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MindMap Gallery Analysis of geometric composition of planar member system

Analysis of geometric composition of planar member system

A mind map about structural mechanics, including the basic concepts of geometric composition analysis, computational degrees of freedom, geometric composition rules of member systems, etc.

Edited at 2023-11-27 11:45:49

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Analysis of geometric composition of planar member system

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Outline

Analysis of geometric composition of planar member system

Basic concepts of geometric composition analysis

geometrically invariant system

geometrically variable system

geometrically constant system

geometric transient system

degrees of freedom

Constraint (Contact)

external constraints

Living hinge support (1 constraint)

Fixed hinge support (2 restraints)

Fixed supports (3 constraints)

internal constraints

Single link (1 constraint)

Compound chain rod (m nodes = 2m-3 constraints)

Single hinge (2 constraints)

Compound hinge (connecting m steel sheets = m-1 single hinges = 2 (m-1) constraints)

Instantaneous hinge (virtual hinge)

Compute degrees of freedom

The degree of freedom of the rod is the main body

General arbitrary member system

W=3m-(2h 3g r)

The degree of freedom of the hinge point is the main body

Applicable only to hinged member systems

W=2j-(b r)

Calculation results

W>0, geometrically constant system

W=0, minimum number of constraints

Properly constrained, geometrically invariant system

Improper constraints, geometrically variable system

W=0, redundant constraints

Properly constrained, geometrically invariant system

Improper constraints, geometrically variable system

Geometric composition rules of member systems

binary rule

Adding or removing a binary body in a system will not change the geometric composition of the original system.

two-piece rule

Two steel sheets (which have been determined to be geometrically invariant parts with no redundant connections) are connected by a single hinge and a link rod that does not pass through this hinge, forming a geometrically invariant system without redundant constraints.

The two steel sheets (which have been determined to be geometrically invariant parts with no redundant connections) are connected with three chain rods that are neither completely parallel nor intersecting at the same point, forming a geometrically invariant part without redundant constraints.

Improper placement of constraints

Three-link rods that actually intersect at one point—geometrically constant system

Through three links intersecting at one point on the extension line - geometric transient system

Three parallel but unequal length chain rods—geometric transient system

Three parallel and equal-length chain rods—a geometrically constant system

Three Steel Pieces Rule

The system formed by connecting the three steel pieces (which have been determined to be geometrically invariant parts without redundant connections) with three hinge points that are not on the same straight line is geometrically invariant and has no redundant constraints.

The three steel plates are connected two by two with three pairs of chain rods. If the rotation centers of the three instantaneous hinges formed by the three pairs of chain rods are not on the same straight line, a geometrically invariant system will still be formed.

Determine whether three hinges are collinear

Improper placement of constraints

A virtual hinge at infinity

The virtual hinge at infinity of the parallel chain rod is not parallel to the line connecting the other two hinges - geometrically invariant system

The virtual hinge at infinity of the parallel chain rod is parallel to the line connecting the other two hinges - geometric transient system

The parallel chain rods forming the virtual hinge at infinity are of the same length and parallel to the line connecting the two real hinges and are of the same length - a geometrically constant system

Two virtual hinges at infinity

If the two pairs of parallel chain rods forming a virtual hinge are not parallel to each other, then the three hinges are not collinear - a geometrically invariant system

If two pairs of parallel chain rods forming a virtual hinge are parallel to each other, then the three hinges are collinear - a geometric transient system

If two pairs of parallel chain rods forming a virtual hinge are parallel to each other and of equal length, then the three hinges are collinear - a geometrically constant system

Three virtual hinges at infinity

The virtual hinge is formed by three pairs of parallel chain rods in any direction, and the three hinges are collinear - a geometric transient system

The three pairs of parallel chain rods forming a virtual hinge are each of equal length - a geometrically constant system

Geometric composition analysis methods and applications

Start from the basics

Starting from the steel sheet inside the system

Some tips on geometric composition analysis

When the system has dyads, the dyads can be removed in sequence.

When the system and the foundation are connected by three chain rods that are neither parallel to each other nor intersect at one point, the supporting chain rods can be removed and only the geometric composition of the upper system itself can be analyzed.

When the system and the foundation are connected by three chain rods, the foundation is generally regarded as an independent steel piece, and the geometric composition of the entire system (including the foundation) is analyzed.