This document presents a numerical approach for analyzing eccentrically loaded rectangular footings resting on soil. The approach is implemented using Microsoft Excel and VBA programming. It can handle any loading condition, from no eccentricity to one-way or two-way eccentricity. The approach satisfies equilibrium conditions and finds the pressure distribution by iteratively adjusting the position of the neutral axis until volume and center of gravity errors are minimized. Examples are presented where the approach achieves over 99.9% accuracy compared to the true solutions.
This document describes 9 numerical calculation methods for analyzing piled rafts, ranging from simple to more complex models. The simplest is the linear contact pressure method (Method 1), which assumes a linear distribution of contact pressures without interaction between the foundation and soil. Winkler's model (Methods 2-3) represents the soil as elastic springs, with pile reactions proportional to settlement. The continuum model (Methods 4-9) considers interaction between all foundation elements and layered soil, represented as a continuum. The most complex method is the modulus of compressibility method for layered soil (Methods 6-9). Finite element analysis is used except for one rigid raft method. The key equations for each method are presented.
Template-Based Paper Reconstruction from a Single Image is Well Posed when th...pigei
1) The document presents a method for reconstructing a 3D model of a piece of paper from a single 2D image by using a template, when the paper deformations result in parallel rulings like pages of a book.
2) It shows that under the assumption of parallel rulings, the problem can be reduced to reconstructing 2D points given a pair of 1D cameras, making the problem well-posed.
3) The paper formulates the reconstruction as a functional optimization problem that is solved numerically to recover the 3D shape and pose of the paper. Experimental results demonstrate applications like texture replacement and occlusion handling.
The document discusses concurrent lines in triangles. It defines concurrent lines as three lines in a plane that intersect at a single point. It then examines the angle bisectors, perpendicular bisectors of sides, altitudes, and medians of a triangle, and proves that in each case the three lines are concurrent. Specifically, it shows that the three angle bisectors intersect at the triangle's incentre, the three perpendicular bisectors intersect at the circumcentre, the three altitudes intersect at the orthocentre, and the three medians intersect at the centroid.
Advanced mathematical analysis of chassis integrated platform designed for un...Dr.Vikas Deulgaonkar
The present work deals with advanced mathematical stress analysis of a platform integrated structure mounted on vehicle chassis designed for unconventional type of loading pattern. The perceptible loading cases in the present analysis comprise static load and its effect on the platform/structure by usage of simple shear force & bending moment diagrams. Deflection analysis using conventional Macaulay’s method invokes the structures suitability for the transportation. Present analysis accentuates on the design stage aspects of the platform as this research is a step in proposed doctoral study. A different type of combination of longitudinal and cross members in platform/frame design is formulated. Present design is anticipated after analysis of all possible combinations& orientations of longitudinal and cross members. Determination of section properties of longitudinal and cross members of the platform & deduction of elementary stress based on the unconventional load pattern are the fundamental steps in design and analysis of structure. Peculiarity of this analysis is the usage of combined section modulus of three members for computation of stress. Present research provides a tool that can be used prior to computer aided design and finite element analysis.
This document presents an analysis of orthotropic reinforced concrete slabs with long side openings using the affine theorem and yield line method. Ten possible yield line failure patterns are considered for slabs with continuous, simply supported, two long sides continuous, and two short sides continuous edge conditions. Virtual work equations are formulated for each failure pattern. Numerical examples are provided to illustrate the governing failure patterns for different slab geometries and support conditions. The affine theorem is used to transform orthotropic slab properties into equivalent isotropic properties to simplify the analysis.
Math cad vm-001 stress-strain transformationsJulio Banks
This document describes three approaches to teaching stress-strain transformations: theoretical calculations, experimental measurements, and finite element analysis. It analyzes an L-shaped aluminum beam subjected to a load to determine stresses at a point. Method I calculates stresses theoretically. Method II uses strain measurements from a rosette gauge to determine stresses through equations. Method III uses finite element analysis. The results are compared to reinforce understanding of stress-strain concepts through different approaches.
Effect of Surface Roughness on Contact Pressureijsrd.com
In engineering problems surface roughness is often neglected for the sake of simplicity of model and to reduce time, but surface roughness plays an important role in many engineering applications like MEMS, Thermal Contact Conductance, and Insulation etc. In this paper, modeling of surface roughness is done in Ansys 13 and effect of surface roughness on contact pressure is shown. The rough surface model created as array of asperities with the same radius of asperities and same asperity height, which is assumed to be following Gaussian distribution. The modeling is done for surface roughness 1, 5, 10, 15 and 20 μm. The effect of surface roughness on contact pressure is shown by comparison of contact pressure of 1 and 5 μm roughness. Physical dimension of the two body contacting each other is 1 X 1 X 6 mm. The pressure applied on the body is 1.8MPa with consideration of other body with fix support.
This document describes 9 numerical calculation methods for analyzing piled rafts, ranging from simple to more complex models. The simplest is the linear contact pressure method (Method 1), which assumes a linear distribution of contact pressures without interaction between the foundation and soil. Winkler's model (Methods 2-3) represents the soil as elastic springs, with pile reactions proportional to settlement. The continuum model (Methods 4-9) considers interaction between all foundation elements and layered soil, represented as a continuum. The most complex method is the modulus of compressibility method for layered soil (Methods 6-9). Finite element analysis is used except for one rigid raft method. The key equations for each method are presented.
Template-Based Paper Reconstruction from a Single Image is Well Posed when th...pigei
1) The document presents a method for reconstructing a 3D model of a piece of paper from a single 2D image by using a template, when the paper deformations result in parallel rulings like pages of a book.
2) It shows that under the assumption of parallel rulings, the problem can be reduced to reconstructing 2D points given a pair of 1D cameras, making the problem well-posed.
3) The paper formulates the reconstruction as a functional optimization problem that is solved numerically to recover the 3D shape and pose of the paper. Experimental results demonstrate applications like texture replacement and occlusion handling.
The document discusses concurrent lines in triangles. It defines concurrent lines as three lines in a plane that intersect at a single point. It then examines the angle bisectors, perpendicular bisectors of sides, altitudes, and medians of a triangle, and proves that in each case the three lines are concurrent. Specifically, it shows that the three angle bisectors intersect at the triangle's incentre, the three perpendicular bisectors intersect at the circumcentre, the three altitudes intersect at the orthocentre, and the three medians intersect at the centroid.
Advanced mathematical analysis of chassis integrated platform designed for un...Dr.Vikas Deulgaonkar
The present work deals with advanced mathematical stress analysis of a platform integrated structure mounted on vehicle chassis designed for unconventional type of loading pattern. The perceptible loading cases in the present analysis comprise static load and its effect on the platform/structure by usage of simple shear force & bending moment diagrams. Deflection analysis using conventional Macaulay’s method invokes the structures suitability for the transportation. Present analysis accentuates on the design stage aspects of the platform as this research is a step in proposed doctoral study. A different type of combination of longitudinal and cross members in platform/frame design is formulated. Present design is anticipated after analysis of all possible combinations& orientations of longitudinal and cross members. Determination of section properties of longitudinal and cross members of the platform & deduction of elementary stress based on the unconventional load pattern are the fundamental steps in design and analysis of structure. Peculiarity of this analysis is the usage of combined section modulus of three members for computation of stress. Present research provides a tool that can be used prior to computer aided design and finite element analysis.
This document presents an analysis of orthotropic reinforced concrete slabs with long side openings using the affine theorem and yield line method. Ten possible yield line failure patterns are considered for slabs with continuous, simply supported, two long sides continuous, and two short sides continuous edge conditions. Virtual work equations are formulated for each failure pattern. Numerical examples are provided to illustrate the governing failure patterns for different slab geometries and support conditions. The affine theorem is used to transform orthotropic slab properties into equivalent isotropic properties to simplify the analysis.
Math cad vm-001 stress-strain transformationsJulio Banks
This document describes three approaches to teaching stress-strain transformations: theoretical calculations, experimental measurements, and finite element analysis. It analyzes an L-shaped aluminum beam subjected to a load to determine stresses at a point. Method I calculates stresses theoretically. Method II uses strain measurements from a rosette gauge to determine stresses through equations. Method III uses finite element analysis. The results are compared to reinforce understanding of stress-strain concepts through different approaches.
