It is clear that the pipes considered in this paper are slender objects in terms of the ratio of axial length to the average diameter of the cross-sections. Beam models exploit this fact by reducing the object to a one-dimensional line along the axial dimension—possibly enriching its unknown displacement by additional unknowns—so that in contrast to three-dimensional descriptions where a volume has to be considered and discretized the computational effort can be drastically reduced.
Beam models can be derived by inserting an ansatz on how beam-like objects can deform into 3D equations of continua and subsequently averaging these equations over the cross-section. We need some assumptions to derive the ansatz where from one-dimensional unknowns along the centerline of the beam the displacement of the whole three-dimensional beam object is constructed. The classical ones are that the cross-sections do not change their shape and remain planar. Now, a first major decision has to be made: are the cross-sections allowed to rotate independently of displacements of the centerline (Timoshenko-like theories) or do they have to be perpendicular at all times (Bernoulli hypothesis)? This question can only be answered by looking at the real spatial dimensions and the material of the objects that are going to be modeled. The steel pipes have a rather large diameter, see Sect.
2 , in comparison to their characteristic length so that shear deformation may play an important role. Therefore, we use a Timoshenko-like ansatz:
$$u^{\mathrm {full}}(x)= u(x_1) + \left(\varLambda (x_1) - I_3\right)x_A.$$
Here
\(u(x_1) \in \mathbb {R}^3\) describes the displacement of the centerline, the vector
\(x_A\in {{\mathrm{span}}}\{E_2, E_3\}\) lies in the cross-section of the beam,
\(x_1\) parameterizes its centerline (see also Fig.
12 ),
\(\varLambda \in \mathbb {R}^{3\times 3}\) characterizes the rotation of the cross-section and
\(I_n\) denotes the
\(n\times n\) identity matrix. Now there are two extreme cases of beam models:
The geometrically exact (GE) beam, obtained by taking \(\varLambda \in SO(3)\) , e.g. parameterized by its rotation vector \(\phi \in \mathbb {R}^3\) (see Ibrahimbegovic 1997 2004
The linear Timoshenko (TS) beam, obtained by linearizing \(\varLambda (\phi )\in SO(3)\) so that \(u^{\mathrm {full}}(x)=u(x_1)+\phi (x_1)\times x_A\) together with the linear continuum equations.
We will focus on the static behavior since the industrial standards which are going to be constraints in the shape and material optimization problem (see Sect.
4 ) involve only static solutions of linear Timoshenko beams. In addition the more sophisticated constitutive laws that will introduce degradation and damage effects over time are mainly driven by slow long-lasting processes, so analysis in a quasi-static setting seems reasonable. Emergency stops where a power plant must shut down as fast as possible can initiate damaging processes in the piping systems due to the shock of pressure loss, but these are not considered because of the fundamentally different physical mechanisms.
Fig. 12 Left: Beam with local basis \(\{E_1,E_2,E_3\}\) embedded in space by inertial frame \(\{e_1,e_2,e_3\}\) and starting point \(v\) . Centerline indicated as dashed line along \(E_1\) -direction. Right: Annular cross-section and its dimensions
For initially straight prismatic beams of length
L , oriented along the
\(x_1\) -axis, we obtain the following apparently similar but in terms of complexity very different systems of differential equations, both holding for
\(x_1 \in (0,L)\) :
$$\begin{array}{ll} {\text {TS}}\, {\text{beam}} &{}{\text {GE}}\, {\text{beam}}\\ \left\{ \begin{array}{ll} 0 = N_{\mathrm {el}}' + \bar{N}\\ 0 = M_{\mathrm {el}}' + E_1 \times N_{\mathrm {el}} + \bar{M} \end{array}\right. &{} \left\{ \begin{array}{ll} 0 = (\varLambda N_{\mathrm {el}})' + \bar{N}\\ 0 = (\varLambda M_{\mathrm {el}})' + (E_1 + u') \times (\varLambda N_{\mathrm {el}}) + \bar{M} \end{array}\right. \end{array} $$
(21)
\(N_{\mathrm {el}}\) and
\(M_{\mathrm {el}}\) are the internal forces and moments whereas
\(\bar{N}\) and
\(\bar{M}\) are given external forces and moments. All quantities are given in the local basis of the beam
\(\{E_1, E_2, E_3\}\) ,
\((\cdot )'\) denotes differentiation w.r.t.
\(x_1\) and
\(\times\) is the cross-product.
