The Plane Problem

[J. Appl. Math & Phys. (ZAMP) 23, 795804, 1972]

Introduction

The following is concerned with a consistent one-dimensional treatment of the class of beam problems dealing with the plane deformation of originally plane beams.

Our principal result is a system of non-linear strain displacement relations which is consistent with exact one-dimensional equilibrium equations for forces and moments via what is considered to be an appropriate version of the principle of virtual work.

Having a consistent system of equilibrium and strain displacement equations it is further necessary to stipulate, or rather to establish by means of an appropriate set of physical experiments, an associated system of constitutive equations. We discuss the nature of this aspect of the problem, including a solution of its linearized version, but without arriving at the solution of the general problem.

The principal novelty of the present results is thought to be a rational incorporation of transverse shear deformation into one-dimensional finite-strain beam theory. A case may be made that the theory, with this effect incorporated, is of a more harmonious form than the corresponding classical theory, where account is taken of finite bending and stretching, while at the same time it is postulatedfollowing Euler and Bernoullithat the transverse shearing strain is absent, with the corresponding force being a reactive force.

As an application of the general work a solution is given of the problem of circular ring buckling, including consideration of the effects of axial normal strain and of transverse shearing strain on the value of the classical Bresse-Maurice Lévy buckling load.

Kinematics of Beam Element

We consider an element ds of a one-dimensional beam with equations x = x(s) and y = y(s) before deformation. We designate the tangent angle to the beam curve byM⁰ and write cos M⁰ = xc(s) and sin M⁰ = yc(s), where primes indicate differentiation with respect to s. We note that M⁰ is also the angle between the normal to the beam curve and the y-axis.

Due to deformation the points x = x(s) and y = y(s) of the undeformed beam curve are changed to x(s) + u(s) and y(s) + v(s). We now assume that transverse elements which were originally normal to the beam curve do not necessarily remain so but end up enclosing an angle 1/2SF with this curve. At the same time we designate the angle enclosed by such an element and the y-axis by M. We then have a geometrical situation as shown in Figure 1. We note in particular, in addition to the angle F, the relative change of length e of the beam curve element ds, and the

Fig. 1.

change of the angle M⁰ into an angle M, and we read from the deformed beam element, as relations between F,M, e, u and v,

Dynamics of Beam Element

We now consider the deformed beam element, with normal and shear forces N and Q and with a bending moment M, in accordance with Figure 2. Together with this we assume force load intensities p^x and p^y and a moment load intensity m, per unit of undeformed beam curve length, also in accordance with Figure 2.

Fig. 2.

We then read from Figure 2 as component equations of force equilibrium in the directions of x and y,

At the same time we obtain as equation of moment equilibrium

We note, for future use, the possibility of deducing from (2a, b) the relations

where n = p^xcos M + p^y sin M and q = p^ycos Mp^xsin M are components of load intensity in the directions of N and Q, respectively.

Constitutive Equations

We postulate that the material of the beam is elastic and that we have the existence of axial and transverse force strains H and J and of a bending strain N, in such a way that constitutive equations for beam elements may be written in the form

We are ignorant, at this point, not only in regard to the form of the functions g in (4), but also in regard to definitions for the components of strain H,J and N which enter into the constitutive equations (4).*

Defining Equations for Strain

In order to obtain equations for strain we consider a virtual work equation of the form

and we stipulate, as Principle of Virtual Work, that equation (5) be equivalent to the dynamic equations (2) and (3) in the interior of the interval ( s¹, s²), given that GH, GJ and GN are appropriate expressions for virtual strains.

Since we know the form of the dynamic equations but do not at this point know expressions for virtual strains we use equation (5), in conjunction with (2) and (3), to deduce expressions for virtual strains.

Introduction of (2) and (3) into equation (5) gives a relation of the form

*However, we expect that H|e, J|F and N|McMc0, for sufficiently small strain.

and in this we may now consider N, Q and M as arbitrary differentiable functions of s.

In order to utilize (6) we integrate by parts, thereby eliminating all derivatives of N, Q and M as well as the boundary terms on the right. In this way we obtain

The arbitrariness of N, Q and M means that (7) implies the virtual strain displacement relations

It remains to take the step from virtual strain displacement relations to actual strain displacement relations.

One of these actual strain displacement relations follows directly from equation (9) in the form

A correspondingly simple derivation of expressions for H and J is clearly not possible through direct use of (8a, b). Remarkably, we may obtain H and J by using (8a, b) in conjunction with the geometrical relations (1). To do this we observe that equations (1) imply the following relations between virtual quantities

We now use (11a, b) in order to eliminate Guc and Gvc in (8a, b). In this way we obtain

The form of (12a, b) is such that we can now go from virtual strains to actual strains. The results are

Having (13a, b) we can further express H and J in terms of u, v and M. Introduction of (13a, b) into (1a, b) gives first and then, by inversion

We finally note the possibility of rewriting the moment equilibrium equation (3) somewhat more simply with the help of the strain components H and J as in (13), in the form

Observations on the Problem of Experimentally Derived Constitutive Equations

In order to see the nature of the problem of experimentally establishing the nature of the functions g in equations (4) we consider the problem of an originally straight beam, with x = s, y = 0 and M⁰ = 0, fixed at the end x = 0 and subject to given displacements u(a) = u^a, v(a) = v^a and M(a) = M^a at the other end. We assume absent distributed loads and have then from equations (2a, b)

where X^a and Y^a are two constants of integration the mechanical significance of which is evident.

