Numerical Methods in Soil Mechanics 03.PDF

Numerical Methods in Soil Mechanics 03.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 "RING DEFORMATION"

Structural Mechanics of Buried Pipes

Boca Raton: CRC Press LLC,2000

Figure 3-1 (top) Vertical compression (strain) in a medium transforms an imaginary circle into an ellipse with decreases in circumference and area

(bottom) Now if a flexible ring is inserted in place of the imaginary ellipse and then is allowed to expand such that its circumference remains the same as the original imaginary circle, the medium in contact with the ring

is compressed as shown by infinitesimal cubes at the spring lines, crown and invert

CHAPTER 3 RING DEFORMATION

Deformation of the pipe ring occurs under any load

For most buried pipe analyses, this deformation is

small enough that it can be neglected For a few

analyses, however, deformation of the ring must be

considered This is particularly true in the case of

instability of the ring, as, for example, the hydrostatic

collapse of a pipe due to internal vacuum or external

pressure Collapse may occur even though stress

has not reached yield strength But collapse can

occur only if the ring deforms Analysis of failures

requires a knowledge of the shape of the deformed

ring

For small ring deflection of a buried circular pipe, the

basic deflected cross section is an ellipse Consider

the infinite medium with an imaginary circle shown

in Figure 3-1 (top) If the medium is compressed

(strained) uniformly in one direction, the circle

becomes an ellipse This is easily demonstrated

mathematic ally Now suppose the imaginary circle

is a flexible ring When the medium is compressed,

the ring deflects into an approximate ellipse with

slight deviations If the circumference of the ring

remains constant, the ellipse must expand out into

the medium, increasing compressive stresses

between ring and medium See Figure 3-1 (bottom)

The ring becomes a hard spot in the medium On

the other hand, if circumference of the ring is

reduced, the ring becomes a soft spot and pressure

is relieved between ring and medium In either case,

the basic deformation of a buried ring is an ellipse —

slightly modified by the relative decreases in areas

within the ring and without the ring The shape is

also affected by non-uniformity of the medium For

example, if a concentrated reaction develops on the

bottom of the ring, the ellipse is modified by a flat

spot Nevertheless, for small soil strains, the basic

ring deflection of a flexible buried pipe is an ellipse

Following are some pertinent approximate

geometrical properties of the ellipse that are

sufficiently accurate for most buried pipe analyses

Greater accuracy would require solutions of infinite

series

Geometry of the Ellipse

The equation of an ellipse in cartesian coordinates,

x and y, is:

a2x2 + b2y2 = a2b2

where (See Figure 3-2):

a = minor semi-diameter (altitude)

b = major semi-diameter (base)

r = radius of a circle of equal circumference The circumference of an ellipse is p(a+b) whi c h reduces to 2pr for a circle of equal circumference

In this text a and b are not used because the pipe industry is more familiar with ring deflection, d Ring deflection can be written in terms of semi-diameters a and b as follows:

d = D/D = RING DEFLECTION (3.1)

where:

D = decrease in vertical diameter of ellipse from

a circle of equal circumference,

= 2r = mean diameter of the circle — diameter to the centroid of wall cross-sectional areas,

a = r(1-d) for small ring deflections (<10%),

b = r(1+d) for small ring deflections (<10%)

Assuming that circumferences are the same for circle and ellipse, and that the vertical ring deflection

is equal to the horizontal ring deflection, area within the ellipse is Ae = Bab; and

Ae= pr2 (1 - d2) The ratio of areas within ellipse and circle is:

Ar= A e/ A o = ratio of areas

See Figure 3-3

Figure 3-2 Some approximate properties of an ellipse that are pertinent to ring analyses of pipes where d is the ring deflection and ry and rx are the maximum and minimum radii of curvature, respectively

