How can you use moment distribution to analyze a statically indeterminate structure?

1 Step 1: Identify the joints and members

The first step is to identify the joints and members of the structure, and assign a number to each of them. A joint is a point where two or more members are connected, and a member is a segment of the structure that carries load. You also need to identify the fixed and hinged joints, as they have different degrees of freedom and stiffness. A fixed joint can resist both shear and moment, while a hinged joint can only resist shear. The stiffness of a member is a measure of its resistance to bending, and depends on its length, cross-sectional area, and material properties.

2 Step 2: Calculate the stiffness factors and fixed-end moments

The second step is to calculate the stiffness factors and fixed-end moments of each member. The stiffness factor is the ratio of the stiffness of a member to the sum of the stiffnesses of all the members connected to the same joint. It represents the fraction of the total moment at a joint that is carried by a member. The fixed-end moment is the moment that would be generated at the ends of a member if both ends were fixed, and depends on the load and geometry of the member. You can use tables or formulas to find the stiffness factors and fixed-end moments for common types of members and loads.

3 Step 3: Distribute the moments and balance the joints

The third step is to distribute the moments and balance the joints of the structure. You start by assuming that all the joints are hinged, and assign the fixed-end moments to the members as initial moments. Then, you select a joint and distribute the unbalanced moment to the adjacent members according to their stiffness factors. This will create new moments at the other ends of the members, which will then be distributed to the next joints. You repeat this process until all the joints are balanced, meaning that the sum of the moments at each joint is zero. This is called one cycle of moment distribution.

4 Step 4: Calculate the reactions and displacements

The final step is to calculate the reactions and displacements of the structure. You can use the equilibrium equations to find the support reactions, such as shear forces and axial forces, by summing up the forces along the horizontal and vertical directions. You can also use the compatibility equations to find the displacements, such as rotations and deflections, by relating them to the moments and stiffnesses of the members. You can use tables or formulas to find the displacements for common types of members and loads.

Moment distribution is a useful and simple method to analyze a statically indeterminate structure. It can handle various types of structures, loads, and boundary conditions, and it only requires basic arithmetic operations. However, it can also be tedious and time-consuming, especially for complex structures with many joints and members. Therefore, you may want to use software tools or other methods to check your results and optimize your design.

5 Here’s what else to consider

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