stability and degrees of indeterminacy terje haukaas.pdf

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Stability and Degrees of Indeterminacy

A structure has many characteristics, many of which are properties of the structural members. This document, however, describes some global characteristics that are important in linear static structural analysis. The characteristics are stability and degree of static and kinematic indeterminacy. Stability Stability implies that there are no modes of deformation with zero stiffness. Such modes are often called mechanisms and they make the structure unstable. An unstable structure will collapse even without load. For 2D structural models it is often straightforward to see that a structure is unstable. Figure 1 shows examples of unstable structural models. The two hinges at the top of the frame combined with the pinned supports means that this structure will collapse sideways. Even one hinge would be sufficient to make it unstable. The truss structure in Figure 1 is also unstable; a cross brace is required to make this a useful structure.

Terje Haukaas University of British Columbia www.inrisk.ubc.ca

Stability and Degrees of Indeterminacy Page 2

Figure 1: Unstable frame and truss. Obviously, great care must be exercised to avoid the presence of instabilities in structures. Fortunately, there exist indicators that expose instabilities. The first is a negative degree of static indeterminacy, described shortly. However, this indicator is imperfect. A structure can be statically determinate and still unstable. Another way of detecting instabilities is to model the structure by the stiffness method and attempt to solve the resulting linear system of equations. If the stiffness matrix is singular, i.e., it cannot be inverted, then instability may be the cause.

Degree of Static Indeterminacy A structures degree of static indeterminacy (DSI) exposes the deficit of equilibrium equations compared with the number of unknown internal forces in the structure. In other words, for a statically determinate structure (DSI=0) it is possible to compute the section forces (M, V, N) by equilibrium equations alone. More advanced methods are required for structures that are statically indeterminate. Different engineers have different habits when it comes to determining the DSI. However, every rule boils down to counting the number of unknowns and comparing it with the number of available equilibrium equations. In this document, this calculation is set up as follows: DSI = f m + r( ) e j + h( ) (1) where all variables are non-negative integers with the following meaning: f = forces = number of internal force in each member m = members = number of members r = restraints = number of restraints, i.e., boundary conditions e = equations = number of equilibrium equations per joint j = joints = number of joints h = hinges = number of hinges or other section force releases The number of internal forces, f, in each member depends on the member type. A truss member has only one unknown force: the axial force. Conversely, a frame member in a 2D structural model has three internal forces: axial force, shear force, and bending moment. This number increases from three to six for 3D frame members. Table 1 summarizes the value of f for different structures. The number of equilibrium equations, e, per joint is obtained by counting the orthogonal directions in which equilibrium can be considered. For the typical case of 2D frame structures there are three equilibrium equations per joint: horizontal,

Frame Truss



vertical, and angular equilibrium. For a joint in a 2D structure with only truss members entering, i.e., member without bending stiffness, rotational equilibrium is cancelled. Table 1 summarizes the value of e for different structures. Table 1: Forces per member and equations per joint.

f e 2D truss 1 2 2D frame 3 3 3D truss 1 3 3D frame 6 6 The number of restraints, r, is obtained by counting the number of support reactions. Although rather trivial, Figure 2 provides an overview of the number of unknown reaction forces for different kinds of 2D boundary. The arrows in the figure show the forces. The degrees of freedom will be described later.

Figure 2: Number of unknown forces and degrees of freedom for some 2D joint types. The number of hinges, h, is obtained by counting the number of releases in the structure. Essentially, a hinge releases one or more section forces from being transferred into one or more adjacent member. As a result, h counts the number of section forces that are prevented from transferring into adjacent members. For example, an ordinary hinge in a continuous beam prevents the bending moment

Unknown reaction forces Degrees of freedom

Fixed

Pinned

Roller

Slider'

3" 0"

2" 1"

1" 2"

2" 1"

Support type



from being transferred through the hinge, but the shear and axial forces are transferred regardless of that hinge. Naturally, this reduces the number of unknowns because the bending moment is known to be zero at the hinge. It is emphasized that other releases are possible, where the shear or axial forces is prevented from transferring to an adjacent member. For frames the number of members, m, and joints, j, in a structural model is subjective. However, the subjectivity does not affect the final DSI. Usually, joints are identified wherever there is a boundary condition or a bend or intersection in the structure. If for some reason the analyst places a joint in the middle of a frame member then this increases j and m in a way that leaves DSI unchanged. Internal and External Degrees of Indeterminacy Although practically irrelevant, it is possible to divide the DSI into two categories. The external DSI is the number of boundary conditions, i.e., number of unknown reaction forces minus the number of global equilibrium equations. For 2D structures there are three global equilibrium equations. For 3D structures there are six. As an example, a 2D structure with one fixed support and one pinned support, i.e., five unknown reaction forces, has an external DSI equal to two because there are only three global equilibrium equations. If the total DSI is greater then the rest are internal DSI. Degree of Kinematic Indeterminacy While DSI provides information about unknown member forces the degree of kinematic indeterminacy (DKI) exposes the number of unknown joint displacements and rotations. DSI is a key number in force-based structural analysis methods and DKI is the key figure in displacement-based methods. In fact, the DSI is the size of the flexibility matrix and DKI is the size of the stiffness matrix, for the flexibility methods and the stiffness method, respectively. The DKI is easier to determine than the DSI. Even a computer can do it in a straightforward manner. This is why the stiffness method is implemented in all structural analysis software, while the flexibility method is not. Essentially, DKI counts the number of degrees of freedom (DOFs) of a structure. Each joint, usually called node in displacement-based methods, has a pre-defined number of DOFs. A 2D structure has three DOFs per node, i.e., three possible directions to move: horizontal, vertical, and rotation. Similarly, a 3D structural model has six degrees of freedom per node: three displacements and three rotations. For truss structures the rotational DOFs are neglected altogether because they are associated with zero stiffness from the truss elements. Some structural analysis programs deal with trusses by first keeping all rotational degrees of freedom and later restraining them in the same way as nodes with boundary conditions are restrained. When doing hand calculations there are two exceptions to the rule that every node has equally many DOFs. The first is for nodes with boundary conditions. For example, a fixed node has zero DOFs. Figure 2 provides an overview for 2D structures of the number of DOFs per node for different boundary conditions. The



second exception to the rule appears when axial deformations are neglected in the analysis of frame structures. This is quite common in hand calculations with the classical stiffness method because the axial stiffness of frame members is usually significantly higher than the bending stiffness. Neglecting axial deformations requires careful consideration of the DOFs at each node, which is difficult for a computer. Hence, in computer analysis it is easier to always account for axial deformations. By hand, one simply removes the DOFs that will experience zero displacement when the members do not deform axially.

stability and degrees of indeterminacy terje haukaas.pdf

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