Equivalent thickness bending analogy for integrally stiffened structures

A method of developing the surface contour of a desired tool for use in thermal forming an unformed, integrally stiffened, member of a material which exhibits stress relaxation upon exposure to an elevated temperature to produce a desired complex shaped member after exposure to the elevated temperature, said method comprising the steps of:

• (a) providing a plurality of experimental forming tools having substantially different radii of curvature;
• (b) thermal forming a set of specimens of the material, all of the specimens having the same integral stiffening configuration and being of uniform size, each individual specimen of a set being constrained to a different one of the experimental forming tools;
• (c) cooling all of the specimens to substantially the same temperature;
• (d) after step (c), releasing each of the specimens from restraint;
• (e) after step (d), measuring the radius of the surface of each specimen that was in contact with the forming tool;
• (f) for each specimen, producing a data set of the form (x, y) where x is the forming tool radius and y is the formed member radius;
• (g) providing a stress relaxation curve for the material of the member based upon the initial and final temper conditions of the formed members, the stress relaxation curve being in the form of a mathematical expression;
• (h) for each data set produced in step (f), substituting the tool radius x and formed member radius y into the mathematical expression provided in step (g) and developing a mathematical expression which can be solved for the thickness of an unstiffened, constant thickness, specimen that would achieve the formed radius y when thermal formed in a tool having the tool radius x;
• (i) for each data set produced in step (f), plotting the thickness calculated in step (h) against the formed member radius, with the horizontal axis representing formed radius and the vertical axis representing thickness;
• (j) plotting a plurality of thicknesses for the plurality of specimens;
• (k) joining all of the points so plotted to form an equivalent thickness curve;
• (l) expressing the equivalent thickness curve as a mathematical expression;
• (m) determining from the equivalent thickness curve the thickness of a constant thickness member that yields the same formed radius as the integrally stiffened member when constrained to a forming tool of the same radius;
• (n) using the constant thickness member determined in step (m) to determine the amount of strain that must be retained within the specimen after thermal forming to produce the desired complex shaped member, there being a mathematical relationship between retained strain and the radius of curvature of the desired complex shaped member;
• (o) determining from the stress relaxation curve the value of the applied strain to be applied by the tool to the unformed member during thermal forming to achieve the value of retained strain necessary to produce the desired complex shaped member, there being a mathematical relationship between applied strain and the radius of curvature of a forming tool for forming the desired complex shaped member;
• and
• (p) knowing the applied strain, mathematically calculating the radius of curvature of the tool for forming the desired complex shaped member.

A method of developing the surface contour of a desired tool for use in cold forming an unformed, integrally stiffened, member of a material which exhibits a relationship between a strain applied by a forming operation and a resulting strain after the applied strain has been released, said method comprising the steps of:

• (a) forming a set of specimens of the material, all of the specimens having the same integral stiffening configuration and being of uniform size, each individual specimen of a set being constrained to a different radius of curvature;
• (b) releasing each of the specimens from restraint;
• (c) after step (b), measuring the radius of the surface of each formed specimen;
• (d) for each specimen, producing a data set of the form (x, y) where x is the radius of curvature to which the specimen was constrained in step (a) and y is the formed specimen radius;
• (e) for the material of the specimens, providing a relationship between applied strain and retained strain, the relationship being in the form of a mathematical expression;
• (f) for each data set produced in step (d), substituting the radius of curvature x and formed specimen radius y into the mathematical expression provided in step (e) and developing a mathematical expression which can be solved for the thickness of an unstiffened, constant thickness, specimen that would achieve the formed radius y when restrained to the radius of curvature x, then released from that restraint;
• (g) for each data set produced in step (d), plotting the thickness calculated in step (f) against the formed member radius, with the horizontal axis representing formed radius and the vertical axis representing thickness;
• (h) plotting a plurality of thicknesses for the plurality of specimens;
• (i) joining all of the points so plotted to form an equivalent thickness curve;
• (j) expressing the equivalent thickness curve as a mathematical expression;
• (k) determining from the equivalent thickness curve the thickness of a constant thickness member that yields the same formed radius as the integrally stiffened member when constrained to a forming tool of the same radius;
• (l) using the constant thickness member determined in step (k) to determine the amount of strain that must be retained within the specimen after forming to produce the desired complex shaped member, there being a mathematical relationship between retained strain and the radius of curvature of the desired complex shaped member;
• (m) determining from the mathematical expression of step (e) the value of the strain to be applied to the unformed member during forming to achieve the value of retained strain necessary to produce the desired complex shaped member, there being a mathematical relationship between applied strain and the radius of curvature necessary for forming the desired complex shaped member;
• and
• (n) knowing the applied strain, mathematically calculating the radius of curvature necessary for forming the desired complex shaped member.

A method of developing the surface contour of a desired tool for use in thermal forming an unformed, integrally stiffened, member of a material which exhibits strain relaxation upon exposure to an elevated temperature to produce a desired complex shaped member after exposure to the elevated temperature, said method comprising the steps of:

• (a) thermal forming at least one complex shaped stiffened member of the material in a forming tool;
• (b) cooling to a lower temperature the complex shaped stiffened member;
• (c) after step (b), releasing the complex shaped stiffened member from restraint;
• (d) passing a plurality of imaginary spaced apart planes through the contour of the formed complex shaped stiffened member at spaced apart locations to thereby form a plurality of imaginary cross sectional elements;
• (e) dividing each of the imaginary cross sectional elements into a plurality of imaginary segments, each having a substantially uniform stiffening configuration and a substantially uniform radius of curvature;
• (f) after step (c), measuring the radius of the surface of the formed member at each of the imaginary segments;
• (g) for each imaginary segment, producing a data set of the form (x, y) where x is the forming tool radius and y is the formed segment radius;
• (h) providing a strain retention curve for the material of the complex shaped stiffened member based upon the initial and final temper conditions of the formed member, the strain retention curve being in the form of a mathematical expression;
• (i) for each data set produced in step (g), substituting the tool radius x and formed member radius y into the mathematical expression provided in step (h) and developing a mathematical expression which can be solved for the thickness of an unstiffened, constant thickness, specimen that would achieve the formed radius y when thermal formed in a tool having the tool radius x;
• (j) for each data set produced in step (g), plotting the thickness calculated in step (i) against the formed member radius, with the horizontal axis representing formed radius and the vertical axis representing thickness;
• (k) plotting a plurality of thicknesses for the plurality of complex shaped stiffened members;
• (l) joining all of the points so plotted to form an equivalent thickness curve;
• (m) expressing the equivalent thickness curve as a mathematical expression;
• (n) determining from the equivalent thickness curve the thickness of a constant thickness member that yields the same formed radius as the integrally stiffened member when constrained to a forming tool of the same radius;
• (o) using the constant thickness member determined in step (n) to determine the amount of strain that must be retained within the specimen after thermal forming to produce the desired complex shaped member, there being a mathematical relationship between retained strain and the radius of curvature of the desired complex shaped member;
• (p) determining from the strain retention curve the value of the applied strain to be applied by the tool to the unformed member during thermal forming to achieve the value of retained strain necessary to produce the desired complex shaped member, there being a mathematical relationship between applied strain and the radius of curvature of a forming tool for forming the desired complex shaped member;
• and
• (q) knowing the applied strain, mathematically calculating the radius of curvature of the tool for forming the desired complex shaped member.