THE 1982 WILLIAM M. MURRAY LECTURE

Modeling Architectural Structure: Experimental Mechanics in Historiography and Criticism

Engineering analysis of monumental buildings of the past and present has produced new insights into architectural design and construction

by Robert Mark

Introduction

In The T w o Cul tures , C.P. Snow reported from a sur- vey of some 25 percent of British scientists and engineers in the 1940's and I950's that most "just touched their caps to the traditional [humanistic] culture. [Scientists and engineers] have their own culture, intensive, rigorous, and constantly in action. This culture contains a great deal of argument, usually more rigorous, and almost always at a higher conceptual level than literary argu- ments. '" Rather than dwell upon the much discussed divergence, a convergence of the " two cultures" is the subject of this paper: The use of the "scientific method" and more particularly the application of techniques of experimental mechanics to analyze issues which, until recently, have been treated almost entirely in the realm of literati.

The structural technology underlying all works of architecture and particularly large, monumental buildings is frequently too complex to be readily grasped by most observers, or even by the authors of influential architectural treatises. Thus, it is not unusual for historical judgements to be based on incomplete understanding, and in turn, for architectural design precedents based on past works to be distorted. The historical treatment of two giant Gothic cathedrals, whose construction at Chartres and at Bourges in north-central France began almost simultaneously in 119405, is a case in point. The importance of Chartres Cathedral in the history of architecture is implicit in the emphasis placed on the building in the literature dealing with the Gothic (for example, see Ref. 2). In effect, Chartres became the model for most of the major Gothic cathedrals that followed. Bourges Cathedral, often mentioned as an interesting footnote, has been the subject of but one modern monograph, and that pub- lished only in France. '

As with every other feature of Chartres, much has been written about its structure. A typical observation is that

Robert Mark (SESA MembeO is Professor of Civil Engineering and Architecture, Princeton University, Princeton, NJ 08540.

Paper was presented at 1982 SESA/JSME Spring Meeting hem on Oahu and Maui, HI on May 23-30.

the flying buttresses of Chartres (a relatively new invention at the time of construction) "are the first to have been conceived, not only structurally but also aesthetically, as integral parts of the overall design" (Ref. 2, p. 204). Our study indicated, however, that the buttressing system of Chartres, which we found to weigh some two and one-half times as much as the buttressing of Bourges, was not fully effective. Rather than achieving a masterful integra- tion of structure with style, the architect of Chartres appeared uncertain about his buttressing. And this conclusion was corroborated by the entirely different picture that emerged from a technical study of the structure of the choir at Bourges. Under similar environmental loadings, stress levels in its relatively light structure were found to be low, indicating the general superiority of the Bourges buttressing scheme.' Thus, although the architect of Chartres made a major aesthetic contribution, our analyses showed that where technical matters were concerned, the cathedral's design was far less revolutionary than has been claimed. On the other hand, the structural solution used at Bourges was unique in its time for solving the problem of maintaining the structural integrity of a tall masonry building with far less costly buttress con- struction.

These observations have had some considerable impact on architectural historiography. Indeed, the traditional view of Chartres has been modified in a number of publications over the last decade. Evdn historians not in sympathy with this technical, 'revisionist' perspective must now respond to it. For example, the noted French: architectural historian, Louis Gr0decki, after first re- proaching "technicians [who] always find the structural approach interesting," in his survey of Gothic architecture then acknowledged in a brief description of Bourges Cathedral " that it has recently been observed that com- pared to buildings in the Chartres family, the Bourges structure is remarkably l ight . '"

The scope of the new research is illustrated by examples of several recent studies--in structural archaeology: a reconstruction of the original buttressing of t h e early Gothic building of Notre Dame Cathedral, and the structural basis for the transition from six-part to four- part vaulting in High Gothic cathedrals--and in struc- tural history: a technical commentary on Christopher

Experimental Mechanics �9 361

Fig. 1--Notre Dame Cathedral, Paris. Comparative sections through the nave: (a) archaeological reconstruction of the original Gothic configuration, c. 1180; (b) after 1225; (c) after rebuilding begun in 1845

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Wren's 17th-century design for St. Paul's Cathedral, and an analysis of the early 20th-century Swiss engineer Robert Maillart's reinforced-concrete framing at Chiasso. It will also be demonstrated how the new insights into historic structures can sharpen structural criticism of modern works of architecture. Examples include analyses of the structural design basis of two modern architectural monu- ments: the Kresge Auditorium at M1T and the new Opera House in Sydney, Australia.

Structural Archaeology The interior height of the Cathedral of Notre Dame in

Paris, begun in 1163, exceeded that of all earlier Gothic cathedrals by some eight meters. No doubt, it was this great increase in height which led Notre Dame's designers to employ, for the first time, flying buttresses to support the high clerestory walls of the nave. But unfortunately, we have only indirect evidence for the original configura- tion of this seminal structural device. Massive recon- struction of the cathedral which would alter the entire buttressing system was begun in 1225, and another major reconstruction probably took place in the fourteenth century. Then in the nineteenth century, Eugene Viollet- le-Duc, whose name came to be synonymous with French Gothic restoration of the era, spent over twenty years rebuilding its structure as well as in refurbishing its statuary and interior (Fig. 1).

