A Cantilever Beam Problem with Small Deflections and Perturbed Boundary Data

Abstract

The boundary value problem of a fourth-order beam equation is investigated. We formulate a nonclassical cantilever beam problem with perturbed ends. By determining appropriate values of and estimates for perturbation measurements on the boundary data, we establish an existence theorem for the problem under integral boundary conditions where and is continuous on

1. Introduction and Preliminaries

Beams are one of the main structural elements in construction engineering. One of the objectives of the beam theory is to study the behavior of beams to analyze deformations under loads. The deformation of beams occurs when the beam is under a load, which causes the beam to develop bending moment and shear force. The deformation of the beam is modeled by the fourth-order Euler–Bernoulli equation where represents the deflection of the beam, represents the slope, is the bending moment (torque), is the shear force, and is the load density stiffness. The function is a load on the beam; it is uniformly distributed if the loading is only the weight of the beam, without any further concentrated mass at the free end. The boundary conditions are governed by the particular type of beams under study and the way in which the beam is supported. The most important type of beams that have many useful applications in industry is the cantilever beam, where the beam is fixed (clamped, anchored to a support, or built into a wall) at one end (say at ) and free at the other end (say ). The fixed end must have zero transitional and zero rotational motions, whereas the free end must have zero bending and zero shearing force. The bending moment is equal to the applied force multiplied by the distance to the point of application; so, it becomes zero at the free end because there is no stress at this end. So, we must have the following boundary conditions

The cantilever beams are widely used in construction engineering and can be found in many structures such as buildings and bridges. Some of the common examples of cantilever beams are aircraft wings, cranes, suspended bridges, balconies, diving board, electronic spring connectors, shelves, basketball backboard, road signs, and many other examples. Because of its importance, the cantilever beam problem received a wide and considerable attention from researchers, and a large number of research about cantilever can be found from a quick literature search [see and the references therein]. In particular, in [1–3], the authors investigated the existence of deflections (solutions) for the case under conditions (2). In [4–6], the case was investigated under conditions (2). In [7–11], the fully fourth-order nonlinear boundary value problem

was investigated under conditions (2). The authors in [8, 9] assumed a Nagumo-type condition on , and [10] assumed a Caratheodory condition on

In all the aforementioned research, the cantilever problem was investigated under the classical condition (2). Some special types of relaxed, restrained, or propped cantilever beams violate these conditions. If the fixed end is loose, the support (or the anchor) is relaxed then or/and If the free end is rolled or pinned support, then . If a concentrated force, or translational elastic spring is attached at the free end, then and while if a concentrated moment, or rotational elastic spring is attached, then If a concentrated mass is placed at the free end, it will develop a shear force of the form , and when this reduces the condition to the classical one.

The present paper deals with the fully fourth-order nonlinear boundary value problem where is continuous on under a perturbed homogeneous conditions, where the two ends are perturbed (the fixed end is slightly relaxed, and the free end is slightly supported). This can be formulated using integral conditions of the form where . Here, satisfies a growth condition with the variable parameters: where and are positive continuous functions on Moreover, letting then we have the following assumptions

Letting where and

Then, we impose the following condition

Condition (13) provides small deflections. The parameter represents the reciprocal of the flexural rigidity which measures the resistance to bend; so, smaller values of indicate large flexural rigidity for the material of the beam, and this causes small deflections when load is applied on the beam. Hook’s law is valid as long as the deflection is small. Problems with large deflections cannot be solved in terms of the linear beam theory—in which Euler-Bernoulli Equation applies, since Hook’s law is no longer valid. When the beam is assumed to be homogeneous, made of high rigid material, and behaves in a linear elastic manner, its deflection under bending is usually small. When thin flexible beams are used, large deflections are expected to occur. Many researchers investigated cantilever beams with large deflections [12–26], and others investigated small deflections [27, 28]. Most of the beams used in industry and constructions (buildings, bridges, aircrafts, etc.) undergo small deflections [29]. This is of utmost importance since large deflections can cause cracks in the beams, and this may eventually lead to disastarous damages. So, best efforts are made to limit deflections in the ceilings and walls and in the design of aircraft, see [29] for more details on beam theory. Small deflections usually occur when either the loaded force is small (hence, is small) or the material of the beam has high flexural rigidity, which implies that is small.

Condition (10) provides small values for to produce the required slight changes to the default boundary settings of a cantilever beam. The functions represent perturbation measurements related to bearings, rollers, springs, or any other mechanical settings that will perturb the boundary data.

