Sectional properties

SectionBuilder evaluates the sectional stiffness and mass properties of the cross-section.

  • A detailed listing of these properties can be found in the output file. Sample output results are shown in figs. 2, 4, 6, 7.
  • To better understand this section, it is necessary to be familiar with the definition of the stiffness and mass properties of beam sections. This involves several concepts such as
    • The decoupling of the general problem into the extension-bending and shear-torsion problems.
    • The centroid and the principal centroidal axes of bending.
    • The shear center and principal axes of shear at the shear center.
    • The center of mass and the principal axes of inertia at the center of mass.

Sectional stiffness properties

1. The sectional compliance matrix

Figure 1. The applied stress resultants

First, SectionBuilder computes the 6×6 sectional compliance matrix of the section. The compliance matrix, S, gives the sectional deformation measures in terms of the sectional stress resultants
ε = SF,
where arrays ε and F, both of size 6×1, store the sectional deformation measures and sectional stress resultants, respectively.

  • The array of sectional deformation measures is defined as
    εT = { ε1, ε2, ε3, κ1, κ2, κ3 }T,
    where ε1 is the sectional axial strain, ε2 and ε3 are the sectional transverse shear strains, κ1 is the sectional twist rate, and κ2 and κ3 are the sectional bending curvatures.
  • The array of sectional stress resultants is defined as
    FT = { F1, F2, F3, M1, M2, M3 }T,
    where F1 is the sectional axial force, F2 and F3 are the sectional transverse shear forces, M1 is the sectional twisting moment, and M2 and M3 are the sectional bending moments. See definition of the sign conventions for these quantities.

Table 1 lists the values of a typical compliance matrix.

F1F2 F3M1 M2M3
ε15.32086e-0090.00000e+0000.00000e+0000.00000e+000 7.72512e-020-1.53876e-019
ε20.00000e+0001.62627e-0084.36195e-0177.89874e-0230.00000e+0000.00000e+000
ε30.00000e+0004.36195e-0171.62627e-008-2.05784e-023 0.00000e+0000.00000e+000
κ10.00000e+0000.00000e+0000.00000e+0009.57387e-0050.00000e+0000.00000e+000
κ27.72512e-0200.00000e+0000.00000e+0000.00000e+000 7.36451e-005-9.83274e-013
κ3-1.53876e-0190.00000e+0000.00000e+000 0.00000e+000-9.83274e-0137.36451e-005
Table 1. The 6×6 sectional compliance matrix, S.

2. The sectional stiffness matrix

Next, SectionBuilder computes the 6×6 sectional stiffness matrix of the section. The stiffness matrix, C, gives the stress resultants in terms of the sectional deformation measures
F = C ε,
where the stiffness matrix is the inverse of the compliance matrix, C = S-1. Table 2 lists the values of a typical stiffness matrix.

ε1ε2 ε3κ1 κ2κ3
F11.87939e+0080.00000e+000 0.00000e+0000.00000e+000 -1.97142e-0073.92684e-007
F20.00000e+0006.14905e+007 -1.64929e-001-5.07315e-011 0.00000e+0000.00000e+000
F30.00000e+000-1.64929e-001 6.14905e+0071.32170e-011 0.00000e+0000.00000e+000
M10.00000e+0000.00000e+000 0.00000e+0001.04451e+004 0.00000e+0000.00000e+000
M2-1.97142e-0070.00000e+000 0.00000e+0000.00000e+000 1.35786e+0041.81295e-004
M33.92684e-0070.00000e+000 0.00000e+0000.00000e+000 1.81295e-0041.35786e+004
Table 2. The sectional compliance matrix, C.

Figure 2 shows a typical print-out of the compliance and stiffness matrices in the output file.

Figure 2. The formatted print-out of the compliance and stiffness matrices in the output file.

3. Fully coupled stiffness matrix

Table 1 and Table 2 list the most general forms of the compliance and stiffness matrices, respectively: in general, these matrices are fully populated, 6×6 matrices. If the matrices are fully populated, the following message will be printed
This section presents coupling between extension-bending and shear-torsion behaviors.

For many sections, however, these matrices split into two, uncoupled, 3×3 matrices, highlighted in red and blue in tables 1 and 2. In such case, the remaining entries in those matrices all vanish.

  1. The terms highlighted in red define the coupled, extension-bending problem.
  2. The terms highlighted in blue define the coupled, shear-torsion problem.

4. The sectional constants of the extension-bending problem

Figure 3. The extension-bending problem

To simplify the analysis of the extension-bending problem, it is convenient to introduce the concept of centroid, a point of coordinates (x2c, x3c) shown in fig. 2. The extension-bending problem involves the following quantities.

  • The three stress resultants: the sectional axial force, F1, and the two bending moments, M2 and M3.
  • The corresponding sectional deformations: the sectional axial strain, ε1, and the two curvatures, κ2 and κ3.

In the study of the extension-bending problem, the axial force is applied at the centroid and the bending moments are evaluated about the centroidal axes b2c and b3c shown in fig. 2.

The sectional constants associated with the extension-bending problem are listed below and are summarized in table 3.