An archaeological reconstruction of the original (1180) flying buttress system of the Notre Dame nave based on drawings by Viollet-le-Duc to record the structure as he found it, and by observation of early stonework in the existing building and in buildings constructed contem- poraneously near Paris has been developed by Wm. W. Clark, an architectural historian at Queens College, CUNY. The most decisive piece of evidence in his re- construction is the trace of an arch in an existing solid wall buttress which Clark believes gives information both for the location and the shape of the original flying buttresses. Clark's archaeological analysis would normally be accepted as complete; but since the original buttressing was crucial for maintaining the stability of the cathedral, we concluded that quantitative structural analysis should also be performed in order to substantiate the archaeolo- gical reconstruction. A 1 : 100-scale photoelastic model of a typical bay 'frame' of the cathedral, representing the piers, buttresses, lateral walls, and ribbed vaulting, was

therefore tested under simulated gravity and wind loadings (Fig. 2).

Structural modeling is rarely used by architectural designers in the United States, mainly because the pro- fessional f ee structure for architectural engineering discourages the employment of specialized consultants? Because of a rash of costly modifications to reduce the effects of sway in tall buildings subjected to high winds, however, the profession has lately been more accepting of wind-tunnel model testing at the design stage. Yet these tests are to gather only wind-pressure distributions which are then fed into a numerical model to predict the dynamic response of the building's structure. 7 Far greater use of physical modeling is found in architectural-structural re- search, particulary for long-span structures, new forms of concrete thin shells, and for nonlinear behavior of con- crete and steel-framed systems. 8'9 Indeed, it was from experience with photoelastic modeling of reinforced- concrete structures in the 1960's that I perceived that the approach could be applied in many instances for historic buildings of unreinforced masonry. In designing such experiments it is assumed that the masonry 'frame' acts as a monolith; in effect, that it is everywhere in a state of compression. As it happens, this assumption coincides also with criteria for successful masonry performance because the tensile strength of medieval mortar is so low [about 2(10) s Pal that structural cracking will accompany any significant level of tension. And indeed, our tests of Gothic structure have indicated that compression does prevail. Tension is present only at local regions, and we have shown in a number of instances that these critical regions require special attention from custodial staffs.'~

Since only general force distributions are sought, no effort was taken to represent the actual cross-seCtions of the building elements of Notre Dame in great detail; even the interaction of three-dimensional vaulting was accounted for by 'equivalent' arches. Dead-weight load distributions were estimated from drawings and photographs of the cathedral. These were modeled at l:100,000-scale by an array of point loadings while the model was subjected to a stress-freezing, thermal cycle. Wind-load distributions were derived from long-time (100-year, for Paris) local meteoro- logical records, theoretical wind-velocity profiles for the assumed terrain of the building site in the medieval era, and wind-tunnel test data from modeling similar building configurations (see Ref. 10, p. 22-24). Wind loadings were also modeled by an array of point loadings at

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1:30,000-scale, representing the effect of wind speed of 115 km/hr at an elevation corresponding to that of the 45-m high cathedral roof peak.

Based on the combination of these loadings, the scaled model results indicated that the cathedral structure would have been subjected to moderate levels of compressive stress [less than 4(10) 6 Pa] throughout, except for two local regions of tension on the windward side of the buttressing system designated in Fig. 2. Under the action of high winds, cracking of the mortar would have occurred in both of these regions (indicated tension is three to five times that of the tensile strength of medieval mortar). But because of the highly localized nature of the indicated tension, it is doubtful that major damage to the masonry would have followed, .provided that the custodial staff repaired the joints promptly enough after a storm to pre- vent more general deterioration.

Our conclusion then, is that the archaeological recon- struction of the original Gothic configuration with its seminal flying buttresses is feasible; the structure would not have been devoid of problems, but it could have been maintained. Yet the analysis also serves to reveal another stimulus for the massive, thirteenth-century campaign of reconstruction of Notre Dame. It has been generally assumed that this was undertaken to 'modernize' the cathedral, to open up the clerestory walls and bring in considerably more light than was permitted by the earlier wall structure. But now we may speculate that problems of maintenance of the building structure could also have compelled the cathedral chapter to consider reconstruction. By 1225, builders had amassed much additional experience from constructing a number of even taller churches with flying buttresses; hence, a revised design would have been far more certain of achieving success.

High Gothic Vault ing

A second example of structural archaeology concerns the sudden shift from the use of six-part to four-part vaulting in High Gothic architecture. With few exceptions prior to the year 1200, square-planned, six-part vaults were used in the high bays of all the larger churches, including Notre Dame; after 1200, only rectangular- planned, four-part vaults covered the soaring interior spaces (Figs. 3 and 4). A causal relationship between the development of the raised High Gothic clerestory supported by flying buttresses and the change in vault configuration can thus be accepted prima facie, yet the texts on Gothic architecture are rather vague on this point.

