Displacement field calculations are necessary for many structural steel engineering problems such as cold expansion of holes, embedment of bolts and rivets, and installation and maintenance of external devices. To this end, rigorous closed form analytical displacement solution is obtained for structural steel open-hole plates with in-plane loading. The material of the model is considered to be elastic perfectly plastic obeying the von Mises yield criterion with its associated flow rule. On the basis of this solution, two simplified engineering formulae are proposed and carefully discussed for practical engineering purposes. Graphical representations of results show validity of each formula as compared with rigorous solution and other studies.

Displacement field calculation around stress concentrators in steel structural members is on-going engineering task which is necessary for a number of reasons. It provides the basis for specific industrial codes and regulations such as AISC or Eurocodes, makes substantial part of commercial software based on finite element method (FEM), and proves validity of initial stress solutions. In a large number of cases, structural elements utilized in civil, mechanical, and aerospace engineering require major repair or are out of service due to ductile failures originating at holes. To this end, two widely applied technological processes are of great importance, namely, cold expansion of holes to improve fatigue life of structural members [

In cylindrical coordinate system

Geometrical model of a circular plate under internal pressure in cylindrical coordinate system.

The following boundary conditions in stress should be satisfied:

To conduct further analysis, the dimensionless parameters may be introduced

In the outer elastic zone, taking into account the condition of continuity of stresses at the elastic-plastic boundary, the stress-displacement solution may be also analytically defined:

In general procedure (which is mathematically and physically rigorous) exploited in the present research, the total strain in the inner plastic zone is assumed to be the sum of elastic and plastic portions. The elastic portion is obtained from Hooke’s law (

The total displacement along the radius is the sum of (

As it follows from the procedure of deducing (

Another simplification of general displacement solution (

Radial displacement distributions in plastic zone for medium pressure.

The same type of displacement distributions is shown in Figure

Radial displacement distributions for completely plasticized material due to high pressure.

It can be seen from both figures that, for

Dependence of radial displacement in plastic zone on material compressibility for medium pressure.

Dependence of radial displacement in completely plasticized plate on material compressibility.

For medium or lower loads (Figure

Displacement distributions versus radius for medium pressure calculated on the basis of various engineering approximations.

As it can be seen from this figure, the difference between rigorous solution (

The analyses presented in this study are based on the rigorous formula (

Dependence of radial expansion on practical values of pressure load.

It can be seen from Figure

Taking into account that, in real engineering applications, the stress state near the hole is rather complex, the proposed simplified solutions can be effectively used in the preliminary design stage to assist three-dimensional complete numerical modeling usually applied to most engineering structures. Based on a general analytical elastic perfectly plastic displacement analysis presented, elastic-plastic boundaries in plate structures and permanent enlargement of open holes due to in-plane loading are easily assessable. These data are required to select optional material/geometrical parameters such as yield limit, elastic modulus, Poisson’s coefficient, and dimensional ratios for real fastener techniques and predict fatigue crack growth life for cold-worked holes [

On the basis of rigorous analytical procedure developed for radial displacements, two simplified engineering formulae for steel structural applications are proposed and carefully analyzed. The first approximate formula (

In general, for structural steel plate applications, consideration of elastic strains in the inner plastic zone leads to lower values of radial displacements especially for smaller values of Poisson’s ratio, closer to the edge of the hole and upon higher external loading. It should be also reminded, however, that, in contrast to the displacement calculation procedures, the related strain analysis, that is, determination of strain fields, must be always conducted using rigorous formula (

The author declares that there is no conflict of interests regarding the publication of this paper.