Abstract:
A method of manufacturing an elongate element ( 10 ) using a punching operation assumes a polynomial relationship between punch depth (d punch ) and neutral axis, with the constants being a polynomial function of plastic deformation of the beam. Using finite element analysis, a relationship between the required plastic deformation, the second moment of area of the element and the neutral axis of the element can be derived.

Description:
RELATED APPLICATION 
     This application claims priority to GB 1114438.3 filed 22 Aug. 2011, the entire contents of which is incorporated by reference. 
     BACKGROUND OF INVENTION 
     The present invention is concerned with a method of manufacturing elongate components. More particularly, the present invention is concerned with a method of manufacturing elongate aircraft components such as aircraft wing stringers by inducing plastic deformation by a differential contact three point bending operation. 
     Components such as aircraft stringers need to be shaped to the wing aerodynamic profile. In order to achieve this, stringers are plastically deformed by differential contact three point bending. By “differential contact three point bending” we mean a process by which the stringer is supported at two spaced positions on a first side. A punch is applied therebetween from a second, opposite side in order to induce a bending moment to cause local plastic deformation. Because the stringer deforms as the punch progresses, the contact area with the punch and the supports may change. This is not a classical three point bending load case (the forces are not point loads) and as such is defined as “differential contact”. 
     One problem with this method is that the stringer will spring back after the bending moment is released due to its elasticity. Known methods of forming are non-predictive. The operator will attempt to estimate the amount of punch movement required to provide a given plastic deformation. The punch is applied to the stringer by the estimated value and the plastic deformation is measured once the punch is retracted. If the level of deformation is too low, the operator will estimate a further punch distance and reapply the punch. Successive bending operations are applied at the same position until the required deformation is achieved. The punch information is then stored and applied to the next component, and so on until the required “first time” punch movement is refined to a satisfactory degree. 
     A typical 18 m stringer will be have up to 250 punching locations along its length. As such it is desirable to reduce the number of punching operations at each station. 
     Should the stringer be overdeformed (i.e. undergo too much deformation when punched), scrapping the part is not feasible as such parts are very expensive. As such, an inverse bending moment (again, estimated by the operator) is applied to the component to reverse the deformation. Such repeated and reverse application of plastic deformation to the stringer can cause problems such as work hardening and fatigue. 
     Because of the complex and changing geometry of the stringers used in the aerospace sector, prediction of the stringer plastic deformation by analytical methods is not appropriate. 
     An alternative is to use numerical simulation, such as finite element analysis (FEA) to predict the deformation of the stringer at each punch location. Not only would the analysis of a single stringer need to be repeated at each punch stage (in order to arrive at the required plastic deformation), but because of the changing cross section of the stringer along its length, these analyses would need to be carried out for each discrete punch position. This would be extremely time consuming and costly with respect to computing resource. 
     SUMMARY OF THE INVENTION 
     It is an aim of the present invention to provide an improved method of manufacture which mitigates the “trial and error” method of the prior art, whilst utilising the benefits of numerical simulation without undue burden. 
     According to the invention there is provided a method of manufacturing an elongate component comprising the steps of:
         providing a punching apparatus configured to apply a differential contact three point bending load to the component by advancing a punch at a punch location between two supports by a punch distance,   calculating a punch travel (d punch ) based on the second moment of inertia of the component at the punch location (I xx ), the location of the neutral axis distance of the component at the punch location (Y) and the required plastic deformation (d plastic ) of the component at the punch location,   bending the elongate component at the punch location by moving the punch by the punch travel.       

     Preferably the punch travel is calculated assuming that the required plastic deformation (d plastic ) is related to the second moment of inertia of the component at the punch location (I xx ), and the neutral axis distance at the punch location (Y) by an nth order polynomial of the form: 
                 d   plastic       I   xx       =       ∑     i   =   0     n     ⁢       B   i     ⁢     Y   i               
where B, are functions of the punch travel (d punch ).
 
     Preferably: 
               B   i     =       ∑     j   =   1     m     ⁢         BB   ij     ⁡     (       d   punch     -     d   elastic       )       j             
where BB ij  are constants.
 
     Preferably which n=m=2; i.e. the polynomials are quadratic. 
     Preferably the punch travel is calculated from the expression: 
               d   punch     =       [       (             ±       [                 I   xx     ⁡     (         BB     1   ⁢   b       ⁢     Y   2       +       BB     2   ⁢   b       ⁢   Y     +     BB     3   ⁢   b         )       2     +               4   ⁢     I   xx     ⁢       d   plastic     ⁡     (         BB     1   ⁢   a       ⁢     Y   2       +       BB     2   ⁢   a       ⁢   Y     +     BB     3   ⁢   a         )               ]         -                 [         BB     1   ⁢   b       ⁢     Y   2       +       BB     2   ⁢   b       ⁢   Y     +     BB     3   ⁢   b         ]     ⁢     I   xx             )       2   ⁢       I   xx     ⁡     (         BB     1   ⁢   a       ⁢     Y   2       +       BB     2   ⁢   a       ⁢   Y     +     BB     3   ⁢   a         )           ]     -     d   elastic             
where BB 1a , BB 1b  etc are constants.
 
     Preferably d elastic  is calculated analytically from the cross section of the stringer at the punch location. 
     Constants BB 1a , BB 1b  etc are preferably calculated statistically from a representative sample of numerical simulations, which may be finite element analyses. 
    
