Static Indeterminacy

A structure is statically indeterminate when it can no longer be solved using only equilibrium equations. There is a distinction between:

• Only support reactions can be determined (externally statically determinate)

• Internal forces can be determined (internally statically determinate)

If equilibrium equations are not sufficient, a structure is statically indeterminate. The degree of static indeterminacy is expressed as the degree of static indeterminacy.

It is necessary to determine the degree of static indeterminacy to solve these structures using the force method.

These two categories are discussed together in sections 4.5.2 and 4.5.3 of the book Mechanica: Evenwicht (Hartsuijker and Welleman, 2015). For trusses, the analysis is simplified as described in section 9.2.2 of the book Mechanica: Evenwicht (Hartsuijker and Welleman, 2015). Although the equations shown with \(r\), \(v\), and \(e\) can be effective for simple structures, they lead to incorrect results for complex structures. An approach that always works is shown below for separately determining external and internal static indeterminacy.

Determining the Degree of External Static Indeterminacy

For the calculation of external static indeterminacy, follow these steps:

Example

Fig. 16 Example structure

As an example, we determine the external static indeterminacy of this structure.

1. Split the structure into rigid parts that can rotate independently of each other when separated from the supports. Draw the free-body diagram of these hinged parts. The support reactions and any external forces act on these parts. At the hinged connection, two unknown forces act: horizontal and vertical. These forces at the connection have an equal and opposite reaction force on the adjoining part.

   Example

   

   Fig. 17 Split structure

   The structure can be divided into two rigid, hinge-connected parts. The supports are replaced by support reactions, and the hinged connection by a horizontal and vertical force (and reaction forces). The external force acts only on one of the two parts.

2. Count the number of unknown forces: the support reactions and connection forces in the hinges (do not count the reaction forces separately)

   Example

   

   Fig. 18 Number of unknown forces

   There are a total of 6 support reactions and 2 connection forces. Note: the connection forces are shown twice, but these forces are equal at both ends of the connection, so they can be counted as one.

3. Count the equilibrium equations: 3 equilibrium equations per rigid part of the structure

   Example

   

   Fig. 19 Number of equilibrium equations

   There are two rigid parts, so 6 equilibrium equations

4. The degree of static indeterminacy is the number of support reactions + connection forces - number of equilibrium equations

Example

The degree of external static indeterminacy for this example is \(6 + 2 - 6 = 2\).

Determining the Degree of Internal Static Indeterminacy

For the calculation of internal static indeterminacy, follow these steps:

Example

Fig. 20 Example structure

As an example, we determine the internal static indeterminacy of this structure.

1. Split the structure into all separate nodes and members, and draw the free-body diagram for all nodes, considering which members and supports are connected to the nodes:

   • At a hinged connection and free/hinged ends, no bending moments act

   • On a member connected to a roller support, only a force perpendicular to the roller direction acts.

   • From a two-force member, only a normal force acts

   Example

   

   Fig. 21 Free-body diagrams of nodes

   The structure consists of 4 nodes. Unknown support reactions act on nodes A and B. Node A is a hinged end, so no bending moments act there. In node B, bending moments can occur from BC and BD and be in equilibrium, although the support is a hinge. On node C, no bending moment acts from bar DB due to the hinged connection. Node D is hinged, so no bending moments act there.

2. Draw the free-body diagram for the members: draw the reaction forces on the members resulting from the forces on the nodes.

   Example

   

   Fig. 22 Free-body diagrams of members

   From Fig. 21, the reaction forces on the members can be drawn.

3. Count the number of unknown forces: support reactions and section forces (do not count the reaction forces separately)

   Example

   

   Fig. 23 Number of unknown forces

   From Fig. 21, the number of unknown forces can be counted. There are 4 support reactions and 20 section forces.

   These section forces can also be counted in the free-body diagrams of the members from Fig. 22:

   

   Fig. 24 Number of unknown section forces

4. Count the equilibrium equations: 1 equilibrium equation per two-force member, 3 equilibrium equations per general member, 1 equilibrium equation for a roller hinge, 2 equilibrium equations per hinged node, and 3 equilibrium equations per general node.

   Example

   

   Fig. 25 Number of equilibrium equations per member

   All members are general members. That gives 12 equilibrium equations for the members.

   

   Fig. 26 Number of equilibrium equations per node

   Of the nodes, two are fully hinged; for the rest, the moment sum is also important. That gives 10 equilibrium equations.

   In total, there are 22 equilibrium equations

5. The degree of static indeterminacy is the number of support reactions + section forces - number of equilibrium equations

   Example

   The degree of internal static indeterminacy for this example is \(4 + 20 - 22 = 2\).

Instructions from lecture

This topic is presented in a lecture available from 0:34:40 - 1:00:40 here in Dutch for TU Delft students.

Exercises

• Exercises 4.11 - 4.22, from chapter 4 of the book Mechanics: Evenwicht (Hartsuijker and Welleman, 2015). Ignore the questions about kinematic determinacy. Answers are available on this website.

• Exercise 9.6, from chapter 9 of the book Mechanics: Evenwicht (Hartsuijker and Welleman, 2015). Ignore the questions about kinematic determinacy. Answers are available on this website.