A method for computing statically indeterminate structures that successively approximates the exact solution. When using the general force method or the displacement method to analyze a statically indeterminate structure (see Static Analysis of Member Structure), a system of linear equations must be established and solved. If the unknown number is large, the calculation work will be quite heavy. On the basis of the displacement method in 1930, H. Cross proposed the moment distribution method which does not need to solve the equations but successively approximates.
A rigid frame with four bars of equal cross section (Figure a, in figure i represents the linear stiffness of each bar, i = EI / l , E is the elastic modulus of the material, I is the moment of inertia of the section of the rod , and l is the length of the rod), if the point O effect of an external pair MO , so that the unit of the node O occurs angle ? 0 = 1, the deformable structure is shown in a dashed line, as shown in FIG. appropriate moment B (both angle and bending moment Take clockwise direction as positive.
It is customary to call the rotating end of each rod the proximal end and the other end the distal end.) It can be seen from the figure that the moment MO acting on point O will be shared by the proximal end of each bar and transmitted to the distal end. The relationship between the bending moment at the distal end and the bending moment at the proximal end of each rod is called the transmission coefficient C. For a rod of equal section, it is only related to the support of the distal end. The transmission coefficient of each rod is CO 1? 1/2, CO2 = 0, CO3? -1, CO4 = 0.
According to the principle of static equilibrium, MO must be equal to the sum of the proximal bending moments, that is, MO = 4 i 1 + 3 i 2 + i 3 , as shown in Figure c. The proximal bending moment when a unit angle of rotation occurs at the end of the rod is called the rotational stiffness K i of the rod, and the sum of the rotational stiffness of each rod
is called the rotational stiffness of node
O. figure according to any one of the proximal end of the bending moment can be expressed as a
called the distribution coefficient
DI . It is shown that when the point
MO when actuated, the
lever I assigned to the torque stiffness of the lever is rotated
KI to node
rotational stiffness ratio in
question. It can be seen that when point
O is subjected to any given external moment
M , the proximal bending moment of any rod is
M i O =
D i M ; the distal bending moment is
C iM Oi =
??The basic idea of the moment distribution method: ?Fix the node, add a rigid arm to the O node to control the rotation, calculate the fixed end bending moment of each end of the rod caused by the load, and act on the fixed end of each rod at a node The algebraic sum of the bending moment is called the unbalanced moment; ?Relax the node, cancel the rigid arm that does not exist, let the node rotate and obtain the distributed moment of each rod according to the unbalanced moment distribution coefficient; ?Transmit torque,
According to the distribution torque and the transmission coefficient of each rod, it is transmitted to the distal end of each rod, and each transmission torque is obtained. Follow this rule, distribute, transfer and repeat the calculations until the rod end torque value is obtained with sufficient accuracy. Finally, the final moment of the bar is equal to the sum of the fixed final moment, the distribution moment and the transmission moment.
For rigid frames with lateral displacement, the methods developed by the moment distribution method can also be used for the design, such as the design of single-span rigid frames without shear force distribution, and the moment distribution method with equations shear force balance additions, but its range of application is limited or not very convenient, so the iterative method is often used for the general rigid frame with lateral displacement.