Not surprisingly, stylistic explanations predominate in the art-historical literature. Implicit in all of these is the understanding that the use of six-part vaults arose from the common practice of alternating the size and shape of the nave piers. Since the number of vaults ribs which spring from the piers with six-part vaulting is alternately one and three, this system is claimed to be a more logical visual complement to alternating piers. By the same reasoning, the stylistic theories attribute the adoption of four-part vaulting to the introduction of uniform, non- alternating piers."

Constructional explanations based on the premise that erecting centering for four-part vaults was simpler than for six-part vaults are also unconvincing. For example, Vi011et-le-Duc maintained that the primary reason that the six-part plan was abandoned was because the square, six-part bay required a diagonal rib which was much longer than the transverse ribs. This imparted construc-

Fig. 2--Dark-field isochromatics in windward portion of reconstructed nave-section model under simulated wind Ioadings

tional problems in that the arches of the transverse ribs had to be very acute or stilted in order [o attain the same heights as the keystones of the diagonal ribs. ~ Others have concluded that centering was more difficult to erect for four-part than for six-part vaulting, but that four- part vaults had other overriding advantages such as lighter weight, which accounts for their adoption. Lighter vaul- ting implies less thrust on the walls and buttressing and therefore lighter construction throughout. It also offers constructional savings because it needs less centering for erection and of course requires a smaller amount of finished stone.

Three-dimensional photoelastic and finite-element model studies of medieval ribbed vaulting, carried out at Prince- ton, were originally undertaken to determine the structural role of the vault rib. '3'4 In the course of these studies it was also found that the weight of a six-part vault was significantly less than the weight of four-part vaults covering the same area. Indeed, it became evident that the lighter weight of the six-part configuration is largely owed to the fact that it carries fewer ribs than its four-part equivalent. I n any event, the finding that the thirteenth- century builders, who generally favored light construction, would choose to construct heavier vaults over increasingly slender piers and walls in the tallest churches did nothing to clarify the enigma surrounding the abrupt change in vaulting form.

A constructional constraint that arose during a pre- viously unconsidered phase of vault erection provided the basis for a new theory to explain the change. Consider first the salient structural feature of Gothic vaulting: the 'focusing' of the distributed forces within the vaults at the points of vault support on the clerestory wall. There are three components of this focused force resultant at the springing: a downward, vertical component equal to a

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Fig. 3--Bourges Cathedral. Six-part vaulting, c. 1200

fraction of the total weight of the vaulting and its ribbing, and supported by the clerestory walls which are in turn carried by the piers of the nave arcade; an outward, horizontal component tending to overturn the clerestory wall, but resisted in the mature Gothic church by flying buttresses; and a longitudinal, horizontal component against the adjacent bay along the axis of the church. This last force is ordinarily stabilized by the adjacent bay of vaulting whose longitudinal component acts in the opposite direction to that of its neighbor. In effect, the completed bays of vaulting 'lean' against each other.

From this brief description of the vault supporting mechanism, it is evident that the skeletal form of the developed Gothic church can handily support any reasonable form of vaulting, six-part or four-part. A different condition is present, however, during the con- struction of the vaults which we may assume was carried out, one bay at a time, on movable centering. Since the erection of the vaulting was necessarily preceded by the erection of the piers, walls and flying buttresses, the vertical weight and the outward, horizontal vault force components after the vault centering is removed are resisted by the same structural elements as in the finished church. The longitudinal component, though, must at this stage be supported by the clerestory wall since there was not yet in place an adjacent bay to stabilize it. And as the springing of the vaults was carried further upward from the base of the clerestory in the later buildings, coping with this longitudinal thrust became a more crucial problem during construction.

The modeling of Bourges six-part vaulting indicated a longitudinal force component of 19(10) 4 N; for the slightly larger bays of Cologne Cathedral's four-part vaulting, the longitudinal thrust was found to be 9(10) 4 N, more than.a fifty-percent force reduction compared with equivalent six-part vaulting (Ref. 10, p. 117). The magnitude of these force components might be better appreciated when they are compared with the calculated, outward-acting

Fig. 4--Chartres Cathedral. Four-part vaulting, c. 1200

force components: 14(10) 4 N for Bourges, and 16(10) 4 N for Cologne--and also compared with the elaborate but- tressing systems used to contain the outward force com- ponents. (Note that these outward-acting components are given at the edge of one bay only; in the completed building, the thrusts of two bays act on the wall, doubling the total outward force.) The constructional problems presented by the intensity of these longitudinal forces do not appear to have been acute in the early Gothic churches as the vault springing could be anchored in the typically massive wall below the clerestory. The countering of this force only became a major problem with the demand for greater clerestory height and the accom- panying fenestration. Hence, the crux of our argument is that the Gothic builders needed to select a vaulting system with considerably less longitudinal thrust.

