Stability

MEE1004 - Mechanics of Materials: Lecture 15

Static equilibrium

Achieved when all forces and moments exerted on an object are balanced.

Stability

When there are a sufficient number of constraints (available force/moment reactions) to ensure that static equilibrium can be maintained.

A stable object is able to maintain equilibrium regardless of the loading conditions.

Mechanism

An object that has fewer constraints than the number of degrees of freedom (DOF).

Degrees of Freedom for a 2D Object without constraints

3

Degrees of Freedom for a 3D object without constraints

6

Number of remaining degrees of freedom for the object with constraints

1

Statically Determinate Structures

An object might be stable and statically determinate (if the constructions are proper) when it has the same number of constraints as degrees of freedom.

DOF = 3, Constraints: r = 3

Statically Indeterminate Structures

An object might be stable and statically indeterminate (if the constraints are proper) when it has more constraints than degrees of freedom.

DOFs = 3, Constraints: r = 4.

Improper Constraints

An object with a sufficient number of constraints for stability may still be unstable in cases of improper constraints. Occurs when equilibrium is not ensured.

Truss

A series of rigid bodies joined together by pin joints.

Typical assumptions in truss design & analysis

• The weight of members can be neglected (it is small compared to the applied loads),

• The joints are modelled as freely-rotating pin joints (they have little resistance rotation),

• External loads are only applied at joints.

Shape of Truss

Defined by the location of each pin. For stability, the distance between pin joints must be constant.

Maxwell’s Stability Criterion

m + r ≥ 2j

m: the number of truss members (bars)

r: number of reactions (from the supports)

j: number of joints between members

When m + r < 2j

Structure is unstable and termed a mechanism.

When m + r = 2j

The structure might be stable and statically determinate (if the constraints are proper)

When m + r > 2j,

The structure might be stable and statically indeterminate, or redundant (if the constraints are proper).

Improperly Connected Trusses

Obvious cases - no triangles in the structure

Less obvious cases - there will be loading scenarios where it is impossible to satisfy all of the equilibrium conditions (i.e. one of the force / moment summations will be non-zero).

Types of constraints

Pin supports - r = 2

Roller support - r = 1

Forces in a Truss

Truss members are only subjected to axial forces and not transverse forces

These axial forces are equal and opposite (either tensile or compressive).

They only exert reaction forces in the longitudinal direction of the truss members

FBDs of Truss Joints / Members

All members are assumed to be in tension so the member reactions on each joint point away from the joint.

For each joint there are only 2 equilibrium conditions that can be usefully applied.

Moment equilibrium cannot be used to solve for unknown reactions.

Member of Joints

A strategy for determining the tensile or compressive forces acting on each member of a truss using FBDs of the truss joints.

Method of Joints Tips

1. Choose a suitable location to start. Often it is best to start by determining the support reactions.

2. Continue the analysis at subsequent locations making use of the forces determined at the previous location.

3. Include reaction (axial force) from each truss member that is connected to that perpendicular joints.

4. Only two conditions for static equilibrium may be usefully applied (horizontal and vertical forces).

5. Always assume members are in tension, so that a positive value indicates a tensile force.

6. State whether forces are tensile or compressive.