If the beams are linearly elastic and have a doubly symmetric cross-section the constitutive law is:
$$\begin{aligned} \begin{array}{ll} {\text {TS}}\, {\text{beam}} &{} {\text {GE}}\, {\text{beam}}\\ \left\{ \begin{array}{l} N_{\mathrm {el}} = N = C_N (u' + E_1\times \phi )\\ M_{\mathrm {el}} = M = C_M \phi ' \end{array}\right. &{} \left\{ \begin{array}{l} N_{\mathrm {el}} = N = C_N \varLambda ^T\left(u' - (\varLambda - I_3) E_1\right)\\ M_{\mathrm {el}} = M = C_M {{\mathrm{vec}}}(\varLambda ^T \varLambda ') \end{array}\right. \end{array} \end{aligned}$$
(22)
In the elastic case the elastic response
\(N_{\mathrm {el}}\) ,
\(M_{\mathrm {el}}\) of course coincides with the theoretical stress resultants
N ,
M that relate the total strain linearly to the response. The operator
\({{\mathrm{vec}}}(\cdot )\) extracts the axial vector from a skew symmetric matrix. The matrices
\(C_N\) and
\(C_M\) encode material and geometric properties of the beam and are defined as
$$C_N := {{\mathrm{diag}}}(EA_c,GA_s,GA_s), \qquad C_M := {{\mathrm{diag}}}(GI_t,EI,EI),$$
(23)
where
\({{\mathrm{diag}}}(\cdot )\) indicates the diagonal elements of a square matrix. The occurring quantities describe the material with elastic and shear modulus
E and
G , as well as geometric properties of the cross-section with area of cross-section
\(A_c\) , corrected shear area
\(A_s\) and second and polar moment of area
I and
\(I_t\) . The pipes considered in this paper have circular (specifically annular) cross-sections, so there is just one second moment of inertia
\(I=\int x_2^2{\text {d}}x_A=\int x_3^2{\text {d}}x_A = \frac{\pi }{4}(r_{\mathrm {o}}^4-r_{\mathrm{i}}^4)\) and torsional constant
\(I_t = 2I\) . The following shear correction is used (Cowper
1966 ):
$$A_s = k A_c,\quad k := \frac{6(1+\nu )(1+m^2)^2}{(7+6\nu )(1+m^2)^2 + (20+12\nu )m^2},\quad m := \frac{r_{\mathrm{i}}}{r_{\mathrm{o}}}<1$$
(24)
with
\(\nu\) being the Poisson’s ratio and
\(r_{\mathrm{i}}\) ,
\(r_{\mathrm{o}}\) being the inner radius and outer radius of the pipe; see Fig.
12 .
In addition to the linear elastic behavior, the thermo-elastic effects are to be captured by the model. This can be done by the reduction of a three-dimensional thermo-elastic continuum as in the elastic case (Jones
1966 ). Since we consider only steady states we simply need to couple the effects of the temperature
\(\vartheta\) on the elastic stress resultants
\(N_{\mathrm {el}}\) and
\(M_{\mathrm {el}}\) . Coupling in the other direction would only be possible due to vibrations of the structure. It is further assumed that the temperature in the system is constant during steady-state operation. Therefore, it is also constant across the cross-sections and along the pipes and there is no need to solve the heat equation on the graph defining the piping system. Then the temperature couples only in the first component of the inner forces
\(N_{\mathrm {el}}\) of the beam:
$$N_{\mathrm {el}} = N - \alpha _H EA_c (\vartheta - \vartheta _0) E_1.$$
(25)
In (
25 ) the actual temperature
\(\vartheta\) affects the normal forces only if it differs from the reference temperature
\(\vartheta _0\) at which the structure is stress-free. If the structure gets hotter, i.e.
\(\vartheta > \vartheta _0\) , the resulting normal force is decreased, which means the beam tends to lengthen, where
\(\alpha _H\) is the heat expansion coefficient. The dependence of the elastic (and shear) modulus on the temperature is indirectly modeled: since there are only two isothermal test cases with
\(\vartheta = \vartheta _{\mathrm{h}} = 600\,^{\circ }\hbox {C}\) (operational temperature: called the hot-load case) and
\(\vartheta = \vartheta _0 = 20\,^{\circ }\hbox {C}\) (no thermal effects: called the cold-load case). The corresponding elastic moduli
\(E_{\mathrm{h}}, E_0\) can then be chosen in advance.
