## Abstract

In the framework of classical plasticity, even when limit multipliers and collapse mechanisms associated with different loads independently acting on a solid or structure are known, not much can be inferred on the limit multiplier of the combined loading. Frame structures under the action of dead loads and seismic forces, soil–foundation interaction problems, tunnels under a variety of loads, deepwater pipelines subject to bending and pressure constitute only a few selected examples for which some sort of superposition rule, as well as bounding techniques, would be extremely useful. The present paper introduces a set of theorems for bounding limit multipliers for combined loads. In particular, ranging from a minimum knowledge about the critical state under a particular loading to a reasonable guess of the kinematics of the problem under combined loads, more and more refined bounds for the overall limit multiplier are derived. The results, to the best of the authors’ knowledge, are novel and a few examples showing their practical value are presented and discussed.

## 1. Introduction

An important aspect of structural analysis, especially for ultimate safety assessment or design, consists in evaluating the maximum load that the structure can sustain. This is at the heart of many structural codes, and the earthquake which has recently shaken the central regions of Italy, wrecking homes and causing a considerable death toll, is just the most recent example of the utmost importance of a reliable evaluation of the collapse load in many structural engineering problems. On account of the uncertainty of many parameters, the use of methods based on the classic limit analysis, which avoids computationally expensive and often uncertain time-stepping analyses, represents a useful and valuable approach. The distinctive feature of classical limit analysis is the determination of the load factor or, in practice, of its upper and/or lower bounds at which a critical event occurs, namely plastic collapse. Therefore, in the past years, the limit analysis research field has been the focus of intensive research efforts. In general, the most up-to-date formulations are derived within an optimization problem framework, aiming to take advantage of the latest mathematical developments in nonlinear convex programming algorithms. Nevertheless, in spite of the rapid evolution in computer performance, determining accurate collapse load estimates can still represent a significant computational effort. Additionally, as it is the case in most plasticity problems, the principle of superposition of loads does not hold true in the framework of classical limit analysis. Such a principle is extremely useful in the treatment of many practical engineering problems and has been widely exploited in the past two centuries on the basis of linearized strain and constitutive laws and linear equilibrium equations for the stress state. In fact, in the linear theory of elasticity, the response of a body under the action of any given combination of loads is known once the solutions of the particular load cases are available. In the classical theory of plasticity, owing to both non-uniqueness of the stresses in strains and nonlinearity of the incremental elastoplastic process until the collapse, superposition cannot be applied. However, in engineering applications, a large number of situations pose the problem of evaluating the failure under combined loads, and the cases where superposition cannot be applied are of great importance. Although critical loads are often available for each loading condition separately, serious difficulties can be instead encountered in the computation of the combined critical loading, where combined loading can be simply interpreted as the occurrence of more than one independent exertion factor (Zyczkowski 1981).

Some results are available for shakedown problems (König 1987), which ensure the stabilization of plastic deformations under several loading histories, but no information is provided about combined critical loading, when unlimited plastic deformation takes place while the different loads remain constant in time, notwithstanding the knowledge of the critical loading for each loading case.

A superposition procedure has been recently employed by Puzrin & Randolph (2001) with reference to particular yield criteria and in the framework of upper bound limit analysis. They investigated the implications of combining two kinematically admissible velocity fields and showed that, under certain conditions, the superposition of two different collapse mechanisms can be used as a means of improving the upper bound collapse load for a defined loading. Thus, they make reference to superposition in a different way from what is meant in the present work and, moreover, the applicability of their findings is limited by the strict assumptions at the base of the treatment.

Frame structures under the action of dead loads and seismic forces, soil–foundation interaction problems, tunnels under a variety of loads, deepwater pipelines subject to bending and pressure constitute a few selected examples for which some sort of superposition rule would be extremely useful. In all these cases, bounding techniques can be extremely helpful. From a purely engineering standpoint, lower bounds to the limit carrying capacity are generally more relevant than upper bounds, since in many practical applications safety factors are needed. However, upper bounds may be employed to estimate the inaccuracy with respect to the actual limit multiplier.

This said, on the sole basis of the tools of the classical theory of limit analysis and making resort to a set of inequalities, the present paper introduces a set of theorems for bounding limit multipliers for combined loads. These findings, to the best of the authors’ knowledge, are novel and a few examples showing their practical value are presented and discussed.

