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SALINAS
"In the Midst of a Loneliness": The Architectural History of the Salinas Missions |
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APPENDIX 2:
SEVENTEENTH CENTURY MISSION CHURCH ROOF BEAMS:
A Structural Analysis
William R. Gafford, Ph.D
and
James E. Ivey
A preliminary analysis of the beams that supported the roof of the nave and transepts of the mission churches of Our Lady of the Immaculate Conception (Nuestra Señora de la Purísima Concepción) at the Quarai unit, and San Gregorio II at the Abó unit of Salinas National Monument, was performed using data supplied by Mr. James Ivey of the National Park Service office in Santa Fe, New Mexico, and by on-site measurements to supplement information supplied by Mr. Ivey. This activity was done during the months of March through June, 1988, for the main purpose of determining whether the members supporting the roofs of these churches were just adequately structurally designed, whether they were moderately overdesigned, heavily overdesigned, or dangerously underdesigned.
Since there were no trained structural engineers available for consultation during the planning of Spanish colonial mission churches of the seventeenth century, the franciscan friar in charge of such planning and construction was forced to rely on his own past experience perhaps coupled with that of older and more experienced builders. The results were astounding! Calculations have shown that the beams supporting the roofs of the mission churches at Quarai and Abó were very close to the sizes that would be required had the structures been built in accordance with current building codes!
These calculations were based on current procedures for the design (selection of size) of solid wood timbers for such applications. Certain assumptions were made, particularly regarding the material that was used for these major roof beams, by Mr. Ivey; an estimate of the probable maximum snow load that would occur in that region of the state; and the size of tree branches that were used for "latillas" (very small secondary beams) that span between the main beams.
Input items used in the calculations below have been identified as "factual" or "assumed" as the case may be, with the "factual" items having been supplied by Mr. Ivey and by on-site measurements, and the "assumed" items having been supplied by the writer and verified by Mr. Ivey and others as being reasonable.
NUESTRA SENORA DE LA PURISIMA CONCEPCION DE QUARAI
Nave Roof Beams
The following are the input items used for the mathematical analysis of the roof beams of the nave of the mission church at Quarai:
Beam Size: 10.5" x 12" (factual)
Span (L) of Beams: 27 feet (factual)
Beam spacing: 24" on center (factual)
Corbels: 2, stacked at each end of beams (factual)
Corbel size: 10.5 x 12" (factual)
Material: Ponderosa (or lodgepole) pine (factual)
Roof live load: 25 pounds per square foot (assumed)
Weight of roofing: 56 pounds per square foot (assumed)
Weight of latillas 4 pounds per square foot (assumed)
Ultimate bending stress: 8000 lb. per sq. in. (assumed average)1
Following are abbreviations used:
E elastic modulus of the material
Fact actual bending stress
fb allowable bending stress
Fb average ultimate bending stress
ft foot or feet
ft-lb moment, foot-pounds
I moment of inertia of the section
in inch or inches
in-lb moment, inch-pounds
L span, feet
lb pound or pounds
Mmax maximum bending moment
plf pounds per linear foot
psf pounds per square foot
psi pounds per square inch
Sact actual section modulus
Smin minimum section modulus
sq ft square foot or square feet
W total load per beam
w load per linear foot of beam
Calculations
The first condition investigated assumes the beam to be simply-supported at its ends with a span of 13.5 feet, which is the clear distance between ends of the longest corbels under each end of the beam. This condition was investigated initially because it provides the worst condition of bending and beam deflection, and a positive result would clearly indicate that the roof beams were not dangerously underdesigned.
