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Wood Armer for combined bending & in plane forces
(Morley's Equivalent Sandwich analogy)
For a slab subject to a moment field
(Mx, My, Mxy) and
a stress field (Nx, Ny, Nxy), the Wood Armer
calculation may be combined with the Clark-Nielsen
calculation using the "equivalent sandwich analogy"
proposed by Morley.
For a full explanation and derivation of the formulae, the
reader is referred to either:
-
"Concrete slabs: analysis and design" L.A Clark
and R.J Cope (Elsevier Applied Science)
-
"Concrete Bridge Design to BS5400" L.A Clark
(Construction Press) Chapter 5 (section entitled
"Reinforced Concrete Plates") and Appendix A
By
reference to Chapter 7 of "Concrete slabs: analysis and
design", the Morley calculation may be carried out
by following the procedure below. All calculations proceed after the determination of the
moment/stress field Mx, My, Mxy, Nx, Ny, Nxy.
Top:
NxT=(Mx+Nx*dT)/d, NyT=(My+Ny*dT)/d, -NxyT=(-Mxy-Nxy*dT)/d
Bottom:
NxB =(-Mx+Nx*dB)/d, NyB=(-My+Ny*dB)/d, -NxyB =(Mxy-Nxy*dB)/d
where
d=lever arm between the layers of reinforcement=h-cB-cT
and dB=distance from centroid of section to bottom
reinforcement=h/2-cB
and dT=distance from centroid of section to top
reinforcement=h/2-cT
The
Morley calculation is followed by the Clark-Nielsen
calculation so that the final output is Nx(T), Nx(B),
Ny(T), Ny(B), Fc(T) and
Fc(B). In
this calculation, however, the procedure is applied twice, once
for top stresses and once for bottom stresses (NxT not equal to
NxB, NyT not equal to NyB etc).
A
simple example may be used to demonstrate the Morley & Clark
Nielson calculation.
This example is a thin rectangular slab on knife-edge
supports and fixed in translation at one edge. A point load applied to an opposing corner and a UDL
applied to the whole plan area of the slab.
Properties:
-
Rectangular
surface, plan dimensions length 16 units, width 10 units
-
Mesh attributes: Any quadrilateral shell element (QSI4
elements used in subsequent calcs)
-
regular mesh of element size 1 unit
-
Geometric attributes: thickness 0.2 units
-
Material attributes: E=1E6, poissons ratio=0.3
Supports:
-
fixed in translation (X, Y, Z) on bottom edge
-
fixed in translation (Z only) on all other edges
Loading attributes:
-
Structural load, Concentrated in Y direction 100 units total
-
Structural load, global distributed Z direction –1.0unit/unit
area
Download
Morley example model (LUSAS mdl file)
Moment and stress field from LUSAS
Modeller, extracted at 4
nodes for calculations to be checked explicitly:
|
Node
|
29
|
50
|
80
|
128
|
|
Nx
|
-76.922
|
2.714
|
0.034
|
2.879
|
|
Ny
|
-33.280
|
74.810
|
33.701
|
13.785
|
|
Nxy
|
-11.310
|
-8.011
|
-0.157
|
-0.987
|
|
Mx
|
0.034
|
0.003
|
-2.659
|
-8.198
|
|
My
|
0.113
|
0.000
|
-1.599
|
-4.706
|
|
Mxy
|
-4.029
|
-2.994
|
2.245
|
-0.174
|
Calculated Morley stresses by hand, determined from the
moment and stress field (Mx, My, Mxy, Nx, Ny, Nxy) using the
procedure explained above.
|
Node
|
29
|
50
|
80
|
128
|
|
NxT
|
-38.289
|
1.373
|
-13.279
|
-39.548
|
|
NyT
|
-16.076
|
37.403
|
8.856
|
-16.639
|
|
-NxyT
|
25.801
|
18.976
|
-11.147
|
1.365
|
|
NxB
|
-38.632
|
1.341
|
13.314
|
42.427
|
|
NyB
|
-17.204
|
37.407
|
24.845
|
30.424
|
|
-NxyB
|
-14.491
|
-10.965
|
11.304
|
-0.379
|
Calculated Clark
Nielsen
stresses determined from the equivalent sandwich analogy
stresses (NxT, NyT, NxyT, NxB, NyB, NxyB), using the Clark-Nielsen
equations.
|
Node
|
29
|
50
|
80
|
128
|
|
Nx(T)
|
0
|
20.349
|
0
|
0
|
|
Ny(T)
|
1.310
|
56.378
|
18.213
|
0
|
|
Nx(B)
|
0
|
12.306
|
24.618
|
42.806
|
|
Ny(B)
|
0
|
48.372
|
36.150
|
30.803
|
Download
spreadsheet calculations (MS Excel format)
Results from LUSAS
Modeller, extracted at the same 4 nodes
for comparison to the hand calculations undertaken:
|
Node
|
29
|
50
|
80
|
128
|
|
Nx(T)
|
0
|
20.349
|
0
|
0
|
|
Ny(T)
|
1.310
|
56.378
|
18.213
|
0
|
|
Nx(B)
|
0
|
12.306
|
24.618
|
42.806
|
|
Ny(B)
|
0
|
48.372
|
36.150
|
30.803
|
By inspection the results tabulated above agree closely with
those derived by hand calculation and demonstrate that the
calculations for this example are satisfactory.
It should be noted that Wood Armer
moments are not required for design when Nx(T), Ny(T),
Nx(B) and Ny(B) have been calculated using Morley followed
by the Clark-Nielsen
procedure. However,
since many design codes are based on the moment capacity
of sections, Wood Armer
moments are also reported by LUSAS so that the engineer
can decide how in-plane forces should be dealt with based
on the load effects.
Other Wood Armer
related topics
|
