Numerical Methods in Soil Mechanics 08.PDF

Numerical Methods in Soil Mechanics 08.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 STIFFNESS"

Structural Mechanics of Buried Pipes

Boca Raton: CRC Press LLC,2000

Figure 8-1 Typical stress-strain diagram from laboratory tests showing:

E = modulus of elasticity = slope within the elastic zone

E" = stiffness = slope at any point on the graph,

or the average slope of the cord in some stress range such as A to B

Figure 8-2 Comparison of a leaf spring and a cylindrical spring showing the stiffness F/D for each For the ring, F/D is called pipe stiffness and EI/D3 is called ring stiffness

CHAPTER 8 RING STIFFNESS

It is noteworthy that analyses of ring deformation

and ring deflection, Chapters 3 and 7, require a

property of material called stiffness Stiffness is

defined as resistance to deflection Figure 8-1

shows a typical stress-strain diagram The abscissa

is strain which is deflection per unit length The

ordinate is stress which is load per unit area and

which is resistance of the material to strain Such

diagrams come from laboratory tests Stiffness of

the material is the slope, E, of the stress-strain

diagram at any particular point — at any particular

stress Stress-strain diagrams can be provided for

shearing stress-strain as well as normal stress-strain

The initial linear portion of the diagram is the elastic

zone within which stress causes no permanent

deformation The material rebounds elastically The

slope, which is constant, is called the modulus of

elasticity

Notation:

E = modulus of elasticity or stiffness,

E = slope of the stress-strain diagram,

s = normal stress (or shearing stress t ),

e = normal strain (or shearing strain g),

I = centroidal moment of inertia of the

cross-sectional area (of the wall per unit length in

the case of a pipe),

L = length (as of a leaf spring),

D = mean diameter of the ring,

t = wall thickness of the ring (or cross-

sectional area A per unit length of pipe)

F = concentrated load (diametral line load

per unit length in the case of a pipe)

The concept of stiffness is easily extended to a

spring See Figure 8-2 The leaf spring on the left

is deflected a vertical distance by load F The

spring stiffness is the slope of the F/ diagram and

is equal to F/ = 48EI/L3 See texts on mechanics

of materials An interesting comparison can be

drawn between the leaf spring and a circular spring,

or ring, on the right The leaf spring is analogous to

the circular spring which, from Appendix A, has a

s pring stiffness of F/ = 53.77EI/D3 (Sp)ring

stiffness is similar in form to the leaf spring stiffness

It contains the term, EI/D3, which shows up in every analysis of ring deformation and deflection EI/D3 is

an important form of spring stiffness For a pipe with a rectangular wall cross section, I = t3/12, the ring stiffness becomes F/D = 4.48E/(D/t)3 D/t is another form of ring stiffness Stiffness can be expressed in a variety of ways:

EQUIVALENT QUANTITIES NOMENCLATURE F/D Pipe Stiffness

= 53.77 (EI/D3) Ring Stiffness, EI/D 3

= 4.48 E(t/D)3 where D/t = m-TERM

= 4.48 E/(DR-1)3 where DR = (OD)/t

= Dimension Ratio

Units of stiffness can be reconciled by noting that both I and F are per unit length of pipe In terms of basic dimensions, stiffnesses are:

STIFFNESS RING = (EI/D3) FL-2(L4/L)L-3 = FL-2 PIPE = (F/D) (F/L)L-1 = FL-2

FL-2 is the correct dimension for stiffness of material Many properties of materials have the dimension FL-2 Check out, for example, strength, and bulk modulus The m-term (D/t) and the dimension ratio (DR) are not properties of material until multiplied by E, which has the dimension FL-2

In pipe analysis, EI is sometimes referred to as wall stiffness In fact, EI is not a true stiffness because its dimension is not FL-2

Beyond the zone of elasticity, stiffness E is still the slope of the F/D diagram However, it is no longer

a constant From the stress-strain diagram of Figure 8-1, if the material is stressed to its ultimate where the slope is zero, it loses all stiffness and simply flows In the case of pipe stiffness, F/D often extends beyond the zone of elasticity This is true,

in particular, of plastic pipes — including metals which act as plastics after reaching yield stress For some materials F/D is affected by temperature and/or time Pipe stiffness, F/D, is preferred by plastic pipe industries because it can

Figure 8-3 Two methods of testing for pipe stiffness F/)

