The convexity about ultimate bearing hypersurfaces of structures

Abstract

The relationship between the convexity on the ultimate bearing surface of a structure and the second-order effects of loads is discussed. All of generalized non-overload forces acted on a structure forms a convex set when ignoring the second-order effects (coupling effects between the generalized forces). It is true also when the Hessian matrix composed of the second-order partial derivatives on the hypersurface about the ultimate bearing of the structure is negative definite. The outward convexity is kept when the surface is expressed by certain dimensionless parameters. A series of properties based on the convexity are pointed out. Some applications in the analysis of bearing capacity of structures were illustrated with examples. The study shows that an evaluation about the bearing capacity state of a complex structure can be made on the basis of several points on the surface of the ultimate bearing of the structure.

Keywords

service / limit load / ultimate bearing surface / convexity / generalized force / generalized displacement / frame structure

Introduction

Ultimate bearing capacities of structures (or members) are not only significance for mechanical theory but also for the foundation of modern structure design theory. Especially in the limit state design method based on the reliability ultimate bearing capacities of structures (or members) are much more important. It is no exaggeration to say that any essential understanding of the structure limit state, even if is extremely limited, may result in some of the fundamental change of design method. This article attempts to discussion about the limit state of the structures.

The convexity of bearing surfaces

Let R be the resistance of a structure, and let A be the action on the structure. Then service status function G is defined as follows

(1) G=A-R,

For a structure given, the service status function G can be expressed by generalized force vector P = (P₁, P₂,…, P_n) as

(2)

where the set of total interior points of hypersurface G(P) = 0 in vector space of real (n + 1)-tuples be denoted by W and its boundary \partial \Omega (see Fig. 1). Without loss of generality, can be limited to the discussion of P_i≥0 (i = 1, 2, …, n). For "P0∈ \partial \Omega and "P∉(W∪ \partial \Omega ) in e-neighborhood O(P₀, e), neglecting the higher-order epsilon, we have

(3) G(P_{\mathrm{1}},P_{\mathrm{2}},...,P_n)={\displaystyle \frac{\partial G}{\partial P_i}}d{P}_i=\nabla G·(\mathbf{P}-{\mathbf{P}}_{\mathrm{0}})>\mathrm{0},

where G be the gradient of the function G at the point P0 and operation“•”be the inner product in the n-dimensional linear space. When B ∉ (W∪ \partial \Omega ) located at the normal line ∇G (see Fig. 1), apparently the Eq. (3) still holds. Therefore it renders the outward normal line on the hypersurface G (P) = 0 positive (as shown in Fig. 1). If considering the second-order effect (The coupling effect between Pi which is geometric nonlinear effect essentially.), then have

(4) G(P_{\mathrm{1}},P_{\mathrm{2}},...,P_n)=\nabla G·(\mathbf{P}-{\mathbf{P}}_{\mathrm{0}})+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\displaystyle \frac{{\partial }^{\mathrm{2}}G}{\partial P_i\partial P_j}}d{P}_{i}d{P}_j>\mathrm{0},

or

(5) \nabla G·(\mathbf{P}-{\mathbf{P}}_{\mathrm{0}})>-{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\displaystyle \frac{{\partial }^{\mathrm{2}}G}{\partial P_i\partial P_j}}d{P}_{i}d{P}_j.

In which the Einstein summation convention [ 1] be used. Inequality (3) reveals that if ignoring the second-order effect the hypersurface about the ultimate bearing of the structure is outward convex (see Fig. 1). Moreover Inequality (5) reveals that the bearing surface of a structure must be outward convex when the matrix Hessian (composed of all the second-order partial derivatives) on the surface be negative definite. Therefore we may draw the following conclusion [ 2, 3].

All of generalized non-overload forces acted on a structure forms a convex set (W∪ \partial \Omega ) when ignoring the second-order effects (coupling effects between the generalized forces).

All of generalized non-overload forces acted on a structure forms a convex set (W∪ \partial \Omega ) when the Hessian matrix composed of the second-order partial derivatives on the hypersurface about the ultimate bearing of the structure is negative definite.

