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1 Introduction

1.1 Plastic Analysis of Structures

Many materials, e.g., most of metals, have distinct, plastic properties, i.e., they are ductile, see, e.g., [10, 23]. Even after the stress intensity attains the yield point stress, such materials can deform considerably without breaking. This implies that if the stress intensity at a certain point of a hyperstatic structure reaches the critical (yield) value, the structure does not necessarily fail or deform excessively. Instead, a certain amount of stress redistribution takes place and some further load increments can be supported. Structural failure does not occur before a kinematic mechanism of unconstrained plastic flow develops. Thus, the actual load-carrying capacity of a structure is higher (in some cases quite considerably) than that derived from classical elastic analysis. A crucial question for the engineer designing structures like buildings, bridges, etc., or structural components is to which extent a plastic deformation is permissible without leading to a failure of the structure, the component, resp., with respect to the expected load and material strength conditions. Applying standard methods, the load carrying capacity is determined using a certain code with general rules for safety evaluations. The use of such general rules may be very expensive in the safety evaluation and design of a structure. On the other hand, safety assessment and design based on stochastic optimization techniques, taking into account the available knowledge about random parameter variations, reduce the expected total project costs (primary costs, e.g., costs of construction, plus recourse cots, e.g., strengthening costs) considerably. Consequently, this way one obtains more robust (safe) information about the maximum load factors, hence, the carrying capacity, as well as about robust optimal designs. A further big advantage of stochastic optimization methods is the possibility of updates of the maximum load factors and robust optimal designs based on inspection, sampling and other posterior information about the probability distribution of the random parameters. For elastic-perfectly plastic materials, the ultimate load condition corresponding to complete collapse of the structure can be obtained through application of a pair of dual theorems [14, 18, 23]:

  1. (ST)

    Static Theorem (lower bound or safe theorem) If any stress distribution throughout the structure can be found which is everywhere in equilibrium internally and balances the external loads, and at the same time does not violate the yield condition, those external loads will be carried safely by the structure.

  2. (KT)

    Kinematic Theorem (upper bound or unsafe theorem)

    Collapse occurs if a collapse mechanism, fulfilling the compatibility condition, exists such that the work done by the external loads is larger than the corresponding internal plastic work.

    Limit analysis is concerned [5, 8, 10, 14, 16, 18, 19, 22, 24, 33, 34, 36, 37, 40, 41]

    with establishing the strength of a structure, i.e., its capacity for the supporting of loads. Hence, using the plastic ductility of structural materials in improving the design of structures, limit analysis is not concerned with deformation: it can not therefore provide the load carrying capacity for a structure with elements that have a limited ductility or deformability, nor for a structure which becomes unstable because of the displacements induced by plastic deformation, see [10, 13, 18, 24].

1.2 Limit (Collapse) Load Analysis of Structuresas a Linear Programming Problem

Assuming that the material behaves as an elastic-perfectly plastic material [17, 23, 32] a conservative estimate of the collapse load factor λ T is based [5–8, 10, 13, 14, 22, 33, 40] on the following linear program:

$$\mbox{ maximize }\lambda $$
(8.1a)

s.t.

$$\begin{array}{rcl}{ F}^{L}& \leq & F \leq {F}^{U}\end{array}$$
(8.1b)
$$\begin{array}{rcl} CF& =& \lambda {R}_{0}.\end{array}$$
(8.1c)

Here, (8.1c) is the equilibrium equation of a statically indeterminate loaded structure involving an m ×n matrix C = (c ij ), m < n, of given coefficients c ij , 1 ≤ i ≤ m, 1 ≤ j ≤ n, depending on the undeformed geometry of the structure having n 0 members (elements). After taking into account the given support (boundary) conditions, we may suppose that rank C = m. Furthermore, R 0 is an external load m-vector, and F denotes the n-vector of internal forces and bending-moments in the relevant points (sections, nodes or elements) of lower and upper bounds F L, F U.

For a plane or spatial truss [25, 38] we have that n = n 0, the matrix C contains the direction cosines of the members, and F involves only the normal (axial) forces moreover,

$${F}_{j}^{L} := {\sigma }_{ yj}^{L}{A}_{ j},{F}_{j}^{U} := {\sigma }_{ yj}^{U}{A}_{ j},j = 1,\ldots ,n(= {n}_{0}),$$
(8.2)

where A j is the (given) cross-sectional area, and σ yj L, σ yj U, respectively, denotes the yield stress in compression (negative values) and tension (positive values) of the j-th member of the truss. In case of a plane frame, F is composed of subvectors [38],

$${ F}^{(k)} = \left (\begin{array}{l} {F}_{1}^{(k)} \\ {F}_{2}^{(k)} \\ {F}_{3}^{(k)} \end{array} \right ) = \left (\begin{array}{l} {t}_{k} \\ {m}_{k}^{+} \\ {m}_{k}^{-} \end{array} \right )\!,$$
(8.3a)

where F 1 (k) = t k denotes the normal (axial) force, and \({F}_{2}^{(k)} = {m}_{k}^{+},{F}_{3}^{(k)} = {m}_{k}^{-}\) are the bending-moments at the positive (“right”), negative (“left”) end of the k-th member with respect to a certain chosen orientation of the members. In this case F L, F U contain – for each member k – the subvectors

$${ F}^{(k)L} = \left (\begin{array}{c} {\sigma }_{yk}^{L}{A}_{k} \\ - {M}_{kpl} \\ - {M}_{kpl} \end{array} \right ),{F}^{(k)U} = \left (\begin{array}{c} {\sigma }_{yk}^{U}{A}_{k} \\ {M}_{kpl} \\ {M}_{kpl} \end{array} \right ),$$
(8.3b)

resp., where M kpl , k = 1, …, n 0, denotes the plastic moments (moment capacities) [17, 32] given by

$${M}_{kpl} = {\sigma }_{yk}^{U}{W}_{ kpl},$$
(8.3c)

and W kpl = W kpl (A k ) is the plastic section modulus of the cross-section of the k-th member (beam) with respect to the local z-axis. In order to omit instabilities, such as buckling, σ yk L can be selected by

$${\sigma }_{yk}^{L} := -{\kappa }_{ k}{\sigma }_{yk}^{U}$$
(8.3d)

with a certain reduction factor κ k .

For a space frame [25, 38], corresponding to the k-th member (beam), F contains the subvector

$${F}^{(k)} := {({t}_{ k},{m}_{kT},{m}_{k\bar{y}}^{+},{m}_{ k\bar{z}}^{+},{m}_{ k\bar{y}}^{-},{m}_{ k\bar{z}}^{-})}^{T},$$
(8.4a)

where t k is the normal (axial) force, m kT the twisting moment, and \({m}_{k\bar{y}}^{+},{m}_{k\bar{z}}^{+}\), \({m}_{k\bar{y}}^{-},{m}_{k\bar{z}}^{-}\) denote four bending moments with respect to the local \(\bar{y},z\)-axis at the positive, negative end of the beam, respectively. Finally, the bounds F L, F U for F are given by

$$\begin{array}{rcl}{ F}^{(k)L}& =& \big{(}{\sigma }_{ yk}^{L}{A}_{ k},-{M}_{kpl}^{\bar{p}},-{M}_{ kpl}^{\bar{y}},-{M}_{ kpl}^{\bar{z}},-{M}_{ kpl}^{\bar{y}},-{M}_{ kpl}^{z}{\big{)}}^{T}\ \end{array}$$
(8.4b)
$$\begin{array}{rcl}{ F}^{(k)U}& =& \big{(}{\sigma }_{ yk}^{U}{A}_{ k},{M}_{kpl}^{\bar{p}},{M}_{ kpl}^{\bar{y}},{M}_{ kpl}^{\bar{z}},{M}_{ kpl}^{\bar{y}},{M}_{ kpl}^{\bar{z}}{\big{)}}^{T},\end{array}$$
(8.4c)

where [17, 32]

$${M}_{kpl}^{\bar{p}} := {\tau }_{ yk}{W}_{kpl}^{\bar{p}},{M}_{ kpl}^{\bar{y}} := {\sigma }_{ yk}^{U}{W}_{ kpl}^{\bar{y}},{M}_{ kpl}^{\bar{z}} := {\sigma }_{ yk}^{U}{W}_{ kpl}^{\bar{z}},$$
(8.4d)

are the plastic moments of the cross-section of the k-th element with respect to the local twisting axis, the local \(\bar{y}-,\bar{z}\)-axis, respectively. In (8.4d), \({W}_{kpl}^{\bar{p}} = {W}_{kpl}^{\bar{p}}(x)\) and \({W}_{kpl}^{\bar{y}} = {W}_{kpl}^{\bar{y}}(x),{W}_{kpl}^{\bar{z}} = {W}_{kpl}^{\bar{z}}(x)\), resp., denote the polar, axial modulus of the cross-sectional area of the k-th beam and τ yk denotes the yield stress with respect to torsion; we suppose that τ yk = κτk σ yk U with a reduction factor κτk . Moreover, x denotes the r-vector of design variables, see Sect. 8.1.1 for more details.

Remark 8.1.

Possible plastic hinges [17, 24, 32] are taken into account by inserting appropriate eccentricities e kl > 0, e kr > 0, k = 1, …, n 0, with e kl , e kr < < L k , where L k is the length of the k-th beam.

Remark 8.2.

Working with more general yield polygons [1, 40, 42], the stress condition (8.1b) is replaced by the more general system of inequalities

$$H{({F}_{d}^{U})}^{-1}F \leq h.$$
(8.5a)

Here, (H, h) is a given ν ×(n + 1) matrix, and F d U : = (F j Uδ ij ) denotes the n ×n diagonal matrix of principal axial and bending plastic capacities

$${F}_{j}^{U} := {\sigma }_{ y{k}_{j}}^{U}{A}_{ kj},{F}_{j}^{U} := {\sigma }_{ y{k}_{j}}^{U}{W}_{{ k}_{j}pl}^{\kappa j},$$
(8.5b)

where kj, κj are indices as arising in (8.3b)–(8.4d). The more general case (8.5a) can be treated by similar methods as the case (8.1b) which is considered here.

