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Mathematics in civil engineering: set theory, algebra and analysis
By Ir Lincoln W H LAU

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Prologue

 

Mathematics is a vital tool in civil engineering modelling. For instance, the Euler’s critical load formula for buckling of columns was developed by Leonhard Euler, the renowned Swiss mathematician, in the 18th century. In this Feature Story, some advanced mathematical concepts and their uses in civil engineering modelling will be shared, succinctly, with the readers.

 

Self Photos / Files - Screenshot 2024-12-05 142803

Collaboration between civil engineer and mathematician

 

Rough set theory (as a methodology for decision making)

 

Suppose set S is defined as a collection of objects: for example, the conditions of concrete samples collected at various building locations by the owner. Each object has several “condition” attributes, such as the age of concrete, environmental exposure conditions (nominal values: industrial, non-industrial), extent of cracked concrete or exposed rebar (nominal values: large, medium, small area), severity of rebar corrosion, extent of Alkali-Aggregate Reaction (AAR), location of concrete samples, and so forth.

 

For the purpose of determining whether concrete repair is necessary for an object, each object shall be associated with a “decision” attribute assuming a nominal value such as (i) “need to repair” or (ii) “no need to repair”. The objects and their attributes (including condition and decision) can be expressed in an information table. The set S is taken as an information system.

 

Indiscernibility relation is the relation between two objects or more, where all the nominal values are identical in relation to a subset of considered attributes. For example, if only environmental exposure condition is considered, since there are only two nominal values for this attribute, there will be two indiscernibility elementary sets.

 

If the decision attribute is considered, there will be two indiscernibility elementary sets, namely “need to repair” set, and “do not need to repair” set. Sometimes there are objects from each of these two elementary sets that have all the same condition attributes. It can be that their decision attributes are ascribed by different decision makers. These objects will be classified as inconclusive and possibly in need of repair.

 

From set S, we can derive a target set S* containing objects recommended for concrete repair by the building owner to the engineer for decision. A lower approximate of set S* is defined as containing objects in need of repair. An upper approximate of set S* is defined as containing objects possibly in need of repair, including those inconclusive objects. The pair of upper and lower approximate sets is called Rough Set of S*.

 

For some objects in set S*, their condition and decision attributes are same and redundant. Therefore, only one object is sufficient to represent these objects. Some condition attributes are correlated and thus dependent. Hence, some unnecessary attributes can be eliminated.

 

For the set S*, after removing redundant objects and eliminating correlated attributes, the resulting set of condition attributes is called a Reduct. For each object in the Reduct, there will be a decision rule. Hidden patterns may be discovered also.

 

The use of rough set and related theories can be found, for example, in the following literature:

 

  1. Wang L, Yuan C C, Lu D G, Zhang S H (2006). ‘Application of rough sets in high-rise building structure’s knowledge discovery’. Journal of Harbin Institute of Technology, 38(12), pp. 2073–2076.
  2. Gorsevski P V and Jankowski P (2008). ‘Discerning landslide susceptibility using rough sets’. Computers Environment and Urban Systems , 32(1), pp. 53-65.
  3. Qu J, Bai X, Gu J, Taghizadeh-Hesary F, and Lin J (2020).‘Assessment of rough set theory in relation to risks regarding hydraulic engineering investment decisions’. Mathematics, 8(8), 1308.

 

Fuzzy logic (as a methodology for decision making)

 

In Boolean logic, the truth values of variables can only be the integer values “0” or “1”. But fuzzy logic deals with the variables’ many-valued truth values, which may be any real number between “0” and “1”. The fuzzy logic value reflects the degree of uncertainty.

 

For example, in Boolean logic, the answer to the question “Is Sam honest?” is yes (“1”) or no (“0”). But in fuzzy logic, the answers can be from a 4-member set {extremely honest - “1”, very honest - “0.8”, honest occasionally - “0.4”, extremely dishonest - “0”}.

