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Thomas U. Walter's 1859 drawing shows a section of the dome. Suspended above the eye of the inner dome is Constantino Brumidi's fresco The Apotheosis of Washington. The concept of a double dome with a large painting incorporated between the two segments was inspired by the dome of the Panthéon, in Paris, which Walter had visited in 1838.

Examining the

CAPITOL DOME

A comprehensive structural analysis of the United States Capitol's dome reveals that the dome is sound and capable of resisting all reasonably anticipated loadings.

Eric C. Stovner, P.E., S.E., Emmanuel E. Velivasakis, PE, Charles H. Thornton, PE, Glenn G. Thater, and Christopher P. Pinto, I.E.

In 1990, Congress authorized a capital program for the renovation of the dome of the United States Capitol to address concerns arising from a perceived increase in exterior cracks and water infiltration. As part of the master plan initiated by Alan M. Hantman, the architect of the Capitol, to ensure the soundness and longevity of the dome, LZA Technology, a division of the Thornton-Tomasetti Group, headquartered in New York City, was commissioned in 1998 to provide structural engineering consultation services. The primary focus of LZA's work was an analysis of the dome's structural elements using state-of-the-art computer-assisted techniques and methods. The analysis was undertaken to assess the structural soundness of the dome prior to implementation of a four-year, $44-million program devised for its restoration by Hoffman Architects, headquartered in North Haven, Connecticut. The Office of the Architect of the Capitol is overseeing the project.

Construction of the first capitol dome-made of timber-was begun in 1818, bringing to fruition President George Washington's intent to crown the Capitol with a grand dome. In the early 1850s Thomas U. Walter, the fourth architect of the Capitol, was commissioned to design and supervise the expansion of the U.S. Capitol. In 1855 he proposed replacing the timber dome with a larger dome that would be in architectural scale with the expanded building. Because of the preexisting supporting rotunda masonry walls, the new dome-three times the height of the original dome-would have to be limited in weight. The designers chose to use cast iron as the primary building material, an appropriate selection since it can be cast with cutouts or voids in areas where material is not required for structural purposes. This reduces the overall weight of the structure. Cast iron also enabled the designers to customize shapes for the members and provided superior fire resistance. The castings were constructed within close tolerances, and their quality is excellent. Interestingly, the total cost of the new iron dome was approximately the same as that of the smaller timber dome erected 40 years earlier: roughly $1 million.

Construction of the dome began in 1857 under the supervision of Montgomery Meigs, a military engineer. Although work was halted at times during the Civil War, President Lincoln-according to some accounts-made it a point of pride for the Union to complete the dome despite the ongoing conflict. The 20 ft (6.1 m) tall Statue of Freedom, which caps the dome, was installed following the dome's completion in 1863.

The dome structure, together with its inner dome, its outer cupola, most of its outer skin, and its lower skirt, is cast iron. The boilerplates, tie rods, hangers, and skirt needle beams are wrought iron. (The skirt needle beams, rolled in 1857, were among the first generation of I-beam shapes.) From the highest point of the Statue of Freedom-which commands the top of the dome-to the top of the supporting wall, the dome is approximately 199 ft (60 m) in height. The diameter of the structure at the base is approximately 100 ft (30 m). The top of the statue is approximately 290 ft (88 m) above grade, which compares to St. Paul's in London and St. Isaac's in St. Petersburg, among the largest domes in the world. The main framing of the dome consists of 36 meridional arched half-ribs, arranged as inverted "wishbones" and tied together with circumferential hoop rings. These ribs transmit loads down to the cylindrical wall of the rotunda, which is constructed of bonded brick and stone masonry and is approximately 5 ft (1.5 m) thick. The rib sections-and most of the hoops as well-are trussed with chord depths of approximately 4 ft (1.2 m). The rib sections are approximately 10 ft (3 m) tall and spaced at approximately 9 ft (2.7 m) centers toward the bottom of the dome. The heaviest cast section used in construction weighs approximately 10 tons (9.1 Mg).

