CHAPTER 5. Results and Analysis. 5.1 Four Point Bend Test. This chapter presents the comparison of shear distribution between the numerical

Transcription

1 CHAPTER 5 Results and Analysis This chapter presents the comparison of shear distribution between the numerical models and the analytical models. The effect of geometric non-linearity in the way the structure carries the load will be discussed in detail. Localized effect at the loading points and support will also be considered. 5.1 Four Point Bend Test Four-point bend test is essentially a two dimensional problem, the shear pattern only changes along the beam span axis. The element forces in the global Y direction on the cross sections of selected beam spans are summed and plotted versus the beam span. The reason to collect the forces with respect to the global Y-axis is that it is the force that responsible for the beam deflection in four-point bend test and thus comparable to the total shear force in the theoretical model. Figure 5.1 shows the beam spans and expected shear patterns of important points such as the support and loading points. Since four-point bend test is a symmetric problem, the shear distribution plot ends at the mid-point of the sandwich beam. From the simple shear load versus beam span plot we know that the total shear load is constant at 2 P between the support and the loading point. Total shear is expected to go to zero beyond the loading point. 7

2 P 2 Results and Analysis P 2 Shear P 2 Support ( mm) Loading point (73.25 mm) Line of symmetry (146.5 mm) Beam Span FIGURE 5.1 Beam span and shear overview of the four-point bend test. From a simple free body diagram, as shown on Figure 5.2, we can see the expected shear distribution on all three layers of the sandwich beam. Figure 5.2 shows a cut of the left hand end of the sandwich beam and the corresponding forces and moments. From the simple free body diagram, we know that without shape change, the total shear carried by the cut cross section should be equal to the load at the support, 2 P. The shear load on each layer of the sandwich beam, V, is expected to be in the negative Y direction with respect to the global Y-axis of the software, Y G. 71

3 Y G X G M V P 2 FIGURE 5.2 Free body diagram showing shear direction with respect to the global coordinate system Top Face Sheet Shear Distribution The load steps analyzed are 1,N, 2,N, 3,N, 4,N, 5,N, 6,N, 6,6N, and 7,N total load. Shear load distribution comparisons between the finite element model and the theoretical model for the top face sheet are presented from Figure 5.3 to Figure 5.1. In general, the shear carried by the face sheet agrees well with the theoretical distribution near the support region. The further away the beam span is from the support, the more discrepancy observed between the two models shear distribution. The difference between the shear distributions of the two models also becomes bigger as the applied load increases. 72

4 Global Y Force, RTF(Yg) (N) FIGURE Top face sheet shear distribution for 1, N applied load. Global Y Force, RTF(Yg) (N) FIGURE 5.4 Top face sheet shear distribution for 2, N applied load. Global Y Force, RTF(Yg) (N) FIGURE 5.5 Top face sheet shear distribution for 3, N applied load. 73

5 Global Y Force, RTF(Yg) (N) FIGURE 5.6 Top face sheet shear distribution for 4, N applied load. Global Y Force, RTF(Yg) (N) FIGURE 5.7 Top face sheet shear distribution for 5, N applied load. Global Y Force, RTF(Yg) (N) FIGURE 5.8 Top face sheet shear distribution for 6, N applied load. 74

6 Global Y Force, RTF(Yg) (N) FIGURE 5.9 Top face sheet shear distribution for 6,6 N applied load. Global Y Force, RTF(Yg) (N) FIGURE 5.1 Top face sheet shear distribution for 7, N applied load. The theoretical model shows constant shear load carried by the top face sheet between the support and the loading point for all load steps. The dip on the theoretical shear distribution curves close to the loading point is due to the inability of the equations used to fully describe the boundary condition. The above-mentioned dip occurs consistently in every load step, which indicates that this is caused by the formulation of the model. The numerical model s top face sheet shear distribution shows a general trend of decreasing negative resultant load at locations further away from the support. At locations closer to the loading point, the resultant load becomes positive. For larger 75

