Plain Concrete Tunnel Lining-Design Concepts

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1 TRANSPORTATON RESEARCH RECORD Plain Concrete Tunnel Lining-Design Concepts RENHARD GNLSEN Discussed in tis paper are domestic and foreign concepts for te design of plain concrete tunnel lining. Comparisons are made in terms of te respective design capacity assumed by te concepts. A new concept, tailored to U.S. tunnel design practice, wic builds on an advanced approac to defining te capacity of plain concrete tunnel lining, is developed. Te first concept is te standard approac ncorporated n te Deutsces nstitute fiir Normung (DN) building code (DN 1045). Tis concept is built on strengt design criteria similar to tose considered n American Concrete nstitute (AC) Code 318, Strengt Design. Te appllcablllty of tis design concept to unreinforced concrete tunnel lining as been proven extensively during te last decade. Partial cracking of te concrete, a caracteristic of strengt design, s restricted by te code to te extent tat stability and safety requirements are observed. Te second concept discussed is incorporated in te AC Code 318.1, Building Code Requirements for Structural Plain Concrete. n contrast to DN 1045, te AC code is based on working stress provisions, wic tus precludes cracking of te concrete. A comparison of te permissible lining forces of te two concepts s presented. A new Combined Design Concept, based on discussions of AC and DN 1045, is developed. Tis concept is specifically tailored to te design of underground lining and incorporates U.S. design practices. AC Building Codes 318 and are used for reference. Te concept used in te design of a tunnel lining sould reflect boundary conditions associated wit te sape of te tunnel and te construction concept used. Traditionally, tunnel sapes and load assumptions, linked to construction metods, often required a lining tat could witstand bending moments. n tese cases, consideration of axial forces in te lining is most often of minor design importance. However, researc sows tat axial trust, rater tan bending, sould be te primary factor wen competitive lining is required Te promotion of axial liner trust as proven effective. Tis concept as been verified troug model studies performed by various autors and confirmed troug teoretical analysis. An arced tunnel geometry and a igly flexible lining are sown to be te crucial factors in te design of economic structures. f tese criteria are satisfied, axial trust in te lining, rater tan bending, will govern design requirements, provided tat adequate construction concepts are implemented. Reinforcement or use of a ticker liner, oterwise required to accommodate bending moment, can frequently be omitted. Consequently, a design concept is required tat is tailored to tin, unreinforced, arced tunnel lining. To account for te specific boundary conditions of tunnel linings, it is necessary Law/Geoconsult nternational, nc., 1140 Hammond Drive, Suite 5250-E, Atlanta, Ga tat tis design concept differ significantly from te existing American Concrete nstitute (AC) Building Code for Plain Concrete. Like many existing building codes, te AC code is primarily tailored for surface structures. Te Deutsces nstitut filr Normung (DN) and te AC codes for unreinforced concrete design are discussed in tis paper, and a new design concept is developed and proposed for use in U.S. tunnel design. Neverteless, critical evaluation by potential users will be crucial to furter development of te concept to enance meeting practical design requirements. Basically, te proposed concept is applicable to concrete linings regardless of weter te concrete is cast in place, precast, or pneumatically applied However, some caracteristics of precast or pneumatically applied concrete (sotcrete) are not specifically dealt wit ere. Tese are primarily te andling and installation requirements of precast elements and te plastic beavior of green sotcrete wen subjected to early loading from te ground Te corresponding relief of stress concentrations represents safety reserves in addition to tose discussed in tis paper. Also, it sould be noted tat tis discussion is of te sizing of te lining only. Oter important factors of te design processload assumptions, selection of a calculation model, and te like-are not dealt wit. Unreinforced linings of underground structures ave been constructed trougout istory. Examples are numerous, ranging from ancient tunnels to today's linings of underground structures for water, conveyance, and sewage. Also, conditions similar to tose of unreinforced linings prevail in segmental (inged) precast linings. However, it was not until te introduction of te New Austrian Tunneling Metod (NATM) tat te use of unreinforced cast-in-place concrete liners became standard practice for igway and transit structures. First applications of NATM for igway tunnels, including te use of unreinforced concrete lining, date back to te early 1960s. Lined unreinforced tunnels in Austria, Switzerland, and Japan were te basis for researc on and experience wit tis innovative lining concept. Economic considerations and experience gained abroad prompted oter countries to review teir traditional design standards. n te Federal Republic of Germany, for instance, extensive use of NATM for subway systems called for te development of an analytical approac to unreinforced tunnel lining. Ti.s resulted in te definition of a new design standard, incorporated in te German building code (DN 1045) (1). Effective January 1984, te DN code for unreinforced concrete was incorporated in te general provisions for te design