## 2. Framework

### (a) Perfectly plastic behaviour

The fundamental theory of plasticity and limit analysis was pointed out more than 50 years ago (Hill 1950; Prager & Hodge 1951). According to its foundations, in the present work, it is postulated that (Maugin 1992):

(i) a strain energy

*φ*(*ϵ*^{el}) function rules the elastic behaviour;(ii) a convex yield domain

*Y*exists in the Westergaard principal stress space and is defined by the condition 2.1(iii) a normal flow rule 2.2where is a scalar parameter, holds true.

Also, the flow rule is assumed to be associated and defines the plastic potential as .

Equation (2.2) can also be stated in a variational form, on the basis of the Hill–Mandel dissipation principle, i.e.
2.3and the Drucker–Ilyushin *stability* postulate—that is, for any strain cycle—can be obtained as a direct consequence of equations (2.2) and (2.3) (Ilyushin 1948; Drucker 1988).

On account of the small-strain hypothesis, the following decomposition formula is also assumed to hold true
2.4where stands for the domain occupied by the elastoplastic body. ∂*Ω*_{t} and ∂*Ω*_{u} denote the boundary regions where tractions and displacements are assigned, respectively.

Summarizing, it is 2.5 being the stiffness tensor in the linearly elastic range.

### (b) The Greenberg minimum principle and the theorems of limit analysis

In the limit analysis, a stress field ** s**(

**) that obeys the field and boundary equilibrium equations is called**

*x**statically admissible*. The set of all the statically admissible stress fields is called

*S*.

The *plastically admissible* stress fields belong to the convex domain *Y* ,
2.6The set Σ is defined as .

A displacement/velocity field is called *kinematically admissible* if it is mathematically well behaved (e.g. continuous and piecewise continuously differentiable) and obeys the external and internal constraints, if any. The set of kinematically admissible velocity fields is called *K*.

The set of plastically admissible *velocity fields* can be introduced as
2.7where ** t** is the boundary tractions and, without loss of generality, the hypothesis of the absence of body forces is postulated. Then, the set

*V*is defined as .

The classical theorems of limit analysis provide upper and lower bounds on the loads under which an elastic perfectly plastic body reaches a critical state, i.e. a state in which large increases in plastic deformation become possible with little, if any, increase in load (Lubliner 1990; Khan & Huang 1995). This state is called *unrestricted plastic flow* and the loading state at which it becomes possible is called *ultimate* or *limit loading*. In a state of unrestricted plastic flow, elasticity may be ignored and therefore a treatment based on rigid-plastic behaviour is valid for limit analysis theorems.

The state of unrestricted plastic flow can be found on the basis of the Greenberg minimum principle (Drucker *et al*. 1952; Washizu 1982) by minimizing the total dissipation functional, ,
2.8among the admissible velocity fields is the applied loading and *λ* is the load multiplier. Given that the actual dissipated plastic power is
2.9since
2.10the value of the limit loading, *λ**t*^{0}—and therefore the load multiplier, *λ*—is given by Rayleigh’s quotient
2.11where ** v** represents the actual velocity field at the collapse.

On this basis, the lower and the upper-bound theorems of limit analysis can be established.

#### (i) Lower-bound theorem

Let *λ*_{s} be a load multiplier. If , i.e. if at least one stress field ** σ***∈Σ exists, then the following inequality holds true
2.12Therefore,

*λ*

_{s}is a

*safety factor*.

#### (ii) Upper-bound theorem

Let *λ*_{k} be a load multiplier. If a kinematically and plastically admissible velocity field exists such that
2.13then it is
2.14

Therefore, *λ*_{k} is an *overload factor*.

Formulae (2.12) and (2.14) provide upper and lower bounds for the actual plastic multiplier *λ*,
2.15In the following, a compact notation will be adopted, that is
2.16and the subscript ‘(…)’, i.e.
2.17will indicate the specific case under consideration.

Without loss of generality, the absence of body forces is postulated.

## 3. Limit multiplier bounds for combined loads

As stated before, the scope of the present work is to establish some theorems which can yield relevant bounds on the overall limit multiplier in case of combined loading (figure 1).

In simple words, say that for an elastic perfectly plastic body two different loading conditions are prescribed on ∂*Ω*_{t}, and . If the corresponding limit multipliers *λ*_{1} and *λ*_{2} (*λ*_{1}≤*λ*_{2}) are determined, then what can be inferred about the limit multiplier *λ*_{+} corresponding to the combined loading ?

A first theorem, which can directly provide an overload factor with minimal information about the mutual dissipation of the single loading cases, can be immediately stated.

