EXAMPLE 16 racked Slab Analysis RAKED ANALYSIS METHOD The moment curvature diagram shown in Figure 16-1 depicts a plot of the uncracked and cracked conditions, Ψ 1 State 1, and, Ψ State, for a reinforced beam or slab. Plot A-B--D shows the theoretical moment versus curvature of a slab or beam. The slope of the moment curvature between points A and B remains linearly elastic until the cracking moment, Mr, is reached. The increase in moment curvature between B- at the cracking moment, Mr, accounts for the introduction of cracks to the member cross-section. The slope of the moment curvature between point -D approaches that of the fully cracked condition, Ψ State, as the moment increases. Since the moments vary along the span of a slab or beam, it is generally not accurate to assign the same cracked section effective moment of inertia along the entire length of a span. A better approach and the one recently added to the program is to account for the proper amount of curvature for each distinct finite element of the slab or beam that corresponds to the amount of moment being applied to that element. After the moment curvatures are known for each element, the deflections can be calculated accordingly. This verification example will compare the results from Example 8.4, oncrete Structures, Stresses and Deformations, Third Edition, A Ghali, R Favre and M Elbadry, pages 85-89, with the results obtained from. Both the calculations and the analysis use the cracked analysis methodology described in the preceding paragraphs. PROBLEM DESRIPTION The slab used in this example has dimensions b = 0.3 m and h = 0.6 m. The slab spans 8.0 m and has an applied load of 17.1 KN/m. EXAMPLE 16-1
Figure 16-1 Moment versus curvature for a reinforced slab member Figure 16- One-Way Slab EXAMPLE 16 -
GEOMETRY, PROPERTIES AND LOADING Slab thickness h = 0.65 m Slab width b = 0.3 m lear span L = 8.0 m oncrete Ultimate Strength h f = 30 MPa oncrete cracking strength f cr =.5 MPa Modulus of elasticity, onc. Es = 30 GPa Modulus of elasticity, Steel Ec = 00 GPa Poisson s ratio ν = 0. Uniform load w = 17.1 KN/m reep coefficient ϕ ( t,t 0 ) =.5 Free shrinkage ε S ( t,t 0 ) = 50E-6 c Note: The concrete cracking strength of f cr =.5 MPa was used in this example using the Run menu > racking Analysis Option command. TEHNIAL FEATURES OF TESTED racked Slab Analysis RESULTS OMPARISON calculated the displacements using a Nonlinear racked Load ase (see Figure 16-1). The first nonlinear load case was calculated without creep and shrinkage effects and the second nonlinear load case included creep and shrinkage effects. Table 16-1 shows the results obtained from compared with the referenced example. Table 16-1 omparison of Results ASE AND FEATURE TESTED Mid-Span Displacement No reep / Shrinkage (m) Mid-Span Displacement with reep / Shrinkage (m) INDEPENDENT RESULTS RESULTS DIFFERE NE 14.4 mm 13.55 mm 5.90% 3.9 mm 4.51 mm.51% EXAMPLE 16-3
