The order of degrees of freedom (DOFs) is shown in Figure 1.1: Order of Degrees of Freedom.

Figure 1.1: Order of Degrees of Freedom

The stiffness matrix in element coordinates is (Przemieniecki):

(1–1)

where:

A = cross-section area (input as AREA on R command)

E = Young's modulus (input as EX on MP command)

L = element length

G = shear modulus (input as GXY on MP command)

Ix = input torsional moment of inertia (input as IXX on RMORE command)

Jx = polar moment of inertia = Iy + Iz

az = a(Iz,φy)

ay = a(Iy,φz)

bz = b(Iz,φy)

fz = f(Iz,φy)

fy = f(Iy,φz)

Ii = moment of inertia about direction i (input as Iii on R command)

The consistent mass matrix (LUMPM,OFF) in element coordinates LUMPM,OFF is (Yokoyama):

(1–2)

where:

Mt = (ρA+m)L(1-εin)

ρ = density (input as DENS on MP command)

m = added mass per unit length (input as ADDMAS on RMORE command)

εin = prestrain (input as ISTRN on RMORE command)

Az = A(rz,φy)

Ay = A(ry,φz)

Bz = B(rz,φy)

Fz = F(rz,φy)

Fy = F(ry,φz)

The mass matrix (LUMPM,ON) in element coordinates is:

(1–3)

The element gyroscopic damping matrix is the same as for PIPE16.

The pressure and temperature load vector are computed in a manner similar to that of BEAM3.

The element coordinates are related to the global coordinates by:

(1–4)

where:

{u} = vector of displacements in global Cartesian coordinates

[T] is defined by:

(1–5)

where:

S3 = sin (θ)

C3 = cos (θ)

X1, etc. = x coordinate of node 1, etc.

Lxy = projection of length onto X-Y plane

d = .0001 L

θ = user-selected adjustment angle (input as THETA on R command)

If a third node is given, θ is not used. Rather C₃ and S₃ are defined using:

Then,

(1–6)

(1–7)

(1–8)

(1–9)

and

(1–10)

(1–11)

The x and • refer to vector cross and dot products, respectively. Thus, the element stiffness matrix in global coordinates becomes:

(1–12)

(1–13)

(1–14)

(1–15)

( is defined in Large Strain).

The centroidal stress at end i is:

(1–16)

where:

Fx,i = axial force (output as FX)

The bending stresses are

(1–17)

(1–18)

where:

My,i = moment about the element y axis at end i

Mz,i = moment about the element z axis at end i

tz = thickness of beam in element z direction (input as TKZ on R command)

ty = thickness of beam in element y direction (input as TKY on R command)

The maximum and minimum stresses are:

(1–19)

(1–20)

The presumption has been made that the cross-section is a rectangle, so that the maximum and minimum stresses of the cross-section occur at the corners. If the cross-section is of some other form, such as an ellipse, the user must replace Equation 1–19 and Equation 1–20 with other more appropriate expressions.

For long members, subjected to distributed loading (such as acceleration or pressure), it is possible that the peak stresses occur not at one end or the other, but somewhere in between. If this is of concern, the user should either use more elements or compute the interior stresses outside of the program.