This historical photo shows the twin-shelled iron dome, which was constructed between 1857 and 1863, near completion.

In addition to the main rib framing, elaborate arrangements of cast-iron members support the inner and outer shells of the dome and give it its distinctive shape. The inner dome is an ornamental architectural feature-a coffered, truncated shell-that is hung from the ribs and bracketed at the peristyle pillars and provides an oculus view up to The Apotheosis of Washington-Constantino Brumidi's fresco painted as a tribute to George Washington-on the underside of the canopy, which is hung from the tops of the ribs. At the top of the dome, the tholos is framed with 12 columns, on top of which stands the 19.5 ft (5.9 m) Statue of Freedom. The dome elements are bolted together through cast lugs. At the base of the dome, the colonnade is approximately 10 ft (3 m) wider than the rotunda wall below-accomplished by large cantilevering brackets at the top of the wall. The brackets support the colonnade above and are concealed by a curtain of cast iron, which is referred to as the skirt.

Structural elements and loading attachments (hangers et cetera) frame into other elements at common joints and thus form a trussed, or axial force, system. Local, or secondary, bending occurs within individual elements (rib sections II to VIII) since such elements are not completely triangularized within their length. These secondary bending stresses are small since the nontriangularization occurs within the member length. The nontriangularization was a designed optimization of material because it entailed shorter total lengths of cast material.

LZA engineers performed visual condition surveys of all representative structural component types. The superstructure is in good condition and exhibits no significant deterioration. Visual observation of nonstructural elements revealed deterioration of certain elements, particularly exterior balcony floor plates and balustrade rails, but this does not affect the overall structural integrity of the dome. Some evidence of potential distress in the form of cracks in the rotunda masonry walls was observed, and LZA installed an electronic crack- and temperature-monitoring system on the walls to evaluate the significance of these cracks.

While a traditional masonry dome is a three-dimensional continuous structure, an iron-ribbed dome can be accurately idealized as discrete, linear elements. The manner in which the dome's structural and attached elements are detailed permits the dome to be modeled using linear elements. The dome's structural elements are primarily linear truss members with connectors symmetrical to work points (lines of force action), and the dome's attached elements-cupola, cladding, canopy, and inner dome-are all detailed to preclude independent shell behavior.

LZA developed a decision tree and matrix to select the computer analysis program most suitable for the ribbed truss dome. Criteria included maximum model size and computing speed, accuracy of matrix math and effects of internal renumbering of nodes, ease of inputting tapered and unusual member shape properties, and graphical output capabilities. LZA chose a prerelease version of LARSA, produced by LARSA, Inc., of New York City, which uniquely provided most of the desired criteria. Initially the run time for the model used took three and a half hours on a 300 MHz Pentium II PC with a 256 Mbyte RAM. LZA engineers developed a method implemented by LARSA that optimized the hardware memory allocation and trimmed the run time to three hours. LARSA provided support and a first-generation matrix numerical solver technique that cancels out zeros at unrestrained degrees of freedom and reduces the bandwidth, resulting in a run time of just seven minutes.

Thomas U. Walter's drawing, above, shows a section of the tholos and the Statute of Freedom.

LZA created the geometry for the three-dimensional structural analysis model based on a thorough review of drawings of the dome executed by its designer, Thomas U. Walter, and site visits. The analytical computer model includes the bottom of the cast-iron brackets that bear on the rotunda wall and extends to the top of the tholos at the underside of the statue pedestal. The model represents the primary structural elements of the dome, with the exception of the skirt elements that carry their own weight onto bearing walls below the roof of the Capitol. There are approximately 7,700 nodes and 16,000 finite- element members in LZA's structural analysis model.

