This is a preprint. Preprints are preliminary reports that have not undergone peer review. They should not be considered conclusive, used to inform clinical practice, or referenced by the media as validated information.
Background Stalk lodging (breaking of agricultural plant stalks prior to harvest) is a multi-billion dollar a year problem. Stalk lodging occurs when bending moments induced by a combination of external loading (e.g. wind) and self-loading (e.g. the plant’s own weight) exceed the bending strength of plant stems. Previous biomechanical plant stem models have investigated both external loading and self-loading of plants, but have evaluated them as separate and independent phenomena. However, these two types of loading are highly interconnected and mutually dependent. The purpose of this paper is twofold: (1) to investigate the combined effect of external loads and plant weight on the displacement and stress state of plant stems / stalks, and (2) to provide a generalized framework for accounting for self-weight during mechanical phenotyping experiments used to predict stalk lodging resistance.
Results A method of properly accounting for the interconnected relationship between self-loading and external loading of plants stems is presented. The interconnected set of equations are used to produce user-friendly applications by presenting (1) simplified self-loading correction factors for a number of common external loading configurations of plants, and (2) a generalized Microsoft Excel framework that calculates the influence of self-loading on crop stems. The effect of self-loading on the structural integrity of wheat is examined in detail. A survey of several other plants is conducted and the influence of self-loading on their structural integrity is also presented.
Conclusions The self-loading of plants plays a potentially critical role on the structural integrity of plant stems. Equations and tools provided herein enable researchers to account for the plant’s weight when investigating the flexural rigidity and bending strength of plant stems.
Yield losses due to stalk lodging (breakage of crop stems or stalks prior to harvest) are estimated to range from 5–20% annually [1,2]. Despite a growing body of literature surrounding the topic of stalk lodging in wheat, barley, oats, and maize [3–7], a detailed study on the interconnected relationship between external loading (e.g. wind) and self-loading (e.g. plant weight) on stalk bending strength has not been reported. Previous biomechanical plant stem models have examined the influence of morphology, material, and weight on stem failure, while others have separately analyzed the effects of externally induced bending forces (e.g., wind) on stem failure [3,4,8–15]. However, the effects of self-weight and external loads on stem failure are inextricably connected. The bending moment induced from self-weight is a function of the distance between the plant’s base and its center of gravity. As external loads displace the center of gravity away from the base of the stem, the bending moment induced from self-weight increases. Although previous studies have independently looked at external loads [4,5,16] and self-weight loading , the authors are not aware of previous investigations into the interplay between these factors.
Calculating the total applied bending moment (MTOTAL), resulting from the combination of external bending loads (Mext) and bending loads induced from self-weight (Mint) is necessary to understand stem strength and stalk lodging resistance. Stalk lodging resistance is commonly estimated through the use of mechanical phenotyping equipment [e.g.,]. Such equipment typically applies an external load to plant stems and measures plant deflection. The externally applied moment (Mext) and plant displacement (𝛿) data are then analyzed to determine the flexural stiffness of the stem (EI) or the bending strength of the stem. For example see :
However, the effect of self weight on plant deflection is ignored or assumed to be negligible in these analyses. In some instances researchers have removed leaf blades, grain heads, and all plant biomass above the grain prior to mechanical testing to limit the effect of self-weighting. However, it is unclear if such measures are adequate. To more accurately characterize flexural stiffness and stalk bending strength the contribution of self weight to plant deflection and to the induced bending moment MTOTAL needs to be calculated.
The purpose of this paper is therefore twofold: (1) to investigate the combined effect of external loads and plant weight on the displacement and stress state of plant stems, and (2) to provide a generalized framework for accounting for self-weight during mechanical phenotyping experiments used to predict stalk lodging resistance.
Box 1: Glossary of Terms
Externally applied force
Geometric coefficient for applied moments
Geometric coefficient for applied forces
Location where loading is applied
Externally applied moment
Total induced moment from externally applied forces and moments
Total induced moment from self-loading
Total applied moment (sum of Mext and Mint)
Position where deflection is being calculated
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  and the theory of superposition of loading , 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 :
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: