Substructural Analysis Essay

1. Introduction

With the development of transportation infrastructure, urban road network density is increasing, and road intersections are worsening. Urban overpasses are an effective way to solve this problem. In the design and planning of urban overpasses, curved bridges can make road alignment smooth and beautiful. They also reduce the effect on ground transportation and play an importation role in relieving urban traffic congestion [1,2]. Considering restrictions of topography, space, and other factors, irregular crotches with variable width, slope, and curvature are usually adopted to connect main bridges and ramps. Irregularly-shaped bridges, such as the one shown in Figure 1a, become the main structural form of urban overpass. However, a number of urban overpasses with irregular crotches, such as the one shown in Figure 1b, have been destroyed while in operation in China [3]. It indicates that design methods and techniques are not mature in practice. Research on the modal characteristics of irregularly-shaped bridges can provide references for irregularly-shaped bridge design, and gradually improve the safety operations of these structures.

The cross-section of an irregularly-shaped bridge mainly adopts thin-walled box girders [4]. It increases the complexity of analyzing the modal characteristics of irregularly-shaped bridges due to the combinational effect of bending and torsion in thin-walled box girders. A number of studies have been conducted on the calculation principles and analysis methods for the modal characteristics of curved thin-walled box girder bridges [5,6,7,8,9]. However, the main girder and ramp mutually contact and restrict each other at the crotch of an irregularly-shaped bridge. Its dynamic characteristics present a complicated coupling effect with bending and torsion, which are different from the conventional curved structure.

Finite element analysis provides an effective way to analyze the modal characteristics of this structure. Yoon et al. [10] proposed a curved beam element with seven degrees of freedom at each node, and put forward the formula of element stiffness and mass matrices to analyze the modal characteristics of thin-walled curved beams. Sapountzakis et al. [11] dispersed the curved beam by straight line elements—considering warping and shear deformation—to calculate the structural modal characteristics, which verified the accuracy of the curved beam model to replace broken lines. Lu et al. [4] analyzed the modal characteristics of irregularly-shaped bridges using shell elements. A comparative analysis with a grillage method is performed to prove the accuracy of a proposed method. However, the complex structure is divided into a large number of units with multiple degrees of freedom, which would seriously affect the computational efficiency and accuracy of the finite element method. Especially, when the bridge’s parameters change, the finite element model reconstruction increases the analysis’ complexity. A substructure method is widely used because it can minimize the size of matrices, and reduces the expense of computational time. The basic idea of the substructure method is to decompose the complex structure into several substructures according to structural features. In addition, analysis on each respective substructure is performed. Overall modal characteristics can be obtained by comprehensively assembling all the substructures according to boundary conditions [12]. If the local design parameter is changed in the structural model, only the modal characteristics of the corresponding substructure should be recalculated. It could effectively improve operational efficiency and achieve overall analysis of the complex structure.

As an important component of substructure methods, the dynamic substructure method can take advantage of the modal characteristics of each subsystem with a simple calculation process to finally obtain the modal characteristics of the whole structure. Since the 1960s, the dynamic substructure method has been developing rapidly [13]. Especially in recent years, this method has been increasingly applied in the field of Bridges. Biondi et al. [14] presented a substructure approach for analyzing the dynamic response of the train–rails–bridge system. It regards the train, rail, and bridge deck as three substructures, and simultaneously computes the dynamic responses of the train, rail, and bridge to analyze vehicle–bridge dynamic interaction. Li et al. [15] proposed the damage identification strategy based on the dynamic substructure method, which divided the complex structure into several substructures, and identified the structural damage based on the dynamic response of the substructures under moving load excitation. Kong et al. [12] demonstrated a new substructure approach to analyzing the vehicle-induced vibration of long-span hybrid cable-stayed bridges. It divided the bridge into many substructures with a reasonable length and condensed the substructures model in detail with refined mesh into super elements. It used the dynamic substructure mode synthesis method to analyze the vehicle-induced dynamic response under deterministic flows.