2 16 and construction of tunnels witin te German railway network (e.g., ig-speed rail line, Hannover to Wilrzburg). Tis step is significant given te exceptionally large dimensions of te tunnels involved. Te need for a U.S. standard approac was encountered during preparation of te NATM designs for Le Mounl Lebanon Tunnel, Stage 1, Ligt Rail Transit System, Pittsburg, Pennsylvania, and te Weaton Station and tunnels of te Glenmont Route of te Wasington, D.C., subway system. Te design concept developed ere was first used for te design of tese sections. t offers cost-effectiveness and meets te conventional safety requirements of te U.S. standards. Te autor is confident tat te concept represents an important step forward in te evolution of American tunnel design. DESGN ACCORDNG TO THE GERMAN STANDARD (DN 1045) For te design of bot reinforced and unreinforced structural members, only te strengt concept is specified in DN 1045 (1). Te working stress model is used only to ceck deformation beavior under service load. Trust capacity of te tunnel lining is defined using simplified analytical tools. Te equations used are contained in Manual 220 (2) tat accompanies te DN Code. Ultimate Trust in te Lining Te ultimate tluust capacity of unreinforced concrete is expressed (2, 3) as P, = (Ag) (p,) (1-2 e/) P 0 = (Ag) (p,) were P, = ultimate tluust at given (computed) eccentricity, Po = ultimate tluust at zero eccentricity, e '"' M P eccentricity of tluust resultant in te lining, Ag = (b) () area of te overall liner cross section, b = unit widt of te lining, = overall tickness of te lining, and P, = design strengt for concrete specified in DN 1045, Table 12. (la) (lb) 13wn is minimum strengt value of any test series performed on cubes (DN 1045, Table 1). J3, ::: (0.85) (0.80) (J3wn) = 0.7 Pwn (2) 13, values account for te reductions of prism strengt from cube strengt and faligue effects (4). Equation 2 is valid for P, up Lo 17.5 MN/m 2 (2,536 psi). For iger-strengt concrete, an additional reduction of Pwn is required. Trust capacity (P. in Equation 1) is analyzed in Figure 1 according to te effect of different design assumptions. P. values are indicated as percentages of P 0 (e/ = 0). 100 BO.. 60 ~ TRANSPORTATON RESEARCH RECORD A., 8., C DESGN ASSUMPTONS (S) "-- ' ' ' ' 0.4 o.s f FGURE 1 Trust capacity of unreinforced lining according to different design assumptions. Concept A is simplified design stress distribution according to Figure 2, bottom, and considered for Equation 1. Concept B is parabolic stress distribution according to Figure 2, bottom, used in te strengt design for reinforced concrete components (DN 1045). n Concepts A and B, te eccentricity input parameter (e) is conservatively based on discounting te effect of concrete tension cracking: e = e 1 = en. 1 (see Figure 2, bottom) were e 1 is te eccentricity of liner trust resulting from linear elastic calculations and en. 1 is te eccentricity of liner trust after partial cracking (nonlinear beavior) of te lining. Concept C is stress distribution according to Concept B. n addition, a decrease of trust eccentricity due to nonlinear beavior (cracking) is considered (5): e 1 > en. 1 (see Figure 2). Figure 1 sows tat te conservative design assumptions (A and B) lead to similar results and tus justify te simplified assumption (A) for determining lining capacity. Restriction on Trust Eccentricity and Crack Dept According to DN 1045 cracking from service loads must not exceed te centroid of te concrete section. Tis is to be acieved by limiting te trust resultant's eccentricity (e) to 0.3. n comparison, te parabolic stress distribution in Figure 2, bottom, corresponds to an eccentricity of (1, 4). Te difference between te two eccentricity values is considered negligible. Tis relationsip between te eccentricity of te trust resultant and crack dept applies to ultimate compression only (parabolic stress distribution, Figure 2, bottom). Working stress considerations (linear triangular stress distribution, Figure 2, top) lead to a smaller crack dept (x). Te eccentricity value (e) used in Equation 1 usually relates to working stress caracteristics. Terefore te design approac of DN 1045, Equation 1 linked to ultimate stress assumptions, is conservative and eliminates te requirement to ceck te crack dept at working stress level.

