# Elastica of Non-Prismatic and Nonlinear Elastic Cantilever Beams under Combined Loading
## 1 Introduction  
Elastica problems are frequently encountered in many areas of modern structural engineering. The first study on elastica was published by [PERSON] [1], who studied the large deformed shapes of a slender rod. The elastica problem of geometric and material nonlinearity has been discussed by many researchers in recent years. Additionally, non-prismatic beams are commonly used in engineering practice for their economic, aesthetic, and optimization benefits. A number of studies have been published on these topics. For example, the elastica behavior of non-prismatic cantilever beams fabricated from nonlinear elastic materials, such as Ludwick-type materials, under combined loading has been studied over the past few decades. Representative works in the same area as this study are listed and briefly discussed below. [PERSON] and [PERSON] [2] studied the elastica of buckled bars represented by moment-curvature relationships based on the hyperbolic tangent law. [PERSON] and [PERSON] [3] investigated elastica under tip point loading, where the beam material followed the Ramberg-Osgood stress-strain law. [PERSON] and [PERSON] [4, 5] computed the elastica of prismatic beams subjected to point loading and tip couple loading, respectively, for beams obeying [PERSON]'s constitutive law. [PERSON] and [PERSON] [6] computed the elastica of prismatic beams fabricated from Ramberg-Osgood-type materials under tip couple loading. [PERSON] and [PERSON] [7] conducted considerable research on the elastica of prismatic and non-prismatic flexible beams using the method of equivalent systems. [PERSON] [8] studied the elastica of prismatic beams obeying [PERSON]'s constitutive law that were subjected to combined loading consisting of uniform loading and tip point loading. [PERSON] [9] computed the elastica of rectangularcantilever beams fabricated from Ludwick-type materials under combined loading based on different arc length assumptions. [PERSON] et al. [10] investigated the elastica of non-prismatic beams under tip couple loading that obeyed the generalized [PERSON]'s constitutive law. [PERSON] et al. [11] studied the large deflection of a nonlinear, elastic, asymmetric, Ludwick cantilever beam subjected to point loading at the free end. [PERSON] et al. [12] investigated the large deflection of curved elastic beams fabricated from Ludwick-type materials subjected to uniform loading and concentrated vertical loading at the free end. [PERSON] et al. [13] studied a close-form solution for the nonlinear deflection of non-straight, Ludwick-type beams subjected to point loading at the free end using Lie symmetry groups. [PERSON] and [PERSON] [14] computed the elastica of non-prismatic cantilever columns fabricated from Ludwick-type materials that were subjected to axial point loading. In all the works discussed above, uniform loading, tip point loading, and tip couple loading were not examined simultaneously in a combined loading system. The elastica problem considered in the present paper does not absolutely satisfy the superposition principle. Therefore, it is important to include as many individual loading cases as possible in a combined loading system because the net response to multiple stimuli is not equal to the sum of the responses to individual stimuli. This study aimed to examine the elastica of a cantilever beam subjected to a combination of three kinds of loads, namely uniform loading, tip point loading, and tip couple loading, simultaneously. The loading system used in this study can generate a total of seven loading cases that have not been covered in the literature. The cross-section of the non-prismatic beam is solid and rectangular with a width and height that vary linearly along the beam axis. The nonlinear beam material obeys [PERSON]'s constitutive law. For the analyses of elastica in this study, the following assumptions were made. The neutral axis is inextensible, the effects of Poisson's ratio and transverse shear deformation are negligible, and the strain caused by bending is small.