## Theorem 3.1 (A sufficient condition for λ_{+} ≤ λ_{1} (or λ_{+} ≤ λ_{2}))

*If the ‘mutual dissipation’* *(or* *is non-negative, then the limit multiplier λ*_{+} *for the combined loading cannot exceed λ*_{1} *(or λ*_{2}*), that is*
3.1

It is worth proving only the first relationship in equation (3.1), given that demonstrating the second one follows exactly the same steps. By virtue of the principle of virtual power (PVP) (Maugin 1980), the limit multiplier of the combined loading can be written as 3.2where is the velocity field at the critical state under the sole traction . Since it is 3.3once again by virtue of PVP, it is found that the following equation holds true 3.4

The stability postulate equation (2.3) implies that
3.5where the inequality *α*≥0 depends on the fact that, being and, by hypothesis, , then it is in order to be *λ*_{+}>0.

Hence, inequality (3.3) becomes 3.6and the theorem is proved.

On the other hand, with the scope of pursuing safety factors, a whole set of bounds for the combined loading can be proved on the basis of a basic set of inequalities. In order to do so, some lemmas are first established.

## Lemma 3.2 (A sufficient condition for λ_{+} ≥ λ_{1})

*Let* *be the velocity field at the critical state, with* *and* *acting at the same time. Then it is*
3.7

In fact, first of all, since , it is and equation (3.7) implies .

The limit multiplier *λ*_{+} with reference to the actual velocity field at the critical state is
3.8

Under the hypothesis (3.7), it is 3.9and by virtue of the stability postulate, it follows 3.10which proves the lemma.

## Lemma 3.3 (A sufficient condition for λ_{+} ≥ λ_{2} ≥ λ_{1})

*Under the same conditions of lemma 3.2, it is possible to prove that*
3.11*the line of reasoning being the same as before.*

## Lemma 3.4 (A necessary condition for λ_{+} ≤ λ_{1})

*Let* *be the velocity at the critical state with two different loadings,* *and* *, acting on the elastoplastic body. Said λ*_{+} *the limit multiplier for the combined loading, it is*
3.12

In fact, by virtue of the Greenberg minimum principle, it is
3.13and both sides of equation (3.13) are positive quantities. Therefore, on the basis of the hypothesis, it can be written
3.14and, given the stability postulate and the PVP, it results
3.15Since the limit multiplier is *λ*_{1}>0, it follows that
3.16which proves the first statement in equation (3.12). By applying the same line of reasoning with *λ*_{2} in place of *λ*_{1}, the second statement can be easily proved, that is,
3.17On the ground of the previous lemmas, the anticipated theorems bounding the value of the limit multiplier for combined loading on the sole basis of the limit multipliers of the component loads are now proved. These results can also be viewed as some sort of superposition theorems for the limit multipliers and, for the maximum clarity of the presentation, two different theorems will be proved: the first for the case of two-component loadings and the second for the case of *n*-component loadings.

## Theorem 3.5 (A lower bound for the overall limit multiplier λ_{+} in the case of two loadings)

*Let* *be the velocity at the critical state with two different loadings,* *and* *, acting on the elastoplastic body. Said λ*_{+} *the limit multiplier for the combined loading, it is*
3.18

In order to prove equation (3.18), it can be noticed that
3.19In fact, three different cases may occur:
3.20given that the case is not acceptable, being in contrast with the condition of plastically admissibility of the velocity at the critical state, that is . Therefore, by virtue of lemmas 3.2 and 3.3, i.e. equations (3.7) and (3.11), both relationships (I) and (II) in equation (3.20) imply that *λ*_{+}≥*λ*_{1} and equation (3.19) holds true.

From the third of equations (3.20), the following topological property can be deduced (figure 2), 3.21being the set of kinematically and plastically admissible velocity fields .

Thus, this relationship and the PVP allow us to state that
3.22where *λ*_{k1} and *λ*_{k2} are the limit multipliers corresponding to the loading conditions and , respectively. It follows that
3.23and, by adding inequalities (3.23), it is
3.24which proves the theorem, since
3.25being *λ*_{2}≡*kλ*_{1}≥*λ*_{1} and *m*=1+*k*^{−1}>1.

