Continous Beams And One Way Slabs
CONTINUOUS BEAMS AND ONE-WAY SLABS
The ACI Code gives approximate formulas for finding shear and bending moments in continuous beams and one way slabs. A summary list of these formulas follows. They are equally applicable to USCS and SI units. Refer to the ACI Code for specific applications of these formulas.
For Positive Moment
End spans
If discontinues end is unrestrained wl 2 n / 11
If discontinues end is integral with the support wl 2 n / 14
Interior spans wl 2 n / 16
For Negative Moment
Negative moment at exterior face of first interior support
Two spans wl 2 n / 9
More spans wl 2 n / 10
Negative moment at other faces of interior supports wl 2 n / 11
Negative moment at face of all supports for slabs with spans not exceeding 10ft (3m) and (b) beams and girders where the ratio of sum of column stiffness to beam stiffness exceeds 8 at each end of the span wl 2 n / 12
Negative moment at interior faces of exterior supports, for members built integrally with their supports
Where the support is a spandrel beam or grider wl 2 n / 24
Where the support is a column wl 2 n / 16
Shear Forces
Shear in end members at first interior support 1.15 wl n / 2
Shear at all other supports wl n / 2
End Reactions
Reactions to a supporting beam, column, or wall are obtained as the sum of shear forces acting on both sides of the support.
American Society for Testing and Materials (ASTM) specifications cover renforcing steel. The most important proper- ties of reinforcing steel are
1. Modulus of elasticity E s , lb/in 2 (MPa)
2. Tensile strength, lb/in 2 (MPa)
3. Yield point stress f y , lb/in 2 (MPa)
4. Steel grade designation (yield strength)
5. Size or diameter of the bar or wire
The tensile strength of concrete is used in combined-stress design. In normal-weight, normal-density concrete the tensile strength can be found from
___
f r = 7.5Ö f ’ c lb / in 2 in USCS units
___
f r = 0.7Ö f ’ c MPa in SI units
The modulus of elasticity of concrete E c adopted in modified form by the ACI Code can be calculated by the formula given below:
A trial batch of concrete can be tested to determine how much concrete is to be delivered by the job mix. To determine the volume obtained for the job, add the absolute volume V a of the four components—cements, gravel, sand, and water.
Find the V a for each component from
V a = W L / ( SG ) W u
where V a = absolute volume, ft 3 (m 3 )
W L = weight of material, lb (kg)
S G = specific gravity of the material
W u = density of water at atmospheric conditions (62.4 lb/ft 3 1000 kg /m 3 )
Then, job yield equals the sum of V a for cement, gravel, sand, and water.
Water/ Cemetitous Materials Ratio
The water / cementitious (w/c) ratio is used in both tensile and compressive strength analyses of Portland concrete cement. This ratio is found from
w / c = w m / w c
where w m = weight of mixing water in batch, lb (kg); and w c = weight of cementitious materials in batch, lb (kg).
The ACI Code lists the typical relationship between the w/c ratio by weight and the compressive strength of concrete. Ratios for non-air-entrained concrete vary between 0.41 for a 28-day compressive strength of 6000 lb/in 2 (41 MPa) and 0.82 for 2000 lb/in 2 (14 MPa). Air-entrained concrete w/c ratios vary from 0.40 to 0.74 for 5000 lb/in 2 (34 MPa) and 2000 lb/in 2 (14 MPa) compressive strength, respectively. Be certain to refer to the ACI Code for the appropriate w/c value when preparing designs or concrete analyses.
Further, the ACI Code also lists maximum w/c ratios when strength data are not available. Absolute w/c ratios by weight vary from 0.67 to 0.38 for non-air-entrained concrete and from 0.54 to 0.35 for air-entrained concrete. These values are for a specified 28-day compressive strength f ’ c in lb/in 2 or MPa, of 2500 lb/in 2 (17 MPa) to 5000 lb/in 2 (34 MPa). Again, refer to the ACI Code before making any design or construction decisions.
Maximum w/c ratios for a variety of construction conditions are also listed in the ACI Code. Construction conditions include concrete protected from exposure to freezing and thawing; concrete intended to be watertight; and concrete exposed to deicing salts, brackish water, seawater, etc
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