Abstract:
A method and arrangement for improving tomographic determinations by means of radiation, specially suitable for steel bars in concrete. The method comprising irradiating said body with penetrating radiation, recording said radiation transmitted through said body in recording means; providing a reference system with a plurality of independently identified and individualized reference elements made of a high density radiation-absorbent material, such reference elements being arranged in an orderly manner; identifying the above mentioned measurements; determining irradiation times; and determining the position and size of objects within a body based on the information recorded on the recording means used for measurement.

Description:
FIELD OF APPLICATION 
     The present invention relates to methods for analyzing the three-dimensional structure of a body opaque to visible light, using penetrating radiation, such as X-rays or gamma-rays, whose intensity is attenuated as a function of the product of the thickness times the density of the traversed matter. Particularly, the invention is intended for the analysis of structures, and especially for the determination of the position and diameter of steel bars in reinforced concrete structures, such as beams, columns, and slabs. 
     Experimental work carried out to this date has employed gamma-radiation sources. However, it is possible to apply the same method to cases in which X-ray tubes, linear accelerators, or other radiation sources are used, for the same as well as for other applications. 
     The present invention comprises a method for performing an automatic and accurate determination of parameters that are essential for a three-dimensional analysis, such as the position of the source and that of the transmitted-radiation recording means; an optimal arrangement for improving image quality when the recording means comprises a film which is sensitive to variations in radiation intensity; an arrangement for the tomographic determination of reinforcement bars in very thick pieces of concrete using low energy radioactive sources; and a procedure for obtaining a tomographic result from such measurements. 
     DESCRIPTION OF THE PRIOR ART 
     Tomographic techniques are extensively used in medicine. The present method is particularly suited, without limitation, for reinforced concrete, and may be extended to other applications. In this field, X-rays generated by conventional tubes, gamma-rays emitted by radioactive sources or high energy electromagnetic radiation generated by linear accelerators have been used for obtaining two-dimensional images of radiation transmitted through structural pieces of reinforced concrete, either using radiation-sensitive films (conventional or digital radiographic plates) (ref. 3), radiation-to-visible light fluorescent conversion screens (JP 61-254837, U.S. Pat. No. 6,333,962) or scintillation- or solid state-type radiation detectors (U.S. Pat. No. 5,933,473). The two-dimensional information obtained through the application of these techniques is useful for detecting metallic elements, the presence of corrosion therein, voids and cracks in a piece of concrete. The three-dimensional problem, that is, the determination of the position and diameter of steel bars in reinforced concrete structures, has also been dealt with, although without much progress so far (ref. 1, 2, 3, U.S. Pat. No. 5,828,723) except for the works described in ref. 4. Some of the problems associated with this type of measurements have not yet been solved. These problems are: the difficulties for acquiring data with proper accuracy regarding the position of the source and of the recording means required for the tomographic mapping of steel bars; the low contrast of images, and hence the difficulties involved in interpreting them, due to the intensity of scattered radiation, especially in the case of concrete pieces of considerable size; and the limitation in range of concrete thicknesses that may be studied using low energy portable sources. 
     The procedure for the tomographic determination of reinforcement bars, that is, for determining the position and diameter of steel bars in a reinforced concrete structure, consists, in a first step, in arranging the radiation source in a first position on one side of the structural element under study and the recording means on the other side approximately opposite to the location of the source during irradiation, such that the section under examination is comprised within the volume of a pyramid of height equal to the distance between the source and the recording means and base equal to the area covered by the recording means. In a second step, the element under examination is exposed to radiation emitted by the source during a period of time long enough for the recording means to receive the dose of transmitted radiation required to obtain an optimal contrast between the radiation transmitted through the steel bars and the radiation that has not traversed the steel bars. The amount of irradiation time is determined as a function of the product of the thickness times the density of the element under examination. In subsequent steps, the procedure is repeated, with the source, and sometimes the recording means, located at different positions such that the volume under examination is similar to that in the first step. Finally, the information recorded by the recording means is analyzed for all the measurements referred to a given structural piece, using a mathematical algorithm to obtain the position and diameter of steel bars inside the volume covered by the measurements. 
     In order to apply said mathematical algorithm it is necessary to know the positions of both the source and the recording means for the different measurements, referred to a coordinate system fixed to the structural piece under study. 
     The present invention improves the prior art regarding the above-mentioned requirement by providing:
         a) the introduction of reference elements generating fiducial marks for greater accuracy, reliability and automation potential.   b) an arrangement of the recording means consisting in placing such means at a certain distance from the structural piece under study, thereby improving the quality of the information obtained. The optimum value for this distance is calculated using a Montecarlo-type simulation program specially developed for this purpose (ref. 6)   c) an arrangement of—elements to filter the scattered radiation, optimized by the above-mentioned Montecarlo-type program (ref. 6).   d) a methodology for analyzing the information recorded in the recording means for solving the tomographic problem, that is, the three-dimensional mapping of steel bars in a reinforced concrete structure that substantially improves the previous art.       

