Local PWB and Component Bowing of an Assembly Subjected to a Bending Moment

1994 ◽  
Vol 116 (2) ◽  
pp. 92-97 ◽  
Author(s):  
D. B. Barker ◽  
Sidharth
Keyword(s):  
Cross Sections ◽  
Closed Form Solution ◽  
Bending Moment ◽  
Form Solution ◽  
Printed Wiring Board ◽  
Axial Forces ◽  
Beam Equations ◽  
Wiring Board ◽  
Component Assembly ◽  
Array Format

An analytical model is developed for determining the bowing of a component mounted on a printed wiring board (PWB) that is subjected to a bending moment. The model assumes a uniform elastic attach, between the component and the board. The elastic attach is assumed to transmit axial forces and restrain cross-sections of the component against rotation. The closed form solution to the beam equations directly determines the bowing of the component and the board. The solution is then used for computing the forces and moments, and hence, stresses in the leads that can occur in static or vibrational loading of a PWB/component assembly. The present analysis applies to electronic components with uniformly distributed leads in an array format, such as some PGA components, or to the class of components with parallel rows of leads such as a DIP or a SOIC. To demonstrate the solution and whether or not the rotational stiffness of the component leads needs to be considered, three different types of packages are analyzed.

Bending and Twisting of Pipes With Strain Hardening

1984 ◽  
Vol 106 (2) ◽  
pp. 188-195 ◽  
Author(s):  
J. H. Lau ◽  
T. T. Lau
Keyword(s):  
Effective Strain ◽  
Closed Form Solution ◽  
Bending Moment ◽  
Small Deformation ◽  
Form Solution ◽  
Simultaneous Action ◽  
Deformation Analysis ◽  
Engineering Practice ◽  
Strain Diagram ◽  
Interaction Curves

A closed-form solution is presented for the small deformation analysis of a straight thin-walled circular cylinder subjected to the simultaneous action of bending and twisting moments. Dimensionless interaction curves and charts which relate the variables, bending moment, curvature, maximum effective strain, twisting moment, and shear strain are also provided for engineering practice convenience. The average stress-strain diagram of the cylinder is described by two straight lines. The result presented herein is not only a good approximation of a wide class of piping materials, but also provides a standard tool for estimating the accuracy of different direct schemes such as numerical integration, finite-difference, and finite-element methods.

Comparison of Static and Dynamic Stiffness for Beams Undergoing Flexural and Torsional Loading With an Intermediate Mass

2000 ◽  
Author(s):  
Arnoldo Garcia ◽  
Arnold Lumsdaine ◽  
Ying X. Yao
Keyword(s):  
Closed Form ◽  
Natural Frequency ◽  
Cross Sections ◽  
Dynamic Stiffness ◽  
Closed Form Solution ◽  
Frequency Equation ◽  
Form Solution ◽  
Torsional Loading ◽  
Intermediate Mass ◽  
Form Equation

Abstract Many studies have been performed to analyze the natural frequency of beams undergoing both flexural and torsional loading. For example, Adam (1999) analyzed a beam with open cross-sections under forced vibration. Although the exact natural frequency equation is available in literature (Lumsdaine et al), to the authors’ knowledge, a beam with an intermediate mass and support has not been considered. The models are then compared with an approximate closed form solution for the natural frequency. The closed form equation is developed using energy methods. Results show that the closed form equation is within 2% percent when compared to the transcendental natural frequency equation.

scholarly journals Green's functions for unsymmetric composite laminates with inclusions

2020 ◽  
Vol 476 (2233) ◽  
pp. 20190437
Author(s):  
Chia-Wen Hsu ◽  
Chyanbin Hwu
Keyword(s):  
Composite Laminates ◽  
Green's Functions ◽  
Green’S Functions ◽  
Closed Form Solution ◽  
Bending Moment ◽  
Form Solution ◽  
Displacement Discontinuity ◽  
Computational Time ◽  
Out Of Plane ◽  
Logarithmic Functions

It is known that the stretching and bending deformations will be coupled together for the unsymmetric composite laminates under in-plane force and/or out-of-plane bending moment. Although Green's functions for unsymmetric composite laminates with elliptical elastic inclusions have been obtained by using Stroh-like formalism around 10 years ago, due to the ignoring of inconsistent rigid body movements of matrix and inclusion, the existing solution may lead to displacement discontinuity across the interface between matrix and inclusion. Due to the multi-valued characteristics of complex logarithmic functions appeared in Green's functions, special attention should be made on the proper selection of branch cuts of mapped variables. To solve these problems, in this study, the existing Green's functions are corrected and a simple way to correctly evaluate the mapped complex variable logarithmic functions is suggested. Moreover, to apply the obtained solutions to boundary element method, we also derive the explicit closed-form solution for Green's function of deflection. Since the continuity conditions along the interface have been satisfied in Green's functions, no meshes are required along the interface, which will save a lot of computational time and the results are much more accurate than any other numerical methods.