Effect of Surface Roughness on Contact Pressureijsrd.com
In engineering problems surface roughness is often neglected for the sake of simplicity of model and to reduce time, but surface roughness plays an important role in many engineering applications like MEMS, Thermal Contact Conductance, and Insulation etc. In this paper, modeling of surface roughness is done in Ansys 13 and effect of surface roughness on contact pressure is shown. The rough surface model created as array of asperities with the same radius of asperities and same asperity height, which is assumed to be following Gaussian distribution. The modeling is done for surface roughness 1, 5, 10, 15 and 20 μm. The effect of surface roughness on contact pressure is shown by comparison of contact pressure of 1 and 5 μm roughness. Physical dimension of the two body contacting each other is 1 X 1 X 6 mm. The pressure applied on the body is 1.8MPa with consideration of other body with fix support.
Prediction of Small Size Pneumatic Tyres for Tyre Traction PropertiesIDES Editor
An analytical approach for determining tyre force
and moment characteristics due to pure slips has been
presented. The analytical model predicts these forces and
moments in terms of easily measurable tyre parameters such
as contact patch area, lateral, longitudinal, and vertical
stiffnesses of tyre. These analytical predictions show good
agreement with existing measured data. The model is suitable
for study of vehicle dynamic simulations.
Stress reduction using semi elliptical slots in axially loaded plate having c...IAEME Publication
This document analyzes stress reduction in an axially loaded plate with a circular hole using semi-elliptical slots. Finite element analysis is used to study how the maximum stress is affected by varying the geometric parameters of the elliptical slots. Stress reduction of 42% is achieved for optimized parameters of a=100mm, b=20mm, l=20mm. Relations for the optimum parameters are determined as a=2.5D, b=0.5D, l=0.5D, where D is the diameter of the circular hole. The semi-elliptical slots provide higher stress reduction than auxiliary circular holes.
Estimation of IRI from PCI in Construction Work ZonesIDES Editor
Roughness is good evaluator of performance of road.
This paper presents a case study of IRI (International
Roughness Index) estimation at NH 67 during four laning of
Trichy - Tanjavur section. An attempt has been made to
evaluate the IRI of construction work zones using Levenberg-
Marquardt back-propagation training algorithm. A MATLAB
based model is developed, and the data from the case study are
used to train and test the developed model to predict IRI. The
models’ performances are evaluated through Correlation
coefficient (R2) and Mean Square Error (MSE).
Deflection of structures using double integration method, moment area method, elastic load method, conjugate beam method, virtual work, castiglianois second theorem and method of consistent deformations
Smoothing of the Surface Estimates from RadarclinometryIRJET Journal
This document summarizes a study that aimed to improve the accuracy of estimating surface heights from radar images using a shape-from-shading technique called radarclinometry. Radarclinometry derives surface heights from variations in shading detected in radar images, but is limited by ambiguity from uncertain radar backscatter properties. The study applied a smoothness constraint to the surface height estimates to remove this ambiguity. The constraint enforced gradual variation in surface normals across the image. Evaluating the technique on a RADARSAT-1 radar image, the study found the final surface height estimates were more accurate after 100 iterations of applying the smoothness constraint, reducing errors compared to estimates without the constraint.
Study of Structural Behaviour of Gravity Dam with Various Features of Gallery...IDES Editor
The size and shape of opening in dam causes the
stress concentration, it also causes the stress variation in the
rest of the dam cross section. The gravity method of the analysis
does not consider the size of opening and the elastic property
of dam material. Thus the objective of study is comprises of
the Finite Element Method which considers the size of
opening, elastic property of material, and stress distribution
because of geometric discontinuity in cross section of dam.
Stress concentration inside the dam increases with the opening
in dam which results in the failure of dam. Hence it is
necessary to analyses large opening inside the dam. By making
the percentage area of opening constant and varying size and
shape of opening the analysis is carried out. For this purpose
a section of Koyna Dam is considered. Dam is defined as a
plane strain element in FEM, based on geometry and loading
condition. Thus this available information specified our path
of approach to carry out 2D plane strain analysis. The results
obtained are then compared mutually to get most efficient
way of providing large opening in the gravity dam.
Effect of foundation flexibility on dynamic behaviour of asymmetric building ...eSAT Journals
Abstract In general the seismic design of building frame structures the designers will consider only the results of fixed base condition the effect of flexibility is ignored. In post-earthquake study the framed structure reveals that the interaction of soil and foundation plays an important role in damage of the building frame structures. In this regard a literature survey has been done on frame structures supported on various foundations such as isolated, combined, raft & pile foundations. To examine the literature revels the few investigations were done on asymmetric building frame structure is supported on isolated footing. So in this paper is an attempt to the study of dynamic behavior of asymmetric building frame structure is supported on isolated footings. The modeling and analysis is done by using “finite element method software” SAP2000 VERSION 14, by considering the different soil conditions, (soft, medium, hard) different soil parameters (passion’s ratio, young’s modulus, dynamic shear modulus) different height ratio’s, different span ratio’s & fixed base conditions. The response of the building frame structure is obtained in terms of fundamental natural period, lateral displacement and seismic base shear. Keywords: Soil structure interaction, Fundamental natural period, Base shear, Lateral displacement….
Pseudo static passive response of retaining wall supporting c-φ backfilleSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
This document discusses the design of tower foundations. It notes that tower foundations make up 20-40% of total tower costs. The document presents a computer program written in BASIC to optimally design tower foundations. The program uses concepts like calculating the uplift resistance provided by soil weight in an inverted cone or pyramid shape. It considers factors like soil type, depth, and dimensions to calculate safety factors against uplift and sliding. The overall goal of the program is to aid in producing economical and reliable tower foundation designs.
Final Report Turbulant Flat Plate AnsysSultan Islam
- The document describes a computational fluid dynamics (CFD) simulation of turbulent flow over a flat plate using ANSYS CFX.
- The simulation aims to validate results against experimental data from NASA and analyze sensitivity of skin friction coefficient and velocity profiles.
- The flat plate geometry, meshing approach, and boundary conditions are described based on the NASA and Caelus experiments.
- Results for velocity profiles and skin friction coefficients along the plate are presented and validated against experimental trends.
- Grid convergence and sensitivity to turbulence models are analyzed, with the SST and k-epsilon models showing similar results.
This document summarizes a study on the nonlinear response of composite steel-concrete box girder bridges. A 3D finite element model was developed incorporating shear connector slip. Parametric analyses were performed on prototype 2-lane and 4-lane bridges with varying geometry under IRC loading conditions. Load distribution factors were proposed as a simplified design method based on results from the finite element analyses. The model was validated against published experimental data on a composite beam test specimen.
BEARING CAPACITY OF SQUARE FOOTING RESTING ON STABILIZED SAND DUNES.SunilKumar586753
This document summarizes a study on determining the bearing capacity of square footings resting on stabilized sand dunes using the PLAXIS software. The study analyzed different data sets with varying fly ash percentages and curing times to determine the cohesion, internal friction angle, and ultimate bearing capacity. The methodology involved creating soil models, assigning properties, applying loads, and plotting load-settlement graphs. Bearing capacities were calculated using the double tangent method and were found to increase with higher fly ash percentages and curing times. The data set with 30% fly ash content and 14 day curing time produced the highest bearing capacity of 680 kN/m^2.
This document discusses stress and strain analysis. It defines stress at a point and introduces the stress tensor. The stress tensor is symmetric. Principal stresses are the maximum and minimum normal stresses. Mohr's circle can be used to determine stresses on any plane through a point by graphically representing the transformation of stresses between planes. The principal planes contain no shear stress and maximum shear stress planes are 45 degrees from principal planes.
Application of Elastic Layered System in the Design of RoadIJERA Editor
This document discusses the application of elastic layered system theory in road design. It can be used to calculate asphalt pavement thickness, analyze stress in cement concrete pavements, and calculate load stress in porous concrete bases. The elastic layered system assumes layers are made of homogeneous, isotropic materials and loads are applied as uniform circles. It provides a simple model that generally reflects actual stress conditions despite differences from reality. The document provides examples of using the theory to calculate pavement thickness and analyze deflection and stress at different points in the pavement structure.
This document discusses the analysis of shear stresses in beams. It begins by explaining how bending causes sliding between layers in a built-up beam, making it weaker than a solid beam. It then derives the formula for calculating the horizontal shear stress in a beam based on the vertical shear force and moment of inertia. The derivation shows that horizontal and vertical shear stresses are equal. It applies the formula to a rectangular beam section, showing the shear stress is highest at the neutral axis and distributed parabolically. The assumptions and limitations of treating shear stress as uniform across a section are also addressed.
This document contains a list of 6 students enrolled in a civil engineering course at Birla Vishvakarma Mahavidyalaya Engineering College. It also contains a leveling document prepared by an unknown author on the topic of leveling for the course. The leveling document defines leveling as a surveying method that establishes elevations along a reference line. It describes the primary uses of leveling for infrastructure like highways, canals, sewers, and more. The document also outlines leveling procedures, including establishing stations, taking intermediate foresights, and using turning points.