Figure 3-3 Ratio of areas, Ar = Ae /Ao

(Ae within an ellipse and A o within a

circle of equal circumference) shown

plotted as a function of ring deflection

What of the assumption that the horizontal and

vertical ring deflections are equal if the

circumferences are equal for circle and ellipse? For

the circle, circumference is 2pr For the ellipse,

circumference is (b+a)(64-3R4)/(64-16R2), where R

is approximately R = (b-a)/(b+a) Only the first

terms of an infinite series are included in this

approximate ellipse circumference See texts on

analytical geometry Equating circumferences of the

circle and the ellipse, and transforming the values of

a and b into vertical and horizontal values of ring

deflection, dy and dx, a few values of dy and the

correspondin g dx are shown below for comparison

Deviation

dy (%) dx (%) (dy - dx)/dy

For ring deflections of d = dy = 10%, the

corresponding dx is less than 10% by only

4.8%(10%) = 0.48% This is too small to be

significant in most calculations such as areas within

the ellipse and ratios of radii

Radii of curvature of the sides (spring lines) and the

top and bottom (crown and invert) of the ellipse are:

rx = r (1 - 3d + 4d2 - 4d3 + 4d4 - )

ry = r (1 + 3d + 4d2 + 4d3 + 4d4 + )

For ring deflection less than d = 10%, and neglecting

higher orders of d,

rx = r (1 - 3d), and ry = r (1 + 3d)

However, more precise, and almost as easy to use,

are the approximate values:

rx = a2 / b = r (1 - d)2 / (1 + d)

ry = b2 / a = r (1 + d)2 / (1 - d)

An important property of the ellipse is the ratio of radii rr = ry/rx, which is:

rr= (1 + d)3 / (1 - d)3 (3.2) where:

rr = ratio of the maximum to minimum radii of curvature of the ellipse See graph of Figure 3-4

Measurement of Radius of Curvature

In practice it is often necessary to measure the radius of curvature of a deformed pipe This can be done from either inside or outside of the pipe See

Figure 3-5 Inside, a straightedge of known length L

is laid as a cord The offset e is measured to the curved wall at the center of the cord Outside, e can

be found by laying a tangent of known length L and

by measuring the offsets e to the pipe wall at each end of the tangent The average of these two offsets is the value for e Knowing the length of the cord, L, and the offset, e, the radius of curvature of the pipe wall can be calculated from the following equation:

r = (4e2 + L2)/8e (3.3)

It is assumed that radius of curvature is constant within cord length L The calculated radius is to the surface from which e measurements are made Example

An inspection reveals that a 72-inch corrugated metal pipe culvert appears to be flattened somewhat

on top From inside the pipe, a straightedge (cord)

12 inches long is placed against the top, and the mid-ordinate offset is measured and found to be 11/32 inch What is the radius of curvature of the pipe ring

at the top?

From Equation 3.3, r = (4e2 + L2)/8e Substituting in values and solving, ry = 52.5 inches which is the average radius within the 12-inch cord on the inside

of the corrugated pipe On the outside, the radius is greater by the depth of the corrugations

d (%) rr

3 1.197

4 1.271

5 1.350

6 1.434

8 1.618

10 1.826

12 2.062

15 2.476

20 3.375

Figure 3-4 Ratio of radii,

rr= ry/rx = (1+d)3/(1-d)3,

(ry and rxare maximum and

minimum radii, respectively,

for ellipse) shown plotted

as a function of ring deflection d

Figure 3-5 Procedure for calculating the radius of curvature of a ring from measurements of a cord of length

L and the middle ordinate e

Ring Deflection Due to Internal Pressure

When subjected to uniform internal pressure, the

pipe expands The radius increases Ring deflection

is equal to percent increase in radius;

d = Dr/r = DD/D = 2pre/2pr = e

where:

d = ring deflection (percent),

Dr and DD are increases due to internal pressure,

r = mean radius,

D = mean diameter,

e = circumferential strain,

E = modulus of elasticity = s/e

s = circumferential stress = Ee = Ed

But s = P'(ID)/2A, from Equation 2.1,

where:

P' = uniform internal pressure,

ID = inside diameter,

A = cross sectional area of wall per unit length

Equating the two values for s , and solving for d,

Figure 3-6 Quadrant of a circular cylinder fixed at

the crown A-A-A with Q-load at the spring line,

B-B-B, showing a slice isolated for analysis

Ring Deformation Due to External Loading

Computer software is available for evaluating the deformation of a pipe ring due to any external loading Analysis is based on the energy method of virtual work according to Castigliano Analysis provides a component of deflection of some point B

on a structure with respect to a fixed point A It is convenient to select point A as the origin of fixed coordinate axes — the axes are neither translated nor rotated See Appendix A

Example Consider the quadrant of a circular cylinder shown

in Figure 3-6 It is fixed along edge A-A-A, and is loaded with vertical line load Q along free edge

