The behavior of beam elements is governed by the beam-section stiffness matrix and the section mass matrix. A generalized nonlinear beam section (SECTYPE,,GENB) accommodates only diagonal terms in the cross-section stiffness matrix defining the relationship between generalized stress and generalized strains; it is therefore suitable only for beams with homogeneous and isotropic materials where the coupling between different generalized strains is small. For composite beams made of dissimilar constituent materials, or homogeneous beams with orthotropic material and material orientation not parallel to the beam axis, the coupling between different generalized strains can be significant and generally leads to full cross-section stiffness matrix.
The preintegrated composite beam section (SECTYPE,,COMB,MATRIX) is an abstract cross section type that enables you to define a fully populated but symmetrical cross-section stiffness and mass matrix directly. You can use preintegrated composite beam sections when using BEAM188 or BEAM189 elements, provided that linear elastic material behavior is acceptable.
The full cross-section stiffness relates the generalized-stress to generalized-strain in the following form:
where
| N = Axial force |
| M1 = Bending moment in plane XZ |
| M2 = Bending moment in plane XY |
| τ = Torque |
| S1 = Transverse shear force in plane XZ |
| S2 = Transverse shear force in plane XY |
| Bm = Warping bi-moment |
| ɛ = Axial strain |
| κ1 = Curvature in plane XZ |
| κ2 = Curvature in plane XY |
| χ = Twist of the cross section |
| γ1 = Transverse shear strain in plane XZ |
| γ2 = Transverse shear strain in plane XY |
| Bk = Warping bi-curvature |
| Sij(T) (where i = 1,7 and j = i,7) = Stiffness constants in the upper triangle of the cross-section stiffness matrix as a function of temperature |
| T = the current temperature |
With a unit beam length, the section mass matrix relates the resultant forces and torques to accelerations and angular accelerations as follows (applicable to the local element coordinate system):
where
| Nx = Axial resultant force |
| Ny = Resultant force in beam Y coordinate direction |
| Nz = Resultant force in beam Z coordinate direction |
| Mx = Torque about the beam axes |
| My = Torque about the beam Y coordinate direction |
| Mz = Torque about the beam Z coordinate direction |
| Ax = Axial acceleration |
| Ay = Acceleration in beam Y coordinate direction |
| Az = Acceleration in beam Z coordinate direction |
| Rx = Angular acceleration about beam axes |
| Ry = Angular acceleration about beam Y coordinate direction |
| Rz = Angular acceleration about beam Z coordinate direction Cij(T) (where i=1,7 and j = i,7) = mass or moment of inertia terms in the upper triangle of the cross-section mass matrix as a function of temperature |
| T = Current temperature |
Each of the following commands specifies a particular component quantity necessary for defining a preintegrated composite beam section:
Table 12.3: Commands for Specifying Preintegrated Composite Beam Section Data
| Command | Quantity Defined and Data Specified |
|---|---|
| CBMX[a] |
Cross-section stiffness matrix. Upper triangle of 6 x 6 or 7 x 7 matrix. Row and column 7 are used only when KEYOPT(1) = 1. |
| CBMD[a] |
Cross-section mass matrix. Upper triangle of 6 x 6 matrix. |
| CBTMP[a] |
Temperature specification for a subsequent matrix.
|
| CBTE[a] |
Thermal expansion coefficient
|
[a] Repeatable for six independent temperatures.
Temperature dependencies (T) — You can define each of the preintegrated composite beam section data components as temperature-dependent. It is possible to specify up to six temperatures (T) by reissuing any command as necessary. If you issue a command for a temperature specified earlier, the most recent data supersedes the previous value.
Each component (axial, bending, torque, transverse shear, and warping) of a preintegrated beam section definition is input directly as a 7 x 7 matrix. The terms of the matrix are defined via the CBMX command.
The following input example uses the CBMX command to define a (temperature-independent) parameterized cross-section stiffness matrix for a circular solid cross section with unit radius:
Exx = 30e6 Nuxy = 0.3 Gxy = 0.5*Exx / (1+Nuxy) Radius = 1 Pi = 3.14 Xarea = Pi * Radius**2 Izz = Pi/4 * Radius**4 Iyy = Pi/4 * Radius**4 J = Izz + Iyy sectype,1,comb,matrix cbmx,1, Exx*Xarea cbmx,2, Exx*Iyy cbmx,3, Exx*Izz cbmx,4, Gxy*J cbmx,5, Gxy*Xarea cbmx,6, Gxy*Xarea cbmx,7, 0
You can define preintegrated composite beam sections when using element BEAM188 or BEAM189 elements. The following conditions apply:
The section data defined by each command listed in Table 12.3: Commands for Specifying Preintegrated Composite Beam Section Data is associated with the section most recently defined (SECTYPE).
