# A SDoF Response Model for Elastic–Plastic Beams under Impact at Any Point on the Span

## Abstract

**:**

## 1. Introduction

## 2. SDoF Model for Low-Velocity Heavy-Mass Impact

_{0}and a low velocity V

_{0}at a point l

_{1}from the left-hand support, as shown in Figure 1. Without loss of generality, the dimension l

_{1}is taken as smaller than l

_{2}= l − l

_{1}. The contact width between the wedge and the beam is assumed to be negligibly small in comparison to the beam length. The beam is assumed to be made of elastic-perfectly plastic material, with the mass density ρ, Young’s modulus E, the yield stress σ

_{0}and Poisson’s ratio ν.

#### 2.1. Loading Stage

_{p}= σ

_{0}H

^{2}B/4 and N

_{p}= σ

_{0}HB are the fully plastic bending moment and fully plastic axial force, respectively, for the beam with a rectangular-shaped cross-section.

**Regime I**represents pure elastic deformations developed in the beam with the shape function w

_{1}(x) approximated as

^{3}/12 is the section moment of inertia and W is the displacement of the beam at the loading location x = 0.

_{0}< M

_{N}, M

_{1}< M

_{N}and M

_{2}< M

_{N}, where M

_{N}is the bending moment at a plastic hinge related to axial force N:

_{1}> M

_{0}> M

_{2}for l

_{1}< l

_{2}, which suggests that the initiation of plastic yielding occurs at the left support x = −l

_{1}. Thus, the terminating condition for Regime I is M

_{1}= M

_{T}at the inception of yield t = t

_{1y}.

_{1y}, the critical displacement W

_{1y}, obtained by substituting Equations (6) and (7) into Equation (4), and the instantaneous velocity $\dot{W}\left({t}_{1y}\right)$ are used as the initial condition for Regime II.

**Regime II**represents the elastic–plastic deformations developed in the beam when the plastic hinge has formed at the left support x = −l

_{1}.

_{1}, l

_{2}and 0 can be calculated via Equations (13), (8) and (9), respectively. Then, when substituting the corresponding values of M and N into Equation (3), the beam’s resistance during this regime can be obtained as

_{0}< M

_{N}, M

_{1}= M

_{N}and M

_{2}< M

_{N}. This regime continues until a critical displacement W

_{2y}is attained to define the plasticity hinge formation at x = 0 for M

_{0}> M

_{2}. Thus, the terminating conditions for this regime are M

_{0}= M

_{N}at t = t

_{2y}. When the terminating conditions for Regime II have been reached, the response of the beam is about to enter the next regime, with the initial conditions, namely W = W

_{2y}and $\dot{W}=\dot{W}\left({t}_{2y}\right)$.

**Regime III**represents the elastic–plastic deformations developed in the beam when the plastic hinges have formed at the left support and the impact point, and plastic yielding is about to occur at the right point.

_{2}are obtained via Equations (6) and (9), respectively, while the bending moments M

_{0}and M

_{1}are obtained by Equation (13). Hence, the beam’s resistance during this regime can be formulated as

_{0}= M

_{N}, M

_{1}= M

_{N}and M

_{2}< M

_{N}. This regime continues until the plasticity hinge forms at the support x = l

_{2}, implying that the terminating condition for this regime is M

_{2}= M

_{N}at time t = t

_{3y}.

_{3y}when reaching the terminating condition of Regime III is attained from Equation (4) with the substitution of Equations (6) and (9). The axial force N = N

_{3y}at W = W

_{3y}is obtained by Equation (6), which is used as the initial value to determine the dependence of the axial force on the displacement of the beam in Regime IV.

**Regime IV**represents the elastic–plastic deformations developed in the beam when the plastic hinges have formed at the supports and the loading point.

