The finite element method is an approaching method. This means that the results will never be exact and can be made better and better, within reason, by sub-dividing the element mesh. Accurate results are dependent on the modeling of the problem.
The following pages are studies of finite element mesh fineness and its relationship to accurate stress and deflection results. These studies are meant to be an aide to help you select appropriate mesh fineness for a structure you are trying to model. These studies will also answer the "why" many people ask when told they must use a "mesh" of elements to model a structural item (such as a shear wall) instead of using one giant element. Obviously these studies only give an overview of some basic elements and the engineer must be the final judge as to whether a specific finite element model is a good reflection of the "real" structure.
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Shear Wall Properties L = 240 in Area = 1440 in2 B = 12 in H = 120 in. E = 4000 ksi n = 0.30 G = 1538.5 ksi
P = 15,000 kip |
I = BH3/12 Δ = PL3/3EI + K = P / Δ |
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Element |
1x1 |
1x2 |
2x2 |
2x4 |
4x8 |
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Deflection |
4.54 |
8.07 |
8.26 |
10.43 |
11.29 |
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Error |
62% |
33% |
31% |
13% |
6% |
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Stiffness |
3304.0 |
1858.7 |
1816.0 |
1438.2 |
1328.6 |
Note on Methodology:
| Floor | Shear | Moment | Elements |
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4 |
9.99k |
100.05k-ft |
P49-P52 |
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3 |
20k |
300.08k-ft |
P33-P36 |
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2 |
30k |
600k-ft |
P17-P20 |
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1 |
40k |
999.98k-ft |
P1-P4 |
Shown above are the analysis results of a 4-story shear wall. This example is for a straight shear wall, however the method and results are valid for box, channel, or any other shear wall shapes.
The RISA-2D files that were used to obtain these results are included as “4X1WALL.R3D” and “4X4WALL.R3D”. The 10 kip story loads were applied uniformly across each story. This was done to more accurately model loads being applied to the wall from a rigid or semi-rigid floor.
The story shears at each level were calculated as the sum of the FX corner forces. The story moments at each level are calculated from the FY corner forces as shown below:
Mi = (Fyouter
The story shears were calculated as shown below from the corner forces. See the screen shot close up of the FX corner forces on the next page.
| Story | Sum FX Corner Forcers at Story Level (k) | Shear | Moment |
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4 |
[0.462 + 0.786 + 1.77 + 1.98] * 2 = 9.996 |
9.9k |
P49-P52 |
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3 |
[1.2 + 1.65 + 3.45 + 3.7] * 2 = 20.000 |
20k |
P33-P36 |
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2 |
[1.9 + 2.5 + 5.14 + 5.46] * 2 = 30.000 |
30k |
P17-P20 |
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1 |
[11.76 – 0.32 + 6.09 + 2.47] * 2 = 40.000 |
40k |
P1-P4 |
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For the graphical display of the corner forces, there are 4 corner forces shown for each plate. This is similar to a beam element which has 2 member end forces. To get the story shear at any line, just sum up all the FX corner forces along the line. |
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Level 4 Global FX Corner Forces |
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To get the story moment at any line, just sum the moments obtained by multiplying the Fy corner forces along a line, times the moment arm such as the distance of each Fy force to the center of the wall. |
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Level 4 Global FY Corner Forces |
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A quicker way to calculate story forces is to use
the Internal
Force Summation tool to have the program automatically calculate the
global forces that pass through a given elevation of the shear wall. To
do this, just click on the 
icon that appears below
the selection toolbar
For additional advice on this topic, please see the RISA Tips & Tricks webpage at risa.com/post/support. Type in Search keywords: Story Shears.
The story moments are calculated as the FY Corner forces times their moment arms. The forces are symmetrical so each force on one side is multiplied by twice the arm length.
| Story | Sum FY Corner Forces * Moment Arm (k) | Story Moment | Elements |
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4 |
[3.92 k * 15' + (2.48 k + 3.02 k) * 7.5'] = 100.05 k' |
100.1 kip-ft |
P49-P52 |
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3 |
[12.33 k * 15' + (7.9 k + 7.45 k) * 7.5'] = 300.075 k' |
300.1 kip-ft |
P33-P36 |
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2 |
[24.93 k * 15' + (16.5 k + 13.64 k) * 7.5'] = 600.00 k' |
600 kip-ft |
P17-P20 |
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1 |
[45.81 k * 15' + (25.16 k + 16.55 k) * 7.5'] = 999.975 k' |
999.9 kip-ft |
P1-P4 |
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Horizontal Deflection at Top |
0.028 in. |
0.033 in. |
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Shear @ A-A |
10.82 kips |
10.53 kips |
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Shear @ B_B |
24.2 kips |
23.08 kips |
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Shear @ C_C |
33.74 kips |
33.05 kips |
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Shear @ D-D |
45.26 kips |
47.35 kips |
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Reactions at E |
10.8 kips |
10.53 kips |
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Reactions at F |
24.2 kips |
23.06 kips |
This is an example of a typical concrete shear wall with penetrations for windows and doors of various sizes. The files for the models are WALLPEN1.R3D (coarse mesh) and WALLPEN2.R3D (fine mesh). No theoretical solution results are given to compare with, however the two finite element densities are compared to observe the rate of convergence to the “true” answer. The shears at the various lines are computed by adding up the X corner forces for the element corners closest to the lines. The horizontal deflection is for the top of the wall. A very rigid link is added to the top of the wall to simulate the effect of a concrete horizontal diaphragm. This has the effect of stiffening the walls and spreading the load uniformly across the top of the wall. The load is applied as a uniform load of 3.0 kips/ft. across the top of the wall. The total width of the wall is 38 ft, so the total applied load is 114 kips. The total height of the wall is 18 ft.
The “coarse” mesh on the left is an example of the minimum finite element mesh that should be used to model this type of wall. Notice that the course mesh gives good results for the wall shears and reactions. The overall deflection of the coarse mesh is off by about 15% from the “fine” mesh. The coarse mesh tends to give too much stiffness to the slender walls around the loading door opening on the left, this can be seen in the larger reactions at points E and F as well in the horizontal deflections. The fine mesh on the right shows that the slender wall sections are more flexible than shown by the coarse mesh and thus the reactions and wall shears are reduced for the slender wall sections.
Shown below is a common modeling problem with plate elements. Since plates only have connectivity at their corner nodes, the applied load at middle joint connects to the plates below the joint, but not to the one above it. Because of this lack of connectivity, you see the joint "pushing through" the plate edge above in the plotted deflected shape.
The proper way to hand this type of mesh is with one of the mesh transitions described in the following section.
For additional advice on this topic, please see the RISA Tips & Tricks webpage at risa.com/post/support. Type in Search keywords: Plate Connectivity.
Shown below are two methods for transitioning from an area with a fine mesh to an area with a larger / coarser mesh.
Three to One Two to One
Shown below are two methods for transitioning from an area with a rectangular mesh into an area with a radial mesh.
Square to Round Round to Square
For additional advice on this topic, please see the RISA Tips & Tricks webpage at risa.com/post/support. Type in Search keywords: Plate Mesh.
