1. General
In FEM-Design, the edge connections are a powerful tool to get the forces from an edge of a plate or a wall. Plates and walls are called shells in the analysis and also in this article. The edge connections can show both real calculated forces, average them over some finite elements and show resultants – just like the line supports show the reactions (figure 1). However, there has been debate over how to interpret the force reactions, especially the direction of the force in them.
Figure 1 - Different ways to show connection forces
2. Edge connection coordinate directions
Each connection has its own coordinate system – the axes, that indicate the x’, y’ and z’ directions of the connection. The logic for the coordinates of a connection are as follows:
• y’ direction – is always pointing away from the edge, perpendicular to it and it is in the plane of the shell element
• x’ direction – is always along the edge. Its direction (left or right) is depending on the direction of the z’ direction (figure 2)
• z’ direction – is always pointing against of the shell’s z’ direction, and thus is also perpendicular to the shell. Based on y’ and z’ directions, the x’ direction can be found
Figure 2 - Edge connection directions
Because of how the z’ axis of the connection is oriented, the x’ directions of the edge connections of a shell always produce a counter-clockwise pattern when looked against the z’ coordinate of the shell (figure 3). It is easy to remember this from the right-hand-screw-rule.
Figure 3 - Rule of thumb for connection directions
In case of the internal holes, these are counter-intuitively arranged in a clockwise pattern (figure 4). It makes more sense if we would pretend the hole to be cut from the outside (like cutting paper with with scissors, as in figure 4).
Figure 4 - Shells with holes
The direction of the edge connection cannot be manually changed, but user can swap the top and bottom faces of a shell which in turn will change all the connections’ directions – this swapping function is located in the Modify Region tool (figure 5). The connections will still follow the rule that the z’ axis of the connection is pointing against the z’ axis of the shell (figure 3).
Figure 5 - Swapping top and bottom
3. Positive and negative values of a connection
As it is with the supports, the sign of the value is depending on the position of the force. When the force is on the positive side of the edge’s coordinate system, then the value is positive and vice versa (figure 6). The direction of the arrows, both the distributed result and the resultant, are always towards the zero of the coordinate system or more generally – we can think that the arrow is pointing against the corresponding axis – so the positive z’ value is pointing against positive z’ axis of the connection (not along it).
Figure 6 - Results direction and sign
4. Decoding the direction of the arrow
The direction of the arrow can be thought of like a reaction to the external load in the connection. For example, in figure 7 below, the right wall is pushed down by the force. The edge connection in the left wall is trying to resist the downward force, thus it shows the arrows pointing up. Both the distributed small arrows and the resultant arrow follow the same logic. The values in the figure are negative, because the direction of the arrows (the reaction force inside the connection) goes along the x’ axis of the edge connection’s coordinate system, and based on the previous paragraph, the value must then be negative (positive was against the axis and negative was along the axis).
Figure 7 - Example of a wall
5. How to decipher the table results for later use
The table results are sometimes needed for some external calculations. In FEM-Design, we can take out the table or list of the results for edge connections. The direction of the force is needed to understand how they are located in the model.
Also, it is necessary to note that the numbering for the edges is continues for one shell – so shell with two edge connections has numbers CE.1 and CE.2 (not something random like CE.12 and CE.56). It is against some common belief, that each side has it’s corresponding number – they have not - the top edge of the wall is not always 3 for example. The edges are numbered counter-clockwise if looked against the z’ axis of the shell – like the local x’ axis of the connection it uses the same right-hand-screw-rule. Similarly, the inner holes are numbered clockwise. The numbering starts from the edge of the shell that was first drawn when the shell was drawn (not the edge connection that we added first).
In case of walls that are drawn with the regular line tool, the order of the sides are bottom, one side, top, the other side – so in case we have connection in the top and bottom only, then bottom would be first and the top would be second. But if we would only have two side connections, then one of them will be first (the one that is the first when we move counter-clockwise from the bottom). In figure 8 below, there are some examples of how the numbers are generated for walls.
Figure 8 - Wall connection numbering
In case of plates, the order begins from the side that was drawn first (figure 9).
Figure 9 - Plate connection numbering
In the example below (figure 10), there are four walls. Two of them are facing us, the other two are facing away from us (look at the local coordinate systems for the walls). This means that we have different orientation of edge connections.
Also, some walls do not have connections on all sides, since FEM-Design will automatically remove edges on the free sides, so care must be taken when adding edges and interpreting the results.
