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What is Force Matrix Method?

According to the basic principle of the force method, the method of calculating the internal force and displacement of a structure using a matrix as a mathematical tool is one of the structural matrix analysis methods.
??The structural matrix analysis method needs to discretize the structure into a finite number of elements for the calculation. The unit forms commonly used in the matrix force method are simply supported type and cantilever type. These two types of units are relatively simple, and the simple supported type is especially common.

When the element supports a non-nodal load, it can be replaced by an equivalent nodal load. The method is to treat the boundary nodes between the units as fixed to obtain the solid end reaction force and then act on the nodes in the opposite direction.
??The matrix force method is based on the force method.

When calculating a statically indeterminate structure, the basic system and the unknown basic force must be selected. There are two selection methods: one is selected by the calculator according to the specific situation of the structure, and calculated on the basis of the artificially selected basic system; the other is to combine the force method with knowledge of the range in linear. algebra, and first establish The Knot Equilibrium Equation is then used to automatically separate the redundant basic unknown forces using Jordan’s elimination method.

This method of analysis is called the range force method. Because the above method is more closely integrated with the force method, it is easier to understand and commonly used.
??When both the original load and the unknown basic force are considered external forces, the force matrix of the node
, Column of the structure of the matrix substantially unknown forces
and the basic unit of unknown force (end of the bar of force) of
the matrix relationship
as follows:
P
+
X
?????????(1)
P and
X respectively represent the binding force
and the basic structure of unknown forces
affect the basic matrix of the internal forces of the system.
??Elementary unknown force
The stem end with the corresponding
displacement ratio
m is the flexibility matrix of the unassembled structure, which is equal to the flexibility matrix
i ) of each unit
as the diagonal matrix of the subblock. And the
displacement direction end of the displacement rod
and the load
node (the node comprises a substantially unknown forces and the displacement force in the load direction
P and the
relation
X for
(3)
= [
P ?
X ]
T ;
is the transformation matrix from the displacement of the end of the bar to the displacement of the load direction of the node. The principle of virtual work, available
X- ]
T .
??According to the strain coordination condition corresponding to the basically unknown force direction,
= 0, we can obtain in the formula ??????
??In formula (6)
X is called the flexibility matrix of the assembled structure, that is, coefficient matrix?in the basic equation of the general force method
, and
is the free term matrix
P in the basic equation of the method overall strength
, so formula (6 ) is the basic equation of the Matrix Expression force method. From (6) can be obtained
, in (1) and (4), to give the basic unknown unit forces
and nodal loads displacement direction
P . Column matrix Accrued
, can be determined from the force means equilibrium conditions all column matrix rod ends
??(9)
the unit of substantially all unknown forces transforms the rod end matrix from the power unit. The actual bar end force matrix is
​​a fixed end force matrix
f which should be caused by the non-nodal load of the superimposed element in equation (9)
. The
real rod end force matrix of ith element must be
???????(10)
??The steps of the force matrix method to calculate the bar end force are: ?select the basic system and the unknown basic force; ?split the element and find the charge of the equivalent node; ?calculate the flexibility matrix of the element
I , and constitute an
m ; ? obtaining
X- ,
P , by (7) (8) equation
P ; ? by formula (9) we obtain the force of all the ends of the rod
, so that equation (10) real Rod force of the rod of the end
a .
??When using the matrix force method to calculate the displacement of a statically determinate structure, let the basic unknown force be X = 0 in formula (4) , and the formula for the node displacement of the statically determinate structure be the load direction can be obtained as
???????(11)
??In the analysis of statically indeterminate structures, since the basic unknown of the matrix force method is the redundant force, it is more appropriate to calculate the statically indeterminate structure with a small number of times. However, it is difficult to compile large-scale general purpose programs suitable for various structures using the force matrix method. Therefore, the matrix shift method with unified unitary form of the basic system is often used for analysis today.
?Puziminiski, translated by Wang Derong et al.: “Matrix Structure Analysis Theory”, National Defense Industry Press, Beijing, 1974. (JSPrzemieniecki, Matrix Structural Analysis Theory, McGraw-Hill, New York, 1968.)

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