SUMMARY / RELATED TOPICS

In structural engineering, the flexibility method called the method of consistent deformations, is the traditional method for computing member forces and displacements in structural systems. Its modern version formulated in terms of the members' flexibility matrices has the name the matrix force method due to its use of member forces as the primary unknowns. Flexibility is the inverse of stiffness. For example, consider a spring that has Q and q as its force and deformation: The spring stiffness relation is Q = k q where k is the spring stiffness, its flexibility relation is q = f Q. Hence, f = 1/k. A typical member flexibility relation has the following general form: q m = f m Q m + q o m where m = member number m. Q m = vector of member's characteristic deformations. F m = member flexibility matrix which characterises the member's susceptibility to deform under forces. Q m = vector of member's independent characteristic forces; these independent forces give rise to all member-end forces by member equilibrium.

Q o m = vector of member's characteristic deformations caused by external effects applied to the isolated, disconnected member. For a system composed of many members interconnected at points called nodes, the members' flexibility relations can be put together into a single matrix equation, dropping the superscript m: q M × 1 = f M × M Q M × 1 + q M × 1 o where M is the total number of members' characteristic deformations or forces in the system. Unlike the matrix stiffness method, where the members' stiffness relations can be integrated via nodal equilibrium and compatibility conditions, the present flexibility form of equation poses serious difficulty. With member forces Q M × 1 as the primary unknowns, the number of nodal equilibrium equations is insufficient for solution, in general—unless the system is statically determinate. To resolve this difficulty, first we make use of the nodal equilibrium equations in order to reduce the number of independent unknown member forces; the nodal equilibrium equation for the system has the form: R N × 1 = b N × M Q M × 1 + W N × 1 where R N × 1: Vector of nodal forces at all N degrees of freedom of the system.

B N × M: The resulting nodal equilibrium matrix W N × 1: The vector of forces arising from loading on the members. In the case of determinate systems, matrix b is square and the solution for Q can be found from provided that the system is stable. For statically indeterminate systems, M > N, hence, we can augment with I = M-N equations of the form: X i = α Q j + β Q k +... I = 1, 2... I The vector X is the so-called vector of redundant forces and I is the degree of statical indeterminacy of the system. We choose j, k... α, β such that X i is a support reaction or an internal member-end force. With suitable choices of redundant forces, the equation system augmented by can now be solved to obtain: Q M × 1 = B R R N × 1 + B X X I × 1 + Q v ⋅ M × 1 {\displaystyle \mathbf _=\mathbf _\mathbf _+\mathbf _\mathbf _+\mathbf _\qquad \qquad \qquad \mat

Raúl Adalberto Mondesí is a Dominican-American professional baseball shortstop for the Kansas City Royals of Major League Baseball. He is the first player in modern MLB history to make his debut in the World Series, doing so with the Royals in Game 3 on October 30, 2015, against the New York Mets. Mondesí signed with the Kansas City Royals on July 27, 2011, receiving a signing bonus of \$2 million, he made his professional debut in 2012 with the Idaho Falls Chukars of the Rookie-level Pioneer League. He finished the season, hitting.290/.346/.386 with three home runs in 50 games. Prior to the 2013 season, Mondesí was ranked by Baseball America as the team's seventh best prospect, he played the 2013 season with the Lexington Legends of the Class A South Atlantic League. On May 27 he hit for the cycle, he finished the season hitting.261/.311/.361 with seven home runs. After the season, he was named the Class A Lexington Player of the Year by the Royals. Prior to the 2014 season, Mondesí was considered one of the top prospects in baseball.

He was ranked by Baseball America as the team's third best prospect. He was ranked by MLB.com as the 38th best prospect overall. Mondesi spent the 2014 season with the Wilmington Blue Rocks of the Class A-Advanced Carolina League, played for the Peoria Javelinas of the Arizona Fall League after the regular season. In 2015, Mondesi played for the Northwest Arkansas Naturals of the Class AA Texas League, he appeared in the 2015 All-Star Futures Game. The Royals added Mondesi to their active roster for the 2015 World Series, he became the first player since Bug Holliday in 1885 to make his major league debut during a championship series, when he pinch hit during Game 3 for pitcher Danny Duffy and struck out against New York Mets pitcher Noah Syndergaard. He won his first World Series ring on November 2, 2015, when the Kansas City Royals won Game 5 over the New York Mets. Mondesi began the 2016 season in Double-A. On May 10, Mondesi was suspended 50 games after testing positive for clenbuterol, banned by Major League Baseball.

His suspension was 80 games, but was reduced due to the substance having been ingested via cold medicine. On August 16, 2016, Mondesi hit his first major league home run against Detroit Tigers pitcher Justin Verlander. Mondesi became the youngest Royal to hit a home run since Clint Hurdle in 1978. Mondesi ended the season hitting.185/.231/.281 with nine stolen bases in 47 games. Mondesi spent the majority of the 2017 season in AAA, he appeared in 25 games in batting.170 /.214 /.245 with five stolen bases. After beginning the 2018 season in the minors, Mondesi was recalled during the month of June to play shortstop regularly. In 75 games, he batted.276/.306/.498 and hit 14 home runs with 32 stolen bases. In 2019, Mondesi hit 9 home runs with 62 RBI in 102 games. Mondesí is the son of former MLB player Raúl Mondesí. Known as Adalberto when he signed with the Royals, Mondesí went by Raúl Mondesí from 2015 to 2018, he was born in Los Angeles, while his father played for the Los Angeles Dodgers. He grew up in the Dominican Republic.

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