Empirical relation to estimate the reduction of root fillet stress in spur ge...eSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
CEE 213—Deformable Solids The Mechanics Project Arizona Stat.docxcravennichole326
CEE 213—Deformable Solids The Mechanics Project
Arizona State University CP 1—Axial Bar
Computing Project 1
Axial Bar
The computing project Axial Bar concerns the solution of the problem of a prismatic bar
in a uniaxial state of stress with a variable load. The goal is to write a MATLAB program
that will allow the solution for a variety of load distributions and for all possible bounda-
ry conditions (i.e., fixed or free at either end).
The theory needed to execute this project is contained in the set of notes (entitled CP 1—
Axial Bar) that accompany this problem statement. Those notes provide an introduction
to each aspect of the computation required to solve the problem. The general steps are as
follows:
1. CP 1.1. Develop a routine based upon Simpson’s Rule to numerically integrate
the applied loads and moments of the applied loads. This code segment will pro-
duce the quantities I0 and I1 that are mentioned in the CP1 notes. To get this part
working it would be a good idea to get your code to integrate some functions that
you can do easily by hand (e.g., the constant or linear functions). This step will
be referred to as CP1.1, which is the first benchmark with an intermediate due
date for this project. The main deliverable is the working code for Simpson’s
Rule, verified for several functions.
2. CP 1.2. Develop a routine to set up and solve the system of equations that allow
for the determination of the state variables (u and N) at both ends of the bar. This
step will require some logic to make it work easily for different boundary condi-
tion cases (it should cover all of them). Debug your code with a problem that you
can solve by hand (e.g., bar fixed at one end with a uniformly distributed load).
This step will be referred to as CP 1.2, which is the second benchmark with an
intermediate due date for this project. The main deliverable is a working code
that does the Simpson integration for I0 and I1 and then forms and solves the ap-
propriate matrix equation to find the end state.
3. Develop a routine to integrate the governing equations from the left end to the
right end using generalized trapezoidal rule to do the integration numerically.
Store the results at each step along the axis and provide a plot of the applied load
p, the axial displacement u, and the net axial force N as functions of x. Note that
the number of generalized trapezoidal rule segments does not have to be the same
as the number of Simpson segments. This step completes the code for CP 1. In-
clude all three parts in the final report.
1
CEE 213—Deformable Solids The Mechanics Project
Arizona State University CP 1—Axial Bar
4. Structure your code so that you can easily change the loading function. Include
simple load forms (constant, linear ramp up from left to right, linear ramp down
from right to left, trapezoidal distribution, sinusoidal distribution, and a patch
load over an interior part of the rod—from x=a ...
An Analytical Method for Static Earth Pressure Distribution against Rectangul...IJERA Editor
Analytical methods for computing the lateral earth pressure against tunnel is vastly used by engineers all over the
world. Conventional analytical methods compute the lateral pressure in either active or passive state while the
stress state usually falls between these two boundaries in many practical cases. Furthermore, using these
boundary coefficients lead to either overestimated or underestimated results in design. Thus, a modified method
based on the strain increment theory for calculating the lateral pressure against rectangular tunnels is presented
herein to consider the amount of lateral deformation at each depth. First, the results for different values of
overburden depth, friction angle and wall mobilized angle are investigated. Then comparative finite element
analyses were performed to examine the effectiveness of the method. According to this study, the pressure
pattern is completely nonlinear especially at the corners of tunnel lining. In fact, the pressure increases
nonlinearly to about three times of the value at top. Lateral earth pressure decreases with the increase of friction
angle which is in good agreement with finite element results. Overall, the pressure patterns derived by this
method for shallow depths (less than tunnel height) are almost the same as those computed by finite element
method.
The document provides an analysis of the slope design for an open pit operation submitted by Andrews Surface-Design LTD. West Country Minerals conducted an independent review using software like DIPS, RocLab, RocPlane, and Slide. The analysis examined potential failure mechanisms like planar, wedge, and toppling failures. It also analyzed the stability of individual benches and the overall slope. The review found that some benches would fail without support but adding tension bolts could achieve the target safety factors, particularly if higher bolt tension is used for seismic conditions. Overall, the analysis found the slope design to be stable but recommended some amendments to bolt support and further sensitivity analysis.
Prediction of Small Size Pneumatic Tyres for Tyre Traction PropertiesIDES Editor
An analytical approach for determining tyre force
and moment characteristics due to pure slips has been
presented. The analytical model predicts these forces and
moments in terms of easily measurable tyre parameters such
as contact patch area, lateral, longitudinal, and vertical
stiffnesses of tyre. These analytical predictions show good
agreement with existing measured data. The model is suitable
for study of vehicle dynamic simulations.
Stress reduction using semi elliptical slots in axially loaded plate having c...IAEME Publication
This document analyzes stress reduction in an axially loaded plate with a circular hole using semi-elliptical slots. Finite element analysis is used to study how the maximum stress is affected by varying the geometric parameters of the elliptical slots. Stress reduction of 42% is achieved for optimized parameters of a=100mm, b=20mm, l=20mm. Relations for the optimum parameters are determined as a=2.5D, b=0.5D, l=0.5D, where D is the diameter of the circular hole. The semi-elliptical slots provide higher stress reduction than auxiliary circular holes.
Estimation of IRI from PCI in Construction Work ZonesIDES Editor
Roughness is good evaluator of performance of road.
This paper presents a case study of IRI (International
Roughness Index) estimation at NH 67 during four laning of
Trichy - Tanjavur section. An attempt has been made to
evaluate the IRI of construction work zones using Levenberg-
Marquardt back-propagation training algorithm. A MATLAB
based model is developed, and the data from the case study are
used to train and test the developed model to predict IRI. The
models’ performances are evaluated through Correlation
coefficient (R2) and Mean Square Error (MSE).
Deflection of structures using double integration method, moment area method, elastic load method, conjugate beam method, virtual work, castiglianois second theorem and method of consistent deformations
Smoothing of the Surface Estimates from RadarclinometryIRJET Journal
This document summarizes a study that aimed to improve the accuracy of estimating surface heights from radar images using a shape-from-shading technique called radarclinometry. Radarclinometry derives surface heights from variations in shading detected in radar images, but is limited by ambiguity from uncertain radar backscatter properties. The study applied a smoothness constraint to the surface height estimates to remove this ambiguity. The constraint enforced gradual variation in surface normals across the image. Evaluating the technique on a RADARSAT-1 radar image, the study found the final surface height estimates were more accurate after 100 iterations of applying the smoothness constraint, reducing errors compared to estimates without the constraint.
Study of Structural Behaviour of Gravity Dam with Various Features of Gallery...IDES Editor
The size and shape of opening in dam causes the
stress concentration, it also causes the stress variation in the
rest of the dam cross section. The gravity method of the analysis
does not consider the size of opening and the elastic property
of dam material. Thus the objective of study is comprises of
the Finite Element Method which considers the size of
opening, elastic property of material, and stress distribution
because of geometric discontinuity in cross section of dam.
Stress concentration inside the dam increases with the opening
in dam which results in the failure of dam. Hence it is
necessary to analyses large opening inside the dam. By making
the percentage area of opening constant and varying size and
shape of opening the analysis is carried out. For this purpose
a section of Koyna Dam is considered. Dam is defined as a
plane strain element in FEM, based on geometry and loading
condition. Thus this available information specified our path
of approach to carry out 2D plane strain analysis. The results
obtained are then compared mutually to get most efficient
way of providing large opening in the gravity dam.
Effect of foundation flexibility on dynamic behaviour of asymmetric building ...eSAT Journals
Abstract In general the seismic design of building frame structures the designers will consider only the results of fixed base condition the effect of flexibility is ignored. In post-earthquake study the framed structure reveals that the interaction of soil and foundation plays an important role in damage of the building frame structures. In this regard a literature survey has been done on frame structures supported on various foundations such as isolated, combined, raft & pile foundations. To examine the literature revels the few investigations were done on asymmetric building frame structure is supported on isolated footing. So in this paper is an attempt to the study of dynamic behavior of asymmetric building frame structure is supported on isolated footings. The modeling and analysis is done by using “finite element method software” SAP2000 VERSION 14, by considering the different soil conditions, (soft, medium, hard) different soil parameters (passion’s ratio, young’s modulus, dynamic shear modulus) different height ratio’s, different span ratio’s & fixed base conditions. The response of the building frame structure is obtained in terms of fundamental natural period, lateral displacement and seismic base shear. Keywords: Soil structure interaction, Fundamental natural period, Base shear, Lateral displacement….