1.3 Plastic and Elastic Design of Structures

In the plastic design of trusses and frames [22, 26, 27, 29, 34, 36] having n 0 members, the n-vectors F L, F U of lower and upper bounds

$${F}^{L} = {F}^{L}({\sigma }_{ y}^{L},{\sigma }_{ y}^{U},x),{F}^{U} = {F}^{U}({\sigma }_{ y}^{L},{\sigma }_{ y}^{U},x),$$
(8.6)

for the n-vector F of internal member forces and bending moments F j , j = 1, …, n, are determined [13, 22] by the yield stresses, i.e., compressive limiting stresses (negative values) \({\sigma }_{y}^{L} = \big{(}{\sigma }_{y1}^{L},\ldots ,{\sigma }_{y{n}_{0}}^{L}{\big{)}}^{T}\), the tensile yield stresses \({\sigma }_{y}^{U} = \big{(}{\sigma }_{y1}^{U},\ldots ,{\sigma }_{y{n}_{0}}^{U}{\big{)}}^{T}\), and the r-vector

$$x = {({x}_{1},{x}_{2},\ldots ,{x}_{r})}^{T}$$
(8.7)

of design variables of the structure. In case of trusses we have that, cf. (8.2),

$$\begin{array}{rcl} {F}^{L}& =& {\sigma }_{ yd}^{L}A(x) = A{(x)}_{ d}{\sigma }_{y}^{L}, \\ {F}^{U}& =& {\sigma }_{ yd}^{U}A(x) = A{(x)}_{ d}{\sigma }_{y}^{U},\end{array}$$
(8.8)

where n = n 0, and σ yd L, σ yd U denote the n ×n diagonal matrices having the diagonal elements σ yj L, σ yj U, respectively, j = 1, …, n, moreover,

$$A(x) = \Big{[}{A}_{1}(x),\ldots ,{A}_{n}(x){\Big{]}}^{T}$$
(8.9)

is the n-vector of cross-sectional area \({A}_{j} = {A}_{j}(x),j = 1,\ldots ,n\), depending on the r-vector x of design variables x κ, κ = 1, …, r, and A(x) d denotes the n ×n diagonal matrix having the diagonal elements A j = A j (x), 1 < j < n.

Corresponding to (8.1c), here the equilibrium equation reads

$$CF = {R}_{u},$$
(8.10)

where R u describes [22] the ultimate load [representing constant external loads or self-weight expressed in linear terms of A(x)].

The plastic design of structures can be represented then [1, 2] by the optimization problem

$$\min G(x),$$
(8.11a)

s.t.

$$\begin{array}{rcl} & & {F}^{L}({\sigma }_{ y}^{L},{\sigma }_{ y}^{U},x) \leq F \leq {F}^{U}({\sigma }_{ y}^{L},{\sigma }_{ y}^{U},x)\end{array}$$
(8.11b)
$$\begin{array}{rcl} & & CF = {R}_{u}\end{array}$$
(8.11c)
$$\begin{array}{rcl} & & \quad x \in D,\end{array}$$
(8.11d)

where G = G(x) is a certain objective function, e.g., the volume or weight of the structure, and C ⊂ IR + denotes the convex set of admissible design vecotrs x.

Remark 8.3.

As mentioned in Remark 8.2, working with more general yield polygons, (8.11b) is replaced by the condition

$$H{[{F}^{U}{({\sigma }_{ y}^{U},x)}_{ d}]}^{-1}F \leq h.$$
(8.11e)

For the elastic design we must replace the yield stresses σ y L, σ y U by the allowable stresses σ a L, σ a U and instead of ultimate loads we consider service loads R s . Hence, instead of (8.11a–d) we have the related program

$$\min G(x),$$
(8.12a)

s.t.

$$\begin{array}{rcl} & & {F}^{L}({\sigma }_{ a}^{L},{\sigma }_{ a}^{U},x) \leq F \leq {F}^{U}({\sigma }_{ a}^{L},{\sigma }_{ a}^{U},x),\end{array}$$
(8.12b)
$$\begin{array}{rcl} & & CF = {R}_{s},\end{array}$$
(8.12c)
$$\begin{array}{rcl} & & {x}^{L} \leq x \leq {x}^{U},\end{array}$$
(8.12d)

where x L, x U still denote lower and upper bounds for x.

2 Plane Frames

For each bar \(i = 1,\ldots ,B\) of a plane frame with member load vector \({F}_{i} = {({t}_{i},{m}_{i}^{+},{m}_{i}^{-})}^{T}\) we consider [37, 41] the load at the negative, positive end

$${F}_{i}^{-} := {({t}_{ i},{m}_{i}^{-})}^{T},{F}_{ i}^{+} := {({t}_{ i},{m}_{i}^{+})}^{T},$$
(8.13)

respectively.

Furthermore, for each bar/beam with rigid joints we have several plasticcapacities: The plastic capacity N ipl L of the bar with respect to axial compression, hence, the maximum axial force under compression is given by

$${N}_{ipl}^{L} = \vert {\sigma }_{ yi}^{L}\vert \cdot {A}_{ i},$$
(8.14a)

where σ yi L < 0 denotes the (negative) yield stress with respect to compression and A i is the cross sectional area of the ith element. Correspondingly, the plastic capacity with respect to (axial) tension reads:

$${N}_{ipl}^{U} = {\sigma }_{ yi}^{U} \cdot {A}_{ i},$$
(8.14b)

where σ yi U > 0 is the yield stress with respect to tension. Besides the plastic capacities with respect to the normal force, we have the moment capacity

$${M}_{ipl} = {\sigma }_{yi}^{U} \cdot {W}_{ ipl}$$
(8.14c)

with respect to the bending moments at the ends of the bar i.

Remark 8.4.

Note that all plastic capacities have nonnegative values.

Using the plastic capacities (8.14a–c), the load vectors \({F}_{i}^{+},{F}_{i}^{+}\) given by (8.13) can be replaced by dimensionless quantities

$$\begin{array}{rcl} \hspace{34.14322pt} {F}_{i}^{L-}& :=&{ \left ( \frac{{t}_{i}} {{N}_{ipl}^{L}}, \frac{{m}_{i}^{-}} {{M}_{ipl}}\right )}^{T},{F}_{ i}^{U-} :={ \left ( \frac{{t}_{i}} {{N}_{ipl}^{U}}, \frac{{m}_{i}^{-}} {{M}_{ipl}}\right )}^{T}\mbox{ (8.15a,b)} \\ {F}_{i}^{L+}& :=&{ \left ( \frac{{t}_{i}} {{N}_{ipl}^{L}}, \frac{{m}_{i}^{+}} {{M}_{ipl}}\right )}^{T},{F}_{ i}^{U+} :={ \left ( \frac{{t}_{i}} {{N}_{ipl}^{U}}, \frac{{m}_{i}^{+}} {{M}_{ipl}}\right )}^{T}\hspace{38.41139pt} \mbox{ (8.15c,d)}\\ \end{array}$$

for the negative, positive end, resp., of the ith bar.

Remark 8.5 (Symmetric yield stresses under compression and tension).

In the important special case that the absolute values of the yield stresses under compression ( < 0) and tension ( > 0) are equal, hence,

$$\begin{array}{rcl}{ \sigma }_{yi}^{L}& =& -{\sigma }_{ yi}^{U}\end{array}$$
(8.16a)
$$\begin{array}{rcl}{ N}_{ipl}^{L}& =& {N}_{ ipl}^{U} =: {N}_{ ipl}.\end{array}$$
(8.16b)

The limit between the elastic and plastic state of the elements is described by the feasibility or yield condition: At the negative end we have the condition

$$\hspace{119.50148pt} {F}_{i}^{L-}\in {K}_{ i},{F}_{i}^{U-}\in {K}_{ i}\mbox{ (8.17a,b)}$$

and at the positive end the condition reads

$$\hspace{119.50148pt} {F}_{i}^{L+} \in {K}_{ i},{F}_{i}^{U+} \in {K}_{ i}.\mbox{ (8.17c,d)}$$

Here, K i , K i ⊂ IR 2, denotes the feasible domain of bar “i” having the following properties:

  • K i is a closed, convex subset of IR 2.

  • The origin 0 of IR 2 is an interior point of K i .

  • The interior K̊ i of K i represents the elastic states.

  • At the boundary ∂K i yielding of the material starts.

Considering, e.g., bars with rectangular cross sectional areas and symmetric yield stresses, cf. Remark 8.5, K i is given by K i = K 0, sym , where [19, 21]

$${K}_{0,\mbox{ sym}} =\{ {(x,y)}^{T} : {x}^{2} + \vert y\vert \leq 1\}$$
(8.18)

where \(x = \frac{N} {{N}_{pl}}\) and \(y = \frac{M} {{M}_{pl}}\), (see Fig. 8.1). Note that the symbols x, y, (x, y), resp., denote in this Sect. 8.2 just real variables, a point in the real plane.

Fig. 8.1
figure 1_8

Domain K 0, sym with possible approximations

Full size image

In case (8.18) and supposing symmetric yield stresses, the yield condition (8.17a–d) reads

$$\begin{array}{rcl}{ \left ( \frac{{t}_{i}} {{N}_{ipl}}\right )}^{2} + \left \vert \frac{{m}_{i}^{-}} {{M}_{ipl}}\right \vert & \leq & 1,\end{array}$$
(8.19a)
$$\begin{array}{rcl}{ \left ( \frac{{t}_{i}} {{N}_{ipl}}\right )}^{2} + \left \vert \frac{{m}_{i}^{+}} {{M}_{ipl}}\right \vert & \leq & 1.\end{array}$$
(8.19b)

Remark 8.6.

Because of the connection between the normal force t i and the bending moments, (8.19a,b) is also called “M–N-interaction”.

If the M–N-interaction is not taken into account, K 0, sym is approximated from outside, see Fig. 8.1, by

$${K}_{0,\mbox{ sym}}^{u} :=\{ {(x,y)}^{T} : \vert x\vert ,\vert y\vert \leq 1\}.$$
(8.20)

Hence, (8.17a–d) are replaced, cf. (8.19a,b), by the simpler conditions

$$\begin{array}{rcl} \vert {t}_{i}\vert & \leq & {N}_{ipl}\end{array}$$
(8.21a)
$$\begin{array}{rcl} \vert {m}_{i}^{-}\vert & \leq & {M}_{ ipl}\end{array}$$
(8.21b)
$$\begin{array}{rcl} \vert {m}_{i}^{+}\vert & \leq & {M}_{ ipl}.\end{array}$$
(8.21c)

Since the symmetry condition (8.16a) does not hold in general, some modifications of the basic conditions (8.19a,b) are needed. In the non symmetric case K 0, xm must be replaced by the intersection

$${K}_{0} = {K}_{0}^{U} \cap {K}_{ 0}^{L}$$
(8.22)

of two convex sets K 0 U and K 0 L. For a rectangular cross-sectional area we have

$${K}_{0}^{U} = \left \{{(x,y)}^{T} :\ x \leq \sqrt{1 - \vert y\vert },\ \vert y\vert \leq 1\right \}$$
(8.23a)

for tension and

$${K}_{0}^{L} = \left \{{(x,y)}^{T} : -x \leq \sqrt{1 - \vert y\vert },\ \vert y\vert \leq 1\right \}$$
(8.23b)

compression, where again \(x = \frac{N} {{N}_{pl}}\) and \(y = \frac{M} {{M}_{pl}}\) (see Fig. 8.2).