Based on different solutions to the problem, dynamic substructure methods can be divided into the component mode synthesis (CMS) method, the interface displacement synthesis method, the migration substructure method, and the super element method. CMS is widely used as the most mature theory among them [16,17,18]. Based on the differences between some treatments to the interface, CMS can be divided into the fixed interface component mode synthesis method, the free interface component mode synthesis method, and the hybrid component mode synthesis method. The free interface component mode synthesis method proposed by Hou has been developing rapidly and is widely used [19,20,21,22]. It does not contain the interface node displacement coordinates in the comprehensive equation, which is convenient to combine and verify with the experimental mode technique. The double coordinate free-interface mode synthesis method possesses outstanding advantages in structural modal analysis with high calculation accuracy.

The free-interface mode synthesis method is gradually applied to the modal analysis of bridges. However, there is very little investigation on the modal characteristics of irregularly-shaped bridges based on this method. In this paper, the double coordinate free-interface mode synthesis method is adopted to analyze the modal characteristics of an irregularly-shaped bridge. Ramp curve radius, crotches diaphragm beam stiffness, cross-section feature, and bearing conditions are selected as feature parameters. Their influence on the modal characteristics of an overall irregularly-shaped bridge is analyzed. The analysis can provide a reference for design and can improve the safety standard. However, there is a long distance from the parameters’ impact on the modal characteristics of the irregularly-shaped bridge to the impact on dynamic behaviours. The critical analysis on the corresponding structural dynamic response beyond modal analysis will be a matter for future studies. Additionally, the parameters’ impact on the dynamic responses of irregularly-shaped bridges considering nonlinear behaviours is also important [23,24]. Pushover analysis on the irregularly-shaped bridge will be conducted in future studies to enrich the conclusion [25].



Philosophical logic

“Curry’s Paradox” (with Jc Beall), new article in the Stanford Encyclopedia of Philosophy (first published 2017).

“LP, K3 and FDE as Substructural Logics”, in Logica Yearbook 2016, ed. P. Arazim and T. Lavička (College Publications, 2017).

“The Very Idea of a Substructural Approach to Paradox”, forthcoming in Synthese

“Naive Structure, Contraction and Paradox”Topoi 34 (2015): 75-87

Rejecting structural contraction has been proposed as a strategy for escaping semantic paradoxes. The challenge for its advocates has been to make intuitive sense of how contraction might fail. I offer a way of doing so, based on a “naive” interpretation of the relation between structure and logical vocabulary in a sequent proof system. The naive interpretation of structure motivates the most common way of blaming Curry-style paradoxes on illicit contraction. By contrast, the naive interpretation will not as easily motivate one recent noncontractive approach to the Liar paradox.

“Validity and Truth-Preservation”(with Julien Murzi), in Unifying the Philosophy of Truth, ed. T. Achourioti, K. Fujimoto, H. Galinon, and J. Martínez-Fernández (Springer, 2015)

The revisionary approach to semantic paradox is commonly thought to have a somewhat uncomfortable corollary, viz. that, on pain of triviality, we cannot affirm that all valid arguments preserve truth. We show that the standard arguments for this conclusion all break down once (i) the structural rule of contraction is restricted and (ii) how the premises can be aggregated—so that they can be said to jointly entail a given conclusion—is appropriately understood. In addition, we briefly rehearse some reasons for restricting structural contraction.

“Validity Curry Strengthened”, Thought 2 (2013): 100-107

Several authors have argued that a version of Curry’s paradox involving validity motivates rejecting the structural rule of contraction. This paper criticizes two recently suggested alternative responses to “validity Curry.” There are three salient stages in a validity Curry derivation. Rejecting contraction blocks the first, while the alternative responses focus on the second and third. I show that the distinguishing feature of validity Curry, as contrasted with more familiar forms of Curry’s paradox, is that paradox arises already at the first stage. Accordingly, blocking the second or third stages won’t suffice for resolving the paradox.

“Deflating Logical Consequence”,Philosophical Quarterly 61 (2011): 320-42

Deflationists about truth seek to undermine debates about the nature of truth by arguing that the truth predicate is merely a device that allows us to express a certain kind of generality.  I argue that a parallel approach is available in the case of logical consequence.  Just as deflationism about truth offers an alternative to accounts of truth’s nature in terms of correspondence or justification, deflationism about consequence promises an alternative to model-theoretic or proof-theoretic accounts of consequence’s nature.  I then argue, against considerations put forward by Hartry Field and Jc Beall, that Curry’s paradox no more rules out deflationism about consequence than the Liar paradox rules out deflationism about truth.

