Curling Stress Equation for Transverse Joint Edge of a Concrete Pavement Slab Based on Finite-Element Method Analysis

Transcription

1 TRANSPORTATION RESEARCH RECORD Curling Sress Equaion for Transverse Join Edge of a Concree Pavemen Slab Based on Finie-Elemen Mehod Analysis TATSUO NISHIZAWA, TADASHI FUKUDA, SABURO MATSUNO, AND KENJI HIMENO In he design of concree pavemen, curling sresses caused by he emperaure difference beween he op and boom surfaces of he slab should be calculaed a he ransverse join edge in some cases. However, no such equaion has been developed in he pas. Accordingly, a curling sress equaion was developed based on sress analysis using he finie-elemen mehod (FEM). In his FEM analysis, a concree pavemen and is ransverse join were expressed by means of a hin plae Winkler foundaion model and a spring join model, respecively. Muliregression analysis was applied o he resuls of he FEM numerical calculaion and, consequenly, a curling sress equaion was obained. Afer comparing he calculaed resuls of he equaion wih curling sress equaions developed in he pas, i was confirmed ha he equaion was valid and pracical. In he hickness design of concree pavemen in Japan, combined sress (load sress and curling sress) is used o esimae faigue damage o concree slab (1). Therefore, he esimaion of curling sress is very imporan in he design. Wesergaard (), Kelley (3) and Iwama (4) have sudied curling sress heoreically and experimenally. In heir sudies, however, hey used very limied condiions. In Japan, equaions for he load sress and curling sress a he longiudinal join edge are presened in he Japan Road Associaion s manual on cemen concree pavemen (1). According o our sudies (5,6), however, sresses a he ransverse join edge are criical under some condiions. Therefore, a curling sress equaion ha can be applied o he ransverse join edge should be developed. We have aemped o develop a curling sress equaion based on sress analysis using he finie-elemen mehod (FEM) and muliregression analysis. The curling sress equaion developed in his sudy is compared wih previously proposed equaions. FEM ANALYSIS Formulaion for FEM The equilibrium equaion of he FEM model aking hermal sress ino accoun can be expressed as follows: T. Nishizawa, Ishikawa Naional College of Technology, Ishikawa, Japan. T. Fukuda, Tohoku Universiy, Sendai, Japan. S. Masuno, Sao Road Company, Ld., Asugi, Japan. K. Himeno, Hokkaido Universiy, Sapporo, Japan. ( K + J) d = fs + fv + f q K = siffness marix of concree slabs; J = siffness marix of joins or cracks; d = nodal displacemen vecor; f s = exernal force vecor; f v = self weigh load vecor; f = hermal load vecor; and q = subbase reacion force vecor. The hermal load vecor due o hermal srain in a concree slab is wrien as follows: f B = srain-displacemen marix; D e = elasic sress-srain marix; = hermal srain vecor; and d = displacemen vecor due o hermal srain. = is wrien as follows: d i d The curling displacemen a ih node can be esimaed by he following equaion. w = T e B D ε dv = B d i w i w wi i = θxi = y θyi w i x 3M he x = y i + i 4 3 M = αe Tz dz V ( ) h / h / z = coordinae along he slab deph; α = coefficien of hermal expansion of concree; (1) () (3) (4) (5)