## Corollary 3.6 (A general lower bound for the overall limit multiplier λ_{+})

*Even if no information is available on the value of λ*_{2}*, that is, λ*_{2}*|{λ*_{2}*≥λ*_{1}*, λ*_{2}*∈R*^{+}*}, the following inequality always holds true*
3.26

In fact, since *m* is bounded, equation (3.26) straightforwardly follows from relationships (3.25) and (3.19), i.e.
3.27It is worth noticing that inequalities (3.25) and (3.26) can be interpreted as relatives of Gershgorin circle theorem (Varga 2004). In fact, as the Gershgorin theorem may be used to bound the spectrum of a square matrix, inequalities (3.25) and (3.26) can be employed to obtain an estimate of the actual limit multiplier in the presence of the combined loads and to establish a safety factor with regard to the critical state. Indeed, as a consequence of equation (3.27), it is easy to prove, for example, that
3.28In other words, by virtue of equation (3.28), the combined loading
3.29can be considered *safe.*

## Corollary 3.7 (A partition technique for obtaining lower bounds)

*The limit multiplier λ for an elastoplastic body under a certain load* *t*^{0} *can be bounded by partitioning* *t*^{0} *in a set of loading conditions such that*
3.30*and whose limit multipliers* *are known. Indeed, the following inequality is valid*
3.31*Moreover, even if* *is unknown, provided* *, it is*
3.32

In fact, by applying theorem 3.5 to any arbitrary partitioning of the loading *t*^{0}(provided the limit multipliers of the component loadings are known), equation (3.31) is simply obtained by choosing the largest value among the lower bounds for *λ*. Then inequality (3.32) holds true by virtue of corollary 3.6.

It is worth highlighting the value of corollary 3.7 in the assessment of the degree of safety of an applied loading in all those cases when the solution of the elastoplastic problem cannot be easily pursued. In fact, many problems in engineering are amenable to be partitioned in two or more simple loading cases and the proposed result allows the evaluation of the limit response of an elastoplastic body under the action of any given combination of loads, once the limit multipliers of the particular load cases are available.

Finally, the extension of theorem 3.5 and of its corollaries to the case of different loading conditions is given.

## Theorem 3.8 (A lower bound for the overall limit multiplier λ_{+} in the case of *n* loadings)

*If an elastoplastic body is subject to* *different loadings on its boundary ∂ Ω*

_{t}

*, say*3.33

*whose limit multipliers are known and it is*3.34

*the following inequality holds true*3.35

*Moreover, even if the multipliers*

*are unknown, the following inequality is applicable*3.36

Given hypotheses (3.33) and (3.34), a partition of *t*^{0}_{+} such that
3.37can be introduced, with
3.38Even if both the limit multipliers *λ*_{>} and *λ*_{<}, corresponding to the loads *t*^{0}_{>} and *t*^{0}_{<}, respectively, are unknown, the results by lemmas 3.2 and 3.3 can be applied to write
3.39Since the loading *t*^{0}_{>} can be regarded as a combined loading case, it is possible to apply theorem 3.5 with respect to its limit multiplier, *λ*_{>}, and obtain
3.40Moreover, being any limit multiplier a positive number by definition, the following inequality must also be true
3.41thus it is
3.42By virtue of this result and of corollary 3.7, it is also
3.43and, on account of the equality , the theorem is proved.

It is the case of noticing that inequality (3.25) resembles the Reuss lower-bound formula for the stiffness of a linearly elastic n-phase composite (Maugin 1992), with the limit multipliers in place of the elastic moduli. As a consequence, it can be inferred that in case the limit load multiplier for each individual loading pattern is not exact but it is estimated using lower- or upper-bound limit load theorems, the resultant limit multiplier evaluated by means of the proposed procedure will be affected by an additional degree of approximation. Trivially, if the individual load multipliers are upper bounds of the actual ones, inequality (3.25) will provide a combined load multiplier which cannot be guaranteed to be a safety factor. The opposite happens in case of lower bounds.

No correspondence can be found, finally, with the Voigt upper bound.

## 4. A summary of the procedures for bounding the limit multiplier in the case of combined loadings

On the basis of the findings of the previous section, it is finally possible to summarize all the procedures to bound the limit multiplier in the case of combined loads, showing their practical value. The presentation is restricted, for the sake of clarity, to the case of two different loadings but it can be directly extended to the case of *n* loadings by virtue of theorem 3.8.

