Hardy Cross
Encyclopedia
Hardy Cross, 1885–1959, born in Nansemond County, Virginia
, was a U.S. structural engineer
and the developer of the moment distribution method
for structural calculation of large buildings. The method was in general use from c.1935 until c.1960 when it was gradually superseded by other methods. It made possible the efficient and safe design
of many reinforced concrete buildings during an entire generation.
, who would later become a Celtic studies
scholar. Both studied at Norfolk Academy
. He obtained a BS
in civil engineering
from the Massachusetts Institute of Technology
in 1908, and then joined the bridge
department of the Missouri Pacific Railroad
in St. Louis
, where he remained for a year, after which he returned to Norfolk Academy
in 1909. After a year of graduate study at Harvard he was awarded the MCE degree
in 1911. Hardy Cross developed the moment distribution method while working at Harvard university.
of civil engineering at Brown University
, where he taught for seven years. After a brief return to general engineering practice, he accepted a position as professor of structural engineering
at the University of Illinois at Urbana-Champaign
, in 1921. At the University of Illinois Hardy Cross developed his moment distribution method and influenced many young civil engineers. His students at Illinois had a hard time arguing with him because he was hard of hearing. He left Illinois in 1937 to become the chair of the civil engineering department at Yale University
, a position from which he retired in 1953.
Accurate structural analysis
of large reinforced concrete building frames in the 1950s was a formidable task. It is a tribute to the engineering profession, and to Hardy Cross, that there were so few failures. When engineers had to compute the stresses and deflections in a statically indeterminate
frame, they inevitably turned to what was generally known as the "moment distribution" or "Hardy Cross" method. In the moment distribution method, the fixed-end moments in the framing members are gradually distributed to adjacent members in a number of steps such that the system eventually reaches its natural equilibrium
configuration. However the method was still an approximation but it could be solved to be very close to the actual solution.
The Hardy Cross method is essentially the Jacobi iterative scheme
applied to the displacement formulation of structural analysis.
Today the "moment distribution" method is no longer commonly used because computers have changed the way engineer
s evaluate structures, and moment distribution programs are seldom created nowadays. Today's structural analysis software is based on the flexibility method
, direct stiffness method
or finite element methods
(FEM).
Another Hardy Cross method
is also famous for modeling flows in complex water supply network
s. Until recent decades, it was the most common method for solving such problems.
He received numerous honors. Among these were an honorary
Master of Arts
degree from Yale University
, the Lamme Medal of the American Society for Engineering Education (1944), the Wason Medal for Most Meritorious Paper of the American Concrete Institute
(1935), and the Gold Medal of the Institution of Structural Engineers
of Great Britain (1959).
[Cross 1949:2]
Virginia
The Commonwealth of Virginia , is a U.S. state on the Atlantic Coast of the Southern United States. Virginia is nicknamed the "Old Dominion" and sometimes the "Mother of Presidents" after the eight U.S. presidents born there...
, was a U.S. structural engineer
Structural engineer
Structural engineers analyze, design, plan, and research structural components and structural systems to achieve design goals and ensure the safety and comfort of users or occupants...
and the developer of the moment distribution method
Moment distribution method
The moment distribution method is a structural analysis method for statically indeterminate beams and frames developed by Hardy Cross. It was published in 1930 in an ASCE journal. The method only accounts for flexural effects and ignores axial and shear effects...
for structural calculation of large buildings. The method was in general use from c.1935 until c.1960 when it was gradually superseded by other methods. It made possible the efficient and safe design
Design
Design as a noun informally refers to a plan or convention for the construction of an object or a system while “to design” refers to making this plan...
of many reinforced concrete buildings during an entire generation.
Early life
Cross was born to Virginia planter Thomas Hardy Cross and his wife Eleanor Elizabeth Wright. He had an elder brother, Tom Peete CrossTom Peete Cross
Tom Peete Cross was an American Celticist and folklorist.-Education and career:Cross did his undergraduate education at Hampden–Sydney College, receiving his B.A. in 1899. He went on to Harvard University to pursue an M.A. and Ph.D....
, who would later become a Celtic studies
Celtic Studies
Celtic studies is the academic discipline occupied with the study of any sort of cultural output relating to a Celtic people. This ranges from linguistics, literature and art history archaeology and history, the focus lying on the study of the various Celtic languages, living and extinct...
scholar. Both studied at Norfolk Academy
Norfolk Academy
Norfolk Academy is an independent co-educational day school in Norfolk, Virginia. Chartered in 1728, it is the oldest secondary school in Virginia and the eighth oldest in the United States...
