Return to Answer

The beam equations \eqref{1} and \eqref{2} "admit a vector formulation" in the sense that they can be rigorously deduced from the 3D theory of (nonlinear) elasticity. I'm not aware of a precise reference dealing specifically with those two examples, but the Encyclopedia of Physics entry [1] by Antman, which also alludes briefly to the deduction of \eqref{1} in §26, example $\delta$, page 698, presents two different methods for solving the problem, namely the projection methods ([1], §11, pp. 660-663) and the asymptotic ([1], §13, pp. 664-665). Antman gives a fairly complete (even if somewhat dates) surveys of the theories of beams, rods and all continua with 1D behavior, considering also the history of the subject. Well, my two cents.

Edit after the downvote. After I saw the OP and my answer downvoted, I decided to add a few words of explanation. According to Stuart Antman ([1], §1, p. 641),

A theory of rods² or, equivalently a one-dimensional theory of solids is a characterization of the behavior of slender three-dimensional solid bodies by a set of equations having the parameter of a curve and the time as the only independent variables.

_{²We use "rod" as a generic name for "arch", "bar", "beam", "column", "ring", "shaft", etc. We employ rod both in the intuitive sense of a slender body and in several precise mathematical senses. The meaning will be clearer from the context.}

Said that, by carefully reading the question and its motivation, one sees that the techniques shown in [1] exactly suit the needs of the Asker: his search for a particular 3D equation for the models \eqref{1} and \eqref{2} is probably useless since it would be unsuitable for the study of a non-rectilinear beam, while the general procedures described in [1] where the equations describing a 1D medium are deduced from the standard, general 3D equations by applying them to particular constitutive equations, are applicable for beams described by a general spatial curve.

Reference

[1] Stuart S. Antman, "The Theory of Rods", in Flügge, S. & Truesdell, C. (Eds.) Festkörpermechanik/Mechanic of Solids, Handbuch der Physik/Encyclopedia of Physics, Vol. VI part 2, Springer-Verlag, 1972, 641-703.

The beam equations \eqref{1} and \eqref{2} "admit a vector formulation" in the sense that they can be rigorously deduced from the 3D theory of (nonlinear) elasticity. I'm not aware of a precise reference dealing specifically with those two examples, but the Encyclopedia of Physics entry [1] by Antman (which also alludes briefly to the deduction of \eqref{1} in §26, example $\delta$, page 698) presents two different classes of methods for solving the problem, namely projection methods ([1], §11, pp. 660-663) and the asymptotic method ([1], §13, pp. 664-665). Antman gives a fairly complete (even if somewhat dated) survey of the theories of beams, rods and all continua with 1D behavior, considering also the history of the subject. Well, my two cents.

Edit after the downvote. After I saw the OP and my answer downvoted, I decided to add a few words of explanation. According to Stuart Antman ([1], §1, p. 641),

A theory of rods² or, equivalently a one-dimensional theory of solids is a characterization of the behavior of slender three-dimensional solid bodies by a set of equations having the parameter of a curve and the time as the only independent variables.

_{²We use "rod" as a generic name for "arch", "bar", "beam", "column", "ring", "shaft", etc. We employ rod both in the intuitive sense of a slender body and in several precise mathematical senses. The meaning will be clearer from the context.}

Said that, by carefully reading the question and its motivation, one sees that the techniques shown in [1] exactly suit the needs of the Asker: his search for a particular 3D equation for the models \eqref{1} and \eqref{2} is probably useless since it would be unsuitable for the study of a non-rectilinear beam, while the general procedures described in [1] where the equations describing a 1D medium are deduced from the standard, general 3D equations by applying them to particular constitutive equations, are applicable for beams described by a general spatial curve.

Reference

[1] Stuart S. Antman, "The Theory of Rods", in Flügge, S. & Truesdell, C. (Eds.) Festkörpermechanik/Mechanic of Solids, Handbuch der Physik/Encyclopedia of Physics, Vol. VI part 2, Springer-Verlag, 1972, 641-703.

The beam equations \eqref{1} and \eqref{2} "admit a vector formulation" in the sense that they can be rigorously deduced from the 3D theory of (nonlinear) elasticity. I'm not aware of a precise reference dealing specifically with those two examples, but the Encyclopedia of Physics entry [1] by Antman, which also alludes briefly to the deduction of \eqref{1} in §26, example $\delta$, page 698, presents two different methods for solving the problem, namely the projection methods ([1], §11, pp. 660-663) and the asymptotic ([1], §13, pp. 664-665). Antman gives a fairly complete (even if somewhat dates) surveys of the theories of beams, rods and all continua with 1D behavior, considering also the history of the subject. Well, my two cents.

Edit after the downvote. After I saw the OP and my answer downvoted, I decided to add a few words of explanation. According to Stuart Antman ([1], §1, p. 641),

A theory of rods² or, equivalently a one-dimensional theory of solids is a characterization of the behavior of slender three-dimensional solid bodies by a set of equations having the parameter of a curve and the time as the only independent variables.

_{²We use "rod" as a generic name for "arch", "bar", "beam", "column", "ring", "shaft", etc. We employ rod both in the intuitive sense of a slender body and in several precise mathematical senses. The meaning will be clearer from the context.}

Said that, by carefully reading the question and its motivation, one sees that the techniques shown in [1] exactly suit the needs of the Asker: his search for a particular 3D equation for the models \eqref{1} and \eqref{2} is probably useless since it would be unsuitable for the study of a non-rectilinear beam, while the general procedures described in [1] can is applicable for beams described by a general spatial curve.