    
     
       SUMMARY OF THE DRAWINGS 
       A method in accordance with the present invention will now be described with reference to the accompanying figures in which: 
         FIG. 1   a  is a view of a stringer undergoing a punching operation; 
         FIG. 1   b  is a close-up view of the stringer of  FIG. 1   a , pre-punching; 
         FIG. 1   c  is a close-up view similar to that of  FIG. 1   b  during the punching operation; 
         FIG. 1   d  is a close-up view similar to  FIGS. 1   b  and  1   c  post punching; 
         FIG. 2   a  is a cross section view of an I-stringer; 
         FIG. 2   b  is a table of various stringer cross-section geometries; 
         FIG. 3   a  is a table of results of various punching simulations; 
         FIG. 3   b  is a graph of the results of  FIG. 3   a;    
         FIG. 4   a  is a graph of the quadratic function of punch movement B1; 
         FIG. 4   b  is a graph of the quadratic function of punch movement B2; and, 
         FIG. 4   c  is a graph of the quadratic function of punch movement B3. 
     
    
    
     DETAILED DESCRIPTION OF INVENTION 
     Turning to  FIG. 1   a , an elongate aircraft wing stringer  10  is shown having an upper surface  12  and a lower surface  14 . As shown in  FIG. 1   b , the example stringer is an I-beam having a top flange  24  and a bottom flange  26 . 
     The stringer  10  is supported at its lower surface  14  on two space supports  16 ,  18  supported on ground  20 . A punch  22  can be moved in a vertical direction in order to deform the stringer  10  between the two supports  16 ,  18 . Such punching apparatuses are known and will not be described in detail here. 
     In order to deform the stringer to the profile of a desired aircraft wing, the punch  22  is pushed downwards with a force F by a punch deflection d punch . d punch  is defined as the amount by which the punch  22  is advanced from a starting position in contact with the top surface  12  of the stringer  10  to the position of  FIG. 1   c.    
     As shown in  FIG. 1   c , as the punch  22  is advanced, the stringer  10  deforms both elastically and, eventually, plastically. The total deformation of the stringer at the point of contact with the punch is d punch . 
     Turning to  FIG. 1   d , the punch  22  has been retracted and the stringer  10  will spring back by an elastic punch limit, d elastic , leaving a final punched deformation of d plastic . As mentioned above, the method of obtaining the required d plastic  is to progressively deform the beam, repeating the steps of  FIGS. 1   c  and  1   d , recording both d punch  and d plastic  until the desired deformation is met. Subsequent forming operations at that point along the beam are then used to refine this method until a suitable d punch  is found for the required d plastic . 
     Stringer cross sections vary along their length in both size and shape. The prior art iterative process must therefore be carried out for each individual punch location, of which there are many. 
     Assuming that the second moment of inertia and the neutral axis of the stringer between the supports  16 ,  18  is constant, d plastic  can be calculated as follows: 
                       d   plastic       I   xx       =         B   1     ⁢     Y   2       +       B   2     ⁢   Y     +     B   3               (   1   )               
where:
     I xx =Second moment of inertia of the beam cross section,   Y=Neutral axis of beam cross section,   B1=Quadratic function of punch movement, d punch  (see below)   B2=Quadratic function of punch movement, d punch  (see below)   B3=Quadratic function of punch movement, d punch  (see below)
 
 B   1   =BB   1a ( d   punch   −d   elastic ) 2   +BB   1b ( d   punch −d plastic )  (2a)
 
 B   2   =BB   2a ( d   punch   −d   elastic ) 2   BB   2b ( d   punch   −d   elastic )  (2b)
 
 B   3   =BB   3a ( d   punch   −d   elastic ) 2   +BB   3b ( d   punch   −d   elastic )  (2c)
 
where:
 
BB 1a , BB 1b  etc are material dependent constants to be determined.
   

     This series of equations is indeterminate. Therefore a numerical solution is used. Values for the various constants (BB 1a , BB 1b  etc) are derived from a number of selective numerical finite element analysis simulations. An example of a range of representative I-stringer geometries is shown in  FIGS. 2   a  and  2   b . The I-stringer shown in  FIG. 2   a  has an upper flange having a thickness TF, a first portion to one side of the web having a length BFA and a second portion to the other side of the web having a length BFF; a web having a first thickness TWF at a first location and a second thickness TWA at a second location; and a lower flange having a thickness TA, a first portion to one side of the web having a length BAA and a second portion to the other side of the web having a length BAF. The I-stringer of  FIG. 2   a  has a height H. The lower flange has a length a 1  and a thickness a 2  as shown in  FIG. 2   a . The upper flange has a length a 3  and a thickness a 4 , as shown in  FIG. 2   a . The web has a height of a 5  and a thickness a 6  as shown in  FIG. 2   b.    
     Once a representative number of FEA simulations have been run, say 8 different values of d punch  from 4 to 11 mm for each of the cross sections listed in  FIG. 2   b , the results can be summarised as shown in  FIG. 3   a  and plotted as shown in  FIG. 3   b , with values of c plastic /I xx  vs Y. These plots, for each constant value of d punch  are thereby representative of equation (1) above. 
     From this analysis, a series of curves are retained such that the constants BB 1a , BB 1b  etc can be plotted against (d punch −d elastic ). (NB the elastic punch limit d elastic  is determined analytically). The least squares method used to calculate the values of BB 1a , BB 1b  etc. 
     This allows the relationship between d punch  and each of B 1 , B 2 , B 3  to be plotted (see  FIGS. 4   a  to  4   c ) and hence values of BB 1a , BB 1b  etc to be determined by least squares regression. 
     Once all of the constants have been determined, the value of d punch  as a function of required plastic deformation d plastic  can be predicted using the following relation: 
     
       
         
           
             
               d 
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     In addition, the present invention provides a method of manufacturing using a prediction of the punch force required by the process. 
     Variations of the above embodiment fall within the scope of the present invention.