The Axially Loaded Circular Cylinder

1962 ◽  
Vol 29 (2) ◽  
pp. 318-320
Author(s):  
H. D. Conway
Keyword(s):  
Exact Solution ◽  
Fourier Transform ◽  
Circular Cylinder ◽  
Closed Form ◽  
Cross Sections ◽  
Closed Form Solution ◽  
Form Solution ◽  
Concentrated Force ◽  
Transform Method ◽  
Fourier Transform Method

Commencing with Kelvin’s closed-form solution to the problem of a concentrated force acting at a given point in an indefinitely extended solid, a Fourier transform method is used to obtain an exact solution for the case when the force acts along the axis of a circular cylinder. Numerical values are obtained for the maximum direct stress on cross sections at various distances from the force. These are then compared with the corresponding stresses from the solution for an infinitely long strip, and in both cases it is observed that the stresses are practically uniform on cross sections greater than a diameter or width from the point of application of the load.

Nonlinear Analysis of a Class of Inversion-Based Compliant Cross-Spring Pivots

2021 ◽  
Author(s):  
Shiyao Li ◽  
Guangbo Hao ◽  
Yingyue Chen ◽  
Jiaxiang Zhu ◽  
Giovanni Berselli
Keyword(s):  
Nonlinear Analysis ◽  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Geometric Parameters ◽  
Contributing Factors ◽  
Characteristic Analysis ◽  
Center Shift ◽  
Axial Forces ◽  
Constraint Model

Abstract This paper presents a nonlinear model of an inversion-based generalized cross-spring pivot (IG-CSP) using the beam constraint model (BCM), which can be employed for the geometric error analysis and the characteristic analysis of an inversion-based symmetric cross-spring pivot (IS-CSP). The load-dependent effects are classified in two ways, including structure load-dependent effects and beam load-dependent effects, where the loading positions, geometric parameters of elastic flexures, and axial forces are the main contributing factors. The closed-form load-rotation relations of an IS-CSP and a non-inversion-based symmetric cross-spring pivot (NIS-CSP) are derived with consideration of the three contributing factors for analyzing the load-dependent effects. The load-dependent effects of IS-CSP and NIS-CSP are compared when the loading position is fixed. The rotational stiffness of the IS-CSP or NIS-CSP can be designed to increase, decrease, or remain constant with axial forces, by regulating the balance between the loading positions and the geometric parameters. The closed-form solution of the center shift of an IS-CSP is derived. The effects of axial forces on the IS-CSP center shift are analyzed and compared with those of a NIS-CSP. Finally, based on the nonlinear analysis results of IS-CSP and NIS-CSP, two new compound symmetric cross-spring pivots are presented and analyzed via analytical and FEA models.

REFINED BEAM THEORIES BASED ON A UNIFIED FORMULATION

2010 ◽  
Vol 02 (01) ◽  
pp. 117-143 ◽  
Author(s):  
ERASMO CARRERA ◽  
GAETANO GIUNTA
Keyword(s):  
Differential Equations ◽  
Cross Section ◽  
Cross Sections ◽  
Order Approximation ◽  
Three Dimensional ◽  
Closed Form Solution ◽  
Approximation Order ◽  
Form Solution ◽  
Geometrical Parameters ◽  
Beam Theories

This paper proposes several axiomatic refined theories for the linear static analysis of beams made of isotropic materials. A hierarchical scheme is obtained by extending plates and shells Carrera's Unified Formulation (CUF) to beam structures. An N-order approximation via Mac Laurin's polynomials is assumed on the cross-section for the displacement unknown variables. N is a free parameter of the formulation. Classical beam theories, such as Euler-Bernoulli's and Timoshenko's, are obtained as particular cases. According to CUF, the governing differential equations and the boundary conditions are derived in terms of a fundamental nucleo that does not depend upon the approximation order. The governing differential equations are solved via the Navier type, closed form solution. Rectangular and I-shaped cross-sections are accounted for. Beams undergo bending and torsional loadings. Several values of the span-to-height ratio are considered. Slender as well as deep beams are analysed. Comparisons with reference solutions and three-dimensional FEM models are given. The numerical investigation has shown that the proposed unified formulation yields the complete three-dimensional displacement and stress fields for each cross-section as long as the appropriate approximation order is considered. The accuracy of the solution depends upon the geometrical parameters of the beam and loading conditions.

scholarly journals Closed form solution in buckling optimization problem of twisted rod

