Box 1: Glossary of Terms
Term
|
Definition
|
𝛿
|
Horizontal deflection
|
EI
|
Flexural stiffness
|
F
|
Externally applied force
|
fM
|
Geometric coefficient for applied moments
|
fF
|
Geometric coefficient for applied forces
|
h
|
Height
|
L
|
Location where loading is applied
|
M
|
Externally applied moment
|
Mext
|
Total induced moment from externally applied forces and moments
|
Mint
|
Total induced moment from self-loading
|
MTOTAL
|
Total applied moment (sum of Mext and Mint)
|
P
|
Position where deflection is being calculated
|
S
|
Section modulus
|
w
|
Weight
|
W
|
Weight-induced moment
|
σbending
|
Bending stress
|
Derivation of Closed Form Solution
To determine the contribution of self weight to the mechano-stability of plant stems, we must first derive a closed form solution for the internal bending moment of the stem (MTOTAL). Figure 1 depicts the free body diagram of a plant stem with an arbitrary loading applied at two locations. This stem depicts two weights (w) (e.g. stem weight, grain weight), as well as two externally applied loads (F) and two externally applied moments (M).
As the stem deflects, the moments induced from self-weights will increase as a function of the deflection of the stem. For the weight (w) at each location, we can calculate the induced moment from self-weight (W) as the product of the weight and the weight’s offset (i.e., the deflection of the stem at the location of the weight (𝛿)). Thus for the two locations shown in Figure 1, we have:
It should be noted that Equations 1, 2 and 6 assume that the maximum moment induced by self-loading is applied to the entire length of the stem. For more detail, see the Limitations section.
However, the offsets (𝛿1 and 𝛿2) in equations 2 and 3 are not known and are a function of the externally applied moments and forces. Using engineering theory for beam deflection [18] and the theory of superposition of loading [17], we can calculate the deflection of the stem at height h1 (i.e., location 1) as a function of the applied forces, applied moments, and weight-induced moments. Equation 4 shows this calculation, where the first row of equation 4 concerns loads, moments and weights at location 1 (i.e., at height h1) and the second row of equation 4 concerns forces, moments and weights at location 2 (i.e., at height h2).
Similarly, we can write the deflection of the stem at h2 as:
Thus we have four linearly independent equations (Equations 2 through 5) allowing us to solve for four unknown values (W1, W2, 𝛿1, 𝛿2).
Furthermore, Equations 2 through 5 can be generalized to account for any number of locations (n) along the length of the stalk. First, for any loading location L, at a height hL along the stalk, deflected by 𝛿L, Equations 2 and 3 can be generalized as:
Next, Equations 4 and 5 can be generalized by noting that each force, moment or weight (F, M, or W, shown in bold in Equations 4 and 5) is multiplied by a geometric coefficient. The geometric coefficient for each term is a function of the height where the deflection is measured and the height at which the loading is applied. This geometric coefficient can be denoted as either ƒF (for forces) or ƒM (for applied moments or weight-induced moments). As such, for any location P at a height of hP, the deflection 𝛿P is calculated by summing the product of each load, moment or weight (F, M, or W) and its corresponding geometric coefficient (ƒF or ƒM) at every loading location (from L=1 to L=n). Note that this geometric coefficient assumes a constant flexural stiffness (EI), as discussed in the Limitations section. Thus the generalized form of Equations 4 and 5 can be written as:
Equation 7 can now be consolidated into a fully generalized form of:
Where the geometric coefficients for the forces and moments are defined as [18]:
Equations 6 through 9 can also be put into a generalized matrix form. From Equations 6 and 8 we see that for any number of weights at any number of locations (n), we will have 2n unknown values (𝛿1, 𝛿2, … 𝛿n, W1, W2, .. Wn), and 2n linearly independent equations. By rearranging these equations and converting them to matrix notation we can write:
Where the first matrix in the equation is a square matrix of size 2n x 2n, and the second and third matrices in the equation are column matrices of size 2n x 1. Within the square matrix, the top left and bottom right n x n submatrices (shown in green text) are identity matrices, the bottom left n x n submatrix (shown in blue text) is a diagonal matrix of the negative weights (-w), and the top right n x n submatrix (shown in orange text ) is the negative geometric coefficients of the weight-induced moments, as calculated by Equation 10. We can then solve this matrix equation to calculate the deflections and weight-induced moments:
We can now look at the total moment (MTOTAL) of any cross-section along the length of the stem. In particular, MTOTAL can be written as a function of hP and hL, by considering all of the loads that are applied to the stem above the cross-section of interest (i.e, for hL ≥ hP),
Finally, we can write the bending stress of the stem in terms of the internal moment and the section modulus of the cross-section (S(hP)):
Finite Element Modeling and Data Triangulation
As a form of data triangulation [18] to confirm the closed form solution presented above, a series of 768 non-linear finite element models of plant stems were developed. The stems were modeled as 2-noded linear beam elements, fixed at their base. The models were developed in Abaqus/CAE 2019 [19,20] and analyzed in Abaqus/Standard 2019 using a direct, full Newton solver [19,20]. Model development and post-processing were automated through a series of custom Python scripts. Stems were modeled with a weight at height h1, applied force at height h2, and moment at height h3. A full parametric sweep of elastic moduli, moment of inertia, heights, moments, weights, and forces were performed. Table 1 describes the parameter space for the models. Values of input parameters to the models were based on previous studies of plant stem material properties [21, 22]. The applied moment (Mtotal) and the deflection of each finite element model was then compared to the deflection and applied moment calculated using the closed form solution methods presented above.
Table 1: The parameter space of the finite element analyses. Models were developed for the minimum and maximum values (n=2), and evaluated every 2N of applied load from 0N to 10N (n=6). A total of 768 finite element models were evaluated.
Value
|
E (N/mm2)
|
I (mm4)
|
h1 (mm)
|
h2 (mm)
|
h3 (mm)
|
M (Nmm)
|
W (N)
|
F (N)
|
Minimum
|
1.00E+03
|
1.00E+04
|
800
|
400
|
100
|
0
|
0
|
0
|
Maximum
|
1.00E+08
|
1.00E+05
|
1200
|
700
|
300
|
2000
|
2
|
10
|
n =
|
2
|
2
|
2
|
2
|
2
|
2
|
2
|
6
|
Analyzing the Effect of Self-Loading on Wheat
The closed form solution method was applied to vertically-partitioned wheat biomass data to determine the effects of self loading in wheat. Biomass data was collected from a commercially available wheat (Triticum aestivum) variety during the 2018 growing season in Saskatoon, Saskatchewan. As depicted in Figure 2 the biomass of wheat stems, leaves, and spikes were sampled and weighed every 10 cm along the length of the plants. Planting density was approximately 1.3 million plants per hectare with a 30.5 cm (12”) row spacing.
Biomass data were gathered from a 68 cm x 68 cm square of wheat in the center of a 122 cm wide plot. The 68 cm x 68 cm square contained an average of 366 stems. A total of five samples of biomass data were taken periodically from July 27 to August 29, 2018. The same plot was used for all sampling dates with enough space left between samples to have undisturbed wheat in each subsequent sample. A square guide was placed over the middle rows of the plot to indicate the sampling area and any plants outside of the guide were then removed. Biomass was harvested in 10 cm layers measured from the ground with the highest layer collected first (topmost layer varied in size depending on total plant height). All plant matter from a single layer was harvested, weighed in the field to obtain wet-basis biomass, and bagged to be dried later. The samples were oven dried at 65oC for a minimum of 48 hours to obtain the dry-basis biomass.