3 Gnilsen M WORKNG STRESS X 0.4 (MAX) ULTMATE., ~:'V l r ~~+ D~ 1 / Br - Br T 0.6 (MN) SMPLFED DESGN STRESS DSTRBUTON (CONSDERED FOR EO. 1) e1 en. (DESGN ASSUMPTON) (6). Like most local codes, te AC code as been developed primarily to address requirements of aboveground structures not underground structures. Consequently, working stress criteria only are provided for te design of unreinforced concrete members. Working Stress Criteria Te principles of te working stress design concept are observed in AC Tus a linear elastic stress distribution in te lining is assumed. Bot compression and tensile stresses must not exceed permissible values. Corresponding trust limitations can be derived from stress restrictions in Section of AC and Section 7.8 of AC (7): wit respect to limitations on compression and fbu() - f,,,, S ftu (5) 17 (4) PARABOLC STRESS DSTRBUTON -.l..-o..'< wit regard to te permissible tensile stress of te concrete were f au = compression due to axial trust, fbu(t) = tension due to bending, and fbu(c) = compression due to bending. + f X 0.5 (MAX) 0.6 ~(1 2%l (MN)! FGURE 2 Different design assumptions at permissible eccentricity e = 0.3 : working stress design assumption and transition to ultimate stress level (top) and ultimate stress design assumption of DN 1045 (bottom). Safety Considerations To account for irregularities in te structure, load uncertainties, and computational inaccuracies, a factor of safety (FS) is considered in arriving at a permissible trust value: (3a) (3b) Safety Considerations Te index (u) indicates tat trust and bending, factored by U, are considered for Equations 4 and 5. U is load factor according to AC 318. All permissible stress values are governed by Section 6.2 of te code: fc,. = permissible bending compression, fcu. = permissible axial compres.sion, and f,,, = permissible bending tension. Te permissible stress ifperm), induced by service load (P 11 ) equals <J,J ($)/U. Terefore te factored permissible stresses as used in Equations 4 and 5 [i.e., induced by te factored load (P.,)] equal (f,j ($)./ 11 is nominal strengt and$ is 0.65, strengt reduction factor, according to AC Permissible Trust n te Lining Permissible trust values (P perm). considering compression limits, can ten be derived from Equation 4: were FS is 2.5, as specified for unreinforced concrete components in DN DESGN ACCORDNG TO AC were Sis b 2 /6, section modulus. By transformation, (6a) (6b) Unreinforced concrete members are covered by te Building Code Requirements for Structural Plain Concrete (AC 318.1) Permissible trust values wit respect to tensile capacity can be derived from Equation 5:

4 18 TRANSPORTATON RESEARCH RECORD 1150 and by transformation, Pperm = J,j[(e/S) - (l/a 8 )] U (7a) (Tu) Equations 6b and 7b require te factored permissible stress values [(f,.) Gil contained in AC 318.1, Section 6.2, and AC , Table 7.1. Te nominal strengt values included in tese parameters are broken out. Nominal strengt ifn) according to AC for flexure is f,,. =Jc' (compression) (8) ~ 60 E C> ALL CURVES: F.S. ~ 2.50 L~:'~"' J,,. = 5((/) 1 / 2 (tension) For axial compression it is fcin = 0.6 Jc' (9) (10) nstead of using Equations 6b and 7b, P perm can be calculated using tese nominal strengt values: Pperm = (0.6//) (f/) (0.65/[(f//A 8 ) + e(0.6f/s)] U (11) substituting Equation 6b and substituting Equation 7b. COMPARSON OF DN 1045 AND AC (12) Comparison of te two codes evaluates te impact of different design assumptions. Bot concepts are expected to lead to structurally sound underground linings. However, te d~gree of conservatism implicit in te design assumptions varies, and tis potentially affects te economy of te underground structure. Te same factor of safety, FS = 2.50, is used for eiter concept to eliminate te effect of explicitly imposed safety margins. According to AC 318.1,!perm = f,. ($/U) (13a) were qi is 0.65 (6) and U is 1.63, leading to FS = 2.5. Te value of 1.63 represents average load conditions; compared wit te full range of U from 1.4 to 1.7, /perm = (0.65/1.63) fn = 0.40 f,. According to DN 1045, /perm = ~r/2.5 ~r = 0.56 ~w (8) fc' = 0.85 ~ws (13b) (14a) ~... = fw = cube strengt, averaged from a test series according to DN By substitution,!perm = 0.66///2.5 = 0.26fc' (14b) FGURE 3 Comparison of permissible trust according to AC and DN Permissible trust values given in Table 1 and sown in Figure 3 are determined from Equations 1 and 3, and 11 and 12, respectively. Values are indicated as percentages of P perm at zero trust eccentricity determined according to DN Plain concrete provisions according to AC lead to a considerably smaller design capacity tan do tose calculated according to DN Tis reflects te tensile stress limitations and te observation of a minimum eccentricity value specified in te AC code. However, it must be noted tat many local codes developed from DN 1045 call for consideration of a minimum "unavoidable" eccentricity as well. % DEVELOPMENT OF THE COMBNED DESGN General According to te Guidelines for Tunnel Lining Design of te Underground Tecnology Researc Council (9), "structural codes sould be used cautiously" because "most codes ave been written for above-ground structures on te basis of assumptions tat do not consider ground lining interaction." Te guidelines furter state: "Blind application of structural design codes is likely to produce limits on te capacity of linings tat are not warranted in ligt of te substantial contributions from te ground and te important influence of construction metods on bot te capacity and cost of linings." Similar statements ave been made by oter autors and organizations (5, 10, 11). Careful evaluation of te applicability of existing strncniral codes to te design of tunnel liners was necessary. Comparison of te German and te U.S. standards for unreinforced concrete design indicates tat te strengt model used in DN 1045 allows greater use of te lining capacity tan do te AC working stress criteria. t is widely acknowledged tat te strengt criteria can represent a sound basis for te design of unreinforced concrete tunnel lining. Neverteless, it is understandable tat use of foreign codes (i.e., te German DN code) is severely limited in te United States. Terefore a new concept needed to be developed to better reflect typical U.S. situations.

5 Gnilsen 19 TABLE 1 PERMSSBLE THRUST VALUES ECCENTRCTY AC DN 1045 e/ EO. 11 EO. 12 EO EO (92) % % ( 92) % 100 % ( 20) ( 0) MNMUM ECCENTRCTY ACCORDNG TO AC / SECTON Tis design concept uses design parameters and safety definitions incorporated in AC 318 (12). n addition, elements of te German standard are used, tus making use of a proven tool for applying strengt criteria to unreinforced concrete design. Figure 4 sows te sources used in developing te design approac. Elements of two codes are combined to arrive at te Combined Design Concept. Teir applicability is subject to te extent of concrete cracking tat corresponds to te eccentricity of te trust resultant. For eccentricities up to a limit defined later, a strengt design concept, referred to as Modified Strengt Design, is used. For eccentricities tat exceed te defined limit, a working stress concept is applied to te uncracked lining portion, tus averting furter cracking of te concrete lining. assumptions made for stress and strain in te concrete. Deviating from AC 318, owever, te developed design concept considers te potential of tunnel liners to render reinforcement (5, JO, 11) unnecessary. According to DN 1045 and AC 318, a simplified rectangular compression stress figure is recommended for design purposes unless test results call for oter distributions. Simplified rectangular stress figures are derived differently in te design concepts sown in Figure 5. DN 1045 and te Modified Strengt Design define smaller ultimate compression stress blocks tan are granted by AC 318 (Figure 5, rigt). Corresponding ultimate strain assumptions are sown in Figure 5, left. Te correlation between tese ultimate stress-strain assumptions and working stress conditions as been sown in Figure 2. Te effect on te conservatism implicit in te design assumptions is analyzed as follows. According to AC 318, c = 3 [ (/2) - e] (15a) AC 318.1: AC 318: PLAN CONCRETE- RENFORCED CONCRETE- WORKNG