# Deflections Of Statically Indeterminate Structures

When analysing and then designing a rigid jointed, statically indeterminate structure, I can solve for all the forces and moments using the Hardy Cross (Moment Distribution) Method.

However, I am unsure about how to solve the deflections on the beams and columns. I need these values in order to design based on serviceability limit states and verify deflection checks.

Can you please demonstrate (or point out) a method to quickly and effectively find the displacements in a multi-storey, statically indeterminate structure.

• Whats wrong with FEM? Oct 1, 2019 at 18:11
• Thanks for your response. I cannot use FEM software as I do not have access to it. I also prefer solving using hand calculations. I know there must be a method as it would have been used before computer programs. I would greatly appreciate it if you could point out a particular method.
– Amit
Oct 1, 2019 at 22:18
• Well you can use tables and superposition methods. But this was why FEM was invented because there was no generally easy and fast tool for this. FEM can be done by hand too if you wish, and free fem applications exist. Oct 2, 2019 at 4:39
• Can you please elaborate on what you mean by superposition methods and tables. I would appreciate it if you could point out a few. I am mainly interested in the hand calculations and that is why I am not applying FEM.
– Amit
Oct 2, 2019 at 7:00

To rigorously solve for deflections of indeterminate structures, the most feasible method is the Direct Stiffness Method (aka matrix methods). This is generally the approach finite element method software packages (FTool, SAP2000, RISA, etc.) are implementing when dealing with line elements.

For structures with a relatively small number of degrees of freedom it is possible to implement the direct stiffness method and arrive at a solution (somewhat) by hand. In my matrix methods class at university we generally solved the matrices using a calculator or by implementing a routine coded in MATLAB. So really, we were doing exactly what the FEM software does but more slowly and/or transparently.

Prior to the advent of computers (including programmable calculators) engineers would have used approximate methods of analysis to ballpark deflections. One very simple example of this might be that if we know the load on a beam we can determine an upper-bound deflection by considering it to be simply supported. When doing a hand check of computer calculations, engineers today will still take this approach. That is, we make simplifying assumptions and get a back-of-the-envelope feel for whether the computer solution is appropriate.

Computer-based analyses are fairly essential for the design of today's structures by today's design codes. The goal of hand calculations is rarely to exactly duplicate the computer solution. Rather, the goal is to use hand methods and engineering judgement to validate the computer solution.

• Thanks for the answer. Just to further enquire, if I am doing hand calculations and attempting to find the deflection on a fixed-fixed beam, I can idealise it as a simply supported with concentrated moments on each side? Then I can find the deflections as normal? What about when the beam is within the frame? Are these results conservative?
– Amit
Oct 2, 2019 at 22:06
• @Amit, a fixed-fixed beam is identical to a simply-supported beam with concentrated moments at the ends, so yes, you could do that. As to how conservative these results are, I'd recommend asking a separate question, since that depends on the type of structure (frame, truss, etc), materials and arguably even country/state.
– Wasabi
Oct 2, 2019 at 23:51
• Ok, Thankyou for the responses.
– Amit
Oct 3, 2019 at 1:00