Draw probable plots of the shearing forces Q(ξ) and moments M(ξ) along the beam. Compare with the sym- metrical case, where the forces are both equal to +1000kN, r(x) = const.

TESTING QUESTIONS AND EXERCISES to Lectures no. 1,2,3,4

Question: Why the use of the Winkler model is recommended only if thickness H of deformable soils under foundation is “small”, say H/B < 1,0÷1,5 ?

Question: Antisymmetric beam of the length 2π with the ends at ±π is loaded by two opposite forces ±1000kN applied at the cross-sections ±π/2. Subsoil reaction r(x) is linear along the beam, r(-x) = r(x), of course r(0) = 0.

Draw probable plots of the shearing forces Q(ξ) and moments M(ξ) along the beam. Compare with the sym- metrical case, where the forces are both equal to +1000kN, r(x) = const.

Question: Is it true that for a finite Winkler beam 6 virtual Bleich forces provide higher accuracy than 4 such forces? Justify.

Question: Is it true that a half-finite Winkler beam can never be solved by making use of only one virtual force?

Justify.

Question: Two silt layers are found under a foundation beam which is B=1,0m wide (B<< L); the upper one 1,5m thick is stronger I

= 0,07, the lower one 2,0m thick is much weaker I

= 0,39. In GEO bearing capacity calculations it was found that the lower soil layer has a greater margin of safety than the stronger upper one.

Explain how this can happen (there are two reasons).

Question: Plot of bending moments along the beam loaded by a concentrated force has a sharp edge under the force. Explain why.

Question: Draw probable solutions y(x), M(x), Q(x) along a beam on elastic subsoil which has a perfect hinge under the concentrated force P, like in Fig.

Question: Draw mining corrections ∆M, ∆Q, ∆r along a beam of the length L for -L/2 < x < L/2 if R > 0 happens (convex case). Pay special attention to signs of the corrections. Why

∆ = 0?

Question: the H.Bleich method for beams and plates on the Winkler subsoil:

Present one basic common feature and one major difference.

Question: Is this true that the relations Q = -EI⋅d

y(x)/dx

yields from the Euler-Bernoulli hypothesis? Justify.

Question: Underground mining exploitation at a depth H in 2D (long mining front) causes continuous deformations of the surface. Which situation is more dangerous for a CE-infrastructure on the surface:

H=400m or H=800m? There is no other difference neither in type of rocks (tgβ), thickness of the mineral deposit (h), nor mining technology (a) etc.

Question: A certain CE-object is sensitive to mining-caused tilt T. Present two (different) possible measures to be undertaken on the exploitation level which can reduce the potential danger (2D case), so can reduce the tilt.

Question: Present a situation in which the biggest losses are caused by uniform mining subsidences on a certain vast area.

Question: Present a situation in which a danger is caused by compressive mining strains i.e. ε < 0.

Question: How many unknowns operate in the ZEM_SIN algorithm, if the beam is subdivided into 17 calculation segments?

Question: The beam is subdivided into 21 calculation segments in ZEM_SIN. Is this true that y

< y

? How about y

> w

? And maybe w

= w

?

−∞

P

C

y( ξ )

ξ 0

+∞

Question: The fundamental solution of the first kind y(ξ) for the Winkler beam appears to be an even function (symmetrical). Explain how this yields a conclusion that moments M(ξ) are even, too.

Question: The fundamental solution of the second kind y(ξ) for the Winkler beam loaded by a moment M at ξ=0 appears to be an odd function. Justify.

Question: What’s the use of the tilted object rectification in mining area protection?

Question: a rigid smooth foundation beam is loaded by a vertical force P [kN/m]

acting on a small eccentricity e

.Draw probable plots of subsoil reactions along the width B for:

a) the Winkler subsoil model

b) the Pasternak subsoil model

c) the elastic half-space.

Calculation job:

Hints:

1) there are 3x5 identical square segments 1m x 1m distinguished on the founding level – 6 real footings in this number; F

can be used to model interactions between the segments if the superposition law is assumed;

action of „b” on „a” or of „C” on „B” etc. uses F

= 0,511 action of „c” on „A” or of „a” on „C” uses F

= 0,225

action of „B” on „B” or of „c” on „c” etc. uses F

= 2,974 and so on.

(some values are not used here, like F

, F

, F

, F

…, because there is no such pair of footings).

2) of course, focus on the columns’ heads level (2.5m above the footings), where the head settlements are equal.

Extra Question:

What will change if 4 outer footings subside of ∆

(mining effects) and 2 central ones of ∆

?

A rectangular tank is placed on a 2m x 4m rigid plate, under which there are 2x3 identical vertical steel columns of the length (height) H = 2.5m and stiffness EA = 500 MN.

Each column rests on an identical 1m x 1m prefabricated square footing. There are two symmetry axes,

and the load is constant q = 0.150 MPa.

Calculate the axial forces in the columns [MN].

It is the elastic half-space, ν = 0,3 and E

= 30 MPa.