Fig. 13 Typical creep behavior of a specimen under constant axial load. Three phases of creep: decelerating creep rate (I), approximately constant minimal creep rate (II), accelerating creep up to fracture (III). The constant offset of the strain curve is the elastic part of the total strain
We could incorporate even more sophisticated constitutive laws in the beam model by considering the continuous 3D analog and reducing it once again. However, recent publications, e.g. Gruttmann et al. (
2000 ) and Mata et al. (
2007 ), instead reconstruct the 3D strain and stress states at certain points of the cross-section, evaluate the 3D constitutive law at these points and then integrate over the cross-section to obtain the beam stress resultant. We choose a rather phenomenological approach so that the constitutive law is able to reproduce certain creep damage effects, see e.g. Naumenko et al. (
2010 ) and Altenbach et al. (
1999 ), such as the creep curve shown in Fig.
13 . We define the total beam strain
\(\varSigma\) due to smallness assumptions as the sum of the elastic strain, thermal strain and creep strain:
$$\varSigma = \varSigma _{\rm{el}} + \varSigma _{\rm{th}} + \varSigma _{\rm{cr}}.$$
(26)
The elastic response
\(F_{\rm{el}}^T = (N_{\rm{el}}^T, M_{\rm{el}}^T)\) of the GE beam is then calculated as
$$\begin{aligned} F_{\rm{el}} := C(\varSigma - \varSigma _{\rm{th}} - \varSigma _{\rm{cr}}) := \left(\begin{array}{lll} C_N &\quad{} 0\\ 0 &\quad{} C_M \end{array} \right) \left[\left( \begin{array}{ll} \varLambda ^T\left(u' - (\varLambda - I_3) E_1\right)\\ {{\mathrm{vec}}}(\varLambda ^T \varLambda ') \end{array}\right) - \varSigma _{\rm{th}} - \varSigma _{\rm{cr}}\right] . \end{aligned}$$
(27)
Then mimicking the mechanisms of continuous creep laws we define the evolution of the creep strain. Beginning with phase II we use the power creep law or Norton–Bailey law that relates the minimum creep rate
\(\dot{\varepsilon }_{\rm{cr}}\) to the applied load to a certain power
\(m_1\) . Clearly, for this multi-axial purpose we must define a scalar effective quantity
\(E_{\rm{el}}\) to condense the current stress state:
$$\dot{\varepsilon }_{\rm{cr}}= a_{\rm{cr}} E_{\rm{el}}^{m_1} \quad E_{\rm{el}} := \sqrt{F_{\rm{el}}^T F_{\rm{el}}}$$
(28)
$$\dot{\varSigma }_{\rm {cr}}= d_{\rm{cr}}\, \dot{\varepsilon }_{\rm {cr}}\frac{F_{\rm {el}}}{E_{\rm {el}}}.$$
(29)
The next step is to model the decreasing creep rate of phase I by a decreasing hardening term in the source of the creep rate:
$$\dot{\varepsilon }_{\rm {cr}}= a_{\rm {cr}}\left[ 1 + c_{\rm {cr}}\exp \left( -\frac{\varepsilon _{\rm {cr}}}{k_{\rm {cr}}}\right) \right] E_{\rm {el}}^{m_1}.$$
(30)
The last step is to incorporate a scalar damage variable
\(\omega \in [0,1)\) that triggers accelerated creep of phase III and sits in the denominator of the sources driving the creep rate in the describing equations. The final creep damage constitutive model is as follows:
$$\begin{aligned} \begin{aligned} \dot{\varepsilon }_{\rm {cr}}&= a_{\rm {cr}}\left[ 1 + c_{\rm {cr}}\exp \left( -\frac{\varepsilon _{\rm {cr}}}{k_{\rm {cr}}}\right) \right] \left( \frac{E_{\rm {el}}}{1-\omega }\right) ^{m_1},\\ \dot{\varSigma }_{\rm {cr}}&= d_{\rm {cr}}\, \dot{\varepsilon }_{\rm {cr}}\frac{F_{\rm {el}}}{E_{\rm {el}}},\\ \dot{\omega }&= b_{\rm {cr}}\frac{E_{\rm {el}}^{m_2}}{(1-\omega )^{m_3}}. \end{aligned} \end{aligned}$$
(31)
Here,
\(a_{\rm {cr}}, b_{\rm {cr}}\) and
\(d_{\rm {cr}}\) are material parameters. The above quantities are all functions of space
x and time
t . All the above internal variables are coupled to the beam kinematics by the elastic response
\(F_{\rm {el}}\) [see (
27 )], which has to fulfill differential system (
21 ) for every time
t . For the kinematics the time is also only an internal variable since the system is equilibrated for given creep strain
\(\varSigma _{\rm {cr}}(t)\) . The initial conditions are
$$\left. \left(\begin{array}{l} \varepsilon _{\rm {cr}}\\ \varSigma _{\rm {cr}}\\ \omega \end{array}\right)\right| _{t=0} = 0.$$
(32)
Clearly, if the material parameters
\(a_{\rm {cr}}\) and
\(b_{\rm {cr}}\) are positive
\(\omega\) is monotonically increasing over time, as is
\(\varepsilon _{\rm {cr}}\) . The material parameters
\(m_2\) and
\(m_3\) control the effects of the load and the damage status on the growth of the damage variable. The creep strain always evolves in the direction of the elastic response, a typical relaxation effect, and therefore might increase and shrink in cyclical stress situations.
