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Clark-Nielsen (Wood-Armer for in plane forces)

The Wood-Armer equations resolve bending and twisting moments for the purpose of RC slab design.  Similar equations may be derived to resolve the forces required to resist an in plane force triad consisting of two in plane forces per unit length (Nx, Ny) and an in plane shear force per unit length (Nxy).  These equations are generally known as the Clark-Nielsen 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" Appendix A, the Clark-Nielsen calculation may be carried out by following the procedure below.  All calculations proceed after the determination of the in-plane stress field Nx, Ny, Nxy.  For an elastic section subject only to in-plane forces, the stress field is:

Top: NxT=Nx/2, NyT=Ny/2, NxyT=Nxy/2
Bottom: NxB=Nx/2, NyB=Ny/2, NxyB=Nxy/2

Note that although the sign convention for Nxy differs between LUSAS and "Concrete Bridge Design to BS5400", Nxy always appears in the equations below as a squared or an absolute value and thus no amendment is required to take account of the sign convention.  The equations are described in a form that can be translated directly to a spreadsheet format:

Top reinforcement

• Determine the generalized Clark-Nielsen forces:
  NxT1=NxT+2*NxyT*cotø+NyT*(cotø)^2+ABS((NxyT+NyT*cotø)/sinø)
  NøT1=NyT/((sinø)^2)+ABS((NxyT+NyT*cotø)/sinø)

• Suppose NxT1<0; top layer is in X-direction compression
  NxT2=0
  NøT2=(1/(sinø)^2)*(NyT+ABS((NxyT+NyT*cotø)^2/(NxT+2*NxyT*cotø+NyT*(cotø)^2)))

• Suppose NøT1<0; top layer is in ø-direction compression
  NxT3=NxT+2*NxyT*cotø+NyT*(cotø)^2+ABS((NxyT+NyT*cotø)^2/NyT)
  NøT3=0

• Select the appropriate values from the above for output.  Note that if NxT1<0 and NøT1<0, then no top reinforcement is required
  Nx(T)=IF(NøT1<0,IF(NxT3<0,0,NxT3),IF(NxT1<0,0,NxT1))
  Nø(T)=IF(NxT1<0,IF(NøT2<0,0,NøT2),IF(NøT1<0,0,NøT1))

Bottom reinforcement

• Since NxT=NxB, NyT=NyB, NxyT=NxyB in this case, the top and bottom layer equations are identical.

Concrete forces (top)

• Determine the generalized Clark-Nielsen concrete force
  FcT1=IF(FcTI<0,-2*FcTI*(cotø-cscø),-2*FcTI*(cotø+cscø))
  where FcTI=NxyT+NyT*cotø

• Suppose NxT1<0; top layer is in X-direction compression
  FcT2=((NxT+NxyT*cotø)^2+(NxyT+NyT*cotø)^2)/(NxT+2*NxyT*cotø+NyT*(cotø)^2)

• Suppose NøT1<0; top layer is in ø-direction compression
  FcT3=NyT+(NxyT^2/NyT)

• Suppose NxT1<0 and NøT1<0, there is no cracking and "concrete force parallel to the crack" has no meaning.  Reasonable value to give is the second (most compressive) principle stress:
  FcTP=(NxT+NyT)/2-SQRT(((NxT-NyT)/2)^2+NxyT^2)

• Select the appropriate values from the above for output. 
  Fc(T)=IF(AND(NøT1<0,NxT1<0),FcTP,IF(NøT1<0,FcT3,IF(NxT1<0,FcT2,FcT1)))

Concrete forces (bottom)

• Since NxT=NxB, NyT=NyB, NxyT=NxyB in this case, the top and bottom layer equations are identical.

A simple example may be used to demonstrate the Clark-Nielsen calculation for orthogonally placed (minimised area) reinforcement.  The example used here is a a shell subjected to an unsymmetrical tensile load; a thin rectangular slab fixed in translation at one edge and with a point load applied to an opposing corner. No bending field (Mx, My, Mxy) will be generated, while a stress field will be generated where various combinations of positive and negative stresses for Nx, Ny and Nxy exist, enabling the verification of the Clark-Nielsen calculations for a variety of conditions.  The calculations for Top and Bottom face reinforcement will produce the same result and so a single hand calculation will verify both results.

• 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 on bottom edge
• Loading attributes: Structural load, Concentrated in Y direction 100 units total

Download Clark Nielsen example model (LUSAS mdl file)

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

Calculation of Clark-Nielsen stresses by hand, determined from the stress field (Nx, Ny, Nxy) using the procedure explained above.  Download spreadsheet calculations (MSExcel format)

Clark-Nielsen stresses 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 Clark-Nielsen calculations for this example are satisfactory.

Other Wood Armer related topics

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