Statically Determinate Structure

How to determine whether a structure (beam, frame, truss) is statically determinate. Additionally, you will learn about possible cases of supports that we encounter in mechanics tasks.

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A static structure is a structure that is stationary. This means that it cannot move under the influence of external forces. If it moves, it will be a mechanism.

The branch of mechanics in which we deal with the motion of bodies is kinematics and dynamics. Statics is the part of mechanics in which the elements remain at rest, and for this to be the case, several conditions must be met.

In this entry:

1. Supports. Types and symbols

2. Degree of static indeterminacy

3. Statically determinate structure

4. Statically indeterminate structure

5. Mechanisms

Supports. Types and symbols

For the structure to be stationary, we must support it. In mechanics tasks, you will most often encounter the following supports. The drawings show support reactions and symbols for individual supports:

Fig.1 Fixed support (not movable)

Fig.2 Movable support (sliding)

Fig.3 Restraint (bracket)

Fig.4 Vertical skate

Fig.5 Horizontal skate

Fig.6 Joint

All the supports mentioned above can be found in the application Beam solver  where you can test ways of supporting beams.

Static arrangements can be divided into two types:

1. Statically determinate structure - the degree of static indeterminacy of such a structure is zero.

2. Statically indeterminate structure - the degree of static indeterminacy of such a structure is greater than zero.
   

Degree of static indeterminacy

To calculate the degree of static indeterminacy of a given structure, you can use the formula below. This formula is best used when calculating beams and frames.

N = R - J - 3

where:

N - degree of static indeterminacy

R - number of support reactions. That is, the sum of all reactions for our supports

J - number of internal joints - if there are none P=0

3 - number of equilibrium equations. In static structure it is 3

For truss we will use the formula below:

2w = p + r

where:

w - number of truss nodes

p - number of truss bars

r - number of support reactions

A truss is determinate if the above relationship is satisfied. So twice the number of nodes must be equal to the sum of the bars and the number of reactions.

Statically determinate structures

A statically determinate structure is a structure for which we can calculate support reactions. For example, our beam or will be statically determinate if we are able to calculate all reactions using three equilibrium equations. The degree of static indeterminacy in such structure is zero.

N=0

Fig.7 Statically determinate beam

Using the formula above

N=R-J-3 => N=3 - 0 - 3 = 0

where:

R=3 - sum of all reactions for our supports

J=0 number of internal joints, none

3 - number of equilibrium equations

And for a simple truss, the calculation will look like this:

2w = p + r => 2*5 = 7 + 3 => 10 = 10

where:

w = 5 - number of nodes

p = 7 - number of links

r = 3 - number of support reactions

Fig.8 Statically determinate truss

Statically indeterminate structure

A statically indeterminate structure is a structure for which we are unable to calculate the support reactions. The degree of static indeterminacy in such structure is greater than zero. The number of unknown support reactions is greater than the number of equilibrium equations.

N > 0

Fig.9 Statically indeterminate beam

where:

R=5 - sum of all reactions for our supports

J=0 - number of internal joints, none

3 - number of equilibrium equations

The beam is twice statically indeterminate - ( result=2)

Such a structure is too rigid and cannot work. Under the influence of load or temperature change, the beam cannot move, or in other words, cannot work.

Mechanisms

The last type of structure that I will discuss are structures that can move - mechanisms. The degree of static indeterminacy for such structure is negative.

N < 0

Fig.10 Beam with the possibility of movement - mechanism

N=R-J-3 => N=2 - 0 - 3 = -1

That's the end of the topic of static indeterminacy, thank you and I invite you to browse other entries 😊

PS. All beam images used in the post were generated in beam solver.