7 applied loads, the shear resultant changes sign closer to the support. This is contrary to the predicted shear distribution based on Figure 5.2, which shows that all layers of the sandwich structure have a negative resultant load in Y G direction. This discrepancy is caused by geometric non-linearity. Figure 5.11 shows the free body diagram of a section of the top face sheet in its original and deformed composition. With geometric non-linearity considered, the numerical software collects the element forces in Y G axis. Therefore, as shown on Figure 5.12, the software is actually summing the Y G component of the shear force, V TF (Yg), and the Y G component of the membrane force, N TF (Yg). The membrane effect of the top face sheet is so large that N TF (Yg) overcomes TF (Yg) V on the cross section, resulting in positive resultant force in the Y G direction on the surface, R TF (Yg). The slope of the beam increases at locations further away from the support, causing N TF (Yg) to become increasingly significant. Increase in the applied load causes a larger slope on the beam as a whole and larger membrane forces in the face sheets. As a result, the sign change of R TF (Yg) from negative to positive occurs closer to the support point with the increase of the applied load (Table 5.1). This shows that if geometric nonlinearity is taken into account, membrane forces in the top face sheet, or global bending from the whole structure point of view, plays a significant role in carrying the shear load and resisting the beam deflection. Once the R TF (Yg) curve moves upward on the shear distribution graphs, it is the indication that the membrane forces have started affecting the load-carrying method of the beam. 76

8 Y G Original Composition X G Deformed Composition V TF N TF N TF ( Xg ) V TF V TF (Yg) N TF (Yg) N TF V TF ( Xg ) FIGURE 5.11 Breakdown of forces into global axes directions. V TF (Yg) + = N TF (Yg) R TF (Yg) Y G Component of Shear + Y G Component of Membrane Force = Resultant Force in Y G Direction FIGURE 5.12 Components of the resultant force in the Y G direction. 77

9 Table 5.1 Beam locations of the membrane forces domination initiation. Applied Load (N) Beam Locations (mm) Core Yielding Initiation 1, 45 No 2, 39 No 3, 33 No 4, 29 No 5, 26 No 6, 2 Yes 6, Yes 7, 16 Yes It is important to note that the shear carrying effect of the membrane forces becomes significant long before core yielding initiates. However this effect only increases significantly at the initiation of core yielding. In order to quantify the dominance of N TF (Yg) over V TF (Yg) in carrying the load, the cross section at mm from the support is selected for analysis. The reason of this selection is because the above-mentioned location has the highest positive R TF (Yg) in the region between the support and the loading point. The equation used to quantify this domination at a certain beam span is stated as below: ( i) R ) ( i 1) P( i) P( i 1) TF ( Yg) TF ( Yg Average Membrane Dominance = ( 1% ) R (5.1) Where P is the applied load and i corresponds to the number of the load step. This equation finds the average dominance of the membrane forces over shear forces in the Y G 78

10 direction, as each additional load is applied. Table 5.2 shows the average membrane dominance over shear forces at each load increment. Table 5.2 Average membrane dominance within each load interval Load Step (58.25mm beam location). Applied Load (N) Resultant Load (N) Core State Initial 1, Elastic End 2, Elastic Initial 2, Elastic End 3, Elastic Initial 3, Elastic End 4, Elastic Initial 4, Elastic End 5, Elastic Initial 5, Elastic End 6, Plastic Initial 6, Plastic End 6, Plastic Initial 6, Plastic End 7, Plastic Load Interval (N) 1, 1, 1, 1, 1, 6 4 Average Membrane Dominance (%) For example, in the 1, N load increment between 5, N and 6, N, there is average membrane forces dominance over shear of about 9.64%. Table 5.2 clearly shows that the dominance of the membrane forces over shear forces increases significantly after the initiation of core yielding. The yielding of the core results in large deflection and hence significant geometric non-linearity effect. This finding shows that the argument made by the higher order theory that the face sheets resist deflection 79

11 through bending rigidity about their own neutral axes is not valid when geometric nonlinearity is considered. The additional load carried by the sandwich beam after core yielding is actually carried through bending rigidity of the entire structure. A zigzag pattern is observed at the region close to the loading point. Again this is due to the geometric non-linearity effect. Consider the free body diagram shown in Figure Y G Original Composition Applied Load, 2 P X G Deformed Composition N TF R TF (Yg) V TF V TF (Yg) N TF V TF N TF (Yg) V TF ( Xg ) N TF ( Xg ) FIGURE 5.13 Schematic of localized load and breakdown of the forces into global directions. 8