. He obtained a BS
Bachelor of Science
A Bachelor of Science is an undergraduate academic degree awarded for completed courses that generally last three to five years .-Australia:In Australia, the BSc is a 3 year degree, offered from 1st year on...
in civil engineering
Civil engineering
Civil engineering is a professional engineering discipline that deals with the design, construction, and maintenance of the physical and naturally built environment, including works like roads, bridges, canals, dams, and buildings...
from the Massachusetts Institute of Technology
Massachusetts Institute of Technology
The Massachusetts Institute of Technology is a private research university located in Cambridge, Massachusetts. MIT has five schools and one college, containing a total of 32 academic departments, with a strong emphasis on scientific and technological education and research.Founded in 1861 in...
in 1908, and then joined the bridge
Bridge
A bridge is a structure built to span physical obstacles such as a body of water, valley, or road, for the purpose of providing passage over the obstacle...
department of the Missouri Pacific Railroad
Missouri Pacific Railroad
The Missouri Pacific Railroad , also known as the MoPac, was one of the first railroads in the United States west of the Mississippi River. MoPac was a Class I railroad growing from dozens of predecessors and mergers, including the St. Louis, Iron Mountain and Southern Railway , Texas and Pacific...
in St. Louis
St. Louis, Missouri
St. Louis is an independent city on the eastern border of Missouri, United States. With a population of 319,294, it was the 58th-largest U.S. city at the 2010 U.S. Census. The Greater St...
, where he remained for a year, after which he returned to Norfolk Academy
Norfolk Academy
Norfolk Academy is an independent co-educational day school in Norfolk, Virginia. Chartered in 1728, it is the oldest secondary school in Virginia and the eighth oldest in the United States...
in 1909. After a year of graduate study at Harvard he was awarded the MCE degree
Academic degree
An academic degree is a position and title within a college or university that is usually awarded in recognition of the recipient having either satisfactorily completed a prescribed course of study or having conducted a scholarly endeavour deemed worthy of his or her admission to the degree...
in 1911. Hardy Cross developed the moment distribution method while working at Harvard university.
Career
Cross next became an assistant professorProfessor
A professor is a scholarly teacher; the precise meaning of the term varies by country. Literally, professor derives from Latin as a "person who professes" being usually an expert in arts or sciences; a teacher of high rank...
of civil engineering at Brown University
Brown University
Brown University is a private, Ivy League university located in Providence, Rhode Island, United States. Founded in 1764 prior to American independence from the British Empire as the College in the English Colony of Rhode Island and Providence Plantations early in the reign of King George III ,...
, where he taught for seven years. After a brief return to general engineering practice, he accepted a position as professor of structural engineering
Structural engineering
Structural engineering is a field of engineering dealing with the analysis and design of structures that support or resist loads. Structural engineering is usually considered a specialty within civil engineering, but it can also be studied in its own right....
at the University of Illinois at Urbana-Champaign
University of Illinois at Urbana-Champaign
The University of Illinois at Urbana–Champaign is a large public research-intensive university in the state of Illinois, United States. It is the flagship campus of the University of Illinois system...
, in 1921. At the University of Illinois Hardy Cross developed his moment distribution method and influenced many young civil engineers. His students at Illinois had a hard time arguing with him because he was hard of hearing. He left Illinois in 1937 to become the chair of the civil engineering department at Yale University
Yale University
Yale University is a private, Ivy League university located in New Haven, Connecticut, United States. Founded in 1701 in the Colony of Connecticut, the university is the third-oldest institution of higher education in the United States...
, a position from which he retired in 1953.
Accurate structural analysis
Structural analysis
Structural analysis is the determination of the effects of loads on physical structures and their components. Structures subject to this type of analysis include all that must withstand loads, such as buildings, bridges, vehicles, machinery, furniture, attire, soil strata, prostheses and...
of large reinforced concrete building frames in the 1950s was a formidable task. It is a tribute to the engineering profession, and to Hardy Cross, that there were so few failures. When engineers had to compute the stresses and deflections in a statically indeterminate
Statically indeterminate
In statics, a structure is statically indeterminate when the static equilibrium equations are insufficient for determining the internal forces and reactions on that structure....
frame, they inevitably turned to what was generally known as the "moment distribution" or "Hardy Cross" method. In the moment distribution method, the fixed-end moments in the framing members are gradually distributed to adjacent members in a number of steps such that the system eventually reaches its natural equilibrium
Mechanical equilibrium
A standard definition of static equilibrium is:This is a strict definition, and often the term "static equilibrium" is used in a more relaxed manner interchangeably with "mechanical equilibrium", as defined next....
configuration. However the method was still an approximation but it could be solved to be very close to the actual solution.
