Numerical Methods in Soil Mechanics A.PDF

Numerical Methods in Soil Mechanics A.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 "CASTIGLIANO’S EQUATION"

Structural Mechanics of Buried Pipes

Boca Raton: CRC Press LLC,2000

Figure A-1 Virtual load q on a bridge useful for calculating the deflection of the bridge at the location

of q, in the direction of q, due to the truck load Q

Figure A-2 Free-body-diagrams for writing moment M for the Q-load (left) and moment m for the virtual

q-load (right) for a ring

APPENDIX A CASTIGLIANO'S EQUATION

A powerful method for calculating deflection is the

method of virtual work Consider the beam

(bridge) of Figure A-1 We would like to know

the deflection of the beam at midspan due to a

truck load Q located as shown I locate myself at

midspan before the truck reaches the bridge I am

the virtual load, q Comes now the truck As it

reaches the location shown, I feel myself being

lowered by the deflection of the bridge at midspan

due to the truck load Q I do virtual work because

of deflection, D, of the bridge by the truck My

virtual work is qD But this virtual work acts on

the bridge which stores my virtual work as

(potential) virtual energy The stored virtual

energy is mdq, where m is the moment at any

point on the bridge due to the virtual load, and dq

is the change in curvature of the bridge at that

point due to the truck load But dq = Mdx/EI So

the virtual stored energy is mMdx/EI Equating

virtual work to virtual stored energy,

qD =fo mMdx/EI

where

q = virtual load, (Let q = 1.)

m = moment in the beam due to load q,

M = moment in the beam due to load Q,

E = modulus of elasticity of material,

I = moment of inertia of the cross section of

the beam about its horizontal neutral

surface,

L = length of the beam

Because the virtual load was set equal to unity, the

resulting equation for deflection of the bridge at

midspan due to the truck is,

D=f o mMdx/EI

integrated over the length of the beam M is the

equation of the moment as a function of x to be

integrated over the length of the beam x is

distance from an assumed origin of axes The

dummy moment, m, is the equation of moment as

a function of q throughout the length of the beam

in terms of x measured from the same origin of

axes Clearly, the two moment equations require analyses of two separate free-body-diagrams of the beam

Deflection by virtual work is not limited to a straight beam It can be a curved beam or a ring for which equations for moments M and m are written in terms of radius and angle, q Deflections can also be found for shear ing loads and thrust as well as moments In fact, because energy is scalar, the deflection due to moments, thrust and shear can be found by adding the energies of all of the load elements

Castigliano observed that it is us ually easier to apply the virtual work equation by use of the Leibnitz rule which allows differentiation under the integral sign as follows:

D=f o (M/EI)( M/ q)ds CASTIGLIANO EQUATION where

D = deflection of the beam at the location of

the virtual load q in the direction of q q is

a differential that approaches zero at the location where the deflection is to be calculated (q can also be a differential moment for calculating the angle of rotation of the beam at the location of q.),

M = equation of the moment at any point due

to both q and Q,

EI = stiffness of the beam,

ds = differential distance along the beam (dx

for the beam)

The advantage of Castigliano over virtual work is

a reduction of analysis Only one free-body-diagram is required with both the virtual load and the applied load in place For small deflections, Castigliano's equation is well adapted to rings because M can be written for the applied loads and pressures plus the virtual load in terms of stiffness EI, angle q, and ds EI is constant, and

ds = rdq where r is the radius of the ring See

Figure A-2

©2000 CRC Press LLC

Figure A-3 F-load on a ring showing a quadrant of the ring as a free-body-diagram for evaluation of the moment at A by Castigliano's equation

Figure A-4 Free-body-diagram of a quadrant of a ring subjected to an F-load, showing the notation for evaluation of the vertical deflection of A with respect to B

Because of symmetry, it is convenient to consider

half of the ring

Example 1

What is the moment at A due to an F-load on a

ring? The free-body-diagram is a quadrant shown

in Figure A-3 Let Q = F/2 for convenience

Moment MA is an unknown in addition to the three

reactions at B The quadrant is statically

indeterminate to the first degree Therefore, an

equation of deflection is required in addition to the

three equations of static equilibrium In this case,

it is possible to find MA by means of a single

equation of deflection Note that the relative

rotation of A with respect to B is zero, i.e., yA/B =

0 Applying m at A in the direction of rotation of

yA/B, from Castigliano,

yA/B = (M/EI)( M/ m)rdq = 0

M = Qsinq - MA - m, where m 0

M/ m = -1

Substituting into Castigliano's equation,

0 = (Qrsinq-MA)dq = [Qrcosq+MAq],

from the limits 0 < q < p/2

Substituting the limits, MA = 2Q/p

Example 2

Knowing MA, what is ring deflection, d, of

Example 1 due to the F-load? For a

free-body-diagram, use the quadrant redrawn in Figure A-4

d = D/D = (yA/B)/r; where yA/B is vertical

displacement of A with respect to B due to the

half load, Q = F/2 From Castigliano, yA/B =

(M/EI)( M/ q)rdq

M = (F/2 + q)rsinq - 2Q/p; where q 0

M/ q = r sinq

Substituting into Castigliano's equation,

yA/B = (Qr3/EI) (sin2q - (2sinq)/p)dq

yA/B = Qr3/EI[q/2 - (sin2q)/4+2cosq)/p], from the limits 0 < q < p/2

Substituting limits, yA/B = (Qr3/EI)(p/4 - 2/p) But d = (yA/B)/r, so d = 0.0186Fr2/EI This is one of the solutions in Table A-1, a summary of moments, thrusts, and ring deflections for rings subjected to typical loads

Example 3

A ring from a profile rib HDPE pipe is cut and loaded as shown in the sketch below The objective is to find any possibility for long term wall buckling at B due to constant load, F What

is the gap, 2x, as a function of load, F? The right half of ring AB is analyzed by Castigliano's equation,

x = (M/EI)( M/ p)rdq

M = (F+p)r(1-cosq); where p 0

M/ p) = r(1-cosq)

x = (Fr3/EI) (1-2cosq+cos2q)dq

Integrating,

x = (Fr3/EI)[q-2sinq+q/2+(sin2q)/4]

Substituting limits, 0 < q < p,

x = (Fr3/EI)(3p/2)

©2000 CRC Press LLC

Table A-1 Mechanical Analyses of Thin-wall Rings With Symmetrical Loads

©2000 CRC Press LLC

Table A-2 Circular Arcs (Partial Circular Rings)