Searching for contact forces
at the contact of the structure with a subsoil
EXAMPLE 0:
no interaction between the structure and the subsoil.
Statically determined beam,
simply supported, no settlements of the supports.
For a continuous uniform loading q:
2 L R q
RA B
0
B
A u
u
3
3 2 3 2
q
1 L
a L 2 a EI 1
24 a L u q
For a concentrated force R:
L R b RA
,
L R a RB
0
B
A u
u
22 22
R
1 L
b L 1 a EI 6
L b a u R
Statically determined beam, simply supported, spring supports.
For a continuous uniform loading q:
2 L R q
RA B
,
A A
A A
2 C
L q C u R
,
B B
B B
2 C
L q C u R
L a C 2
L q L b C 2
L q L a L 2 a EI 1 24
a L u q
B 3 A
3 2 3 2
q
1
For a concentrated force R:
L R b RA
,
L R a RB
A A
A A
L C
b R C u R
,
B B
B B
L C
a R C u R
q [kN/m]b a
1
RA [kN] RB [kN]
A B
R [kN]
a b
1
RA [kN] RB [kN]
A B
R [kN]
a b
1
RA [kN] RB [kN]
A B
CA CB
q [kN/m]
b a
1
RA [kN] RB [kN]
A B
CB CA
Symbols:
- uniform load q, [kN/m],
-
Rireactions, concentrated force R, [kN], - spring support stiffness C, [kN/m], - beam length L= a+b, [m],
-
u1vertical displacement of a considered point "1" on the beam, [m],
- EI means a beam bending stiffness.
L a C L
a R L b C L
b R L b L 1 a EI 6
L b a u R
B 2 A
2 2 2 R
1
Conclusion:
taking into account the subsoil deformations does not affect neither the forces on contact with the subsoil nor the internal forces in the beam - it only affects the settlements of the beam.
The last two components in the displacements u
1q, u
1Rdescribe the settlement/tilt of the rigid beam on the springs.
EXAMPLE 1:
the simplest interaction
between the structure and the subsoil.
Foundation beam rests on three independent spring supports.
The task is statically undetermined with one unknown.
Solution – The Force Method.
At the point "1" the contact of the beam with the subsoil is removed - there is a "gap", and point "1" splits virtually into two points:
1' = point "1" but on the beam,
1" = point "1" but on the support element.
Now, the beam on the other two supports A-B is statically determined as in Example 0.
1)
The load q on the statically determined beam "compresses the gap" and the upper point 1’
would be below point 1”, i.e. virtual "penetration" of the concrete beam into the support happens.
If the springs are independent then the "penetration" at the point "1" has two reasons - as for u
1qin Example 0:
a) deflection of the beam itself on the fixed undeformable supports (due to EI < ), b) settlement/tilt of the beam as a rigid body caused by settlements of the supports
"A" and "B" due to the action of q
(effect of the spring deformation for stiffness C
A, C
B< ).
Note:
if the subsidence independence of supports cannot be accepted, one more factor would appear: the unloaded point 1" on the cut-support would still settle due to the load of neighbouring supports R
Aand R
Bcaused by q (typical "neighbour effect" for the elastic half-space); this fact should be taken into account in point b).
2)
A pair of balanced forces R
1applied respectively at the bottom of the statically
determined beam and on the top of the support "opens the gap", i.e. the virtual points 1’
and 1” move away from each other (effects of q are not considered here).
If the springs are independent then the appearance of virtual "gap" at point "1" has three reasons - a little like in the expression u
1Rin Example 0:
a) a deflection upwards of the beam itself on the fixed undeformable supports (EI < ) caused by the force R
1acting upwards - see the first component in u
1Rin Example 0,
q [kN/m]
a b
1
R1 [kN] RB [kN]
A B
CB CA C1 R1 [kN]
q a b
RA RB
A B
CB
CA C1 R1
1
1’
1”
b) displacement/tilt of the beam as a rigid body caused by force R
1due to lifting of supports "A" and "B" by support-reactions caused by force R
1, cf. second and third components in u
1R,
c) additional self-subsidence of the support "1" from downward force R
1.