2 36 TRANSPORTATION RESEARCH RECORD 155 T = emperaure disribuion in he slab; h = slab hickness; E = Young s modulus of concree; x i = x coordinae a ih node; and y i = y coordinae a ih node. Because emperaure disribuion along he slab deph is assumed o be linear in his sudy, M can easily be esimaed from he emperaure difference beween he op and boom of he slab. Then T e f = B D B dv = K d On he oher hand, he relaionship beween subbase reacion and deflecion can be expressed as follows: U is he flexible marix of he subbase. Solving Equaion 7 for q, we can obain Subsiuing Equaions 6 and 8 ino Equaion 1, he following equaion is obained: Then we divide he displacemen vecor ino he loading and curling componens as follows: (6) (7) (8) (9) (10) d e is he loading displacemen vecor. Subsiuing Equaion 10 ino Equaion 9, he following siffness equaion can be obained. ( ) ( ) K + J + U 1 1 d = f + f J + U d (11) Equaion 11 is he basic siffness equaion for FEM considering he hermal srain. Subbase Model The Winkler foundaion model is used and modified o consider parial suppor of he subbase. In his model, when some nodes of a concree slab lif up from a subbase surface, he subbase suppor will be los a hese nodes. This can be formulaed as follows: q = q = subbase reacion force; w = deflecion; = gap due o curling deformaion; and k = modulus of subbase reacion. w 0 V U q = d 1 q = U d ( ) K + J d = fs + fv + f q 1 = f + f + K d U d d = de + d s v e s v ( ) ( ) ( 0 ) k w w0 for w w0 0 0 for w w < 0 (1) Since he incorporaion of Equaion 1 ino Equaion 7 makes Equaion 11 nonlinear, he Newon-Raphson mehod is used o solve he nonlinear equaion. Transverse Join Model Load ransfer across a ransverse join is modeled by a se of hree elasic springs shear, bending and orsion springs (7 ) and can be expressed as follows: f f = K d d l u J l u K J (13) f l, f u = resulan force vecors a loaded and unloaded nodes, respecively; d l, d u = displacemen vecors a loaded and unloaded nodes, respecively; and κ w 0 0 = 0 κ κ n κ w, κ, κ n are spring consans of shear, bending, and orsion springs, respecively. Load ransfer of normal join srucures is carried mainly by shear force. Momen ransfer is supposed o exis only a very small cracks, such as ransverse cracks in coninuously reinforced concree pavemens. Furhermore, according o our sudy, orsion load ransfer hardly exiss and can be negleced in sress analysis. Thus, he effecs of κ and κ n are examined in he following secion. Combined Sress ( ) In he design of concree pavemen in Japan, combined sresses in a concree pavemen slab subjeced o boh raffic loads and curling deformaions are calculaed by simply adding he load sress and he curling sress, which are esimaed separaely. However, if a wheel load applies o a posiion subbase suppor is los because of he curling deformaions, he calculaion resuls would be incorrec. To examine he combined sress condiion a a ransverse join edge, FEM calculaions were conduced using he inpu daa conained in Table 1 and a mesh layou as illusraed in Figure 1. A load of 5 f was applied a he ransverse join edge as he uniform square load was disribued over a 30-cm 30-cm area. Figure shows curling deformaions and sress disribuions in he concree slabs. The concree slabs deform like bowls. Because he concree slabs are long (8 m), sresses are no uniform in he concree slabs. Longiudinal sresses are much greaer han ransverse ones in he inerior par of he concree slabs. In he middle of he ransverse join edge, large ransverse sresses can be observed. Le σ, σ l, and σ c be a curling sress, a loading sress, and a combined sress, respecively, due o he combinaion of loading and curling deformaions. Figure 3 depics he relaionship among hese

3 Nishizawa e al. 37 TABLE 1 Inpu Daa for Calculaing Combined Sress sresses calculaed by FEM and emperaure difference. Alhough curling sress increases when he emperaure difference increases, i is no proporional. This effec occurs because increasing he curling deformaion reduces he number of nodes aached o he subbase and, hus, reduces resrain due o he subbase reacion. Alhough σ c is a lile greaer han σ + σ l in he range of large emperaure difference, he difference beween σ c and σ + σ l is much less han anicipaed. WARPING STRESS EQUATION Range of Values of Parameers To develop he curling sress equaion by means of muliregression analysis, parameers ha have a significan effec on he curling sress should be seleced and FEM numerical calculaions should be conduced varying he values of hese parameers. Concree pavemens for normal road condiions were considered and he range of values of he parameers as indicaed in Table was se. FEM numerical calculaions were conduced for 19 cases combining hese values, and curling sresses in he longiudinal and ransverse join edges were calculaed for each case. Curling Sress Calculaed by FEM Longiudinal Join Edge Figures 4 7 illusrae he effecs of parameers on he curling sresses in he longiudinal join edge. Values of he parameers ha are no presened in he figures are se o he values underlined in Table. From hese figures, he following remarks can be made: Young s modulus of concree has lile effec on curling sresses. FIGURE 1 Mesh layou.

