The Rman Panthen One f the mst impressive structures f Rman antiquity is the Panthen (never t be cnfused with the Parthenn in Athens). Its supervising architect was the Rman emperr Hadrian. It was built between A.D. 118 and 128 and dedicated t all Rman gds (in Greek, pan = all, then = f the gds). Its elevatin, the term fr a representatin f a facade, is shwn in Figure 2.42. The entrance hall is a large prtic in the Greek Crinthian style. The prtic frnts a large cylindrical structure that is capped by a hemispherical dme. The exterir f the cylindrical structure cnsists f flat Rman brick, carefully laid ut, rw upn rw. The dme f the Panthen was regarded t be a shape f ideal perfectin that cnveyed bth beauty and pwer. Plate 5 shws the spectacular Figure 2.42. Antine Desgdetz, engraving f the elevatin f the Panthen. Frm Les edifices antiques de Rme, Claude-Antine Jubert, Paris, 1779 1
Plate 5. Givanni Pal Panini, The Interir f the Panthen, 1737. Oil n canvas. Detrit Museum f Art 2
interir. Its circular flr has a diameter f 142 feet and the interir f the dme rises abut 142 feet abve the flr at its highest pint. At the tp is a circular pening, r culus (in Latin, culus = eye), with a diameter f abut 24 feet t let in light and air. Other than the entrance, the culus is the sle surce f natural light fr the interir. Ntice the circular arrays f rectangular indentatins, r cffers, in the ceiling. The attentin t detail in the cmpsitin f the interir with its framed statues and sets f fluted Crinthian clumns and pilasters (the rectangular vertical elements flanking pairs f clumns) in Plate 5 stands in cntrast t the Greeks greater fcus n external space and frm. Befre the Rmans culd build the massive dme f the Panthen, they had t have sme understanding f the structural challenges that a dme presents and they had t respnd t them. The shell f a dme is its structural part. Classically, shells f dmes are made f masnry r cncrete. Figure 2.43 shws hw the shell can be thught f as a cmpsite f arches btained by slicing the dme vertically thrugh its central axis. These arches, just like thse studied earlier, generate utward hrizntal frces. Because the arches f the shell taper as they rise, they weigh central axis circular base Figure 2.43. The shell f a dme as a cmpsite f arches less cmparatively, s that the magnitudes f these hrizntal frces are less than befre. Hwever, unlike the situatin f the aqueduct f Figures 2.33 and 2.34, there is nly the base f the dme and the internal resistance f the shell t cntain them. The vertical crss sectins f a dme thrugh its central vertical axis are called meridians and the hrizntal circular crss sectins are called hps. They are depicted in Figure 2.44. A hp is a ring-shaped, hrizntal slice f the shell. Tw frces are at wrk n the hps. One is the push f the base f the dme prpagated up alng the rigid shell. The ther is the weight f the dme abve the hp pushing dwn. Tward the tp, the weight is nt a factr and the push f the structure frm belw dminates. This puts the hp under cmpressin, in the same way that the keystne f an arch is under cmpressin. But farther dwn, the weight f the shell abve exceeds the upward push frm belw. The utward push f the hrizntal cmpnents f this excess frce puts the hps (thse frm the base t abut tw thirds f the way up the shell) under tensin. See Figure 2.44. This tensin is called hp stress. The First Principle f Structural Architecture tells us that unless the shell is able t resist this tensin, the shell will expand alng the the hps, s that cracks will develp alng the meridians (that can 3
central axis hps base O base meridians Figure 2.44. Hp stress n the shell f a dme lead t structural failure f the dme in extreme cases). The Rmans built the Panthen ut f cncrete and intermittent curses f bricks. Cncrete was a building material that was relatively easy t build with, but like brick r stne, it had little tensile strength. S it facilitated the cnstructin f the Panthen, but it had relatively little capacity t cntain the hp stress within the shell f the dme. Hwever, cncrete culd be made lighter r heavier simply by adding lighter r heavier aggregate, meaning stne and masnry materials, t the mix. The Rmans used this t advantage. By making cncrete with the light vlcanic rck pumice fr the upper sectins, they reduced bth the weight f the dme and therefre the hp stress. The cncrete f the dme weighs abut 81 punds per cubic ft fr mst f the shell and abut 100 punds per cubic ft fr the sectin f the shell abve the supprting cylindrical wall. The circular culus at the tp nt nly supplies light and air, it