and to two half-circles at the ends, radius r = BD = 1m. The beam is loaded by tensile mining actions on the hori

GEOENGlNEER|NG-FOUNDAT|ONS (M.Sc. Study in CE) Final Evajuation Test #1

Y) _=O

Calculation Task #2: a 14m long symmetrical foundation beam can be split to ^a rectangular part BxL1=2mxl2m

and to two half-circles at the ends, radius r = BD = 1m. The beam is loaded by tensile mining actions on the hori

zontal contact level which act along the beań. Ultimate shearing stress t equals @ = 30kPa with the pre-ultimate interval xo :3.0m. Two veńical beam sides are free offrictional forces, Zt= 0. Draw shapes of t(.x) and Z(x).

C#1 (15 min, max 7pts.)

C#2 (l0 min, max 4pts.) _{DATE of the TEST:}

Q#1 (5 min, max 3pts.) ^Student's name: ł,tr BPZAVAĘ

Q#2 (5 min, max 3pts.) Student's lD number: 3' +O(

Q#3 (5 min, max 3pts.)

Today's Student's CODE in the format AB12: MBEtr'

TOTAL (40 min, max 20pts.) FlNAL SCORE:

Attention: note that completely wrong answer may by evaluated with negative points !

Cąlculation Task #I: simple modelling of a long vertical pile uses a horizontal (rotated) half_infinite beam ot Winkler subsoil. The beam is loaded only by a force Po applied to the end Ę:0, no concentrated momęnt Mo

at this end. The general solution to be used fo. _Ę : _xlLw > 0, L* _{_ d4} ^.BI(B .6) is as follows:

1) Formulate all boundary conditions, (2pts.)

2) Find all the constants Ci, (2pts.)

3) Present graphically the following three solutions:

y(Ę): settlements _[m], (]pt.)

M(Ę): bending moments [kNm], (Lpt.)

Q@: ^shearing forces [kN], (Ipt.) along the entire beam, 0 ^< Ę. ^*.

Hint: app|y the Euler-Bernoulli condition.

Ouestion #l: a very rigid foundation beam is loaded eccentrically by a vertical force P _[kN/m]

applied on the left side from the center. Contact of the beam with the subsoil is smooth.

Draw in one figure probable shapes of vertical contact stresses under foundation for 3 different cases of the beam-subsoil interaction:

a) ^the Winkler subsoil model with a concave mining subsidence (R < 0).

b) ^the Winkler subsoil model with no mining deformations,

c) ^the Winkler subsoil model with a convex mining subsidence (R> 0).

The eccentriciĘ e is "small" and the mining radius _|R| is "very large".

Mark a)b)c) for each curve.

Ouestion #2: answer 6 questions with only one of the letters T=TRUE or F=FALSE, respectively:

(]pt. for 4 correct answers, 2pls. for 5 correct answers, 3pts. for 6 correct answers; _,

nó written justification is Óxiected here - just ^mark T oiF; no negative points wili&gned here)

. the Pastemak model ofthe elastic subsoil uses 3 parameters ... ^F

. when a long mining fiont (in 2D) approaches a building, compressive strains appear first (e<0) ...F

. ₄ virtual forces ł ,Tz,T,Tł ^of Blęichare necessary to solve any infinite beam ...,. ,.,..,...F

. in ZEM-SIN algońt}rm for prismatic beams with 13 identical segments ńere ^is always _}\1> yn andwl'= r, ^{, . . . . .} T

. in ZEM-SIN algorińm for prismatic beams witłr

7 identical segments there is alwa}s 1t ⁼ }1 ^{. . . , . . . . . .}

^F

. _{in ZEM-SIN} algorithm for beams which undergo mining subsidences Ą+4 ^where d: ^const, numerical solutions ri do not depend on the value ofd

Ouestion ^#3: infinite beam on the Winkler subsoil has a perfectly fixed support (no spring element) at a distance s from the point _Ę :0 being loaded by P.

Draw probable solutions for the vertical displacement curve _Jp, moments M and shearing forces Qfor ^-oo < Ę ^< +*.

Yrs):o

Pay special attention to the supported cross-section and the loadęd place.

Calculate values of tensile forces Z [kN]

acting at three distinguished cross-sections:

,,: tT,,lh,3o Zc:ZB ^* Z,Xr, 30

Zr:Zc ⁺ 3,ł, ^3o 7v

'!t ^1,1

= p=-ń"p,,ń=o

,, JrĆ;Z: ^{*c,,, F n. 17 *7\}

'-roĄ-rP

h,, 'N ⁼

l3 -t_

t-P

rył=-P, Ą:o

:O T

Q: ^{*-El 'Y,,,,}

ła,(Cr,ę: _bś\r, ^,LŃ= ł|"

.?'.W --T

= ^{P" L,j}

-3 óv dy-

-)9

f(Ę) e-Ę,sin

e-Ę.cosł ę*Ę,cńtł ę*Ł.sin{,/

df/dŁ -e-Ę,sinĘ + e-Ę.cosĘ -e-Ę,sinĘ - e-Ę.cosĘ e,\.cox/ę

^s.stnĘ e*Ę,sinĘ/#cost

dzfldŁ2 -2ę-Ę.cosĘ l/ _2ę-Ę.sinŁ L' _ft*Ę.siną ŹX*;ą

d3fldąs 2e-Ę,sin,+2ę-Ę.cos, -Zę-Ę,sĘ a 29-ĘĘ6Ę1| -de* .sin|- _{Ze* .cośŁ} 2ę*,ósĘ- 2e*Ę.sinŁ

'V^

O1 l\ i i,=/ [_ >

L}ą % ^{,l l}

t r--o +El .C3,(-l,

lOs Z-El

!-, ^{=po ,l- T=o}

L-U lb_l8-19-Winter

Wł

l;,,*rł

ó"