Home Courses Articles Watch! Tools Downloads Request ----------------------------------------------------- Soil Mechanics > Physical Properties of Soil > Soil Phase Relationships > Friction Angle of Soils + Typical Values > Cohesion Intercept of Soils + Typical Values > Lateral Earth Pressure > At-Rest State > Rankine's Lateral Earth Pressure > Coulomb's Lateral Earth Pressure > Slope Stability > Effects of Water on Slope Stability > Infinite Slope Analysis > Circular Arc Failure of Slope Analysis > Critical Failure Surface > Geotechnical Laboratory and In-Situ Testing Methods > 101 How To Plan A Construction Site Soil Investigations Program > 101 How to Write a Geotechnical Report > Determination of Water Content > Specific Gravity of Soils > Advanced Soil Mechanics Topics > Soil Liquefaction: Factor of Safety, Calculations, and Simplified Procedures Foundation Analysis and Design > Settlement of Shallow Foundations > Settlement types, definitions, and general equation > Immediate settlement computations > Primary Consolidation > Secondary Compression > Bearing Capacity of Shallow Foundations > Terzaghi's Method > Meyerhof's Method > Brinch Hansen's Method > Bearing from SPT Number > Effect of Ground Water Table > Bearing Capacity of Piles (Deep Foundations) > Ultimate Tip Resistance of Piles > Shaft Resistance of Piles > Sheet-pile Walls: Cantilevered and Anchored > Cantilever Sheet Pile Walls Penetrating Sandy Soils > Anchored Sheet Pile Walls Penetrating Sandy Soils > Factors of Safety for Cantilevered Sheet Pile Walls Load Calculation > Structural Loading > Dead Load vs Live Load > Load Combinations Reinforced Concrete Design > General Topics of Concrete Material and Design > What Is Concrete? > Concrete Properties > Section Properties of Reinforcing Bars & Cement Types > Load Combinations of Concrete Design > Design of Concrete Members > Reinforced Concrete Beam Design > Flexural Design of Reinforced Concrete Beams > Serviceability of Reinforced Concrete Beams > Shear Design of Reinforced Concrete Beams Structural Steel Design > General Topics of Steel Material and Design > Steel Structures > Connections in Steel Structures > Cold Formed Steel Sections Construction > Elements of construction > Construction Site Layout Planning Elements > All Types of Roofs And Their Details > Types of Foundations From Construction Point of View > All Types of Foundation Materials Timber Design > Design of Timber Members > Design of Sawn Timber Beams or Joists > Design of Sawn Timber Columns and Compressive Members Masonry Design Finite Elements Method > Basics of Finite Elements > Introduction to Finite Elements And the Big Picture > One-dimensional Bars/Springs > Plane Trusses Transportation Surveying > Instruments and Distance Measurements > Introduction to Surveying > Distance Measurement > Leveling, Bench marks, Turning points > Leveling > Horizontal and Vertical Angles, Azimuths, Traverses, Closure Error > Angle Measurement > Topographic Data, Base Mapping > Topographic Surveys > Creating a Plan and Profile Introduction Concepts & Formulas Videos Solved problems Download Files One-dimensional Bars/Springs Courses > Finite Elements Method > Basics of Finite Elements > One-dimensional Bars/Springs For so many Finite Element Codes in MATLAB including 1D, 2D, and 3D codes, trusses, beams, solids, large deformations, contact algorithms and etc. you can visit matlab-fem.com

Introduction on One-dimensional Bars/Springs :

A large number of practical problems are governed by the one-dimensional boundary value problem (1D BVP) of the following form:

Since the differential equation is of second order, for a unique solution, there must be at least two specified boundary conditions.

Concepts and Formulas of One-dimensional Bars/Springs:

Governing differential equation (ODE):
The general governing differential equation for axial deformation of bars/springs is:

where A(x) is the area of cross section and can vary along the length; E is the elastic modulus; and q(x) is the applied distributed load in the axial direction.

Types of boundary conditions
The boundary conditions of the following forms may be specified at one or more points:

1- Essential boundary conditions (EBC):

2- Natural boundary conditions (NBC):

Explicit equations for a two-node linear element:
The interpolation functions are simple linear functions of x. Furthermore, if we assume that k,p, and q are constant over an element, then it is easy to carry out integrations and write explicit formulas for element equations:

the position of element nodes:

Interpolation functions (shape functions):

and consequently derivatives of shape functions:

Thus, the element stiffness matrix is:

and the explicit equations for a linear element:

Watch! (Videos):
Solved sample problems of One-dimensional Bars/Springs:

Download Files: Read also: Share:

Follow our official Facebook page (@civilengineeringbible) and Twitter page (@CivilEngBible) and do not miss the best civil engineering tools and articles!