DIRECT DESIGN METHOD FOR PRESTRESSED CONCRETE SLABS Chen-Hwa Wang Associate Professor of Civil Engineering and Mechanics The Catholic University of America Washington, D.C. Although several methods such as with maximum bending moment moment-balancing( 1 , load-balanc- may be considered as the control ing( 2 ) and structural membrane section. However, a continuous slab theory (31 can be applied to design has several maximum moment peaks of slabs, none of these is a direct in the middle portion of spans and design method. At present, the load- over supports. Therefore, selection balancing method is widely applied of a control section which has the because it is simpler than the others. largest maximum moment may not It is a trial and analysis approach necessarily provide the optimum de- for obtaining the expected stresses sign. Conventional reinforcing steel and behavior in a slab by the proper or discontinuous tendons for resist- determination of slab thickness, bal- ing partial moment stress at several anced load, eccentricity, and applied sections with moments higher than prestressing. Since this approach is at the control section becomes more time consuming and difficult to a practical and economical. Hence, new design engineer in practice, the selection of the control section this paper presents a straightfor- really depends on the type of struc- ward method and procedures for de- ture and engineering judgment sign of slabs with consideration of whether or not to use mild steel re- the selection of control sections and inforcement or local prestressing in distribution of prestressing. Illustra- the design. In any case, the selection tive examples are also included for of a control section in each direction practical application. of a two-way slab should be con- sidered independently. CONTROL SECTIONS Control sections, which are the DIRECT DESIGN basis of the direct design method, The direct design method is de- should be first selected considering veloped based on the load-balancing maximum moment and the economy analysis for obtaining the desired of the entire slab; the behavior of minimum and/or allowable maxi- the slab under design load can then mum compressive stresses at the ex- be predicted. treme fibers of the control sections. For a simple span, the section It is governed by the appropriate 62 PCI Journal A direct method for design of slabs, instead of the usual process of trial and analysis, is introduced considering desired behavior and necessary distribution of prestressing in the slabs. Direct design equations are developed, procedure is outlined, and illustrative problems are included. computation of the balanced load control sections, using continuous and the amount and eccentricity of draped tendons. However, the ideal- the applied prestressing force de- ized configuration as shown in Fig. pending on the load and boundary 1 is used in the direct design method conditions of the slab. When the for simplicity. The effect of pre- dead load is significantly smaller stressing due to the portion of ten- than the computed balanced load, don that is concave downward is concrete stresses should be investi- minor and may be neglected or com- gated for possible overstress at the puted( 4 > depending on the curva- initial stages of the member under ture of the tendon and the qualified prestressing and dead load. If the professional judgment. allowable stress at that stage is ex- DESIGN EQUATIONS ceeded, a thicker slab or conven- The direct design equations for tional reinforcement should be pro- slabs under uniformly distributed vided to prevent tensile cracking. loads are presented below and their Prestressing is uniformly provided derivations, based on the load-bal- across the whole width of slab and ancing analysis, are given in the is proportional to the moment in the Appendix. rt __ _^--- rt ^ a^rs4.gl ^endo^ !ft /1 L Fig. 1. Idealized tendon configuration June 1968 63 In general, the slab thickness is where B is the moment coefficient, determined by its moment stress, M = BWL2, depending on the slab stiffness, or soundproof and fire rat- boundary condition, W is the total ing requirements, depending on the design load applied, psf or plf, and type of structure. However, if a F is measured in kips/ft. or kips. fully employed control section is ex- Similarly, for a two-way slab, Fig. pected, the slab thickness, t, should 2, the balanced load and the effec- be computed as tive prestressing in the direction "a" are