     The use of fiducial marks produced by radiation-absorbing reference objects that are recorded on a radiographic film or digital recording means for image aligning or calibrating purposes is well known in medical radiography (ref. 7) as well as in industrial X-ray determinations (ref. 5). In the first case, reference elements are incorporated into the patient&#39;s body in order to solve the problem derived from the change of position of the patient in different measurements. In the case of industrial X-ray determinations, for example, a sequence of radiation-absorbing bars of known dimensions is placed on the upper portion of a reinforced concrete slab in order to determine the thickness of the slab. 
     In the present invention, the use of fiducial marks serves the double purpose of improving both the accuracy and reliability of tomographic determinations of the location and size of steel bars in a reinforced concrete structure. In contrast to other industrial X-ray applications, the objects whose images are to be recorded on the radiography are generally located at a considerable distance from the recording means. Therefore, it is possible to obtain position data through stereoscopic reconstruction, by taking two or more exposures with the radiation source placed in different locations. For the reconstruction to be accurate it is important to know precisely the position of both film and source with respect to the structure under study. 
     SUMMARY OF THE INVENTION 
     In the method of the present invention, fiducial marks are recorded on the recording means, for example a radiographic plate, by placing radiation-absorbent reference objects on frames located on the radiation source side and/or the recording means side, in accurately known positions. The purpose of this is to minimize the need of manual measurements and recordings during fieldwork that are prone to error. In this way the information required for the tomographic determination is automatically recorded on the recording means for use in the subsequent computational analysis, thus preventing inaccuracies and errors characteristic of manual determinations, as well as others of the previous art, such as the position of the radiographic film inside the cassette where it is placed in order to avoid its exposure to light. 
     In addition to reference elements, in the method of the present invention the frame on the recording-means side includes an arrangement for positioning the recording means at an optimum distance from the structural element being examined. The frame also allows the addition of adequate filters for attenuating the effect of scattered radiation that, in the case of reinforced concrete, severely limits the quality of images. Both optimum distance and optimum filters are determined by means of a computer program specially developed for this purpose. Finally, the method comprises a specially developed procedure for carrying out the tomographic analysis, taking into account the fiducial marks and the conditions of frames and supports. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a perspective view of the “source assembly” used in an arrangement for determining the position and size of steel reinforcement bars in reinforced concrete slabs and walls; 
         FIG. 2  is a cross-sectional diagram of an arrangement used for the study of reinforcement bars in slabs and walls, using an “external source”; 
         FIG. 3  is a cross-sectional diagram of an arrangement used for the study of reinforcement bars in reinforced concrete structures, using an “internal source”; 
         FIGS. 4(   a ),  4 ( b ) and  4 ( c ) illustrate two arrangements for the study of reinforcement bars in the lower central section of a beam and an arrangement for the study of reinforcement bars in a column; 
         FIGS. 5(   a ),  5 ( b ) and  5 ( c ) are perspective views of the rack used for the study of steel reinforcement bars in beams and columns, in open and closed positions; 
         FIG. 6  shows the position of the rack on a beam; 
         FIG. 7  is a diagram showing source, rebars, reference elements and plate arrangement assumed for calculating the errors involved in using the method of the present invention to determine the position of steel reinforcement bars in a structure. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     The method of the present invention includes data collection equipment to be used in the field (the “field work”) and a procedure for the analysis of such data to obtain the tomographic result. The equipment can be arranged in different ways according to the characteristics of the structural element under study, according to the description given below. 
     First the general aspects of the method are described and then an explanation is given about the manner in which the method is applied to different specific cases (inspection of slabs and walls, of beams and columns and other cases.) 
     The field equipment that is an object of the present invention comprises, in the first place, an assembly of devices on the radiation-source side, designated “source assembly”. In the case of measurements using an “external source”, that is, with the radiation source located externally to the piece under study, the “source assembly” comprises a support for the radiation source (X-ray tube, linear accelerator, radioactive source or other) with its corresponding collimator and a first rack, hereinafter called “rack I”, detachably attached to said support, containing the reference elements. The “source assembly” can be provided with shielding elements. The rack comprises a frame made of a lightweight but strong material (such as, for example, Al), of rectangular shape or otherwise, two rigid plates having the same shape of the frame attached to said frame so as to cover both of its sides, such plates being made of poor radiation-absorbent material (such as for example some kind of plastic). The “source assembly” is fixed to one side of the body to be examined. If measurements are to be carried out using an “internal source”, the method is applicable only if a radioactive tablet is used as a radiation source, and in such case the “source assembly”becomes simply a holder for the source container/projector and an “extension” that is introduced within the piece under study through a hole drilled therein, so that the radioactive tablet can be placed in different positions within the body under examination. 