Support for the new theory is evident in the manner in which six-part vaults were deployed. I n every major Gothic church possessing square-planned, six-part vaulting, the vaults spring from a massive section of wall below the clerestory. Furthermore at Bourges, the largest and last of the great Gothic churches with six-part vaulting, heavy iron chains are placed just below the clerestory of the choir (Ref. 3, pp. 82-83). Since these chains provide no reinforcement to the masonry in the vertical or lateral directions, the only reasonable explanation for their placement is that they were intended to help contain the longitudinal thrust of the vaults during construction. Further, it has been noted that the nave of Bourges, constructed in a later thirteenth-century campaign, had considerably thicker walls without the metal reinforcement, apparently compensating for its absence (Ref. 3, p. 129).

The fact that four-part vaulting exhibits less than half the longitudinal thrust of six-part vaulting must have been understood by the master who raised the vaults at Chartres.

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Its rectangular four-part vaults springing from a point above the base of the clerestory allowed a dramatic increase in the height of the clerestory as well as in the potential for light to illuminate the interior of the cathedral. The entire clerestory at Chartres is over 14 m high--about the same height as the nave arcade--and the vaults spring from nearly 4 m above the base of the base of the clerestory, a remarkable departure from earlier designs.

The observation that it was impractical to have six-part vaulting spring from a point above the base of the cleres- tory also sheds new light on the comparison of the designs of Chartres and Bourges, It had appeared that Bourges rather than Chartres might have become the model for the great High Gothic cathedrals that followed (Ref. 4, p. 93). But now it is clear that the supremacy of the Chartres model was assured because the six-part vaults of Bourges would not permit an enlarged cleres- tory. On the other hand, the flexibility of the Chartres design, particularly of its four-part vaulting, could satisfy the Gothic urge for additional height and light. The adoption of four-part vaulting allowed the structure of the mature churches to become more truly skeletal. And while the change had important stylistic implications, it also provides additional evidence of the Gothic designer's understanding of the disposition of structural forces during the construction of the giant buildings, a subject that has only begun to receive the attention that it deserves in the study of Gothic architecture.

These perceptions of the influence of structural behavior on building design for which there is almost no primary documentation, except for the existing buildings, are made evident only by the application of techniques of structural analysis. The same approach, for reinterpreting the architectural forms chosen by designers of buildings of more recent date, is illustrated in the following section.

Structural History Christopher Wren, the architect of St�9 Paul's Cathedral,

London, is generally portrayed in the literature on archi- tecture as "one of the three or four greatest Englishmen �9 . . fit to rank as an artist with Shakespeare . . . while [his contemporary] Isaac Newton reckoned him among the three greatest geometers of his day�9 ' ' l ' Wren's fame is derived from a career as professor of astronomy at London and at Oxford, a founder and later, president of the Royal Society, and as well, his appointment as sur- veyor general (Royal architect) of England. Yet, one of the most perplexing aspects of Wren's career for historians who have searched through the documents of the era is the elusiveness of demonstrable connections between his science and his architecture.

Wren's first major architectural commission, for the Sheldonian Theatre at Oxford University (begun 1662) where he created the then-unique, 21-m clear-span audi- torium, marked the beginning of his prominence and the belief that his architecture was in substantial debt to his science. Photoelastic modeling of the trusses used by Wren to support the Theatre ceiling, however, led us to the conclusion that his truss design was not much im- proved from a medieval, king-post roof truss. '6 It came nowhere close to the structural efficiency of triangulated trusses which had already been used in earlier bridge designs. And further studies of both Wren's writings and his buildings did nothing to clarify the science/architecture enigma; but in the process of examining St. Paul's Cathe- dral in London, another generally accepted architectural shibboleth was laid bare.

Fig. 5--St. Paul's Cathedral, London. Dome, 1708

Innovation with structure was not an outstanding characteristic of the architecture of the Renaissance, Perhaps this was true because the publication of building plans, which was a new development, inspired emulation rather than experimentation. ~7 Large Renaissance dome structures were an exception, however. The three greatest domes raised in Western Europe during the era are at the cathedral in Florence, constructed by Brunelleschi (1420-34); at St. Peter's Cathedral, Rome, derived from models by Michelangelo (d. 1564) and constructed by Giacomo della Porta and Domenico Fontana (1585-90); and at St. Paul's Cathedral (1705-08), (Fig. 5)�9

All of these projects derived inspiration from the 45-m diameter concrete dome of the Roman Pantheon (AD 120-124)--with one major difference that completely alters their structural basis�9 The exterior profile of the Pantheon dome is fiat and well buttressed by a massive ring of masonry, while the profiles of the Renaissance domes were intended above all other considerations to be imposing. They are raised on tall drums above already high building structures. Both the domes at Florence and at Rome, of similar span to the Pantheon, were con- structed of brick and stone with double shells to enhance stability and to reduce the height of the interior profile so that their undersides do not appear at the end of a darkened cylinder. Both domes, which also support heavy central lanterns, are also plagued with structural problems because even though their profiles are pointed (Florence more so than Rome), they are not adequately buttressed, Meridianal cracking became evident as the domes spread outward~ Brunelleschi had tried to avoid this problem by

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Fig. 6--Section through St. Paul's dome (J. Gwyn, 1756)

placing wooden chains as well as iron reinforcement within the dome at Florence; and at Rome a total of ten massive iron chains have been placed around the dome over time. Yet these steps have also not proved entirely adequate because of the huge mass of the dome masonry that engenders the gigantic hoop forces.