Fig. 8.2
figure 2_8

Feasible domain as intersection of K 0 U and K 0 L

Full size image

In case of tension, see (8.17b,d), from (8.23a) we obtain then the feasibility condition

$$\begin{array}{rcl} \frac{{t}_{i}} {{N}_{ipl}^{U}}& \leq & \sqrt{1 - \left \vert \frac{{m}_{i }^{- }} {{M}_{ipl}}\right \vert }\end{array}$$
(8.24a)
$$\begin{array}{rcl} \frac{{t}_{i}} {{N}_{ipl}^{U}}& \leq & \sqrt{1 - \left \vert \frac{{m}_{i }^{+ }} {{M}_{ipl}}\right \vert }\end{array}$$
(8.24b)
$$\begin{array}{rcl} \left \vert \frac{{m}_{i}^{-}} {{M}_{ipl}}\right \vert & \leq & 1\end{array}$$
(8.24c)
$$\begin{array}{rcl} \left \vert \frac{{m}_{i}^{+}} {{M}_{ipl}}\right \vert & \leq & 1.\end{array}$$
(8.24d)

For compression, with (8.17a,c) and (8.23b) we get the feasibility condition

$$\begin{array}{rcl} - \frac{{t}_{i}} {{N}_{ipl}^{L}}& \leq & \sqrt{1 - \left \vert \frac{{m}_{i }^{- }} {{M}_{ipl}}\right \vert }\end{array}$$
(8.24e)
$$\begin{array}{rcl} - \frac{{t}_{i}} {{N}_{ipl}^{L}}& \leq & \sqrt{1 - \left \vert \frac{{m}_{i }^{+ }} {{M}_{ipl}}\right \vert }\end{array}$$
(8.24f)
$$\begin{array}{rcl} \left \vert \frac{{m}_{i}^{-}} {{M}_{ipl}}\right \vert & \leq & 1\end{array}$$
(8.24g)
$$\begin{array}{rcl} \left \vert \frac{{m}_{i}^{+}} {{M}_{ipl}}\right \vert & \leq & 1.\end{array}$$
(8.24h)

From (8.24a), (8.24e) we get

$$-{N}_{ipl}^{L}\sqrt{1 - \left \vert \frac{{m}_{i }^{- }} {{M}_{ipl}}\right \vert }\leq {t}_{i} \leq {N}_{ipl}^{U}\sqrt{1 - \left \vert \frac{{m}_{i }^{- }} {{M}_{ipl}}\right \vert }.$$
(8.25a)

and (8.24a), (8.24e) yield

$$-{N}_{ipl}^{L}\sqrt{1 - \left \vert \frac{{m}_{i }^{+ }} {{M}_{ipl}}\right \vert }\leq {t}_{i} \leq {N}_{ipl}^{U}\sqrt{1 - \left \vert \frac{{m}_{i }^{+ }} {{M}_{ipl}}\right \vert }.$$
(8.25b)

Furthermore, (8.24b), (8.24f) and (8.24c), (8.24g) yield

$$\begin{array}{rcl} \vert {m}_{i}^{-}\vert & \leq & {M}_{ ipl}\end{array}$$
(8.25c)
$$\begin{array}{rcl} \vert {m}_{i}^{+}\vert & \leq & {M}_{ ipl}.\end{array}$$
(8.25d)

For computational purposes, piecewise linearizations are applied [30] to the nonlinear conditions (8.25a,b), see also [3, 15]. A basic approximation of K 0 L and K 0 U is given by

$${K}_{0}^{Uu} :=\{ {(x,y)}^{T} : x \leq 1,\vert y\vert \leq 1\} =\;]\infty ,1] \times [-1,1]$$
(8.26a)

and

$${K}_{0}^{Lu} :=\{ {(x,y)}^{T} : x \geq -1,\vert y\vert \leq 1\} =\; [-1,\infty ] \times [-1,1]$$
(8.26b)

with \(x = \frac{N} {{N}_{pl}}\) and \(y = \frac{M} {{M}_{pl}}\) (see Fig. 8.3).

Fig. 8.3
figure 3_8

Approximation of K 0 by K 0 Lu and K 0 Uu

Full size image

Since in this approximation the M–N-interaction is not taken into account, condition (8.17a–d) is reduced to

$$\begin{array}{rcl} - {N}_{ipl}^{L}& \leq & {t}_{ i} \leq {N}_{ipl}^{U}\end{array}$$
(8.27a)
$$\begin{array}{rcl} \vert {m}_{i}^{-}\vert & \leq & {M}_{ ipl}\end{array}$$
(8.27b)
$$\begin{array}{rcl} \vert {m}_{i}^{+}\vert & \leq & {M}_{ ipl}.\end{array}$$
(8.27c)

2.1 Yield Condition in Case of M − N-Interaction

2.1.1 Symmetric Yield Stresses

Consider first the case

$${\sigma }_{yi}^{U} = -{\sigma }_{ yi}^{L} =: {\sigma }_{ yi},\ i = 1,\ldots ,B.$$
(8.28a)

Then,

$${N}_{ipl}^{L} := \vert {\sigma }_{ yi}^{L}\vert {A}_{ i} = {\sigma }_{yi}^{U}{A}_{ i} =: {N}_{ipl}^{U},$$
(8.28b)

hence,

$${N}_{ipl} := {N}_{ipl}^{L} = {N}_{ ipl}^{U} = {\sigma }_{ yi}{A}_{i}.$$
(8.28c)

Moreover,

$${M}_{ipl} = {\sigma }_{yi}^{U}{W}_{ ipl} = {\sigma }_{yi}{W}_{ipl} = {\sigma }_{yi}{A}_{i}\bar{{y}}_{ic},\ i = 1,\ldots ,B,$$
(8.28d)

where \({\overline{y}}_{ic}\) denotes the arithmetic mean of the centroids of the two half areas of the cross-sectional area A i of bar i.

Depending on the geometric from of the cross-sectional areal (rectangle, circle, etc.), for the element load vectors

$${ F}_{i} = \left (\begin{array}{c} {t}_{i} \\ {m}_{i}^{+} \\ {m}_{i}^{-} \end{array} \right ),i = 1,\ldots ,B,$$
(8.29)

of the bars we have the yield condition:

$$\begin{array}{rcl}{ \left \vert \frac{{t}_{i}} {{N}_{ipl}}\right \vert }^{\alpha } + \left \vert \frac{{m}_{i}^{-}} {{M}_{ipl}}\right \vert & \leq 1\quad \mbox{ (negative end)}&\end{array}$$
(8.30a)
$$\begin{array}{rcl}{ \left \vert \frac{{t}_{i}} {{N}_{ipl}}\right \vert }^{\alpha } + \left \vert \frac{{m}_{i}^{+}} {{M}_{ipl}}\right \vert & \leq 1\quad \mbox{ (positive end)}.&\end{array}$$
(8.30b)

Here, α > 1 is a constant depending on the type of the cross-sectional area of the ith bar. Defining the convex set

$${K}_{0}^{\alpha } := \left \{\left(\begin{array}{r}x\\ y\end{array}\right) \in {\mathsf{I\hspace{-1.49994pt}R}}^{2} : \vert x{\vert }^{\alpha } + \vert y\vert \leq 1\right \},$$
(8.31)

for (8.30a,b) we have also the representation

$$\begin{array}{rcl} \left (\begin{array}{c} \frac{{t}_{i}} {{N}_{ipl}} \\ \frac{{m}_{i}^{-}} {{M}_{ipl}} \end{array} \right ) \in {K}_{0}^{\alpha }\quad \mbox{ (negative end)}\end{array}$$
(8.32a)
$$\begin{array}{rl}\left (\begin{array}{c} \frac{{t}_{i}} {{N}_{ipl}} \\ \frac{{m}_{i}^{+}} {{M}_{ipl}} \end{array} \right ) \in {K}_{0}^{\alpha }\quad \mbox{ (positive end)}.\end{array}$$
(8.32b)

2.2 Piecewise Linearization of K 0 α

Due to the symmetry of K 0 α with respect to the transformation

$$x \rightarrow -x,y \rightarrow -y,$$

K 0 α is piecewise linearized as follows.

Starting from a boundary point of K 0 α, hence,

$$\left (\begin{array}{c} {u}_{1} \\ {u}_{2}\end{array} \right ) \in \partial {K}_{0}^{\alpha }\quad \mbox{ with}\quad {u}_{ 1} \geq 0,\ {u}_{2} \geq 0,$$
(8.33a)

we consider the gradient of the boundary curve

$$f(x,y) := \vert x{\vert }^{\alpha } + \vert y\vert - 1 = 0$$

of K 0 α in the four points

$$\left (\begin{array}{c} {u}_{1} \\ {u}_{2}\end{array} \right ),\left (\begin{array}{c} - {u}_{1} \\ {u}_{2} \end{array} \right ),\left (\begin{array}{c} - {u}_{1} \\ - {u}_{2} \end{array} \right ),\left (\begin{array}{c} {u}_{1} \\ - {u}_{2} \end{array} \right ).$$
(8.33b)

We have

$$\begin{array}{rcl} \nabla f\left (\begin{array}{c} {u}_{1} \\ {u}_{2}\end{array} \right )& =& \left (\begin{array}{c} \alpha {u}_{1}^{\alpha -1} \\ 1 \end{array} \right )\end{array}$$
(8.34a)
$$\begin{array}{rcl} \nabla f\left (\begin{array}{c} - {u}_{1}\\ {u}_{2 } \end{array} \right )& =& \left (\begin{array}{c} - \alpha {(-(-{u}_{1}))}^{\alpha -1} \\ 1 \end{array} \right ) = \left (\begin{array}{c} - \alpha {u}_{1}^{\alpha -1} \\ 1 \end{array} \right )\end{array}$$
(8.34b)
$$\begin{array}{rcl} \nabla f\left (\begin{array}{c} - {u}_{1} \\ - {u}_{2} \end{array} \right )& =& \left (\begin{array}{c} - \alpha {(-(-{u}_{1}))}^{\alpha -1} \\ - 1 \end{array} \right ) = \left (\begin{array}{c} - \alpha {u}_{1}^{\alpha -1} \\ - 1 \end{array} \right )\end{array}$$
(8.34c)
$$\begin{array}{rcl} \nabla f\left (\begin{array}{c} {u}_{1} \\ - {u}_{2} \end{array} \right )& =& \left (\begin{array}{c} \alpha {u}_{1}^{\alpha -1} \\ - 1 \end{array} \right ),\end{array}$$
(8.34d)

where

$$\begin{array}{rcl} \nabla f\left (\begin{array}{c} - {u}_{1} \\ - {u}_{2} \end{array} \right )& =& -\nabla f\left (\begin{array}{c} {u}_{1} \\ {u}_{2}\end{array} \right )\end{array}$$
(8.35a)
$$\begin{array}{rcl} \nabla f\left (\begin{array}{c} {u}_{1} \\ - {u}_{2} \end{array} \right )& =& -\nabla f\left (\begin{array}{c} - {u}_{1} \\ {u}_{2} \end{array} \right ).\end{array}$$
(8.35b)

Furthermore, in the two special points

$$\left (\begin{array}{c} 0\\ 1 \end{array} \right )\quad \mbox{ and}\quad \left (\begin{array}{c} 0\\ -1 \end{array} \right )$$

of ∂K 0 α we have, cf. (8.34a), (8.34c), resp., the gradients

$$\begin{array}{rcl} \nabla f\left (\begin{array}{c} 0\\ 1 \end{array} \right )& =& \left (\begin{array}{c} 0\\ 1 \end{array} \right )\end{array}$$
(8.36a)
$$\begin{array}{rcl} \nabla f\left (\begin{array}{c} 0\\ -1 \end{array} \right )& =& \left (\begin{array}{c} 0\\ -1 \end{array} \right ).\end{array}$$
(8.36b)

Though \(f(x,y) = \vert x{\vert }^{\alpha } + \vert y\vert - 1\) is not differentiable at

$$\left (\begin{array}{c} 1\\ 0 \end{array} \right )\quad \mbox{ and}\quad \left (\begin{array}{c} - 1\\ 0 \end{array} \right ),$$

we define

$$\begin{array}{rcl} \nabla f\left (\begin{array}{c} 1\\ 0 \end{array} \right )\!\!\!\!& := \left (\begin{array}{c} 1\\ 0 \end{array} \right )&\end{array}$$
(8.36c)
$$\begin{array}{rcl} \nabla f\left (\begin{array}{c} - 1\\ 0 \end{array} \right )& := \left (\begin{array}{c} - 1\\ 0 \end{array} \right ).&\end{array}$$
(8.36d)

Using a boundary point \(\left (\begin{array}{c} {u}_{1} \\ {u}_{2}\end{array} \right )\) of K 0 α with u 1, u 2 > 0, the feasible domain K 0 α can be approximated from outside by the convex polyhedron defined as follows.

From the gradients (8.36a–d) we obtain next to the already known conditions(no M–N-interaction):

$$\begin{array}{rcl} \nabla f{\left (\begin{array}{c} 0\\ 1 \end{array} \right )}^{T}\left (\left (\begin{array}{c} x\\ y \end{array} \right ) -\left (\begin{array}{c} 0 \\ 1 \end{array} \right )\right )& =&{ \left (\begin{array}{c} 0\\ 1 \end{array} \right )}^{T}\left (\begin{array}{c} x\\ y - 1 \end{array} \right ) \leq 0 \\ \nabla f{\left (\begin{array}{c} 0\\ -1 \end{array} \right )}^{T}\left (\left (\begin{array}{c} x\\ y \end{array} \right ) -\left (\begin{array}{c} 0 \\ - 1 \end{array} \right )\right )& =&{ \left (\begin{array}{c} 0\\ -1 \end{array} \right )}^{T}\left (\begin{array}{c} x\\ y + 1 \end{array} \right ) \leq 0 \\ \nabla f{\left (\begin{array}{c} 1\\ 0 \end{array} \right )}^{T}\left (\left (\begin{array}{c} x\\ y \end{array} \right ) -\left (\begin{array}{c} 1 \\ 0 \end{array} \right )\right )& =&{ \left (\begin{array}{c} 1\\ 0 \end{array} \right )}^{T}\left (\begin{array}{c} x - 1\\ y \end{array} \right ) \leq 0 \\ \nabla f{\left (\begin{array}{c} - 1\\ 0 \end{array} \right )}^{T}\left (\left (\begin{array}{c} x\\ y \end{array} \right ) -\left (\begin{array}{c} - 1 \\ 0 \end{array} \right )\right )& =&{ \left (\begin{array}{c} - 1\\ 0 \end{array} \right )}^{T}\left (\begin{array}{c} x + 1\\ y \end{array} \right ) \leq \end{array}$$
(0.)

This yields

$$\begin{array}{rcl} y - 1& \leq & 0 \\ -1(y + 1)& \leq & 0 \\ x - 1& \leq & 0 \\ -1(x + 1)& \leq & \end{array}$$
(0)

or

$$\begin{array}{rcl} \vert x\vert & \leq & 1\end{array}$$
(8.37a)
$$\begin{array}{rcl} \vert y\vert & \leq & 1.\end{array}$$
(8.37b)

Moreover, with the gradients (8.34a–d), cf. (8.35a,b), we get the additional conditions

$$\begin{array}{rcl} & & \nabla f{\left (\begin{array}{c} {u}_{1} \\ {u}_{2}\end{array} \right )}^{T}\left (\left (\begin{array}{c} x\\ y \end{array} \right ) -\left (\begin{array}{c} {u}_{1} \\ {u}_{2}\end{array} \right )\right ) \\ & & \qquad ={ \left (\begin{array}{c} \alpha {u}_{1}^{\alpha -1} \\ 1 \end{array} \right )}^{T}\left (\begin{array}{c} x - {u}_{1} \\ y - {u}_{2} \end{array} \right ) \leq 0\mbox{ (1st quadrant)} \end{array}$$
(8.38a)
$$\begin{array}{rcl} & & \nabla f{\left (\begin{array}{c} - {u}_{1} \\ - {u}_{2} \end{array} \right )}^{T}\left (\left (\begin{array}{c} x\\ y \end{array} \right ) -\left (\begin{array}{c} - {u}_{1} \\ - {u}_{2} \end{array} \right )\right ) \\ & & \qquad = -{\left (\begin{array}{c} \alpha {u}_{1}^{\alpha -1} \\ 1 \end{array} \right )}^{T}\left (\begin{array}{c} x + {u}_{1} \\ y + {u}_{2} \end{array} \right ) \leq 0\qquad \mbox{ (3rd quadrant)} \end{array}$$
(8.38b)
$$\begin{array}{rcl} & & \nabla f{\left (\begin{array}{c} - {u}_{1}\\ {u}_{2 } \end{array} \right )}^{T}\left (\left (\begin{array}{c} x\\ y \end{array} \right ) -\left (\begin{array}{c} - {u}_{1} \\ {u}_{2} \end{array} \right )\right ) \\ & & \qquad ={ \left (\begin{array}{c} - \alpha {u}_{1}^{\alpha -1} \\ 1 \end{array} \right )}^{T}\left (\begin{array}{c} x + {u}_{1} \\ y - {u}_{2} \end{array} \right ) \leq 0\qquad \mbox{ (2nd quadrant)} \end{array}$$
(8.38c)
$$\begin{array}{rcl} & & \nabla f{\left (\begin{array}{c} {u}_{1} \\ - {u}_{2} \end{array} \right )}^{T}\left (\left (\begin{array}{c} x\\ y \end{array} \right ) -\left (\begin{array}{c} {u}_{1} \\ - {u}_{2} \end{array} \right )\right ) \\ & & \qquad = -{\left (\begin{array}{c} - \alpha {u}_{1}^{\alpha -1} \\ 1 \end{array} \right )}^{T}\left (\begin{array}{c} x - {u}_{1} \\ y + {u}_{2} \end{array} \right ) \leq 0\quad \mbox{ (4th quadrant).} \end{array}$$
(8.38d)

This means

$$\begin{array}{rcl} \alpha {u}_{1}^{\alpha -1}x - \alpha {u}_{ 1}^{\alpha } + y - {u}_{ 2}& =& \alpha {u}_{1}^{\alpha -1}x + y -\left (\alpha {u}_{ 1}^{\alpha } + {u}_{ 2}\right ) \leq 0\end{array}$$
(8.39a)
$$\begin{array}{rcl} \hspace{-50.00008pt}-\left (\alpha {u}_{1 }^{\alpha -1 }x + \alpha {u}_{ 1}^{\alpha } + y + {u}_{ 2}\right )& =& -\left (\alpha {u}_{1}^{\alpha -1}x + y + \left (\alpha {u}_{ 1}^{\alpha } + {u}_{ 2}\right )-\right )\! \leq \! 0\quad \end{array}$$
(8.39b)
$$\begin{array}{rcl} -\alpha {u}_{1}^{\alpha -1}x - \alpha {u}_{ 1}^{\alpha } + y - {u}_{ 2}& =& -\alpha - {u}_{1}^{\alpha -1}x + y -\left (\alpha {u}_{ 1}^{\alpha } + {u}_{ 2}\right ) \leq 0 -\end{array}$$
(8.39c)
$$\begin{array}{rcl} \alpha {u}_{1}^{\alpha -1}x - \alpha {u}_{ 1}^{\alpha } - y - {u}_{ 2}& =& \alpha - {u}_{1}^{\alpha -1}x - y -\left (\alpha {u}_{ 1}^{\alpha } + {u}_{ 2}\right ) \leq 0.\end{array}$$
(8.39d)

With

$$\alpha {u}_{1}^{\alpha }+{u}_{ 2} = \nabla f{\left (\begin{array}{c} {u}_{1} \\ {u}_{2}\end{array} \right )}^{T}\left (\begin{array}{c} {u}_{1} \\ {u}_{2}\end{array} \right ) =: \beta ({u}_{1},{u}_{2})$$
(8.40)

we get the equivalent constraints

$$\begin{array}{rcl} \alpha {u}_{1}^{\alpha -1}x + y - \beta ({u}_{ 1},{u}_{2})& \leq & 0 \\ \alpha {u}_{1}^{\alpha -1}x + y + \beta ({u}_{ 1},{u}_{2})& \geq & 0 \\ -\alpha {u}_{1}^{\alpha -1}x + y - \beta ({u}_{ 1},{u}_{2})& \leq & 0 \\ \alpha {u}_{1}^{\alpha -1}x - y - \beta ({u}_{ 1},{u}_{2})& \leq & \end{array}$$
(0.)

This yields the double inequalities

$$\begin{array}{rcl} \vert \alpha {u}_{1}^{\alpha -1}x + y\vert & \leq \beta ({u}_{ 1},{u}_{2})&\end{array}$$
(8.41a)
$$\begin{array}{rcl} \vert \alpha {u}_{1}^{\alpha -1}x - y\vert & \leq \beta ({u}_{ 1},{u}_{2}).&\end{array}$$
(8.41b)

Thus, a point \(u = \left (\begin{array}{c} {u}_{1} \\ {u}_{2}\end{array} \right ) \in \partial {K}_{0}^{\alpha },\ {u}_{ 1} > 0,{u}_{2} > 0\), generates therefore the inequalities

$$\begin{array}{rcl} & -1 \leq x \leq 1&\end{array}$$
(8.42a)
$$\begin{array}{rcl} & -1 \leq y \leq 1&\end{array}$$
(8.42b)
$$\begin{array}{rcl} & -\beta ({u}_{1},{u}_{2}) \leq \alpha {u}_{1}^{\alpha -1}x + y \leq \beta ({u}_{1},{u}_{2})&\end{array}$$
(8.42c)
$$\begin{array}{rcl} & -\beta ({u}_{1},{u}_{2}) \leq \alpha {u}_{1}^{\alpha -1}x - y \leq \beta ({u}_{1},{u}_{2}).&\end{array}$$
(8.42d)

Obviously, each further point \(\hat{u} \in \partial {K}_{0}^{\alpha }\) with \(\hat{{u}}_{1} > 0,\hat{{u}}_{2} > 0\) yields additional inequalities of the type (8.42b,d).

Condition (8.42a–d) can be represented in the following vectorial form:

$$\begin{array}{rcl} & -\left (\begin{array}{c} 1\\ 1 \end{array} \right ) \leq I\left (\begin{array}{c} x\\ y \end{array} \right ) \leq \left (\begin{array}{c} 1\\ 1 \end{array} \right )&\end{array}$$
(8.43a)
$$\begin{array}{rcl} & -\beta ({u}_{1},{u}_{2})\left (\begin{array}{c} 1\\ 1 \end{array} \right ) \leq H({u}_{1},{u}_{2})\left (\begin{array}{c} x\\ y \end{array} \right ) \leq \beta ({u}_{1},{u}_{2})\left (\begin{array}{c} 1\\ 1 \end{array} \right )\!,&\end{array}$$
(8.43b)

with the matrices

$$I = \left (\begin{array}{cc} 1&0\\ 0 &1 \end{array} \right ),\ H({u}_{1},{u}_{2}) = \left (\begin{array}{cc} \alpha {u}_{1}^{\alpha -1} & 1 \\ \alpha {u}_{1}^{\alpha -1} & - 1 \end{array} \right )\!.$$
(8.44)

Choosing a further boundary point \(\hat{u}\) of K 0 α with \(\hat{{u}}_{1} > 0,\hat{{u}}_{2} > 0\), we get additional conditions of the type (8.43a).

Using (8.42a–d), for the original yield condition (8.32a,b) we get then the approximative feasibility condition:

  1. 1.

    Negative end of the bar

    $$\begin{array}{rl} - {N}_{ipl} \leq {t}_{i} \leq {N}_{ipl}\end{array}$$
    (8.45a)
    $$\begin{array}{rl}- {M}_{ipl} \leq {m}_{i}^{-}\leq {M}_{ ipl}\end{array}$$
    (8.45b)
    $$\begin{array}{rl} - \beta ({u}_{1},{u}_{2}) \leq \alpha {u}_{1}^{\alpha -1} \frac{{t}_{i}} {{N}_{ipl}} + \frac{{m}_{i}^{-}} {{M}_{ipl}} \leq \beta ({u}_{1},{u}_{2})\end{array}$$
    (8.45c)
    $$\begin{array}{rl}- \beta ({u}_{1},{u}_{2}) \leq \alpha {u}_{1}^{\alpha -1} \frac{{t}_{i}} {{N}_{ipl}} - \frac{{m}_{i}^{-}} {{M}_{ipl}} \leq \beta ({u}_{1},{u}_{2}).\end{array}$$
    (8.45d)

    2. Positive end of the bar

    $$\begin{array}{rcl} - {N}_{ipl}& \leq & {t}_{i} \leq {N}_{ipl} \end{array}$$
    (8.45e)
    $$\begin{array}{rcl} -{M}_{ipl}& \leq & {m}_{i}^{+} \leq {M}_{ ipl} \end{array}$$
    (8.45f)
    $$\begin{array}{rcl} -\beta ({u}_{1},{u}_{2})& \leq & \alpha {u}_{1}^{\alpha -1} \frac{{t}_{i}} {{N}_{ipl}} + \frac{{m}_{i}^{+}} {{M}_{ipl}} \leq \beta ({u}_{1},{u}_{2}) \end{array}$$
    (8.45g)
    $$\begin{array}{rcl} -\beta ({u}_{1},{u}_{2})& \leq & \alpha {u}_{1}^{\alpha -1} \frac{{t}_{i}} {{N}_{ipl}} - \frac{{m}_{i}^{+}} {{M}_{ipl}} \leq \beta ({u}_{1},{u}_{2}). \end{array}$$
    (8.45h)

Defining

$${ \Gamma }^{(i)} := \left (\begin{array}{ccc} \frac{1} {{N}_{ipl}} & 0 & 0 \\ 0 & \frac{1} {{M}_{ipl}} & 0 \\ 0 & 0 & \frac{1} {{M}_{ipl}} \end{array} \right ),\ {F}_{i} = \left (\begin{array}{c} {t}_{i} \\ {m}_{i}^{+} \\ {m}_{i}^{-} \end{array} \right )\!,$$
(8.46)

conditions (8.45a–h) can be represented also by

$$\begin{array}{rcl} -\left (\begin{array}{c} 1\\ 1 \\ 1 \end{array} \right )& \leq & {\Gamma }^{(i)}{F}_{ i} \leq \left (\begin{array}{c} 1\\ 1 \\ 1 \end{array} \right )\end{array}$$
(8.47a)
$$\begin{array}{rcl} -\beta ({u}_{1},{u}_{2})\left (\begin{array}{c} 1\\ 1 \\ 1\\ 1 \end{array} \right )& \leq & \left (\begin{array}{ccc} \alpha {u}_{1}^{\alpha -1} & 1 & 0 \\ \alpha {u}_{1}^{\alpha -1} & - 1& 0 \\ \alpha {u}_{1}^{\alpha -1} & 0 & 1 \\ \alpha {u}_{1}^{\alpha -1} & 0 & - 1 \end{array} \right ){\Gamma }_{i}{F}_{i} \leq \beta ({u}_{1},{u}_{2})\left (\begin{array}{c} 1\\ 1 \\ 1\\ 1 \end{array} \right )\!.\ \qquad \end{array}$$
(8.47b)

Multiplying (8.45a,c,d,g,h) with N ipl , due to

$$\frac{{N}_{ipl}} {{M}_{ipl}} = \frac{{\sigma }_{yi}{A}_{i}} {{\sigma }_{yi}{W}_{ipl}} = \frac{{\sigma }_{yi}{A}_{i}} {{\sigma }_{yi}{A}_{i}\bar{{y}}_{ic}} = \frac{1} {\bar{{y}}_{ic}},$$
(8.48)

for (8.45a,c,d,g,h) we also have

$$\begin{array}{rcl} - \beta ({u}_{1},{u}_{2}){N}_{ipl}& \leq & \alpha {u}_{1}^{\alpha -1}{t}_{ i} + \frac{{m}_{i}^{-}} {\bar{{y}}_{ic}} \leq \beta ({u}_{1},{u}_{2}){N}_{ipl}\end{array}$$
(8.49a)
$$\begin{array}{rcl} -\beta ({u}_{1},{u}_{2}){N}_{ipl}& \leq & \alpha {u}_{1}^{\alpha -1}{t}_{ i} -\frac{{m}_{i}^{-}} {\bar{{y}}_{ic}} \leq \beta ({u}_{1},{u}_{2}){N}_{ipl}\end{array}$$
(8.49b)
$$\begin{array}{rcl} -\beta ({u}_{1},{u}_{2}){N}_{ipl}& \leq & \alpha {u}_{1}^{\alpha -1}{t}_{ i} + \frac{{m}_{i}^{+}} {\bar{{y}}_{ic}} \leq \beta ({u}_{1},{u}_{2}){N}_{ipl}\end{array}$$
(8.49c)
$$\begin{array}{rcl} -\beta ({u}_{1},{u}_{2}){N}_{ipl}& \leq & \alpha {u}_{1}^{\alpha -1}{t}_{ i} -\frac{{m}_{i}^{+}} {\bar{{y}}_{ic}} \leq \beta ({u}_{1},{u}_{2}){N}_{ipl}.\end{array}$$
(8.49d)

2.3 Approximation of the Yield Condition by Using Reference Capacities

According to (8.31), (8.32a,b) for each bar \(i = 1,\ldots ,B\) we have the condition

$$\left (\begin{array}{c} \frac{t} {{N}_{pl}} \\ \frac{m} {{M}_{pl}} \end{array} \right ) \in {K}_{0}^{\alpha } = \left \{\left (\begin{array}{c} x\\ y \end{array} \right ) \in {\mathsf{I\hspace{-1.49994pt}R}}^{2} : \vert x{\vert }^{\alpha } + \vert y\vert \leq 1\right \}$$

with \((t,m) = ({t}_{i},{m}_{i}^{\pm }),\ ({N}_{pl},{M}_{pl}) = ({N}_{ipl},{M}_{ipl})\).

Selecting fixed reference capacities

$${N}_{i0} > 0,{M}_{i0} > 0,i = 1,\ldots ,B,$$

related to the plastic capacities N ipl , M ipl , we get

$${\left \vert \frac{{t}_{i}} {{N}_{ipl}}\right \vert }^{\alpha } + \left \vert \frac{{m}_{i}^{\pm }} {{M}_{ipl}}\right \vert ={ \left \vert \frac{{t}_{i}} {{N}_{i0}} \cdot \frac{1} {\frac{{N}_{ipl}} {{N}_{i0}} } \right \vert }^{\alpha } + \left \vert \frac{{m}_{i}^{\pm }} {{M}_{i0}} \cdot \frac{1} {\frac{{M}_{ipl}} {{M}_{i0}} } \right \vert \!.$$

Putting

$${\rho }_{i} = {\rho }_{i}(a(\omega ),x) :=\min \left \{\frac{{N}_{ipl}} {{N}_{i0}} , \frac{{M}_{ipl}} {{M}_{i0}} \right \}\!,$$
(8.50)

we have

$$\frac{{\rho }_{i}} {\frac{{N}_{ipl}} {{N}_{i0}} } \leq 1, \frac{{\rho }_{i}} {\frac{{M}_{ipl}} {{M}_{i0}} } \leq 1$$

and therefore

$${ \left \vert \frac{{t}_{i}} {{N}_{ipl}}\right \vert }^{\alpha }+\left \vert \frac{{m}_{i}^{\pm }} {{M}_{ipl}}\right \vert ={ \left \vert \frac{{t}_{i}} {{\rho }_{i}{N}_{i0}}\right \vert }^{\alpha }\cdot {\left \vert \frac{{\rho }_{i}} {\frac{{N}_{ipl}} {{N}_{i0}} } \right \vert }^{\alpha }+\left \vert \frac{{m}_{i}^{\pm }} {{\rho }_{i}{M}_{i0}}\right \vert \cdot \left \vert \frac{{\rho }_{i}} {\frac{{M}_{ipl}} {{M}_{i0}} } \right \vert \leq {\left \vert \frac{{t}_{i}} {{\rho }_{i}{N}_{i0}}\right \vert }^{\alpha }+\left \vert \frac{{m}_{i}^{\pm }} {{\rho }_{i}{M}_{i0}}\right \vert.$$
(8.51)

Thus, the yield condition (8.30a,b) or (8.32a,b) is guaranteed by

$${\left \vert \frac{{t}_{i}} {{\rho }_{i}{N}_{i0}}\right \vert }^{\alpha } + \left \vert \frac{{m}_{i}^{\pm }} {{\rho }_{i}{M}_{i0}}\right \vert \leq 1$$

or

$$\left (\begin{array}{c} \frac{{t}_{i}} {{\rho }_{i}{N}_{i0}} \\ \frac{{m}_{i}^{\pm }} {{\rho }_{i}{M}_{i0}} \end{array} \right ) \in {K}_{0}^{\alpha }.$$
(8.52)

Applying the piecewise linearization described in Sect. 8.2.1 to condition (8.52), we obtain, cf. (8.45a–h), the approximation stated below. Of course, conditions (8.45a,b,e,f) are not influenced by this procedure. Thus, we find

$$\begin{array}{rcl} - {N}_{ipl}& \leq & {t}_{i} \leq {N}_{ipl}\end{array}$$
(8.53a)
$$\begin{array}{rcl} -{M}_{ipl}& \leq & {m}_{i}^{-}\leq {M}_{ ipl}\end{array}$$
(8.53b)
$$\begin{array}{rcl} -{M}_{ipl}& \leq & {m}_{i}^{+} \leq {M}_{ ipl}\end{array}$$
(8.53c)
$$\begin{array}{rcl} -\beta ({u}_{1},{u}_{2})& \leq & \alpha {u}_{1}^{\alpha -1} \frac{{t}_{i}} {{\rho }_{i}{N}_{i0}} + \frac{{m}_{i}^{-}} {{\rho }_{i}{M}_{i0}} \leq \beta ({u}_{1},{u}_{2})\end{array}$$
(8.53d)
$$\begin{array}{rcl} -\beta ({u}_{1},{u}_{2})& \leq & \alpha {u}_{1}^{\alpha -1} \frac{{t}_{i}} {{\rho }_{i}{N}_{i0}} - \frac{{m}_{i}^{-}} {{\rho }_{i}{M}_{i0}} \leq \beta ({u}_{1},{u}_{2})\end{array}$$
(8.53e)
$$\begin{array}{rcl} -\beta ({u}_{1},{u}_{2})& \leq & \alpha {u}_{1}^{\alpha -1} \frac{{t}_{i}} {{\rho }_{i}{N}_{i0}} + \frac{{m}_{i}^{+}} {{\rho }_{i}{M}_{i0}} \leq \beta ({u}_{1},{u}_{2})\end{array}$$
(8.53f)
$$\begin{array}{rcl} -\beta ({u}_{1},{u}_{2})& \leq & \alpha {u}_{1}^{\alpha -1} \frac{{t}_{i}} {{\rho }_{i}{N}_{i0}} - \frac{{m}_{i}^{+}} {{\rho }_{i}{M}_{i0}} \leq \beta ({u}_{1},{u}_{2}).\end{array}$$
(8.53g)

Remark 8.7.

Multiplying with ρ i we get quotients \(\frac{{t}_{i}} {{N}_{i0}} , \frac{{m}_{i}^{\pm }} {{M}_{i0}}\) with fixed denominators.

Hence, multiplying (8.53c–g) with ρ i , we get the equivalent system

$$\begin{array}{rcl} - {N}_{ipl}& \leq & {t}_{i} \leq {N}_{ipl}\end{array}$$
(8.54a)
$$\begin{array}{rcl} -{M}_{ipl}& \leq & {m}_{i}^{-}\leq {M}_{ ipl}\end{array}$$
(8.54b)
$$\begin{array}{rcl} -{M}_{ipl}& \leq & {m}_{i}^{+} \leq {M}_{ ipl}\end{array}$$
(8.54c)
$$\begin{array}{rcl} -\beta ({u}_{1},{u}_{2}){\rho }_{i}& \leq & \alpha {u}_{1}^{\alpha -1} \frac{{t}_{i}} {{N}_{i0}} + \frac{{m}_{i}^{-}} {{M}_{i0}} \leq \beta ({u}_{1},{u}_{2}){\rho }_{i}\end{array}$$
(8.54d)
$$\begin{array}{rcl} -\beta ({u}_{1},{u}_{2}){\rho }_{i}& \leq & \alpha {u}_{1}^{\alpha -1} \frac{{t}_{i}} {{N}_{i0}} -\frac{{m}_{i}^{-}} {{M}_{i0}} \leq \beta ({u}_{1},{u}_{2}){\rho }_{i}\end{array}$$
(8.54e)
$$\begin{array}{rcl} -\beta ({u}_{1},{u}_{2}){\rho }_{i}& \leq & \alpha {u}_{1}^{\alpha -1} \frac{{t}_{i}} {{N}_{i0}} + \frac{{m}_{i}^{+}} {{M}_{i0}} \leq \beta ({u}_{1},{u}_{2}){\rho }_{i}\end{array}$$
(8.54f)
$$\begin{array}{rcl} -\beta ({u}_{1},{u}_{2}){\rho }_{i}& \leq & \alpha {u}_{1}^{\alpha -1} \frac{{t}_{i}} {{N}_{i0}} -\frac{{m}_{i}^{+}} {{M}_{i0}} \leq \beta ({u}_{1},{u}_{2}){\rho }_{i}.\end{array}$$
(8.54g)

Obviously, (8.54a–g) can be represented also in the following form:

$$\begin{array}{rcl} \left \vert {t}_{i}\right \vert & \leq & {N}_{ipl}\end{array}$$
(8.55a)
$$\begin{array}{rcl} \left \vert {m}_{i}^{-}\right \vert & \leq & {M}_{ ipl}\end{array}$$
(8.55b)
$$\begin{array}{rcl} \left \vert {m}_{i}^{+}\right \vert & \leq & {M}_{ ipl}\end{array}$$
(8.55c)
$$\begin{array}{rcl} \left \vert \alpha {u}_{1}^{\alpha -1} \frac{{t}_{i}} {{N}_{i0}} + \frac{{m}_{i}^{-}} {{M}_{i0}} \right \vert & \leq & \beta ({u}_{1},{u}_{2}){\rho }_{i}\end{array}$$
(8.55d)
$$\begin{array}{rcl} \left \vert \alpha {u}_{1}^{\alpha -1} \frac{{t}_{i}} {{N}_{i0}} -\frac{{m}_{i}^{-}} {{M}_{i0}} \right \vert & \leq & \beta ({u}_{1},{u}_{2}){\rho }_{i}\end{array}$$
(8.55e)
$$\begin{array}{rcl} \left \vert \alpha {u}_{1}^{\alpha -1} \frac{{t}_{i}} {{N}_{i0}} + \frac{{m}_{i}^{+}} {{M}_{i0}} \right \vert & \leq & \beta ({u}_{1},{u}_{2}){\rho }_{i}\end{array}$$
(8.55f)
$$\begin{array}{rcl} \left \vert \alpha {u}_{1}^{\alpha -1} \frac{{t}_{i}} {{N}_{i0}} -\frac{{m}_{i}^{+}} {{M}_{i0}} \right \vert & \leq & \beta ({u}_{1},{u}_{2}){\rho }_{i}.\end{array}$$
(8.55g)

By means of definition (8.50) of ρ i , system (8.55a–g) reads

$$\begin{array}{rcl} \left \vert {t}_{i}\right \vert & \leq & {N}_{ipl}\end{array}$$
(8.56a)
$$\begin{array}{rcl} \left \vert {m}_{i}^{-}\right \vert & \leq & {M}_{ ipl}\end{array}$$
(8.56b)
$$\begin{array}{rcl} \left \vert {m}_{i}^{+}\right \vert & \leq & {M}_{ ipl}\end{array}$$
(8.56c)
$$\begin{array}{rcl} \left \vert \alpha {u}_{1}^{\alpha -1} \frac{{t}_{i}} {{N}_{i0}} + \frac{{m}_{i}^{-}} {{M}_{i0}} \right \vert & \leq & \beta ({u}_{1},{u}_{2})\frac{{N}_{ipl}} {{N}_{i0}}\end{array}$$
(8.56d)
$$\begin{array}{rcl} \left \vert \alpha {u}_{1}^{\alpha -1} \frac{{t}_{i}} {{N}_{i0}} + \frac{{m}_{i}^{-}} {{M}_{i0}} \right \vert & \leq & \beta ({u}_{1},{u}_{2})\frac{{M}_{ipl}} {{M}_{i0}}\end{array}$$
(8.56e)
$$\begin{array}{rcl} \left \vert \alpha {u}_{1}^{\alpha -1} \frac{{t}_{i}} {{N}_{i0}} -\frac{{m}_{i}^{-}} {{M}_{i0}} \right \vert & \leq & \beta ({u}_{1},{u}_{2})\frac{{N}_{ipl}} {{N}_{i0}}\end{array}$$
(8.56f)
$$\begin{array}{rcl} \left \vert \alpha {u}_{1}^{\alpha -1} \frac{{t}_{i}} {{N}_{i0}} -\frac{{m}_{i}^{-}} {{M}_{i0}} \right \vert & \leq & \beta ({u}_{1},{u}_{2})\frac{{M}_{ipl}} {{M}_{i0}}\end{array}$$
(8.56g)
$$\begin{array}{rcl} \left \vert \alpha {u}_{1}^{\alpha -1} \frac{{t}_{i}} {{N}_{i0}} + \frac{{m}_{i}^{+}} {{M}_{i0}} \right \vert & \leq & \beta ({u}_{1},{u}_{2})\frac{{N}_{ipl}} {{N}_{i0}}\end{array}$$
(8.56h)
$$\begin{array}{rcl} \left \vert \alpha {u}_{1}^{\alpha -1} \frac{{t}_{i}} {{N}_{i0}} + \frac{{m}_{i}^{+}} {{M}_{i0}} \right \vert & \leq & \beta ({u}_{1},{u}_{2})\frac{{M}_{ipl}} {{M}_{i0}}\end{array}$$
(8.56i)
$$\begin{array}{rcl} \left \vert \alpha {u}_{1}^{\alpha -1} \frac{{t}_{i}} {{N}_{i0}} -\frac{{m}_{i}^{+}} {{M}_{i0}} \right \vert & \leq & \beta ({u}_{1},{u}_{2})\frac{{N}_{ipl}} {{N}_{i0}}\end{array}$$
(8.56j)
$$\begin{array}{rcl} \left \vert \alpha {u}_{1}^{\alpha -1} \frac{{t}_{i}} {{N}_{i0}} -\frac{{m}_{i}^{+}} {{M}_{i0}} \right \vert & \leq & \beta ({u}_{1},{u}_{2})\frac{{M}_{ipl}} {{M}_{i0}}\end{array}$$
(8.56k)

Corresponding to Remark 8.7, the variables

$$\begin{array}{rlrlrl} {t}_{i},{m}_{i}^{+},{m}_{ i}^{-},{A}_{ i}\mbox{ or }x & & \end{array}$$

enters linearly. Increasing the accuracy of approximation by taking a further point \(\hat{u} = (\hat{{u}}_{1},\hat{{u}}_{2})\) with the related points \((\hat{{u}}_{1},-\hat{{u}}_{2}),(-\hat{{u}}_{1},\hat{{u}}_{2}),(-\hat{{u}}_{1},-\hat{{u}}_{2})\), we obtain further inequalities of the type (8.55c–g), (8.56c–g) respectively.

3 Stochastic Optimization

Due to (8.1c), (8.10), (8.11b), (8.12b), the 3B – vector

$$\begin{array}{rcl} F = {({F}_{1}^{T},\ldots ,{F}_{ B}^{T})}^{T}& &\end{array}$$
(8.57a)

of all interior loads fulfills the equilibrium condition

$$\begin{array}{rcl} CF = R& &\end{array}$$
(8.57b)

with the external load vector R and the equilibrium matrix C.

In the following we collect all random model parameters [31, 35, 39], such as external load factors, material strength parameters, cost factors, etc., into the random ν-vector

$$a = a(\omega ),\ \omega \in (\Omega ,\mathcal{A},\mathcal{P}),$$
(8.58a)

on a certain probability space \((\Omega ,\mathcal{A},P)\).