 

There are fuzzy logic operators corresponding to those of Boolean logic. For example, the Boolean “AND (x,y)” corresponds to the Fuzzy “Min(x,y)”. The Boolean “OR(x,y)” corresponds to the Fuzzy “Max(x,y)”. Additionally, the Boolean “NOT(x)” corresponds to the fuzzy “1-x”.

 

The fuzzy system can be used in decision making with the following methodology:

 

  1. Fuzzify all input values of a system into fuzzy membership functions;
  2. Apply rules in the rule database to compute the fuzzy output functions to get the truth values;
  3. De-fuzzify the fuzzy output truth values to get quantifiable output values; and
  4. Make decision based on the quantifiable output values above.

 

Fuzzy logic has been used, for example, in the following literature:

 

  1. Ganguli R (2016). ‘Uncertainty handling using Fuzzy Logic in Structural Health Monitoring’. 8th International Symposium on NDT in Aerospace (AeroNDT 2016). 22(1). Bangalore, India. November.
  2. Liao H and Plebankiewicz E (2021). ‘Applications of fuzzy technology in civil engineering and construction management’. Journal of Civil Engineering and Management, 27(6), pp. 355-357.
  3. Madanda V C, Sengani F, and Mulenga F K (2023). ‘Applications of Fuzzy Theory-Based Approaches in Tunnelling Geomechanics: a State-of-the-Art Review’. Mining Metallurgy & Exploration, 40, pp. 819-837.

 

Self Photos / Files - 2Advanced mathematical tools have their place in structural health monitoring

 

Nonlinear analysis

 

A nonlinear analysis in structural engineering is one where a nonlinear relation holds between applied forces and displacements. Nonlinear effects can originate from geometrical nonlinearity (namely, large deformations, buckling), material nonlinearity (namely, elasto-plastic material), and contact force.

 

When two non-rigid bodies encounter each other, the geometry at the contact region changes until the force or stress in the system comes to an equilibrium condition. This process introduces a nonlinearity which is dealt by contact elements. This type of nonlinearity is called contact nonlinearity.

 

The stiffness matrices of the structures governed by non-linear relations will not be constant during loading, and a different solver is needed.

 

Non-linear buckling analysis is a static method which accounts for material and geometric nonlinearities (P-Δ and P-δ), perturbations, geometric imperfections, and so forth. (Readers who have taken the relevant undergraduate courses should have been taught, in them, what a linear buckling is. On the other hand, post buckling analysis allows us to determine the load carrying capacity of a structure after buckling. The analysis is of course nonlinear and can be static or dynamic.)

 

For nonlinear dynamic analysis, we cannot use response spectrum analysis, which depends on superposition and is linear in nature. In addition, some structural components can yield during dynamic actions. Thus, inelastic properties and deformation (plastic hinge rotation for example) are called into play. Moreover, stiffnesses degradation under cyclic loading and initial elastic stiffness are also involved.

 

To tackle non-linearity in mathematical formulation, we need to linearise. There is a profusion of linearisation methods, such as piecewise approximations and Taylor’s Theorem. It is a subject that has undergone much research.

 

Coupled with the finite element method, non-linear analysis is used recently, for example, in the following literature:

 

  1. Engen M, Hendriks M A N, Øverli J A, and Åldstedt E (2017). ‘Non-linear finite element analyses applicable for the design of large reinforced concrete structures’. European Journal of Environmental and Civil Engineering, 23(11), pp. 1381-1403.
  2. Saeed H H and Abed H (2023). ‘Nonlinear finite element Analysis of laterally loaded piles in Layered Soils’. Electronic Journal of Structural Engineering, 23(3), pp.1-5.

 

Eigenvalue problem

 

The related eigenvalue problem can occur in structural dynamics. By solving it, engineers can obtain the natural frequencies and critical modes of vibration. They can also assess the dynamic behaviour of structures under different loading conditions.

 

Eigenvalue analysis also predicts the theoretical buckling strength of a structure which is idealised as elastic, as in linear buckling analysis.