“Expressibility and the Liar’s Revenge”,Australasian Journal of Philosophy 89 (2011): 297-314

There is a standard objection against purported explanations of how a language L can express the notion of being a true sentence of L. According to this objection, such explanations avoid one paradox (the Liar) only to succumb to another of the same kind. Even if L can contain its own truth predicate, we can identify another notion it cannot express, on pain of contradiction via Liar-like reasoning. This paper seeks to undermine such ‘revenge’ by arguing that it presupposes a dubious assumption about the linguistic expression of concepts. Successful revenge would require that there be a notion other than truth that plays the same role with respect to concept-expression that truth is naturally thought to play before we are confronted with the Liar paradox.

“The Rationale Behind Revision-Rule Semantics”,Philosophical Studies 129 (2006): 477-515

According to Gupta and Belnap, the “extensional behavior” of ‘true’ matches that of a circularly defined predicate.  Besides promising to explain semantic paradoxicality, their general theory of circular predicates significantly liberalizes the framework of truth-conditional semantics.  The authors’ discussions of the rationale behind that liberalization invoke two distinct senses in which a circular predicate’s semantic behavior is explained by a “revision rule” carrying hypothetical information about its extension. Neither attempted explanation succeeds.  Their theory may however be modified to employ a relativized notion of extension.  The resulting contextualist semantics for ‘true’ construes circularity as a pragmatic phenomenon.


Philosophy of language

“Logical Expressivism and Logical Relations”, in From Rules to Meanings: New Essays on Inferentialism, ed. O. Beran, V. Kolman, and L. Koreň (Routledge, 2018).

According to traditional logical expressivism, logical operators allow speakers to explicitly endorse claims that are already implicitly endorsed in their discursive practice – endorsed in virtue of that practice’s having instituted certain logical relations. Here, I propose a different version of logical expressivism, according to which the expressive role of logical operators is explained without invoking logical relations at all, but instead in terms of the expression of discursive-practical attitudes. In defense of this alternative, I present a deflationary account of the expressive role of vocabulary by which we ascribe logical relations.

“Assertoric Force Perspectivalism: Relativism Without Relative Truth”Ergo 1 (2014): 139-168.

According to relativist accounts of discourse about, e.g., epistemic possibility and matters of taste, the truth of propositions must be relativized to nonstandard parameters. This paper argues that the central thrust of such accounts should be understood independently of relative truth, in terms of a perspectival account of assertoric force. My point of departure is a stripped-down version of Brandom’s analysis of the normative structure of discursive practice. By generalizing that structure, I make room for an analogue of the “assessment sensitivity” MacFarlane characterizes in terms of relative truth. I argue that my reformulation supplies a stronger rationale for the most distinctive feature of MacFarlane’s brand of relativism, its account of when speakers ought to retract assertions. Furthermore, I show that the view usually regarded as a “moderate” alternative to MacFarlane’s “radical” relativism requires the more radical deviation from an absolutist account of assertoric force.

“Linguistic Function and Content: Reflections on Price’s Pragmatism”,  Philosophical Quarterly 64 (2014): 497-506

Huw Price proposes a strategy for dissolving ontological puzzles through a pragmatist account of our conceptual activity. Here I consider the proper place for conceptual content in Price’s pragmatism. Price himself rules out any explanatory role for content, just as he rules out any explanatory role for representational notions such as reference and truth. I argue that the cases are disanalogous, and that he offers no good reasons for avoiding explanatory appeal to content. Furthermore, I argue that doing so is incompatible with his pragmatist project.

“Naïve Truth-Conditions and Meaning”,Philosophical Quarterly 58 (2008): 265-77

Critics of explaining meaning in terms of truth-conditions have tended to charge proponents with misconceptions regarding truth.  I argue that the “naive” version of the truth-conditional theory that best accounts for its resilience fails on a different and more basic ground, namely a circularity arising from the contingency of meaning.  One reason this problem has been overlooked is a tendency (noted by Dummett in a different connection) to assimilate the naive truth-conditional theory to an idealized verificationism.

