Sectional properties

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

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.

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.

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.

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.

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.

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.

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.