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Wood-Armer calculation (for plate elements)

It is reasonably common in bridge design that RC slabs must be designed to resist a combination of moments (Mx, My) and a twisting moment (Mxy) using orthogonal reinforcement.  In Concrete magazine (February 1968), R H Wood proposed a procedure for "The Reinforcement of Slabs in Accordance with a Pre-determined Field of Moments".  The following correspondence included additional equations derived for skew slabs (principle moment directions inclined to the reinforcement directions).  Together the equations are known as the Wood Armer equations.

For a full explanation and derivation of the formulae, the reader is referred to either:

1. "Concrete slabs: analysis and design" L.A Clark and R.J Cope (Elsevier Applied Science)
2. "Concrete Bridge Design to BS5400" L.A Clark (Construction Press) Chapter 5 (section entitled "Reinforced Concrete Plates") and Appendix A

By reference to "Concrete Bridge Design to BS5400" Chapter 5 (section entitled "Reinforced Concrete Plates") and Appendix A, the Wood-Armer calculation may be carried out by following the procedure below.  All calculations proceed after the determination of the moment field Mx, My, Mxy.  The equations have been amended to suit the sign convention for stress results from LUSAS, and are described in a form that can be translated directly to a spreadsheet format:

Top (hogging) reinforcement

• Determine the generalized Wood-Armer moments:
  MxT1=Mx+2*Mxy*cotø+My*(cotø)^2+ABS((Mxy+My*cotø)/sinø)
  MøT1=My/((sinø)^2)+ABS((Mxy+My*cotø)/sinø)

• Suppose MxT1<0; section is in X-direction sagging
  MxT2=0
  MøT2=(1/(sinø)^2)*(My+ABS((Mxy+My*cotø)^2/(Mx+2*Mxy*cotø+My*(cotø)^2)))

• Suppose MøT1<0; section is in Y-direction sagging
  MxT3=Mx+2*Mxy*cotø+My*(cotø)^2+ABS((Mxy+My*cotø)^2/My)
  MøT3=0

• Select the appropriate values from the above for output.  Note that if MxT1<0 and MøT1<0, then no top reinforcement is required
  Mx(T)=IF(MøT1<0,IF(MxT3<0,0,MxT3),IF(MxT1<0,0,MxT1))
  Mø(T)=IF(MxT1<0,IF(MøT2<0,0,MøT2),IF(MøT1<0,0,MøT1))

Bottom (sagging) reinforcement

• Determine the generalized Wood-Armer moments:
  MxB1=Mx+2*Mxy*cotø+My*(cotø)^2-ABS((Mxy+My*cotø)/sinø)
  MøB1=My/((sinø)^2)-ABS((Mxy+My*cotø)/sinø)

• Suppose MxB1>0; section is in X-direction hogging
  MxB2=0
  MøB2 =(1/(sinø)^2)*(My-ABS((Mxy+My*cotø)^2/(Mx+2*Mxy*cotø+My*(cotø)^2)))

• Suppose MøB1>0; section is in Y-direction hogging
  MxB3=Mx+2*Mxy*cotø+My*(cotø)^2-ABS((Mxy+My*cotø)^2/My)
  MøT3=0

• Select the appropriate values from the above for output.  Note that if MxB1<0 and MøB1<0, then no bottom reinforcement is required
  Mx(B)=IF(MøB1>0,IF(MxB3>0,0,MxB3),IF(MxB1>0,0,MxB1))
  Mø(B)=IF(MxB1>0,IF(MøB2>0,0,MøB2),IF(MøB1>0,0,MøB1))

A simple example may be used to demonstrate the Wood Armer calculation for orthogonally placed (minimised area) reinforcement.  A simply supported rectangular slab in subjected to a uniform pressure: no in-plane forces (Nx, Ny, Nxy) will be generated and a cursory check of the primary results ahead of the Wood Armer calculations is possible.  The example will generate a generally sagging field of moments, however by reversal of the load, all the Wood Armer calculations for orthogonally placed (minimised area) reinforcement carried out by LUSAS may be checked if necessary.

• Rectangular surface, plan dimensions length 16 units, width 10 units
• Mesh attributes: Any quadrilateral plate or 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, poisson's ratio=0.3
• Supports: pinned on all 4 sides viz.
• Left & right = fixed in translation (X,Y,Z) fixed in rotation (X only)
• Top & bottom = fixed in translation (X,Y,Z) fixed in rotation (Y only)
• Loading attributes: Structural load, global distributed Z direction 1.0unit/unit area

Download Wood Armer example model (LUSAS mdl file)

Cursory Check on primary results

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

Calculation of Wood Armer moments by hand, determined from the moment field (Mx, My, Mxy) using the procedure explained above.  Download spreadsheet calculations (MSExcel format)

Wood Armer Moments from LUSAS Modeller, extracted at the same 4 nodes for comparison to the hand calculations undertaken:

By inspection the results tabulated above agree closely with those derived by hand calculation and demonstrate that the Wood Armer calculations for this example are satisfactory.

Other Wood Armer related topics

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