Numerical Methods in Soil Mechanics E.PDF

Numerical Methods in Soil Mechanics E.PDF Numerical Methods in Geotechnical Engineering contains the proceedings of the 8th European Conference on Numerical Methods in Geotechnical Engineering (NUMGE 2014, Delft, The Netherlands, 18-20 June 2014). It is the eighth in a series of conferences organised by the European Regional Technical Committee ERTC7 under the auspices of the International Society for Soil Mechanics and Geotechnical Engineering (ISSMGE). The first conference was held in 1986 in Stuttgart, Germany and the series has continued every four years (Santander, Spain 1990; Manchester, United Kingdom 1994; Udine, Italy 1998; Paris, France 2002; Graz, Austria 2006; Trondheim, Norway 2010). Numerical Methods in Geotechnical Engineering presents the latest developments relating to the use of numerical methods in geotechnical engineering, including scientific achievements, innovations and engineering applications related to, or employing, numerical methods. Topics include: constitutive modelling, parameter determination in field and laboratory tests, finite element related numerical methods, other numerical methods, probabilistic methods and neural networks, ground improvement and reinforcement, dams, embankments and slopes, shallow and deep foundations, excavations and retaining walls, tunnels, infrastructure, groundwater flow, thermal and coupled analysis, dynamic applications, offshore applications and cyclic loading models. The book is aimed at academics, researchers and practitioners in geotechnical engineering and geomechanics.

Anderson, Loren Runar et al "STRESS ANALYSIS"

Structural Mechanics of Buried Pipes

Boca Raton: CRC Press LLC,2000

APPENDIX E STRESS ANALYSIS

Figure E-1 shows a stressed medium with an

infinitesimal cube, Op Op is the

free-body-diagram for stress analysis A maximum of six

pairs of stresses can act on Op, three normal stresses

(direct stresses) and three couples (shearing stress

pairs) The stresses all occur in pairs in

equilibrium Notation is as follows:

sx = normal stress in the x-direction acting on a

yz-plane,

sy = normal stress in the y-direction acting on a

yz-plane,

sz = normal stress in the z-direction acting on

an xy plane,

tx = each of two equal and opposite shearing

couples acting about an x-axis in

yz-planes,

ty = each of two equal and opposite shearing

couples acting about a y-axis in xz-planes

tz = each of two equal and opposite shearing

couples acting about a z-axis in xy planes

The resultant normal and shearing stresses on any

plane that passes through the cube can be found by

a stress circle proposed by Otto Mohr and found in

texts on mechanics of solids The Mohr circle is

applied separately to each of the three orthogonal

views of the cube The Mohr circle, Figure E-1,

applies to a view of the xy-plane (front view) The

two couples are shown as shearing stress pairs on

surfaces of the cube that are at right angles to each

other The shearing couple on the top and bottom

is equal and opposite to the shearing couple on the

sides in order to satisfy equilibrium conditions

Because the couples are equal, the subscript is

dropped for shearing stress t Analysis comprises

three steps

A The Mohr stress circle is plotted by first drawing

the stress axes, s and t , as shown Three points

establish the stress circle on the axes From the

stresses on the infinitesimal cube, points on the

stress axes are plotted as follows:

1 Plot the normal and shearing stress point, (s , t ) acting on the y-plane,

2 Plot the normal and shearing stress point, (sy, t ) acting on the x-plane,

3 Locate the center of the circle on the s -axis

By simply connecting the two plotted points, the center of the circle is located on the s -axis Contrary to popular sign convention, compression

is positive for normal stress; and counterclockwise

is positive for couples (shearing stresses) Molecular bonding forces hold the material together in compression Tension is simply a reduction in compressive bonds in the material