12 Localized loading causes crushing of the foam core at the region directly below the loading point. This then results in significant localized deformation of the top face sheet. The face sheet begins to act like a membrane, picking up load in the axial direction. The NTF term of the top face sheet now is in tension close to the loading point. The membrane effect is significantly larger than the shear effect, causing the N term to TF (Yg) be significantly higher then V TF (Yg). As a result, the resultant load, R TF (Yg), tends to follow the direction of the axial load, with negative resultant before the loading point, and positive resultant after. Global Y Force, RTF(Yg) (N) Applied Load (N) FIGURE 5.14 Load carried by top face sheet at loading point (73.25 mm). Figure 5.14 shows the load carried by the top face sheet just before the loading point (73.25 mm from the support). It should be noted that at the loading point, there are two factors that decide the load carrying method of the top face sheet. These two factors are: localized membrane effect that causes tension in the top face sheet, and membrane effect from global bending that causes compression in the top face sheet (Figure 5.15). The shear load carried by the top face sheet at the loading point becomes increasingly negative from 1, N to 6, N load step (linear range). This shows that the localized membrane effect is more significant than the global bending membrane effect in the 81

13 linear range. The negativity of the top face sheet s shear component seems to go down at the initiation of core yielding. This could be because the large deflection caused by the initiation of core yielding increases the significance of the membrane effect from global bending. This contributes to the compressive load on the top face sheet s cross section. This compressive load cancels off some of the localized membrane effect tension created on the top face sheet at the loading point. As a result, the top face sheet s resultant shear decreases in negativity. Localized Effect (Tension) Global Bending (Compression) FIGURE 5.15 Factors affecting the top face sheet shear distribution at the loading point Bottom Face Sheet Shear Distribution The bottom face sheet shear distributions for both analytical and numerical approach are presented in Figures 5.16 to The numerical model results do not show the localized effect at the support clearly because the large partitions in that region failed to show the rapid change of the R BF (Yg). R BF (Yg) is the global Y resultant load of the bottom face sheet. The localized effect at the support will be discussed more in later section. 82

14 Global Y Force, RBF(Yg) (N) FIGURE Bottom face sheet shear distribution for 1, N applied load. Global Y Force, RBF(Yg) (N) FIGURE 5.17 Bottom face sheet shear distribution for 2, N applied load. Global Y Force, RBF(Yg) (N) FIGURE 5.18 Bottom face sheet shear distribution for 3, N applied load. 83

15 Global Y Force, RBF(Yg) (N) FIGURE Bottom face sheet shear distribution for 4, N applied load. Global Y Force, RBF(Yg) (N) FIGURE Bottom face sheet shear distribution for 5, N applied load. Global Y Force, RBF(Yg) (N) FIGURE Bottom face sheet shear distribution for 6, N applied load. 84

16 Global Y Force, RBF(Yg) (N) FIGURE Bottom face sheet shear distribution for 6,6 N applied load. Global Y Force, RBF(Yg) (N) FIGURE Bottom face sheet shear distribution for 7, N applied load. The theoretical model basically shows the same shear distribution as the top face sheet, having constant shear starting from the support and vanish to zero shear at the loading point. The spike found near the loading point of the theoretical shear distribution is again due to the inability of the theoretical formulation to fully describe the boundary conditions of four-point bending. The numerical results are very different from the theoretical results as expected. Instead of showing a constant R BF (Yg) component along the region between the support and the loading point, the numerical results show an increasingly negative R BF (Yg) further away from the support. After BF (Yg) R component reaches a maximum 85

17 negative value, its magnitude decreases as the loading point is approached. This again is due to the geometric non-linearity effect. The shape change of the sandwich beam causes the structure to pick up load through global bending. As a result of the membrane effect, the bottom face sheet experiences tension in the axial direction. Further away from the support, the deformation and slope of the deformed structure increases. The bigger the slope of the structure, the more membrane forces affect the outcome of the R BF (Yg). The N component of the bottom face sheet contributed BF (Yg) greatly to the R BF (Yg) in the negative direction, causing an increasingly negative R BF (Yg). Closer to the loading point, the sandwich beam tends to decrease in slope and becomes nearly horizontal after the loading point. This decrease in angle reduces the membrane effect and thus a drop in negativity of RBF (Yg) near the loading point is observed. The observations and findings made from the bottom face sheet complement well with the arguments made for the top face sheet shear distribution for the global bending scenario. Bottom face sheet regions near to the support are expected to behave like the top face sheet regions near to the loading point. Localized foam core crushing is expected to cause significant geometric non-linearity in bottom face sheet near the support point. This geometric non-linearity will then affect the load carrying method of bottom face sheet near to the support. To verify this assumption, another numerical model was created. Smaller partitions were made near the support region in order to capture the anticipated rapid RBF (Yg) change due to the localized effect. Since only qualitative comparison is needed, only two load steps were run on this further partitioned model: 1, N (before core yielding) and 6,6 N (after core yielding). 86