Even though he had never been to Italy, the problems of the St. Peter's dome were probably known to Wren as his confidant John Evelyn had visited the Roman cathe- dral. Although the span of the St. Paul's dome was to be but three-quarters that of St. Peter's, Wren's concern may have contributed to his failure to make a decision on the dome design from the many schemes that were worked out until very late in the construction of the cathedral, which began in 1675.

Wren's design (Fig. 6) is elegant both visually and tech- nically. He has succeeded in creating the profile .of a majestic outer dome with a lightweight structure of lead- sheathed timber supported by a thin chain-reinforced brick cone--which also supports a stone lantern of some 850 tons--and a separate, light, inner brick dome well suited to interior spatial needs. In contrast to the normal structural action of a high dome, the cone is actually stabilized by the heavy central lantern. Instead of pro- ducing pernicious bending, the central loading insures that the cone acts mainly in compression. Indeed, the form of

the cone and the outward tapering of the supporting drum below it has led to speculation (although we could find no substantiative documentation) that Wren was guided by experiments with catenaries performed by his colleague Robert Hooke at the Royal Society. In contrast to St. Peter's, Wren employed a single iron chain that proved sufficient to maintain the integrity of his relatively light- weight structure. Finite-element studies of the dome con- figuration indicate that stresses within the supporting masonry are generally low under both gravity and wind forces and that Wren's chain was well placed to fulfill its structural role. Nevertheless, we concluded that the success of the structure was mainly owed to a well- developed sense of practical design, unrelated to any analytical technique.

Perhaps more important for questions of modern architectural design than the enigma concerning the possible connection between Wren's science and his architecture is the ironic response by certain architectural theorists of the 19th and 20th centuries who criticize Wren's dome, characterizing it as "dishonest" structure. These have even suggested that the timber outer covering should be removed .to reveal its "structurally honest," load-carrying cone.'8 Criticism of this nature lays bare the absurdity of carrying the modern theme of 'architectural morality' to its logical end. The Renaissance high dome

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Fig. 7--Warehouse shed, Chiasso, Switzerland, 1924

is, after all, primarily a work of monumental sculpture, and Wren's dome seems ever more visually effective than Michelangelo's, while his engineering of its structure was far superior.

Reinforced-concrete Framing

A more contemporary illustration of the structural approach for historical interpretation is provided by the reinforced-concrete constructions of the 20th-century engineer, Robert Maillart. These are often admired by contemporary critics for their "organic forms" and their "constructive art" as is, for example, the framing of a shed roof attached to a warehouse in Chiasso on the Swiss-Italian border which Maillart designed in 1924 (Fig. 7). The frame has been characterized as being "similar to forms found in nature because it is consciously derived from the same principles of mechanics as the corresponding natural structures follow unconsciously.'"9 The author of this analogy, to reinforce his hypothesis, then represented the frame together with the longitudinal section of the wing-bone of an eagle.

Another, somewhat more critical observation of the Chiasso frame was that it is "an interesting example of a technically correct, function-determined form, which nevertheless is unconvincing. The form of the supports and of the lower chords seems willful, in the manner of jugenstil art, because the action of the forces, which are expressed exactly, is too complicated to be intuitively grasped. ''2~ Yet despite many such published observations, as with the works of Wren,, the actual technical basis of Maillart's forms had until quite recently received but scant attention. My colleague David Billington has studied the technical development of Maillart's bridge designs~'; and I, together with a former colleague, John Abel and a former student, James Chiu, analyzed the Chiasso frame. 22

Contrary to the bemusement of the second critic, the derivation of the form of the frame is quite straight- forward, at least to an engineer. If the moment connection at the columns is neglected, the horizontal span of the frame may be assumed to act like a simple beam over- hanging its supports and carrying a uniformly distributed roof loading as shown in Fig. 8(a). The shape of the corresponding bending moment diagram is given in Fig. 8(b). The depth of an efficient structural system to

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Fig. 8--Development of shed structural form: (a) simply supported beam loading; (b) bending- moment diagram; (c) constant-force truss; (i:l) gabled, constant-force truss

replace the beam should follow the beam-bending moment diagram, as shown in Fig. 8(c), to maintain a constant level of axial forces in the chords. Rigid vertical members between the upper and lower chords (similar to those found in a Vierendeel truss) help to carry shear forces and to transmit roof loadings to the lower chord. In the final stage of design, the system is 'bent ' at its center to con- form to a gabled roof, where the overlapping structure [dashed in Fig. 8(d)] is merely eliminated.

