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Schematic representation of a continuum structure
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Redundancy in structures can provide load carrying capacity for the structure under multiple load cases, damaged conditions, or different configurations. Since these different load paths arise due to some changes in conditions, they are also referred to as alternate paths. The damage tolerance of a structure is a function of its ability to develop...
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Citations
... Although the Michell structures are the optimized lattice layouts that transfer the load from points of application to points of support (i.e., load paths), the relationship between load paths and optimized lattice layouts has not been established. Understanding the load flow within a structure provides valuable insight about the performance and efficiency of the structure, and it can serve as a tool to measure the structural functionality of a design [17][18][19][20][21][22][23]. However, the load flow and load paths show the existing structural functionality and are not necessarily optimized for a given structure. ...
... A square plate under an axial load (Fig. 1) is selected to test the observation of Eqs. (17)(18)(19)(20)(21). According to Eq. (21), the determinant of the stress tensor does not change during the optimization when only traction boundary conditions are applied. ...
The connection between topology optimization and load transfer is established in this work. The load transfer functions are used as an intermediate variable for topology optimization. This approach uses the total variation to minimize different objective functions such as the norm of the stress tensor and deviatoric principal stress subjected to equilibrium. To attain the topology of the microstructure, the principal load paths that follow the optimized principal stress directions are calculated. The principal vector field has singularities that are removed by an interpolation scheme. The optimal periodic microstructure is constructed using the load functions and the microstructures’ dimensions. The first advantage of this scheme is that using the load functions reduces the number of equilibrium constraints from two to one and reduces the number of variables from three stress tensor components to two load functions, leading to computational cost savings. The second advantage is that the nonlinear elliptic partial differential equations derived from the total variation equations are solved using the Gauss–Newton method, which has a quadratic convergence, speeding up the convergence toward the optimal structure. The third feature of the load-path-based optimization method is that the equilibrium and optimization problems are solved simultaneously.
... In this process, some regions within the structure bear more load than do other regions (Harasaki and Arora 2001). A load path may also be thought of as the trajectory or distribution of the applied load from the point of application to the supports (Venkataraman et al. 2009). This definition is reminiscent of topology optimization, through which the optimal distribution of material between the load and the supports that minimizes some performance measure, such as displacement (Bendsøe 1989;Bendsoe and Sigmund 2003), is sought. ...
Historic masonry monuments experience aging and accumulated damage, which consequently degrade both their stiffness and strength. Stiffness degradation can be quantified through nondestructive testing and has been widely used to determine the onset of structural damage by monitoring the changes in monuments' dynamic behavior. For justifying and configuring repair and rehabilitation activities, strength degradation (i.e., loss in load-carrying capacity) provides a more relevant indicator than does stiffness degradation. However, strength degradation is also significantly less feasible to assess because it requires the use of destructive testing. In this paper, the authors estimate strength degradation by establishing an empirical relationship between loss in structural redundancy due to damage and the corresponding loss in load-carrying capacity. Here, structural redundancy is represented as the ability of the structure to accommodate the applied load through alternative load paths. Loss in structural redundancy is approximated as the disturbance in the load path due to the development and progression of cracks. The proposed redundancy measure is demonstrated on a historic masonry coastal fortification considering differential settlement of supports.
... Optimization of the compliant mechanism uses binary variables to indicate the presence or absence of a load path. This method may have a limited range of applications due to the difficulties in presenting the load paths by connectivity for continuum structures [14]. In this paper, definition and formulations for load path functions are introduced and the theoretical and computational foundation of an efficient and robust load path algorithm based on the load path functions are established. ...
This paper proposes a load path function method for load path visualization and load flow calculation in plane elasticity problems. The load path functions are directly derived from 2D equilibrium equations in the absence of body forces. The stresses are written in terms of the load path functions and the load flow is calculated using the load path function contours. A set of Poisson's equations in terms of load path functions and stress components is developed and an efficient numerical procedure to solve them is discussed. Numerical results show the proposed load path function method is able to define the load paths and obtain the load flow using an Airy stress function for problems with available closed form solution, and also can be easily integrated into the numerical approaches to calculated load paths for complex problems.
With the increasing application of additive manufacturing (AM) in the industry, more attention has been paid to the performance of the additively manufactured part. How to maximize the structural efficiency of a part to achieve the desired performance has become the focus and challenge of design for additive manufacturing (DFAM). To give full play to the complex-geometry, multi-material, multi-scale, and multi-function manufacturing capabilities of AM, new approaches in DFAM have to be explored. The relationship between stress and growth in biology indicates that naturally evolved perfect structures are all optimal responses to their applied force. This work discusses a novel method to DFAM at the macroscale, mesoscale and microscale from a force-flow perspective, aiming to achieve an organic integration of force-flow based design with AM-driven manufacturing. Firstly, the characteristics of force-flow and its embodiment in nature and engineering are analyzed. Then, an overview of topology optimization, lattice/cellular structure design and infill pattern design are respectively provided in the combination of force-flow and AM. Finally, the future development directions of DFAM based on force-flow are proposed according to the limitations of current research.
The paper proposes a load function method for load path visualization and load flow calculation in plane elasticity problems. The load functions are directly derived from 2D equilibrium equations. The stresses are written in terms of the load functions and the load flow is calculated using the load function contours. The load functions are defined using generalized Beltrami representation. A constructive proof is given for the existence of load functions. A set of Poisson's equations in terms of load functions and stress components are developed and an efficient numerical procedure to solve them is discussed. Numerical results show the proposed load function method is able to define the load paths and obtain the load flow using an Airy stress function for problems with available closed form solution, and also can be easily integrated into the numerical approaches to calculate load paths for complex problems.