Reference

[1] Stuart S. Antman, "The Theory of Rods", in Flügge, S. & Truesdell, C. (Eds.) Festkörpermechanik/Mechanic of Solids, Handbuch der Physik/Encyclopedia of Physics, Vol. VI part 2, Springer-Verlag, 1972, 641-703.

The beam equations \eqref{1} and \eqref{2} "admit a vector formulation" in the sense that they can be rigorously deduced from the 3D theory of (nonlinear) elasticity. I'm not aware of a precise reference dealing specifically with those two examples, but the Encyclopedia of Physics entry [1] by Antman, which also alludes briefly to the deduction of \eqref{1} in §26, example $\delta$, page 698, presents two different methods for solving the problem, namely the projection methods ([1], §11, pp. 660-663) and the asymptotic ([1], §13, pp. 664-665). Antman gives a fairly complete (even if somewhat dates) surveys of the theories of beams, rods and all continua with 1D behavior, considering also the history of the subject. Well, my two cents.

Edit after the downvote. After I saw the OP and my answer downvoted, I decided to add a few words of explanation. According to Stuart Antman ([1], §1, p. 641),

A theory of rods² or, equivalently a one-dimensional theory of solids is a characterization of the behavior of slender three-dimensional solid bodies by a set of equations having the parameter of a curve and the time as the only independent variables.

_{²We use "rod" as a generic name for "arch", "bar", "beam", "column", "ring", "shaft", etc. We employ rod both in the intuitive sense of a slender body and in several precise mathematical senses. The meaning will be clearer from the context.}

Said that, by carefully reading the question and its motivation, one sees that the techniques shown in [1] exactly suit the needs of the Asker: his search for a particular 3D equation for the models \eqref{1} and \eqref{2} is probably useless since it would be unsuitable for the study of a non-rectilinear beam, while the general procedures described in [1] where the equations describing a 1D medium are deduced from the standard, general 3D equations by applying them to particular constitutive equations, are applicable for beams described by a general spatial curve.

Reference

[1] Stuart S. Antman, "The Theory of Rods", in Flügge, S. & Truesdell, C. (Eds.) Festkörpermechanik/Mechanic of Solids, Handbuch der Physik/Encyclopedia of Physics, Vol. VI part 2, Springer-Verlag, 1972, 641-703.

The beam equations \eqref{1} and \eqref{2} "admit a vector formulation" in the sense that they can be rigorously deduced from the 3D theory of (nonlinear) elasticity. I'm not aware of a precise reference dealing specifically with those two examples, but the Encyclopedia of Physics entry [1] by Antman, which also alludes briefly to the deduction of \eqref{1} in §26, example $\delta$, page 698, presents two different methods for solving the problem, namely the projection methods ([1], §11, pp. 660-663) and the asymptotic ([1], §13, pp. 664-665). Antman gives a fairly complete (even if somewhat dates) surveys of the theories of beams, rods and all continua with 1D behavior, considering also the history of the subject. Well, my two cents.

Reference

[1] Stuart S. Antman, "The Theory of Rods", in Flügge, S. & Truesdell, C. (Eds.) Festkörpermechanik/Mechanic of Solids, Handbuch der Physik/Encyclopedia of Physics, Vol. VI part 2, Springer-Verlag, 1972, 641-703.

The beam equations \eqref{1} and \eqref{2} "admit a vector formulation" in the sense that they can be rigorously deduced from the 3D theory of (nonlinear) elasticity. I'm not aware of a precise reference dealing specifically with those two examples, but the Encyclopedia of Physics entry [1] by Antman, which also alludes briefly to the deduction of \eqref{1} in §26, example $\delta$, page 698, presents two different methods for solving the problem, namely the projection methods ([1], §11, pp. 660-663) and the asymptotic ([1], §13, pp. 664-665). Antman gives a fairly complete (even if somewhat dates) surveys of the theories of beams, rods and all continua with 1D behavior, considering also the history of the subject. Well, my two cents.

Edit after the downvote. After I saw the OP and my answer downvoted, I decided to add a few words of explanation. According to Stuart Antman ([1], §1, p. 641),

A theory of rods² or, equivalently a one-dimensional theory of solids is a characterization of the behavior of slender three-dimensional solid bodies by a set of equations having the parameter of a curve and the time as the only independent variables.

_{²We use "rod" as a generic name for "arch", "bar", "beam", "column", "ring", "shaft", etc. We employ rod both in the intuitive sense of a slender body and in several precise mathematical senses. The meaning will be clearer from the context.}

Said that, by carefully reading the question and its motivation, one sees that the techniques shown in [1] exactly suit the needs of the Asker: his search for a particular 3D equation for the models \eqref{1} and \eqref{2} is probably useless since it would be unsuitable for the study of a non-rectilinear beam, while the general procedures described in [1] can is applicable for beams described by a general spatial curve.

Reference

[1] Stuart S. Antman, "The Theory of Rods", in Flügge, S. & Truesdell, C. (Eds.) Festkörpermechanik/Mechanic of Solids, Handbuch der Physik/Encyclopedia of Physics, Vol. VI part 2, Springer-Verlag, 1972, 641-703.