Fixed End Support ( 2D & 3D) Reactions in Structural Analysis

Reactions are forces that are created in the supports of a structure, when these supports restrain the displacement at that point. If the support do not restrain the movement at that point, there are no reactions. Based on this, there are several types of supports provided to the structure.

Fixed End Support ( 2D & 3D) Reactions in Structural Analysis

All the components of a structure are connected and linked using supports and connections as shown in fig.1 below. In the figure, you can see a beam supported using two columns at the ends of the beam.

Fig.1. Supports in a Framed Building Structure
Image Credits: ASCE Library

You can read different types of supports in detail in the article: Types of Supports in Structural Analysis - Boundary Conditions

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Fixed End Support Reactions in 2D

Hereafter, we are going to represent the number of  reactions at each support as r_e

Consider a beam AB with one end is fixed at 'A' and the end "B" is free subjected to a load at an angle of θ. The load results in the deflection of point B.

We will discuss the reactions at 'B' and Á' to understand the how the reactions are originated in the supports.

Reactions At End B

1. Under the action of load ‘P’, the end A remains as such but the end B deflects.
2. Deflection at point B results in a displacement of Δy a non-zero value and corresponding Θb ( slope at B).
3. There are no reactions at B, as the end is not restrained. The end is free to move.

Support Reactions @ B = r_e =0;

Reactions At End A

1. The fixed end ‘A’restricts the movement of end BEAM along x direction, as a result of which displacement along x-axis is zero. Hence, Change in distanceΔx = 0;
2. Similarly, The end ‘A’restricts the displacement along y-axis, hence Δy = 0;
3. Unlike, end B, the end A do not allow any deflection i.e (Δy = 0), and hence the slope at A is θA = 0; 

Hence at End A, (Fixed End) the Horizontal, vertical and rotational movement is restrained; Δx= 0; Δy = 0; θA = 0 that leads to reactions Rx, Ry and M. Where Rx is the reaction caused along x-direction, Ry is the reaction along Y-direction and the rotation resisted creates reaction in the form of moment M.

Support Reactions @ Fixed End A = r_e = ( Rx , Ry, and M) = 3 nos

Here, the moment created 'M' is Mz about the Z-axis. 

Fixed End Support Reactions in 3D

The above explained cantilever beam can be picturized in 3D as shown below. Here, reactions are formed along all the three directions. Hence, the number of support reactions = 6;

They are Rx, Ry, Rz, Mx, My, and Mz.

Fixed End Support Reactions in 3D

As shown in the 3D fixed support figure, the longitudinal axis of the element is along X-direction and Y and Z axis are lateral axis. 

The moment created about the longitudinal axis of the structure ( in this case, about X-axis is Mx ) would result in the twisting of the structure. This moment is called as Twisting Moment.

The moment created about the lateral axis of the structure ( in this case, the lateral axis are Y and Z axis, and My & Mz) would result in the bending of the structure. This moment is called as bending moment.

The moment experience by the element in 2D in before case is Mz which is bending moment.

Also Read: Hinged End Support ( 2D & 3D) Reactions in Structural Analysis