A first value of the safety factor can be immediately obtained from expressions (3.25) and (3.26), on the sole basis of the values of all or, at least one, of the limit multipliers of the particular loading, i.e. 4.1However, the additional knowledge of the kinematics at the critical state of the particular loadings, straightforwardly provides, by virtue of theorem 3.1, the following overload factors 4.2Furthermore, if some basic information is available or can be easily inferred about the kinematics of the critical state under combined loads, then lemmas 3.2 and 3.3 allow us to establish that 4.3which constitute safety factors. It is important to note that 4.4and, as a consequence, if the safety factors in equation (4.3) are available, they constitute better ones than those from equation (4.1).

In other words, ranging from a minimum knowledge about the critical state of the particular loading to a reasonable guess of the kinematics of the problem under combined loads, more and more refined bounds for the overall limit multiplier can be derived by virtue of the presented results, as summarized in table 1.

## 5. Examples

A few examples showing the actual value of the proposed formulae in engineering problems are finally presented and discussed in this section. The examples are chosen in order to present quantitative estimates of the actual carrying capacity of common structural problems and also to depict the regions which set the limits of the plastic interaction surfaces. The use of upper bounds to estimate the error with respect to the actual limit multiplier is also shown.

### (a) Example 1: frame structure under the action of combined horizontal and vertical point loads

With reference to figure 3, the frame structure can be subject to two different loading conditions, that is a horizontal force *H*^{0}, a vertical force *V*^{0} and a combined loading given by both the previous forces exerted on the structure at the same time.

The positions of possible plastic hinges can be immediately located and, therefore, all the collapse mechanisms are known (Massonet & Save 1980). Thus, the following limit multipliers are obtained
5.1where is the plastic moment, *σ*_{Y} is the yield stress and *W*^{pl} is the plastic section modulus of the cross section, which is assumed the same for all the elements. Theorem 3.5 gives the following lower bound, *λ*_{L}, for the combined loading limit multiplier, *λ*_{+},
5.2Inequality (5.2) can be trivially verified since .

Making reference to the collapse mechanisms illustrated in figure 3, it is also immediately recognized that the mutual dissipation, as defined by equation (3.1), is positive,
5.3where and denote the velocity fields at the critical states associated with the vertical and horizontal point loads, respectively, and is the corresponding angular velocity. As a consequence, application of theorem 3.1 leads to an upper bound, *λ*_{U}, for the combined limit multiplier, *λ*_{+}, that is
5.4where
5.5

### (b) Example 2: limit-carrying capacity of an elliptical cross section subject to combined bending

Figure 4 shows the elliptical cross section of a cylinder subject to combined bending and *b*≥*a* are the main semi-diameters along the principal axes {*x*_{1},*x*_{2}}.

By assuming that, within the cross section, the evolution of the normal stress from the purely elastic to the fully plastic state yields a bi-rectangular distribution of the yield stress *σ*_{Y}, with a discontinuity around the neutral axis
5.6the plastic bending can be easily computed by taking into account the static moments of the regions delimited by the neutral axis (Zyczkowski 1981). Indeed, after some algebraic calculations, it is easy to obtain
5.7Thus, the corresponding plastic multipliers are defined as follows:
5.8The interaction curve can be also obtained by equating to *M*_{+}. This leads to write, in the space of the generalized stresses , the equation
5.9which plots an ellipse.

A lower bound, *λ*_{L}, for the limit multiplier of the combined loading, *λ*_{+}, can be obtained by virtue of theorem 3.5. Indeed, equation (3.25) allows us to write the inequality
5.10Then, substitution of equation (5.8) into equation (5.10) gives
5.11which is trivially satisfied if the physically manifest hypotheses and are assumed to hold true.

In order to obtain an upper bound, *λ*_{U}, reference can be made to theorem 3.1. The mutual dissipation, as indicated in equation (3.1), is indeed equal to zero for both the first and the second loading conditions, being the bending moments coaxial with the principal axes {*x*_{1},*x*_{2}} of the cross section. The mutual dissipation (3.1) written in terms of generalized stresses (bending moments and generalized plastic strains (rate of plastic curvatures , gives
5.12Thus, the most accurate upper bound *λ*_{U} is given by
5.13

### (c) Example 3: thin-walled tube subject to torque and tension

A thin-walled tube, with mean radius *R* and thickness *t*, is subject to the action of a torque and an axial force, *N*^{0}. This case represents one of the basic testing arrangements to investigate yield criteria (Johnson & Mellor 1973) and therefore it can be considered to constitute a ‘gold standard’ for testing the material response to combined stresses. A fundamental prerequisite is that the axial and shear stresses can be considered constant along the cylinder axis, as well as in the hoop and radial directions on account of the thickness of the wall.

