Stress resultants definition

@STRESS_RESULTANTS_DEFINITION {
@STRESS_RESULTANTS_NAME {StrResName} {
@STRUCTURE_TYPE {StrType}
@LOADING {
@SCALING_FACTOR {sca}
@FORCES {F1, F2, F3}
@MOMENTS {M1, M2, M3}
}
@LOADING {
@SCALING_FACTOR {sca}
@FORCES {N1, N2, N12}
@MOMENTS {M1, M2, M12}
@SHEAR_FORCES {Q1, Q2}
@DERIVATIVES {d1, d2}
}
@LOADING {
...
}
@APPLIED_AT_CENTERS {FlagCenter}
@APPLICATION_POINT_LOCATION {x2a, x3a}
@COMMENTS {CommentText}
}
}

Introduction

External loads are applied to beam, plates, and shells, and consequently, stress resultants develop at any point in the structure. Dymore is a finite element based tool for the comprehensive analysis of flexible multibody system. In particular, it features geometrically exact beam and plate/shell elements. These geometrically exact elements predict the dynamic response of the structure and evaluate time histories of stress resultants at any point along the beam's span or over the plate's mid-surface. Given these stress resultants, SectionBuilder evaluates the three-dimensional stress field at any point of the beam's cross-section. The Dymore/SectionBuilder interface explains this process in more details.

The three-dimensional state of stress at any point of the beam's cross-section depends on the stress resultants at that point along the beam's span.

Flag StrType can take two values only.

  1. If StrType = BEAM, the stress resultants defined here correspond to those acting on the cross-section of a beam. This case is discussed in section stress resultants for beams.
  2. If StrType = PLATE, the stress resultants defined here correspond to those acting on the normal material line of a plate or shell.This case is discussed in section stress resultants for plates and shells.

Stress resultants for beams

Figure 1. Stress resultants acting on the beam's cross-section.

For beams, the stress resultants over a typical cross-section are defined by means of a force array and of a moment array.

  1. The components of the force vector acting on the beam's cross-section and resolved along axes b1, b2, and b3 are denoted F1, F2, and F3, respectively, as shown in fig. 1. F1 is the axial force and F2 and F3 the transverse shear forces acting along unit vectors b2, and b3, respectively. These forces are applied at point O, the origin of the axis system.
  2. The components of the moment vector acting on the beam's cross-section and resolved about axes b1, b2, and b3 are denoted M1, M2, and M3, respectively, as shown in fig. 1. M1 is the twisting moment or torque, and M2 and M3 the bending moments about unit vectors b2 and b3, respectively.
  3. The force array is defined as FT = {F1, F2, F3}.
  4. The moment array is defined as MT = {M1, M2, M3}.

It is possible to change the point of application of the loading to another point of the cross-section, or to refer these forces and moments to the centroid and shear center of the section.

Notes

  1. For beams, a single @LOADING section should appear that contains the following input only: a scaling factor, sca, a force array, F, and a moment array, M.
  2. The scaling factor, sca, is a multiplicative factor that will affect both the force and the moment arrays. (Default value: sca = 1.0.)
  3. The force array, F, defines the forces acting on the cross-section. (Default value: FT = {0.0, 0.0, 0.0}.)
  4. The moment array, M, defines the moments acting on the cross-section. (Default value: MT = {0.0, 0.0, 0.0}.)
  5. If FlagCenter = YES, the axial force, F1, is applied at the centroid and the transverse shear forces, F2 and F3, are applied at the shear center, as depicted in fig. 2. (Default value: FlagCenter = NO)
  6. It is sometimes convenient to be able to apply the loads at an arbitrary point of the cross-section. If the application point is defined, the forces, F1, F2, F3, will be applied at point A with coordinates x2a and x3a, as depicted in fig. 2.
Figure 2. Left figure: forces are applied at point A, with coordinates x2a and x3a.
Right figure: transverse shear forces are applied at the shear center, axial force at the centroid.

Stress resultants for plates and shells

Figure 1. Stress resultants acting acting on a differential element of the plate.

For plates or shells, the stress resultants over a differential element of the structure are defined defined by means of an in-plane load array, a transverse shear force array, and a moment array.

  1. The components of the force vector acting on a face normal to unit vector b1 and resolved along unit vectors b1, b2, and b3 are denoted N1, N12, and Q1, respectively, as shown in fig. 1. The components of the force vector acting on a face normal to unit vector b2 and resolved along unit vectors b1, b2, and b3 are denoted N12, N2, and Q2, respectively, as shown in fig. 1. These forces are applied at point O, the origin of the axis system, typically located at the plate's mid-plane.
  2. The components of the moment vector acting on a face normal to unit vector b1 and resolved along unit vectors b1 and b2 are denoted M12 and M2, respectively, as shown in fig. 1. The components of the moment vector acting on a face normal to unit vector b2 and resolved along unit vectors b1 and b2 are denoted M1 and - M12, respectively, as shown in fig. 1.
  3. The in-plane force array is defined as NT = {N1, N2, N12}.
  4. The transverse shear force array is defined as QT = {Q1, Q2}.
  5. The moment array is defined as MT = {M1, M2, M12}.

Notes

  1. For plates and shells, multiple @LOADING sections can appear that contains the following input only: a scaling factor, sca, an in-plane force array, F, a transverse shear force array, Q, a moment array, M, and a definition of the derivative level.
  2. The scaling factor, sca, is a multiplicative factor that will affect the in-plane force, transverse shear force, and moment arrays. (Default value: sca = 1.0.)
  3. The in-plane force array, N, defines the in-plane forces acting on the normal material line. (Default value: NT = {0.0, 0.0, 0.0}.)
  4. The transverse shear force array, Q, defines the transverse shear forces acting on the normal material line. (Default value: QT = {0.0, 0.0}.)
  5. The moment array, M, defines the moments acting on the normal material line. (Default value: MT = {0.0, 0.0, 0.0}.)
  6. The loading arrays defined in each section define the loading or its spatial derivatives. If d1 = 0 and d2 = 0, the in-plane force array provides the loading at that point of the plate, N. If d1 = i and d2 = j, the in-plane force array provides the spatial derivative of the loading at that point, dN/(dx1i dx2j). (Default value: d1 = 0 and d2 = 0.)