868 fully mature hybrid maize stem specimens (16 hybrids, 2 replicates, ~27 samples per plot on average) were subjected to: (1) in vivo bending tests to failure [17, 28], (2) morphological measurements of internode lengths, and (3) rind penetration tests to measure the cross-sectional morphology at each internode [11, 29, 30]. The morphological data were analyzed and used to predict the failure location of the stalks. The actual failure locations were then compared to the predicted failure locations.
The 16 hybrids were developed by crossing B73 or Mo17 with a set of 8 genetically diverse inbred lines. These hybrids were developed to produce a broad range and stalk bending strength and flexural stiffness phenotypes. The hybrids were planted at Clemson University Calhoun Field Laboratory during summer 2019. The soil type in Clemson-CFL is Toccoa soil sandy loam to fine sandy loam. The hybrids were grown in a Randomized Complete Block Design with two replications. In each replication, each hybrid was planted in two-row plots with row length of 4.57 m and row-to-row distance of 0.76 m with a targeted planting density of 70,000 plant ha−1. The experiment was surrounded by non-experimental maize hybrids on all four sides to prevent any edge effects. To supplement nutrients, 56.7 kg ha−1 nitrogen, 86.2 kg ha−1 of phosphorus and 108.9 kg ha−1 potassium were added at the time of soil preparation, and additional 85 kg ha−1 nitrogen was applied 30 days after emergence. Standard agronomic practices were followed for crop management.
Bending Tests & Sample Preparation
Stalks were submitted to in vivo bending tests to failure using a DARLING device  at physiological maturity (38 to 42 days after anthesis). The DARLING device has been shown to produce the same failure patterns as naturally lodged stalks  and is highly predictive of the lodging propensity of different varieties . Only healthy looking and intact plants were tested during our study. The plants that are diseased or malformed (goose-necked, barren, mechanically damaged, etc.) were excluded from the study. Prior to testing, the tassel, leaf blades, and ear were removed, and the stalk was cut just above the ear bidding node. These steps were taken as our lab has found them to be “best practices”. In particular these preparatory steps mitigate oscillations in the stalk during testing and prevent interactions with neighboring plants. Oscillations and interactions with neighboring plants are two primary sources of experimental error in bending strength measurements. After preparing each stalk as outlined above the plant was loaded at the ear bidding node and deflected until failure.
After the in vivo bending test with the DARLING device, the stalks were cut at ground level and transferred to a greenhouse for drying. The greenhouse was maintained at ~38 degrees Celsius (100 degrees Fahrenheit) for 45 days at 35% RH to facilitate slow drying of the stalks and prevent the hollowing of the pith due to rapid drying. Stalks were dried to enable storage of samples without the tissues molding or rotting. Stalks were then stored in an air conditioned laboratory space until morphological measurements could be obtained.
Internode lengths of each specimen were measured with a ruler. The location of stalk failure was also measured with a ruler. Other morphology measurements were taken at the midspan of each internode of every specimen. In particular, caliper measurements were used to obtain the minor and major diameters of each internode. Rind penetration tests were used to obtain the rind thickness of each internode and to calculate the Integrated Puncture Score [11, 29, 30]. Rind penetration tests were performed using an Instron universal testing machine. In particular, a probe was forced through the specimen at a rate of 25 mm/s, and the resulting force-displacement curve was analyzed using a custom MATLAB algorithm to calculate the rind thickness (t) of the stalk cross-section. The probe was 2 mm in diameter with 45 degree chamfer on its end such that the diameter at the tip of the probe was 1 mm. Additional details on the rind penetration test protocol are documented in [11, 29, 30].
Predicting Failure Location
The failure location of each stalk was predicted using Equation 1. In particular, this equation was used to calculate the bending stress at each internode. The distance x was calculated by measuring the distance from the bottom of each internode of interest to the ear-bidding node, which is where the stalk was loaded during the in vivo bending test. The section modulus S was calculated by approximating the stalk cross section as a hollow ellipse. This method assumes that the rind is the primary load-bearing structure of the stem, and has previously been shown to be an accurate assumption [6, 7, 11]. The section modulus was therefore calculated using the minor diameter (d), major diameter (D), and rind thickness (t) .
Due to the destructive nature of in vivo bending tests, direct measurements of some morphological characteristics were not obtainable at every internode. However, given the relatedness among morphological characteristics of the stalk we were able to infill these missing data via imputation techniques. In particular, we make use of the factorial analysis for mixed data (FAMD) approach  implemented by the imputeFAMD in the missMDA package in R [33–37]. In our implementation of this technique, we first aggregated all of the morphological characteristics (major diameter, minor diameter, and rind thickness) and integrated puncture score into a single data set; i.e., for each stalk we have a total of 4 features at each internode. We then imputed the missing values using the FAMD approach, with predictions being made based on 6 principle components; for further discussion see .
Failure Location Prediction Model
Once the imputation process was complete, internode specific stresses for the first six internodes were computed per Equation 1 and Equation 2. These stress values were max-normalized within each stalk. Proceeding in this fashion allows us to make comparisons across stalks; i.e., the feature under study herein is the proportion of the maximum stress that each internode experienced. The max normalized stress value was then paired with the failure status of the internodes, and a logistic regression model was fit, entering each of the max normalized stress predictors as a first order variable.
In analyzing these stems, we can also pose the question: how much stronger could the stem be without increasing its total structural mass (i.e., how much stronger could the stem be if the structural biomass were more optimally distributed along the length of the stalk?). Said another way: what is the theoretical performance improvement of these stalks if one were to actively breed for uniformly distributed internodal stresses?
As can be observed in Equation 1, as we move down the stem from the canopy to the ground, a larger external moment (Fx) is applied to each subsequent internode. Without any morphological differences in internodes, this would result in each basal internodes being more highly stressed than apical internodes, and failure would almost always occur at the bottom most internode. However, in practice, the cross-section of each stem increases as we move from the canopy to the ground to counteract the increased bending load. However, the change in cross-section (i.e. section modulus, S) is not always exactly proportional to the change in loading, and thus we see that some internodes are more highly stressed than others. Thus, there exists a theoretically optimized stem design, in which the section modulus and external loading vary at the same rate down the stem, resulting in every internode experiencing the same maximum stress. This theoretical stem morphology is considered optimized, meaning that reduction in structural biomass anywhere on the stem will result in a higher maximum stress.
To investigate this, a custom optimization algorithm was developed in Matlab (MATLAB R2019a). For each stem, an iterative procedure was performed to re-distribute the structural mass between internodes until all the internodes were equally stressed. Figure 2 depicts a block diagram of the code, and the pseudo code is as follows:
1. Calculate the stresses of all internodes
2. Reduce the volume of the least-stressed internode
a. Find the least stressed internode.
b. Decrease the major radius, minor radius, and rind thickness by 1%. This ensures that the ratios between cross-sectional morphological properties remain consistent.
c. Calculate the resulting change in the structural volume of the internode.
3. Increase the volume of the most-stressed internode
a. Find the least stressed internode.
b. Solve for the increase in major radius, minor radius, and rind thickness that (1) increases the volume of the internode by the amount reduced in Step #2, and (2) maintains the ratios between cross-sectional morphological properties.
4. Check to see if all internode stresses are within 5% of each other. If not, iterate beginning at Step #2