Figure 5 shows an element extracted from the wall at a point is the tensile stress induced by *N*^{0} and is the shear stress induced by , with {*r*,*ϑ*,*x*_{3}} being a cylindrical coordinate system where *x*_{3} is the cylinder axis. Within this framework, it is
5.14When the stress field is characterized by the normal and shear components only, say {*σ*,*τ*}, the principal stresses at ** P** are
5.15where {

*x*

_{I},

*x*

_{II},

*x*

_{III}} are the principal axes of stress. Thus,

*σ*

_{Y}the yield stress of the material according to the von Mises criterion 5.16the plastic domain for any combination of the loadings

*N*

^{0}and is given by an ellipse in the plane {

*σ*/

*σ*

_{Y},

*τ*/

*σ*

_{Y}} (figure 6

*a*), whose equation is 5.17Equation (5.17) and figure 6 immediately suggest that the limit multipliers {

*λ*

_{σ},

*λ*

_{τ}} can be written as 5.18and, by following the same line of reasoning, the limit multiplier for the combined loading is 5.19Theorem 3.5 states that a safety load multiplier

*λ*

_{L}can be found on account of the following inequality 5.20Substitution of equations (5.16) and (5.17) in equation (5.20), after some algebraic manipulations, allows to verify that 5.21In the range the per cent difference between the actual plastic multiplier

*λ*

_{+}and its lower bound

*λ*

_{L}, that is, 5.22attains a maximum at , i.e. 30 per cent. Figure 7 shows the behaviour of the per cent error (5.22) over the range .

Finally, on account of theorem 3.5, an upper bound *λ*_{U} for the combined loading limit multiplier *λ*_{+} can also be obtained. In fact, since the stress is constant, the strain rates at the critical state for both the simple loading cases are constant, too (figure 6*b*)
5.23where *σ*_{σ}=[*σ*_{I}=*σ*_{Y},*σ*_{II}=0,*σ*_{III}=0]^{T}, . The parameters have the meaning summarized in equation (2.5), and are obviously different from the limit multipliers.

By invoking the PVP, the sign of the external mutual dissipations can be deduced from that of the internal ones, that is,
5.24Equation (5.24) points out that both the mutual dissipations are non-negative, so that, by virtue of equation (3.1), it has to be
5.25On account of equations (5.18) and (5.19), the inequalities in equation (5.25) are easily verified. On the other hand, the best choice for the upper bound, *λ*_{U}, of *λ*_{+} depends on the ratio *ψ*_{0}=*τ*^{0}/*σ*^{0}. A study of equation (5.18) leads to
5.26and therefore
5.27where the normalized multipliers *Λ*_{(⋯ )}=*λ*_{(⋯ )}/*λ*_{σ} are employed to call off the common factor . Figure 7 shows the effectiveness of the obtained bounds.

### (d) Example 4: simply supported beam under the action of *n* forces

This final example shows an elementary but significant application of theorem 3.8.

Figure 8 shows a simply supported beam subject to an arbitrary but finite number of forces, such that
5.28Each force, located at the position *x*_{k}=*L*(*k*−1/2)/*n* from the left end of the beam, is affected by a certain multiplier, *λ*_{k}. The reaction at the support on the left can be written as
5.29Therefore, for the generic *k*th load case and for the combined loading, the limit multipliers are, respectively,
5.30where is the plastic bending. Without loss of generality, the analytical expression of *λ*_{+} has been obtained with reference to an arbitrary odd *n*, being a plastic hinge located at *L*/2. Therefore, by virtue of theorem 3.8, a lower bound, *λ*_{L}, can be easily obtained for an arbitrary number *n* of forces as
5.31

A comparison of the results found for *λ*_{L} and *λ*_{+} immediately points out that inequality (5.31) always holds true, provided that *n*≥1. Also, as can be expected, the per cent error can be estimated over the whole range of *n* and the following result is obtained
5.32

## 6. Conclusions

The study has presented some theorems that yield relevant bounds on the overall limit multiplier in the case of combined loads. The results can be regarded as a sort of rule of superposition of the load multipliers in classical limit analysis. The findings appear of theoretical interest and have also been shown to be useful in cases of combined loading. It has been pointed out that determining upper and lower bounds to the combined loading multipliers may allow a fairly accurate quantitative estimate of the actual carrying capacity of several structural problems.

## Footnotes

- Received May 4, 2009.
- Accepted September 21, 2009.

- © 2009 The Royal Society