As be well known, in the classic plastic mechanics, plastic limit load curves (or surfaces) of structures are outward convex. While the buckling load limit curves of structures with high slenderness ratio are slightly outward concave [ 4, 5]. According to the theorems above, a reasonable explanation of this phenomenon may be made: in the traditional plastic mechanics, usually the intensity of the discussion is to consider the physical nonlinear damage problem, without considering the geometric nonlinearity (or second-order effect). Because of ignoring the second-order effect, so the limit load curves (surfaces) have always the convexity. In the buckling ultimate load analysis, the second-order effects cannot be ignored. The non-convexity indicates that the Hessian matrix on the buckling limit load curves (surfaces) is at least semi-positive. The plastic limit load curves (surfaces) for reinforced concrete structures also show outward convex [ 6].

Due to the convexity of a set, if the non-overload generalized forces form a convex set, for "Pi ∈ (W∪ \partial \Omega ) and "ai∈[0, 1] which satisfy

(6) {\displaystyle \sum _{i=\mathrm{1}}^m{a}_i=\mathrm{1}},

then

(7) a_i{\mathbf{P}}_i\in \Omega \cup \partial \Omega .

Thus, apparently the following conclusion is obtained.

For a structure given, if the non-overload generalized forces construct a convex set (W∪ \partial \Omega ), for "Pi ∈ (W∪ \partial \Omega ) the linear combination of P_i which satisfies Eqs. (6) and (7) must not be overload.

Bearing surfaces expressed by dimensionless parameters

Letting e_i (i = 1, 2,…, n) are the regularization bases vectors and Introducing {\Phi }_i={\phi }_i{e}_i (i = 1, 2,…, n;non Einstein summation), without loss of generality, any n-dimensional vector P can be expressed by the linear combination of {\Phi }_i as P = P_ie_i = {\xi }_i{\Phi }_i , apparently {\xi }_i = Pi /j_i (i = 1, 2, …, n) is a set of dimensionless parameters. {\Phi }_i can be called dimensionless orthogonal basis vector. We have

\begin{array}{l}G（P_{\mathrm{1}},P_{\mathrm{2}},\dots ,P_n）·(\mathbf{P}-{\mathbf{P}}_{\mathrm{0}})=（{\displaystyle \frac{\partial G}{\partial P_{\mathrm{1}}}},{\displaystyle \frac{\partial G}{\partial P_{\mathrm{2}}}},\dots ,{\displaystyle \frac{\partial G}{\partial P_n}}）(P_{\mathrm{1}}-P_{\mathrm{01}},P_{\mathrm{2}}-P_{\mathrm{02}},\dots ,P_n-P_{\mathrm{0}n})\\ =({\phi }_{\mathrm{1}}{\displaystyle \frac{\partial G}{\partial P_{\mathrm{1}}}},{\phi }_{\mathrm{2}}{\displaystyle \frac{\partial G}{\partial P_{\mathrm{2}}}},\dots ,{\phi }_n{\displaystyle \frac{\partial G}{\partial P_n}})({\displaystyle \frac{P_{\mathrm{1}}-P_{\mathrm{01}}}{{\phi }_{\mathrm{1}}}},{\displaystyle \frac{P_{\mathrm{2}}-P_{\mathrm{02}}}{{\phi }_{\mathrm{2}}}},\dots ,{\displaystyle \frac{P_n-P_{\mathrm{0}n}}{{\phi }_n}})\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em},\\ =({\displaystyle \frac{\partial G}{\partial {\xi }_{\mathrm{1}}}},{\displaystyle \frac{\partial G}{\partial {\xi }_{\mathrm{2}}}},\dots ,{\displaystyle \frac{\partial G}{\partial {\xi }_n}})({\xi }_{\mathrm{1}}-{\xi }_{\mathrm{01}},{\xi }_{\mathrm{2}}-{\xi }_{\mathrm{02}},\dots ,{\xi }_n-{\xi }_{\mathrm{0}n})=\nabla G({\xi }_{\mathrm{1}},{\xi }_{\mathrm{2}},\dots ,{\xi }_n)·(\psi -{\psi }_{\mathrm{0}})\end{array}

where the \psi and {\psi }_{\mathrm{0}} are vectors P and P₀ expressed separately by {\Phi }_i . In the same way also have