Pseudo static passive response of retaining wall supporting c-φ backfilleSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
This document discusses the design of tower foundations. It notes that tower foundations make up 20-40% of total tower costs. The document presents a computer program written in BASIC to optimally design tower foundations. The program uses concepts like calculating the uplift resistance provided by soil weight in an inverted cone or pyramid shape. It considers factors like soil type, depth, and dimensions to calculate safety factors against uplift and sliding. The overall goal of the program is to aid in producing economical and reliable tower foundation designs.
Final Report Turbulant Flat Plate AnsysSultan Islam
- The document describes a computational fluid dynamics (CFD) simulation of turbulent flow over a flat plate using ANSYS CFX.
- The simulation aims to validate results against experimental data from NASA and analyze sensitivity of skin friction coefficient and velocity profiles.
- The flat plate geometry, meshing approach, and boundary conditions are described based on the NASA and Caelus experiments.
- Results for velocity profiles and skin friction coefficients along the plate are presented and validated against experimental trends.
- Grid convergence and sensitivity to turbulence models are analyzed, with the SST and k-epsilon models showing similar results.
This document summarizes a study on the nonlinear response of composite steel-concrete box girder bridges. A 3D finite element model was developed incorporating shear connector slip. Parametric analyses were performed on prototype 2-lane and 4-lane bridges with varying geometry under IRC loading conditions. Load distribution factors were proposed as a simplified design method based on results from the finite element analyses. The model was validated against published experimental data on a composite beam test specimen.
BEARING CAPACITY OF SQUARE FOOTING RESTING ON STABILIZED SAND DUNES.SunilKumar586753
This document summarizes a study on determining the bearing capacity of square footings resting on stabilized sand dunes using the PLAXIS software. The study analyzed different data sets with varying fly ash percentages and curing times to determine the cohesion, internal friction angle, and ultimate bearing capacity. The methodology involved creating soil models, assigning properties, applying loads, and plotting load-settlement graphs. Bearing capacities were calculated using the double tangent method and were found to increase with higher fly ash percentages and curing times. The data set with 30% fly ash content and 14 day curing time produced the highest bearing capacity of 680 kN/m^2.
This document discusses stress and strain analysis. It defines stress at a point and introduces the stress tensor. The stress tensor is symmetric. Principal stresses are the maximum and minimum normal stresses. Mohr's circle can be used to determine stresses on any plane through a point by graphically representing the transformation of stresses between planes. The principal planes contain no shear stress and maximum shear stress planes are 45 degrees from principal planes.
Application of Elastic Layered System in the Design of RoadIJERA Editor
This document discusses the application of elastic layered system theory in road design. It can be used to calculate asphalt pavement thickness, analyze stress in cement concrete pavements, and calculate load stress in porous concrete bases. The elastic layered system assumes layers are made of homogeneous, isotropic materials and loads are applied as uniform circles. It provides a simple model that generally reflects actual stress conditions despite differences from reality. The document provides examples of using the theory to calculate pavement thickness and analyze deflection and stress at different points in the pavement structure.
This document discusses the analysis of shear stresses in beams. It begins by explaining how bending causes sliding between layers in a built-up beam, making it weaker than a solid beam. It then derives the formula for calculating the horizontal shear stress in a beam based on the vertical shear force and moment of inertia. The derivation shows that horizontal and vertical shear stresses are equal. It applies the formula to a rectangular beam section, showing the shear stress is highest at the neutral axis and distributed parabolically. The assumptions and limitations of treating shear stress as uniform across a section are also addressed.
This document contains a list of 6 students enrolled in a civil engineering course at Birla Vishvakarma Mahavidyalaya Engineering College. It also contains a leveling document prepared by an unknown author on the topic of leveling for the course. The leveling document defines leveling as a surveying method that establishes elevations along a reference line. It describes the primary uses of leveling for infrastructure like highways, canals, sewers, and more. The document also outlines leveling procedures, including establishing stations, taking intermediate foresights, and using turning points.
Empirical relation to estimate the reduction of root fillet stress in spur ge...eSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
CEE 213—Deformable Solids The Mechanics Project Arizona Stat.docxcravennichole326
CEE 213—Deformable Solids The Mechanics Project
Arizona State University CP 1—Axial Bar
Computing Project 1
Axial Bar
The computing project Axial Bar concerns the solution of the problem of a prismatic bar
in a uniaxial state of stress with a variable load. The goal is to write a MATLAB program
that will allow the solution for a variety of load distributions and for all possible bounda-
ry conditions (i.e., fixed or free at either end).
The theory needed to execute this project is contained in the set of notes (entitled CP 1—
Axial Bar) that accompany this problem statement. Those notes provide an introduction
to each aspect of the computation required to solve the problem. The general steps are as
follows:
1. CP 1.1. Develop a routine based upon Simpson’s Rule to numerically integrate
the applied loads and moments of the applied loads. This code segment will pro-
duce the quantities I0 and I1 that are mentioned in the CP1 notes. To get this part
working it would be a good idea to get your code to integrate some functions that
you can do easily by hand (e.g., the constant or linear functions). This step will
be referred to as CP1.1, which is the first benchmark with an intermediate due
date for this project. The main deliverable is the working code for Simpson’s
Rule, verified for several functions.
2. CP 1.2. Develop a routine to set up and solve the system of equations that allow
for the determination of the state variables (u and N) at both ends of the bar. This
step will require some logic to make it work easily for different boundary condi-
tion cases (it should cover all of them). Debug your code with a problem that you
can solve by hand (e.g., bar fixed at one end with a uniformly distributed load).
This step will be referred to as CP 1.2, which is the second benchmark with an
intermediate due date for this project. The main deliverable is a working code
that does the Simpson integration for I0 and I1 and then forms and solves the ap-
propriate matrix equation to find the end state.
3. Develop a routine to integrate the governing equations from the left end to the
right end using generalized trapezoidal rule to do the integration numerically.
Store the results at each step along the axis and provide a plot of the applied load
p, the axial displacement u, and the net axial force N as functions of x. Note that
the number of generalized trapezoidal rule segments does not have to be the same
as the number of Simpson segments. This step completes the code for CP 1. In-
clude all three parts in the final report.
1
CEE 213—Deformable Solids The Mechanics Project
Arizona State University CP 1—Axial Bar
4. Structure your code so that you can easily change the loading function. Include
simple load forms (constant, linear ramp up from left to right, linear ramp down
from right to left, trapezoidal distribution, sinusoidal distribution, and a patch
load over an interior part of the rod—from x=a ...
An Analytical Method for Static Earth Pressure Distribution against Rectangul...IJERA Editor
Analytical methods for computing the lateral earth pressure against tunnel is vastly used by engineers all over the
world. Conventional analytical methods compute the lateral pressure in either active or passive state while the
stress state usually falls between these two boundaries in many practical cases. Furthermore, using these
boundary coefficients lead to either overestimated or underestimated results in design. Thus, a modified method
based on the strain increment theory for calculating the lateral pressure against rectangular tunnels is presented
herein to consider the amount of lateral deformation at each depth. First, the results for different values of
overburden depth, friction angle and wall mobilized angle are investigated. Then comparative finite element
analyses were performed to examine the effectiveness of the method. According to this study, the pressure
pattern is completely nonlinear especially at the corners of tunnel lining. In fact, the pressure increases
nonlinearly to about three times of the value at top. Lateral earth pressure decreases with the increase of friction
angle which is in good agreement with finite element results. Overall, the pressure patterns derived by this
method for shallow depths (less than tunnel height) are almost the same as those computed by finite element
method.
The document provides an analysis of the slope design for an open pit operation submitted by Andrews Surface-Design LTD. West Country Minerals conducted an independent review using software like DIPS, RocLab, RocPlane, and Slide. The analysis examined potential failure mechanisms like planar, wedge, and toppling failures. It also analyzed the stability of individual benches and the overall slope. The review found that some benches would fail without support but adding tension bolts could achieve the target safety factors, particularly if higher bolt tension is used for seismic conditions. Overall, the analysis found the slope design to be stable but recommended some amendments to bolt support and further sensitivity analysis.
A closed form solution for stress concentration around a circular hole in a lIAEME Publication
1) The document presents a closed-form solution to determine stress concentration around a circular hole in an infinite plate with linearly varying stress.
2) The equation developed can determine stress field around the hole without need for computationally intensive numerical methods.
3) Results from the closed-form solution are compared to finite element analysis and show close agreement.
