METHOD FOE ESTIMATING THE STRENGTH OF BODIES AND

STRUCTURES

Field of the Invention

The present invention relates in general to the field of structural analysis. In particular, the present invention relates to the analysis of strength of bodies and structures.

Background of the Invention

Publications and other reference materials referred to herein, including references cited therein, are incorporated herein by reference in their entirety.

Although the terms, "body", "machine part" and "structure" generally denote entities having different construction from each other, they will be used interchangeably herein, and refer to the object upon which load forces are applied.

One of the important factors that is taken into consideration when designing a structure is the maximum force that may be applied to the structure at specific points or regions, and at specific directions, before collapse occurs.

Traditionally, structures are designed for some specific loading conditions at particular regions. These loading conditions are known or estimated according to the functionality of the structure. This leaves open the possibility of the application of unpredicted loads, or the application of forces at unexpected areas on the structure, when functioning in its real environment.

It is therefore an object of the present invention to provide a method for determining the strength of a structure in a way that will consider all possible loading conditions.

It is an additional object of the present invention to provide a method for determining explicitly how the strength of a structure (for all possible loading conditions) depends on the geometric shape of the structure, and the material it is made of.

It is yet an additional object of the present invention to provide a method for determining the strength of homogeneous perfectly plastic structures under all possible loading conditions.

Additional objects and advantages of the present invention will become apparent as the description proceeds.

Summary of the Invention

The present invention relates to a method for determining the maximum value of an arbitrary distribution of external loading for a homogeneous elastic-plastic body that will allow the body to maintain its mechanical integrity. The method comprises determining the elastic limit, F ₀ , of the material comprising the body; determining a purely geometrical property, C, of the body, wherein C is obtained by an expression of the form

; and, setting the maximum external loading at less than

CY ₀ to maintain the mechanical integrity of the body.

When the external loading consists of an external traction field, C is f w dA obtained by the expression of the form — = sup , _n , _n , — ^!

1>H dV

When the external loading consists of an external traction field and a body force field, C is obtained by the expression of the form

-

When said body is a discrete structure or a discrete model of a continuous

structure, C is obtained by an expression in the form — c = sup ^F _w - _s ^w μ( _W )|

Brief Description of the Figures

In the drawings:

-Fig. 1 illustrates a truss under tension;

-Fig. 2 illustrates the safety factor as a function of the ratio of length to height;

-Fig. 3 illustrates the load capacity ratio as a function of the ratio of length to height;

-Fig. 4 illustrates the structure of Fig. 1, having diagonal elements that are thicker than the horizontal and vertical elements;

-Fig. 5 illustrates the load capacity ratio as a function of the ratio between the thickness of the diagonal elements and the thickness of the horizontal and vertical element.

Detailed Description of the Preferred Embodiments

According to conventional methods of structural analysis, the strength of a structure is determined by considering a finite number of known or

- A -

anticipated forces that will act on the structure. In limit analysis of perfectly plastic structures, the strength analysis is carried out by computing a number called the factor of safety, or limit analysis factor. This number indicates by how much the assumed load distribution at the assumed locations and directions can be multiplied before collapse of the structure occurs. The present invention provides a method for estimating the strength of structures made from a given homogenous plastic material, solely according to the spatial geometry of the body, and taking into account all distribution of forces and their directions, that will be applied thereto.

According to the present invention, a maximal positive number, C, exists for a homogenous isotropic elastic perfectly plastic body ω such that the body will not collapse under any external traction field, t, bounded by CY ₀ , where F ₀ is the empirically measured elastic limit (also known as the yield stress, and denoted σ _r ). C is referred to herein as the load capacity ratio.

Thus, while the limit analysis factor of the theory of plasticity (see Kamenjarzh, J., Limit analysis of solids and structures, Boca Raton, FIa.: CRC Press (cl996)) pertains to a specific distribution of external loading, the load capacity ratio is independent of the distribution of external loading, and implies that the body will maintain its mechanical integrity, i.e. no collapse will occur, for any field t on the boundary δω as long as

< CY _n

wherein t _max is the maximum of the magnitude of the external loading. Specifically,

Collapse will occur for some t whose maximum (essential supremum) is larger than CY ₀ .

Similarly, a load capacity ratio exists if, in addition to boundary traction t , there is also a body force field b acting on the body. In this more general case the structure will not collapse if both the magnitudes of t and b are smaller than CY ₀ at all points where the loads are applied.

A load capacity ratio also exists for discrete models of structures such as trusses and frames where discrete loads may be applied at a finite number of nodes. In this case, the structure will not collapse if all the components of the loads acting at the various nodes are smaller than CF ₀ .