{\displaystyle \frac{{\partial }^{\mathrm{2}}G（P_{\mathrm{1}},P_{\mathrm{2}},\dots ,P_n）}{\partial P_i\partial P_j}}d{P}_{i}d{P}_j={\displaystyle \frac{{\partial }^{\mathrm{2}}G（{\xi }_{\mathrm{1}},{\xi }_{\mathrm{2}},\dots ,{\xi }_n）}{\partial {\xi }_i\partial {\xi }_j}}d{\xi }_{i}d{\xi }_j

So the following conclusion was proved.

The convexity of the surface G({\xi }_{\mathrm{1}},{\xi }_{\mathrm{2}},\dots ,{\xi }_n)=\mathrm{0} expressed by dimensionless parameters {\xi }_i (i = 1, 2,…, n) keeps invariably.

Introducing the norm in n-dimensional space as follows

\Vert \mathbf{P}\Vert =\Vert {{\displaystyle P}}_i{{\displaystyle \mathbf{e}}}_i\Vert =\sqrt{(\mathbf{P}\cdot \mathbf{P})}=\sqrt{{\displaystyle \sum _{i=\mathrm{1}}^n{{\displaystyle p}}_i^{\mathrm{2}}}},

and letting

(8) \rho =\Vert a_i{\mathbf{F}}_i\Vert ({\mathbf{F}}_i\in \mathrm{\Omega }\cup \partial \mathrm{\Omega }),

(9) \Pi =\{\mathbf{P}\left|\Vert \mathbf{P}\Vert \right|\le \rho \}.

If non-overload generalized forces construct a convex set and Q∈ \Pi , then Q∈ W∪ \partial \Omega in terms of the norm measurement.

Let F_i = {\Phi }_i = {\phi }_i{e}_i ∈(W∪ \partial \Omega ) (i = 1, 2, …, n;non Einstein summation), Eq. (8) becomes

(10) p=\Vert {{\displaystyle a}}_{\mathrm{1}}{{\displaystyle \mathrm{\Phi }}}_i\Vert =\sqrt{{\displaystyle \sum _{i=\mathrm{1}}^n{{\displaystyle a}}_i^{\mathrm{2}}{{\displaystyle \phi }}_i^{\mathrm{2}}}}.

If

(11) \mathrm{0}\le {\xi }_i\le a_i(i=\mathrm{1},\mathrm{2},\mathrm{...},n),

or

(12) {\displaystyle \sum _{i=\mathrm{1}}^n{{\displaystyle \xi }}_i^{\mathrm{2}}}\le {\displaystyle \sum _{i=\mathrm{1}}^n{{\displaystyle a}}_i^{\mathrm{2}}},

and observed Eq. (10), we have

\Vert \mathrm{P}\Vert =\Vert {{\displaystyle P}}_i{{\displaystyle \mathrm{e}}}_i\Vert =\Vert {{\displaystyle \xi }}_i{{\displaystyle \mathrm{\Phi }}}_i\Vert =\Vert {\displaystyle \sum _{i=\mathrm{1}}^n{{\displaystyle \xi }}_i{{\displaystyle \phi }}_i{{\displaystyle e}}_i}\Vert =\sqrt{{\displaystyle \sum _{i=\mathrm{1}}^n{{\displaystyle \xi }}_i^{\mathrm{2}}{{\displaystyle \phi }}_i^{\mathrm{2}}}}\le \sqrt{{\displaystyle \sum _{i=\mathrm{1}}^n{{\displaystyle a}}_i^{\mathrm{2}}{{\displaystyle \phi }}_i^{\mathrm{2}}}}=p.

Therefore the following conclusion is held.

