Reinforced Concrete Flexural Members

Reinforced Concrete Flexural MembersConcrete is by nature a continuous materialOnce concrete reaches its tensile strength ~400 psi, concrete will crack.Stress in steel will be ~ 4000 psi.

Design CriteriaServiceabilityCrack width limitsDeflection limitsStrength must provide adequate strength for all possible loads

As area of steel in tension zoneAsarea of steel in compression zoned distance from center of tension reinforcement to outermost point in compressiond distance from center of compression reinforcement to outermost point in compression

Strain and Stress in Concrete BeamscStressM = Tjd = Cjd where j is some fraction of the effective depth, dT = Asfs at failure, T = AsFyC = T = force in As and concrete

Stress in Concrete at UltimateACI 318 approximates the stress distribution in concrete as a rectangle 0.85fc wide and a high, where a = 1c.

Cconcrete = 0.85fcabw

Csteel = As fs

Asfy = 0.85fcabw + As fs

Definitions1 shall be taken as 0.85 for concrete strengths fc up to and including 4000 psi. For strengths above 4000 psi, 1 shall be reduced continuously at a rate of 0.05 for each 1000 psi of strength above 4000 psi, but 1 shall not be taken less than 0.65.

bw = width of web

fs = stress in compression reinforcement (possibly fy)

With No Compression SteelAsfy = 0.85fcabw

For most beams, 5/6 j 19/20

Moment Equationrecall, M = Tjd = Cjd and T = AsFy = 0.9 for flexure

Mu Mn=0.9Tjd = 0.9Asfyjd

substituting 5/6 j 19/20

0.75Asfyd Mu 0.85Asfyd

Reinforcement RatioCompression reinforcement ratioReinforcement ratio for beams

Design EquationsFor positive moment sections of T-shaped beams, and for negative moment sections of beams or slabs where b.For negative moment sections where b and for positive moment sections without a T flange and with b. For intermediate cases where b < < b regardless of the direction of bending.

Balanced Reinforcement Ratio, bTo insure that steel tension reinforcement reaches a strain s fy/Es before concrete reaches = 0.003 (steel yields before concrete crushes) the reinforcement ratio must be less than b. Where b is the balanced reinforcement ratio or the reinforcement ratio at which the steel will yield and the concrete will crush simultaneously.For rectangular compression zones (negative bending)For positive bending (T-shaped compression zone) reinforcement ratio is usually very low (b very large)b = effective flange width, least of:bw + half distance to the adjoining parallel beam on each side of the web the span length of the beambw + 16 hf

Balanced Reinforcement RatioNote: if > b can add compression reinforcement to prevent failure due to crushing of concrete.

Depth of Beam for Preliminary DesignThe ACI code prescribes minimum values of h, height of beam, for which deflection calculations are not required.

Preliminary Design Values 5/3 b practical maximum reinforcement ratio

For typical d/bw ratios:

Beam AnalysisACI 318 Approximate Moments and Shears

Compression ReinforcementIf > b must add compression reinforcement to prevent failure due to crushing of concrete

Crack ControlFor serviceability, crack widths, in tension zones, must be limited.ACI 318 requires the tension reinforcement in the flanges of T-beams be distributed over an effective flange width, b, or a width equal to 1/10 span, whichever is smaller. If the effective flange width exceeds 1/10 the span, additional reinforcement shall be provided in the outer portions of the flange.

Flexure Design Example p. 21 notesThe partial office building floor plan shown had beams spanning 30 ft and girders spanning 24 ft. Design the slab, beams, and girders to support a live load of 80 psf and a dead weight of 15 psf in addition to the self weight of the structure. Use grade 60 reinforcing steel and 4000 psi concrete. 30 ft30 ft30 ft30 ft24 ft24 ft24 ft

Reinforcing Steel