     Secondly, the equipment comprises a second rack, hereinafter called “rack II”, located on one of the sides of the piece under study opposite to the source. Rack II comprises one or more planes. Each plane has a frame with covering plates and reference elements such as in rack I. Rack II may also comprise devices to identify each of the measurements, scattered radiation filters, holders for the recording means and a holder for a “gammameter” (which is described below). 
     The reference elements are made of a high density material (such as Pb or W) and are arranged so that the radiation emitted by the source traverses them before reaching the recording means. The reference elements can be rods, beads or similar objects, small beads being most preferred for the reasons set forth below. These reference elements are located on the plates that are fixed to the rack&#39;s frame. 
     The shielding mentioned above in the description of the “source assembly” can be a plate made of lead, having a thickness of, for example 4 mm, surrounding the frame of rack I, said arrangement having the purpose of absorbing radiation scattered in angles above and near 90 degrees through the first concrete layers traversed by radiation. 
     The device to identify each particular measurement may consist in a support fixture for letters made of an absorbent material (Pb, W, etc.) to identify the specific field work being performed, which support fixture stays unchanged during the field work, while a second support fixture containing a numbering system to identify each measurement increases by one unit upon each successive measurement. This device can be integrated into rack II in order to facilitate the procedure of identifying of each measurement, thus improving the prior art. 
     The scattered radiation filter can be housed in between both plates in rack II, so as to achieve an optimum ratio between the intensity of direct radiation and that of scattered radiation reaching the recording means, since said ratio affects the contrast of the image of the object being examined (for example steel bars) on the recording means. The composition and thickness of the elements of said scattered radiation filter are selected through the use of a simulation program, specially designed for this purpose (ref. 6). 
     The “gammameter” is a plate of lightweight material supporting several devices sensitive to radiation, whose purpose is to obtain remote readings of the radiation doses on different spots of the covered area as recorded by the recording means during measurements, thus permitting to assess the optimal irradiation time for the measurement. 
     Once the “source assembly” and rack II have been deployed, the radiation source is turned on and irradiation commences. In the case of a gamma radiation source, the radioactive tablet is moved to the desired position. In the “external source” mode, the tablet is positioned in a collimator made of absorbent material, which is part of the “source assembly”, so that radiation may particularly “illuminate” the volume under study, within a solid angle approximately equal or slightly larger than the solid angle defined by the source and the area of the recording means. In the case of a gamma radiation source, in the “internal source” mode, the tablet is moved by means of an extension of the irradiation mechanism, which is introduced into a hole drilled into the piece under examination, thus allowing a measurement in the volume corresponding to the solid angle defined by the source and the area of the recording means. 
     Irradiation takes place during a specific period of time according to the doses indicated by the “gammameter”, and to the table setting forth the relationship between irradiation time versus the doses for different combinations of sources and recording means. 
     Once the measurement has ended, if the recording means is of the cumulative type (radiographic plate, film or digital), which requires “off-line” reading, it is removed. A new recording means is conveniently set in place in rack II; the measurement numeral is increased by one unit; and a new measurement is carried out with the source in a new position, either changing the position of the “source assembly” or not. 
     After a given structural “sector” has been inspected, and the necessary number of measurements has been made, a tomographic determination is carried out. The analysis procedure, which is part of the method of the present invention, includes a computer program comprising the following steps:
         a) Entering size and orientation data of the section of the piece being measured (beam, column, etc.), object of the tomographic analysis. A main coordinate system, fixed to the structure under study, is defined to which the results of the analysis will be referred.   b) Indicating which side of the piece under study the recording means has been placed on in each of the measurements.   c) Identifying the fiducial marks on the recording means. The program performs a least-square adjustment to determine with high accuracy the position of said fiducial marks in the coordinate system of the recording means. The accuracy achieved by means of this procedure is improved when the fiducial marks are circles or ellipses, corresponding to spherical reference elements (see below a note about the accuracy that can be achieved with this method). The position of the fiducial marks determines: i) the position of the source in each measurement with respect to the recording means and to the main coordinate system and, ii) the position of the recording means with respect to the main coordinate system and with respect to the position of the recording means in the rest of the measurements made on the same section of the structure.   d) Determining data pairs corresponding to the contour of the steel bar projections in different sections, called “contour pair”. The number of required pairs depends on the orientation and changes in direction of the bar projections.   e) Calculating the “shadow cones” defined by the position of the source and each contour pair, obtained for all measurements performed for the same section of the piece under study.   f) Determining shadow cone intersections corresponding to actual bars, taking into account the “contour pairs” for the different cross-sectional measurements performed all along the piece under examination.   g) Producing a technical report with the tomography result, that is, the number, position and diameter of each bar in the examined sections.       

