Consistent Deformations - Force Method | Index of Examples | CCE Homepage |

*Consistent Deformations - Force Method*

Indeterminate Beam with Moment Reaction as Redundant

problem statement

Using the method of consistent deformations, determine the reactions, moments and shears under the loading conditions shown. The modulus of elasticity (E) and the moment of inertia (I) are constant for the entire beam.

Figure 1 - Beam structure to analyze

- determine the degree of indeterminacy

The structure is statically indeterminate to the first degree (r = 4, e = 3, n = r-e = 4-3 = 1).

- select redundant and remove restraint

To solve for this single degree of indeterminacy, the structure has to be reduced to a statically determinate and stable structure. This is done by removing a redundant support.

In this example, the moment reaction at support A is selected as a
redundant to remove in order to obtain a __primary determinate structure__.

Figure 2 - Primary structure

- determine reactions and draw moment diagram for the primary structure

Calculate the support reactions of the primary structure.

The resulting system, (_{A0} indicates the resulting deflection or deformation
at the location of the removed redundant for the primary structure).

Figure 3 - Support reactions

Determine the moment diagram M_{0} due to the applied loads on the
primary structure.

In this example, the cantilever method is used to develop the moment diagram. (See a Virtual Work Cantilever Example for a complete description of this step)

Figure 4 - M_{0} - Moment diagram due to applied loads

- calculate deformation at redundant

Using the virtual work method, calculate the rotational
translation of support A that correponds to the redundant M_{A}.
Remove all loads an apply a unit moment in the direction of the redundant, draw the moment
diagram, m_{a}, and sketch the deflected shape due to the unit moment.

The resulting system, (*f*_{aa} is the deformation caused by the unit load).

Figure 5 - Primary structure with unit load applied and resulting
deflected shape

Figure 6 - Moment diagram m_{a} with M_{A} = 1 ft-k

Calculate the rotational translation corresponding to the redundant M_{A} at support A
using the following equation:

Calculate the deformation at the redundant, _{A0}.
Use the method of virtual work, calculate the areas on the M_{0} diagram and multiply
each area by the corresponding heights, h_{i}, measured at the centroid of this area
on the m_{a} diagram:

Area No. | Area/EI (A) (k-ft^{2})/EI |
Height (h) on m_{a} diagram (k-ft) |
A_{i}*h_{i} (k^{2}-ft^{3})/EI |

A_{01} |
2/3 x 20 x 100/EI = 1333.33/EI | -1/2 | -666.67/EI |

A_{02} |
1/2 x 20 x -36/EI = -360/EI | -1/3 | 120/EI |

A_{03} |
1/2 x 6 x -36/EI = -108/EI | 0 | 0 |

Total = Q(_{A0}) = |
-546.67/EI |

Therefore, with Q = 1 ft-k;

*Q*(_{A0}) = -546.67 (k^{2}-ft^{3})/EI

_{A0} = -546.67 (k-ft^{2})/EI

Calculate the flexibility coefficient, *f*_{aa}, by determining the deformation
of the primary structure when subjected to the redundant load, M_{A} = 1 ft-k.

Again, using the method of virtual work, calculate the areas on the m_{a} diagram and
multiply by the corresponding heights, h_{i}, measured at the centroid of each area:

Area No. | Area/EI (A) (k-ft^{2})/EI |
Height (h) on m_{a} diagram (k-ft) |
A_{i}*h_{i} (k^{2}-ft^{3})/EI |

A_{11} |
1/2 x 20 x -1/EI = -10/EI | -2/3 | 6.67/EI |

Total = Q(f_{aa}) = |
6.67/EI |

Therefore, with Q = 1 ft-k;

*Q*(*f*_{aa}) = 6.67 (k^{2}-ft^{3})/EI

*f*_{aa} = 6.67 (k-ft^{2})/EI

- write consistent deformation equation

The consistent deformation equation that corresponds to the redundant M_{A} (the
moment reaction at support A) is:

_{A0} + *f*_{aa} * M_{A} = 0 (1)

This equation is set equal to zero since the fixed support at A does not allow any rotation.

- solve consistent deformation equation

Using equation (1), solve for M_{A}:

-546.67 (k-ft^{2})/EI + 6.67 (k-ft^{2})/EI * M_{A} = 0

M_{A} = 82

Multiply the unit moment, Q, at M_{A} by 82 to get the final reaction.
The positive answer indicates that the reaction is in the direction of the applied unit moment.

- determine support reactions

Impose the value of the calculated M_{A} along with the other applied loads on the
original structure. Calculate the remaining reactions using the three static equilibrium equations,
(F_{x} = 0,
F_{y} = 0 and
M = 0).

Figure 7 - Beam with support reactions

- draw moment, shear, and axial load diagrams

Shear:

Figure 8 - Final shear diagram

Moment:

Figure 9 - Final moment diagram

Deflected Shape:

Figure 10 - Deflected shape

Contact Dr. Fouad Fanous for more information.