Figure 8-4 Plots of data from two parallel plate tests on a 4-inch PVC sewer pipe

be measured by a parallel plate test To perform the

test, a length of pipe, usually longer than one

diameter, on a flat surface is F-loaded as shown in

Figure 8-3 As load F is applied in increments,

corresponding deflections, D, are measured The

plot of F vs D provides F/D values (pipe stiffness)

within any load limits based on temperature and time

(rate of loading) of the test A similar test is the

three-edge-bearing (TEB) test See Figure 8-3

Double supports on the bottom position the pipe For

purposes of analysis, the TEB test is equivalent to

a parallel plate test It is the basis of design of rigid

pipes Plastic pipes are designed by pipe stiffness

defined as slope of the secant from the origin to the

point of five percent ring deflection on the F vs

D plot

Figure 8-4 is a plot of data from two parallel plate

tests on a 4"PVC sewer pipe The pipe stiffness

(slope F/D) is not constant Ring deflection is

usually limited by specification to a maximum of 5%,

which justifies the slope of a secant from the origin

to five percent strain; i.e., F/D = 85 psi

Based on values for F/D, plastic pipe industries can

evaluate the DR-term, the stiffness ratio, Rs, etc

For example, if E must be modified to serve in a

different temperature than the parallel plate test, an

adjusted value can be found for stiffness ratio, Rs =

E'D3/EI, in predicting ring deflection

Other pipe industries have their reasons for using

F/D The seams in riveted pipes or lock seams in

spiral pipes allow enough slippage to affect EI/D3

The stiffness of mortar lined and/or coated pipes is

affected by hairline cracking of the mortar

Reinforced concrete pipes defy analysis of Rs, et c

Plastic pipe industries favor the use of dimension

ratio, DR, which is defined as the ratio of average

outside diameter to minimum wall thickness The

outside diameter is held constant in the extrusion

machine Wall thickness is varied for the class of

pipe (wall strength) to be produced Steel pipe

industries favor the use of the m-term which is

defined as the ratio of the wall thickness to the

mean diameter Steel industries often use the

inverse D/t-term, which is called ring flexibility

Ring stiffness (EI/D ) is preferred by the steel industries E is a constant Values for I can be calculated Because D and t describe pipes, D/t finds its way into much ring deflection analysis For corrugated pipes, tables of values are published for moments of inertia, I With I, E, and D known, the ring stiffness, EI/D3, can be calculated Parallel plate tests, or TEB tests, on corrugated pipes are a check against calculated ring stiffness, and often reveal differences If ring stiffness is critical, tests can verify or modify the calculated values Lengths

of test pipes should be greater than one diameter Short test sections tend to twist, especially spirally corrugated pipes

Example 1

Figure 8-5 shows the results of parallel plate tests on spiral corrugated pipe What is the difference between calculated ring stiffness and measured ring stiffness? From the F-D plot, F/D = 100 psi From the relationship, F/D = 53.77(EI/D3)

Measured EI/D3 = 1.9 psi

Calculated EI/D3 = 3.7 psi from the following:

E = 30(106) psi

D = 24.57 inches from Figure 8-5

I = 1.892(10-3) in4/in from the AISI Handbook

o f S t e e l D r a i n a g e & H i g h w a y Construction Products.

The measured value of ring stiffnes s is only about half the calculated theoretical value for this pipe Example 2

Suppose the ring of Example 1 is deflected so much that the graph of Figure 8-5 is no longer linear Ring stiffness is the slope of the tangent to the F-D plot at the anticipated ring deflection If the ring deflects over a range, the ring stiffness is the slope of the cord between the ends of the range of ring

deflection It must be pointed out, Ring deflection

based only on elastic ring stiffness is not a pertinent performance limit.

Figure 8-5 Parallel plate test of a 24 D, 2-2/3x1/2 corrugated steel, spiral lock-seam pipe.

Figure 8-6 Three-edge-bearing test of a cement-mortar-lined/cement-mortar-coated thin-wall steel pipe (CML/CMC) The lining is spun centrifugally on the inside The coating is shot-crete with wire reinforcing

8-1 Figure 8-4 is an F-D diagram for a 4-inch PVC

sewer pipe DR30 Compare the measured pipe

s tiffness F/D and pipe stiffness calculated from

published values of E = 400 ksi to 500 ksi

8-2 Figure 8-6 is an F- diagram for CML/CMC

pipe which was handled with care before testing

At each drop of the load-deflection diagram,

cracking was audible and hair cracks appeared

What should be the pipe stiffness for design

purposes? What conditions should be specified?

(F/D = 450 psi) 8-3 Figure 8-7 is the result of a parallel plate test

on spiral rib steel pipe What is pipe stiffness?

(F/D = 72 psi)

8-4 Using average D, calculate ring stiffness EI/D for a reinforced concrete pipe if:

ID =60 inches = inside diameter,

OD = 72 inches = outside diameter,

Es = 30(106) psi = steel modulus,

Ec = 2(106) psi = concrete modulus

Reinforcing steel compris es 3/8-inch rebar hoops rods in the center of the wall spaced at 3 inches Assume that concrete takes no tension

8-5 Calculate ring stiffness of the pipe for Problem 8-4 if two cages of 3/8-inch rods are spaced at 3 inches, but located 1.0 inch from the inside and outside surfaces Assume concrete takes no tension What about average D?

THIS PROBLEM SHOWS WHY RCP PIPE IS DESIGNED BY TEB TESTS AND F/D — NOT

BY RING STIFFNESS, EI/D 3

plot will pass through the origin

Figure 8-7 Load deflection diagram for a spiral-rib pipe showing an initial zero correction such that the linear