In this exposition we have .made a number of simpli- ficatiol~s, including neglecting the interaction of the columns with the horizontal span, that alter the simplified

Experimental Mechanics �9 367

moment diagram [Fig. 8(b)] and the local bending stresses caused by shear forces. Differences between the form of the actual frame and the sketch [Fig. 8(d)] were, no doubt, largely derived from Maillart's consideration of these forces. It is mainly his handling of these structural design details that were determined in a more detailed analysis. Both photoelastic and numerical (STRUDL) modeling were used for the stucly in order to provide a cross-check on the results. Since our interest was in the structure's 'service' behavior, the assumpnon of linear elasticity was made for both models. In-plane structural behavior arising from interaction with the concrete roof or with lateral supporting struts and webs was considered negligible. Only the effect of the additional weight of these structures on the frame has been taken into account, and footings were assumed to be completely fixed.

The results from the two models were in substantial agreement. Almost uniform interference orders in the upper and lower chords of the photoelastic model (Fig. 9) confirm that the general form of the truss does indeed follow the actual moment diagram. The transition from span to column has been well handled, as evidenced by the fairly constant interference order along the inside edge of the upper portion of the column (Fig, 9, pt, b) which scales to produce a stress level of 20(10) 6 Pa in the 25-cm wide, planar, prototype. The magnitude of this stress is high and Maillart's solution was to develop the transverse stiffener diaphragm along the inside column (Fig. !0), in effect creating a stout T-section to reduce the stress to reasonable levels

The scaled maximum compressive stress in the upper chords is 13(!0) ~ Pa (corresponding to Fig. 9. Pt. c) but in the shed's actual monolithic construction, the concrete roof accepts much of the chord compression, reducing the actual stress levels [o a fraction of the indicated value. The location of the points of inflection in the columns (corresponding to Fig. 9, pt. d) corresponds with the elevation of the shed storage platform, which suggests that the visible portion of the columns above ground could have been tapered to a narrow section at platform

level, creating an even more aesthetically pleasing structure. The form of the actual columns gives no hint of this force distribution. Maillart was undoubtedly aware from approximate analysis* that the columns would contain a point of inflection; however, he probably recognized that a stouter section would be required here to resist possible damage by impact from cargo-handling equipment.

These results imply that the design of the shed frame is even more radical than has been generally understood. Maillart took advantage of the w h o l e structure in using the thin, plate-like, transverse members in helping to . . . . . . . . . . . longitudinal forces (with respect to the axis of . . . . . . . . . . In fact, the 'stressed-skin construction' of the frame is found also in Maillart's strikingly light reinforced- concrete bridges. The logic of Maillart's structural con- ception is easily re,~ealed by analysis, as it was in the case of Wren's dome. Unfortunately, such structural logic is not always present m modern architecture, as the following illustrations will reveal only too well.

Structural Criticism

There is no dearth of technical commentary in archi- tectural literature; however, the basis for much of the observation of structural function is questionable. Analogy is frequently used as with the cited comparison of the eagle wing-bone to the Maillart frame. But a more serious problem is the typically grossly simplified analysis that finds acceptance because of its supposed 'logic.' For example, such analysis gave birth to the widely held view that compressive stress-resultants in the webs of ribbed vaults are directed to the groin-intersection between the webs, and that the heavily loaded groins thereby require

*It is unfikely that Maillart wouM have undertaken to predict completely the frame's behavior smce one o f the earliest practical techniques for cmalysis o f such statically indeterminate frames, the moment-distribution method, was published by Hardy Cross in 1930, six years after the date o f the frame destgn. 23

Fig, 9--Light-field isochromatics in Chiasso frame model under simulated dead-weight loading

368 �9 October 1982

ribs for reinforcement. This contention has been dispelled by our recent photoelastic and finite-element model studies which demonstrated that in fact, the compressive trajec- tories flow through the webs directly to the vault springing (Refs. 13 and 14).

The problem is further compounded for some con- temporary architecture where the building form appears to be derived from technical exigency. Such an example is Eero Saarinen's Kresge Auditorium on the MIT campus constructed in 1955 (Fig. 11) which evoked a range of published reaction mainly to its appearance, but with general acceptance of its 'function-determined form.' In point of fact, the auditorium design would seem to have been conceived solely for visual (sculptural) effect without consideration of the appropriateness of its structure. The cast-in-place shell, of 49-m span between supports, is formed from the intersection of two orthogonal meridians and the equator of a 34-m radius sphere, yielding exactly one-eighth of the spherical surface. The spherical segment, triangular in plan, was originally intended to be of con- stant thickness and supported at the three corners only by massive foundations. Yet the presence of large bending forces after the removal of the forming demanded the addition of major reinforcement including the redesign of the window mullions to support the shell edges. The outcome still was not wholly successful. Over the years, deformation of the shell surface brought about deteriora- tion of its lead roof covering, causing flooding in heavy rain and corrosion of the steel reinforcement that neces- sitated closing the building for a full year, in 1979, to allow major structural repairs, 2'