18 The results for the bottom face sheet s shear distribution for these two load steps are shown in Figures 5.24 and 5.25 respectively. From Figures 5.24 and 5.25, the zigzag pattern that was observed near the loading point at the top face sheet is again observed at the region close to the support at bottom face sheet. This zigzag pattern is caused by similar localized membrane effect that was observed on the top face sheet. Local crushing of the foam core causes increased geometric non-linearity in the bottom face sheet. This causes the bottom face sheet to act like a string, picking up the load through the membrane action. The membrane component is much more significant than the shear component, hence the direction of R BF (Yg) changes according to the change of angle of the bottom face sheet. This finding complements the arguments made for the top face sheet s shear distribution near the localized load and further confirms the significance of geometric non-linearity on the thin members of the sandwich structure. Global Y Force, RBF(Yg) (N) FIGURE 5.24 Bottom face sheet shear distribution for 1, N applied load (refined partition). 87

19 Global Y Force, RBF(Yg) (N) FIGURE 5.25 Bottom face sheet shear distribution for 6,6 N applied load (refined partition) Core Shear Distribution The core shear distribution for the various load steps analyzed is shown in Figures 5.26 to The general trend of the numerical shear distribution is very similar to the distribution predicted by the analytical model. Both numerical and analytical models show constant shear along the region between the support and the loading point. Although the shape change effect is significant in the face sheets, this effect does not appreciably affect the global Y force carried by the core, R C(Yg). The relatively low Young s modulus of the core failed to create any significant load transfer in the core s axial direction during shape change. 88

20 Global Y Force, RC(Yg) (N) FIGURE 5.26 Core shear distribution for 1, N applied load. Global Y Force, RC(Yg) (N) FIGURE 5.27 Core shear distribution for 2, N applied load. Global Y Force, RC(Yg) (N) FIGURE 5.28 Core shear distribution for 3, N applied load. 89

21 Global Y Force, RC(Yg) (N) FIGURE 5.29 Core shear distribution for 4, N applied load. Global Y Force, RC(Yg) (N) FIGURE 5.3 Core shear distribution for 5, N applied load. Global Y Force, RC(Yg) (N) FIGURE 5.31 Core shear distribution for 6, N applied load. 9

22 Global Y Force, RC(Yg) (N) FIGURE 5.32 Core shear distribution for 6,6 N applied load. Global Y Force, RC(Yg) (N) FIGURE 5.33 Core shear distribution for 7, N applied load. A slight discrepancy between the numerical and analytical models can be observed at the loading and support points. Higher order theory predicts constant shear even at the loading and the support points whereas the numerical model predicted less shear load carried by the core near to those regions of the beam. This is because the face sheet, as a result of the localized membrane effect phenomena, picks up a significant amount of load. Therefore when geometric non-linearity is taken into account, the face sheets will pick up a significant amount of load, almost equal to the load carried by the core at the loading and support points in some cases. The numerical model with refined partitions was analyzed again and the shear 91

23 distributions for the two load steps considered (1, N and 6,6 N) are shown on Figures 5.34 and From the refined partitions near the support, the gradual shear change can be observed clearly near the support point for the numerical model. Global Y Force, RC(Yg) (N) FIGURE 5.34 Core shear distribution for 1, N applied load (refined partition). Global Y Force, RC(Yg) (N) FIGURE 5.35 Core shear distribution for 6,6 N applied load (refined partition). 5.2 Hydromat Test System The primary goal of this part of the research is to find out how the load is transferred to the face sheets when the core begins to yield, with geometric non-linearity 92