A closed form solution for stress concentration around a circular hole in a lIAEME Publication
This document presents a closed-form solution for determining stress concentration around a circular hole in an infinite plate with linearly varying stress. The plate is subjected to a tensile stress that varies linearly from the top edge to the bottom edge. An equation is derived for the stress field in polar coordinates using stress functions. Boundary conditions are applied at the hole edge and at a large distance from the hole. A solution is obtained for the stresses around the hole in terms of constants and the original varying stress field, without requiring numerical methods. The results are compared to finite element analysis and show close agreement.
Implementation of finite volume method in creeping flow around a circular cyl...eSAT Journals
Abstract
A Finite Volume Method have been performed in simulation of creeping flow around a circular cylinder contained between plates. By adopting the SIMPLE algorithm the governing equations are solved together with Papanastasious regularization. Apparent viscosity is calculated on each iteration by using forward difference operation. Yield surfaces are studied over the range of Oldroyd number 10≤Od≤〖10〗^4. The model results are found to be in good agreement with obtained results of the other method.
Keywords: Creeping flow, Finite Volume Method, Yielded and unyielded zone, Bingham number.
Implementation of finite volume method in creeping flow around a circular cyl...eSAT Journals
Abstract
A Finite Volume Method have been performed in simulation of creeping flow around a circular cylinder contained between plates. By adopting the SIMPLE algorithm the governing equations are solved together with Papanastasious regularization. Apparent viscosity is calculated on each iteration by using forward difference operation. Yield surfaces are studied over the range of Oldroyd number 10≤Od≤〖10〗^4. The model results are found to be in good agreement with obtained results of the other method.
Keywords: Creeping flow, Finite Volume Method, Yielded and unyielded zone, Bingham number.
This document presents information on the bending of single-notched bars. It discusses:
1. The bending behavior depends on the notch angle ψ - for ψ<32.7°, the sliplines are straight meeting the surface at 45° forming a fan shape. For ψ>32.7°, the sliplines are curved circular arcs connecting the stress-free surfaces.
2. Equations are provided to calculate the yield moment, constraint factor, slipline geometry parameters like radii and heights based on the notch angle ψ.
3. A table shows the calculated geometrical parameters and constraint factors for different notch angles ψ.
This document describes a finite element analysis of a thin plate with different hole geometries under tension. Circular, elliptical, and rectangular holes were modeled in the center of a plate. Theoretical stress concentrations were calculated and compared to FE model results. A convergence study determined an optimal mesh size of 25mm. Results from full and quarter plate models showed good agreement, with stress value deviations within 2% and displacement deviations within 1%.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
The document discusses the double integration method for determining beam deflections. It defines beam deflection as the displacement of the beam's neutral surface from its original unloaded position. The differential equation relating the bending moment, flexural rigidity, and slope of the elastic curve is derived. This equation is integrated twice to obtain expressions for the slope and deflection of the beam in terms of the bending moment and constants of integration. Several examples are provided to demonstrate solving for the slope and maximum deflection of beams under different loading conditions using this method.
This document summarizes a mechanical engineering project to optimize the design of a pressure transducer beam for use with a strain gage. The objectives were to design a beam shape that provides equal areas of tension and compression to allow for temperature compensation, areas of constant strain to facilitate gage placement, and a smooth shape to avoid stress concentrations. An iterative process using ANSYS finite element analysis software was used to modify an initial beam cross-section design until the strain plot along the top of the beam met project requirements for sensed pressure levels and strain readings.
The document presents an analytical method for determining stress distributions in pin-loaded orthotropic plates. The method assumes a rigid pin, clearance between the pin and plate, and a constant coefficient of friction. Numerical results are shown for normal, tangential and shear stresses on the cavity for different composite layups. The method can predict stresses for varying clearances and a perfectly fitting pin. Experimental validation and improvements to contact modeling are recommended.
This document summarizes a senior project analyzing the ultimate strength of aircraft structures through testing and analysis. A group of students analyzed and tested beams with different cross-sectional shapes to study local buckling effects on strength, following methods from a paper by their advisor Dr. Todd Coburn. The project involved analyzing critical cross-sections using plastic bending analysis and Cozzone's method. It also developed a hybrid procedure accounting for material non-linearity, flange stability, and other factors influencing ultimate strength. The procedure determines strain distributions and calculates ultimate moments based on stress-strain curves for each section element.
This document presents the development of a state space equation approach to obtain the three-dimensional solution for thick orthotropic plates with symmetric clamped-free edges. The state space method allows for an exact solution that satisfies all equations of elasticity and accounts for all elastic constants. The system matrix, which is a key part of the state space solution, is derived for symmetric clamped-free boundary conditions. Modal expansions are used to develop sixth-order differential equations governing the transverse displacement, whose solutions provide the stresses and displacements across the plate thickness.
This document discusses the derivation of stress equations for curved beams. It outlines the assumptions made, defines relevant terms like the neutral axis and centroidal axis, and derives an equation showing that the stress distribution in a curved beam is hyperbolic. It provides the maximum stress equations and notes they apply for pure bending. Formulas are also provided for calculating stresses in common beam cross sections like rectangular, trapezoidal, T-shaped, and round beams.
IRJET-Fatigue Life Estimation of Machine ComponentsIRJET Journal
This document discusses fatigue life estimation of machine components. It begins with an introduction to fatigue failure, which occurs when fluctuating stresses cause cracks or fractures over many load cycles, even when maximum stresses are below ultimate strength. It then describes three main approaches to estimating fatigue life: stress-life, strain-life, and fracture mechanics. The majority of the document focuses on applying the stress-life approach using a stress-strain diagram to determine the number of cycles before failure based on applied stresses. It provides equations for calculating alternating and mean stresses under different loading types and for determining an equivalent stress under combined loading. The document concludes by outlining the procedural steps to use these methods and equations to estimate the fatigue life of a machine component.
61617, 129 AMP12.081 GO MultipartPage 1 of 6httpsedu.docxalinainglis
6/16/17, 1'29 AMP12.081 GO Multipart
Page 1 of 6https://edugen.wileyplus.com/edugen/shared/assignment/test/qprint.uni
Attempts: Unlimited
Print by: SALEH HUSSAIN
ENGR 213-001 Sp17 / HW CH 12
*P12.081 GO Multipart
*Part 1
Correct
Consider a point in a structural member that is subjected to plane stress. Normal and shear stress magnitudes
acting on horizontal and vertical planes at the point are Sx = 190 MPa, Sy = 110 MPa, and Sxy = 85 MPa.
Assume .
Construct Mohr’s circle for this state of stress on paper and use the results to answer the questions in the
subsequent parts of this GO exercise.
For this Mohr’s circle, point x, which represents the state of stress on the x face of the stress element, should
appear:
*Part 2
Below and to the left of the circle center.
Above and to the left of the circle center.
Below and to the right of the circle center.
Above and to the right of the circle center.
6/16/17, 1'29 AMP12.081 GO Multipart
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Attempts: Unlimited
Attempts: Unlimited
Attempts: Unlimited
Correct
For the given state of stress at a point in a structural member, determine the center C and the radius R of
Mohr’s circle.
Answers: C = MPa, R = MPa.
Answer *1: the tolerance is +/-2%
Answer *2: the tolerance is +/-2%
*Part 3
Correct
Determine the principal stresses ( ) for this state of stress.
Answers: MPa, MPa.
Answer *1: the tolerance is +/-2%
Answer *2: the tolerance is +/-2%
*Part 4
Incorrect
Use the coordinates of point x and the center C of the circle to determine the acute central angle. The other
central angle is 180° minus the acute angle.
For this Mohr’s circle, determine the magnitude of the central angle between:
(a) point x and the point representing the principal stress .
(b) point x and the point representing the principal stress .
Answers:
(a) Angle = °,
(b) Angle = °.
Answer *1: the tolerance is +/-2%
Answer *2: the tolerance is +/-2%
-150 93.94
-56.058 -243.941
-65 *1
-115 *2
6/16/17, 1'29 AMP12.081 GO Multipart
Page 3 of 6https://edugen.wileyplus.com/edugen/shared/assignment/test/qprint.uni
Part 5
Correct
Angles measured in Mohr’s circle are double angles . The orientation of the principal planes in the x–y
coordinate system is defined by an angle . If the acute angle determined in Part 4 is used for , which
sketch correctly defines the orientation of the principal planes?
6/16/17, 1'29 AMP12.081 GO Multipart
Page 4 of 6https://edugen.wileyplus.com/edugen/shared/assignment/test/qprint.uni
Attempts: Unlimited
Attempts: Unlimited
*Part 6
Correct
(a) Determine the magnitude of the maximum in-plane shear stress.
(b) Determine the normal stress that acts on planes of maximum in-plane shear stress. Give the stress value
including sign if any.
Answers:
(a) MPa.
(b) MPa.