Tributary area per beam = (span) (spacing)
= (13.5) (2.0) = 27.0 sq ftW = (Tributary area) (Total load psf)
= (27.0) (85.0) = 2295 lb + dead weightDead weight = (weight per foot) (span)
= (26.25) (13.5) = 354 lbW = 2295 + 354 = 2650 lb
w = W/L = 2650/13.5 = 196.25 plf
Mmax = (w)(L)2/8 = (196.25)(13.5)2/8
= 4471 ft-lbs = 53,650 in-lbSmin = Mmax/Fb = 53,650/8000 = 6.71 in3
Sact = (b)(d)2/6 = (10.5)(12)2/6 = 252 in3
Fact = Mmax/Sact = 53,650/252.0 = 212.9 psi
Safety factor = Fb/Fact = 8000/212.9 = 37.6
Current safety factor = Fb/fb = 8000/850
= 9.41Deflection = (5)(W)(L)3/(384)(E)(I)
= (5)(2650)(162)3/(384)(1,100,000)(1512)
= 0.09 inDeflection allowed = L/360 = 162/360 = 0.45 in
The second condition investigated assumes the beam to be partially fixed at its extreme ends, due to the combined action of the double corbels under each end and the mass of masonry placed on top of the beam. This condition is more realistic than the first investigated, inasmuch as the beams, in addition to the support conditions just described, were tightly packed into their sockets in the masonry walls with clay.
Tributary area per beam = (27.0)(2.0) = 54.0 sq ft
W = (54.0)(85.0) + dead weight
Dead weight = (26.25)(27.0) = 709 lb
W = 4590 + 709 = 5299 lb
w = 5299/27 = 196.25 plf
Mmax = (w)(L)2/10 = (196.25)(27)2/10
= 14,307 ft-lbs = 176,680 in-lbFact = Mmax/Sact = 176,680/252.0 = 681.3 psi
Safety factor = Fb/Fact = 8000/681.3 = 11.74
Current safety factor = Fb/fb = 8000/850
= 9.41Deflection = (W)(L)3/(109)(E)(I)
=(5299)(324)3/(109)(1,100,000)(1512) = 0.99 inDeflection allowed = L/360 = 324/360 = 0.90 in
A close investigation of the results of calculations for the second assumed condition, which is considered to be more realistic than the first for the reasons stated above, shows that the apparent safety factor is slightly in excess of that expected by current building codes, indicating that the beams were approximately the correct size for bending. Scrutinizing the probable maximum deflection, which would occur at the center of the span, it would appear that the deflection is slightly in excess of that limited by current building codes. However, the value L/360 is somewhat arbitrary and is based on the probability that minimum damage to attached plaster ceilings would occur if the deflection were limited to this value. If no such ceiling existed, as was the case of the mission church at Quarai, a normal deflection of L/240 would be permitted by current building codes, because this magnitude of deflection would not be visually noticeable. However, since this mission church and others constructed at about the same time had adobe and plaster surfaces on top of the latillas and mating, a deflection value of about L/300 is not unrealistic.
Under this condition:
Deflection allowed = L/300 = 324/300 = 1.08 in which is slightly in excess of the probable maximum value of 0.99 in.
It would appear, therefore, that deflection is the governing factor in the selection of beam sizes for the roof of the mission church at Quarai, and not bending, which is normally the initial assumption in the investigation or design of roof beams.
Transept Roof Beams
The beams supporting the roof of the transept portions of the mission church at Quarai were investigated in a manner similar to that used for the nave roof beams, assuming that the ends of the beams were partially restrained against rotation as in the second investigation of the roof beams. The following mathematical analysis was used:
Beam Size: 9" x 11" (factual)
Span (L) of beams: 25 feet (factual)
Spacing: 24 inches on center (factual)
Tributary area per beam = (25.0)(2.0) = 50.0 sq ft
W = (50.0)(85.0) = 4250 + dead weight
W = 4250 + (20.63)(25.0) = 4766 lb
w = W/L = 4766/25 = 190.63 plf
Mmax = (w)(L)2/10 = (190.63)(25)2/10
= 11,914 ft-lb = 142,973 in-lbFact = Mmax/Sact = 142,973/181.5 = 788 psi
Safety factor = Fb/Fact = 8000/788 = 10.16
Current safety factor = Fb/fb = 8000/850 = 9.41
Deflection = (W)(L)3/(109)(E)(I)
= (4766)(300)3/(109)(1,100,000)(998) = 1.08 inDeflection allowed = L/300 = 300/300 = 1.0 in
The deflection expected of these beams, 1.08 inch, is slightly in excess of the value that would probably be allowed by current building codes; however, it would undoubtedly be acceptable. The results of this investigation clearly show that the beams used to support the roof of the transept of the mission church at Quarai were approximately the same size that would be required were the church built in conformance with current building codes.