OMPUTER FILES: S16.FDB ONLUSION The results show an acceptable comparison with the independent results. ALULATIONS: Design Parameters: Es = 00 GPa, Ec = 30 GPa, h = 0.65 m, b = 0.3m, As = 80 mm, As = 70 mm, enter of reinf. at 0.05 m Span = 8.0 m, Uniform Load = 17.1 KN/m Figure 16-3 Slab ross-section ase 1 Nonlinear cracked slab analysis without creep and shrinkage 1.1 Transformed Uncracked Section Properties: Area, A = 0.07m Y = 0.319m I, transformed = 7.436E-03 m 4 1. Transformed racked Section Properties: Area, A = 0.07 m = 0.145 m I, cracked = 1.809E-03 m 4 EXAMPLE 16-4
1.3 racked Bending Moment, Mr = 3.3E-03.5 E6 = 58.3 KN-m 1.4 Interpolation coefficient, ζ 1 ββ 58.3 = 1 = 1 1.0 = 0.8 M 136 where β 1 = 1.0 and β = 1.0 1.5 urvature: State1: Uncracked 136E-06 Ψ 1 = = 6E-06 / m 9 30 7.436E-03 M r State: Fully racked 136E-06 Ψ = = 506E-06 / m 9 30 1.809E-03 Interpolated curvature: Ψ = 1 ζ Ψ + ζ Ψ = 1 0.8 (6E-06 / m) + 0.8 506E-06 = 157E-06 / m ( ) ( ) ( ) ( ) m 1 1.6 Slab urvature: Figure 16-4 Span-urvature Diagram 1.7 Deflection: By assuming a parabolic distribution of curvature across the entire span (see the Mean urvature over Entire Span plot in Figure 16-4), the deflection can be calculated as, EXAMPLE 16-5
8 Deflection = 0.00157 00 = 14.4 mm (See Table 16-1) 96 ase Nonlinear cracked slab analysis with creep and shrinkage.1 Aged adjusted concrete modulus, E ( tt) ( 0 ) ( tt) E t 30e9, 0 = = = GPa 1 + Xϕ, 1+ 0.8(.5) Where X ( t,t 0 ) = 0.8 ( Program Default), ϕ ( t,t 0 ) Figure 16-5 below) ES 00 n = = = 0 E ( t,t ) 0 0 =.5 (aging coefficient, see. Age-adjusted transformed section in State1: A 1 = 0.07 m NA = 0.344m from top of slab 3 4 I 1 = 8.74 m y = 0.00m, distance from top of slab to the centroid of the concrete area A = 0.1937 m, area of concrete 3 4 I 6.937 = m, moment of inertia of A about NA I 3 r = = 35.34 m A 3 I 6. 937 κ1 = = = 0. 795, curvature reduction factor 3 I 8. 74.3 Age adjusted transformed section in State: A = 0.0701m NA = 0.33m from top of slab 3 4 I = 4.77 m y = 0.161m, distance from top of slab to the centroid of the concrete area EXAMPLE 16-6
A area of concrete = 0.0431m, 3 4 = 1.190 m, I moment of inertia of A about NA I 3 r = = 7.6 m A 3 I 1. 190 κ = = = 0. 78 3 I 4. 77.4 hanges in curvature due to creep and shrinkage: State 1, hange in curvature between period t 0 to t, y y Deltaψ = κ ϕ( t,t0) ψ ( t0) + ε0( t0) + εs ( t,t0) ε r r 6 6 0. 00 6 0. 00 = 0. 795. 5 6 + 8 ( 50 + ) 35. 34x 35. 34 6 = 199 / m The curvature at time t (State 1) 6 6 1 ( t) ( 6 199) x Ψ = + / m = 1909x / m 3 3 State, hange in curvature between period t 0 to t, y y Deltaψ = κ ϕ( t,t0) ψ ( t0) + ε0( t0) + εs ( t,t0) ε r r 6 6 0. 161 6 0. 161 = 0. 78. 5 506 + ( 50 + ) 7. 6x 7. 6 6 = 148 / m The curvature at time t (State ) 6 6 ( ) ( 506 148) Ψ t = + / m = 3754 / m 3 3 Interpolated curvature: 6 6 6 Ψ t = ( 1 ζ ) Ψ1( t) + ζ Ψ ( t) = 1 0.91 (1909 ) + 0.91 3754 = 3584 / m ( ) ( ) ( ) EXAMPLE 16-7
.5 Deflection at center at time, t: By assuming a parabolic distribution of curvature across the entire span, the deflection can be calculated as, 8 Deflection 0.003584 00 3.90 mm 96 = = (See Table 16-1).6 The Load ase Data form for Nonlinear Long-Term racked Analysis: The reep oefficient and Shrinkage Strain values must be user defined. For this example, a shrinkage strain value of 50E-6 was used. Note that the value is input as a positive value. Figure 16-5 Load ase Data form for Nonlinear Long-Term racked Analysis EXAMPLE 16-8