The folded plate is a very complex structure. Classical methods of analyzing folded plates involve laborious calculations. Moreover, there is no single classical method available for general analysis of folded plates. They depend on the shape of folded plates. Some common methods are – Simpson’s, Whiteney’s, Iterative, Three-shear equations etc. However, if you use FEA, then the analysis itself is quite simple and the labor involves only preparing the model correctly and interpreting the result properly. Your analysis programs must have ‘plate’ elements to successfully analyze folded plates. Modeling the folded plates sometimes can be really tricky. We shall discuss various aspects of folded plate analysis with the following example as shown in figure 30-1. The front view is shown in the figure and the length of the structures is 12 m as shown in figure 30-2 with mesh for sufficiently accurate result.
C G D Thickness 150 mm 3 m B Thickness 300 mm E 1.5 m A F 3 m 4.5 m 3 m Figure 30-1
The frame is loaded with 1.5 kPa uniformly distributed load in addition to the self-weight. The folded plate is made of concrete. If you use your analysis program’s graphics editor for input, the quickest way to create the model is: (1) draw the model first using frame element so that the shape like as of figure 35-1 (2) then copy the frames over 12 m distance (3) now draw the plates (4) after the plates are drawn, delete the frame members since they were drawn here only to ease the model (5) now mesh the plates (6) apply boundary condition, in this problem, the sides of the plates are fixed (7) apply plate thickness, material properties etc. (8) apply surface loads on plates BC, CD and DE.
After you apply the loads, visualization should look like as that of figure 30-3. If you draw the plates in wrong orientation, the load direction may appear awkward as shown in figure 30-4. If it happens, delete the particular plate and redraw again in the opposite direction compared to previous case. For this reason, it is a good practice to mesh the plates, after you have applied the loads
tempted to change the load into –1.5 kPa for plate DE, however, it may create problem while interpreting the output.
Figure 30-2 Figure 30-3 Figure 30-4 Fixed on side 12 m Fixed on side
Are you wondering whether we can take advantage of the symmetry? Of course we do. In fact, only 1/4th of the structure needs to be analyzed. The appropriate boundary conditions are shown in figure 30-5 with respect to the global axes shown in the same figure.
all free Z rotation fixed Y Z X all fixed Figure 30-5
To assign proper boundary conditions accurately, you must visualize the deflected shape of the structure yourself in your mind before performing the actual analysis. Of course, you may bypass this mental exercise by analyzing the whole structure rather than taking advantages of the symmetry. You may wonder why I did not give you theoretical result the above folded plate analysis. Well, theoretical calculations are also based on certain simplified assumptions, which may not completely valid many cases (See Section 3, for example). I admit that you need some benchmark problems to compare your analysis output, still you should start relying on your FEA output to gain confidence! After performing the analysis, you will get following stress components – σx, σy, σz, τxy, τyz, τzx in global axes direction and σx, σy, τxy in local axes direction along with maximum principal normal stress σ, minimum principal normal stress σ and maximum principal shear stress τ. The concept needs clarification.
We have already discussed local axes concept. Now, every ‘plate’ element of the above folded plate may be considered ‘lying’ in a ‘2D plane’ even if it is actually ‘inclined’ in the real structure. For example, a ‘plate element’ of BC may be visualized as shown in figure 30-6. This figure is a specific case of figure 30-5 where stress variation along Z-axis is negligible.
σy τyx = - τxy
y τxy
x
σx
Figure 30-6
The visualization will be more apparent from the figure 30-7, where local axes for the vertical, inclined and horizontal plate elements’ are shown.
Y X Y X Z Global Figure 30-7
A different situation comes when we speak stress components in terms of global axes. Here you’ll find all 6-stress components since we now speak in 3D space. Question: My analysis program doesn’t explicitly show global and local axes stress components. How do I know which convention it is following?
Answer: Difficult to say. Generally, most programs give output with respect to local axes. But there are some programs, which show the result with respect to local axes for some type of ‘elements’ and with respect to global axes for some other type of ‘elements’! Really confusing! See your programs’ manuals for details. However, you better solve some benchmark problems (with known answer) to check.
Question: Shall I provide the reinforcement on the basis of forces on global axes or local axes?
Figure 30-9 These analysis outputs are from Visual Analysis.
As a second example, consider another type of folded plate as shown in figure 30-10.
Figure 30-10
The span of plate (along X axis) is 10 m and thickness of all four plates is 100 mm. The plates are made of concrete and are acted by 3-kPa downward (i.e. along –Z direction) load perpendicular to the surface of the plates. Sides (leftmost and rightmost edges but not middle edge) of the plates are fixed. The global X stress (σx in N/m2) after analysis (in Algor) is shown in figure 30-11.
Figure 30-11 Exercise