     This analysis procedure is characterized by and distinguished from the prior art in that it combines in a simple way the data resulting from different coordinate systems (the coordinate systems for each of the recording means) in a single main coordinate system fixed to the piece under study. 
     The use of the method of the present invention will now be described for particular cases, as non-limiting illustrative examples. 
     In a particular embodiment of the invention, the method is used for the tomographic determination of reinforcement bars in slabs or walls using a  192 Ir source. The equipment comprises, in this case, a “source assembly” fixed to one of the faces of a slab or wall whose reinforcement bars are to be measured, and a “uniplanar” rack II fixed to the opposite side of the slab or wall. In general, although not always, measurements of slabs and walls are carried out in the “external source” mode. 
     An example of this embodiment for the analysis of a slab using an “external source”is illustrated in  FIGS. 1 and 2 . The “source assembly” is shown in  FIG. 1  and comprises a rack I  101  consisting of a frame  102  and two plates  103 , one upper and one lower plate, onto which reference elements  104  are attached in accurately preset positions; a source support composed of two columns  105 , a crossmember  106 , a collimator support  107  and a collimator  108 . The size of this assembly corresponds to the experimental needs and conditions. The case shown in this example is prepared for use with radiographic plates, for example, of 35×43 cm, and, accordingly, the frame  102  of rack I  101  is, for example, of 39×49 cm. The two columns  105  are detachably inserted into the frame  102  of rack I  101 . The columns  105  support a crossmember  106 , which may be located at two preset heights. A support  107  for the collimator  108 , into which the radioactive source is introduced, slides horizontally on the crossmember  106 . The support  107  of the collimator  108  may be fixed to the crossmember  106  in different preset horizontal positions  109  by means of an anchoring bolt. The crossmember  106  crosses rack I  101  diagonally, passing through its center. The shielding  112  that may be used to shield tangential radiation and increase radiological protection is shown partially. The frame  102  and the crossmember  106  are preferably made of Al and the columns  105  are preferably made of steel. 
     The assembly may be attached to the slab or wall under study using different procedures. One of them comprises, for example, the use of two suction cups  110 , attached on opposite sides of the frame  102  of rack I  101 , fixing the whole assembly onto smooth floors or pavings; another procedure comprises the use of screws affixed in holes  111  drilled into frame  102  of rack I  101  for this purpose. 
       FIG. 2  shows a cross-sectional view of an arrangement adapted to be used to study a structure  203  consisting of a reinforced concrete slab, subfloor and floor, with an “external source”, wherein rack I  101  is placed on structure  203  and rack II  205  is placed under said structure  203 . For the analysis according to this arrangement, the source is located at both positions  201  and  202 ; said positions being selected by a calculation that takes into account thickness  203  and density, in this case, of the slab, subfloor and floor assembly. The horizontal and vertical positions of the source in both measurements of the same portion of the slab (or wall) must be determined so as to optimize the following conditions:
         a) the largest possible effective inspection volume  204  should be covered in both measurements.   b) the largest possible effective recording area in the radiographic plate (placed inside housing  209 ) should be obtained, taking into account the sensitivity range of the plate and the difference of radiation paths incident between the borders and the center of the plate.   c) the duration of each measurement should be kept to a minimum.       

