In accordance with the basic principles of the displacement method, a matrix is used to calculate the internal force and displacement. It is one of the structural matrix analysis methods, and its basic unknown is the displacement of the node, because the matrix displacement method is more suitable for the preparation of general calculation formulas than the matrix force method, it has been used more widely.
The structural matrix analysis method first discretizes the structure into a finite number of units and then synthesizes the original structure, so it also belongs to the finite element method. The commonly used element shape of the array displacement method is a straight rod. For curved bars, such as arch structures, although curved bars can also be used as units, the analysis of the elements is cumbersome.
For the sake of simplicity, it can be treated as a dashed line, with each straight line segment treated as a unit. When the element supports a non-nodal load, it can be replaced by an equivalent nodal load. The method is to use the boundary node between the units as the solid end to obtain the solid end reaction force, and then act on the node in the opposite direction.
According to the geometric compatibility condition (deformation condition) after the structure is deformed, the displacement matrix of the node
The rod end of the displacement matrix and
there is a relationship between the formula
represents
for the
transformation matrix .
??Rod end displacement matrix
The rod end of the matrix with a
force ratio
is between
M
m is called the stiffness matrix of the unassembled structure, which is equal to the stiffness matrix
(i) of each element
as the diagonal matrix of the subblock. Its elements can be obtained directly according to the reaction force caused by the unit displacement of the node. Since the element coordinates are not necessarily the general structure coordinates, the obtained element stiffness matrix
(i) must be transformed into the element stiffness matrix under the general coordinates by coordinate transformation.
??According to the joint force and the rod end force converging at the node to maintain a balanced relationship, the rod end force can be obtained
AND the
node force relationship
is
????? ????(3)
a rod end of the power array
to force the array node
transforms the array. According to the principle of virtual work,
T can be obtained
.
??According to the three formulas above, we can obtain
??????????
K
?????????(4)
K =
T
m
??????(5)
Equation (5)
K is called the stiffness matrix of the assembled structure or the general stiffness matrix.
The method to obtain the total stiffness matrix K by formula (5) is called the stiffness method. Because the displacement transformation matrix
The order of is quite high and a large number of storage units is required in the calculation. Therefore, when combining the overall stiffness matrix, the direct stiffness method is often used to directly transfer the elements of the element stiffness matrix to
K. This method involves transferring the same feet in each unit. The target elements are added directly to form the overall stiffness matrix. In the element stiffness matrix, for the near-end node stiffness matrix coefficient
k jj , since all elements gathered at node
j can contribute, there may be several elements added in the overall stiffness matrix, that is say
,
All units of node j . Since it does not need to be calculated by formula (5) and is convenient to calculate, its application is more extensive than the stiffness method.
??Since the displacement of the node in the support constraint direction is usually zero or a known value, all nodes can be displaced
It is divided into two parts, one is the displacement
r which is not constrained by the support
and the other is is the displacement of node
R along the constrained direction of support
. So (4) becomes
Expand the above formula to get
???????(7)
???????(8)
When
R = 0, formula (7) becomes:
r =
r
r?????????(7 ?)
r is the stiffness matrix of the assembled structure corresponding to the displacement that is not restrained by the support. In fact, it is the matrix of coefficients K in the basic equation of the general displacement method
. This matrix can also be obtained by directly inverting the flexibility matrix. And
r consisting of the basic matrix equation is the generally shift method term
(typically in the shift method,
on the same side of the equation, and therefore
r with
the difference between a symbol). Therefore (7′) is the matrix expression of the basic equation of the displacement method.
??Can be obtained by (7) or (7 ?)
r. The rod end force can be obtained by the formulas (1) and (2)
. The actual rod end force
a must be superimposed on the fixed end force
f caused by the non-nodal load on the element
. The real end force of the
i-th unit rod must be
a
(yo) =
(yo)
(yo) +
(yo)????????(9)
??The steps of the matrix displacement method for Calculate the force at the end of the bar are: ? Divide the element and find the equivalent node charge; ? Find the stiffness matrix of the element.
(i) and transform it into the global coordinate stiffness unit matrix; ?Calculate the global stiffness matrix K by formula (5) or the direct stiffness method
; ?calculate
ry
r ; ?calculate the node by displacement formula (7 ‘)
r , and then calculate the rod end force from equations (1) and (2)
. The actual force at the end of the rod must superimpose on
f, which is determined by equation (9).