The load capacity ratio depends only on the geometry of the body, and the expression for it is of the form

Here, W is a vector space of virtual displacements, or virtual velocity fields, of the body or structure, |w| is a semi-norm on W depending on the type of loading considered, and ||^(w)|| is a norm on the infinitesimal strain fields, ε(w) , associated with the virtual displacements.

In the particular case where only boundary traction t is considered, C may be obtained by an expression of the form

Here, LD(Q) ₀ is the space of incompressible vector fields w that satisfy the displacement boundary conditions on a subset F ₀ of the boundary and for which the corresponding linear strains ε(w) are assumed to be integrable; T ₁ is the subset of the boundary that is not supported, and γ ₀ is the trace mapping assigning the boundary value γ _D (w) to any w e LD(Q) _D . The magnitude of a strain matrix should be evaluated using the dual norm to the norm on the space of stress matrices, or deviatoric components of stress matrices, which is used as a failure criterion. Thus, for this case

W dV .

For the traditional von-Mises yield criterion, ε(w) is the Frobenius norm for matrices.

In the case where body force fields are also considered the only adaptation needed is

Therefore, C may be obtained by an expression of the form

w dV

~ svφ _weW{ω)j:ι

For the case of a discrete structure, W is a finite dimensional vector space and so is the space of strain fields. The strain mapping ε is represented by a matrix A so

and the norms are defined on finite dimensional vector spaces. This form should also be used for discrete models, such as finite element models, of continuous bodies.

We now describe some additional mathematical details involved with the expression of the load capacity ratio. We consider the case of a continuous body acted upon by an external boundary traction field as an example. The other cases considered above are analogous. Let ω represent the region occupied by the body in space so the body is supported on a part F ₀ of its boundary and let t be the external surface traction acting on the part F ₇ of its boundary. The set ω is assumed to be open, bounded, and its boundary is assumed to be smooth. Furthermore, it is assumed that F, and F ₀ are disjoint open subsets of the boundary whose closures cover the boundary, and that their closures intersect on a smooth curve.

It is noted that yield criteria in plasticity usually use norms on the deviatoric component of the stress matrix. Thus, for the application to plastic bodies as considered here, the yield condition implies that we

consider only the deviatoric component of the stress which induces a norm on the space of strain matrices corresponding to incompressible vector fields LD(ω) _D . The specific norm depends on the particular yield condition used, e.g., the traditional Tresca or von Mises yield criteria.

Let, |«£-(w)| = [ If(W)IJF ^" be the I ^! -norm of an incompressible strain field,

and let ||w| be the I ¹ -norm on the boundary T ₁ , of the boundary values of the vector field w .

It can be shown (see Segev, K., Load capacity of bodies, International Journal of Non-Linear mechanics 42 (2007) 250-257) that with such a choice of norm on the space of matrices, the load capacity ratio C is given by

where the supremum is taken over all w e LD(Q) _n .

Clearly, the supremum may be approximated using various numerical methods and in particular, finite dimensional approximation spaces of LD(Cl) ₀ .

While some embodiments of the invention have been described by way of illustration, it will be apparent that the invention can be carried into practice with many modifications, variations and adaptations, and with the use of numerous equivalents or alternative solutions that are within the scope of persons skilled in the art, without departing from the spirit of

the invention or exceeding the scope of the claims. Additionally, parts of the description that are out of ambit of the appended claims do not constitute part of the claimed invention.

Example

The following example shows the relevance of the load capacity ratio in analysis and design of bodies and structures.

Figure 1 shows a simple structure (truss) under tension with P = I. AU other quantities such as the yield stress are normalized to value 1. Figure 2 shows the factor of safety (by how much can one multiply the load before the structure reaches its limit state) as a function of the ratio of the length b, to the height, a of the structure. It is noted that as the ratio b/a increases, the factor of safety increases and tends to the limit of 2. This is natural as the diagonal elements become approximately aligned with the horizontal elements and with the external force. Note that according to the prior art, the "factor of safety" is determined only for the known loads P which the structure is planned to sustain.

Figure 3 shows the Load Capacity Ratio as a function of the ratio b/a. It is noted that by definition, as the yield stress is unity, the load capacity is identical to the maximal possible load (at any point, and direction) that may be applied to the structure without collapse. The graph indicates that the load capacity decreases as the ratio b/a increases. This agrees with engineering intuition that longer structures will be sensitive to transversal loads. In fact, for a very long structure the load capacity tends to zero.

Next, consider the case where an engineer attempts to strengthen the structure by using diagonal elements of cross section area A ₁ which is

larger than the cross section A ₀ of the other elements as in Figure 4. In this example we considered the case where the ratio of length to height is 1. The values of the Load Capacity Ratio as a function of the ratio AjA ₀ are shown in Figure 5.

It is noted that increasing the thickness of the diagonals increases the load capacity up to a certain value (about 0.7). However, the graph becomes horizontal and any further increase of the thickness does not contribute the strength of the structure.