Average settlement w

of the footing k, which is d

x b

= 1m x 1m, caused by the footing i which is d

x b

= 1m x 1m and is loaded by a force R

[MN] can be expressed as follows:

= ∙ _{∙ ∙} ∙ F ki

Where the (symmetrical) influence coefficients F

= F

for settlements are presented in Table below; note that for these dimensions and distances, numbers k and i mean simply distances between centers of footings along x,y, expressed in meters.

For simplification assume

∙ ∙ = 0.01 m/MN.

q=150kPa

„A”

„a”

„B”

„b”

„C”

„c”

2m 2m

2m

Table. Influence coefficients F

for identical square segments

Use

asymptotic values as …

Calculation job:

Solve a half-infinite beam loaded by a force P at the distance of π/2 from a free end of the beam, i.e. find both T

= ? and T

= ? Fundamental solution to be used is presented on the right.

Calculation job :

Calculation job:

A foundation beam has a trapezoidal shape: length L=16m, width B

= 1,0m at the left end, B

= 2,0m at the right one. Find maximal mining force Z

[kN] caused by tensile mining ε and draw probable shape of Z(x).

Ignore Z

on both sides. Take pre-ultimate distance x

= 0 and θ= 30 kPa.

Calculation job:

Draw plot of a mining tensile force Z [kN] along the beam of L=16m and next calculate Z

. Take x

=2.0m, θ = 30kPa. The beam is symmetrical but is not prismatic. It consists of 3 intervals:

B=2.0m in the central part 0±4.0m and B=1.5m near the ends, 4.0m long each.

Assume no Z

forces on both sides of the beam.

ξ =

2"#$ _% ∙ & ^ξ ∙ cos ξ + sin ξ , ξ =

2"#$ _% ∙ & ^ξ ∙ −2 sin ξ . ξ = $ _%

4 ∙ & ^ξ ∙ cos ξ − sin ξ 0 ξ = −

2 ∙ & ^ξ ∙ cos ξ P

+∞

π / 2

π / 4

π/ ⁴ T 2

T 1

P

y(ξ)

ξ 0

- ∞

π/2

+ ∞

-π/2

P

The Winkler beam, continuous and doubly-infinite, is loaded by two forces P at ξ

= 0 and ξ

= +π/2.

At the cross-section ξ = -π/2 there is also a perfect support, so no settlement happens there. Solution for the standard unsupported beam (fundamental solution of the first kind) is presented above for 0 < ξ < ∞ in the local system of coordinates (odd/even curves).

By making use of these functions, calculate the bending moment M at this supported cross-section, so M(-π/2) = ? Next plot probable the solutions y, M, Q.

Hint:

at -π/2 apply a vertical force R in place of the support and find

its value which is required by the context; useful functions are

given in the previous job.

Calculation job: The subsoil consists of 2 sublayers of the thicknesses H

and moduli E

.

Using the method of equivalent settlements, estimate the elasticity effective modulus E

for a homogenised strata of thickness 12.0m instead of the real 2m+4m = 6m thick subsoil. Assume B = 2.0m.

Read the required coefficients from the table.

BEDROCK Thickness H2 = 4m, Es2 = 100MPa Thickness = 2m, Es1 = 25MPa

B q

Zo=0

Calculation job:

It was estimated that the ceiling of a tunnel will be deformed (downward deformation, settlement) of ∆

= const(x) = 3cm on the interval AB=8m at the depth h

= 19m under the ground level, so between x

= 0 and x

= 8. This way, nonuniform settlements of the existing foundation beams C, D will happen on the founding level h

= 1m.

Calculate such additional “mining” settlements of both foundation beams C, D, w

= ?cm and w

= ?cm.

For soil above the tunnel assume the angle of subsidence propagation β = π/4 + ϕ/2 = 62

, tgβ = 1,8

And the radius of mining influences r = H/tgβ, dispersion parameter of the subsidences basin σ = r/√(2π) ≅ 0,4·r.

Porosity of the soil mass above the tunnel will be constant, more or less, so a ≈ 1.

The complete subsidence curve takes the form w

(x) = ∆

·Φ(-(x-x

)/σ), if initial subsidences ∆

occurs along (-∞;x

) at the depth H under a protected level (but here they will occur only on AB = (x

,x

) in this case).

The Laplace integral Φ(s) is presented in the table below.

The points marked in Fig. have the following coordinates (x;y): A(0;19), B(8;19), C(8;1), D(12;1).

s = -∞ -4 -3 -2 -1 -½ 0 ½ 1 2 3 4 +∞

Φ(s) = 0 ≈0 0,001 0,023 0,159 0,309 ½ 0,691 0,841 0,977 0,999 ≈1 1

Hint:

(x

;x

) = (-∞;x

) − (-∞;x

)

thus superpose 2 functions of the type w

(x) to get an equation for the profile of „mining subsidences” w(x)

which is concentrated only on finite (x

;x

);

this is the 2D situation of course.

A B

(0;0)

C D

8m

1m

h

19m

18m w(x)

3cm

x

4m

z/B= 0 1 2 3 4 6 8 ∞

ω = 0 0,50 0,85 1,10 1,30 1,48 1,65 2,15 Average settlement of the beam of the width B resting on n ≥ 1 elastic layers can be calculated as:

where z

= 0