[Proc. 5th Intern. Congr. Appl. Mech. pp. 8892, 1938]
1
Introduction
Tension field theory has been developed by H. Wagner1 to describe the state of stress in a certain type of thin-walled structures after buckling has set in. It substitutes for the non-linear problem of a large-deflection theory of elasticity a simplified linear theory which in a number of practically important cases yields results in good agreement with experimental data.
The reasoning leading to tension field theory is best explained by means of a definite example. Consider a strip of thin sheet supported perpendicular to the plane of the sheet and acted upon by uniform shear along the edges in the plane of the sheet.
Up to a certain intensity of the shear load, a uniform plane state of stress is produced in the sheet. If the load is increased beyond that intensity buckling occurs.
However if the distance of opposite edges of the sheet is kept constant the shear load can have an intensity many times that of the buckling load without failure of the structure as such. What happens is that wrinkles are formed in the sheet, the wave length of which decreases with increasing load, and that the sheet is mainly stressed in tension in the direction along the wrinkles while the compressive stress perpendicular to the wrinkles which causes the wrinkling becomes small compared with the tensile stress. Neglecting this compressive stress and also the bending stresses induced by the deformation out of the plane of the sheet against the tensile stress, one has to study types of plane states of stress for which at each point the only principal stress component which is different from zero is a tension, while the strain component perpendicular to the direction of the non-vanishing principal stress component is not uniquely determined by the stress at that point.
In the present paper Wagner's formulation of the problem is modified in such a way, that it appears as a special case of a more general problem in plane stress, in itself of interest and hitherto not considered. It deals with the theory of elasticity of anisotropic media, the curvilinear anisotropy of which depends on the boundary conditions of the problem.
In this new form the tension field is analyzed by straightforward calculus avoiding the lengthy geometrical considerations of Wagner.
As an application of the theory a solution of the following problem is here given: A flat sheet of circular ring form, stiffened along inner and outer edges, is stressed by twisting one edge with respect to the other, the axis of the applied torque being perpendicular to the plane of the sheet. This is the first solved problem in tension field theory where non-parallel tension lines occur.
1H. Wagner, Zeitschr. Flugtechnik u. M otorluftschiffahrt 1929.
2
Formulation of the General Problem
In this paragraph is given the mathematical formulation of the following problem: To determine the state of stress in a material of such curvilinear anisotropy that the axes of elastic symmetry are always tangent to the lines of principal stress. The anisotropy of such a material thus depends on the conditions which are prescribed along its boundaries.
Tension field theory is obtained as a particular case of this problem in which one of the two different moduli of elasticity has the value zero.
Introducing as system of coordinates [,K the orthogonal system of the lines of principal stress in which the line element has the form
the following system of equations has to be solved.
1. The equations of equilibrium for the principal stress components V[ and VK
2. The stress-strain relations. Calling u and v the displacements in [ and K-direction, E[ and EK the moduli of elasticity and G the modulus of rigidity and assuming no lateral contractions, these relations are
3. The relation between h1 and h2 which expresses the fact that the system ([,K) is orthogonal
It seems difficult to solve this system if not either E[ = EK (isotropy) or one of the E's say Geometrically the relation (7) means that the lines K = const. are straight (Figure 1).
Fig. 1.
Furthermore it follows from (2) that
where g is an arbitrary function.
From (4) follows
where g1 and g2 are arbitrary.
For the calculation of the displacements u and v two cases are conveniently distinguished.
(a) The case that the straight lines K = const. are parallel, and hence also the lines [ = const. are straight. Then g1(K) = 0 and by scaling appropriately one may put g2(K) = 1.
The displacements result from Eqs. (3) in the form
Here h and k are two more arbitrary functions.
(b) The case that the lines K = const. are not parallel. In this case a change of scale makes possible to put g1(K) = 1 which means that the variable K is identical with the varying angle D between the lines K = const. and a fixed straight line.
From (3) results then
In (8), (9) and (10) the four unknown quantities V[, h2, u and Q are expressed in terms of four arbitrary functions g, g2, h and k.
Since these arbitrary functions depend on the variable K whereas the boundary conditions are given in terms of a fixed coordinate system, say a Cartesian system (x, y), relations must be established between the system ([,K) and the system (x, y). For this purpose the following equations must be used
The integration of this system of equations is effected by a geometrical consideration. Since the lines K = const. are straight, one may write
This introduced into (11) gives for m and n
and from (13)
Hence
where dD/dK = g1(K)
In (8), (9), (10) and (15) there is given the general solution of the tension field problem in terms of four arbitrary functions which have to be determined by boundary conditions.
4
The Stresses in a Sheet of Circular Ring Form Wrinkled under the Influence of Shear Stresses Acting along the Edges
(Figure 2) As an application of the preceding general results this problem is solved rigorously. The tension lines start from the inner edge, and because of symmetry, each makes along the inner edge the same angle E with the radius from the origin. This angle E cannot be assumed but must be determined by means of the boundary conditions.
Introducing as independent variables [ and K the distance [ along the tension lines from an origin whose position depends on the angle K of this line with the x-axis, one finds the following relations between the coordinates x, y, the polar
Fig. 2.
coordinates r,M, and the coordinates [,K
For the stresses one obtains from (8) the principal stress
and the radial and tangential stresses
For the displacements in radial and tangential direction ur and uM results
Equations (18) and (19) are the most general expressions for stresses and displacements compatible with the assumed tension lines. If one is only interested in a stress distribution independent of the angle variable M = KU + S/2, that is, independent of K, these expressions are simplified to
and
For the determination of the four constants R0 = r0 sin E, g0, h0 and k0 serve two boundary conditions at the inner edge r = r0 and two conditions at the outer edge r
= r1. It shall be assumed that the shear load is applied by means of two stiffening rings of cross sectional area A0 and A1 which have moduli of elasticity E0 and E1. The two rings are necessary to prevent the sheet from collapsing after wrinkling has started.
Assuming that the tangential displacement of the outer edge is zero and of the inner edge equal to and expressing the fact that the radial deformation of the stiffening rings under the influence of the uniformly distributed radial load has to equal the radial deformation of the sheet along its edges, one has the conditions
Further discussion of the problem is here restricted to the case of rigid stiffening rings, so that in (23) and (25) one puts
That is, ur(r1) = 0 and ur(r0) = 0. Conditions (22) and (23) lead with (21) to
From (24) and (25) follows
Equations (27), (28) and (29) are three linear homogeneous equations for g0, h0, k0 which cannot all be zero. Therefore the determinant of this system has to vanish, that is
Fig. 3.
Developing and introducing the expressions for h2(r) and R0 from (16) this becomes the following equation to determine the angle E
which angle thus is seen to depend on the ratio r0/r1.
Since (32) cannot be solved explicitly, E has been determined numerically as function of r0/r1 and the result plotted in Figure 3. It is seen that E varies between 45°
and 90°; if the diameter of the hole approaches the diameter of the outer edge, the sheet behaves approximately like a straight strip and E equals 45°; if the diameter of the hole is small compared with the diameter of the sheet the tension lines depart under right angles from the radius vector. For very small r0/r1where the numerical
Fig. 4.
evaluation of (32) is inconvenient the following approximate relation holds
Having obtained E, one also knows g0, and with that the stresses in terms of the tangential edge displacement .
Of especial interest is the ratio between radial stress and tangential stress along the edges since the radial stress for prescribed shear determines the dimensions of the stiffening rings. This ratio is, according to (20),
The values of the ratios (34) are plotted as function of r0/r1 in Figure 4.