One of the basic methods for the analysis of statically indeterminate structures (see static analysis of rod structure) with the redundant unknown force corresponding to the redundant connection as the basic unknown.
To expose these redundant unknown forces, the basic structure must sever or eliminate the redundant connections, and then replace their restraining effects with the corresponding internal forces or reaction forces. For the continuous beam shown in Figure 1a, the unknown reaction can be used after the intermediate support is removed, the force X 1 replaces the original bearing constraints, thus transforming the original structure into a statically determined geometrically invariant structure. , which is called the basic structure of the force method.
If we can try to determine the unknown excess force, the entire calculation can be processed as a statically determinate structure.
??Typical equation?If the redundant unknown force in the basic structure can replace the restraining effect of each redundant connection in the original structure, the two are required to have exactly the same force state and strain state. In a linearly deformed structure, there is a definite relationship between force and strain.
As long as the strain is the same, the state of the force must be the same. The key is how to calculate the basic structure under the unknown excess force and load, and each unknown excess force acting at the point The displacement.
??According to the principle of superposition, the total displacement of any point of the basic structure is equal to the sum of the displacements generated when the unknown excess force and the original load act separately, that is
; Yes ? is used to
represent the displacement caused by the unit force, ?
i 1 =
i 1 , ?
i 2 =
i 2 , …, ?
x n ? in and so on. Since the redundant connections in the original structure are originally continuous, for the basic structure to be consistent with the deformation of the original structure, there should be ?
i 1 +
i 2 +… +
X n ? in + ?
iP = 0 (
i = 1, 2,…,
n ) This set of equations is called the typical equation of the force method. It can also be derived from the principle of minimum virtual force. The leading coefficient of the main diagonal is always positive. The secondary coefficients located in symmetrical positions on both sides of the main diagonal can be positive, negative or zero. From the reciprocal displacement theorem, there is
? k j =
? ji , which can reduce the calculation work by half. The displacement? iP (called the free term) caused by the charge
can also be positive, negative, or zero. The superposition principle can be used to calculate the internal force of the original structure by solving for the unknown excess force from the typical equation. For example, the bending moment M of the original structure
In the formula,
2 , …,
n are the bending moment of the basic structure under the action of an unknown unit force; M p is the bending moment
of the basic structure under the original load.
For the three-fold statically indeterminate structure of a hingeless arch with variable load surface (Figure 2a), the elastic center method can be used. Eliminate all secondary coefficients in the typical equation, first cut at its vertex axis O, use the paired axial force X1, shear force X2, and bending moment X3 as the redundant unknown force (Figure 2b), and then use point O as the origin coordinate, X 1 and X 2 lines of action is used as the x and y coordinate axes, then the bending moment under the action of unit excess force is
??Therefore, when the influence of the axial force and the shear force on the deformation of the section is not considered, the relative angle of rotation of the section O is
s is the length of the microsection along the axis of the arch;
E is the elastic modulus of the material;
I is the moment of inertia of the section of the rod. If 1/
EI is considered as the width of the microsegment,
EI can be considered as the elastic microarea or elastic weight along the arc axis, so
33 can be considered as the elastic microarea drawn along the arc axis The overview plot (Figure 2c). Secondary coefficient
, can be considered as the product of inertia of the general graph of the
x-axis and y-
32 can be considered as the static moments of the general graph of the x-axis and y-
??If the origin of the coordinates moves with the center of gravity of the general graph, it is called the elastic center, and the x and y axes are parallel to the main axis of the general graph, then its static moment and product of inertia are will become zero, so all secondary coefficients will disappear. In this way, the typical equation of the force method is transformed into three independent equations, which greatly simplifies the calculation work, which is called the elastic center method. In order not to change the force and deformation state of the original structure, the section is generally connected to the elastic center with a rigid arm, and three pairs of excess unknown forces X 1 , X 2 , X 3 are decomposed into the elastic center. for calculating.
?Editor-in-Chief Li Liankun: “Structural Mechanics” (Second Edition), Higher Education Press, Beijing, 1984.