4 38 TRANSPORTATION RESEARCH RECORD 155 Transverse Join Edge Figures 8 14 illusrae he effecs of parameers on he curling sresses in he ransverse join edge. From hese figures, he following remarks can be made: Young s modulus of concree and he lengh of a concree slab have very lile effec on curling sresses. Unlike he case of longiudinal join edge, curling sresses in he ransverse join edge are no proporional o he emperaure difference. The widh and hickness of a concree slab and he modulus of subbase reacion have a significan effec. When he emperaure difference becomes large, a par of he concree slab debonds from he subbase. This changes he suppor condiion and he effecs of he srucural parameers appear in he curling sresses. As depiced in Figure 13, here is no effec of shear spring consan on he curling sresses. There is no effec because curling deflecions along he ransverse join are he same beween he concree slabs, and hus he shear spring a he join does no ac. On he oher hand, roaions due o curling deflecions along he ransverse join are no he same. Thus, when he bending spring consan is large, he bending spring resrains he roaions and causes he curling sresses in he concree slabs as depiced in Figure 14. Bu is effec is relaively small. FIGURE Deformaion and sress disribuions in he concree slab. In he case of a long lengh of concree slab (8 m, 10 m), curling sress increases proporionally wih he emperaure difference. In he case of shorer slab (5 m), however, i is no proporional o he emperaure difference because he resrain due o subbase reacion decreases in he shorer slab. The hickness of a concree slab and he modulus of subbase reacion have lile effec on he curling sresses. Based on hese resuls, he emperaure difference, widh and hickness of he concree slab, and modulus of subbase reacion were seleced as variables for he curling sress equaion. The bending spring consan was excluded because of is small effec. Derivaion of Curling Sress Equaion The variables ha affec curling sress were found o be he emperaure difference, widh and hickness of concree slab, and modulus of subbase reacion. Thus, we assumed a curling sress equaion as follows: c1 c c3 c σ = c 0 h B k (14) σ = maximum curling sress in he ransverse join edge (kgf/cm ); = emperaure difference ( C); k = modulus of subbase reacion (kgf/cm 3 ); h = hickness of concree slab (cm); B = widh of concree slab (m); and c 0... c 4 = consans. FIGURE 3 Combined sress calculaed by finie-elemen model (ransverse join edge). Taking he logarihms of boh sides of Equaion 14, i was ransformed o a linear equaion. The consans, c 0... c 4, were deermined by muliregression analysis using he calculaion resuls of he 19 cases menioned earlier. The second column of Table 3 conains he values of he consans obained by he analysis. Since hese values are saisically derived, hey have no physical meaning. Furhermore, in an engineering sense, a simpler equaion would be preferable. Therefore, we modified hese

5 TABLE Inpu Daa for Muliregression Analysis FIGURE 4 Effec of Young s modulus of concree (longiudinal join edge). FIGURE 6 Effec of hickness of concree slab (longiudinal join edge). FIGURE 5 Effec of lengh of concree slab (longiudinal join edge). FIGURE 7 Effec of modulus of subbase reacion (longiudinal join edge).

6 FIGURE 8 Effec of Young s modulus of concree (ransverse join edge). FIGURE 11 Effec of modulus of subbase reacion (ransverse join edge). FIGURE 9 Effec of lengh of concree (ransverse join edge). FIGURE 1 Effec of widh of concree slab (ransverse join edge). FIGURE 10 Effec of hickness of concree slab (ransverse join edge). FIGURE 13 Effec of shear spring consan (ransverse join edge).