_{3y}and to highlight the characteristic that plastic hinges have formed at the supports and the loading point in this regime, the deformation shape function of the beam in this regime is given as

_{3y}at W = W

_{3y}using Mathematica software version 11.3. The differential equation solution is valid from W = W

_{3y}until the establishment of a membrane state, i.e., N = N

_{p}at W = W

_{p}. Thus, the bending moment for a specific displacement W can be given via Equation (13). Consequently, from Equation (3), the beam’s resistance during this regime is determined as

_{3y}, the transverse velocity field is obtained by taking the derivative of w

_{2}(x) defined in Equation (16) with respect to time:

_{3y}and $\dot{W}\left({t}_{3y}\right)$ at t = t

_{3y}. The general conditions for Regime IV are M

_{0}= M

_{1}= M

_{2}= M

_{N}and N < N

_{p}. The terminating conditions for this regime are N = N

_{p}and M = 0 at time t = t

_{p}.

**Regime V**represents that full plastic deformation develops along the beam and a membrane state is reached for impact with sufficiently large kinetic energy. Thus, substituting M

_{0}= M

_{1}= M

_{2}= 0 and N = N

_{p}into Equation (3) gives the following loading path:

_{p}and $\dot{W}\left({t}_{p}\right)$ at t = t

_{p}. At the end of the loading stage, say at t

_{m}, the displacement of the beam reaches its maximum value, and the velocity of the impact system decreases to zero, namely, $W\left({t}_{m}\right)={W}_{m}$ and $\dot{W}\left({t}_{m}\right)=0$, which are also the initial conditions of the unloading stage.

#### 2.2. Unloading Stage

_{1m}, M

_{0m}and M

_{2m}are the bending moments at x = −l

_{1}, 0 and l

_{2}at the peak point W = W

_{m}, respectively; N

_{m}is the axial force at the peak point W = W

_{m}; ΔM

_{1}, ΔM

_{0}and ΔM

_{2}are the change of the bending moments at x = −l

_{1}, 0 and l

_{2}during unloading; ΔN is the change of the axial force during unloading; and W

_{u}is the displacement of the beam at x = 0 during unloading.

_{m}and N

_{m}at the unloading point in Equation (27) can be determined from the theoretical analysis of the loading path, while the changes of the bending moment and the axial force are unknown, which is relevant to the change of the shape function of the beam during the unloading stage.

_{m}− W

_{u}.

_{m}(x) is the shape function at the maximum deflection W

_{m}, and w

_{u}(x) = w

_{m}(x) − Δw(x) is the shape function during unloading.

_{m}≤ W

_{3y}, the first term at the right-hand side of Equation (33), Δε

_{1}, can be determined by using the shape function w

_{1}(x) defined in Equation (5):

_{m}≥ W

_{3y}, Δε

_{1}can be determined by using the shape function w

_{2}(x) defined in Equation (16), expressed as

_{2}, only depends on Δw(x). Thus, Δε

_{2}can be determined when using Equation (28), expressed as

_{m}. Thus, the unloading path from any loading regimes can be determined from Equation (27) when using the values of F

_{m}, T

_{m}, ΔN and ΔM in the corresponding regime.

_{f}, the impact force applied to the beam decreases to zero, and the velocity and displacement at the impact point of the beam are given as

_{r}is the absolute value of the rebound velocity of the striker, and the negative sign denotes its direction being opposite to that of the initial velocity V

_{0}; W

_{f}is defined as the final displacement at the end of the restitution phase.

## 3. Finite Element Model

^{3}is made of an elastic-perfectly plastic material with mass density ρ = 7850 kg/m

^{3}, yield strength σ

_{0}= 300 MPa, Young’s modulus E = 206 GPa and Poisson ratio v = 0.3. The beam is modelled as deformable bodies using C3D8R, an eight-node linear brick element with reduced integration. In order to model a clamped boundary condition, constraints limiting all degrees of freedom are assigned to both ends of the beam.