24 considered. A total of six load steps are analyzed: 17.2 kpa (2.5 psi), 34.5 kpa (5 psi), 51.7 kpa (7.5 psi), 68.9 kpa (1 psi), 86.2 kpa (12.5 psi), and 13.4 kpa (15 psi). Localized yielding in the core close to the edge support begins to occur at 51.7 kpa. However, as shown in chapter four, localized yielding does not cause significant differences in the overall deformation and shear distribution of the sandwich panel as a whole. Localized yielding only causes a change in the shear distribution near the localized region. Actual core yielding due to the applied distributed load occurs at loadings 68.9 kpa and above. However, only 86.2 kpa and 13.4 kpa load steps caused significant core plasticity in the sandwich panel. Core plasticity will be discussed more in the coming sections of this chapter. Figure 5.36 shows the sandwich panel span overview. Analysis and observations are made along several cross sections of the sandwich plate along the X-axis. Simply supported edges Effective Contact Area Support Loading begins (4.6 mm) (114.3 mm) Sandwich Panel Center (34.8 mm) X Z FIGURE 5.36 Panel span overview of a quarter sandwich panel (Top View). 93

25 5.2.1 Total Shear Distribution HTS load scenario is a three dimensional problem. The shear distribution in the sandwich panel varies according to the location of the region along the width and length of the panel. This complicates the visualization of the expected shear load distribution using simple free body diagrams. The total shear distribution between the numerical and analytical models will be compared. It is important to note that the higher order theory presented by Chintala [2] is only valid for a sandwich plate with a completely yielded core. This theory is not applicable to any of the HTS load cases in the numerical model, which have a non-yielded core at lower loads, or partially yielded core at higher loads. The classical sandwich plate theory is therefore used to compare and validate the numerically predicted shear distribution of the plate in the linear range. Comparison between the numerically determined shear distribution and the classical sandwich plate theory distribution was done at all load steps. Using the assumption that in the linear range the core carries the entire shear load, results from equation 2.14 are compared with the total resultant load in the global Y direction, R TOT (Yg), obtained numerically. Figures 5.37 to 5.39 show the total shear resultant comparisons between the numerical and theoretical models in the linear range. 94

26 Global Y Force, RTOT(Yg) (N) Distance from left support (mm) FIGURE 5.37 Total plate shear distribution comparison along X-axis at 17.2 kpa. Global Y Force, RTOT(Yg) (N) Distance from left support (mm) FIGURE 5.38 Total plate shear distribution comparison along X-axis at 34.5 kpa. Global Y Force, RTOT(Yg) (N) Distance from left support (mm) FIGURE 5.39 Total plate shear distribution comparison along X-axis at 51.7 kpa. 95

27 Figure 5.4 shows the percentage difference between the classical sandwich plate theory and numerical model along the X-axis for all considered load steps. The numerical and theoretical models show excellent agreement and the variations are less than 4% from the 17.2 kpa to 68.9 kpa load steps. The small discrepancy in the beginning is believed to be due to the corner lifting of the plate. When a plate is loaded, the four corners of the full plate tend to lift up, which is the main reason for using corner bolts in the actual experiment (Figure 3.5). models have picked up this plate behavior. The boundary conditions applied to the simply supported edges prevented the corner of the plate from lifting up. As a result, the nodes at the lifted region where simply supported boundary condition is applied have a negative resultant global Y element force. This has caused the total shear carried by the analyzed cross section to decrease. This effect was not detected by the classical sandwich plate theory and hence the discrepancy. Discrepancies at other locations along the plate span are believed to be caused by geometric non-linearity. The percentage error begins to grow significantly at 86.2 kpa and 13.4 kpa. Error increases in the non-linear range because classical sandwich plate theory is only applicable in the linear range of the plate behavior. This result demonstrates the necessity of the shear distribution data obtained from the numerical models. 96

28 Error (%) kpa 34.5 kpa 51.7 kpa 68.9 kpa 86.2 kpa 13.4 kpa Distance from left support (mm) FIGURE 5.4 Error percentage between the theoretical and numerical results for various load case along X-axis. Figures 5.41, 5.42, and 5.43 show the propagation of the yielded region with increasing load steps from 68.9 kpa to 13.4 kpa. Note that the scale of the plastic strain color bands has been manually set so that all three load steps have the same scale. The usage of a standardized scale allows better comparisons to be made between the three plastic strain contours. The core plastic strain propagation pattern is similar to the prediction by Mercado and Sikarskie [3]. Although core yielding initiation has occurred at 68.9 kpa, the yielded region is not large and does not affect the shear distribution significantly. This is the reason that the numerical and analytical shear distributions still agree very well at 68.9 kpa. The plastic yielding region expanded tremendously at 86.2 kpa and 13.4 kpa. The increase in the size of the plastic yielding region causes significant discrepancies between the numerical and the analytical results for the 86.2 kpa and 13.4 kpa load steps (Figure 5.4). 97