Nodal geometry for the ribs was developed on the basis of a drawing by Walter entitled "Demonstration of Main Rib." This drawing provides dimensions to the interior and exterior edges of the rib sections. LZA established its nodal geometry by considering the centerlines of the rib chord cruciform cross sections. These dimensions are appropriate for the analysis of a trussed rib structure in which primary tension and compression forces act through the centroids of truss chord members

LZA coordinated with Lucius Pitkin, Inc., of New York City, for a program of metallurgical testing. Nondestructive in situ surface replication and hardness tests were performed at various rib sections and on the boilerplate to characterize the materials and determine their strength. To determine moduli of elasticity, a coupon sample was cut from a rib and a sample was cut from the primary boilerplate. Laboratory tests were also performed on samples of the exterior skin that had been cut for the installation of drainage piping in 1994.

The cast iron is a gray type, the ribs having a ferritic microstructure and the exterior skin a pearlitic microstructure. Based on the hardness tests, the estimated ultimate tensile strength averages approximately +18 kips per square inch, or ksi, (+124 MPa), which has been confirmed by the laboratory testing performed on the coupon. The estimated ultimate compressive strength averages approximately -70 ksi (-483 MPa). Based on laboratory tensile testing of the boilerplate material, its yield strength averages +29.5 ksi (+203 MPa) and its ultimate tensile strength averages +46 ksi (+317 MPa). These strength values correlate with values from 19th-century material testing as described in engineering texts from the period and testing performed in 1956. The density of the rib coupon was measured at 430 pounds per cubic foot, or pcf (6.9 Mg/m3), and the boilerplate sample at 480 pcf (7.7 Mg/m3). These densities are in the typical range for cast iron and wrought iron, respectively, although 450 pcf (7.2 Mg/m3) is closer to the mean value for cast iron. The modulus of elasticity, E, was tested, utilizing strain gauges on a coupon, at an average 10,000 ksi (69,000 MPa) for the cast iron. Though at the lower range of values for cast iron, this is typical for cast iron of this strength.

The dead-load self-weight was calculated by the analysis software using the member areas and material densities input in the model. Superimposed dead loads-such as the inner dome and the exterior skin plates-were calculated from information in the Walter drawings. Dead loads were calculated within close range of the historically published weight of the dome, 8,909,200 lb of iron (4,041,213 kg). An interesting conclusion from the calculation of dead loads is that the percentage of material volume or weight devoted to the superstructure is approximately the same as that devoted to nonstructural elements. In comparison, modern steel-framed structures typically weigh less than a fifth as much as the building elements they support.

Various live loads are possible; the governing live-load condition consists in a buildup of ice on the exterior surface during a cold storm. A distributed live load from a half-inch (12.7 mm) layer of ice was utilized with two cases considered: ice buildup on the entire dome surface and an unbalanced load condition of ice on only half of the dome. The weight of a half-inch (12.7 mm) layer of ice on the entire dome surface is 122,000 LB (55.3 Mg).

Wind loads were calculated in accordance with the principles and climatological velocity and gust response factors set by Building Officials and Code Administrators International (BOCA) and ASCE 7-93 (Minimum Design Loads for Buildings and Other Structures), utilizing the 80 mph (129 km/h) design wind speed specified in the building code for Washington, D.C., with wind pressures applied perpendicular to every rib. The external pressure coefficients appropriate for a large-diameter cylindrical structure such as the dome were derived from research performed for cooling tower structures. The magnitudes of the peak coefficients result in wind pressure, which correlate with magnitudes for conventional buildings calculated to include internal wind pressure.

Because the structural elements of the dome are attached and supported with restraint and are not thermally isolated, changes in temperature will induce stresses in the structure as it expands or contracts. The average daily temperature in Washington is approximately 56°F (13.3°C), the highest temperature ever recorded being 106°F (41.1°C) and the lowest -15°F (26.1°C). Assuming the dome was built during a period of average temperature, the maximum changes in temperature that the dome has been subjected to are a positive 50°F (27.8°C) change and a negative 71°F (39.4°C) change. Such thermal differentials were applied as loads to the entire analysis model, and unbalanced thermal loading conditions were also considered.

This view from the rotunda floor of the inner dome shows Brumidi's Apotheosis.

The various potential load combinations were scrutinized to determine the governing stress condition in the structure. Load combinations were factored in accordance with ASCE 7-93.

All members of the structural analysis model were reviewed for the worst-case stress condition for all load combinations. The vertical reactions for dead plus superimposed dead (service) loads sum to 8.4 million LB (3.8 million kg), which is equivalent to the weight of the dome minus the skirt. As the gravity loads increase toward the bottom of the dome, the ribs increase in cross-sectional size. However, the compression stresses generally increase gradually in the ribs toward the bottom. Stresses range from 0 to -4 ksi (0 to -27.6 MPa).

The maximum horizontal reaction at each rib pillar from gravity loads is about 3 kips (1.36 Mg) and is radially outward (perpendicular to the rotunda wall). Thus, the horizontal reactions exert an inward force at the top of the rotunda wall. Approximately 40 ft (12.2 m) above is the spring line-or the bottom of the arched portion of the ribs-at which the dome tends to thrust outward. A pivoting or double curvature made possible by the flexibility of the tall peristyle pillar construction changes the direction of the horizontal reactions and delivers these reactions inward at the top of the rotunda wall. This is a brilliant redirection of structural forces. Presumably, this was considered by Walter's chief assistant, August G. Schoenborn, who conceived the structural form of the rib and peristyle framing. The rotunda masonry bearing wall, cylindrical in plan, cantilevers above the Capitol's fourth, or attic, floor level. Inward horizontal reactions deriving from gravity loads are ideally suited for the cantilevered unreinforced masonry bearing wall because the cylinder behaves as a compression ring.

There are approximately 7,700 nodes and 16,000 finite-element members in LZA Technology's structural analysis model, shown here in isometric view.

The first hoop from the top that undergoes tension in response to dead and superimposed dead loads is the hoop between sections III and IV. The hoops of solid wrought iron plate have higher forces than the hoops of rigid cross bracing because of the increased areas and stiffnesses of the plates.

The hoop boilerplate at the top of section VIII has maximum compression stresses of -2.2 ksi (-15.2 MPa). A gradual transition of stress levels occurs in the intermediate hoops, with the exception of the boilerplate hoop between sections V and VI, which draws more load than the cross bracing in the hoops at the levels immediately above and below. Displacements are well within generally acceptable limits. The maximum vertical deflection of the dome under dead plus live loads is less than 1/4 in. (6.4 mm).

From lateral loading, the maximum tension stress in the tie rod bracing members that occur between ribs is +7 ksi (+48.3 MPa). Rib and hoop stresses range from 0 to +4 ksi (+27.6 MPa).

The maximum deflection from lateral load is about 1 in. (25 mm) and occurs at the tholos level. Expressed as a ratio to the height of the dome, the lateral deflection is 1/2,500 of the height. Because of the magnitude of dead loads, the dome is not subject to uplift for design lateral loads. .

Load combinations involving peak thermal loads tend to be the governing load combinations for the rib section chord members. Thermal forces increase toward the bottom of the dome and become larger in rib sections II and III. Hoop forces arising from thermal loads are moderate. The largest deformations occur during peak thermal loadings. The largest vertical deflection, about 1 in. (25 mm) downward, occurs in the tholos area and is due to combined dead loads and negative differential thermal loads. The largest lateral deflection, about 1.5 in. (38 mm), occurs near the top of the tholos and is due to combined dead loads and unbalanced positive differential thermal loads. Plots of the crack displacement and temperature test data show a clear and distinct pattern; that is, as the temperature increases the cracks decrease in size and as the temperature decreases the cracks increase in size. Positive thermal loading on the structure causes the cast iron to expand and results in an outward force being applied to the top of the rotunda walls. The approximately 5 ft (1.5 m) thick masonry walls move outward slightly as the thermal loading increases. The movement, although very small, is greatest at the top of the wall and decreases toward the bottom. This movement will tend to close horizontal or diagonal cracks on the exterior face of the wall as the thermal loading increases. The cracks' average movement is 2/1,000 to 3/1,000 in. (51 to 76 µm) per 10°F (5.56°C) change in temperature.