 

For the slope analysis, the 3D limit equilibrium analysis can be reduced to the solution of a generalised eigenvalue problem in which the largest real eigenvalue is just the factor of safety (FoS). This is discussed in Hong Zheng’s 2009 article titled “Eigenvalue Problem from the Stability Analysis of Slope”.

 

The eigenvalue problem can be approximated by the finite element method. This method is discussed in Jaap van der Vegt’s 2021 lecture “Finite Element Methods for Eigenvalue Problems”.

 

Tensor analysis

 

Tensors are simply mathematical objects that can be used to describe physical properties. Scalars and vectors are two examples. But tensors are a generalisation of scalars and vectors when a scalar is a zero rank tensor, and a vector is a first rank tensor. The rank (or order) of a tensor is defined by the number of directions (or the dimensionality of the array) required to describe it. For example, properties that require one direction (first rank) can be described by a 3×1 column vector, and properties that require two directions (second rank tensors) can be described by nine numbers, as a 3×3 matrix. As such, in general an nth rank tensor can be described by 3n coefficients.

 

Tensor is quite useful in 3-D stress analysis. The use of it rather than vector in better modelling rock stress is proved by Ke Gao and J P Harrison in their 2019 article “Examination of Mean Stress Calculation Approaches in Rock Mechanics”. Tensor analysis is further found, for example, in:

 

  • Nguyen L D K, Zhang B, Wang Y, Liu W, Chen F, Mustapha S and Runcie P (2015). ‘On Damage Identification in Civil Structures Using Tensor Analysis’, Lecture Notes in Computer Science, 9077, pp. 459-471.

 

Furthermore, in geotechnical engineering, a moment tensor describes the deformations (or a group of equivalent forces) at the source location that generates seismic waves; A moment tensor is similar to a stress tensor describing the state of stress at a particular point. These deformations can be recovered by seismic waves recorded by sensors. To interpret these data, we need a moment tensor inversion method. Such a method is applied, for example, in:

 

  • Men J J, Zhao X, Zhu L and Wang X D (2017). ‘Moment-tensor-based Acoustic Emission Detection Method for Reinforced Concrete Structure’. Journal of Disaster Prevention and Mitigation Engineering, 5, pp. 822-827, 841.

 

Fractional calculus

 

In ordinary differential calculus, we have 1st order derivative and 2nd order derivative. But are there such things as 1/2 derivative, 1.5 derivative, and so on and so forth? Yes: in fractional calculus, of which they are the subject matter. Fractional calculus is a generalisation of ordinary calculus with its derivatives and integrals of arbitrary real or complex order. Behind it, of course, is a complex theory. Researchers have used it much in the same way they have used complex analysis in modelling, that is, without bothering too much on the physical meaning of fractional calculus.

 

Although conceptually difficult, fractional calculus is used in civil engineering modelling sometimes, for example, in:

 

  1. Erjian W, Hu B, Li J, Cui K, Zhang Z, Cui A, and Ma L (2022). ‘Nonlinear Viscoelastic-Plastic Creep Model of Rock Based on Fractional Calculus’. Advances in Civil Engineering, 6, pp. 1-7.
  2. Bienert J, Maeder M, Marburg S, Chocholaty B and Islam M (2024). ‘Modal analysis on highly damped structures using fractional calculus’. Preprint submitted to Mechanical Systems and Signal Processing.

 

Epilogue

 

At this juncture, it is hoped that the readers, especially those who are practising engineers, have found these concepts rewarding and capable of being transmuted into innovative and creative ideas in daily problem solving. Indeed, there is still a plethora of other mathematical concepts used in civil engineering modelling. My hope is to canvass them in a later article.

 

About the author

 

Ir Lau Wai Hin, Lincoln is a Corporate Member of the HKIE (Civil and Geotechnical Disciplines) who holds a Bachelor of Science in Civil Engineering (honours), a Master of Philosophy in Engineering, and a Postgraduate Certificate in Education (Mathematics), all from The University of Hong Kong. He is now a practising Civil and Geotechnical Engineer in a local consultant firm and has over 25 years of engineering experiences.

 

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