Beam stresses are not available for output; however, the stress resultants are available as ETABLE quantities.
Section offsetting (SECOFFSET) is not available.
The preintegrated stiffness components must yield a positive definite section stiffness at analysis time.
Only the temperature of the beam axis is relevant.
The following input example compares the results of a preintegrated composite beam section to an offset channel section:
/prep7 /com, Approximate material properties of steel in U.S. customary units Exx = 30e6 nuxy = 0.3 gxy = 0.5*Exx / (1+nuxy) /com, Parameterize length and number of elements to more easily study /com, the effects of discretization and length-to-thickness ratio leng = 25 nelem = 1 mp,ex,1,Exx mp,prxy,1,nuxy mp,gxy,1,gxy et,1,188,1,,3 ! Include warping and use the cubic shape function offy = -8 offz = -10 /com, /com, Define a beam channel section. Unequal lengths and thicknesses are /com, used to demonstrate an unsymmetric section with noncoincident /com, centroid and shear center. /com, /com, An arbitrary nodal offset is used to demonstrate coupling between /com, axial and bending stiffness, as well as coupling between shear /com, and torsional stiffness. /com, sectype,1,beam,chan secdata,1,1.5,2, 0.1, 0.15, 0.2 secoffset,user,offy,offz /com, For the channel section, generate elements along the x-axis secnum,1 n,1 *do,i,1,nelem loc = i*leng/nelem n,i+1,loc e,i,i+1 *enddo /com, /com, Retrieve the essential calculated and geometric properties from /com, the channel section /com, *get,xarea,secp,1,prop,area *get,iyy,secp,1,prop,iyy *get,izz,secp,1,prop,izz *get,iyz,secp,1,prop,iyz *get,J,secp,1,prop,tors *get,iw,secp,1,prop,warp *get,scyy,secp,1,prop,scyy *get,sczz,secp,1,prop,sczz *get,cgy,secp,1,prop,cgy *get,cgz,secp,1,prop,cgz *get,offy,secp,1,prop,offy *get,offz,secp,1,prop,offz *get,shcy,secp,1,prop,shcy *get,shcz,secp,1,prop,shcz yc = offy - cgy ! Distance between nodal offset and centroid zc = offz - cgz ys = offy - shcy ! Distance between nodal offset and shear center zs = offz - shcz yssq = ys*ys zssq = zs*zs /com, /com, Adjust the preintegrated torsional stiffness based on nodal offset k44 = Gxy*J k44 = k44 + ys*ys*Gxy*xarea*sczz k44 = k44 + zs*zs*Gxy*xarea*scyy /com, /com, Use the material and channel section properties to define the 7x7 /com, upper triangle of the preintegrated section stiffness matrix /com, sectype,2,comb,matrix cbmx,1, Exx*xarea, -zc*Exx*xarea, yc*Exx*xarea cbmx,2, Exx*iyy+zc*zc*Exx*xarea, -Exx*Iyz-yc*zc*Exx*xarea cbmx,3, Exx*izz+yc*yc*Exx*xarea cbmx,4, k44, -ys*Gxy*xarea*sczz, zs*Gxy*xarea*scyy cbmx,5, Gxy*xarea*sczz cbmx,6, Gxy*xarea*scyy cbmx,7, Exx*iw slist /com, /com, For the preintegrated section, generate elements parallel to /com, the x-axis and offset from the beam with channel section secn,2 n,1001,,10 *do,i,1,nelem loc = i*leng/nelem n,i+1001,loc,10 e,i+1000,i+1001 *enddo /com, /com, Load the beams as cantilevers. Fix one end and apply a /com, transverse load in the z-direction at the free end d,1,all f,nelem+1,fz,-100 d,1001,all f,nelem+1001,fz,-100 fini /sol solve finish /post1 /com, /com, COMPARE RESULTS: All displacements, including warping, compare /com, well between beam and preintegrated sections /SHOW,JPEG prnsol,u prnsol,rot prnsol,warp finish