STRENGTH STRESS DN 1045 : PLAN CONCRETE STRENGTH DN 1045 : RENFORCED CONCRETE STRENGTH MCOMBNED DESGN "-CDC FGURE 4 Derlvatlon of te Combined Design Concept. Modified Strengt Design Based on AC 318 and te German Standard A liner's capability to sustain nonlinear beavior induced by an eccentric trust is best modeled by te strengt design approac. Tis is also indicated by te Alternate Design provisions in AC 318 (6, 12) in wic it is stressed tat "te straigt line teory [as it is used in te working stress metod of AC 318.1] applies only to design members in flexure witout axial load." Suc load conditions do not typically occur in arced structures. Te Modified Strengt Design concept developed ere follows te strengt design concept CAC 318) in te design a = 0.85c = 2.55 [ (/2) - e] (15b) According to DN 1045 and te Modified Strengt Design, a = 2.0 {(/2) - e] (15c) were a is te dept of te rectangular design stress block according to DN 1045 and te Modified Strengt Design. Te conservatism in te Modified Strengt Design compared wit AC 318 regarding te widt of te design stress block is expressed by a/a = 2.55/2.0 = 1.28 n te Modified Strengt Design, ultimate trust capacity is written as P~ = (b) () (0.85//) [1 - (2 e/)] (16a) for e > 0.1 and as P 0 = 0.8 () (b) (0.85.f.:') fore~ 0.1 (16b) Because tese equations are based on tension cracking, teir validity is limited to reacing a permissible crack widt as specified later.

6 20 TRANSPOUATON RESEARCH RECORD 1150 i~ b, +c,t;j e /z c a, ~ 3 ( V1 e, ) a, b, c, REFERENCE : a, AC 318 b, DN 1045 c., "MODFED STRENGTH DESGN" b, + c, a, ~ 2('Yz e) a 0.85 c FGURE 5 Design parameters for strengt design concepts: strain design assumption (left) and ultimate stress design block (rigt). Crack Restrictions Distortion is controlled by limiting trust eccentricity (e), toe= 0.30 of te liner tickness (). n te ultimate stress design, suc eccentricity corresponds to cracking to te centroid of te cross section. n comparison, stresses from service loads lead to smaller crack depts (Figure 2, top). Design Based on AC Beyond Maximum Permissible Cracking Crack restrictions limit te use of te strengt design metod to a maximum eccentricity of 0.3. However, greater eccentricities will not cause furter cracking as long as tension stresses in te uncracked liner portion remain witin te elastic range. A pertinent design approac is provided by te working stress metod of AC Observation of te permissible concrete tension (Equations 9 and 13) yields te permissible values of liner trust at given eccentricities. Figure 6 sows a linear stress distribution as induced by service load (P,J at te critical eccentricity (e = 0.3 ). A comparison wit ultimate stress conditions is sown in Figure 2. Te dept of te compressed liner portion (iicompr) is compr = 3 X (0.2 ) = 0.6 (17) Wen te concrete's tensile strengt is considered, an even larger uncracked liner dept () is actually provided. t is governed by te provision of AC tat te trust (P,J must not cause tensile stresses tat exceed permissible values (6). From Equation 7b it generally follows tat and (18) (19) TENSON: 11 perm L (AC 318.1) x ' ii com r Pn (" Pperml JOMPRESSVE TRESS :s fc perm FGURE 6 Cracking of te llnlng at te strengt design limit, e = 0.3. were AB = b, uncracked portion of te lining pertinent to trust eccentricity e = 0.3 ; = uncracked dept of te lining pertinent to e = 0.3 ; s = b2 /6, section modulus of te uncracked linin~xortion; and fr perm = 5(//) 2 ($/U), according to Equations 9 and 13. Uncracked lining dept (), is determined by iteration from Equation 20 developed from Equation 19 after substitution and transformation. t was demonstrated earlier tat te uncracked dept of te cross section is greater tan 0.6. For most cases, a value of approximately 0.65 will result from te iteration. Te value of P perm considered in Equation 20 pertains to e = 0.3 and is calculated according to Equation 16a wit te additional consideration of a safety factor (FS). [2-1.2 (/)]/b =fr perm/p perm (20) Te trust eccentricity (e) can ten be expressed by :