If vector P={\xi }_i{\Phi }_i satisfies Eq. (11) or Eq. (12) then P\in \Omega \cup \partial \Omega in terms of the norm measurement.

It should be pointed out that ai in equations above should satisfy Eq. (6). Under this restriction it is easy to obtain (see Appendix):

(13) {{\displaystyle \left({\displaystyle \sum _{i=\mathrm{1}}^n{{\displaystyle a}}_i^{\mathrm{2}}}\right)}}_{\min }=\mathrm{1}/n,{{\displaystyle \hspace{1em}\left({\displaystyle \sum _{i=\mathrm{1}}^n{{\displaystyle a}}_i^{\mathrm{2}}}\right)}}_{\max }=\mathrm{1.}

Inserting the maximum value into the Eq. (12) the formula called n-dimensional spherical related relationships that quite widely used be derived follows:

(14) {\displaystyle \sum _{i=\mathrm{1}}^n{{\displaystyle \xi }}_i^{\mathrm{2}}}\le \mathrm{1.}

Since {\phi }_i is the ultimate bearing capacity under the alone action of {\Phi }_i and P_i is the i-th component of P, in most cases. There is {\xi }_i=P_i/{\phi }_i\le \mathrm{1.0} . Equation (14) is often referenced in this intuitive manner. Thus it is not hard to see, the so-called n-dimensional spherical correlation is actually a special case of non-overload set in terms of the norm measurement, in which a certain one a_i takes value 1.0 and the all rest of a_i take values 0.

As be well known, a comprehensive discussion about the surface G(P) = 0 will inevitably involve complex nonlinear mechanics (including geometrical and physical nonlinearity) analysis. Even in today when the computing technology has been developed highly, it is not an easy task. The propositions proved above show that it is possible that an assessment about the service status of a structure withstood complex effects may be made once several points F_i on the surface G(P) = 0 be known. Clearly, the analysis on non-overload by means of norm metric is not entirely equivalent to by means of the convex linear combination expressed in Eqs. (6) and Eq. (7). From the application point of view, the analysis by means of the formulas similar to Eq. (14) perhaps is more suitable than by means of Eq. (6) and Eq. (7).

Examples

Example-1 Analyze the bearing state of a cantilever with cross-sectional area A, plastic resistance moment Z and material yield limit f_y submitted to generalize force P as shown in Fig. 2(a).

If thinking about the strength limit load of the cantilever under the axial pressure to be Af_y, the generalized force F₁ = Af_ye₁ shown in Fig. 2 (b) is a point on the bearing capacity hypersurface G(P) = 0 ( \partial \Omega ) of the cantilever. Similarly, if considering the strength limit load of the cantilever under the bending Zf_y, the generalized force F₂ = Zf_ye₂ shown in Fig. 2(c) is another point on the bearing capacity hypersurface G(P) = 0 ( \partial \Omega ) of the cantilever.

It is possible to neglect second-order effect in the structural analysis for the engineering structures met the normal serviceability (For example, in European Union’s steel structure design standard [ 7] there are certain clauses concerning whether or not to consider the second-order effects in the calculation of frames), i.e., the coupling effects between generalized forces N and M can be neglected. Consequently, the all of non-overload generalized forces construct a convex set. As mentioned above, for any a∈[0, 1], then

a{\mathbf{F}}_{\mathrm{1}}+(\mathrm{1}-a){\mathbf{F}}_{\mathrm{2}}=aA{f}_y{\mathbf{e}}_{\mathrm{1}}+(\mathrm{1}-a)Z{f}_y{\mathbf{e}}_{\mathrm{2}}\subset (\mathrm{\Omega }\cup \partial \mathrm{\Omega }),

Therefore, if exists 0≤a0≤1 renders

\mathbf{P}=N{\mathbf{e}}_{\mathrm{1}}+M{\mathbf{e}}_{\mathrm{1}}=a_{\mathrm{0}}{\mathbf{F}}_{\mathrm{1}}+(\mathrm{1}-a_{\mathrm{0}}){\mathbf{F}}_{\mathrm{2}}=a_{\mathrm{0}}A{f}_y{\mathbf{e}}_{\mathrm{1}}+(\mathrm{1}-a_{\mathrm{0}})Z{f}_y{\mathbf{e}}_{\mathrm{2}},

held, in other words, if the two equations

N=a_{\mathrm{0}}A{f}_y,\hspace{1em}M=(\mathrm{1}-a_{\mathrm{0}})Z{f}_y,

held simultaneously then we have a conclusion clearly that the structure loaded the P is in non-overload state.