_{0}= 30 kg and velocity V

_{0}= 2.0 m/s are assigned to the reference point to ensure a relatively larger ratio of m

_{0}/${m}_{b}^{e}$ and to weaken the influence of the necking phenomenon [19]. The motion of the striker is allowed only in the horizontal direction to transversely impact the beam at a point l

_{1}from the left-hand support. To analyse the influence of the impact location on the overall dynamic behavior of the beam, numerical simulations are performed with four impact locations, i.e., l

_{1}/l = 1/8, 1/4, 3/8 and 1/2.

_{1}/l = 1/2. The mesh sizes of 1 mm, 1.5 mm, 2 mm and 4 mm are tested. The dependence of the characteristic response parameters, namely, the spring-back (i.e., W

_{m}− W

_{f}) and maximum contact force on the mesh size, are plotted in Figure 4. To balance the simulation accuracy and the computational cost, the mesh size is selected as 1.5 mm in the numerical study.

## 4. Validation of the Proposed SDoF Model

#### 4.1. Overall Dynamic Response Histories

_{r}with an opposite direction due to the elastic strain energy stored in the beam. To study the influence of the impact location on the variation of velocity with time, a parameter a

_{v}= (V

_{0}+ V

_{r})/t

_{f}, which represents the average rate of change of velocity over the response time, is introduced here. It is found from Figure 5a that with the increasing l

_{1}/l, a

_{v}becomes smaller and so does its changing rate.

_{1}/l. To illustrate this phenomenon, a fully plastic beam with T = T

_{p}is taken as an example. It is revealed from Equation (26) that as l

_{1}approaches l

_{2}, the slope of the loading path decreases and gets its minimum in the central impact case with l

_{1}= l

_{2}. Likewise, the variation of the slope of the unloading path with l

_{1}/l shows the same trend as that of the loading path, which can be explained from the perspective of the elastic strain energy. With the approximation of the unloading path as a linear function of ΔW, the slope during unloading can be estimated as

_{e}is the elastic strain energy stored in the beam.

_{1}/l, the rebound velocity V

_{r}increases while the maximum impact force F

_{m}decreases, with detailed analysis referring to Section 4.2. Thus, it follows from Equation (38) that the slope of the unloading path K

_{u}decreases with an increasing l

_{1}/l.

#### 4.2. Characteristic Response Parameters of Beam under Impact at Any Point

_{1}/l, which ranges from zero to 1/2.

_{m}= W

_{m}/H and δ

_{f}= W

_{f}/H obtained from the theoretical and numerical predictions, are plotted against λ in Figure 6a. Satisfactory agreement is observed, although the proposed SDoF model predicts a slightly larger deflection with the increase of λ, owing to the neglect of the thinning effect of the beam with a sufficiently larger displacement, as analysed for the displacement-time history curve in Figure 5a. It is shown from Figure 6a that both the maximum and final displacements show an increasing trend with an increase of λ. Meanwhile, the spring-back ΔW = W

_{m}− W

_{f}corresponding to the gap between the δ

_{m}− λ and δ

_{f}− λ curves is seen to increase with an increase of λ and attains its maximum value in the central impact. To explain this phenomenon, a similar analysis to that on the slope of the unloading path can be conducted to approximate the spring-back ΔW, written as

_{u}with λ. Thus, the spring-back of the beam attains its maximum value in the central impact with l

_{1}= l

_{2}and decreases with the increase of l

_{1}/l.

_{1}/l, the maximum impact force decreases, which can be attributed to the decrease of the beam’s resistance. Specifically speaking, the dynamic loading path can be approximated as a linear model due to its characteristics, as illustrated in Figure 6b. Thus, the maximum impact force F

_{m}can be approximated by

_{k}= m

_{0}${V}_{0}^{2}$/2 is the impact kinetic energy of the striker.

_{1}/l, which implies a greater structural resistance mainly owing to the axial force when a relatively large deformation develops in the beam. Thus, it follows from Equation (41) that a larger l

_{1}/l inducing a smaller K will result in a smaller F

_{m}.

