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Friday, August 4, 2017

Understanding Sign Conventions in Structural Analysis


Introduction
Signs conventions is always a pausing point for all civil engineering students being introduced to structural analysis for the first time. The truth is that if you really do not understand it, you cannot make significant progress in mastering structural analysis calculations.


The three most pronounced internal forces analysed for in structural analysis are bending moment, shear force, and axial force. It is very common for people to define and state their sign convention before proceeding in any structural analysis problem. This is due to the variations in selection of positive and negative coordinates.

For instance, most Indian and American text books will adopt the conventional cartesian coordinates system, and plot all positive bending moment upwards, and negative moments downwards, while most British and Scandinavian textbooks will plot positive moment downwards, and negative moments upwards.

However, is there any standard approach?

I will prefer to answer yes, because it is more appropriate to plot bending moment diagrams in the tension zone/fibre of each structural member. This is especially important in reinforced concrete structures where there is need to provide tension reinforcement. For instance, once you look at the bending moment diagram of a continuous beam subjected to uniformly distributed load, you can easily point out and provide bottom reinforcements at the spans, and top reinforcements at the supports.

See example below;



From the image above, you can see the consistency of the bending moment diagram with the typical arrangement of the reinforcement. This is because, the bending moment diagrams were plotted in the tension zone, and of course it means that positive moment was plotted downwards. However, if the cartesian coordinates system was strictly followed, we would have been reversing the placement of the reinforcements with the bending moment diagram.



The same thing also applies for frames. Plot your bending moment diagram in the tension zone of the members of the frame. At my stage and level, once I look at a structure and the loading, I can easily sketch the moment diagram. I do not really need calculations to tell me if it will be positive or negative.

However, there are standard procedures that you must follow in your calculations if your results are to be consistent. By consistent, what I mean is that positive will mean sagging moment, while negative will mean hogging moment. With this, you can plot your diagram directly as you have calculated, without having to interchange signs.

The summary of the procedures is given below.

BEAMS

(a) Bending moment


When you are coming from the left hand side of the structure, all clockwise moments are positive and vice versa.

When you are coming from the right hand side of the structure, all anti-clockwise moments are positive and vice versa.

(b) Shear force

When you are coming from the left, upward forces are positive, while downward forces are negative.

When you are coming from the right, downward forces are positive, while upward forces are negative.

(c) Axial force
Compressive axial forces are negative, while tensile axial forces are positive.


All you have to to is to look at the direction of the force. When coming from the left hand side of beams, axial forces pointing towards the right means that the beam/section is in compression. While when coming from the right, axial forces pointing towards the right means that the beam/section is in tension and vice versa

FRAMES
We are going to use the frame below as an example.


(a) Bending moment
When coming from the left hand side, all clockwise moments are positive, and anti-clockwise moments negative. All you have to do is to look at the point where you are taking your moment and observe the nature of rotation the force will produce. This is valid for both vertical and horizontal forces.

When coming from the right, all anti-clockwise moments are positive.

For frames, we plot positive moment inside, and negative moments outside the frame. See example below.

(b) Shear force 
When coming from the left, upward vertical forces are positive and downward forces are negative. For horizontal forces on columns, forces pointing towards the right produce negative shear forces.

When coming​ from the right, downward vertical forces are positive, while upward vertical forces are negative. Horizontal forces on columns pointing towards the left will produce produce positive shear forces.

We plot positive shear forces outside the frame, and negative shear forces inside the frame.


(c) Axial Force
When coming from the bottom of columns, upward vertical support reactions means that the column is in compression whether you are coming from the left or right.

When coming from the right hand side of beams, forces pointing towards the right means that the beam is in compression and vice versa.

To me, you are free to select any location to plot your axial forces. Some people draw axial force diagrams to coincide with the centerline of the structure. In the classroom where I was trained, we draw negative axial forces outside the frame, and positive axial forces inside the frame. So it's your choice to make.

I hope you find this article useful. If so, I will like you to share it.

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