Thus, since in some cases the vector R of external loads depend also on the design r-vector x, we get

$$R = R\left (a(\omega ),x\right ).$$
(8.58b)

Of course, the plastic capacities depend also on the vectors x and a(ω), hence,

$$\begin{array}{rcl}{ N}_{ipl}^{\Gamma }& =& {N}_{ ipl}^{\Gamma }\left (a(\omega ),x\right ),\ \Gamma = L,U\end{array}$$
(8.58c)
$$\begin{array}{rcl}{ M}_{ipl}& =& {M}_{ipl}\left (a(\omega ),x\right ).\end{array}$$
(8.58d)

We assume that the probability distribution and/or the needed moments of the random parameter vector a = a(ω) are known [2, 28, 31, 42].

The remaining deterministic constraints for the design vector x are represented by

$$x \in \mathcal{D}$$
(8.59)

with a certain convex subset D of IR r.

3.1 Violation of the Yield Condition

Consider in the following an interior load distribution F fulfilling the equilibrium condition (8.58b).

According to the analysis given in Sect. 8.2, after piecewise linearization, the yield condition for the ith bar can be represented by an inequality of the type

$$H{\Gamma }^{(i)}\Big{(}a(\omega ),x\Big{)} \leq {h}^{(i)}\Big{(}a(\omega ),x\Big{)},i = 1,\ldots ,B,$$
(8.60)

with matrices H, Γ(i) = Γ(i)(a(ω), x) and a vector h (i) = h (i)(a(ω), x) as described in Sect. 8.2.

In order to take into account violations of condition (8.60), we consider the equalities

$$H{\Gamma }^{(i)}{F}_{ i} + {z}_{i} = {h}^{(i)},i = 1,\ldots ,B.$$
(8.61)

If

$${z}_{i} \geq 0,\mbox{ for all }i = 1,\ldots ,B,$$
(8.62a)

then (8.60) holds, and the yield condition is then fulfilled too, or holds with a prescribed accuracy.

However, in case

$${z}_{i}\ngeq 0\ \mbox{for some bars }i \in \{ 1,\ldots ,B\},$$
(8.62b)

the survival condition is violated at some points of the structure. Hence, structural failures may occur. The resulting costs Q of failure, damage and reconstructure of the frame is a function of the vectors z i , i = 1, …, B, defined by (8.61). Thus, we have

$$Q = Q(z) = Q({z}_{1},\ldots ,{z}_{B}),$$
(8.63a)

where

$$\begin{array}{rcl} z& :=& {({z}_{1}^{T},{z}_{ 2}^{T},\ldots ,{z}_{ B}^{T})}^{T},\ \end{array}$$
(8.63b)
$$\begin{array}{rcl}{ z}_{i}& :=& {h}^{(i)} - H{\Gamma }^{(i)}{F}_{ i},\ i = 1,\ldots ,B.\end{array}$$
(8.63c)

3.2 Cost Function

Due to the survival condition (8.62a), we may consider cost functions Q such that

$$Q(z) = 0,\mbox{ if }z \geq 0,$$
(8.64)

hence, no (recourse) costs arise if the yield condition (8.60) holds.

In many cases the recourse or failure costs of the structure are defined by the sum

$$Q(z) = \sum\limits_{i=1}^{B}{Q}_{ i}({z}_{i}),$$
(8.65)

of the element failure costs \({Q}_{i} = {Q}_{i}({z}_{i}),i = 1,\ldots ,B\).

Using the representation

$$\begin{array}{rcl}{ z}_{i} = {y}_{i}^{+} - {y}_{ i}^{-},\quad {y}_{ i}^{+},{y}_{ i}^{-}\geq 0,& &\end{array}$$
(8.66a)

the member cost functions Q i = Q i (z i ) are often defined [20, 31] by considering the linear function

$$\begin{array}{rcl}{ q}_{i}^{-T}{y}_{ i}^{-} + {q}_{ i}^{+T}{y}_{ i}^{+}& &\end{array}$$
(8.66b)

with certain vectors \({q}_{i}^{+},{q}_{i}^{-}\) of cost coefficients for the evaluation of the condition z i ≥ 0, z i ≱ 0, respectively.

The cost function Q i = Q i (z i ) is then defined by the minimization problem

$$\begin{array}{rcl} & \min & {q}_{i}^{-T}{y}_{ i}^{-} + {q}_{ i}^{+T}{y}_{ i}^{+}\end{array}$$
(8.67a)
$$\begin{array}{rl} \mbox{ s.t.} {y}_{i}^{+} - {y}_{ i}^{-} = {z}_{ i}\end{array}$$
(8.67b)
$$\begin{array}{rl}{y}_{i}^{-},{y}_{ i}^{+} \geq 0.\end{array}$$
(8.67c)

If \({z}_{i} := {({z}_{i1},\ldots ,{z}_{i\mu })}^{T},{y}_{i}^{\pm } := {({y}_{i1}^{\pm },\ldots ,{y}_{i\mu }^{\pm })}^{T},i = 1,\ldots ,B\) then (8.67a–c) can also be represented by

$$\begin{array}{rcl} & \min & \sum\limits_{l=1}^{\mu }({q}_{ il}^{-}{y}_{ il}^{-} + {q}_{ il}^{+}{y}_{ il}^{+})\end{array}$$
(8.68a)
$$\begin{array}{rcl} & \mbox{ s.t.}& {y}_{il}^{+} - {y}_{ il}^{-} = {z}_{ il},\quad l = 1,\ldots ,\mu \end{array}$$
(8.68b)
$$\begin{array}{rcl} & & {y}_{il}^{-},{y}_{ il}^{+} \geq 0,\quad l = 1,\ldots ,\mu.\end{array}$$
(8.68c)

Since the pairs of variables \(({y}_{il}^{-},{y}_{il}^{+}),l = 1,\ldots ,\mu \), are not connected with each other by the constraints, and the objective function is separabel with respect to these pairs of variables, (8.68a–c) can be decomposed into μ separated minimization problems

$$\begin{array}{rcl} & \min & {q}_{il}^{-}{y}_{ il}^{-} + {q}_{ il}^{+}{y}_{ il}^{+}\end{array}$$
(8.69a)
$$\begin{array}{rcl} & \mbox{ s.t.}& {y}_{il}^{+} - {y}_{ il}^{-} = {z}_{ il}\end{array}$$
(8.69b)
$$\begin{array}{rcl} & & {y}_{il}^{-},{y}_{ il}^{+} \geq 0,\end{array}$$
(8.69c)

for the pairs of variables \(({y}_{il}^{-},{y}_{il}^{+})\), l = 1, …, μ.

Under the condition

$${q}_{il}^{-} + {q}_{ il}^{+} \geq 0,l = 1,\ldots ,\mu ,$$
(8.70)

the following result holds:

Lemma 8.1.

Suppose that (8.70) holds. Then the minimum value function Q il = Q il (z il ) of (8.69a–c)is a piecewise linear, convex function given by

$${Q}_{il}({z}_{il}) :=\max \{ {q}_{il}^{+}{z}_{ il},-{q}_{il}^{-}{z}_{ il}\}.$$
(8.71)

Hence, the member cost functions Q i = Q i (z i ) reads

$$\begin{array}{rcl}{ Q}_{i}({z}_{i})& =& \sum\limits_{l=1}^{\mu }{Q}_{ il}({z}_{il}) = \sum\limits_{l=1}^{\mu }\max \{{q}_{ il}^{+}{z}_{ il},-{q}_{il}^{-}{z}_{ il}\},\end{array}$$
(8.72a)

and the total cost function Q = Q(z) is given by

$$\begin{array}{rcl} Q(z)& =& \sum\limits_{i=1}^{B}{Q}_{ i}({z}_{i}) = \sum\limits_{i=1}^{B} \sum\limits_{l=1}^{\mu }\max \{{q}_{ il}^{+}{z}_{ il},-{q}_{il}^{-}{z}_{ il}\}.\end{array}$$
(8.72b)

3.3 Choice of the Cost Factors

Under elastic conditions the change Δσ of the total stress σ and the change ΔL of the element length L are related by

$$\Delta L = \frac{L} {E}\Delta \sigma ,$$
(8.73a)

where E denotes the modulus of elasticity.

Assuming that the neutral axis is equal to the axis of symmetry of the element, for the total stress Δσ in the upper (“+”) lower (“ − ”) fibre of the boundary we have

$$\Delta \sigma = \frac{\Delta t} {A} \pm \frac{\Delta m} {W} ,$$
(8.73b)

where W denotes the axial modulus of the cross-sectional area of the element (beam).

Representing the change ΔV of volume of an element by

$$\Delta V = A \cdot \Delta L,$$
(8.74a)

then

$$\begin{array}{rcl} \hspace{-12.0pt}\Delta V & =& A \cdot \Delta L = A \cdot \frac{L} {E}\Delta \sigma = A \cdot \frac{L} {E}\left (\frac{\Delta t} {A} \pm \frac{\Delta m} {W} \right ) \\ & =& \frac{L} {E}\Delta t \pm \frac{L} {E} \frac{A} {W}\Delta m = \frac{L} {E}\Delta t \pm \frac{L} {E} \frac{1} {\frac{W} {A} }\Delta m = \frac{L} {E}\Delta t \pm \frac{L} {E} \cdot \frac{1} {\bar{{y}}_{c}}\Delta m,\qquad \end{array}$$
(8.74b)

where \(\bar{{y}}_{c}\) is the cross-sectional parameter as defined in (8.28d).

Consequently, due to (8.73a,b), for the evaluation of violations Δt of the axial force constraint we may use a cost factor of the type

$${\Gamma }_{K} := \frac{L} {E},$$
(8.75a)

and an appropriate cost factor for the evaluation of violations of moment constraints reads

$${\Gamma }_{M} = \frac{L} {E} \cdot \frac{1} {\bar{{y}}_{c}}.$$
(8.75b)

3.4 Total Costs

Denoting by

$${G}_{0} = {G}_{0}\Big{(}a(\omega ),x\Big{)}$$
(8.76a)

the primary costs, such as weighted negative load factors, material costs, costs of construction, etc., the total costs including failure or recourse costs are given by

$$G = {G}_{0}\big{(}a(\omega ),x\big{)} + Q\big{(}z\big{(}a(\omega ),x,F(\omega )\big{)}\big{)}.$$
(8.76b)

Hence, the total costs G = G(a(ω), x, F(ω)) depend on the vector x = (x 1, …, x r )T of design variables, the random vector a(ω) = (a 1(ω), …, a ν(ω))T of model parameters and the random vector F = F(ω) of all internal loadings.