t ` L k(1.8fWl + 4f) ( 1) Wa _ 48kaBaW x 1- 1 + 48(k¢B¢ + kbBb) where t = minimum slab thickness, 2 in. L = span length, ft. Cl f W¢ / (5) W1 = balanced load, psf 8kb 1 + 48kbBb (6) where C k = ratio of the transformed Lb 6B¢LQ eccentricity to slab thick- ness as shown in Fig. 1, 72B¢WL2 and F= k=1—r1-0.5(r2+r3) t[1 + 48(k¢B¢ + kbBb)] x f = design minimum com- — fC¢t2 pressive stress, psi (7) fl= ultimate concrete strength, psi The formulas for balanced load and When the desired minimum com- the effective prestressing in the di- pressive stress is zero, f = 0, the rection "b" can be obtained by sim- slab thickness, in Eq. 1 can be sim- ply changing the subscripts and plified to superscripts from "a" to "b" and "b" to "a" in Eqs. 5 to 7. t = 0.745 L kc (2) Loads on beams supporting two- way slabs are non-uniformly dis- Thicker slabs can be used for any tributed along their lengths. For other reason, but its maximum com- practical simplicity, as accepted in pressive stress in the control section reinforced concrete slab-beam de- is always lower than the allowable sign, a uniform load could be as- one and is inversely proportional to sumed as the square of the thicker slab when f=0. W = W^L¢ + W2L¢ (8a) According to direct design, the 2 3 balanced load, W1, and the effective for load on short beam prestressing, F, in a one-way slab or the beam supports for one-way ,, W = W1L + W2L¢ X 2slabs can be computed as 3 t C 3 — (L¢/Lb)2 1 _ 48kBW + 8k f (8b) ) (L (3) 2 Wl — 1 + 48kB for load on long beam F, = 72BW L2 + 12 f t2 (4) t(1 + 48kB) where residual load W2 = W — W1. 64 PCI Journal Therefore, Eqs. 3 and 4 could also rigidity, or other factors, and then be applied to the direct design of compute the total design load W, beams. psf or plf. From the above equations, the 2. Determine the control section desired minimum compressive stress based upon the moments and the in the control sections, and in any economy of the entire slab. other sections with lower moment 3. Compute an acceptable maximum stress, can be automatically satisfied. k-value based on the minimum The maximum allowable compres- clear coverage for the tendons sive stress can be obtained only when and the assumed slab thickness. the slab thickness satisfies Eqs. 1 Maximum sag is preferred except or 2. For any section with higher at the free ends where the center moment stress than the control sec- of gravity of the concrete should tion additional reinforcement or lo- coincide with the center of grav- cal prestressing must be provided. ity of the steel. 4. Compute the balanced load, W1, DESIGN PROCEDURES for comparison with the dead 1. Assume the dead load, i.e., thick- load of slab and for checking the ness, of slab based on its stress, possibility of reverse stress. Fig. 2. Idealized configuration ma two-way slab June 1968 65 5. Compute the minimum thickness the purpose of getting a watertight of slab by Eq. 1 and compare it roof slab. Compute the required pre- with the assumed slab thickness stressing for f = 4000 psi. to determine whether or not it Since the slab thickness is given, needs to be revised. the total design load of slab is 6. Repeat steps 1 to 5 if slab thick- W=94+75=169 psf. Assuming ness is revised. the section over the middle support 7. Compute effective prestressing, as the control section, its maximum F, and number of tendons and moment is WLL/8 and B = 1/8. spacings. Computing k: r1 =1.47/7.5 = 0.188 8. Compute the necessary reinforce- lowest h/4 in. at % L ment or local prestressing, based from the end support on reinforced or prestressed con- r2= 1.25/7.5=0.166 crete theory, whenever it is re- minimum cover over quired. support APPLICATION r8 =0.5 at free end The use of the direct design equa- k = 1 - 0.188 - 0.5 (0.166 + 0.50) tions will be illustrated by means of = 0.479 the following example problems. Since the desired minimum compres- sive stress is f = 100 psi, from Eq. 3 Example 1. A one-way roof slab has the balanced load is a plan as shown in Fig. 3. The 7½-in, concrete slab weighs 94 psf. W =48X0.479XAX169+8X0.479X100X 1 1+ 48X0.479X1/$ and carries a live load of 75 psf. ( la Minimum coverage for the cables 1 7-5 / is 1-in, measured to the cgs and 30 _ 132 psf the minimum desired compressive and the slab thickness is computed stress in the concrete is 100 psi for by Eq. 1 I! V i+ ;I L=30 Fig. 3. Slab dimensions for Example 1 66 PCI Journal 0.132 Therefore, t=30 0.479(1.8x4+4x0.1) f1 —f2 =100 psi = 30 x 0.19 same as requested. =5.7 in.