7 Gni/sen 21 e = [0.5 - (0.5 - e)] (21) Pperm fore > 0.3 is calculated using Equations 19-21: Pperm = [5({///2] 0.65/[(e/S) - (1/b)] U (22) Equation 22 is reflected in Figure 7. Te grap sows ow P perm increases wit a decreasing trust eccentricity (e). Safety Considerations n accordance wit AC building code requirements for bot reinforced and unreinforced concrete, a permissible trust value must include an explicit safety factor tat is related to te ultimate design capacity of te liner. Load uncertainities are ereby accounted for by a factor (U). Reduction of te ultimate design strengt by <> accounts for imperfections of te structure and computational inaccuracies. A compound safety factor can tus be written: FS = U/<1> (23) r-- 1 e ecc. EQUATON 22 e FGURE 7 Permissible trust versus eccentricity, e > 0.3. Pperm l Explicit and implicit safety components are analyzed elsewere (8). Tey are considered in determining te factors U and <> to be used in te proposed liner design concept. Te evaluation suggests tat a conservative approac will be taken by meeting te safety factor provisions of AC (<>= 0.65 and U = 1.4 to 1.7). Tese parameters provide a compound safety factor of from 2.15 to 2.62 according to Equation 23. Comparison of Design Concepts Te grap sown in Figure 8 reflects te computation of permissible trust values given in Table 2. Altoug Table 2 covers te full range of load factors (1.4 troug 1. 7), U = 1.63 is based on te grapic display. For te Combined Design Concept (<>= 0.65) te factor U = 1.63 corresponds to FS = 1.63/0.65 = 2.5. Values in Table 2 are indicated as percentages of P penn at zero eccentricity determined according to DN Values in parenteses relate to ~ E ~ c.. t; ::i a: - LU...J 00 gj ::!! a: LU c ~ ALL CURVES: U = 1.63!F APPLCABLE) (2) AC 318 (,0= 0.70, F.S. = 2.33) DFFERENCE BETWEEN AC 318,1 AND THE " COMBNED DESGN " (3) "COMBNED DESGN " (,0 = 0.65, F.S. = 2.50) (1) DN 1045, (F.S. = 2.50) AC (,0 = 0.65, F.S. = 2.50) Q t~ Yti FGURE 8 Comparison of permissible trust according to AC 318 ("Strengt Design"), DN 1045, AC 318.1, and te Combined Design Concept.

8 22 TRANSPORTATON RESEARCH RECORD 1150 TABLE 2 DESGN FACTORS Reference to Figure 8 Design Cri teria Tru st Ultimate Wi rl t of Permissible Eccentr ici t y Desi gn Safety Design Stress Trust Design Crincept Notes Strengt Figure e/ 0 lj F S %nf R, l l/2l e J % (1) D N (2) ' (132) 1.4 ( 1. 7) ,43) AC ( 68) (3) " Modified St rengt Design" ' ( 99) 1.4 ( (2.62) ( 49) ' 1 Nole: '1, equivalent to modified slrengt design for eccentricity e :o- O 3. '2, emin = 0.11, according to te slrengt design parameter used [see Figure 5, and refer to equalion 15b in wic a = 2.55 (0 5 - e)); amax = 2.55 ( ) = '3, em n = 0.10 ; according al AC 318.1, Section te maximum load factor (U = 1.7), and oter values relate to minimum safety factors permitted by te codes. For a comparison wit Pperm according to AC refer to Figure 3. Figure 8 indicates permissible trust values subjected to te design parameters (Figure 5) and safety factors used. Te Combined Design Concept (i.e., te Modified Strengt Design for eccentricities smaller tan 0.3 ) yields similar results for concentric trust but allows iger permissible values for eccentric trust in te lining tan does DN A prime purpose of combining two distinctly different concepts in te Combined Design Concept was to extend te applicability of unreinforced concrete design. n particular, te combined concept allows for eccentricities greater tan te strengt design limit (cracking to one-alf of te lining dept; emax = 0.3 ), if te liner trust is small enoug to satisfy working stress criteria imposed on te uncracked portion of te lining dept. Tis provision can be crucial to permitting use of unreinforced concrete were load cases generate low trust values tat increase te eccentricity of te trust resultant (e.g., dead weigt). Figure 8 also makes it possible to evaluate te impact of using te working stress design according to AC over rlifferent eccentricity ranges. P perm is evaluated for eccen tricities greater tan 0.3. t sows tat AC yields sligtly more conservative results if used only for e > 0.3 (Figure 7 and Equation 22). n contrast, te curve describing Equation 12 is based on te unlimited use of AC (i.e., for all eccentricities). Tis results in a iger P perm for e > b. 23? '-f'...