Figure 10 - Walls with connections
There are four possible list (table) results for edge connections:
• Line connection forces – it is the result for each finite element node
• Line connection. Resultants – is one resultant per connection
• Line connection (arc). Resultants. Constant by sign – is the resultants for round edges (that are based on arcs). Resultants are for each positive and negative part
• Line connection (linear). Resultants. Constant by sign – is the resultants for straight edges (that are based on lines). Resultants are for each positive and negative part
Let’s look at the second result in a table form (figure 11). There we can see that the resultant Fx’ (this is force along the edge) for wall 1 connection 1 (W.1.CE.1) is +0,367kN. If we consult the figure 10 for the placement of the connections and direction of the local coordinate system directions for the connection elements, then we can say that the resultant force in the connection is pointing from right to left (since it has a positive value and we have seen in the previous paragraph, that positive value means direction against the coordinate system). In figure 12, we can see that this is indeed exactly like we describe.
Another example is the first connection in third wall (W.3.CE.1). We can see that the Fy’ value is (-24,4)kN. Since the y’ axis of any connection is pointing away from the edge (in this case it is the bottom edge of the plate, so axis is pointing down), we can say that the resultant is also pointing down (since negative value means direction along the coordinate system). We can see from figure 13, that this is indeed the case.
Figure 11 - Results in table form
Figure 12 - Result of the connection visualized
Figure 13 - Result of the connection visualized
6. More examples
In normal structures, the walls are not switching faces like we had in the previous example. So, normally the x’ directions of edge connections of adjacent walls are pointing opposite to each other. This means that if look at the adjacent edge values in the table and they have opposite signs, they are actually pointing in the same direction. And vice versa – if the adjacent edges have results with the same sign (both positive or both negative) then they are pointing in different directions (against each other or away from each other). Moreover, if the values are exactly the same then the total force from one wall is being carried over to the other wall and no force is going to any other element in that connection. When the values are different, then some of the force in the connection is coming/going to some other element that is connected there.
With that knowledge, let’s look at another common structural analysis problem.
In the figure 14 below, there are two walls on top of each other and a plate in the middle. We want to know the force that one wall is carrying over to the other so that we could design the shear connection between two walls and we need to know what force is transferred between the walls and the plate when we do not apply the edge connection to the plate. We consider the plate to be connected to the bottom wall. In the example below it would be easy to do it manually since we have only two connections, but in real models there can be thousands of connections and this is usually something that engineers want to automate in the spreadsheet calculations.
Figure 14 - Example with walls and plate
Here we have three edge connections for walls – bottom wall has two connections and top wall has only one. In this example, we added the plate edge as well, just to compare the values that we calculate manually, but we do not need the connection for the example. We mark down the connection IDs – we are interested in forces between W.1.CE.2 and W.4.CE.1 connections. Let’s look at the resultants for the edge connections in table form (figure 15).
Figure 15 - Results in table form
We can see that both the Fx’ for W.1.CE.2 and W.4.CE.1 are negative, so they go along their own local coordinate systems. We described previously that if adjacent edges have values with the same sign, they are pointing opposite to each other. Also, since the values are different, then there must be some force that is going to the plate (coming from the plate).
In this example, we can say that the force between the walls is 7,5kN (the same that is coming from the top wall). The direction is probably not important for the design of the connection, but if it were, then we could get the direction like this: we see that the top connection’s x’ axis is pointing to the right, the value in the table is negative, this means that the resultant force is pointing along the x’ axis (from left to right). As we saw in the very first example in figure 7, we could consider the direction of the resultant in the connection as a reaction to the external force. The external force would be the bottom wall + plate movement. So, in this connection, the top wall is trying to move to right, while the bottom wall + plate are trying to move to the left.
In order to calculate how much of the force is going to the plate, we need to subtract the value of one connection from the other. To get a meaningful direction for the remaining force let’s subtract the bottom wall’s edge result from the top wall’s edge result. This way we can compare the direction of the remaining force to the top wall edge’s direction (since the top wall’s value is the first operand in the subtraction). So, we would do this operation: (-7,5) – (-31,101) = +23,601kN. We can see that we were right when we compare our result with edge connection of the plate P.1.CE.1 (in figure 15) which has the same absolute value, but a different sign. The sign indeed must be different since the local x’ axis of both connections (top wall and plate) are pointing in the same direction and we established before that two adjacent connections, that are pointing in the same direction, have their resultant forces pointing in the same direction only if the values have opposite signs.
Our calculated value is a positive value according to the top edge’s axis (because the top edge was the first operand in the subtracting operation). Positive value meant opposite to the edge’s axis. Since the top edge’s x’ axis is pointing from left to right, we can say that this remaining force is pointing from right to left. Again, we can consider this resultant force as a reaction to the external forces, so in this connection the external force that is coming from the plate is pointing from left to right – some load is pulling plate to the right and plate tries to grab the connection and pull it to the right with it; and then the connection resists and shows reaction to the left. In the figure 16 below we see exactly how this should look like.
Figure 16 - Result of the connection visualized