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Matrix or Vector | Shape Functions | Integration Points |
---|---|---|

Stiffness Matrix | (Equation 12–4) and (Equation 12–5) | None for elastic case. Same as Newton-Raphson load vector for tangent matrix with plastic case |

Mass and Stress Stiffness Matrices; and Thermal Load and Pressure Load Vectors | (Equation 12–5) | None |

Newton-Raphson Load Vector and Stress Evaluation | Same as stiffness matrix | 3 along the length |

Load Type | Distribution |
---|---|

Element Temperature | Linear thru thickness and along length |

Nodal Temperature | Constant thru thickness, linear along length |

Pressure | Linear along length |

The complete stiffness and mass matrices for an elastic 2-D beam element are given in BEAM3 - 2-D Elastic Beam.

There are three sets of integration points along the length of the element, one at each end and one at the middle.

h is defined as:

h = thickness or height of member (input as HEIGHT on R command) |

The five integration points through the thickness are located at positions y = -0.5 h, -0.3 h, 0.0, 0.3 h, and 0.5 h. Each one of these points has a numerical integration factor associated with it, as well as an effective width, which are different for each type of cross-section. These are derived here in order to explain the procedure used in the element, as well as providing users with a good basis for selecting their own input values for the case of an arbitrary section (KEYOPT(6) = 4).

The criteria used for the element are:

The element, when under simple tension or compression, should respond exactly for elastic or plastic situations. That is, the area (A) of the element should be correct.

The first moment should be correct. This is nonzero only for unsymmetric cross-sections.

The element, when under pure bending, should respond correctly to elastic strains. That is, the (second) moment of inertia (I) of the element should be correct.

The third moment should be correct. This is nonzero only for unsymmetric cross-sections.

Finally, as is common for numerically integrated cross-sections, the fourth moment of the cross-section (I

_{4}) should be correct.

For symmetrical sections an additional criterion is that symmetry about the centerline of the beam must be maintained. Thus, rather than five independent constants, there are only three. These three constants are sufficient to satisfy the previous three criteria exactly. Some other cases, such as plastic combinations of tension and bending, may not be satisfied exactly, but the discrepancy for actual problems is normally small. For the unsymmetric cross-section case, the user needs to solve five equations, not three. For this case, use of two additional equations representing the first and third moments are recommended. This case is not discussed further here.

The five criteria may be set up in equation form:

(14–144) |

(14–145) |

(14–146) |

(14–147) |

(14–148) |

where:

dA = differential area |

y = distance to centroid |

These criteria can be rewritten in terms of the five integration points:

(14–149) |

(14–150) |

(14–151) |

(14–152) |

(14–153) |

where:

H(i) = weighting factor at point i |

L(i) = effective width at point i |

P(i) = integration point locations in y direction (P(1) = -0.5, P(2) = -0.3, etc.) |

The L(i) follows physical reasoning whenever possible as in Figure 14.14: "Beam Widths".

Starting with the case of a rectangular beam, all values of L(i) are equal to the width of the beam, which is computed from

(14–154) |

where:

I_{zz} = moment of inertia (input as IZZ on R command) |

Note that the area is not used in the computation of the width. As mentioned before, symmetry may be used to get H(1) = H(5) and H(2) = H(4). Thus, H(1), H(2), and H(3) may be derived by solving the simultaneous equations developed from the above three criteria. These weighting factors are used for all other cross-sections, with the appropriate adjustments made in L(i) based on the same criteria. The results are summarized in Table 14.4: "Cross-Sectional Computation Factors".

One interesting case to study is that of a rectangular cross-section that has gone completely plastic in bending. The appropriate parameter is the first moment of the area or

(14–155) |

This results in

(14–156) |

**Table 14.4 Cross-Sectional Computation Factors**

Numerical Integration Point (i) | Location thru Thickness (P(i)) | Numerical Weighting Factor (H(i)) | Effective Width (L(i)) | |

Rectangular | Pipe | |||

1 | -.5 | .06250000 | 12I_{zz}/h^{3} | 8.16445t_{p} |

2 | -.3 | .28935185 | 12I_{zz}/h^{3} | 2.64115t_{p} |

3 | .0 | .29629630 | 12I_{zz}/h^{3} | 2.00000t_{p} |

4 | .3 | .28935185 | 12I_{zz}/h^{3} | 2.64115t_{p} |

5 | .5 | .06250000 | 12I_{zz}/h^{3} | 8.16445t_{p} |

Numerical Integration Point (i) | Location thru Thickness (P(i)) | Numerical Weighting Factor (H(i)) | Effective Width (L(i)) | |

Round Bar | Arbitrary Section | |||

1 | -.5 | .06250000 | 0.25341D_{o} | A(-0.5)/h |

2 | -.3 | .28935185 | 0.79043D_{o} | A(-0.3)/h |

3 | .0 | .29629630 | 1.00000D_{o} | A(0.0)/h |

4 | .3 | .28935185 | 0.79043D_{o} | A(0.3)/h |

5 | .5 | .06250000 | 0.25341D_{o} | A(0.5)/h |

where:

P(i) = location, defined as fraction of total thickness from centroid |

I_{zz} = moment of inertia (input as IZZ on R command) |

h = thickness (input as HEIGHT on R command) |

t_{p} = pipe wall thickness (input as TKWALL on R command) |

D_{o} = outside diameter (input as OD on R command) |

A(i) = effective area based on width at location i (input as A(i)
on R command) |

Substituting in the values from Table 14.4: "Cross-Sectional Computation Factors", the ratio of the theoretical value to the computed value is 18/17, so that an error of about 6% is present for this case.

Note that the input quantities for the arbitrary cross-section (KEYOPT(6) = 4) are h, hL(1)(=A(-50)), hL(2)(=A(-30)), hL(3)(=A(0)), hL(4)(=A(30)), and hL(5)(=A(50)). It is recommended that the user try to satisfy (Equation 14–149) through (Equation 14–153) using this input option. These equations may be rewritten as:

(14–157) |

(14–158) |

(14–159) |

(14–160) |

(14–161) |

Of course, I_{1} = I_{3} = 0.0
for symmetric sections.

Alternative to one of the above five equations, (Equation 14–156) can be used and rewritten as:

(14–162) |

Remember that I_{2} is taken about the midpoint
and that I_{zz} is taken about the centroid. The relationship
between these two is:

(14–163) |

where:

The elastic stiffness, mass, and stress stiffness matrices are the same as those for a 2-D beam element (BEAM3 ). The tangent stiffness matrix for plasticity, however, is formed by numerical integration. This discussion of the tangent stiffness matrix as well as the Newton-Raphson restoring force of the next subsection has been generalized to include the effects of 3-D plastic beams. The general form of the tangent stiffness matrix for plasticity is:

(14–164) |

where:

[B] = strain-displacement matrix |

[D_{n}] = elastoplastic stress-strain matrix |

This stiffness matrix for a general beam can also be written symbolically as:

(14–165) |

[K^{B}] = bending contribution |

[K^{S}] = transverse shear contribution |

[K^{A}] = axial contribution |

[K^{T}] = torsional contribution |

where the subscript n has been left off for convenience. As each of
these four matrices use only one component of strain at a time, the integrand
of (Equation 14–165) can be simplified from [B]T[D_{n}][B]
to {B} D_{n} .
Each of these matrices will be subsequently described in detail.

Bending Contribution ([K

^{B}]). The strain-displacement matrix for the bending stiffness matrix for bending about the z axis can be written as:(14–166) where contains the terms of which are only a function of x (see Narayanaswami and Adelman(129)) :

(14–167) where:

L = beam length Φ = shear deflection constant (see COMBIN14 - Spring-Damper) The elastoplastic stress-strain matrix has only one component relating the axial strain increment to the axial stress increment:

(14–168) where E

_{T}is the current tangent modulus from the stress-strain curve. Using these definitions (Equation 14–164) reduces to:(14–169) The numerical integration of (Equation 14–169) can be simplified by writing the integral as:

(14–170) The integration along the length uses a two or three point Gauss rule while the integration through the cross-sectional area of the beam is dependent on the definition of the cross-section. For BEAM23, the integration through the thickness (area) is performed using the 5 point rule described in the previous section. Note that if the tangent modulus is the elastic modulus, ET = E, the integration of (Equation 14–170) yields the exact linear bending stiffness matrix.

The Gaussian integration points along the length of the beam are interior, while the stress evaluation and, therefore, the tangent modulus evaluation is performed at the two ends and the middle of the beam for BEAM23. The value of the tangent modulus used at the integration point in evaluating (Equation 14–170) therefore assumes ET is linearly distributed between the adjacent stress evaluation points.