“Brandom on the Normativity of Meaning”,Philosophy and Phenomenological Research 68 (2004): 141-60

Brandom’s “inferentialism”—his theory that contentfulness consists in being governed by inferential norms—proves dubiously compatible with his own deflationary approach to intentional objectivity.  This is because a deflationist argument, adapted from the case of truth to that of correct inference, undermines the criterion of adequacy Brandom employs in motivating inferentialism.  Once that constraint is abandoned, moreover, the very constitutive-explanatory availability of Brandom’s inferential norms becomes suspect.  Yet Brandom intertwines inferentialism with a separate explanatory project, one that in explaining the pragmatic significance of meaning-attributions does yield a convincing construal of the claim that the concept of meaning is normative.

Early modern philosophy

“Intentionality Bifurcated: A Lesson from Early Modern Philosophy?”, in Contemporary Perspectives on Early Modern Philosophy: Nature and Norms in Thought, ed. M. Lenz and A. Waldow (Springer, 2013).

This paper situates the conclusions of the following two papers (presented here in highly condensed form) in a broader philosophical context. I examine the pressures leading Descartes and Locke to invoke two ways in which thought is directed at objects. For each philosopher, the two kinds of “ofness” come apart as a result of a distinctive rationalist commitment. However, I suggest that understanding why Descartes and Locke are driven to bifurcate intentionality should carry a lesson even for those present-day theorists who reject both Cartesian and Lockean rationalism.

“Objective Being and ‘Ofness’ in Descartes”,Philosophy and Phenomenological Research 84 (2012): 378-418

It is generally assumed that Descartes invokes “objective being in the intellect” in order to explain or describe an idea’s status of being “of something.”  I argue that this assumption is mistaken.  As emerges in his discussion of “materially false ideas” in the Fourth Replies, Descartes recognizes two senses of ‘idea of’.  One, a theoretical sense, is itselfintroduced in terms of objective being.  Hence Descartes can’t be introducing objective being to explain or describe “ofness” understood in this sense.  Descartes also appeals to a pretheoretical sense of ‘idea of’.  I will argue that the notion of objective being can’t serve to explain or describe this “ofness” either.  I conclude by proposing an alternative explanation of the role of objective being, according to which Descartes introduces this notion to explain the mind’s ability to attain clear and distinct ideas.

“Two Kinds of Intentionality in Locke”, Pacific Philosophical Quarterly 91 (2010): 554-86

Ideas play at least two roles in Locke’s theory of the understanding. They are constituents of “propositions,” and some of them “represent” the qualities and sorts of surrounding bodies. I argue that each role involves a distinct kind of intentional directedness. The same idea will in general be an “idea of” two different objects, in different senses of the expression. Identifying Locke’s scheme of twofold “ofness” reveals a common structure to his accounts of simple ideas and complex ideas of substances. A consequence is a distinction among substance sorts parallel to one of his distinctions between primary and secondary qualities.

“‘The Transition from Sensibility to Reason In Regressu‘: Indeterminism in Kant’s Reflexionen,Kant-Studien 92 (2001): 3-12 (the file linked to here is prefaced with an additional figure and commentary summarizing the paper’s main claim)

According to Roman Ingarden, transcendental idealism prevented Kant from “even undertaking an attempt” at elucidating freedom “in terms of the causal structure of the world.” I show that this claim requires qualification. In a remarkable series of Critical-period Reflexionen (5611-4, 5616-9), Kant sketches a defense of the possibility of freedom that differs radically from his published ones by incorporating an indeterministic account of the phenomena. Anticipating Łukasiewicz, he argues that universal causal determination is consistent with an open future: if an action is contingent, there is an infinite regress of determining causes, yet there is a prior time at which this infinite series of causes has not yet commenced. However, he concedes that on this account the unity of experience “cannot fully obtain in the case of free beings.” The fact that Kant even contemplated the indeterministic theory may carry implications for interpreting the argument of the Second Analogy.

“Toward ‘Perfect Collections of Properties’: Locke on the Constitution of Substantial Sorts”,Canadian Journal of Philosophy 29 (1999): 551-93

Locke’s claims about the “inadequacy” of substance-ideas can only be understood once it is recognized that the “sort” represented by such an idea is not wholly determined by the idea’s descriptive content.  The key to his compromise between classificatory conventionalism and essentialism is his injunction to “perfect” the abstract ideas that serve as “nominal essences.”  This injunction promotes the pursuit of collections of perceptible qualities that approach ever closer to singling out things that possess some shared explanatory-level constitution.  It is in view of this norm regulating natural-historical inquiry that a substance-idea represents a sort for which some such constitution serves as the “real essence,” i.e. as that on which all the sort’s characteristic “properties” depend.


