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Wood-Armer for combined bending and 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
dB = 0.5h-cT
dT = 0.5h-cB

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 (for LUSAS version 21)

Download Morley example model (for earlier LUSAS versions)

Moment and stress field from LUSAS Modeller, extracted at 4 nodes for calculations to be checked explicitly:

Component / 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.  

Component / 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.  

Component / 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:

Component / 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


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