Minimizing the expected total costs, we get the following stochastic optimization problem (SOP) of recourse type [20]

$$\begin{array}{rcl} & \min & E\Big{(}{G}_{0}\big{(}a(\omega ),x\big{)} + Q\big{(}z\left (a(\omega ),x,F(\omega )\right )\big{)}\Big{)}\end{array}$$
(8.77a)
$$\begin{array}{rcl} & \mbox{ s.t.}& H{\Gamma }^{(i)}\Big{(}a(\omega ),x\Big{)}{F}_{ i}(\omega ) + {z}_{i}(\omega ) = {h}^{(i)}\Big{(}a(\omega ),x\Big{)}\quad \mbox{ a.s.}, \\ & & \hspace{170.71652pt} i = 1,\ldots ,B\end{array}$$
(8.77b)
$$\begin{array}{rcl} & & CF(\omega ) = R\Big{(}a(\omega ),x\Big{)}\quad \mbox{ a.s.}\end{array}$$
(8.77c)
$$\begin{array}{rcl} & & x \in D.\end{array}$$
(8.77d)

Using representation (8.65), (8.67a–c) of the recourse or failure cost function Q = Q(z), problem (8.77a–d) takes also the following equivalent from

$$\begin{array}{rcl} & \min & E\left ({G}_{0}\Big{(}a(\omega ),x\Big{)} + \sum\limits_{i=1}^{B}\big{(}{q}_{ i}^{-}{(\omega )}^{T}{y}_{ i}^{-}(\omega ) + {q}_{ i}^{+}{(\omega )}^{T}{y}_{ i}^{+}(\omega )\big{)}\right )<EquationNumber>8.78a</EquationNumber> \\ & \mbox{ s.t.}& H{\Gamma }^{(i)}\Big{(}a(\omega ),x\Big{)}{F}_{ i}(\omega ) + {y}_{i}^{+}(\omega ) - {y}_{ i}^{-}(\omega ) = {h}^{(i)}\Big{(}a(\omega ),x\Big{)}\quad \mbox{ a.s.}, \\ \end{array}$$
$$\begin{array}{rcl} & & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad i = 1,\ldots ,B\end{array}$$
(8.78b)
$$\begin{array}{rcl} & & \quad CF(\omega ) = R\left (a(\omega ),x\right )\quad \mbox{ a.s.}\end{array}$$
(8.78c)
$$\begin{array}{rcl} & & \quad x \in D,\ {y}_{i}^{+}(\omega ),{y}_{ i}^{-}(\omega ) \geq 0\quad \mbox{ a.s.},\quad i = 1,\ldots ,B.\end{array}$$
(8.78d)

Remark 8.8.

Stochastic optimization problems of the type (8.78a–d) are called “two-stage stochastic programs” or “stochastic problems with recourse”.

In many cases the primary cost function G 0 = G 0(a(ω), x) represents the volume or weight of the structural, hence,

$$\begin{array}{rcl}{ G}_{0}\Big{(}a(\omega ),x\Big{)}& :=& \sum\limits_{i=1}^{B}{\Gamma }_{ i}(\omega ){V }_{i}(x) \\ & =& \sum\limits_{i=1}^{B}{\Gamma }_{ i}(\omega ){L}_{i}{A}_{i}(x),\end{array}$$
(8.79)

with certain (random) weight factors Γ i = Γ i (ω).

In case

$${A}_{i}(x) := {u}_{i}{x}_{i}$$
(8.80)

with fixed sizing parameters u i , i = 1, …, B, we get

$$\begin{array}{rcl}{ G}_{0}\left (a(\omega ),x\right )& =& \sum\limits_{i=1}^{B}{\Gamma }_{ i}(\omega ){L}_{i}{u}_{i}(x) = \sum\limits_{i=1}^{B}{\Gamma }_{ i}(\omega ){L}_{i}{h}_{i}{x}_{i} \\ & =& c{\left (a(\omega )\right )}^{T}x, \end{array}$$
(8.81a)

where

$$c\left (a(\omega )\right ) :={ \left ({\Gamma }_{1}(\omega ){L}_{1}{u}_{1},\ldots ,{\Gamma }_{B}(\omega ){L}_{B}{u}_{B}\right )}^{T}.$$
(8.81b)

Thus, in case (8.80), G 0 = G 0(a(ω), x) is a linear function of x.

3.5 Discretization Methods

The expectation in the objective function of the stochastic optimization problem (8.78a–d) must be computed numerically. One of the main methods is based on the discretization of the probability distribution P a( ⋅) of the random parameter ν-vector a = a(ω), hence,

$${P}_{a(\cdot )} \approx \mu := \sum\limits_{k=1}^{s}{\alpha }_{ k}{\epsilon }_{{a}^{(k)}}$$
(8.82a)

with

$${\alpha }_{k} \geq 0,\ k = 1,\ldots ,s,\ \sum\limits_{k=1}^{s}{\alpha }_{ k} = 1.$$
(8.82b)

Corresponding to the realizations a (k), k = 1, …, s, of the discrete approximate (8.82a,b), we have the realizations \({y}_{i}^{-(k)},{y}_{i}^{+(k)}\) and F (k), F i (k), k = 1, …, s, of the random vectors \({y}_{i}^{-}(\omega ),{y}_{i}^{+}(\omega ),i = 1,\ldots ,B\), and F(ω). then

$$\begin{array}{rcl} & & E\left ({G}_{0}\big{(}a(\omega ),x\big{)} + \sum\limits_{i=1}^{B}\left ({q}_{ i}^{-T}{y}_{ i}^{-}(\omega ) + {q}_{ i}^{+T}{y}_{ i}^{+}(\omega )\right )\right ) \\ & & \approx \sum\limits_{k=1}^{s}{\alpha }_{ k}\left ({G}_{0}({a}^{(k)},x) + \sum\limits_{i=1}^{B}({q}_{ i}^{-T}{y}_{ i}^{-(k)} + {q}_{ i}^{+T}{y}_{ i}^{+(k)}\right ).\end{array}$$
(8.83a)

Furthermore, the equilibrium equation (8.77b) is approximated by

$$C{F}^{(k)} = R({a}^{(k)},x),\ k = 1,\ldots ,s,$$
(8.83b)

where \({F}^{(k)} := \Big{(}{{F}_{1}^{(k)}}^{T},\ldots ,{{F}_{B}^{(k)}}^{T}{\Big{)}}^{T}\), and we have, cf. (8.78c), the nonnegativity constraints

$$\begin{array}{rcl}{ y}_{i}^{+(k)},\ {y}_{ i}^{-(k)} \geq 0,\ k = 1,\ldots ,s,\ i = 1,\ldots ,B.& &\end{array}$$
(8.83c)

Thus, (SOP) (8.78a–d) is reduced to the parameter optimization problem

$$\begin{array}{rcl} \min & &{ \overline{G}}_{0}({a}^{(k)},x) + \sum\limits_{i=1}^{B}{\alpha }_{ k}\left ({q}_{i}^{-T}{y}_{ i}^{-(k)} + {q}_{ i}^{+T}{y}_{ i}^{+(k)}\right )\\ \mbox{ s.t.}& & H{\Gamma }^{(i)}({a}^{(k)},x){F}_{ i}^{(k)} + {y}_{ i}^{+(k)} - {y}_{ i}^{-(k)} = {h}^{(i)}({a}^{(k)},x), \\ \end{array}$$
(8.84a)
$$\begin{array}{rcl} & & \hspace{85.35826pt} \ i = 1,\ldots ,B,\ k = 1,\ldots ,s\end{array}$$
(8.84b)
$$\begin{array}{rcl} & & C{F}^{(k)} = R({a}^{(k)},x),\ k = 1,\ldots ,s\end{array}$$
(8.84c)
$$\begin{array}{rcl} & & x \in D,\ {y}_{i}^{+(k)},\ {y}_{ i}^{-(k)} \geq 0,\ k = 1,\ldots ,s,\ i = 1,\ldots ,B.\end{array}$$
(8.84d)

A further important class of methods for computing expectations and probabilities, hence, multiple integrals, occurring in stochastic optimization problems, reliability analysis and reliability-based optimal design (RBO), cf. [9, 11, 30, 31], are simulation methods, such as Monte Carlo Simulation (MCS) procedures. Simulation techniques are used especially in cases with only few information about the analytical properties of the underlying technical device, e.g., in case of analytically almost unavailable limit state functions. In principle, MSC is a very simple technique which is widely applicable on the one hand, but may have a very low efficiency of estimation on the other hand. Hence, several improvements were considered in the last time, such as Advanced Monte Carlo Simulation techniques: Variance reduction methods reducing the sampling error, based, e.g., on importance sampling methods, direction sampling, subset simulation, etc., see, e.g., [4]. Further improvements can be obtained by combining these simulation/estimation techniques with Response Surface Methods (RSM) for estimating unknown functions using regression techniques and advanced nonlinear programming procedures (NLP), cf. [12].

3.6 Complete Recourse

According to Sect. 8.3.1, the evaluation of the violation of the yield condition (8.60) is based on (8.61), hence

$$H{\Gamma }^{(i)}{F}_{ i} + {z}_{i} = {h}^{(i)},i = 1,\ldots ,B.$$

In generalization of the so-called “simple recourse” case (8.67a–c), in the “complete recourse” case the deviation

$${z}_{i} = {h}^{(i)} - H{\Gamma }^{(i)}{F}_{ i}$$

is evaluated by means of the minimum value Q i = Q i (z i ) of the linear program, cf. (8.68a–c)

$$\begin{array}{rcl} & \min & {q}^{(i)T}{y}^{(i)}\end{array}$$
(8.85a)
$$\begin{array}{rcl} & \mbox{ s.t.}& {M}^{(i)}{y}^{(i)} = {z}_{ i}\end{array}$$
(8.85b)
$$\begin{array}{rcl} & & \quad {y}^{(i)} \geq 0.\end{array}$$
(8.85c)

Here, q (i) is a given cost vector and M (i) denotes the so-called recourse matrix [20].

We assume that the linear equation (8.85a) has a solution y (i) ≥ 0 for each vector z i . This property is called “complete recourse”.

In the present case the stochastic optimization problem (8.77a–c) reads

$$\begin{array}{rcl} \min E\left ({G}_{0}\Big{(}a(\omega ),x\Big{)} + \sum\limits_{i=1}^{B}{q}^{(i)T}{y}^{(i)}(\omega )\right )\end{array}$$
(8.86a)
$$\begin{array}{rl}\mbox{ s.t.} H{\Gamma }^{(i)}\Big{(}a(\omega ),x\Big{)}{F}_{ i}(\omega ) + {M}^{(i)}{y}^{(i)}(\omega ) = {h}^{(i)}\Big{(}a(\omega ),x\Big{)}\mbox{ a.s., } \\ \hspace{184.9429pt} i = 1,\ldots ,B\end{array}$$
(8.86b)
$$\begin{array}{rcl} & & \quad CF(\omega ) = R\left (a(\omega ),x\right )\quad \mbox{ a.s.}\end{array}$$
(8.86c)
$$\begin{array}{rcl} & & \quad x \in D,\ {y}^{(i)}(\omega ) \geq 0\quad \mbox{ a.s.},\quad i = 1,\ldots ,B.\end{array}$$
(8.86d)

As described in Sect. 8.3.5, problem (8.86a–d) can be solved numerically by means of discretization methods and application of linear/nonlinear programming techniques.