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Macromechanics of a Lamina
Published in Manoj Kumar Buragohain, Composite Structures, 2017
Let us consider a lamina as shown in Figure 4.5a. It is generally orthotropic as the material coordinates and the global coordinates of the lamina are not aligned. Let 1-direction be at an angle θ w.r.t. the x-direction. The sign convention adopted is clockwise positive as per the right-hand rule. Stress components in the global coordinate system and the material coordinate system are shown in Figure 4.5b and c, respectively. We shall first arrive at a transformation of stresses in the global coordinates to those in the material coordinates. Let us then consider a triangular element as shown in Figure 4.5d such that sides AB and BC are normal to the x- and y-directions, respectively, and side AC is normal to the 1-direction. Also, angle ∠ABC is a right angle. Now, let the length of side AC be l such that the lengths of sides AB and BC are l cos θ and l sin θ, respectively.
Arches
Published in A.I. Rusakov, Fundamentals of Structural Mechanics, Dynamics, and Stability, 2020
Internal forces are introduced as the components of the shear force vector V ≡ Vy, axial force vector N ≡ Nz, and the complete moment vector M ≡ Mx. The CSs and positive directions of internal forces arisen due to action of the right part of the arch upon the left part are shown in Figure 4.5a and c. In complicated plane structures, the next sign convention is recommended to determine the shear force: shear force is positive if it rotates the object (the portion of the member) clockwise. The definition of shear force proposed for an arch corresponds to the sign convention.
Force-System Resultants and Equilibrium
Published in Richard C. Dorf, The Engineering Handbook, 2018
Pure bending refers to bending of a beam under a constant bending moment M, which means that the shear force V is zero (because V=dM/dx). Nonuniform bending refers to bending in the presence of shear forces, in which case the bending moment varies along the axis of the beam. The sign convention for bending moments is shown in Figure 5.2; note that positive bending moment produces tension in the lower part of the beam and compression in the upper part.
An Estimate on Distribution of Hysteretic Energy Demand in Seismic Precast Concrete Frame Structures
Published in Journal of Earthquake Engineering, 2021
Bin Du, Zheng He, Guohui Huang
Where, Lb is the span of beam AB; MA and MB are the bending moments at the ends of beam AB, respectively; MC and MD are the bending moments at connections C and D, respectively; θA and θB are the rotations at the left and right end, respectively; θC1 and θC2 are the rotations at the left and right side of connection C, respectively; θD1 and θD2 are the rotations at the left and right side of connection D, respectively; Δ is the relative vertical displacement between the two ends of beam AB; Δ1, Δ2 and Δ3 are the relative vertical displacements of segments AC, CD and DB, respectively, Δ = Δ1+ Δ2+ Δ3. Sign conventions are assigned for the variables associated with bending moments, rotations, deflections and relative displacements.
Force control for path following of a 4WS4WD vehicle by the integration of PSO and SMC
Published in Vehicle System Dynamics, 2018
In this work, a vehicle kinematic model incorporating wheel slip is proposed for the steering controller design. As presented in Figure 2, the vehicle kinematic model is simplified as a bicycle model with two virtual wheels located on the center line of vehicle body, namely front wheel and rear wheel with steering angles of and , respectively. As resultant wheel velocities, and deviate from the wheel centre plane by wheel slip angles and . Considering the vehicle as a rigid body, given an instantaneous centre of rotation (ICR), its projection point (CR) on the vehicle center line can be located as follows: where is positive and is negative as per the sign conventions. The turning radius between ICR and CR is denoted by .
Solutions for lined circular tunnels sequentially constructed in rheological rock subjected to non-hydrostatic initial stresses
Published in European Journal of Environmental and Civil Engineering, 2022
Huaning Wang, Fei Song, Tao Zhao, Mingjing Jiang
Due to the longitudinal advancement of tunnels, the calculation of mechanical response near the tunnel face is a three-dimensional (3 D) boundary-value problem (Alejano et al., 2010; 2012). There exists a longitudinal influence area ahead and behind of the tunnel face, due to its space restraint effect (Chu et al., 2019). The 3 D effect can be approximately taken into account in the plane-strain model by exerting fictitious surface (supporting) forces along the boundary of the tunnel cross-section (Carranza-Torres et al., 2013), as shown in Figure 3. The fictitious surface forces, (), progressively decrease over the increased distance to the tunnel face from () to zero when the tunnel face is at a distance at which it has no influence on the considered section. After the installation of supports, both the fictitious supporting forces from the tunnel face and the supporting forces due to the rock-liner interaction (named the interaction supporting forces in this article), acting on the periphery of the tunnel; with the advancement of tunnels, the fictitious supporting forces decreased to zero while the interaction supporting forces increased from zero to a constant value. The fictitious supporting forces can be expressed as follows (Carranza-Torres et al., 2013): where and are the normal and shear stresses along the tunnel boundary before excavation occurs; is a time-dependent relaxation factor calculated by tunnel convergence, which can be determined by the field measurements or numerical simulations (Chu et al., 2019; Panet, 1995; Paraskevopoulou, 2016; Unlu & Gercek, 2003). According to the aforementioned statement, the problem is cast as a two-dimensional (2 D) plane-strain problem with a lined circular tunnel in a viscoelastic infinite plane subjected to a non-hydrostatic stress, as shown in Figure 1a. Figure 1b,c present the boundary conditions of the rock mass and the liner, respectively. Polar coordinates are employed in the derivation. Sign conventions are defined as positive for tension and negative for compression.