reduces the dme s weight further. (Chapter 7, Vlumes f Spherical Dmes, applies basic calculus t estimate the weight f the dme.) T cntain the utward thrusts f the dme at it base, the Rmans made the supprting cylindrical wall up t 20 feet thick with cncrete that increased in weight frm 100 punds per cubic ft near the tp f the wall t 115 punds per cubic ft at the bttm. The aggregate f the cncrete in the cylindrical wall includes the dense, resilient vlcanic stne basalt. The rigid cncrete shell prpagates the push frm the base upward. The inward cmpnent f this push cunters the tensile stress n the hps in much the same way as the hps f a barrel keep its wden slats (r staves) tgether. The cylindrical wall f the Panthen rests n a substantial fundatin. The cncrete used in the fundatin als cntains basalt and weighs 140 punds per cubic ft (clse t the 150 punds per cubic ft f standard mdern cncrete). The cffering n the inside f the shell shwn in Plate 5 is shallw, serves n structural purpse, and has essentially n impact n the weight f the dme. But the vertical and hrizntal cnfiguratin f ribs that the cffers suggest des resemble the ribbed elements that wuld play an imprtant structural rle in later dmes and vaults. The Rmans ften placed masnry and cncrete masses n tp f the lwer, uter sectins f arches and vaults. These masses were intended t increase the stability f such structures. The Rmans may have intended fr the step rings they built int 4
the lwer part f the dme f the Panthen (see Figures 2.42 and 2.45) t serve such a functin and t cntain hp stress. Hwever, recent studies have indicated that the step rings seem t play n significant rle in this regard. The step rings may als have been put in t facilitate the cnstructin wrk n the shell. The dme was cnstructed with the use f centering. An elabrate frest f timbers reached upward frm the flr f the Panthen t supprt the grwing shell until the cnstructin was cmpleted and the cncrete had set. Figure 2.45 depicts half f a central vertical sectin f the Panthen. It shws the structure f the shell, a sectin f the cylindrical supprting wall, the culus, and the step rings. The vectrs flwing dwn the shell represent the dwnward transmissin f the weight f the dme. Their hrizntal cmpnents generate the hp stress already discussed. The upward-pinting vectrs culus steprings springing line C Figure 2.45. Sectin f the Panthen frm Andrea Palladi s I Quattr Libri dell Architettura (The Fur Bks f Architecture), Venice, 1570 5
represent the supprt f the shell frm its cylindrical base. Their hrizntal cmpnents cunteract hp stress. The arc that is highlighted lies n a circle f radius 1 142 = 71 feet. Its center C is 2 the pint n the centering frm which the builders f the Panthen stretched rpes in all upward directins t guide the spherical shape f the shell during cnstructin. In spite f the effrts t cntrl it, the hp stress in the dme f the Panthen did lead t extensive cracks alng sme meridians f the dme. The distributin f cracks generally crrespnds t penings within the upper parts f the cylindrical wall (sme f these are shwn in Figures 2.42 and 2.45). These penings increase the hp stress in the parts f the shell that rise near them. Nnetheless, the fact that the Panthen has remained standing fr almst 1900 years tells us hw well the Rmans succeeded. The Panthen is ne f the mst imprtant buildings in the histry f architecture. Rman design and cnstructin practices have been very influential. It wuld be hard t imagine tday s architecture withut arches, dmes, and the use f cncrete. Prblems Prblem 1. The vectrs in Figure 2.57 represent frces f the indicated magnitudes and 125 lbs 35 80 lbs 65 45 100 lbs (a) (b) (c) 250 lbs 20 10 170 lbs (d) (e) directins. In each case, draw in the vectrs that represent their hrizntal and vertical cmpnents and cmpute the magnitudes f these cmpnents. Prblem 2. Each f the diagrams in Figure 2.58 represents tw frces and their resultant. Identify the resultant in each case. The magnitudes f sme f the frces (and sme f the angles between 6
135 35 40 30 130 65 45 85 (a) (b) (c) Figure 2.58 55 them) are given. Use the Law f Sines r the Law f Csines t cmpute the magnitudes f the frces that are nt specified. Prblem 3. Refer t Figure 2.59. The vectr A represents the gravitatinal frce acting n sme bject. The vectrs B and C prvide a decmpsitin f A int cmpnents. The vectr D D A B C Figure 2.59 is the hrizntal cmpnent f B. Figure 2.59 appears t shw that the dwnward frce A has a hrizntal cmpnent (namely D). But this can t be since the riginal A acts vertically. Explain the apparent cntradictin. 7