Having defined the physical behavior of a single beam we must describe the behavior of a network of connected beams. The topology of the network is encoded in a graph
\(\mathcal {G}\) , consisting of a finite set of vertices
\(\mathcal {V}\subset \mathbb {R}^3\) , which are given with respect to an inertial global basis
\(\{e_1, e_2, e_3\}\) , and a finite set of edges
\(\mathcal {E}\subset \mathcal {V}\times \mathcal {V}\) . Single edges
\(b\in \mathcal {E}\) are now interpreted as beams with the corresponding physical behavior. In particular, beam
\(b= (v, w)\) connects the different vertices
\(v\) ,
\(w\) , has length
\(L_b= \Vert w- v\Vert\) and fulfills the differential systems (
21 ) and depending on the choice of constitutive behavior either (
22 ) (linear elastic) or (
27 ), (
31 ), (
32 ) (creep-damage-law) with corresponding sets of unknowns
\((u_b, \phi _b)\) (TS),
\((u_b, \varLambda _b)\) (GE) and material matrices
\(C_{N_b}, C_{M_b}\) all given in the local beam basis. This local basis is defined as follows:
$${E_1}_b:= \frac{w- v}{L_b}, \,\, {E_j}_b\cdot {E_1}_b= 0,\,j=2,3 \,\,\wedge \,\, T_b:= \left( {E_1}_b, {E_2}_b, {E_3}_b\right ) \in SO(3).$$
(33)
Using the transformation matrix
\(T_b\) we can express the local unknowns in the global basis and formulate proper transmission conditions for the junctions. We first define
\(\mathcal {M}_v:= \{ b\in \mathcal {E}: v\in b\}\) as the set of beams connected to node
\(v\) and
$$\begin{aligned} x_b^v:= \left\{ \begin{array}{ll} 0, &{} {\text {if}}\,{\text {beam }}\,b\,\,{\text { starts}}\,{\text {in}}\,{\text {node }}\,v\\ L_b, &{}{\text {if}}\,{\text {beam} }\,b\,\,{\text { ends}}\,{\text {in}}\,{\text {node}}\, v \end{array}\right. \quad\quad n_b^v:= \left\{ \begin{array}{ll} -1, &{}{\text {if }}\quad x_b^v= 0\\ 1, &{}{\text {if }} \quad x_b^v= L_b\end{array}\right. \end{aligned}$$
(34)
We assume that pipe sections between the vertices are connected rigidly, since they are connected by welding or the vertex is due to discretization or refinement. We must therefore ensure continuity of displacement and rotation over the joints
$$\begin{aligned} \left. \begin{array}{llcr}{ \text {TS:}} &{} T_bu_b(x_b^v) = T_{\bar{b}} u_{\bar{b}}(x_{\bar{b}}^v) &{} \wedge &{} T_b\phi _b(x_b^v) = T_{\bar{b}} \phi _{\bar{b}}(x_{\bar{b}}^v)\\{ \text {GE:}} &{} T_bu_b(x_b^v) = T_{\bar{b}} u_{\bar{b}}(x_{\bar{b}}^v) &{}\wedge &{} T_b\varLambda _b(x_b^v) T_b^T = T_{\bar{b}} \varLambda _{\bar{b}}(x_{\bar{b}}^v) T_{\bar{b}}^T \end{array}\right\} \quad \forall b, \bar{b}\in \mathcal {M}_v\end{aligned}$$
(35)
as well as balance of forces and moments at nodes that are not fixed by Dirichlet boundary conditions, see Langnese et al. (
1994 ):
$$\begin{aligned} \sum _{b\in \mathcal {M}_v} n_b^vT_bN_{\mathrm {el},b}(x_b^v) = n_v\,\,\wedge \,\, \sum _{b\in \mathcal {M}_v} n_b^vT_bM_{\mathrm {el},b}(x_b^v) = m_v, \end{aligned}$$
(36)
with
\(n_v, m_v\) being nodal loads at
\(v\) . Note that conditions (
36 ) are the same for TS and GE beams, just the respective constitutive law (
22 ) has to be used. Furthermore they contain the Neumann boundary conditions as a special case if a single beam is connected to node
\(v\) . The remaining nodes are of Dirichlet type:
$$\begin{aligned} \left. \begin{array}{llcr} {\text {TS:}} &{} T_bu_b(x_b^v) = u_{D}^v&{}\wedge &{} T_b\phi _i(x_b^v) = \phi _{D}^v\\{ \text {GE:}} &{} T_bu_b(x_b^v) = u_{D}^v&{}\wedge &{} T_b\varLambda _i(x_b^v) T_b^T = \varLambda _{D}^v\end{array}\,\,\right\} \quad \forall b\in \mathcal {M}_v. \end{aligned}$$