Let j and jb denote the reduction factor [11] for compression buckling and for bend buckling respectively. Correspondingly let F₁ = jAf_ye₁ and F₂ = jbWf_ye₂ (as shown in Fig. 2). As mentioned above, the analysis and experiments of members with high slenderness ratio show that the coupling effects between F₁ and F₂ cannot be ignored and it makes the outward convexity vanished.

Example-2 As shown in Fig. 3(a), the load on the framework structure can be expressed as P = pe₁ + qe₂ + se₃. Three points named F₁, F₂ and F₃ (see Fig. 3(b) – Fig. 3 (d)) on the surface G(P) = 0 have been obtained (by structural tests or numerical analysis). The stiffness designed ensures that the second-order effect can be negligible. Try to analyze its service status.

If we can find a_i∈[0, 1] (i = 1, 2, 3) satisfy Eq. (6) and

\mathbf{P}=P{\mathbf{e}}_{\mathrm{1}}+q{\mathbf{e}}_{\mathrm{2}}+s{\mathbf{e}}_{\mathrm{3}}=a_i{\mathbf{F}}_i=a_{\mathrm{1}}P{\mathbf{e}}_{\mathrm{1}}+a_{\mathrm{2}}Q{\mathbf{e}}_{\mathrm{2}}+a_{\mathrm{3}}S{\mathbf{e}}_{\mathrm{3}}

the load P will renders structural non- overload certainly. For example

\mathbf{P}=\mathrm{5.3}(kN/\mathrm{m}){\mathbf{e}}_{\mathrm{1}}+\mathrm{3.6}(kN/m){\mathbf{e}}_{\mathrm{2}}+\mathrm{12.6}(kN/m){\mathbf{e}}_{\mathrm{3}}

{\mathbf{F}}_{\mathrm{1}}=\mathrm{23.04}(kN/m){\mathbf{e}}_{\mathrm{1}},\hspace{1em}{\mathbf{F}}_{\mathrm{2}}=\mathrm{11.61}(kN/m){\mathbf{e}}_{\mathrm{2}},\hspace{1em}{\mathbf{F}}_{\mathrm{3}}=\mathrm{46.96}(kN/m){\mathbf{e}}_{\mathrm{3}}.

Obviously then there are a₁ = 0.23, a₂ = 0.31, a₃ = 0.46 and

{{\displaystyle a}}_{\mathrm{1}}+{{\displaystyle a}}_{\mathrm{2}}+{{\displaystyle a}}_{\mathrm{3}}=\mathrm{0.23}+\mathrm{0.31}+\mathrm{0.46}=\mathrm{1.0},

\mathbf{P}=\mathrm{5}.\mathrm{3}{{\displaystyle \mathbf{e}}}_{\mathrm{1}}+\mathrm{3.6}{{\displaystyle \mathbf{e}}}_{\mathrm{2}}+\mathrm{21.6}{{\displaystyle \mathbf{e}}}_{\mathrm{3}}=\mathrm{0.23}\times \mathrm{23.04}{{\displaystyle \mathbf{e}}}_{\mathrm{1}}+\mathrm{0.31}\times \mathrm{11.61}{{\displaystyle \mathbf{e}}}_{\mathrm{2}}+\mathrm{0.46}\times \mathrm{46.96}{{\displaystyle \mathbf{e}}}_{\mathrm{3}},

i.e., the structure shown Fig. 3 (a) is in non-overload status.

Example-3 A continuous beam structure loaded as shown in Fig. 4(a). Ignore the second-order effect and analyze its service status.