     In order to locate the source according to these conditions, rack I has two possible preset heights and four possible horizontal positions  109  at each side of the center of crossmember  106  separated by a preset distance of, in this example, 1 cm in the direction of crossmember  106 . 
     Also shown in  FIG. 2  is the assembly of reference elements (in this case, small beads)  104 , in rack I  101 , and  207 , in rack II  205 , inserted into plates  103  and  206 , respectively. In the case of rack I  101 , a pair of beads  104  may be placed on the vertical axis of each of the predetermined horizontal positions  109  in crossmember  106  for locating the source, in order to readily visualize the position of the source at each measurement by reading the respective fiducial marks on the recording means. Fiducial marks corresponding to reference elements allow for an accurate determination of the positions of the source and the recording means for each measurement with respect to a fixed coordinate system, which is common for all measurements carried out in the same section. The reference elements that are closest to the source and to the recording means are those that allow for said determination with the least error as regards to the positions of the source and the recording means, respectively. 
       FIG. 2  also shows an arrangement of the scattered radiation filter  208 , of the recording means  209  (for example, a cassette with a radiographic plate and usual amplifying screens therein) and the gammameter  210  in contact with the rear side of the recording means. The upper side of the recording means is in contact with the filter  208 . The space between lower and upper plates  206  is such that, once the recording means is arranged, it will remain at an optimum distance from the structure to be measured, which distance is determined by a simulation carried out with the program of Ref. 6. 
     The procedure for studying the reinforcement bars of slabs or walls consists in the first place, in fixing the “source assembly” on one of the sides of the structure under study by means of one of the above-mentioned alternative procedures. Then the collimator is placed in a first position  201  by moving it horizontally along crossmember  106  to one of the predetermined positions  109  and moving said crossmember vertically to one of the predetermined positions in columns  105 , in such a way that position  201  corresponds to that indicated to meet the above mentioned conditions. Then the second rack II  205  is placed on the opposite side of the structure, using an unexposed recording means  209 . In the following step, irradiation for a predetermined period of time is carried out (for example, according to the indication of the gammameter). At the end of this period of time the source is returned to its shielded container and the recording means  209 , in case it is of the cumulative type (for example, radiographic plate, film or digital), thus requiring “off line” processing, is withdrawn for further processing. 
     This procedure is then repeated, placing the source in a second position  202 , and a new recording means  209  is set in place, in a position similar to the former one. It is not necessary for the recording means positions to be exactly the same in both measurements (as it was in the previous art) because of the reference system of the present invention. 
     Finally, once the programmed measurements for the structure section under study have finished (in this particular example, two radiographic plates), the tomographic determination is carried out, following the procedure outlined above, in order to determine the number of steel bars, their diameter, condition and location inside the structure, by combining the information resulting from all the measurements. 
       FIG. 3  shows a cross-sectional diagram similar to that of  FIG. 2  but with an arrangement that uses the “internal source” mode (see herein below). In this case rack I does not exist, and the method of the present invention applies using reference elements  207  (beads in this case) in plates  206  of rack II. The filter  208  and the “gammameter”  210  shown in  FIG. 2  are not shown here for the sake of clarity, while the recording means is in this case a radiographic plate indicated with a black line with white intervals  301  simulating the fiducial marks or “shadows” produced by the absorption of radiation emitted by the source in the reference elements  207 . The operator only measures and records the position of the rack with respect to some reference point in the structure. The positions of the source and the recording means can then be calculated by knowing the coordinates of the reference beads  207  and measuring the position of their images  301  on the radiographic plate. 
     As in the case of the “external source” mode, without moving the rack a second recording means is then placed on rack II and a new measurement is carried out with the source in a second position. The data from these two measurements, or more if it were the case, are then used to determine the position and size of the reinforcement bars in the slab. This procedure has the advantage of saving time and effort during field work and also allows to avoid errors and hence to achieve a greater accuracy, especially in those cases where simultaneous access to both sides of the structure is not possible, as is frequently the case for slabs or walls in which the source and the recording means are in different rooms. In these cases conventional measurement of the position of the source with respect to the recording means is not trivial. 