SAN GREGORIO II, ABO
Nave Roof Beams
The following are the input items used for the mathematical analysis of the roof beams of the nave of the mission church at Abó:
Beam Size: 12" x 12" (factual)
Span (L) of Beams: 25 feet (factual)
Beam Spacing: 9 feet (factual)2
Corbels: 2, stacked at each end of beams (factual)
Corbel size: 12" x 12" (factual)
Material: Ponderosa (or lodgepole) pine (factual)
Roof live load: 25 pounds per square foot (assumed)
Weight of roofing: 60 pounds per square foot (assumed)
Weight of roof boards: 4 pounds per square foot (assumed)
Ultimate bending stress: 8000 lb. per sq. in. (assumed average)2
Abbreviations used are the same as those for calculations for beams for the mission church at Quarai.
Calculations
The first condition investigated assumes the beams to be partially fixed-end with a span of 25.0 feet, which is the clear distance between walls of the nave. This condition was investigated initially because it provides the probable worst condition of bending and beam deflection due to the long span, and a positive result would clearly indicate that the roof beams were not dangerously underdesigned.
Tributary area per beam = (span) (spacing)
= (25.0)(4.5) = 112.5 sq ftW = (tributary area) (Total load psf)
= (112.5) (89.0) = 10,013 lb + dead weightDead weight = (weight per foot) (span)
= (30.0)(25.0) = 750 lbW = 10,103 + 750 = 10,763 lb
w = W/L = 10,763/25 = 430.5 plf
Mmax = (w)(L)2/10 = (430.5)(25)2/10
= 26,906 ft-lbs = 322,875 in-lbSmin = Mmax/Fb = 322,875/8000 = 40.36 in3
Sact = (b)(d)3/6 = (12) (12)3/6
= 288.0 in3Fact = Mmax/Sact = 322,875/288.0 = 1121 psi
Safety factor = Fb/Fact = 8000/1121 = 7.14
Current safety factor = Fb/fb = 8000/850
Deflection = (W)(L)3/(109)(E)(I)
= (10,763)(300)3/(109)(1,100,000)(1728)
= 1.40 inDeflection allowed = L/360 = 300/360 = .83 in
The probable maximum deflection, 1.40 inches, is nearly two times the value that would be permitted by the current building codes, and would certainly not be permitted.3 A more realistic analysis of the beams would be to assume a somewhat shorter span in the calculations, perhaps the clear span between the ends of the bottom corbels, a total of eighteen feet, still considering the ends to be partially fixed against rotation due to the mass of masonry on top of the beams and the double corbel action supporting their ends.
Tributary area per beam = (18.0)(4.5) = 81.0 sq ft
W = (81.0)(89.0) + dead weight
Dead weight = (30.0)(18) = 540 lb
W = 7209 + 540 = 7749 lb
w = 7749/18 = 431.0 plf
Mmax = (w)(L)2/10 = (431.0)(18)2/10
= 13,948 ft-lbs = 167,378 in-lbFact = Mmax/Sact = 167,378/288 = 581.2 psi
Safety factor = Fb/Fact = 8000/581.2 = 13.77
Current safety factor = Fb/fb = 8000/850
= 9.41Deflection = (W)(L)3/(109)(E)(I)
= (7749)(216)3/(109)(1,100,000)(1728)
= 0.38 inDeflection allowed = L/360 = 216/360 = 0.60 in
A study of the second assumed condition of span and end-support conditions of the nave beams indicates that they would undoubtedly be permitted because of the probable maximum deflection of 0.38 inch compared to the allowed value of 0.60 inch. However, as earlier noted (see the analysis of the roof and transept beams of the mission church at Quarai), a deflection of L/300 would not be unrealistic. Under this condition:
Deflection allowed = L/300 = 0.72 inch which is nearly two times the maximum deflection (0.38 inch) that would be expected for these roof beams.
It would appear again that deflection is the governing factor in the selection of beam sizes for roof of the mission church at Abó, and not bending. This investigation again shows that the beams used to support the roof of the nave of the mission church at Abó (San Gregorio II) were approximately the same size that would be required were the church built in conformance with the current building codes.