B-B-B What is the horizontal deflection of free edge B with respect to fixed edge A? This is a two-dimensional problem for which a slice of unit width can be isolated for analysis Because A is fixed, the horizontal deflection of B with respect to A is xB for which, according to Castigliano:

xB = f (M/EI)(dM/dp)ds (3.5)

xB = displacement of point B in the x-direction,

EI = wall stiffness,

E = modulus of elasticity,

I = centroidal moment of inertia of the cross

section of the wall per unit length of cylinder,

M = moment of force about the neutral axis at C,

p = differential load (dummy load) applied at

point B in the direction assumed for deflection,

ds = differential length along the slice, = rdq

r = mean radius of the circular cylinder

It is assumed that deflection is so small that radius r

remains constant It is also assumed that the

deflection is due to moment M, flexure — not to

shear or axial loads In Figure 3-6, consider arc CB

as a free-body-diagram Apply the dummy load p at

B acting to the right assuming that deflection xB will

be in the x-direction If the solution turns out to be

negative, then the deflection is reversed From the

free-body-diagram CB,

M = Qr(1-cosq) + pr(sinq)

M/ p = r(sinq)

But because p approaches zero (differential),

M = Qr(1-cosq)

ds = rdq

Substituting into Equation 3.5,

xB = (Qr/EI) (1-cosq) r(sinq) rdq

Integrating and substituting in limits of q from 0 to

p/2,

xB = Qr3/2EI

This is one of a number of the most useful

deflections of rings recorded in Table A-1

PROBLEMS

3-1 A plain polyethylene pipe of 16-inch outside diameter and DR = 15 is subjected to internal pressure of 50 psi The surfaces are smooth and cylindrical (not ribbed or corrugated) DR (dimension ratio) = (OD)/t where t = wall thickness Modulus of elasticity is 115 ksi What is the ring deflection? DR is dimension ratio = (OD)/t

(d = 0.28%)

3-2 At ring deflection of 15%, and assuming the pipe cross section is an ellipse, what is the percent error in finding the ratio of maximum to minimum

radii of curvature by means of approximate

Equation 3.2,

rr = (1+d)3 / (1-d)3? (0.066%)

3-3 A 36 OD PVC buried pipeline is uncovered at one location The top of the pipe appears to be flattened A straight edge 200 mm long is laid horizontally across the top and the vertical distances down to the pipe surface at each end of the straight edge are measured and found to be 9.2 and 9.4 mm What is the radius of curvature of the outside surface of the pipe at the crown?

Ry = 542 mm = 21.35 inches)

3-4 Assuming that the ring of problem 3-3 is deflected into an ellipse, approximately what is the ring deflection? Maximum ring deflection is usually limited to 5% according to specifications

3-5 What is the percent decrease in cross-sectional area inside the deflected pipe of problem 3-4 if the ring deflection is d = 5.74%?

(0.33%)

3-6 What is the approximate ratio of maximum to minimum radii, rr, for an ellipse? (rr = 1.8)

3-7 A horizontal, rectangular plate is a cantilever beam loaded by a uniform vertical pressure, P, and supported (fixed) along one edge What is the vertical deflection of the opposite edge? The thickness of the plate is t, the length measured from the fixed edge is L, and the modulus of elasticity is

E Elastic limit is not exceeded Use the Castigliano equation (y = 3PL4 / 2Et3)

3-8 A half of a circular ring is loaded at the crown

by an F-load (load per unit length of the cylinder) The reactions are rollers at the spring lines B, as shown If the wall stiffness is EI, what is the vertical deflection of point A?

(yA = 0.1781 Fr3/EI)

3-9 What is the vertical ring deflection of the hinged arch of problem 3-8 if it is loaded with a uniform vertical pressure P instead of the F-load?

3-10 The top and bottom halves of the circular cylinder of problem 3-8 are symmetrical If the spring lines of the two halves are hinged together, what is the ring deflection due to the F-load and an equal and opposite reaction at the bottom?

(d = 0.1781 Fr2/EI)

3-11 Sections of pipe are tested by applying an F-load For flexible rings, the F-load test is called a parallel plate test What is the ring deflection if elastic limit is not exceeded?

[d = 0.0186F/(EI/D3)D]

3-12 Find EI = f(Q/x) at point B for the ring cut at A and loaded by force, Q (EI = 3p Qr/x)

3.13 A pipe in a casing floats when liquid grout is introduced between pipe and casing Find the moment, thrust and shear at crown and invert

(See Table A-1)