values will therefore be used to estimate the costs of the two adaptation strategies.

It is assumed that an increase in design cover Δc adapt would increase the cost of forms, concrete, reinforcement, finishing and labour by an amount proportional to the extra volume of concrete needed. Since C damage units are €/m 2 of surface area, but C cv is given as per unit volume, then cost of construction (C c ) and C adapt should be converted to cost per surface area exposed to deterioration, and so is corrected for structural member dimension such as slab depth or beam or column width (D). Table 22.3 provides the adaptation costs for various structural elements (per mm of extra cover). The relationships used to evaluate the adaptation costs are provided in [START_REF] Bastidas-Arteaga | Economic Assessment of Climate Adaptation Strategies for Existing RC Structures Subjected to Chloride-Induced Corrosion[END_REF]. This table also presents the adaptation costs for Δc adapt =5 and 10 mm increase in extra cover. (European standard 2004), which is mandatory after 2010 in France, suggests concrete covers for different exposure conditions. For example, there are three exposure zones for marine structures: atmospheric XS1, submerged XS2 and splash and tidal XS3. For the XS3 exposure the recommended concrete cover is 55 mm. This example considers two adaptation strategies implemented at a given adaptation time (e.g., t adapt =2030) defined as follows:

• Adaptation 1: increase existing (Eurocode) design covers by Δc adapt =5 mm for XS3 exposure, or • Adaptation 2: increase existing (Eurocode) design covers by Δc adapt =10 mm for XS3 exposure. It is assumed that if severe concrete cracking is detected when there is an evolution of the construction standard, the concrete cover is increased according to standard recommendations at the time of repair. For example, a 1970 structure (40 mm design cover) damaged in 2015 is repaired to the 2015 specified design cover of 55 mm. This study does not focus on a specific structure. Therefore, there is not information about the characteristics of the concrete used to build the existing structures. Besides, there is no information about the repair materials that will be used in the future. For simplicity, this example assumes that the same concrete was used for the construction and repair of all structures during their lifetime. This concrete has a characteristic compressive strength, f' ck = 35 MPa, as recommended by the Eurocode 2 for XS3 exposure (European standard 2004). Other assumptions are summarised as follows:

• the structural lifetime is T t =100 years,

• the limit crack width for repair is w lim =1 mm,

• rebar diameter is d 0 =16 mm.

• the environmental chloride concentration, C env , corresponds to a XS3 exposure,

• all structural components will be subjected to the same environmental conditions,

• the chloride ingress is one-dimensional,

• the time of adaptation will vary between 2020 and 10 years before the end of the structural lifetime, • the adaptation strategy consists of increasing concrete cover by 5 or 10 mm with respect to standard recommendations, • the time for assessment of costs is t assess = 2013, • the costs of damage are: C damage =263.2€/m 2 , C damage =323€/m 2 , and C damage =353.4€/m 2 for Repairs 1, 2 and 3 (Table 22.2). • the adaptation costs are defined according to Table 22.3, and • the discount rates are 2, 4 and 8%. The probabilistic models used to estimate damage probabilities (p s,i ) are given in Table 22.4. p s,i (Eq. [22.2]) corresponds herein to the probability of severe concrete cracking until reaching the limit crack width w lim . This serviceability limit state allows us to generalise the results. It is assumed that all the random variables are statistically independent. Monte Carlo simulations were used for the assessment of damage probabilities and the propagation of uncertainties throughout the cost-benefit analysis. For more details of the deterioration models used herein see [START_REF] Bastidas-Arteaga | A comprehensive probabilistic model of chloride ingress in unsaturated concrete[END_REF][START_REF] Quinet | L'évaluation socioéconomique des investissements publics. Commissariat Général à la Strategie et à la Prospective[END_REF] and [START_REF] Nguyen | An efficient chloride ingress model for long-term lifetime assessment of reinforced concrete structures under realistic climate and exposure conditions[END_REF]. Concrete tensile strength, f ct MPa normal a 0.53(f c ) 0.5 0.13 [START_REF] Mirza | Statistical Descriptions of Strength of Concrete[END_REF]) Concrete elastic modulus, E c MPa normal a 4600(f c ) 0.5 0.12 [START_REF] Mirza | Statistical Descriptions of Strength of Concrete[END_REF]) a truncated at 0, b truncated at 10mm 22.5.1.2