William R. Gafford
Prof. Emeritus of Civil Engineering
The University of New Mexico
The deflections calculated by Dr. Gafford are under worst conditions. The roof has a live load of 25 pounds per square foot, the equivalent of about twenty inches of snow, the expected maximum for the area. Under dry conditions the weight would be about thirty percent less, and the deflection would decrease about the same amount.
Using Gafford's conclusions above, it is apparent that the corbels are not purely decorative, but an important part of the supporting structure of the roof. The more firmly fixed in the walls the corbels and beams were, and the more weight above their ends in the form of parapets, the better the structure worked.
Some interesting theoretical conclusions may be reached about other missions, using Gafford's calculations. For example, at San Isidro, beams set at two-foot centers, with a breadth of 10 1/2 inches and a depth of twelve inches, with no corbels and no massive parapets to fix their ends, would deflect about 1.6 inches over the twenty-eight foot free span of the nave under a full load of twenty inches of snow. At the same time, the beams add lateral support to the side walls. Such a roof would probably be acceptable to the Franciscans, even if the deflection is a little large by our standards. A slight increase in the beam cross-section would reduce the deflection to less than an inch.
A more interesting case is that of Pecos. Assuming pairs of beams set above the buttresses along the outside of the nave wall, the beam pairs would have been centered at about nine foot intervals, or a spacing of about 4.5 feet per beam, as at Abó. Using square beams of Ponderosa pine one foot on a side over the thirty-nine foot free span of the nave, a maximum live load produces a deflection of a little over eight inches. This is obviously excessive. Pecos must have had substantial corbels, massive parapets, and beams of spruce or fir with a cross-section somewhat larger than one foot. However, even using the one-foot beams, if the beam and corbel ends were well-fixed by massive parapets above the buttresses, then the effective span would have been about twenty-five feet and the deflection about 1 1/2 inches. Good corbelling makes a difficult roofing problem solvable.
It appears that Abó and Quarai were designed and built with the intent to fix the ends of the beams and corbels supporting the roof, rather than for the corbels to be purely decorative.4 This would have achieved the most efficient use of the structural elements of the roof, as well as reducing the size of the necessary beams. The fanciful crenelations of Abó appear to be there to add mass above the ends of the vigas and corbels, thereby allowing them to have a spacing of 4 1/2 feet. How many of the engineering considerations explored by Gafford were known to the Spanish designers of the Salinas missions is unknown. It is quite apparent, however, that they had some rule of thumb that allowed them to estimate the size of the beam needed under various conditions of wood type, span, beam spacing, and corbel design, in such a manner that their results come very close to matching the choices modern builders would make. The designers of the Salinas missions appear to have been the friars stationed at each one. This indicates that the friars must have learned such a rule during their training to become missionaries, along with a large range of other skills. The training programs for friars in the seventeenth century is unknown, but based on the results recorded in the documents and structures of Franciscan New Mexico, it is clear that such training programs deserve research and study.
1 Harmer E. Davis, George Earl Troxell, and Clement T. Wiskocil. The Testing and Inspection of Engineering Materials, third edition (New York: McGraw-Hill Book Company, 1964).
2 Two beams were placed side by side at each center, so that the effective spacing of a beam was 4.5 feet.
3 The beam spacing at Abó is about twice that at Quarai, which doubles the total load per beam.
4 Ross Montomery, in his analysis of the roof structure of Awatovi, assumed that the corbels were effectively useless; see Ross Gordon Montgomery, Watson Smith, and John Otis Brew, Franciscan Awatovi: The Excavation and Conjectural Reconstruction of a 17th-Century Spanish Mission Established at a Hopi Indian Town in Northeastern Arizona, Reports of the Awatovi Expedition, Report No. 3, Papers of the Peabody Museum of American Archeology and Ethnology, Harvard Univesity, Vol. 36 (Cambridge: Peabody Museum, 1949) pp. 240-42. Awatovi was a small, simple church with a nave width of only nineteen feet, but it had adobe walls three feet thick. With well-set beams and a substantial parapet, the corbells would have definitely improved the support of the roof.
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Last Updated: 28-Aug-2006