  • The axial stiffness of the section, S
  • The location of the centroid, (x2c, x3c)
  • The centroidal bending stiffnesses H22c, H33c, and H23c
  • The unit vectors aligned with the principal axes of bending, b2c* and b3c*
  • Angle α between unit vectors b2 and b2c*
  • The principal centroidal bending stifnesses, H22c* and H33c*
Axial stiffnessS 1.87939e+008[N]
Centroid location(x2c, x3c)(-2.08942e-015, -1.04897e-015)[m]
Bending stiffnessH22c1.35786e+004[N.m^2]
Bending stiffnessH33c1.35786e+004[N.m^2]
Bending stiffnessH23c-1.81295e-004[N.m^2]
Principal axisb2*( 0.00000e+000, 7.07111e-001, -7.07103e-001)
Principal axisb3*( 0.00000e+000, 7.07103e-001,  7.07111e-001)
Orientation angleα-4.49997e+001[deg]
Bending stiffnessH22c*1.35786e+004[N.m^2]
Bending stiffnessH33c*1.35786e+004[N.m^2]
Table 3. The sectional bending stiffness constants.

Figure 4 shows a typical print-out of the sectional constants of the extension-bending problem in the output file.

Figure 4. The formatted print-out of the sectional constants of the extension-bending problem in the output file.

5. The sectional constants of the shear-torsion problem

Figure 5. The shear-torsion problem

To simplify the analysis of the shear-torsion problem, it is convenient to introduce the concept of shear center, a point of coordinates (x2k, x3k) shown in fig. 5. The shear-torsion problem involves the following quantities.

  • The three stress resultants: the sectional torque, M1, and the two shear forces, F2 and F3.
  • The corresponding sectional deformations: the sectional twist rate, κ1, and the two transverse shear strain, ε2 and ε3.

In the study of the shear-torsion problem, the sectional torque is applied about the shear center and the lines of action of the transverse shear forces pass through the shear center along unit vectors b2k and b3k shown in fig. 3.

The sectional constants associated with the shear-torsion problem are listed below and are summarized in table 4.

  • The torsional stiffness of the section, H11
  • The location of the shear center, (x2k, x3k)
  • The shear stiffnesses about the shear center K22k, K33k, and K23k
  • The unit vectors aligned with the principal axes of shear, b2k* and b3k*
  • Angle α between unit vectors b2 and b2k*
  • The principal shear stiffnesses, K22k* and K33k*
Torsional stiffnessH11 1.04451e+004[N.m^2]
Shear centre location(x2k, x3k)(2.14943e-019, 8.25031e-019)[m]
Shearing stiffnessK22k6.14905e+007[N]
Shearing stiffnessK33k6.14905e+007[N]
Shearing stiffnessK23k1.64929e-001[N]
Principal axisb2*(0.00000e+000,  7.07107e-001, 7.07106e-001)
Principal axisb3*( 0.00000e+000, -7.07106e-001, 7.07107e-001)
Orientation angleα4.50000e+001[deg]
Shearing stiffnessK22k*6.14905e+007[N]
Shearing stiffnessK33k*6.14905e+007[N]
Table 4. The sectional shearing stiffness constants.

Figure 6 shows a typical print-out of the sectional constants of the shear-torsion problem in the output file.

Figure 6. The formatted print-out of the sectional constants of the shear-torsion problem in the output file.

Sectional mass properties

SectionBuilder computes the 6×6 sectional mass matrix of the section. The mass matrix, M, gives the sectional momenta in terms of the sectional velocities
P = MV,
where arrays P and V, both of size 6×1, store the sectional momenta and sectional velocities, respectively.

  • The array of sectional momenta is defined as
    PT = { p1, p2, p3, h1, h2, h3 }T,
    where p1, p2, and p3 are the components of the linear momentum vectors of the cross-section resolved along unit vectors b1, b2, and b3, respectively, and h1, h2, and h3 are the components of the angular velocity vector of the cross-section about the same unit vectors.
  • The array of sectional velocities is defined as
    VT = { v1, v2, v3, ω1, ω2, ω3 }T,
    where v1, v2, and v3 are the components of the velocity vector of the cross-section resolved along unit vectors b1, b2, and b3, respectively, and ω1, ω2, and ω3 are the components of the angular velocity vector of the cross-section resolved along the same unit vectors.
  • In the study of dynamics problems, it is convenient to introduce the center. The mass constants associated with the dynamics problem are listed below and are summarized in table 5.

    • The mass of the section per unit span of the beam, m00
    • The location of the center of mass, (x2m, x3m)
    • The mass moments of inertia of the section per unit span of the beam about the center of mass m22m, m33m, and m23m
    • The unit vectors aligned with the principal axes of inertia, b2m* and b3m*
    • Angle α between unit vectors b2 and b2m*
    • The principal mass moments of inertia per unit span of the beam, m22m* and m33m*
Mass per unit spanm007.08178e+000[kg/m]
Center of mass location(x2m, x3m)(-7.64979e-019, -1.61038e-018)[m]
Moment of inertiam11m1.02332e-003[kg.m^2/m]
Moment of inertiam22m5.11658e-004[kg.m^2/m]
Moment of inertiam33m5.11658e-004[kg.m^2/m]
Moment of inertiam23m-7.06792e-012[kg.m^2/m]
Principal axisb2*(0.00000e+000, 7.07107e-001, -7.07107e-001)
Principal axisb3*(0.00000e+000, 7.07107e-001,  7.07107e-001)
Orientation angleα-4.50000e+001[deg]
Moment of inertiam22m*5.11658e-004[kg.m^2/m]
Moment of inertiam33m*5.11658e-004[kg.m^2/m]
Table 5. The sectional mass constants.

Figure 6 shows a typical print-out of the sectional mass constants in the output file.

Figure 7. The formatted print-out of the sectional mass constants in the output file.