Transverse Shear Contribution ([K

^{S}]). The strain-displacement vector for the shear deflection matrix is (see Narayanaswami and Adelman(129)):(14–171) A plasticity tangent matrix for shear deflection is not required because either the shear strain component is ignored (BEAM23 and BEAM24) or where the shear strain component is computed (PIPE20), the plastic shear deflection is calculated with the initial-stiffness Newton-Raphson approach instead of the tangent stiffness approach. Therefore, since D

_{n}= G (the elastic shear modulus) (Equation 14–164) reduces to:(14–172) Integrating over the shear area explicitly yields:

(14–173) where A

_{s}is the shear area (see BEAM3 - 2-D Elastic Beam). As is not a function of x in (Equation 14–171), the integral along the length of the beam in (Equation 14–173) could also be easily performed explicitly. However, it is numerically integrated with the two or three point Gauss rule along with the bending matrix [K^{B}].Axial Contribution ([K

^{A}]). The strain-displacement vector for the axial contribution is:(14–174) As with the bending matrix, D

_{n}= E_{T}and (Equation 14–164) becomes:(14–175) which simplifies to:

(14–176) The numerical integration is performed using the same scheme BEAM3 as is used for the bending matrix.

Torsion Contribution ([K

^{T}]). Torsional plasticity (PIPE20 only) is computed using the initial-stiffness Newton-Raphson approach. The elastic torsional matrix (needed only for the 3-D beams) is:(14–177)

The Newton-Raphson restoring force is:

(14–178) |

where:

[D] = elastic stress-strain matrix |

The load vector for a general beam can be written symbolically as:

(14–179) |

where:

and where the subscript n has been left off for convenience. Again,
as each of the four vectors use only one component of strain at a time, the
integrand of (Equation 14–178) can be simplified from
[B]^{T}[D] to
{B} D. The appropriate {B} vector
for each contribution was given in the previous section. The following paragraphs
describe D and for each of the
contributing load vectors.

Bending Restoring Force . For this case, the elasticity matrix has only the axial component of stress and strain, therefore D = E, the elastic modulus. (Equation 14–178) for the bending load vector is:

(14–180) The elastic axial strain is computed by:

(14–181) where:

φ = total curvature (defined below) ε ^{a}= total strain from the axial deformation (defined below)ε ^{th}= axial thermal strainε ^{pl}= axial plastic strainε ^{cr}= axial creep strainε ^{sw}= axial swelling strainThe total curvature is:

(14–182) where {u

^{B}} is the bending components of the total nodal displacement vector {u}. The total strain from the axial deformation of the beam is:(14–183) where:

{u ^{A}} = axial components for the total nodal displacement vector {u}u _{XI}, u_{XJ}= axial displacement of nodes I and J(Equation 14–180) is integrated numerically using the same scheme outlined in the previous section. Again, since the nonlinear strain evaluation points for the plastic, creep and swelling strains are not at the same location as the integration points along the length of the beam, they are linearly interpolated.

Shear Deflection Restoring Force . The shear deflection contribution to the restoring force load vector uses D = G, the elastic shear modulus and the strain vector is simply:

(14–184) where γ

_{S}is the average shear strain due to shear forces in the element:(14–185) The load vector is therefore:

(14–186) Axial Restoring Force . The axial load vector uses the axial elastic strain defined in (Equation 14–181) for which the load vector integral reduces to:

(14–187) Torsional Restoring Force . The torsional restoring force load vector (needed only for 3-D beams) uses D = G, the elastic shear modulus and the strain vector is:

(14–188) where:

γ = total torsional strain (defined below) γ ^{pl}= plastic shear strainγ ^{cr}= creep shear strainThe total torsional shear strain is defined by:

(14–189) where:

θ _{XI}, θ_{XJ}= total torsional rotations from {u} for nodes I, J, respectively.The load vector is:

(14–190) where:

{B ^{T}} = strain-displacement vector for torsion (same as axial (Equation 14–174))

The modified total axial strain at any point in the beam is given by:

(14–191) |

where:

The total curvature and axial deformation strains are adjusted to account for the applied pressure and acceleration load vector terms. The adjusted curvature is:

(14–192) |

where:

φ = [B^{B}]{u^{B}}
= total curvature |

φ^{pa} = pressure and acceleration contribution
to the curvature |

φ^{pa} is readily calculated through:

(14–193) |

M^{pa} is extracted from the moment terms of
the applied load vector (in element coordinates):

(14–194) |

{F^{pr}} is given in BEAM3 - 2-D Elastic Beam and
{F^{ac}} is given in Static Analysis.
The value used depends on the location of the evaluation point:

(14–195) |

The adjusted axial deformation strain is:

(14–196) |

where:

ε = [B^{A}]{u^{A}}
= total axial deformation strain |

ε^{pa} = pressure and acceleration contribution
to the axial deformation strain |

ε^{pa} is computed using:

(14–197) |

where is calculated in a similar
manner to M^{pa}.

From the modified total strain ((Equation 14–191)) the plastic strain increment can be computed (see Rate-Independent Plasticity), leaving the elastic strain as:

(14–198) |

where Δε^{pl} is the plastic strain
increment. The stress at this point in the beam is then:

(14–199) |