The structural concept of the Auditorium was reviewed using a 1 : 144-scale, three-dimensional photoelastic model and a finite-element model of a constant-thickness shell. The analysis confirmed that only the central portion of the Auditorium roof acts truly as a shell; in essence, the central spherical dome is supported by the edge arches, and the in-surface and arch bending are responsible for the cracking and excessive deflection along the edges. 2S

The problems of the Auditorium recall those cited for the Renaissance domes. The profile of the Auditorium is, as were the profiles of St. Peter's and St. Paul's, primarily sculptural rather than structural; but the lesson of Wren's construction which eliminated most of the Renaissance structural problems was not transmitted to modern architecture. This failure resulting from a misunderstanding of the technology of historic architecture played an even far greater role in the debacle of John Utzon's 1956 design for the Sydney Opera House. When the Opera House opened 17 years later, in 1973, it was nine years behind schedule and cost more than 130 million dollars over its original 10 million dollar estimate (despite the relative stable economy of the period). The largest single factor affecting both cost and project time was an early decision made by the designers to construct the roof shells, whose form was in no way derived from any structural premise, to be honestly self supporting.

Much of the additional funding was spent in lengthy engineering studies aimed at constructing the roof of cast- in-place, etliptical-paraboloid, thin shells--and after that scheme was abandoned for lack of feasibility, to construct it of precast, post-tensioned segmental arch ribs joined to

Fig. lO--Chiasso warehouse shed. Construction details

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form spherical folded plates? 6 A much simpler and far less expensive structural system of hidden steel frames could have been used to support the sculptural shells. Indeed, this option was considered by the designers (Fig. 12) and rejected as being "structurally dishonest." Clearly, the elegance of Wren's solution for a similarly sculptural building program went unappreciated.

As an engineer teaching architects, I am tempted to instruct them to abandon sculpture in large-scale projects and to allow building form always to follow that dictated by logical structure (which often produces elegant archi- tectural solutions). Yet as a researcher in architectural history, 1 accept the impulse for designing certain large- scale, 'nonstructural' monuments. Nonetheless, our studies of historic buildings have also demonstrated that there is plenty of precedent for providing support to sculptural buildings with appropriate; efficient structure, not fixed by the building's form. Indeed, this approach is representative of good architecture as well as of good engineering. The Auditorium and the Opera House would have been far more successful from both points of view if their architects had been sensitive to this issue.

Conclusion

Technical analysis of historic building structures has added a new dimension to their study Which augments traditional historiography based on primary documents, archaeological evidence and stylistic analysis. The approach can also be applied to contemporary architecture which needs such technical scrutiny. The absence of p u b l i s h e d technical criticism, for example, no doubt contributed to the recent granting by the American Institute of Architects of a design award to I.M. Pei for his firm's Boston Hancock Tower despite the huilding's major structural defects that caused its windows to pop

out and postponed its occupancy for three and one-half years.*

In addition to furthering understanding of the historic buildings themselves, the studies have revealed new in- sights about early design techniques. The sophistication of much prescientific structural design, as illustrated for example by the use of appropriate forms of Gothic ribbed vaulting as discussed in this paper, led me to the realiza- tion that the early builders must have been employing an experimental method not unlike modern brittle coating. Since large masonry buildings were built over relatively long periods of time, tensile cracking caused either by high winds or the removal of temporary forming were probably observed in the weak lime mortars set between the stone. It was these observations, I believe, that led to early structural innovation (Ref. 10, p. 56). Indeed, t h e

physical separation of designer and builder in contem- porary architecture (and in much other engineering design as well) must contribute markedly to the type of structural failure experienced at the Hancock.

This new approach to architectural history has also un- covered technical precedents which should have important implications for contemporary architecture. For example, it is sometimes argued that monumental buildings had no need for efficient structure because the autocratic rulers of a former era who commissioned them could draw on almost unlimited funds. 28 Yet, our studies have shown that builders from all eras were in fact interested in minimizing the costs of construction which included the expense of obtaining and transporting stone as well as of shaping it and erecting it into place. And furthermore, the

*The AIA jury commented that: "as an object on the skyline, it catches the sun, reflects the sky, and acts as an effective new landmark for downtown Boslon, " but evidently, they did not respond to its technical problems. 27

Fig. l 1--Kresge Auditorium, Cambridge, MA, 1955

3 7 0 �9 October 1982

technical advantages of reducing structural weight that in turn diminishes internal forces in all the supporting members, including foundations, also seems to have been well understood. Wren's structural solution for the dome of St. Paul's Cathedral i s an outstanding though not an unusual example.

Architecture would seem to provide a natural inter- section of the "two cultures," but its technical side has suffered in literary-historical interpretation. A final goal of these studies, combining historical and modern en- gineering analysis for an enhanced view of the architecture of all periods, is to demonstrate the power of a truly interdisciplinary approach in a cultural realm.