. Climate change scenarios

The economic assessment of adaptation measures is widely influenced by time-dependent changes in environmental parameters (temperature, relative humidity (RH)) that are sitespecific. This work focuses on the study of the cost-effectiveness of climate change adaptation strategies for two port locations in France: Saint-Nazaire and Marseille. These cities correspond to different types of climate. Saint-Nazaire is close to the Atlantic Ocean in the Northern part of the country and it has a temperate oceanic climate. Marseille is placed in the South-East of France (Mediterranean coast) and it has a Mediterranean climate that is rather hot and dry. The overall impact of climate change on the future weather of the selected locations was estimated by using data computed by the French general circulation model CNRM-CM5. Figure 22.3 presents the yearly projections of temperature and RH for Saint-Nazaire and Marseille for the selected climate change scenarios (RCP 4.5 and RCP 8.5) since 2005. By comparing climate (before 2005) in both cities, it is noted that Marseille is hotter and dryer than Saint-Nazaire. Climate change projections mainly predict temperature increase without significant changes in RH for both cities until the end of this century. By comparing mean temperatures over the periods (2001)(2002)(2003)(2004)(2005)(2006)(2007)(2008)(2009)(2010) and (2091-2100) for both places, it was found that climate change could increase temperature by approximately 1.5ºC and 3.5ºC for RCP 4.5 and RCP 8.5 scenarios, respectively. Although climate change effects are relatively similar for both cities, the kinematics of chloride ingress and corrosion propagation is affected by the climate specific to each location. Comprehensive deterioration models are therefore useful tools for estimating the effects of specific climate conditions on RC durability.

Climate projections are subjected to considerable uncertainty, and dependent on CO 2 emission scenarios and accuracy of general circulation models (GCM). A comprehensive model of weather (RH and temperature) should be integrated with deterioration models to assess the effects of a changing climate. Given the difficulties of integrating a GCM with deterioration models, a simplified approach for modelling climate is considered in this study [START_REF] Teodorescu | Influence of the Climatic Changes on Wood Structures Behaviour[END_REF]). It accounts for (i) influence of climate change, (ii) seasonal variations, and (iii) random nature of weather within a season. The formulation of this model is detailed in [START_REF] Bastidas-Arteaga | A comprehensive probabilistic model of chloride ingress in unsaturated concrete[END_REF][START_REF] Quinet | L'évaluation socioéconomique des investissements publics. Commissariat Général à la Strategie et à la Prospective[END_REF]. Although the concrete properties and cover are the same for both cities, probabilities of severe cracking are larger for structures built in Saint-Nazaire. As indicated in Section 22.5.1.2, RH is about 8% larger in Saint-Nazaire than Marseille by increasing water content in the capillary pores and chloride diffusion rate. This higher RH will therefore shorten the time to corrosion initiation by increasing the probability of severe cracking. It is also observed the effects of the evolution of construction standards (in 1990 and 2010) that, for a given time after construction, decrease the probabilities of damage.

These results indicate that although increasing design cover is an effective protection for reducing damage induced by chloride ingress, the effectiveness of this measure depends on specific exposure conditions. effects of these climate change scenarios on the probability of severe cracking in also illustrated in Figure 22.4. As expected, probabilities of severe cracking are larger for RCP 8.5 exposure because higher temperatures accelerate chloride ingress and corrosion propagation [START_REF] Bastidas-Arteaga | A comprehensive probabilistic model of chloride ingress in unsaturated concrete[END_REF][START_REF] Nguyen | An efficient chloride ingress model for long-term lifetime assessment of reinforced concrete structures under realistic climate and exposure conditions[END_REF]. The effects of the RCP 8.5 scenario are larger for recent structures because the differences in temperatures between both climate change scenarios announced by general circulation models increase after 2050 (Figure 22.3). As a consequence, for structures built in 1970 the difference of temperature between both climate change scenarios is about 0.7 ºC (in 2070) whereas for recent structures (built in 2010), this difference is 1.7ºC (in 2110). From an engineering point of view, RCP 8.5 and RCP 4.5 climate change scenarios could be interpreted respectively as upper and lower bounds for carrying out sensitivity studies. This point is illustrated later in the assessment of the cost-effectiveness of adaptation measures.