     In order to determine the positions of the source and the recording means (which is necessary for the tomographic determination of reinforcement bars) it is in principle enough to obtain images of two reference elements for each. In practice, however, it is convenient to arrange more than two reference elements and to obtain their corresponding fiducial marks in order to achieve greater accuracy, by averaging the results obtained for individual pairs and also because of the possibility that some of these marks may be superimposed to other marks in the plate, thereby making their interpretation difficult. The possibility of averaging independent values has the advantage of reducing reading errors as well as those errors derived from manufacturing tolerances regarding the predetermined position of the reference elements. 
     In another embodiment of the invention, the method is applied to the tomographic determination of reinforcement bars in beams or columns. In analogous manner as in the previous case, the equipment in this case comprises a “source assembly” fixed on one of the sides of the beam or column whose reinforcement bars should be measured, and a “uni-, bi- or tri-planar” rack II, fixed to another side or sides of the beam or column. 
     An example of this embodiment, applied to the study of a central section of a beam, is illustrated in  FIGS. 4 ,  5  and  6 . 
       FIG. 4  shows schematically two possible arrangements for the study of a beam section and a column section, using a  192 Ir radioactive tablet as a source. In (a) an arrangement of the “external source” mode is shown. The “source assembly” comprises a support  401  for the container/projector of source  402  with its source, fixed to a rack I  403  with a frame and two plates comprising reference elements (similar to rack I  101  in  FIG. 1  and according to the description of the elements shown in  FIG. 1 ). An optional shielding (not shown in  FIG. 4 ). similar to the one shown as reference  112  in  FIG. 1  can be added with the purpose of reducing the intensity of tangential radiation The collimator  404  allows to direct radiation in the desired direction (towards the right hand side in the figure) after the source is displaced from the inside of container  402  to the collimator  404  in order to start the measurement. The rack II  405  consists of two planes, each comprising a frame with plates, reference elements, room for a filter, a support for recording means and, optionally, a gammameter (see  FIGS. 5   a  and  5   b ) in analogous manner to the description for rack II  205  in  FIG. 2 . Said rack is attached to the beam such that both planes remain in contact with two of the beam sides (opposite and adjacent to the side on which the source support is attached) and the intersection of both planes coincides with the edge of the beam opposed to the source side. In the case of “external source”, in which the source is located as shown in  FIG. 4(   a ), one or more measurements (with the source in different positions in the vertical and horizontal directions) are carried out with recording means  406 , located in the recording means supports in the vertical and horizontal planes of rack II  405 . In (b) a similar arrangement but in the “internal source” mode is shown, where the frame  403  and its associated elements are removed from the “source assembly”, and the collimator  404  is replaced by an extension  407  consisting of a tube of suitable material of appropriate diameter (for example approximately 12 mm and 15 mm of internal and external diameter respectively) to allow the displacement of the source therein, which tube is introduced through a hole (for example of about 17 mm in diameter) drilled into one of the lateral sides of the beam at a certain distance from the lower side of the beam (for example about 28 cm if the source being used is  192 Ir, and for example 50 cm if the source is  60 Co). For example, the length of such tube and the depth of the orifice should be approximately equal to the thickness of the beam less approximately 10 cm. When using an “internal source”, N measurements are preferably carried out, (where N is the integer greater and closest to A/10−1, being A the width of the beam section measured in cm). In a first measurement, the source is placed, for example, in position  408 , at about 10 cm from the side of the beam through which the source is inserted and the recording means is placed into the supports of rack  405 , preferably in position  410 , such that its center lies opposite to position  408  of the source. In successive measurements the source and the center of the recording means are placed at, for example, 20, 30 . . . (A−10) cm from the side of the beam through which the source is inserted (positions  409  for the source and  411  for the recording means and successive positions). In (c) an arrangement similar to (b) in the “internal source” mode is shown, for the study of reinforcement bars in a column thicker than established by the applicable standards, for example “Testing concrete”, British Standard 1881, part 205. Recommendations for radiography of concrete, 1986 (ref. 3). In this arrangement a  192 Ir source is used, where an additional plane  412  is added to rack II (as shown in greater detail in  FIG. 5(   c )) and such plane is adjusted so that it contacts the side adjacent