To proceed further we consider the moment equation (3*) as a differential equation for M, by writing

and by considering the constitutive equations involving N and Q partially inverted in the form

so that M = f^M(H,J,N) = f^k(N,Q,M') = g(M,M').

The resultant second-order equation for M must be solved subject to the boundary conditions M(0) = 0 and M(a) = M^a, with which M = M(x; X^a, Y^a,M^a).

Having M we find u and v from (14a, b). The boundary conditions for u and v are satisfied upon setting

We now measure X^a, Y^a and M^a as functions of u^a, v^a,M^a, and of a, giving a set of three relations , etc. The remaining task then is to deduce from the form of these three experimentally determined functions , and the form of the desired three functions g^N,g^Q,g^M in equations (4).

The Linear Case

We consider a range of stresses and strains within which

with a view towards determining the elements , . . ., of the three by three matrix [C].

From equations (17) follow the linearized relations

and the moment equation (3*), again with boundary conditions M(0) = 0 and M(a) = M^a, is reduced to Equations (19) for the translational edge displacements become

In order to solve the problem as stated in (20) to (23) we partially invert (20) in the form

and write (22) in the form , with solution

We then have further, from (24),

and, upon making use of (23a, b),

We now stipulate knowledge of a matrix [B], as a result of experiment, such that

Having (26) to (28) we may then successively determine the elements of the matrix [C*] in terms of the elements of [B]. To see this we write

and have then from the relation M^a = BMNN^a + BMQQ^a + BMMM^a that

from which , and follow in succession in terms of elements of [B].

We next introduce (26c) into (27a, b) and compare the resultant relations with corresponding relations in (28). In this way we obtain the remaining six elements , etc. of the matrix [C*] in terms of the elements of [B].

Finally, having [C*] we find the elements of [C] by returning from (24) to (20).

Buckling of Circular Rings

As an application of the foregoing we consider the classical problem of in-plane buckling of a circular ring of radius R, subject to a uniform normal pressure p. We wish to obtain a buckling-load formula which

incorporates the effects of (1) the symmetrical deformation of the ring prior to the onset of buckling, (2) axial strain associated with the buckling mode, (3) transverse shearing strain associated with the buckling mode. We will be concerned, in particular, with the question of appropriate constitutive equations.

Inspection of Figure 2 indicates that for uniform normal pressure p, per unit of deformed beam curve, we have as expressions for the load intensity components q and n in the force equilibrium equations (2*a, b)

together with an absent moment load intensity m in equation (3*).

We further have, with N as in equation (10) and with RdM⁰ = ds, that Mc = R1 + N. Therewith the equilibrium equations (2*a, b) and (3*) may be written in the form

In complementing (31) by constitutive equations we have no difficulty in deciding that suitable relations involving N and H are of the form

In stipulating a relation involving J we find it necessary to concern ourselves with the question whether J would be determined by the force Q tangential to the deformed cross section or by a force Q^* normal to the deformed centerline. Evidently, we have Q^* given in terms of Q and N by the relation Q^* = Q cos F N sin F or, approximately, by Q^* = Q NJ. If we stipulate that J = BQ^* we arrive at a relation for J in terms of Q and N, of the form J = BQ/(1 + BN).* If we use Q instead of Q^* at the outset we have instead that J = BQ. We may subsume both relations to one of the form

and consider in the end the two limiting cases O = 0 and O = 1.

Having equations (31) and (32) we now consider the stability of the state

for which, evidently, in view of (31b) and (32b)

We now write

and linearize (31) and (32) in terms of Q, M,J,N, N¹ and H¹ so as to have.

*This, together with (32b), is effectively equivalent to constitutive equations of the form Q = (J/B) + (HJ/C) and N = (H/C) + (J2/2C).

Equation (32a) remains as is and equations (32b) and (32c) become*

We now use (32a), (34) and (37) to write (36a, b, c) as a system of equations for N¹, Q and N, as follows

It is evident that (38b), differentiated once, my be written with the help of (38a) and (38c) as one second-order differential equation for Q.

Appropriate solutions, for a complete ring, will be of the form Q = cos ns/R where n = 2, 3, . . . From this follows as the equation for possible values of P,

Equation (39) may be written as a cubic equation for PR2/D, involving axial-strain and transverse shear-strain parameters kH = CD/R2 and kJ = BD/R2. We will here limit ourselves to a discussion of the case kH = 0, with kJ{ k, for which the cubic equation reduces to a quadratic of the form

The smallest positive value of P follows from this for n = 2. We consider in particular the cases O = 1 and O = 0.

When O = 1 we have from (40), in agreement with a recent result by Smith and Simitses**

When O = 0 the solution is

*We note the possibility that C and B, as well as D in equation (32a), may be considered to depend on HP.

**J. Eng. M ech. Div., ASCE 95, EM 3, 559569 (1969).