There are several models to analyze this problem. For example, we can introduce the ultimate load of each concentration force (or moment) acted on the continuous beam separately, i.e., let (non Einstein summation).

F_i=T_i{e}_i,\hspace{1em}{F}_{\mathrm{5}}=M{e}_{\mathrm{5}}\hspace{1em}(i=\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4}),

then the continuous beam load is expressed as

P=t_{\mathrm{1}}{e}_{\mathrm{1}}+t_{\mathrm{2}}{e}_{\mathrm{2}}+t_{\mathrm{3}}{e}_{\mathrm{3}}+t_{\mathrm{4}}{e}_{\mathrm{4}}+m{e}_{\mathrm{5}}.

If the loads meet the proportional loading condition (i.e. the ratio between arbitrary two ti keeps unchanged during the loading process), then the scheme A shown in Fig. 5 can be chosen. This means that the analysis is made by introducing two points on the surface G(P) = 0, so the load of the continuous beam can be expressed as

P=t_{\mathrm{1}}{e}_{\mathrm{1}}+m{e}_{\mathrm{2}}.

If there is a∈[0, 1] so that

P=t_{\mathrm{1}}{e}_{\mathrm{1}}+m{e}_{\mathrm{2}}=a{T}_{\mathrm{1}}{e}_{\mathrm{1}}+(\mathrm{1}-a)M{e}_{\mathrm{2}},

in other words, if

a{T}_{\mathrm{1}}=t_{\mathrm{1}}\hspace{1em}and\hspace{1em}(\mathrm{1}-a)M=m,

hold simultaneously then P will render the structure non-overload certainly.

Similarly, the scheme shown in Fig. 6 may be chosen which means that the analysis is made by introducing three points on the surface G(P) = 0.And no longer gives unnecessary detail.

Conclusion

1) The ultimate bearing surface of a structure is outward convexity when second-order effect of the loads on the structure can be neglected. Therefore all of generalized non-overload forces acted on the structure forms a convex set.

2) The ultimate bearing surface of a structure is outward convexity when the Hessian matrix on the surface is negative definite. Therefore all of generalized non-overload forces acted on the structure forms a convex set also.

3) The outward convexity is kept invariant when the surface is expressed by certain dimensionless parameters.

4) In Conclusion a and b the linear combination of loads can be constructed based on the convexity of the ultimate bearing surface of a structure to analyze the status of service.

5) The evaluation of the service status based on the norm measurement may be more appropriate.

The analysis only based on several basic load effects whether by means of convex linear combination or by means of norm measurement. Apparently, it effectively improves the efficiency of structural analysis. At the same time, since the service status is estimated directly to the whole structure, rather than its components. It may help the structure design method to leap from the checking members to evaluating the structure directly.

Due to the logical symmetry of generalized forces and generalized displacements, the conclusions above keep true when generalized forces are replaced by the corresponding generalized displacements. For example, we have conclusion:

All of generalized non-overload displacements forms a convex set when ignoring the second-order effects (coupling effects between the generalized displacements).

References

[1] Nadir Jeevanjee. An Introducing to Tensors and Group Theory for Physicists. Springer Science+Business Media, 2011

[2] Hiriart-Urruty J B, Lemarechal C. Fundamentals of Convex Analysis. Springer-Verlag Berlin Heidelberg, 2001

[3] Tyrrell Rockafellar R. Convex Analysis. Princeton University Press, 1997

[4] Patrick J D, Peter R K, Gaham W O. Structural Steel Design. Cambridge Press, London, 1988, 221–222

[5] ROBERT ENGLEKIRK. Steel Structures, Johnwiley& Sons. Inc, New York, 1994, 372–375

[6] Spiegel L, Limbrunner G F. Reinforced Concrete Design. Prentice Hall PTR, 1998, 302

[7] European Committee for Standardization. Eurocode 3: Design of steel structures-Part 1–1: General rules. Brussels, 2005

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Higher Education Press and Springer-Verlag Berlin Heidelberg