to the side through which the source is introduced ( FIG. 4(   c )). A hole is preferably drilled on the center of the side through which the source will be introduced if the distance between hole and adjacent sides is, for example, between 20 and 30 cm, or two holes are drilled otherwise, such that each hole is at a distance of, for example, between 20 and 30 cm from either of the sides adjacent to the source side (these values are increased by a factor of about 2.5 if a  60 Co source is employed). In a first measurement the source is introduced, for example, 10 cm within the column. In the case of a single central hole, two recording means located opposite to each other on the planes adjacent to the side through which the source is introduced, and whose centers are opposed to the source, are simultaneously irradiated. Otherwise, a single recording means located at a distance of, for example, between 20 and 30 cm from the hole, on one of the sides adjacent to the side where the source is introduced, is irradiated. As many measurements as necessary are taken displacing the source, for example in 10 cm intervals, up to a depth equal to the dimension of the column in the direction parallel to the hole, less 10 cm. In each measurement the recording means (either one or more, in case of simultaneous measurements) is placed opposite to the source. The measurement(s) in which the source lies, for example, between 20 and 30 cm from the side opposite to the side through which the source is introduced, is (are) carried out placing a recording means on the side opposite to the side where the source was introduced. In the case of two holes, the procedure is repeated with the recording means located on the side opposite to that used in the first series of measurements. 
       FIGS. 5(   a ),  5 ( b ) and  5 ( c ) illustrate arrangements for rack II used in the study of beams and columns as described hereinbefore.  FIGS. 5(   a ) and  5 ( b ) illustrate two perpendicular planes, the main plane  501 , and the secondary plane  502 , fixed to each other, which can be collapsed so to adopt two alternate positions: i) an open position for use during measurement ( FIG. 5(   a )) and ii) a closed position for use during transportation ( FIG. 5(   b )). Additionally, another secondary plane  503  can be added which is perpendicular to the main plane  501 , and which can slide on the main plane frame such that it contacts the free lateral side of the beam or column. Each of the planes  501 ,  502  and  503  are similar to the “uniplanar” rack II  205  of  FIG. 2  and comprises, as well as the latter, a reference system, a housing for a scattered radiation filter, a support for the recording means and a support for a gammameter, such as was described hereinbefore (not all these elements are shown in  FIGS. 5(   a ),  5 ( b ) and  5 ( c ) for the sake of clarity). The position of the centers of the recording means located on planes  501  and  502  can vary by a distance approximately equal to one length of the corresponding plane, whereas in the case of plane  503  this variation is restricted by the fact that one of the edges of the recording means located in the support in this plane is limited by the upper surface of the main plane. The length of the main plane of the prototype shown in  FIG. 6  is determined by the application, and in the example it is, for example, approximately 50 cm, and can be extended using supplements. Also shown in  FIG. 5(   a ) are metal straps  505  that provide a fixing means for rack II to the beam or column, and a screw  504  that is adjusted against the recording means located on plane  502 , when the latter is in a vertical position (as in the case of  FIGS. 4(   a ) and  4 ( b )) to prevent the recording means from sliding downwards. 
       FIG. 6  illustrates the position of a rack II with perpendicular planes in operation position on a beam with recording means  209  on the vertical and horizontal planes (gammameter not shown). The rack II is fixed to the beam or column by any of the alternate procedures described hereinbefore, such that the planes contact the sides of the beam or column and the intersection of planes  501  and  502  coincides with one of the edges of the beam or column. 
     The method of the present invention can also be applied to other studies of structures and other bodies opaque to visible light. Within the field of structures, mention should be made of flexible racks, with the attributes described hereinbefore, for the study of, for example, columns with non-rectangular cross section. The method of the invention can be applied to the study of beam laterals in order to determine the point where certain steel bars change direction in order to resist shear stress near beam supports, or also to the study of the upper sections of beams above beam supports, or of foundations and, in case the thickness of the piece so requires, the “internal source” mode can be applied. The method of the invention has also been used for the study of the inner metal structure of monuments and ornaments. 
