RC Slab Design master class Ian Feltham, Arup

Linear elastic material do not reflect the cracked nature of concrete FE analysis gives stresses in equilibrium with applied loads Structure will have sufficient strength if appropriate reinforcement is provided for FE stresses If slab loaded by out-of-plane loads, it is unsafe simply to provide reinforcement for M x and M y - twisting moment M xy must also be considered For simple orthogonal arrangements, you can design for Wood-Armer |M x |+|M xy | and |M y |+|M yx | moments RC Slab undertakes strength design for reinforcement in any direction, not necessarily orthogonal RC Slab will also design for in-plane loads, either in their own, or in combination with out-of-plane loads Easier to consider in-plane loads first Interpretation of 2D FE analysis results

pxpx pxpx pypy pypy pvpv pvpv pvpv pvpv Reinforcing for in-plane forces fxfx fyfy  s s/tan  To determine f x and f y : Resolve horizontally s/tan .(p x +f x ) = s.p v  f x = p v.tan  - p x Resolve vertically s.(p y +f y ) = s/tan .p v  f y = p v /tan  - p y p x, p y and p v are applied in-plane stressesf x and f y are stresses taken by reinforcement

Reinforcing for in-plane forces s/tan  pypy pypy pxpx pxpx pvpv pvpv pvpv pvpv fxfx fyfy fcfc s/sin  To determine f c : Resolve horizontally s.(p x +f x ) + s/tan .p v = (s/sin .f c ).sin   f c = p x +f x + p v /tan  = p v.tan  + p v /tan  f c = 2p v /sin2  To determine f x and f y : Resolve horizontally s/tan .(p x +f x ) = s.p v  f x = p v.tan  - p x Resolve vertically s.(p y +f y ) = s/tan .p v  f y = p v /tan  - p y f c is the stress in the concrete s 

p x = 5 p y = 0 p v = 5 p x = 0 p y = -5 p v = 5 p x = 5 p y = -5 p v = 5  (degrees) stress (MPa) p x = 0 p y = 0 p v = 5 f s = f x + f y Reinforcing for in-plane forces - effect of varying  Simple approach General approach Provided concrete stress satisfactory

stress in concrete stress in reinforcement tensile strength of concrete Consider tensile stresses in concrete between cracks tensile strength of concrete will vary along bar when the tensile stress reaches the local strength, a new crack will form

 x (compression)  y (compression) f ck f ctk compressive strength of concrete with transverse tension tensile stress in concrete Bi-axial strength of concrete

stress taken by Y reinforcement stress taken by X reinforcement shear stress compressive stress compressive strength of concrete principal tensile stress Reinforcing for in-plane forces - simple approach using Mohr’s circles Note that the stress taken by both the X and Y reinforcement is equal to the principal tensile strength applied stress X (p x,p v ) Y (p y,-p v ) stress in concrete Y X f cd uncracked νf cd cracked

stress taken by Y reinforcement stress taken by X reinforcement shear stress compressive stress compressive strength of concrete principal tensile stress applied stress X (p x,p v ) Y (p y,-p v ) stress in concrete Y (p v,-p v ) X (p v,p v ) Note that the stress taken by the X reinforcement is equal to (p v - p x ) and that taken by the Y reinforcement is equal to (p v - p y ) f cd uncracked νf cd cracked Reinforcing for in-plane forces - general approach using Mohr’s circles

Reinforcing for in-plane forces - general approach formulae 7 equivalent, but more complex, formulae for skew reinforcement

Compression reinforcement in struts centre line of strut horizontal steel vertical steel compressive strain shear strain/2 Principal tensile strain is approximately 3.6 x design strain of reinforcement principal compressive strain   compression steel 40  strain at 20  to strut  strain in vertical steel  strain in horizontal steel  Although compression reinforcement should be avoided, any provided should be within 15  of centre line of strut to ensure strain compatibility

Applied forces and moments resolved into in-plane forces in sandwich layers Layers are not generally of equal thickness Reinforcement requirements for each layer calculated and apportioned to reinforcement positions (considering in- and out-of-plane strain compatibility) Number and thickness of layers adjusted to determine arrangement that gives best reinforcement arrangement Reinforcing for in- and out-of -plane forces MxMx MyMy M xy NxNx NyNy V V

Introduction of faster algorithms for pure in-plane and out-of-plane conditions, which should greatly increase speed Introduction of bands over which reinforcement areas can be averaged Interaction between GSA and AdSec to estimate deflections in slabs RC Slab future developments