7 Nishizawa e al. 41 FIGURE 14 Effec of bending spring consan (ransverse join edge). values o he simple figures found in he hird column of Table 3. Thus, he curling equaion can be wrien as follows: B k σ = h (15) Applying Equaion 15 o he condiions presened in Table, we obained Figure 15, which depics he comparison beween he curling sresses by Equaion 15 and FEM. I can be observed ha he agreemen is quie reasonable. Thus, we concluded ha Equaion 15 is useful for calculaing he curling sress in concree pavemens. FIGURE 15 Comparison beween resuls by he finie-elemen mehod and Equaion 15. B is he widh of a concree slab in meers. Consequenly, he following equaion can be obained: σ ( x ) σ 4 (18) B x = max 1 Figure 17 illusraes he comparison beween values calculaed by Equaion 18 and he FEM. Alhough he values calculaed by Equaion 18 are a lile greaer han hose by he FEM around 1.5 m from he middle of he ransverse join edge i is pracically valid. Curling Sress Disribuion Along he Edge In he hickness design, faigue damage should be calculaed for a poin of wheelpah in a ransverse join edge, wheel loads would mos frequenly pass. Bu he wheelpah is no necessarily in he middle of he join edge. Therefore, a curling equaion ha can give curling sresses any along he join edge is required. For his purpose, we modified Equaion 15. Figure 16 illusraes disribuions of curling sress along he ransverse join edge. Apparenly, he curling sresses in he middle of he join edge are differen from hose in he wheelpah. To ake he curling sress disribuion ino accoun, he following simple quadraic equaion is assumed: COMPARISON BETWEEN PROPOSED AND PREVIOUS EQUATIONS Figure 18 depics he comparison beween he curling equaion proposed in his sudy and previously proposed equaions. The curling equaions ha have been proposed by Kelley and Iwama (3,4) are presened below. Iwama s equaion was based on full-scale experimens and is adoped in he Japan Road Associaion s manual (1) o calculae hermal sresses. σ ( x) = a x + σ max (16) Assuming σ (x) = 0 a he ransverse join corner, he consan, α, can be deermined from Equaion 16 as follows: a = 4 B σ max (17) TABLE 3 Values of Consans FIGURE 16 Curling sress disribuion along he join edge.

8 4 TRANSPORTATION RESEARCH RECORD 155 FIGURE 17 Comparison beween values calculaed by Equaion 18 and he finieelemen mehod. FIGURE 18 equaions. Comparison beween curling TABLE 4 Inpu Daa in he Comparison σ σ Iwama s equaion for longiudinal join edge: = 0.35 C α E w Kelley and Wesergaard s equaion for free edge: Cy α E = (19) (0) C w is a resrain facor, and C y is a facor deermined from he lengh of he concree slab, L, and radius of relaive siffness, l. Values used in he calculaion are conained in Table 4. When he emperaure difference is less han 7 C, curling sresses given by Equaion 15 are greaer han hose given by oher equaions. As he emperaure difference increases, he Equaion 15 curling sresses become less han hose of oher equaions because wih he large emperaure difference, resrain due o he exension of slab and he subbase reacion is less in Equaion 15 han in he oher equaions. CONCLUSION Applying he muliregression analysis o he resuls of FEM numerical calculaions for 19 cases, we developed a curling sress equaion for he ransverse join edge. The equaion proposed in his sudy can give he curling sress any along he ransverse join edge, using widh and hickness of concree slab, modulus of subbase reacion, and emperaure difference. Afer comparing he calculaed resuls of he equaion developed in his sudy wih ohers from he pas, i was confirmed ha he equaion was valid and pracical. The equaion developed in his sudy is limied o he condiions considered in he calculaion, bu i could easily be refined by considering oher condiions in he FEM calculaions and conducing he muliregression analysis again. REFERENCES 1. The Manual for Cemen Concree Pavemen. Japan Road Associaion, 1990.

9 Nishizawa e al. 43. Wesergaard, H.M. Analysis of Sresses in Concree Roads Caused by Variaion of Temperaure. Public Roads, Vol. 8, 197, pp Kelley, E.F. Applicaion of he Resuls of Research o he Srucural Design of Concree Pavemen. Public Road, Vol. 0, Iwama, S. Experimenal Sudies on he Srucural Design of Concree Pavemen. Pavemen Laboraory, Public Works Research Insiue, Minisry of Consrucion, Japan, Fukuda, T., M. Koyanagawa, and S. Murai. Condiion Survey of Concree Pavemens and Is Evaluaion, Proc., 3rd Inernaional Conference on Concree Pavemen Design and Rehabiliaion, Purdue Universiy, Wes Lafayee, Ind., 1985, pp Nishizawa, T., Y. Kajikawa, and T. Fukuda. Effecs of Laeral Disribuion of Heavy Vehicles on Faigue Cracks of Concree Pavemens, Proc., 5h Inernaional Conference on Concree Pavemen Design and Rehabiliaion, Purdue Universiy, Wes Lafayee, Ind., 1995, pp Nishizawa, T., S. Masuno and T. Fukuda. A Mechanical Model for he Raional Design of CRCP, Proc., 3rd Inernaional Conference on Concree Pavemen Design and Rehabiliaion, Purdue Universiy, Wes Lafayee, Ind., 1985, pp Publicaion of his paper sponsored by Commiee on Rigid Pavemen Design.