B The orientation diagram is x and y, representing

x and y planes, shown dotted on the infinitesimal cube The orientation diagram is superimposed on the stress diagram The x-plane is drawn through point (sy,t ) representing the stresses on that x-plane; and the y-plane is drawn through point (sx,

t ) representing the stresses on that y-plane The origin, Op, (called the origin of planes) always falls

on the Mohr stress circle The origin of planes is actually the infinitesimal cube superimposed on the stress diagram Any plane through the Op is a plane through the infinitesimal cube Analysis is the following:

ANY PLANE DRAWN FROM THE ORIGIN OF

INTERSECTS THE MOHR CIRCLE AT THE NORMAL AND SHEARING STRESSES ACTING ON THAT 2-PLANE

The q-plane is correctly oriented with respect to the original x-y coordinate axes of the cube Clearly, maximum and minimum normal stresses (called principal stresses) and maximum shearing stresses can be found on the circle It is common practice to identify the maximum principal stress as the furthest point to the right on the circle, and the minimum principal stress as the furthest point to

Figure E-1 Stress analysis for the front view of infinitesimal cube Op in a stressed medium, showing the three superimposed diagrams: Mohr stress circle, orientation diagram x-y, and strength envelopes

the left Maximum shearing stresses are the highest

and lowest point on the circle The highest is plus

(counterclockwise couple) and the lowest is

negative (clockwise couple)

Of passing interest are some important stress

theorems which can be demonstrated by the Mohr

circle Shearing stresses are zero on the planes of

principal stresses (called principal planes)

Principal stresses (maximum and minimum) act on

planes that are perpendicular to each other

Maximum shearing stresses are equal to the average

of the principal stresses, and act on perpendicular

planes at 45o with the principal planes Planes of

maximum shearing stresses are at right angles to

each other, but one shearing stress is positive and

the other is negative The normal stresses are equal

on both planes of maximum shearing stress The

planes on which stresses act are the dotted lines

from Op to the stress points on the Mohr circle

The numerical values of stresses can be found

either by drawing to scale the Mohr circle and

measuring values, or by trigonometry Following

are some useful trigonometric hints

1 The center of the circle is located on the s

-axis at (sx + σ y)/2

2 The square of the radius of the circle is

[(sx - σ y)2 + τ 2

] by the Pythagorean theorem

3 If a central angle drawn from the center of

the circle intercepts the same arc segment as a

circumferential angle drawn from Op, the central

angle is twice the corresponding circumferential

angle For example, on Figure E-1, the failure angle, b, is half the corresponding central angle of

2b = 90o + j - a Because j and a are known, the failure angle b can be evaluated

C The strength envelopes are the limits of stresses

in the material If stresses increase to the point where the stress circle touches the strength envelope, the material fails on failure plane, b, from

Op to the point of tangency Strength envelopes can be determined in the laboratory by loading the material to failure under tri-axial loads See Figure E-2 A Mohr circle is drawn for each of the failures Tangents to the circles become the strength envelopes Of interest is the theoretical origin at the intersection of the strength envelopes Ideally it represents the sum of all the bonding forces in the material and represents the absolute maximum tensile strength of perfect strands of molecules In fact, perfect strands are unrealistic Nevertheless, in the case of steel, the ideal origin of strength envelopes is so far to the left, that the strength envelopes are essentially horizontal in the range of strengths of steel in typical usage Figure E-3 depicts strength envelopes for steel Because the strength envelopes are essentially parallel, yield stress (failure) in tension is almost the same as in compression Shown on the figure, in solid line, is

a Mohr circle for tension failure With the Op located as shown, failure planes are at 45o It is possible to see these slip planes in some specimens

of failed steel They are called Leuder's lines In typical failures of thin-wall pipes or tanks subjected to excessive internal pressure, the fracture surface is beveled at roughly 45o

Figure E-2 Development of strength envelopes by drawing tangents to Mohr circles plotted from triaxial tests to failure

Figure E-3 Sketch depicting strength evelopes for steel, showing how strength envelopes are parallel because yield stress, sy, is approximately equal in both tension and compression; and showing the 45o