2021 ◽  
Author(s):  
Vladimir Kobelev
Keyword(s):  
Closed Form ◽  
Cross Section ◽  
Cross Sections ◽  
Optimization Problem ◽  
Closed Form Solution ◽  
Form Solution ◽  
Optimal Shape ◽  
Actual Problem ◽  
The Cross ◽  
Simply Connected

Abstract The applications of this method for stability problems are illustrated in this manuscript. In the context of twisted rods, the counterpart for Euler’s buckling problem is Greenhill's problem, which studies the forming of a loop in an elastic bar under torsion (Greenhill, 1883). We search the optimal shape of the rod along its axis. A priori form of the cross-section remains unknown. For the solution of the actual problem the stability equations take into account all possible convex, simply connected shapes of the cross-section. Thus, we drop the assumption about the equality of principle moments of inertia for the cross-section. The cross-sections are similar geometric figures related by a homothetic transformation with respect to a homothetic center on the axis of the rod and vary along its axis. The distribution of material along the length of a twisted rod is optimized so that the rod is of the constant volume T and will support the maximal moment without spatial buckling. The cross section that delivers the maximum or the minimum for the critical eigenvalue must be determined among all convex, simply connected domains. We demonstrate at the beginning the validity of static Euler’s approach for simply supported rod (hinged), twisted by the conservative moment. The applied method for integration of the optimization criteria delivers different length and volumes of the optimal twisted rods. Instead of the seeking for the twisted rods of the fixed length and volume, we directly compare the twisted rods with the different lengths and cross-sections using the invariant factors. The solution of optimization problem for twisted rod is stated in closed form in terms of the higher transcendental functions. In the torsion stability problem, the optimal shape of cross-section is the equilateral triangle.

Plane-Strain Bending With Isotropic Strain Hardening at Large Strains

2010 ◽  
Vol 77 (6) ◽  
Author(s):  
Sergei Alexandrov ◽  
Yeong-Maw Hwang
Keyword(s):  
Strain Hardening ◽  
Plane Strain ◽  
Closed Form Solution ◽  
Bending Moment ◽  
Form Solution ◽  
Isotropic Hardening ◽  
Large Strains ◽  
Rigid Plastic ◽  
Elastic Plastic ◽  
Hardening Law

Finite deformation elastic-plastic analysis of plane-strain pure bending of a strain hardening sheet is presented. The general closed-form solution is proposed for an arbitrary isotropic hardening law assuming that the material is incompressible. Explicit relations are given for most popular conventional laws. The stage of unloading is included in the analysis to investigate the distribution of residual stresses and springback. The paper emphasizes the method of solution and the general qualitative features of elastic-plastic solutions rather than the study of the bending process for a specific material. In particular, it is shown that rigid-plastic solutions can be used to predict the bending moment at sufficiently large strains.

THREE-DIMENSIONAL NON-PRISMATIC BEAM-COLUMNS

2002 ◽  
Vol 02 (03) ◽  
pp. 395-408 ◽  
Author(s):  
R. LEVY ◽  
E. GAL
Keyword(s):  
Cross Sections ◽  
Three Dimensional ◽  
Closed Form Solution ◽  
Form Solution ◽  
Beam Columns ◽  
Straight Beam ◽  
Prismatic Beam ◽  
Principal Directions ◽  
The Finite Difference Method ◽  
Uniform Manner

This paper is concerned with three-dimensional straight beam-columns with no warping whose cross sections vary along the axis in a uniform manner with respect to the principal directions. The basic four coupled differential equations governing the behavior of 3D beam-columns are first rederived using the method of perturbations. These equations are reformulated to include varying cross sections. Finally, a 6 × 6 stiffness matrix (which is sufficient to describe 3D behavior) is computed by solving the equations 6 times for a sequence of appropriate discontinuities. The finite difference method is employed for that purpose. Timoshenko's closed form solution for the buckling load of a tapered column is chosen for comparison with that obtained by the proposed formulation. Effects of twist are also presented.

Creep Deflection of Thick-Walled Piping due to Combined Bending and Internal Pressure

1990 ◽  
Vol 112 (3) ◽  
pp. 251-255 ◽  
Author(s):  
I. Finnie ◽  
M. Shirmohamadi
Keyword(s):  
Internal Pressure ◽  
Closed Form ◽  
Satisfactory Agreement ◽  
Correction Factor ◽  
Closed Form Solution ◽  
Bending Moment ◽  
Form Solution ◽  
Piping System

A closed-form solution is derived for the creep deflection in thick-walled piping subjected to combined internal pressure and bending moment. The solution is limited to the situation usually encountered in practice with sustained gravity loads and support forces in which the additional stresses due to bending are small compared to those due to internal pressure. For this case, it is shown that a simple correction factor may be applied to an elastic computation of pipe deflections to include the effect of creep. Predictions using this factor show satisfactory agreement with observations on a thick-walled piping system which had been in service for 20 years.

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