22.5.2.2. Cost-effectiveness of damage adaptation strategies This section illustrates the probabilistic assessment of the cost-effectiveness of adaptation strategies for existing RC structures. These results were computed for a discount rate r =4% and a damage cost C damage =323€/m 2 (Repair 2) According to Table 22.2, this damage cost corresponds to a medium damage extent that does not require replacement of corroded steel.

A sensitivity study about the effects of discount rate will be presented in section 22.5.2.3.

Figure 22.5 presents the mean BCR for RC slabs in Marseille and Saint-Nazaire under the RCP 4.5 scenario. These results were obtained for slabs built in 2010 and t adapt = 2020. Time of adaptation is taken as 2020 to represent the shortest practical time for a national standard (e.g., Eurocode 2 (European standard 2004)) to consider changes to existing specification, recommend changes, and have changes implemented in future standards. The overall behaviour indicates that the mean of BCR is highly dependent on both the location and the slab size. The mean BCR is lower for Marseille and small slabs. For slabs with D=100 mm in Marseille the mean BCR is lower than 1 indicating that the adaptation strategies are not costeffective. This means that, for the studied material, current design cover (55 mm) is costefficient for Marseille. On the opposite, adaptation strategies are cost-effective for all slab sizes in Saint-Nazaire. Thus, recommendations of current standards and adaptation measures could be more or less adapted to local climate conditions. For both locations, increasing extra cover by 10 mm is less cost-effective than a 5 mm increase in cover. Even if risk reduction ΔR should be higher for Δc adapt =10 mm, the costs associated to this adaptation strategy are larger by reducing the mean BCR. Thus, it is possible to conclude that the mean BCR for the whole structure could be maximised by performing different actions for individual components: (i) optimising the extra cover, (ii) considering different types of adaptation strategies, and/or (iii) business as usual (or do nothing). The mean BCR is also influenced by the year of construction. To explain this relationship, Table 22.5 provides the mean BCR as a function of the construction year for slabs built in Saint-Nazaire and t adapt =2020. It is noted that the BCR is lower than one for older structures and increases for recent ones. A BCR<1 implies that the adaptation measure is not costeffective for old structures and that the existing standards recommendations are most costeffective during the structural lifetime. The increment of BCR is due, on the one hand, to the increase of concrete cover recommended by the standards and/or considered by the adaptation measures. This means that larger concrete cover is more effective for this kind of exposure. On the other hand, larger BCR values are also related to the increase of climate change effects on deterioration rates that justify the implementation of adaptation measures. As mentioned in previous results, the adaptation strategy Δc adapt =10 mm is less cost-effective for all adaptation years. The following results will therefore focus on the repair strategy Δc adapt =5 mm. Table 22.5 also gives the effects of the climate change scenario. It is noted that higher mean BCR are expected for the RCP 8.5 scenario that announces more important changes with respect to actual climate. The differences between results for both scenarios are larger for recent structures because they will be exposed to the largest climate variations that are more pronounced after 2050 (Figure 22.3). These climate variations will induce more corrosion damage by increasing the cost-effectiveness of adaptation strategies. The effect of the time of adaptation on the mean BCR and Pr(BCR>1) for slabs (D=300mm) in Saint-Nazaire, Δc adapt =5 mm and the RCP 4.5 scenario is shown in Table 22.6. It is noted that both the mean BCR and the Pr(BCR>1) decrease when the adaptation year is close to the end of the structural lifetime. Of interest is that the Pr(BCR>1) reaches only slightly above 60% when mean BCR exceeds 4. This illustrates the high variability of damage risks caused by uncertainties of climate projections, and variabilities of material, dimensional and deterioration parameters. These results could be used by an owner/stakeholder to evaluate the benefits and the risks of implementing adaptation strategies at various years. For example, it is observed that mean BCR and Pr(BCR>1) are small for older structures and therefore the owner/stakeholder could prioritise investments in adaptation measures for recent structures. These results could be also used to evaluate the impact of the adaptation year. For example, for structures built in 1990, if the owner/stakeholder decides to postpone the adaptation actions until 2040 the mean BCR is about 1.4 which is still interesting. However, the Pr(BCR>1) for this adaptation time is less than 11% indicating that the risks of having no benefits are larger. 22.7 describes the influence of discount rates (r) on both the mean BCR and the Pr(BCR>1) for slabs (D=300mm) built in several years in Saint-Nazaire and t adapt =2020. It is observed that the mean BCR and Pr(BCR>1) are very sensitive to r and both parameters are larger for small discount rates. This is explained by the fact that small discount rates imply that future costs are larger at present cost by increasing the cost-effectiveness of adaptation measures for repairs close to the end of the structural lifetime. As discussed in Section 22.3.3.2, various governments recommend lower discount rates of about 2% for long-term investments. The BCR analysis therefore shows that the adaptation strategies are more costeffective according to these recommendations.