The recent metallurgical testing substantiated the notion that the ultimate stresses of the dome elements are typical of expected strengths. LZA consulted engineering texts of the 19th century containing descriptions of material testing and calculated or utilized the appropriate factors of safety required to determine suitable allowable stresses. LZA further reviewed the stability characteristics of the dome compression elements. The unstiffened projecting elements, or flanges, typical of the rib and hoop cross-sectional cruciform shapes are sufficiently stout that the allowable compression stresses are not reduced for local buckling considerations. The rib and hoop unbraced lengths were considered by calculating the Euler buckling load of each compression element. For example, the allowable compression stress at the inner chord of rib section V is reduced from -12 ksi (-82.7 MPa) allowable crushing stress to a governing -7 ksi (-48.3 MPa) allowable compression stress caused by buckling. Calculations of the allowable buckling stresses reveal further optimized design: rib sections toward the bottom and hoops toward the top (those members receiving the highest compression forces) are the members with the greatest stoutness.

Defining the factor of safety as the ratio of ultimate failure load to allowable load provides a factor of safety of 6 for the cast iron and a factor of safety of 4 for the wrought iron. For all load combinations, all members are within allowable stresses. Evaluation of the rib and hoop member stresses reveals that approximately 80 percent of the structural members are stressed to less than +3 ksi (+20.7 MPa).

For the wrought iron boilerplates, the maximum tensile stress is in the primary boilerplate at the spring line. The maximum tensile stress is +2.4 ksi (+16.5 MPa), which is well within the allowable tensile stress of +12 ksi (+82.7 MPa). The boilerplate between sections V and VI undergoes compression with a maximum stress of -2.5 ksi (-17.2 MPa), which is within the allowable stress taking into account the buckling strength. Shear stresses were reviewed and found to be minimal, approximately 360 psi (2.48 MPa) maximum. Such stress values are acceptable, since allowable shear stresses are approximately 3 ksi (20.7 MPa) and 12 ksi (82.7 MPa) for cast and wrought iron, respectively.

During a 1956 analysis conducted by the engineering firm Seeley, Stevenson, Value, and Knecht (SSVK), of New York City, significant deterioration was observed in the primary boilerplate upon removal of moldings at the joint between the exterior first-story deck plates and the exterior vertical dome plates. SSVK measured a loss of 25 percent of boilerplate material. As a result, two posttensioned cables were added at the boilerplate level. LZA's three-dimensional analysis indicates that the cables reduce the boilerplate stresses by 600 psi (4.1 MPa), or approximately one-third of the original stresses in the boilerplate. In terms of the magnitude of stress, the loss of material found by SSVK is insignificant, as the ratio of the actual tensile stress to the allowable tensile stress is 20 percent, or if corrected for locations suffering 25 percent deterioration, a stress ratio of 27 percent.

The structural elements of the dome are within their allowable stresses for all applicable loadings. The dome of the United States Capitol is a crowning achievement in the technology of cast iron. Castings were constructed within close tolerances. The quality of the structural castings is excellent. In terms of structural and construction design, the dome is a testament to the talents of scientifically trained minds. The optimization of structural members and the levels of structural redundancy support a structure that is sound and capable of resisting all reasonably anticipated loadings. Thomas U. Walter's accomplishment was a masterful one.

Eric C. Stovner, PE, SE, M.ASCE, is an associate with LZA Technology in Tustin, California. Emmanuel E. Velivasakis, PE, F.ASCE, is a senior vice president, Charles H. Thornton, Ph.D., PE, Hon.M.ASCE, the chairman, Glenn G. Thater a project director, and Christopher I. Pinto, I.E., A.M.ASCE, a senior engineer with LZA Technology in New York.