_{r}increases with an increasing λ induced by an increasing l

_{1}/l, which implies that the more symmetrical load allows for more elastic energy absorbed by the beam. It can be explained as follows. With the aid of Equation (39), a larger l

_{1}/l with a specific kinetic energy can be equivalent to a larger kinetic energy in a central impact case. Besides, it has been demonstrated that a larger impact energy results in a larger elastic strain energy, i.e., a larger rebound velocity [19]. Thus, the rebound velocity attains its maximum value in the central impact corresponding to the largest λ amongst the examined cases with different impact locations.

_{m}and t

_{r}, respectively, with λ are plotted in Figure 6d, which shows that t

_{m}and t

_{r}increase with an increasing λ. This characteristic can be explained from the point of view of momentum. At t = t

_{m}, the impact force reaches its maximum value, and the velocity of the striker decreases to zero. Thus, it follows from the principle of momentum that the initial momentum I

_{0}and the residual momentum I

_{f}of the impact system can be expressed as

_{m}, and the duration during unloading, t

_{r}= t

_{f}− t

_{m}, can be approximated from Equation (42), written as

_{m}and V

_{r}with λ shown in Figure 6b,c, respectively, it is revealed from Equation (43) that t

_{m}and t

_{r}both increase with an increase of λ, which results in an increase of the impact duration t

_{f}.

_{m}and t

_{r}both increase with an increase of λ, which results in an increase of the impact duration t

_{f}.

_{m}, W

_{f}, F

_{m}, COR, t

_{m}and t

_{r}. Table 1 presents the corresponding mean values and the standard deviations of the ratios obtained for the four different values of λ. It is notable that for all the analysed parameters, the mean values are quite close to 1, while the standard deviations approach zero. This demonstrates that the proposed analytical model, although with some simplifying assumptions, can offer a satisfactory approximation of the dynamic response of a beam on condition of a heavy mass with a low impact velocity, at least within the range of parameters examined (the largest value of λ is 4.9). Nevertheless, it should be mentioned that the necking phenomenon can become significant with further increase in the structural deflection due to a larger λ. Thus, one should be extremely cautious and careful when applying this model to the impact cases with comparatively larger initial kinetic energy.

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**A heavy mass m

_{0}with a low velocity V

_{0}striking a clamped beam at a point l

_{1}from the left-hand support.

**Figure 2.**Force and moment analyses on the fully clamped beam quasi-statically loaded at x = 0. M

_{1}, M

_{2}and M

_{0}: the bending moments at the two supports and the loading point of the beam, respectively.

**Figure 5.**Comparisons of response characters between the SDoF and numerical models. (

**a**) Velocity-time history of the striker. (

**b**) Impact force-displacement curve.

**Figure 6.**Dependence of characteristic parameters of an elastic–plastic beam on the impact location. COR = V

_{r}/V

_{0}is defined as the coefficient of restitution. (

**a**) Variation of δ

_{m}and δ

_{f}with λ. (

**b**) Variation of F

_{m}with λ. (

**c**) Variation of COR with λ. (

**d**) Variation of t

_{m}and t

_{r}with λ.

Parameter | Mean | Standard Deviation |
---|---|---|

W_{m} | 1.011 | 0.029 |

W_{f} | 1.005 | 0.044 |

F_{m} | 0.980 | 0.036 |

COR | 0.966 | 0.076 |

t_{m} | 1.029 | 0.021 |

t_{r} | 1.063 | 0.162 |

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**MDPI and ACS Style**

Shi, S.
A SDoF Response Model for Elastic–Plastic Beams under Impact at Any Point on the Span. *Appl. Sci.* **2023**, *13*, 9051.
https://doi.org/10.3390/app13169051

**AMA Style**

Shi S.
A SDoF Response Model for Elastic–Plastic Beams under Impact at Any Point on the Span. *Applied Sciences*. 2023; 13(16):9051.
https://doi.org/10.3390/app13169051

**Chicago/Turabian Style**

Shi, Shiyun.
2023. "A SDoF Response Model for Elastic–Plastic Beams under Impact at Any Point on the Span" *Applied Sciences* 13, no. 16: 9051.
https://doi.org/10.3390/app13169051