Conclusions

The kinematics of the chloride ingress and corrosion propagation mechanisms is highly influenced by the surrounding environmental conditions including climate change that could accelerate or decelerate these processes depending on specific exposure and environmental conditions. Therefore, the results presented in this chapter highlight that a comprehensive probabilistic cost-benefit analysis of adaptation strategies should combine general methods (deterioration models able to account for climate variations, stochastic methods to propagate uncertainties and cost-benefit analysis) with factors specific to each structure (structural characteristics and climate change predictions). The numerical results focused on climate adaptation for existing RC structures built at different years (and therefore under different durability standards), and subjected to two different types of climate in France. The adaptation strategies consisted of increasing the cover recommended in the standards by 5 or 10 mm. The assessment of the mean BCR and the Pr(BCR>1) indicated that although increasing design cover is an effective protection for reducing damage induced by chloride ingress, the cost-effectiveness of this measure depends on specific exposure conditions. An adaptation strategy consisting of increasing design cover by 5 mm was cost-effective for Saint-Nazaire but it was not cost-effective for Marseille. The probabilistic analysis also indicated that the mean BCR and Pr(BCR>1) are dependent on type of structural element, age of construction, adaptation time, damage costs and discount rates. The overall results indicate that the probabilistic framework is well suited to assessing the impact of climate change on RC corrosion damage, and assessing the cost-effectiveness of climate adaptation strategies. Future work will consider other adaptation strategies such as improved concrete quality and coatings.

Figure 22 . 1 -

 221 Figure 22.1 -Proposed framework for determining the cost-effectiveness of adaptation measures

Figure 22 . 2 -

 222 Figure 22.2 -Conceptual description of the probability of damage for various climate change scenarios and repair times

Figure 22 . 3 -

 223 Figure 22.3 -Yearly temperature and relative humidity projections for Saint-Nazaire and Marseille for RCP 4.5 and 8.5 climate change scenarios

Figure 22 . 4 -

 224 Figure 22.4 -Probability of severe cracking for structures built in Saint-Nazaire and Marseille under the RCP 4.5 and RCP 8.5 scenarios

Figure 22 . 5 -

 225 Figure 22.5 -Mean BCR for RC slabs in Marseille and Saint-Nazaire under a RCP 4.5 climate change scenario, for structures built in 2010 and t adapt = 2020

Table 22 .1 -Main characteristics

 22

		of deterioration models considered in this study
	Deterioration stages	Physical phenomena	Influencing environmental
			factors
	Corrosion immunity	Diffusion and convection	Temperature, relative humidity
	Active corrosion	Electrochemistry	Temperature
	Corrosion until severe cracking	Electrochemistry and mechanics Temperature
	22.3.2. Stochastic Assessment of Time-Dependent Damage Risks
	The cumulative distribution function for the time of first damage (corrosion initiation, severe
	concrete cracking, failure) T dam in the period [0, t] for original concrete is:

  ). France used a discount rate of 8% to evaluate public investments from 1985 to 2005. However, following the 2005 Lebègue Report[START_REF] Lebègue | Révision du taux d'actualisation des investissements publics[END_REF], the 'Commissariat Général au Plan' has recommended a 4% discount rate for short term investments and a lower discount rate of 2% for cash flows occurring after more than 30 years[START_REF] Gollier | Pricing the Planet's Future: The Economics of Discounting in an Uncertain World[END_REF]). These discount rates were revised in 2013 by the 'Commissariat Général à la Strategie et à la Prospective' recommending 2.5% and 1.5% discount rates for short term (lifetime lower than 70 years) and long term investments, respectively[START_REF] Quinet | L'évaluation socioéconomique des investissements publics. Commissariat Général à la Strategie et à la Prospective[END_REF].[START_REF] Quinet | L'évaluation socioéconomique des investissements publics. Commissariat Général à la Strategie et à la Prospective[END_REF] also recommends carrying out a sensitivity analysis with a 4.5% discount rate to compare new and old approaches. Countries and institutions worldwide use other discount rates. The European Commission recommends a 5% discount rate

  [START_REF] Peng | Damage risks and cost-benefit analysis of climate adaptation strategies of RC structures in Australia[END_REF]. 22.4.2. Repair and Adaptation Costs 22.4.2.1. Cost of Damage (C damage )

Table 22 .

 22

	2 -Repair strategies and corresponding costs of damage of the Agri-foodstuffs
	terminal at the Port of Nantes Saint-Nazaire
	Description of repair strategy	Type of cost in €/m2
	Reports and	Concrete	Repair Operating	Other	Total
	site	removal	loss	costs
	installation

Table 22 .

 22

	3 -Adaptation costs for slabs
	Structural element	D (mm)	Δc adapt 5 mm (€/m 2 ) Δc adapt 10 mm (€/m 2 )
	Slabs	100	4.23	8.45
	Slabs	300	2.44	4.88
	22.5. Illustrative Example: Adaptation of Existing RC Structures
	22.5.1. Problem Description
	22.5.1.1. Concrete Cover and Exposure
	This example illustrates the probabilistic assessment of the cost-effectiveness of adaptation
	strategies for existing RC structures placed in two coastal cities in France (Saint-Nazaire and
	Marseille) under a splash and tidal exposure. Since this application deals with existing
	structures, it is important to determine which were the standards at the construction time.
	There was an evolution in time of the minimum concrete cover recommended by French
	design standards for maritime exposure. These standards were published as circulars of the

official diary in France. The first circular on 20 October 1906, recommended a minimum concrete cover varying between 15 and 20 mm for the main reinforcement without distinction of the kind of exposure. In 1934, a new circular recommended 35 mm cover for structures close to the sea and 20 mm for structures in land. Concrete cover was increased to 40 mm and 50 mm in 1964 and 1992, respectively for structures close to the sea. Finally, the Eurocode 2

Table 22 .

 22

	4 -Probabilistic models of the random variables
	Variable	Units	Distribution Mean	COV Reference
	Reference chloride diffusion	m 2 /s	log-normal 3×10 -11 0.20	(Duracrete, 2000a; Saetta et
	coefficient, D c,ref					al., 1993; Val & Trapper,
						2008)
	Environmental chloride	kg/m 3	log-normal 7.35	0.20	(Duracrete 2000; Vu and
	concentration, C env Concentration threshold for	wt% cem. normal a	0.5	0.20	Stewart 2000) (Bastidas-Arteaga and

Table 22 .

 22

		5 -Mean BCR for slabs (D=300mm) built in several years in Saint-Nazaire and
		t adapt =2020
	Construction	RCP 4.5		RCP 8.5
	year	Δ cadapt =5mm	Δ cadapt =10mm	Δ cadapt =5mm	Δ cadapt =10mm
	1970	0.81	0.71	0.82	0.71
	1990	3.76	3.41	3.85	3.58
	2010	4.57	4.34	4.74	4.5

Table 22 .

 22 6 -Mean BCR and Pr(BCR>1) (within brackets) for various adaptation years (slabs (D=300mm) built at several years in Saint-Nazaire, RCP 4.5 scenario and Δc adapt =5 mm)

	Construction		Adaptation year
	year	2020	2040	2060	2080
	1970	0.83 (6.1%)	0.05 (0.2%)	0 (0.0%)	0 (0.0%)
	1990	3.64 (43.5%)	1.38 (10.2%)	0.2 (0.7%)	0 (0.0%)
	2010	4.63 (59.0%)	3.85 (44.7%)	1.69 (13.1%)	0.29 (0.9%)
	22.5.2.3. Effect of Discount Rate
	Table

Table 22 .

 22 7 -Mean BCR and Pr(BCR>1) (within brackets) for various discount rates and construction years (slabs (D=300mm) built at several years in Saint-Nazaire, RCP 4.5 scenario and Δc adapt =5 mm)

	Construction Year	r =2%	r =4%	r =8%
	1970	1.32 (5.9%)	0.81 (5.9%)	0.29 (5.4%)
	1990	5.8 (44.6%)	3.76 (43.4%)	1.58 (39.8%)
	2010	7.15 (64.2%)	4.61 (62.3%)	2.15 (53.6%)