Acknowledgments Current support for this research is being provided by

grants from the National Endowment for the Humanities and by the Andrew W. Mellon and the Alfred P. Sloan Foundations. I am indebted to those agencies and also.to many present and former colleagues and students from different academic disciplines within the University who have made vital contributions to the research. I particularly wish to acknowledge the longtime counsel and support given me in these undertakings by my project co-director and colleague in the Department of Civil Engineering;

David P. Billington, and also to thank Huang Yun Sheng, graduate student in the School of Architecture, for his modeling of the Notre Dame reconstruction, and William Taylor, graduate student in the Department of Art and Archaeology, for his initiating the study on the transition of Gothic vaulting.

References 1. Snow, C.P., The Two Cultures and the Scientific" Revolution,

Cambridge University Press, New York, 12-13 (1961). 2. yon Simson, 0., The Gothic Cathedral, Harper, New York (1966). 3. Btanner, R., La eathedrale de Bourges et sa place duns l'architecture

gothique, Tardy, Paris (1962). 4. Mark, R., "'The Structural Analysis o f Gothic ' Cathedrals, "' Sci.

Amer., 227, 90-99 (Nov. 1972). 5. Grodecki, L., Gothic Architecture, Abrahams, New York, 12, 142

(1977). 6. Billington, D.P., Janney, J.R. and Mark, R., Structures Models and

Architects, Princeton Univ., School o f Archit., Princeton, NJ, 54-59 (1963).

7. "Wind Analysis: Preventive Medicine for Cladding Structural Problems, " Engrg. News-Record, 27, 26-30 (March 1980).

8. Bitlington, D.P., Thin Shell Concrete Structures, Second ed., McGraw-Hill, New York, 334-336 (1982).

9. Mark, R., "'Photomechanical Model Analysis o f Concrete Struc- tures," Models for Concrete Structures, Pub. No. SP 24, Amer. Concrete lnstit., Detroit, 187-214 (1970).

10. Mark, R., Experiments in Gothic Structure, MIT Press, Cambridge, /VIA, 119-122 (1982).

11. Stoddard, W., Art and Architecture in Medieval France, Wesleyan Univ. Press, Middletown, CT, 130, 140 and 181 (i966).

12. Viollet-le-Duc, E.E., Rational Building, trans, by G. Huss, MacMillan, New York, 219-222 (1895).

13. Mark, R., Abel, J.F. and O'Neill, K., "Photoelastic and Finite- element Analysis o f a Quadripartite Vault," EXI'ERIMENTAL MECHANICS, 13 (8), 322-329 (Aug. 1973).

14. Alexander, K.D., Mark, R. and Abet, J.F., "'The Structural Behavior o f Medieval Ribbed Vaulting," J. Soc. o f Archit. Historians, XXXVI, 241-251 (Dec'. 1977).

15. Kidson, P., Murray, P. and Thompson, P., A History o f English Architecture, Penguin Books, Harmondsworth, 186 (1965).

16. Dorn, H. and Mark, R., "'The Architecture o f Christopher Wren, "' Sci. Amer., 245, 162-163 (July 1981).

17. Mark, R., "Structural Experimentation in Gothic Architecture, " Amer. Scientist, 66, 548-549 (Sept.-Oct. 1978).

18. Clark, S., "'St. Paul's Cathedral." Observations on Wren's System o f Buttresses for the Dome, Piers, and on Some Other Things," Sir Christopher Wren, A D 1632-1723, Royal lnstit, o f Brit. Archits., London, 71 (1923).

Fig. 12--Sydney Opera House elevations and sections (after Amp); (a) original architectural sketch, 1956; (b) project with concrete thin shells; (c) project with steel framing; (d) post-tensioned concrete arches used in actual construction, completed, 1973

19. Howard, Jr., H.S., Structure, An Architect's Approach, McGraw- Hill, New York, 11-12 (1966).

20. Meyer, P., "'Konstruktion und Sch5"nbeit, '" Schweizerische Bauzeitung, 86, 267 (May 22, 1926).

21. Billington, D.P., The Bridges o f Robert Maillart, Ptqnceton Univ. Press, Princeton, NJ (1979).

22. Mark, R., Chin, J.K. and A b e l J,F., "Stress Analysis o f Historic Structures." Maillart's Warehouse at Chiasso," Tech. and Culture, 15, 49-63 (Jan. 1974).

23. Cross, H., "'Analysis o f Confinuous Frames by Distributing Fixed- End Moments, "Proc. ASCE, 56 (May 1930), and 58 (May 1932).

24. "Hall Reopens After Repairs," Engrg. News-Record, 11, 18 (Sept. 1980).

25. Abel, J.F., "'Structural Analysis and the Humanities, " Conj'. Paper: 3rd Conf. on Matrix Methods in Struct. Mech., Wright-Patterson AFB, OH (Oct. 19-21, 1971).

26. "'Opera House Cost Exceeds Esthnate by $132 Million, " Engrg. News-Record, 5, 13 (April 1973).

2Z "Hancock Tower Garners Architectural Laurel " Engrg. News- Record, 26, 16 (May 1977).

28. Arup, O.N. and Zunz, G.J., "Sydney Opera House, '" Civil Engrg., 50 (Dec. 1971).

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