29 X Z FIGURE 5.41 Plastic strain contour in sandwich core at 68.9 kpa (Top view). X Z FIGURE 5.42 Plastic strain contour in sandwich core at 86.2 kpa (Top view). 98

30 X Z FIGURE 5.43 Plastic strain contour in sandwich core at 13.4 kpa (Top view) Core Shear Distribution Both classical sandwich plate theory and higher order theory assume that the core carries the entire shear load in the linear range. In order to investigate the validity of this assumption a ratio between the core global Y load, R C(Yg), and the total global Y load, R TOT (Yg) in the sandwich structure was examined. This shear ratio was calculated at the partitioned regions along the plate span. The calculated ratios for different load steps at different plate locations are shown on Figure The cross section at 19.5 mm from the left edge is selected to see the changes of the shear ratio with the progression of core yielding. Figure 5.45 depicts the shear ratio change at 19.5 mm plate span for different applied load steps. 99

31 The results show that at any location, the core takes up about 94% or higher shear load of the structure in the linear range. This confirms the validity of the classical assumptions that the core carries majority of the shear load. Geometric non-linearity in this case does not affect the load carrying method of the sandwich panel significantly because the deformation is small relative to the core thickness. The low modulus of elasticity prevents the axial load components of the core to contribute significantly to R TOT (Yg). RC(Yg)/ RTOT(Yg) kpa 34.5 kpa 51.7 kpa 68.9 kpa 86.2 kpa 13.4 kpa Distance from left support (mm) FIGURE 5.44 ly determined core shear ratios along X-axis at various load steps. 1.2 RC(Yg)/ RTOT(Yg) Applied Pressure (kpa) FIGURE 5.45 Core shear ratio at X = 19.5 mm for various load steps. 1

32 In Figure 5.45, it can be seen that the initiation of core yielding has caused the shear ratio of the core to drop. Figure 5.45 shows that shear ratio at X = 19.5 mm drops from more than 98% in the linear range to about 91% at 86.2 kpa and 72% at 13.4 kpa. This shows that once the core begins to have significant plasticity, there is a load transfer from the core to the face sheets. The face sheets carry a significant amount of shear load once core starts to yield. More details about the load carrying method of the face sheets will be presented in next sections Top Face Sheet Shear Distribution From the geometric non-linearity observations made from the beam model, we can expect the top face sheet of the plate to be in compression. When geometric nonlinearity comes into play, the resultant shear within the top face sheet turns out to be positive. RTF(Yg)/ RTOT(Yg) kpa 34.5 kpa 51.7 kpa 68.9 kpa 86.2 kpai 13.4 kpa Distance from left support (mm) FIGURE 5.46 ly determined top face sheet shear ratios along X- axis at various load steps. 11

33 In order to analyze the effect of material non-linearity on the top face sheet s shear distribution, the shear ratio between the top face sheet and the whole structure is analyzed. Figure 5.46 shows the shear ratio of the top face sheet along the X-axis plate span at various load steps. Since the top face sheet shear resultant, R TF (Yg), is in a direct opposite to the total shear resultant, R TOT (Yg), a negative ratio is obtained. The ratio becomes increasingly negative as the load increases. This increase is consistent with the sandwich beam shear distribution for four point bend test. At 13.4 kpa the shear ratio falls out of the pattern and shows a drastic drop. This could be due to the sudden increase in core plasticity that reduces the top face sheet s slope. In other words, due to large core plasticity, the center of the top face sheet becomes more flat than the previous load step. This argument is supported by the apparent sudden drop in the ratio negativity at the loaded region (about 76.2 mm plate span onwards). RTF(Yg)/ RTOT(Yg) Applied Pressure (kpa) FIGURE 5.47 Top face sheet and total shear ratio at X = 19.5 mm for various load steps. Figure 5.47 shows the top face sheet shear ratio change at 19.5 mm from the left edge for various load steps. As expected, the ratio becomes increasingly negative as 12