Condition (8) is a growth condition imposed on the load function This condition generalizes the boundedness condition and the growth condition with constant parameters for some positive constants and

The integral boundary conditions (7) results from perturbing conditions (2) by imposing small measurements related. Note that if the conditions in (7) reduce to (2). The integral boundary conditions have been studied and applied extensively in beam theory by many authors, for example, see [30–41].

The purpose of this paper is to prove the existence of solutions (small deflections) of the cantilever beam boundary value problem (6) under the conditions (7)-(13). This interesting problem has not been studied in research. The result gives an affirmative answer to the question of existence of solutions of (6)-(13), i.e., the existence of small deflections of general types of perturbed cantilever beams under conditions (7)-(13), including the simple cantilever beam problem when

2. Existence and Uniqueness Theorems

The problem (1.6)-(7) can be converted into the following system:

We need the following lemma.

Lemma 1. If with or where then provided

Proof. For we note that Since we have Hence, Using the Cauchy-Schwarz inequality, we obtain Consequently, The proof is complete. Similarly, for we also note that Then, the argument is similar to the proof of the above.☐

Proposition 2. If (8)-(13) hold, then there exists a constant such that for any and any solution to Eq.(6), we have

Proof. Multiplying both sides of the first equation of (15) by where and integrating the resulting equation from 0 to 1, then employing integration by parts with and we obtain Taking into account we have The integrals and can be estimated by means of the Cauchy-Schwarz inequality Thus, Applying now Lemma 1 to the functions and appearing in the right-hand side of this inequality with and respectively, we obtain It follows that Consequently, provided ; that is,
It follows that where Proceeding as before, multiplying both sides of the second equation of (15) by where and integrating the resulting equation from 0 to 1, then employing integration by parts, taking into account and the nonlocal boundary condition we obtain Applying the growth condition (8) to by assuming that with we obtain The integrals appearing in the right-hand side of this inequality can be estimated by means of the inequality: Thus, Applying Lemma 1 to the functions and we obtain From (31), the inequality (37) becomes Choosing , and using the hypothesis (10) with we obtain where Note that This gives Let We see that Thus, and consequently from (31) and (42), we obtain On the other hand, we have Thus, Hence, Similarly, These two inequalities imply the required result and complete the proof of the proposition.☐

The fundamental theorem used in proving the existence of the solution is Schauder’s fixed theorem. In order to make use of this theorem, it is sufficient to present the following lemmas.

Lemma 3. Let be a continuous function. The unique solution of the following initial value problem subject to the nonlocal boundary conditions is given by where is the Green function given by

Proof. Integrating this equation twice, we obtain where are constants of integration. Integrations by parts of the integral with respect to in this equation give We determine and from It follows that Multiplying both sides of this equation by and integrating the resulting from 0 to 1, we obtain Thus, The proof is complete.☐

We also have the following lemma.

Lemma 4. Let be a continuous function. The unique solution of the following initial value problem subjects to the nonlocal boundary conditions is given by where is the Green function given by Thus, problem (15) is equivalent to the following system of integral equations Define the Banach space with norm where Also, define the operator by where

Lemma 5. Under the hypothesis of Proposition 2, there exists such that

Proof. Since and , thus, Using we get For we have Applying the growth condition, and employing the following inequalities that used in the proof of Proposition 2:
, and we obtain Hence, Consider now the closed and convex set ☐

Based on the above results, we have

Lemma 6. For any is contained in

Proof. From Lemma 5, we have and Since we obtain This shows that is contained in
To prove that is compact, we use the Arzela-Ascoli lemma; that is, must be closed, bounded, and equicontinuous. Consequently, has a fixed point by the Schauder’s fixed point theorem.☐

Thus, we have the following theorem.

Theorem 7. There exists a continuous solution which satisfies problem (15) with (8)-(13).

3. Conclusion

The cantilever beam problem modeled by the nonlinear fourth-order equation (6) is investigated under the integral conditions (7) and assuming the conditions (8)-(13). As illustrated in Section 2, the proposed conditions stand for perturbed conditions at the boundary, which occurs if the cantilever beam is not perfectly cantilevered in the sense that the free end is rolled and/or the fixed end is loose. The integral condition generalizes the standard boundary conditions that are usually proposed in literature for a classical cantilever beam problem. Moreover, these boundary conditions are more practical, in the sense that they represent the actual conditions that may arise in a real-world cantilever beam, which is very useful to researchers in construction engineering. The objective of this research is to determine whether small deflections occur in the cantilever beam under perturbed boundary data. The result shows that the solution to the problem exists, which implies that small deflections continue to exist on the beam whether it is perfectly cantilevered or not.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors would like to acknowledge the support of Prince Sultan University, Saudi Arabia, for paying the article processing charges (APC) of this publication. The authors would like to thank Prince Sultan University for their support.