Sectional properties
SectionBuilder evaluates the sectional stiffness and mass properties of the crosssection.
 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 extensionbending and sheartorsion 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
ε^{T} = { ε_{1}, ε_{2}, ε_{3}, κ_{1}, κ_{2}, κ_{3} }^{T},
where ε_{1} is the sectional axial strain, ε_{2} and ε_{3} are the sectional transverse shear strains, κ_{1} is the sectional twist rate, and κ_{2} and κ_{3} are the sectional bending curvatures. 
The array of sectional stress resultants is defined as
F^{T} = { F_{1}, F_{2}, F_{3}, M_{1}, M_{2}, M_{3} }^{T},
where F_{1} is the sectional axial force, F_{2} and F_{3} are the sectional transverse shear forces, M_{1} is the sectional twisting moment, and M_{2} and M_{3} are the sectional bending moments. See definition of the sign conventions for these quantities.
Table 1 lists the values of a typical compliance matrix.
F_{1}  F_{2}  F_{3}  M_{1}  M_{2}  M_{3}  
ε_{1}  5.32086e009  0.00000e+000  0.00000e+000  0.00000e+000  7.72512e020  1.53876e019 
ε_{2}  0.00000e+000  1.62627e008  4.36195e017  7.89874e023  0.00000e+000  0.00000e+000 
ε_{3}  0.00000e+000  4.36195e017  1.62627e008  2.05784e023  0.00000e+000  0.00000e+000 
κ_{1}  0.00000e+000  0.00000e+000  0.00000e+000  9.57387e005  0.00000e+000  0.00000e+000 
κ_{2}  7.72512e020  0.00000e+000  0.00000e+000  0.00000e+000  7.36451e005  9.83274e013 
κ_{3}  1.53876e019  0.00000e+000  0.00000e+000  0.00000e+000  9.83274e013  7.36451e005 
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.
ε_{1}  ε_{2}  ε_{3}  κ_{1}  κ_{2}  κ_{3}  
F_{1}  1.87939e+008  0.00000e+000  0.00000e+000  0.00000e+000  1.97142e007  3.92684e007 
F_{2}  0.00000e+000  6.14905e+007  1.64929e001  5.07315e011  0.00000e+000  0.00000e+000 
F_{3}  0.00000e+000  1.64929e001  6.14905e+007  1.32170e011  0.00000e+000  0.00000e+000 
M_{1}  0.00000e+000  0.00000e+000  0.00000e+000  1.04451e+004  0.00000e+000  0.00000e+000 
M_{2}  1.97142e007  0.00000e+000  0.00000e+000  0.00000e+000  1.35786e+004  1.81295e004 
M_{3}  3.92684e007  0.00000e+000  0.00000e+000  0.00000e+000  1.81295e004  1.35786e+004 
Table 2. The sectional compliance matrix, C.
Figure 2 shows a typical printout of the compliance and stiffness matrices in the output file.
Figure 2. The formatted printout 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 extensionbending and sheartorsion 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.
 The terms highlighted in red define the coupled, extensionbending problem.
 The terms highlighted in blue define the coupled, sheartorsion problem.
4. The sectional constants of the extensionbending problem
Figure 3. The extensionbending problem
To simplify the analysis of the extensionbending problem, it is convenient to introduce the concept of centroid, a point of coordinates (x_{2}^{c}, x_{3}^{c}) shown in fig. 2. The extensionbending problem involves the following quantities.
 The three stress resultants: the sectional axial force, F_{1}, and the two bending moments, M_{2} and M_{3}.
 The corresponding sectional deformations: the sectional axial strain, ε_{1}, and the two curvatures, κ_{2} and κ_{3}.
In the study of the extensionbending problem, the axial force is applied at the centroid and the bending moments are evaluated about the centroidal axes b_{2}^{c} and b_{3}^{c} shown in fig. 2.
The sectional constants associated with the extensionbending problem are listed below and are summarized in table 3.
 The axial stiffness of the section, S
 The location of the centroid, (x_{2}^{c}, x_{3}^{c})
 The centroidal bending stiffnesses H_{22}^{c}, H_{33}^{c}, and H_{23}^{c}
 The unit vectors aligned with the principal axes of bending, b_{2}^{c*} and b_{3}^{c*}
 Angle α between unit vectors b_{2} and b_{2}^{c*}
 The principal centroidal bending stifnesses, H_{22}^{c*} and H_{33}^{c*}
Axial stiffness  S  1.87939e+008  [N] 
Centroid location  (x2c, x3c)  (2.08942e015, 1.04897e015)  [m] 
Bending stiffness  H22c  1.35786e+004  [N.m^2] 
Bending stiffness  H33c  1.35786e+004  [N.m^2] 
Bending stiffness  H23c  1.81295e004  [N.m^2] 
Principal axis  b2*  ( 0.00000e+000, 7.07111e001, 7.07103e001)  
Principal axis  b3*  ( 0.00000e+000, 7.07103e001, 7.07111e001)  
Orientation angle  α  4.49997e+001  [deg] 
Bending stiffness  H22c*  1.35786e+004  [N.m^2] 
Bending stiffness  H33c*  1.35786e+004  [N.m^2] 
Table 3. The sectional bending stiffness constants.
Figure 4 shows a typical printout of the sectional constants of the extensionbending problem in the output file.
Figure 4. The formatted printout of the sectional constants of the extensionbending problem in the output file.
5. The sectional constants of the sheartorsion problem
Figure 5. The sheartorsion problem
To simplify the analysis of the sheartorsion problem, it is convenient to introduce the concept of shear center, a point of coordinates (x_{2}^{k}, x_{3}^{k}) shown in fig. 5. The sheartorsion problem involves the following quantities.
 The three stress resultants: the sectional torque, M_{1}, and the two shear forces, F_{2} and F_{3}.
 The corresponding sectional deformations: the sectional twist rate, κ_{1}, and the two transverse shear strain, ε_{2} and ε_{3}.
In the study of the sheartorsion 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 b_{2}^{k} and b_{3}^{k} shown in fig. 3.
The sectional constants associated with the sheartorsion problem are listed below and are summarized in table 4.
 The torsional stiffness of the section, H_{11}
 The location of the shear center, (x_{2}^{k}, x_{3}^{k})
 The shear stiffnesses about the shear center K_{22}^{k}, K_{33}^{k}, and K_{23}^{k}
 The unit vectors aligned with the principal axes of shear, b_{2}^{k*} and b_{3}^{k*}
 Angle α between unit vectors b_{2} and b_{2}^{k*}
 The principal shear stiffnesses, K_{22}^{k*} and K_{33}^{k*}
Torsional stiffness  H11  1.04451e+004  [N.m^2] 
Shear centre location  (x2k, x3k)  (2.14943e019, 8.25031e019)  [m] 
Shearing stiffness  K22k  6.14905e+007  [N] 
Shearing stiffness  K33k  6.14905e+007  [N] 
Shearing stiffness  K23k  1.64929e001  [N] 
Principal axis  b2*  (0.00000e+000, 7.07107e001, 7.07106e001)  
Principal axis  b3*  ( 0.00000e+000, 7.07106e001, 7.07107e001)  
Orientation angle  α  4.50000e+001  [deg] 
Shearing stiffness  K22k*  6.14905e+007  [N] 
Shearing stiffness  K33k*  6.14905e+007  [N] 
Table 4. The sectional shearing stiffness constants.
Figure 6 shows a typical printout of the sectional constants of the sheartorsion problem in the output file.
Figure 6. The formatted printout of the sectional constants of the sheartorsion 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
P^{T} = { p_{1}, p_{2}, p_{3}, h_{1}, h_{2}, h_{3} }^{T},
where p_{1}, p_{2}, and p_{3} are the components of the linear momentum vectors of the crosssection resolved along unit vectors b_{1}, b_{2}, and b_{3}, respectively, and h_{1}, h_{2}, and h_{3} are the components of the angular velocity vector of the crosssection about the same unit vectors. 
The array of sectional velocities is defined as
V^{T} = { v_{1}, v_{2}, v_{3}, ω_{1}, ω_{2}, ω_{3} }^{T},
where v_{1}, v_{2}, and v_{3} are the components of the velocity vector of the crosssection resolved along unit vectors b_{1}, b_{2}, and b_{3}, respectively, and ω_{1}, ω_{2}, and ω_{3} are the components of the angular velocity vector of the crosssection 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, m_{00}
 The location of the center of mass, (x_{2}^{m}, x_{3}^{m})
 The mass moments of inertia of the section per unit span of the beam about the center of mass m_{22}^{m}, m_{33}^{m}, and m_{23}^{m}
 The unit vectors aligned with the principal axes of inertia, b_{2}^{m*} and b_{3}^{m*}
 Angle α between unit vectors b_{2} and b_{2}^{m*}
 The principal mass moments of inertia per unit span of the beam, m_{22}^{m*} and m_{33}^{m*}
Mass per unit span  m00  7.08178e+000  [kg/m] 
Center of mass location  (x2m, x3m)  (7.64979e019, 1.61038e018)  [m] 
Moment of inertia  m11m  1.02332e003  [kg.m^2/m] 
Moment of inertia  m22m  5.11658e004  [kg.m^2/m] 
Moment of inertia  m33m  5.11658e004  [kg.m^2/m] 
Moment of inertia  m23m  7.06792e012  [kg.m^2/m] 
Principal axis  b2*  (0.00000e+000, 7.07107e001, 7.07107e001)  
Principal axis  b3*  (0.00000e+000, 7.07107e001, 7.07107e001)  
Orientation angle  α  4.50000e+001  [deg] 
Moment of inertia  m22m*  5.11658e004  [kg.m^2/m] 
Moment of inertia  m33m*  5.11658e004  [kg.m^2/m] 
Table 5. The sectional mass constants.
Figure 6 shows a typical printout of the sectional mass constants in the output file.