One of the basic methods of the analysis of statically indeterminate structures (see static analysis of member structures), also known as the displacement method or stiffness method, generally takes the displacement of the node as the basic unknown. The displacement method has two calculation methods, one is to use the basic structure to list typical equations for calculation, and the other is to directly apply the angle displacement equation to establish the static equilibrium equation of a certain node or section in the original structure for calculation. The latter is often called the angle of rotation. Scroll method.
Basic Structure When using the displacement method to compute a statically indeterminate structure, you must first determine the basic unknown, that is, the total number of angular and linear displacements of independent nodes n (as shown in Figure 1a, n = 2 But ignore its axial strain.) Then, correspondingly add additional rigid arms to prevent rotation or additional chain rods to prevent movement at these nodes, so that the structure becomes a collection of a series of discrete parts.
This forms the basic structure of the displacement method (Figure 1b). In general, each discrete part is a statically indeterminate beam of constant cross section.
“The typical equation” is to make the basic structure deformation and internal force equal to the original structure.
The basic structure must carry the same load as the original structure (including temperature changes, support subsidence, etc.), and additional restraints must be displaced the same as the original structure. Since there is no additional constraint on the original structure, the reaction force of the constraint on all additional constraints on the basic structure must equal zero. Based on this, the typical displacement equation method is established:
In the formula, the coefficient
nk represents the reaction moment or reaction force caused by the unit displacement of the i-th additional constraint in the basic structure
due to the unit displacement of the
k-th additional constraint, and the coefficient matrix is Symmetric; the free term
iP represents the ith in the structure
The additional constraint is the reaction moment or the reaction force caused by the load; the basic unknown number
i is the angular displacement or linear displacement of the
i = 1, 2, …,
Obtain the typical coefficients of the equation and the free elements to be drawn when the basic structure of the displacement unit in the respective additional constraint MI FIG under load and MP view and cross section or using the node looking for equilibrium conditions Find out the coefficients and free terms.
Since the members of the basic structure are usually single-span statically indeterminate beams, their fixed-end bending moment formulas under various unit displacements of loads and supports can be derived in advance by the force method or other methods. Such formulas are called angles of rotation. Displacement equation. For example, when the fixed beam at both ends of the constant cross section occurs when the displacement shown in Fig.2 occurs, the angular displacement equation is
[Psi] = [Delta] /
E is a modulus of elasticity of the material;
the I moment of inertia of the cross section;
F. Load Induced Bending Moment a. For bars of variable cross section, the angular displacement equation can also be derived and corresponding diagrams drawn for later use.
The angle of displacement?and the angle of rotation are no longer a necessity for the basic structure of each of MI and MP Fig nor the calculated coefficient for each individual element, and freedom, but directly applied Displacement Equation, the extreme of the rod of each shear or bending moment is expressed as unknown Displacement function of the node. Then each node containing the angular displacement to be computed is intercepted as an insulator.
According to the algebraic sum of the bending moment and the node moment load acting on the node at the proximal end of all rods converging at this node, the algebraic sum must equal zero, and establish the equilibrium equation of the node; then make the cross sections in turn, intercept each insulator containing the node of the linear displacement to be calculated, and enumerate the equilibrium equation of the force projection in the direction of the displacement, that is, the equation of transverse equilibrium, and the equilibrium equation is set up in this way exactly the same as the typical equation.
After solving the typical equations to obtain the basic unknowns xi , the internal force of the structure can be obtained according to the superposition principle or the angular displacement equation.