       FIG. 7  is a simplified scheme of a particular arrangement using an “internal source”, which has been prepared to illustrate quantitatively the accuracy that can be achieved in the tomographic determination of reinforcement bars with the method of the present invention. For the purpose of this description, we only need to take into consideration two source positions  701  and  702 , two steel bars  703  and  704 , reference beads  207  incorporated to upper plate  206  of rack II, in positions (X,Y) indicated in  FIG. 7  (lower plate  206  in rack II, filter  208  and gammameter  210  are not shown in the scheme for the sake of clarity). As described hereinbefore, these beads are very radiation-absorbent elements of a suitable diameter (preferably 2 mm). Under radiation these beads  207  create “shadows” or fiducial marks on the recording means  209 , with the shape of ellipses that are clearer than the background. The reason the spherical shape for these reference elements is preferred is because the “shadow” of a sphere is a circle or an ellipse and these shapes can be adjusted using the least squares method and therefore their position can be determined with greater accuracy than is achievable using other shapes. 
     The accuracy with which the position and diameter of steel bars  703  and  704  can be determined, which constitute the unknown variable of the tomographic problem to be solved, depends on the manufacturing tolerances of all components comprising the reference system and also on the accuracy with which the centers of the bead images and the cross-sectional views of the shadows projected by the steel bars can be extracted from the recording means. The sources of error and its magnitudes, assumed for the purpose of the present calculation, are summarized in the first section of Table I. 
     The calculation of errors associated with the method of the present invention was carried out using realistic computer software that was specially developed for this purpose. Such software simulates the actual situation based on the nominal values for the positions of source, reference beads, steel bars and recording means, while it also takes into account the errors attributable to manufacturing tolerances and reading errors of fiducial marks on the plates. Those errors were estimated on a statistical base, assuming a normal distribution with certain standard deviations. The values and the coordinate system used in this calculation are shown in  FIG. 7 . The steel bars  703  and  704  are assumed to be parallel to the Z axis. Four beads  207  aligned in pairs with the Z axis are also assumed to be located, for example, 10 cm below and above the drawing plane. 
     The lower section of Table I summarizes the results, obtained by simulating two thousand measurements, of the achievable accuracy when using the present method in the determination of the position of unknown steel bars . For the sake of simplicity the calculation did not include the uncertainty of the position of the plate with respect to the rack. This magnitude can be determined very accurately using the fiducial marks of the beads inserted in the lower plate of the rack, which is very close to the plate. The above mentioned errors result basically from the assumption that the position of the source is apriori unknown. Therefore this calculation corresponds to a different arrangement to that described before in connection with  FIGS. 1 and 2 , in which the position of the source was considered as known beforehand. The calculation is therefore especially relevant for the “internal source”cases, for which the position of the source is not accurately known apriori. Despite the aforementioned approximations, the estimated errors in the method of the invention are considered representative for the applications of the invention. 
     
       
         
               
               
             
               
               
               
             
               
             
               
               
               
               
             
           
               
                   
                 TABLE I 
               
               
                   
                   
               
               
                   
                 Errors 
               
               
                   
                 (mm) 
               
               
                   
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 ASSUMPTIONS 
                   
               
               
                   
                 manufacturing error in position X, Z of the beads 
                 0.5 
               
               
                   
                 manufacturing error in the position Y of the beads 
                 0.5 
               
               
                   
                 misalignment in X and Z of plates 206 (FIG. 2) 
                 1 
               
               
                   
                 error in Y of position of plate at the reading place 
                 1 
               
               
                   
                 Position adjustment error of fiducial marks 
                 0.5 
               
               
                   
                 Adjustment error of the position of steel bar images 
                 0.5 
               
             
          
           
               
                 RESULTS 
               
             
          
           
               
                   
                 BAR 1 
                 error in X 
                 0.9 
               
               
                   
                   
                 error in Y 
                 3.9 
               
               
                   
                 BAR 2 
                 error in X 
                 1.2 
               
               
                   
                   
                 error in Y 
                 4.6