34 the load increases, showing a more apparent sign of membrane effect. The negativity of the ratio decreases at 13.4 kpa due to large core plasticity. In order to visualize the membrane effect in the top face sheet, it is useful to know the strain conditions in the top face sheet. The membrane effect in this two dimensional case is much more complicated because now membrane effects occur along both X and Z-axes. Figure 5.48 shows the schematic view of the resultant membrane effect on an element of the top face sheet. + = FIGURE Z-Component X-Component Resultant Resultant membrane effect on an element on the top face sheet. = It is also important to realize that components from each axis have their own deflection angle with respect to the original composition. This factor has caused the visualization of the membrane effect become much more difficult and confusing compared to the beam problem in four-point bending. Therefore, the 68.9 kpa load step was selected to be analyzed numerically, this time with no geometric non-linearity. The numerical model with and without geometric non-linearity can then be compared to find out the effect of geometric non-linearity. The 68.9 kpa load step was selected because at higher loads there is too much localized plasticity near the support when geometric nonlinearity is neglected. This reconfirms the conclusions made from the beam model, where 13

35 face sheets stiffen the structure by carrying additional load through localized membrane effect when geometric non-linearity is considered. Figure 5.49 shows the shear distribution for the top face sheet, with and without geometric non-linearity. The results shows that geometric non-linearity has played a significant effect in determining the actual resultant global Y load, R TF (Yg), on the top face sheet. With geometric non-linearity, the top face sheet is carrying a much higher load in the 1 to 2 mm plate span regions. That coincides with the region where the plate makes the highest deflection angle with respect to its original configuration. Global Y Force, RTF(Yg) (N) Without Geo With Geo Distance from left support (mm) FIGURE 5.49 Top face sheet shear comparison at 68.9 kpa for numerical models with and without geometric non-linearity Bottom Face Sheet Shear Distribution The bottom face sheet of the plate is in tension. Therefore with geometric nonlinearity, the resultant global Y force of bottom face sheet, R BF (Yg), becomes increasingly 14

36 in negative. Again the ratio between the shear at bottom face sheet and the overall shear resultant is analyzed. The ratio results are shown on Figure 5.5. RTF(Yg)/ RTOT(Yg) kpa 34.5 kpa 51.7 kpa 68.9 kpa 86.2 kpa 13.4 kpa Distance from left support (mm) FIGURE 5.5 ly determined bottom face sheet and total shear ratios along X-axis at various load steps. The shear ratio for bottom face sheet has a positive value. The ratio increases at locations closer to the center of the plate. This is because the bottom face sheet increases in tension as it moves closer to the center of the plate. However it is important to note that there is no deflection angle at the plate mid-plane (X = 34.8 mm) and therefore there is no membrane contribution to the global Y resultant of the bottom face sheet, RBF (Yg) at that location. Membrane effect becomes more significant as the applied load increases. A sudden increase in shear ratio for the 86.2 kpa and 13.4 kpa load steps is observed. This is mainly due to the initiation of core yielding that has caused the load transfer from the core to the face sheets. The bottom face sheet carries this additional load through membrane forces. The 13.4 kpa ratio line in Figure 5.5 shows that the ratio pattern is not affected by the flattening of the structure as it was in the top face sheet. This flattening 15

37 could be just a localized phenomenon, which occurs only in the top face sheet region, where the load is applied directly. The core used in this analysis is Airex 63.5, one that is qualified as a soft core. The core could have experienced a change of thickness near the top face sheet region and hence caused the flattening of the region..6 RTF(Yg)/ RTOT(Yg) Applied Pressure (kpa) FIGURE 5.51 Bottom face sheet shear ratio at X = 19.5 mm for various load steps. Further conclusions can be made from the shear ratio plots of the bottom face sheet at a fixed location for various load steps. This type of shear ratio change is shown in Figure The ratio shows a gradual increase throughout with increased changes for the 86.2 kpa and 13.4 kpa load steps. The load transfer to the face sheets due to core plasticity and the geometric non-linearity are the main causes of these increases in the shear ratio. 16

38 Global Y Force, RBF(Yg) (N) Without Geo With Geo Distance from left support (mm) FIGURE 5.52 Bottom face sheet shear comparison at 68.9 kpa for numerical models with and without geometric non-linearity. To look at the membrane effect, again comparison between the numerical models with and without geometric non-linearity is made. This comparison is shown on Figure The resultant global Y load for the bottom face sheet, R BF (Yg), is generally in the negative region. With geometric non-linearity considered, the R BF (Yg) obtained is much lower than the one without geometric non-linearity. This shows the significance of the membrane effect in determining the load carrying method of the structure. Geometric non-linearity could result in the bottom face sheet carrying about more than 3% of the total shear load in the non-linear range. 17