… for beam structures

The general concept of the force method is covered in chapter 2.1 while the force method for beam structures is treated in in chapter 2.2.1 - 2.2.4 and the more specific ‘hoekveranderingsvergelijkingen’ in chapter 3.1 of the book Mechanica, Statisch onbepaalde constructies en bezwijkanalyse (in Dutch) (Hartsuijker and Welleman, 2007). The examples of chapter 3.1 are treated in the force method for frames tructures.

The method of ‘hoekveranderingsvergelijkingen’ has the advantage that it’s very easy to calculate the required rotations using forget-me-nots. However, not for all structures forget-me-nots might be available.

We’ll cover the application to bending structures with the following example.

Example

Fig. 32 Example structure

1. Determine the degree of statical determinacy.

   Example

   For our example, we might be interested in the internal force distribution, so we need to evaluate the degree of internal statical determinacy.

   

   Fig. 33 There are 14 unknown forces.

   

   Fig. 34 There are 13 equilibrium equations

   So this structure is 1st order internally statically indeterminant.

2. Transform the structure in a statical determinant system by releasing releasing a support, splitting the structure at a two-force member or adding hinges: add unknown statically indeterminate forces and displacement constraints for each of the support you released and hinges you added. Be aware that you don’t transform the structure in a (partial) mechanism!

   Example

   There are many options here, of which the most obvious ones a few are shown below:

   Release vertical support at \(\rm{A}\)

   

   Release vertical support at \(\rm{B}\)

   

   This option is not very convenient as there are no forget-me-nots to get the displacement at \(\rm{B}\) for these loads

   Release vertical support at \(\rm{C}\)

   

   Add hinge at \(\rm{B}\)

   

   If only hinges are added, we call this approach ‘hoekveranderingsvergelijkingen’ or ‘gaapvergelijkingen’

   The last option is chosen.

3. Solve for the displacement in terms of the unknown indeterminate forces as you would normally do for a statically determinate structure.

   Example

   We’ve chosen the following statically determinate structure with displacement constraint \(\varphi_{\rm{B}}^{\rm{AB}} \left( M_{\rm{B}} \right) = \varphi_{\rm{B}}^{\rm{BC}} \left( M_{\rm{B}} \right) \):

   

   Fig. 35 The statically determinate structure with displacement constraint

   Using the forget-me-nots, the rotations can be directly be evaluated without evaluating internal forces:

   • \(\varphi_{\rm{B}}^{\rm{AB}} \left( M_{\rm{B}} \right) = \cfrac{4M_{\rm{B}}}{3EI} + \cfrac{200}{3EI}\)

   • \(\varphi_{\rm{B}}^{\rm{BC}} \left( M_{\rm{B}} \right) = -\cfrac{2M_{\rm{B}}}{3EI}\)

4. Use your displacement constraints to solve for the statically indeterminate forces

   Example

   \[\begin{split} \begin{align*} \varphi_{\rm{B}}^{\rm{AB}} \left( M_{\rm{B}} \right) &= \varphi_{\rm{B}}^{\rm{BC}} \left( M_{\rm{B}} \right) \\ \cfrac{4M_{\rm{B}}}{3EI} + \cfrac{200}{3EI} &= -\cfrac{2M_{\rm{B}}}{3EI} \\ M_{\rm{B}} &= -20 \ \rm{kNm} \end{align*} \end{split}\]

Exercises

• Exercises 2.1 - 2.14, 2.23 and 2.25 in chapter 2.3 of the book Mechanica, Statisch onbepaalde constructies en bezwijkanalyse (in Dutch) (Hartsuijker and Welleman, 2007).

• Exercises 3.1 - 3.10, 3.16 - 3.21 in chapter 3.4 of the book Mechanica, Statisch onbepaalde constructies en bezwijkanalyse (in Dutch) (Hartsuijker and Welleman, 2007).

Answers are available on this website for chapter 2 and here for chapter 3.