The Hardy Cross method is essentially the Jacobi iterative scheme
Jacobi method
In numerical linear algebra, the Jacobi method is an algorithm for determining the solutions of a system of linear equations with largest absolute values in each row and column dominated by the diagonal element. Each diagonal element is solved for, and an approximate value plugged in. The process...
applied to the displacement formulation of structural analysis.
Today the "moment distribution" method is no longer commonly used because computers have changed the way engineer
Engineer
An engineer is a professional practitioner of engineering, concerned with applying scientific knowledge, mathematics and ingenuity to develop solutions for technical problems. Engineers design materials, structures, machines and systems while considering the limitations imposed by practicality,...
s evaluate structures, and moment distribution programs are seldom created nowadays. Today's structural analysis software is based on the flexibility method
Flexibility method
In structural engineering, the flexibility method is the classical consistent deformation method for computing member forces and displacements in structural systems...
, direct stiffness method
Direct stiffness method
As one of the methods of structural analysis, the direct stiffness method , also known as the displacement method or matrix stiffness method, is particularly suited for computer-automated analysis of complex structures including the statically indeterminate type...
or finite element methods
Finite element method in structural mechanics
The Finite element method is a powerful technique originally developed for numerical solution of complex problems in structural mechanics, and it remains the method of choice for complex systems. In the FEM, the structural system is modeled by a set of appropriate finite elements interconnected at...
(FEM).
Another Hardy Cross method
Hardy Cross method
The Hardy Cross method is an iterative method for determining the flow in pipe network systems where the inputs and outputs are known, but the flow inside the network is unknown....
is also famous for modeling flows in complex water supply network
Water supply network
A water supply system or water supply network is a system of engineered hydrologic and hydraulic components which provide water supply. A water supply system typically includes:# A drainage basin ;...
s. Until recent decades, it was the most common method for solving such problems.
He received numerous honors. Among these were an honorary
Honorary degree
An honorary degree or a degree honoris causa is an academic degree for which a university has waived the usual requirements, such as matriculation, residence, study, and the passing of examinations...
Master of Arts
Master of Arts (postgraduate)
A Master of Arts from the Latin Magister Artium, is a type of Master's degree awarded by universities in many countries. The M.A. is usually contrasted with the M.S. or M.Sc. degrees...
degree from Yale University
Yale University
Yale University is a private, Ivy League university located in New Haven, Connecticut, United States. Founded in 1701 in the Colony of Connecticut, the university is the third-oldest institution of higher education in the United States...
, the Lamme Medal of the American Society for Engineering Education (1944), the Wason Medal for Most Meritorious Paper of the American Concrete Institute
American Concrete Institute
The American Concrete Institute is a non-profit technical society and standard developing organization . ACI was founded in 1904 and its headquarters are currently located in Farmington Hills, Michigan, USA.- ACI History :1904-1929:...
(1935), and the Gold Medal of the Institution of Structural Engineers
Institution of Structural Engineers
The Institution of Structural Engineers is a professional body for structural engineering based in the United Kingdom. It has 27,000 members in 105 countries. The Institution provides professional accreditation for structural engineers...
of Great Britain (1959).
Hardy Cross Method
Hardy Cross's description of his method follows: "Moment Distribution. The method of moment distribution is this:- Imagine all joints in the structure held so that they cannot rotate and compute the moments at the ends of the members for this condition;
- at each joint distribute the unbalanced fixed-end moment among the connecting members in proportion to the constant for each member defined as "stiffness";
- multiply the moment distributed to each member at a joint by the carry-over factor at the end of the member and set this product at the other end of the member;
- distribute these moments just "carried over";
- repeat the process until the moments to be carried over are small enough to be neglected; and
- add all moments – fixed-end moments, distributed moments, moments carried over – at each end of each member to obtain the true moment at the end."
[Cross 1949:2]