Answer *1: the tolerance is +/-2%
Answer *2: the tolerance is +/-2%
*Part 7
93.941
-150
6/16/17, 1'29.
Happy Birthday to you dear sir please find the attachment of my past but Ivishalyadavreso1111
1. Mohr's circle is a graphical representation of the state of plane stress at a point using a circle constructed from the known normal and shear stresses.
2. Key points on Mohr's circle correspond to the principal stresses and maximum shear stress, with their values determined from the circle's construction.
3. Drawing Mohr's circle involves plotting the known stresses, constructing the circle, and determining critical points that represent maximum and minimum normal and shear stresses on planes through the point of interest.
Low power architecture of logic gates using adiabatic techniquesnooriasukmaningtyas
The growing significance of portable systems to limit power consumption in ultra-large-scale-integration chips of very high density, has recently led to rapid and inventive progresses in low-power design. The most effective technique is adiabatic logic circuit design in energy-efficient hardware. This paper presents two adiabatic approaches for the design of low power circuits, modified positive feedback adiabatic logic (modified PFAL) and the other is direct current diode based positive feedback adiabatic logic (DC-DB PFAL). Logic gates are the preliminary components in any digital circuit design. By improving the performance of basic gates, one can improvise the whole system performance. In this paper proposed circuit design of the low power architecture of OR/NOR, AND/NAND, and XOR/XNOR gates are presented using the said approaches and their results are analyzed for powerdissipation, delay, power-delay-product and rise time and compared with the other adiabatic techniques along with the conventional complementary metal oxide semiconductor (CMOS) designs reported in the literature. It has been found that the designs with DC-DB PFAL technique outperform with the percentage improvement of 65% for NOR gate and 7% for NAND gate and 34% for XNOR gate over the modified PFAL techniques at 10 MHz respectively.
Using recycled concrete aggregates (RCA) for pavements is crucial to achieving sustainability. Implementing RCA for new pavement can minimize carbon footprint, conserve natural resources, reduce harmful emissions, and lower life cycle costs. Compared to natural aggregate (NA), RCA pavement has fewer comprehensive studies and sustainability assessments.
KuberTENes Birthday Bash Guadalajara - K8sGPT first impressionsVictor Morales
K8sGPT is a tool that analyzes and diagnoses Kubernetes clusters. This presentation was used to share the requirements and dependencies to deploy K8sGPT in a local environment.
Advanced control scheme of doubly fed induction generator for wind turbine us...IJECEIAES
This paper describes a speed control device for generating electrical energy on an electricity network based on the doubly fed induction generator (DFIG) used for wind power conversion systems. At first, a double-fed induction generator model was constructed. A control law is formulated to govern the flow of energy between the stator of a DFIG and the energy network using three types of controllers: proportional integral (PI), sliding mode controller (SMC) and second order sliding mode controller (SOSMC). Their different results in terms of power reference tracking, reaction to unexpected speed fluctuations, sensitivity to perturbations, and resilience against machine parameter alterations are compared. MATLAB/Simulink was used to conduct the simulations for the preceding study. Multiple simulations have shown very satisfying results, and the investigations demonstrate the efficacy and power-enhancing capabilities of the suggested control system.
ACEP Magazine edition 4th launched on 05.06.2024Rahul
This document provides information about the third edition of the magazine "Sthapatya" published by the Association of Civil Engineers (Practicing) Aurangabad. It includes messages from current and past presidents of ACEP, memories and photos from past ACEP events, information on life time achievement awards given by ACEP, and a technical article on concrete maintenance, repairs and strengthening. The document highlights activities of ACEP and provides a technical educational article for members.
CHINA’S GEO-ECONOMIC OUTREACH IN CENTRAL ASIAN COUNTRIES AND FUTURE PROSPECTjpsjournal1
The rivalry between prominent international actors for dominance over Central Asia's hydrocarbon
reserves and the ancient silk trade route, along with China's diplomatic endeavours in the area, has been
referred to as the "New Great Game." This research centres on the power struggle, considering
geopolitical, geostrategic, and geoeconomic variables. Topics including trade, political hegemony, oil
politics, and conventional and nontraditional security are all explored and explained by the researcher.
Using Mackinder's Heartland, Spykman Rimland, and Hegemonic Stability theories, examines China's role
in Central Asia. This study adheres to the empirical epistemological method and has taken care of
objectivity. This study analyze primary and secondary research documents critically to elaborate role of
china’s geo economic outreach in central Asian countries and its future prospect. China is thriving in trade,
pipeline politics, and winning states, according to this study, thanks to important instruments like the
Shanghai Cooperation Organisation and the Belt and Road Economic Initiative. According to this study,
China is seeing significant success in commerce, pipeline politics, and gaining influence on other
governments. This success may be attributed to the effective utilisation of key tools such as the Shanghai
Cooperation Organisation and the Belt and Road Economic Initiative.
Introduction- e - waste – definition - sources of e-waste– hazardous substances in e-waste - effects of e-waste on environment and human health- need for e-waste management– e-waste handling rules - waste minimization techniques for managing e-waste – recycling of e-waste - disposal treatment methods of e- waste – mechanism of extraction of precious metal from leaching solution-global Scenario of E-waste – E-waste in India- case studies.
Literature Review Basics and Understanding Reference Management.pptxDr Ramhari Poudyal
Three-day training on academic research focuses on analytical tools at United Technical College, supported by the University Grant Commission, Nepal. 24-26 May 2024
5214-1693458878915-Unit 6 2023 to 2024 academic year assignment (AutoRecovere...
Footing with bi_axial_moments
1. ANALYSIS OF ECCENTRICALLY LOADED
RECTANGULAR FOOTING RESTING ON SOIL
– A NUMERICAL APPROACH
Jignesh V Chokshi, L&T Sargent & Lundy Limited, Vadodara, India
For analysis of isolated rectangular footings with large bi-axial eccentricity, an accurate and
efficient numerical approach satisfying all equilibrium conditions and suitable on computers is
presented in this paper. Microsoft Excel, a cogent tool globally used by structural engineers,
under its VBA programming environment is chosen for programming the numerical approach
and graphically displaying input and results. A generalized program dealing with any
conditions of eccentricities–zero eccentricity, one-way eccentricity or two-way eccentricity– is
developed for analysis of rectangular footings. Several examples, with different eccentricity
conditions are chosen to investigate accuracy of results and verify performance of the
numerical approach implemented in the program.
Introduction
The bearing pressure distribution for rigid isolated footing resting on soil subjected to axial
load and bending moments can be obtained by,
....................................................……………………...……..(1)
In the equation 1,
p = Bearing pressure under footing base at point (x, z),
P = Axial Load; A = Area of Footing,
Mx, Mz = Moment about X–axis and Z–Axis respectively,
Ix, Iz = Moment of Inertia of footing about X–axis and Z–axis
respectively and,
x, z = Coordinates of point at which bearing pressure is to be
calculated.
From the above, eccentricity of loading for footing can be
derived as,
ex = Eccentricity along X-axis from center of gravity of footing = Mz / P
ez = Eccentricity along Z-axis from center of gravity of footing = Mx / P
For isolated rectangular footings, called footings now onwards, when the loading point k(ex,
ez) lies in middle third of the footing, called Kern (shaded area in Fig. 1), magnitude of p is
positive and the soil below footing is said to be in compression. However, if loading point lies
outside the Kern, magnitude of p at few locations in the footing is negative and that portion of
footing is said to be in tension. Since, there exists no mechanism between soil and footing to
Figure 1: Footing Geometry
p
P
A
M x
I x
z+
M z
I z
x+:=
2. resist the tensile stresses, some portion of footing will remain unstressed and the force
equilibrium will occur in the area of footing which remains in contact with soil. Under these
circumstances, bearing pressure at different points of footing will be modified and the line of
zero stresses will shift towards loading point. The portion outside line of zero pressure will be
completely unstressed and is called footing uplift area.
Footings with one-way eccentricity, either ex or ez outside kern, solution to the problem is
simple. However, for footings with two-way eccentricity ex and ez outside Kern, the solution
is not as simple as that for one-way eccentricity. In the available literature, Teng [1] shows
graphical method, charts and related equations; Roark [2] provides tables and Peck [3]
mentions an iterative method for footing with two-way eccentricity. To automate the footing
design process on computer, tables or charts are cumbersome to implement and the information
is very brief. Hence, for computer implementation of footing design process, a numerical
approach is the best choice. A numerical approach is described in the paper, which solves this
problem with tangible accuracy. In this approach, it is assumed that pressure varies linearly,
the footing is rigid and the effect of soil displacement has no effect on the pressure distribution.