Note:
in the case of the elastic half-space (or any other model with lack of independence of supports’ settlements), points b) and c) are coupled; at the same time, one should analyze the effect of three factors: R
1itself and the two reactions applied to the subsoil R
A, R
Bwhich are generated just by this force R
1.
3)In reality, no such “gap” can occur,
i.e. its virtual "opening" due to 1) must be compensated by 2).
This way, the following equation of the Force Method for determining of R
1can be formulated:
u
1q= u
1R1R 1 1 B
1 A
1 2 2 2 1 2
B 3 A
3 2 3 2
1q u
C R L a C L
a R L b C L
b R L b L 1 a EI 6
L b a R L a C 2
L q L b C 2
L q L a L 2 a EI 1 24
a L
u q
so
1 B
2 A 2 2 2
B 3 A
3 2 3 2
1
C 1 L a C L
a L b C L
b L
b L 1 a EI 6
L b a
L a C 2
L q L b C 2
L q L a L 2 a EI 1
24 a L q
R
or
1 3 2 2 B 3
2 2 A 3
2 2 2 2 2
B 3 A 3
3 3 2 2
1
L C
EI L
a L C
EI L
b L C
EI L
b L 1 a L 6
b a
L a L C
EI 2 1 L b L C
EI 2 1 L a L 2 a L 1 24
a L q R
* * * * *
For the simplest symmetrical beam on independent spring supports:
a = b = l = L/2, C
A= C
B= C but C
1– may be different from C
C 2
L q EI
L q 384 u 5
4 q
1
and
1 1 3 1
1R 1
C R C 2
R EI 48
L
u R
From supposed u
1q= u
1Rone obtains R
1and next R
A= R
B= (q·L – R
1)/2:
1 3 3
3 1
L C
EI L
C 2
EI 384
8 C L
EI 384
10
2 q L R
Conclusion:
Interaction between the beam and the subsoil depends on the relative dimensionless stiffness
A= EI/(C·L
3) =
B,
1= EI/(C
1·L
3).
For C
A= C
B= C >> C
1, so relatively C
1 0 there is R
1= 0.
For C
A= C
B= C << C
1, so relatively C 0 there is:
L q 2 2 q L
R1
.
For realistic C
A= C
B= C
1= C there is:
3 3 1
L C 2
EI 3 384
8 C L
EI 384
10
2 q L R
Note that for C
A= C
B= C
1= C + or EI 0 there is
810 2 q L R1
which is a standard solution for 3 fixed supports (undeformable subsoil).
* * * * *
A more general formulation will follow next.
Generalization:
1) the analyzed beam rests on n+2 supports, where n 1 is the numer of unknowns and at the ends there are two supports A and B; note that the supports can be more or less distant but also they can be virtual in that sense that they are to model a continuous contact of the beam with the subsoil (“supports” are neighbours in touch),
2) the load on the upper surface of the foundation is a sequence of concentrated forces, may additionally be a uniform load q (Fig.); usually, these forces Pj are applied at the centers of the computational segments; of course, some of them (or even all) can be zero,
3) displacements of individual supports does not have to be independent (cf. "neighbour effect" for supports on the elastic half-space),
4) following the idea of 1): supports can be virtual computational segments which together with the two end segments give a total beam of continuous contact with the subsoil; if we take equal length of all calculating contact segments, it is L/(n+2); if any force R
A, R
B, R
jis distributed uniformly on its calculation segment then local stresses r
junder foundation are obtained which are the step-wise approximation of the
continuous soil resistance r(x),
5) this nice idea is adopted from a book (in Polish) by Wilhelm Król:
Statyka fundamentów żelbetowych z uwzględnieniem sztywności nadbudowy.
Wydawnictwo Arkady, Warszawa 1964 though it is still older.
q [kN/m]
L
RA [kN] RB [kN]
A B
CB CA
Rj [kN]
Cj Pj[kN/m]