History of 20th century philosophy

“Sellars on the Function of Semantic Vocabulary”British Journal for the History of Philosophy 22 (2014): 792-811

This paper examines two explanations Sellars gives, at successive stages of his career, of how semantic vocabulary (paradigmatically ‘means that…’ and ‘is true if and only if…’) lets us relate linguistic expressions to extra-linguistic reality. Despite their differences, both explanations reveal a distinctive pragmatist approach. According to Sellars, we do not use semantic vocabulary to describe language-world relations. Rather, our taking language to relate to the world is implicit in the moves (inferential or non-inferential) licensed by our semantic assertions. I argue that Sellars’s discussions of the function of semantic vocabulary point to an overlooked position regarding the relation between the concepts of meaning and truth. According to him, the function of meaning ascriptions cannot be explained independently of the function of truth ascriptions. That is because the function of meaning ascriptions essentially involves licensing claims about the world when combined with truth ascriptions. If he is right, this poses a challenge to deflationary accounts of the function of truth talk.

“Intentional Relations and the Sideways-on View: On McDowell’s Critique of Sellars”, European Journal of Philosophy 21 (2013): 300-319

McDowell opposes the view that the intentionality of language and thought remains mysterious unless it can be understood “from outside the conceptual order.”  While he thinks the demand for such a “sideways-on” understanding can be the result of “scientistic prejudice,” he points to Sellars’s thought as exhibiting a more interesting source: a distortion of our perspective “from within the conceptual order.”  The distortion involves a failure on Sellars’s part to see how descriptions from within the conceptual order can present expressions and mental acts as related to extra-conceptual objects (a failure in turn explained by his failure to see how such relations could have normative import).  In this paper, I argue that Sellars’s thought suffers from no such distortion.  If I am right, McDowell’s examination of Sellars has not uncovered a disorder whose treatment might help relieve the desire for a sideways-on view.

“‘Coordinative Definition’ and Reichenbach’s Semantic Framework: A Reassessment”,Erkenntnis 41 (1994) 287-323

Reichenbach’s Philosophy of Space and Time (1928) avoids most of the logical positivist pitfalls it is generally held to exemplify, notably both conventionalism and verificationism.  To see why, we must appreciate that Reichenbach’s interest lies in how mathematical structures can be used to describe reality, not in how words like ‘distance’ acquire meaning.  Examination of his proposed “coordinative definition” of congruence shows that Reichenbachadvocates a reductionist analysis of the relations figuring in physical geometry (contrary to common readings that attribute to him a holistic conventionalism), while embracing a thoroughly holistic understanding of empirical confirmation (contrary to rival operationalist readings).


Book review

Review of Between Saying and Doing: Towards an Analytic Pragmatism by Robert Brandom, Australasian Journal of Philosophy 88 (2010): 367-71

“Curry’s paradox”, as the term is used by philosophers today, refers to a wide variety of paradoxes of self-reference or circularity that trace their modern ancestry to Curry (1942b) and Löb (1955). In this essay, we show how the various Curry paradoxes can be constructed, examine the space of available solutions, and explain some ways Curry’s paradox is significant and poses distinctive challenges.

Building on recent work, I present sequent systems for the non- classical logics LP, K3, and FDE with two main virtues. First, derivations closely resemble those in standard Gentzen-style systems. Second, the systems can be obtained by reformulating a classical system using nonstandard sequent structure and simply removing certain structural rules (relatives of exchange and contraction). I clarify two senses in which these logics count as “substructural.”

This paper aims to call into question the customary division of logically revisionary responses to the truth-theoretic paradoxes into those that are “substructural” and those that are “(fully) structural.” I proceed by examining, as a case study, Beall’s recent proposal based on the paraconsistent logic LP. Beall formulates his response to paradox in terms of a consequence relation that obeys all standard structural rules, though at the price of the language’s lacking a detaching conditional. I argue that the same response to paradox can be given using a consequence relation that preserves detachment rules for a conditional, though at the price of restricting structural rules. The question “Is paradox being blocked by invoking a substructural consequence relation?” is thus ill-posed. The lesson of this example, I conclude, is that there is no useful explication of the idea of a substructural approach to paradox.

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