Equilibrium Conditions
In analysis of eccentrically loaded footings, following equilibrium conditions must comply,
1. Volume of bearing pressure envelope shall be equal to the applied load P,
2. CG of bearing pressure envelope shall coincide with location of applied load P.
For footings having large eccentricities, large area of footing will remain unstressed and hence,
the stability of footing demands special attention. Thus, it is imperative to ensure satisfactory
Factor of Safety against overturning. It is also necessary to keep sufficient area of footing
remaining in contact with soil and bearing pressure not exceeding the allowable bearing
pressure of the soil.
Eccentricity Conditions
For a footing, possible eccentricity conditions can be enumerated as follows:
1. ex = 0 and ez = 0 ; or ex, ez within kern area – Compression on entire base of footing
2. ex > Lx/6 and ez =0 ; ex outside kern – One-way eccentricity along X axis
3. ex = 0 and ez > Lz/6 ; ez outside kern – One-way eccentricity along Z axis
4. ex > 0 and ez > 0 ; ex, ez outside kern – Two-way eccentricity
It shall be noted that, conditions 2, 3 and 4 produces tension on some portion of the footing.
Position of Neutral Axis
For footings with loading point outside Kern, the pressure will vary from negative to positive
below footing base. The points of zero pressure on footing edges can be obtained by
substituting p = 0 and appropriate coordinate of footing edges in Eq. 1. The initial position of
neutral axis can be obtained by connecting a line between two points having zero stresses on
adjacent or opposite edges. However, for static equilibrium to occur, there will be significant
shift of initial neutral axis to its final position.
3. As shown in Figures 1 and 2, following positions of neutral axis can be envisaged.
Case 1: No neutral axis – Compression case (Fig. 1)
Case 2: One end on BC and other end on CD, Pressure at C = 0
Case 3: One end on AB and other end on CD, Pressure at B and C = 0
Case 4: One end on BC and other end on AD, Pressure at C and D = 0
Case 5: One end on AB and other end on AD, Pressure at B, C and D = 0
Case 6: Neutral Axis parallel to Z-axis, Pressure at B & C = 0, Pressure at A= Pressure at D
Case 7: Neutral Axis parallel to X-axis, Pressure at C & D = 0, Pressure at A= Pressure at B
Cases 2 through 5 are for footing with two-way eccentricity and cases 6 and 7 are for footing
with one-way eccentricity.
Numerical Approach
One can imagine that it is almost
impossible to obtain a unified
mathematical equation that solves all of
the above-defined cases. Hence, for
effective solution, the numerical
approach is necessary. For a given size
of footing and loading, the numerical
approach suggested by Peck, et. Al. [3]
is adopted and implemented to obtain
faster and accurate solution. The
numerical procedure essentially works
as follows:
1. Read size of footing and loading.
(P, Mx, Mz, Lx, Lz)
2. Calculate the geometrical
properties of footing.(A, Ix, Iz, ex, ez)
3. Calculate the pressure at corners A,
B, C & D.
4. Obtain initial position of neutral
axis for problems having tension on the
corners.
5. For selected neutral axis, calculate
geometric properties, pressure etc. about neutral axis for portion of footing that remains in
contact with soil.
6. Calculate the volume of pressure diagram envelope.
7. Calculate the center of gravity of pressure diagram envelope.
8. Compare values of P, ex, and ez obtained in step 6 and 7 with input parameters. If
difference is too large, alter the position of neutral axis and repeat step 5 to step 8.
Programming Strategy
The solution methods suggested in the literature are very brief and do not explain a detailed
procedure for implementation of the solution technique on digital computer. A systematic
Figure 2: Positions of Neutral Axis
4. numerical procedure is described here demonstrating each component of the programming
implemented for the solution of the problem. Microsoft Excel with its powerful VBA support
is selected for implementing the numerical procedure on computer. The strategy described
here is for case 2. For other cases, necessary changes are taken care in the generalized
program.
1. Read size of footing and applied forces.
2. Establish the acceptable numerical error in results and limit of number of iterations.
3. Calculate geometrical properties, eccentricities and pressures at each corner of footing.
4. Identify the pressure case of footing from Fig. 2 to know initial position of neutral axis.
5. For cases 1, 6 and 7, simply solve the problem using known method. For cases 2 to 5, find
out the position of points G and H on appropriate edges of footing where p=0.
6. Extend point G on edge AB to locate point E and extend point H on edge AD to locate
point J. Now, the problem is restricted to triangle EAJ, triangle EBG and triangle HDJ.
7. In this method, the iterations are performed in two phases. In the first phase, line EJ - the
neutral axis, will be moved, parallel to EJ, towards point K in subsequent iterations. Select
appropriate step for iteration.
8. For each position of neutral axis EJ, calculate distance Z of loading point K, distance b1
for corner A, b2 for corner B and b4 for corner D normal to neutral axis EJ.
9. Calculate moment of inertia of polygon ABGHDA about its base GH using,
Igh = I(∆EAJ) – I(∆EBG) – I(∆HDJ).
10. Calculate pressure at points A, B and D using pi = ( P x Z x bi ) / ( Igh ). In cases 2 to 7,
pressure at C = 0.
11. Calculate volume and CG of pressure envelop of polygon ABGHD using properties of
triangle and tetrahedron.
12. Compare volume of polygon with applied load P, and center of gravity of pressure envelop
with ex and ez. Calculate percentage error in the achieved solution. If the numerical error
is more than acceptable limit, select another axis EJ at next step and repeat step 8 to 12.
13. Store the positions of neutral axis when
individual error for P, ex and ez is within
acceptable limit. This results in storage of three
positions of line EJ. This means that, at any of
these three positions, error for only one of P, ex or
ez will be within acceptable limits.
14. Terminate further iterations when these three
positions of line EJ are traced. Figure 3 shows the
location of line EJ where individual error for P, ex
and ez is found within acceptable limits. This
completes the first phase of iterations where line EJ
is moved parallel to initial neutral axis. It can be
inferred that the true solution, the unique position
of line EJ where numerical error for P, ex and ez is
simultaneously within acceptable limits, lies within the band bounded by three positions of
neutral axis. To extract the solution band limits, find out the lower-most and upper-most
position of EJ. As shown in Fig. 3, the solution band is bounded by a polygon connected
between points E1, E2, J1 and J2.
Figure 3: Solution Band
5. 15. It was observed that to achieve a tangible accuracy of 99 percentage or better, a slightly
larger band shall be used than originally extracted. The same is implemented in
programming by slightly shifting point E1 on left, E2 on right; J1 downward and J2
upward before initiating second phase of iterations. In the second phase, the objective is to
find the position of EJ where error for P, ex and ez is within limits simultaneously.
16. The second phase of iterations within the newly formulated solution band is initiated by
assuming the neutral axis as a line joining points E1 and J2 (see Fig. 3). Here, the point E1
is pivoted first and second point of neutral axis is altered from J2 to J1 with appropriate
step size. At every position of neutral axis during the iterations, all steps to find out
volume of pressure diagram and CG of pressure envelope are repeated as explained earlier.
Also, the numerical errors for P, ex and ez are calculated to monitor the convergence and
limit on number of iterations is also verified at each step. If the solution is not converged
with the selected pivot, then pivot E1 is shifted at the next step towards E2. The entire
range from E1 to E2 will be pivoted during these iterations, with other end from J2 to J1
until the true solution is found. While iterating within J2 to J1, if the solution diverges, the
program abandons further iterations within J2 and J1 and new pivot point within E1 and
E2 is selected.
17. It shall be noted that, during iterations, the position of line EJ may get changed from one
case to another. For example, at the beginning of the iterations, the position of line EJ
may be representing case 2. However, during subsequent iterations, the position of line EJ
may represent case 3, 4 or 5. The program constantly monitors the case of current neutral
axis and calculates required properties accordingly.
18. For true solution to occur, it is imperative that for a particular position of neutral axis
within solution band, the numerical errors for P, ex and ez, all simultaneously, shall be
within allowable limits. The very first instance of such convergence is reported and
further iterations are abandoned. At this point, essential results such as pressures at A, B
and D, uplift area, position of final neutral axis are reported by the program.
19. Since, solution search is an iterative process; it is expected that there may be other
positions of final neutral axis. It is found that the results of other positions do not vary
much for the desired accuracy, and hence, the accuracy of the first instance of solution is
acceptable for all practical purposes.
Results and Graphics Interface
After successful execution of the program, the following output is generated:
1) The input parameters, 2) position of initial neutral axis, 3) position of final neutral axis, 4)
effective compression area, 5) load and loading point coordinates recovered, 6) maximum
pressures at corners and 7) numerical difference in recovering P, ex and ez.