(37)
Now the whole piping system can be modeled as a beam network with proper constitutive laws and coupling conditions at multiple nodes. The final missing part is the fixation of the pipes on a rigid supporting structure. Three types of hangers are considered in the static case:
Rigid hangers: These are stiff connections to the supporting structure. They are either directly modeled as homogeneous Dirichlet condition for the component of the displacement that is in the direction of the hanger at the relevant node or with beams that have stiff behavior. In the latter case, one end of the rigid hanger is connected to the piping system with the Kirchhoff conditions (36
Constant hangers: These are designed to exert a constant force at a certain point. Again there are two ways to model these hangers. The obvious way is to introduce a dead constant load, i.e. a load that does not depend on the state of the system, as the right-hand side
\(n_v\) of the balance of forces; we used this approach for the optimization problems described in Sects.
3 and
4 . The second way is to define a follower force that changes its direction depending on the movements of the system but not its magnitude:
$$n_v= F_v\frac{v+ u_v- h_v}{\Vert v+ u_v- h_v\Vert }.$$
(38)
Here the hanger connects at node
\(v\) to the piping system with force magnitude
\(F_v\) and is fixed on the suspension structure at
\(h_v\) .
\(u_v\) denotes in slight abuse of notation the displacement of the structure at node
\(v\) in global coordinates.
Spring hangers: These are, as the name suggests, springs that exert a force on the pipes that is proportional to the change of length. They are modeled as trusses which act exactly as springs if a linear elastic material law is used. Again we consider two levels of accuracy: linear and nonlinear trusses with the corresponding differential equations
$$\begin{array}{ll} {\text {linear}}\,{\text {truss}} & {\text {nonlinear}}\,{\text {truss}}\\ \quad 0 = N_{\mathrm {el},1}' + \bar{N}_1 &\quad 0 = \frac{E_1 + u'}{\Vert E_1 + u'\Vert }N_{\mathrm {el},1}' + \bar{N}\\ \quad N_{\mathrm {el},1} = EA_cu_1' & \quad N_{\mathrm {el},1} =EA_c \frac{1}{2} \left(\Vert E_1 + u'\Vert - 1\right) \end{array}$$
(39)
The hangers are coupled to the piping system only via balance of forces at the connecting node since they do not transport torques. If the linear spring hangers are exposed to a distributed load
\(\bar{N}\) that is not purely axial then the resulting rigid body acceleration must be taken into account and therefore the balance of forces must be modified by artificial nodal accelerating forces:
$$\varDelta n_{v} = \varDelta n_{w} = \frac{1}{2}L_b \left(\begin{array}{l} 0\\ \bar{N}_2\\ \bar{N}_3 \end{array}\right),$$
(40)
assuming that
\(\bar{N}_{2,3}\) are constant along the truss element.
In summary, in this section we have described how a hot piping system can be interpreted as a graph whose edges have a defined physical behavior which includes beam kinematics and constitutive laws and are coupled by Kirchhoff-type conditions. The given differential systems allow either a linear elastic or a geometrically and material nonlinear analysis of general piping systems with several types of fixations used in real applications.