Extensive effort is put on the graphical presentation of input and results. Extraordinary
features of Excel chart options are explored and the graphical features of the program includes:
1. Footing Geometry: Size of footing, origin, loading point, Kern, initial neutral axis and
final neutral axis.
2. Bearing Pressure Diagrams: 2D and 3D presentation of contours showing variation of
pressure, after equilibrium conditions are met, over the footing surface. The footing area
is divided into many small parts to produce refined bearing pressure diagram.
6. Verification Examples
Many practical examples were
selected to validate results
produced by the program and
monitor accuracy of the numerical
approach presented here. The
results were compared with input
data and not with solution
obtained from any other reference.
Table 1 shows input data and true
solution for selected problems.
Note that in all problems a
tangible accuracy of 99.9% is
achieved. The table also
demonstrates number of iterations
performed to solve the problem
and run time taken on PC with P4
-1.5GHz processor and 512MB
RAM. Graphical representation
of footing geometry and pressure
distribution diagrams for
examples 1, 2, 3 and 5 are shown
in Fig. 4.1 to 4.8.
Observations and Conclusions
The numerical approach suggested
in this paper produces impressive
results having a tangible accuracy
of 99.9 percentage or better for all
problems under investigation. The
time taken for finding the solution
is computationally economical for
incredible accuracy achieved.
Hence, the numerical approach
presented here can be effectively
implemented to automate the
footing analysis and design.
The use of Excel with its VBA
environment is phenomenally user
friendly and endorses the structural engineers’ acceptance of Excel as a cogent tool for
automating structural design work processes. Even for such a complex problem like footings
with two-way eccentricity, use of Excel is found highly efficient.
Table 1
Verification Problems and Comparison of Results
Problem No
Item
1 2 3 4 5 6
Geometry and Load Data (Units kN and m)
P 278.00 1300.0 1250.0 333.00 2000.0 2000.0
Mx 278.00 162.50 2813.0 150.00 1500.0 1500.0
Mz 250.00 1800.0 750.00 400.00 4000.0 3000.0
Lx 6.00 5.00 6.00 4.00 5.00 5.00
Lz 5.00 2.50 5.00 3.00 2.50 2.50
ex 0.899 1.385 0.600 1.201 2.000 1.500
ez 1.000 0.125 2.250 0.450 0.750 0.750
ex/Lx 0.150 0.277 0.100 0.300 0.400 0.300
ez/Lz 0.200 0.050 0.450 0.150 0.300 0.300
Bearing Pressure at Corners
(Before Modification of Pressure)
PA 28.72 308.00 179.18 102.75 832.00 736.00
PB 12.05 -37.60 129.18 2.75 64.00 160.00
PC -10.18 -100.00 -95.85 -47.25 -512.00 -416.00
PD 6.48 245.60 -45.85 52.75 256.00 160.00
Results obtained by Numerical Method
Case 2 3 4 3 5 5
Step 0.0030 0.0020 0.0030 0.0020 0.0020 0.0010
P’A 32.41 360.24 749.89 146.10 3000.0 1500.0
P’B 11.14 0.00 395.56 0.00 0.00 0.00
P’C 0.00 0.00 0.00 0.00 0.00 0.00
P’D 4.88 265.47 0.00 50.57 0.00 0.00
c 4.624 2.200 4.077 2.923 - -
d 2.976 1.200 4.513 0.888 - -
as % of (Lx x Lz)Contact
Area 77.07 66.01 14.09 52.36 16.00 32.00
Comparison of Results
Precovered 277.97 1300.8 1249.8 333.27 2000.0 2000.0
exrecovered 0.8984 1.3831 0.5996 1.2000 2.0000 1.5000
ezrecovered 1.0008 0.1251 2.2505 0.4503 0.7500 0.7500
(%) Error in
P 0.0088 0.0652 0.0126 0.0804 0.0000 0.0000
ex 0.0647 0.0640 0.0733 0.0838 0.0000 0.0000
ez 0.0822 0.0876 0.0219 0.0737 0.0000 0.0000
Run Time Data
Iterations 1245 640 2081 673 946 1067
Time
(Sec)
7 3 9 3 3 3
7. Example Problem No. 1 ( Case 2 )
Figure 4.1 : Footing Geometry Figure 4.2: Bearing Presure Diagram
Example Problem No. 2 ( Case 3 )
Figure 4.3 : Footing Geometry Figure 4.4: Bearing Presure Diagram
Example Problem No. 3 (Case 4 )
Figure 4.5 : Footing Geometry Figure 4.6: Bearing Presure Diagram
Footings with Two-Way Eccentricity
-3.00
-2.00
-1.00
0.00
1.00
2.00
3.00
-3.00 -2.00 -1.00 0.00 1.00 2.00 3.00
X-Axis: Length of Footing (Lx)
Y-Axis:WidthofFooting(Lz)
Footing Load Point Original_NA Final NA
-3.00
-2.50
-2.00
-1.50
-1.00
-0.50
0.00
0.50
1.00
1.50
2.00
2.50
3.00-
2.50
-2.00
-1.50
-1.00
-0.50
0.00
0.50
1.00
1.50
2.00
2.50
Points along X Axis
PointsalongZAxis
Base Pressure Distribution Diagram - 2D
-2.000-2.000 2.000-6.000 6.000-10.000 10.000-14.000 14.000-18.000
18.000-22.000 22.000-26.000 26.000-30.000 30.000-34.000 34.000-38.000
Footings with Two-Way Eccentricity
-3.00
-2.00
-1.00
0.00
1.00
2.00
3.00
-3.00 -2.00 -1.00 0.00 1.00 2.00 3.00
X-Axis: Length of Footing (Lx)
Y-Axis:WidthofFooting(Lz)
Footing Load Point Original_NA Final NA
-2.50
-2.25
-2.00
-1.75
-1.50
-1.25
-1.00
-0.75
-0.50
-0.25
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50-
1.25
-1.00
-0.75
-0.50
-0.25
0.00
0.25
0.50
0.75
1.00
1.25
Points along X Axis
PointsalongZAxis
Base Pressure Distribution Diagram - 2D
-20.000-20.000 20.000-60.000 60.000-100.000 100.000-140.000
140.000-180.000 180.000-220.000 220.000-260.000 260.000-300.000
300.000-340.000 340.000-380.000
-3.00
-2.50
-2.00
-1.50
-1.00
-0.50
0.00
0.50
1.00
1.50
2.00
2.50
3.00
-2.50
-1.00
0.50
2.00
-2.0
98.0
198.0
298.0
398.0
498.0
598.0
698.0
798.0
Base Pressure Distribution Diagram - 3D 698.000-
798.000
598.000-
698.000
498.000-
598.000
398.000-
498.000
298.000-
398.000
198.000-
298.000
98.000-
198.000
-2.000-
98.000
Footings with Two-Way Eccentricity
-3.00
-2.00
-1.00
0.00
1.00
2.00
3.00
-3.00 -2.00 -1.00 0.00 1.00 2.00 3.00
X-Axis: Length of Footing (Lx)
Y-Axis:WidthofFooting(Lz)
Footing Load Point Original_NA Final NA
8. Example Problem No. 5 (Case 5 )
Figure 4.7 : Footing Geometry Figure 4.8: Bearing Presure Diagram
Acknowledgement
I thank my company M/s. L&T Sargent and Lundy Limited, Vadodara, Gujarat, India, for the
support, encouragement and providing computational facilities for this programming work.
References
1. Foundation Design, Teng W. C., Prentice-Hall Inc., Englewood cliffs, New Jersey.
2. Roark’s Formulas for Stress and Strain, 7th
Edition, Young W. C. and Budynas R. G.,
McGraw Hill, Englewood cliffs, New Jersey.
3. Foundation Engineering, 2nd
Edition, Peck R. B., Hanson W. E., and Thornburn W. H.,
John Wiley and Sons, New York.
-2.50
-2.25
-2.00
-1.75
-1.50
-1.25
-1.00
-0.75
-0.50
-0.25
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
-1.25
-0.50
0.25
1.00
-100.0
300.0
700.0
1100.0
1500.0
1900.0
2300.0
2700.0
3100.0
Base Pressure Distribution Diagram - 3D 2700.000-
3100.000
2300.000-
2700.000
1900.000-
2300.000
1500.000-
1900.000
1100.000-
1500.000
700.000-
1100.000
300.000-
700.000
-100.000-
300.000
Footings with Two-Way Eccentricity
-3.00
-2.00
-1.00
0.00
1.00
2.00
3.00
-3.00 -2.00 -1.00 0.00 1.00 2.00 3.00
X-Axis: Length of Footing (Lx)
Y-Axis:WidthofFooting(Lz)
Footing Load Point Original_NA Final NA