--\. EQUATON 16 l 23 ~ 5 0 ~---'----4 ~ Cl. Q, EQUATON e/ ODFED STRENGTH DESGN ACl318.10N REMANNG UNCRACKED SECTON e = M/P WTH p > o (COMPRESSON) FGURE 9 Design cart for permissible liner trust. compression stress constitute te criterion for P perm Tis is te case for eccentricities of up to 0.3. For eccentricities greater tan 0.3, permissible tension stresses govern P perm According to AC 318 and DN 1045, permissible tension stress induced by service loads is related to compression strengt caracteristics by a nonlinear correlation law. As a result, limits on compression and tension, respectively, diverge wit inc.re.asing concrete strengt. Te grap in Figure 9 is based on a concrete strengt frequently used for tunnel lining design if/ = 3,500 psi). Terefore, for most cases, Figure 9 can be used as a design tool, wic makes it possible to avoid analytical calculations. Also, deviations from oter concrete strengts are negligible for most practical purposes. Summary of te Combined Design Concept Te permissible liner trust according to te design concept developed in tis paper is sown in Figure 9. Te eccentricity (e) of te compression resultant corresponds to a linear elastic stress distribution and is indicated as a variable input parameter. Permissible trust values are given as percentages of te permissible trust at zero eccentricity. Figure 9 is designed from Equations 16a, 16b, 22, and 23 and is valid regardless of specific design data if restrictions on NOTATON a = dept of rectangular design stress block according to AC 318; a = dept of rectangular design stress block according to DN 1045 and te Modified Strengt Design metod; ~g = gross area of overall section; Ag = gross area of uncracked section portion after partial cracking;

9 Gnilsen b = widt of compression face of lining (unit widt); c = distance from extreme compression fiber to neutral axis, design assumption; e = eccentricity of trust in te lining, general; e, = eccentricity of liner trust resulting from linear elastic calculations (service load condition); en = eccentricity of liner trust after partial cracking (nonlinear lining beavior); e = eccentricity of liner trust measured from te centroid of te uncracked liner portion after partial cracking of te unreinforced structure; f c' = compressive strengt of concrete specified in AC 318; f w = average cube strengt of a series according to AC 318; fu = compressive stress in concrete induced by p u general; f,,,, = compression stress induced by Pu at zero eccentricity; fbu(t) = concrete tension stress induced by Mu; compression stress induced by Mu; permissible compressive stress in concrete, general; fcu = fen tj>, permissible compression stress in concrete subjected to Mu; fbu(c) = fc perm = fcim = f,;,, tj>, permissible compression stress in concrete subjected to Pu at e = O; f 1u = fin tj>, permissible tension stress in concrete subjected to Mu; f,. = nominal concrete strengt, general; fcin = nominal concrete compression strengt subject to P at e = 0 according to AC 318.1; f tn = nominal concrete tension strengt subject to M according to AC 318.1; fc,. = nominal concrete compression strengt subject to M according to AC 318.1; f, perm = permissible concrete tension stress induced by P,.; FS = factor of safety; = = M = Mu = p = P,. = overall tickness of lining; minimum tickness of an uncracked lining portion; bending moment, general = P (e); factured moment= Pu (e); axial load, general; nominal axial load (= service load); Pu = factured axial load; pc = concrete compressive force induced by P..; Po = ultimate trust at zero eccentricity; P. = ultimate trust at given eccentricity; ppeml = permissible P,.; s = section modulus of overall section; s = u = section modulus of uncracked section portion; load factor; x = dept of cracked section portion; Pws = fw average cube strengt of a series Pwn = according to DN 1045; Minimum value of any cube strengt test series according to DN 1045; P, = Design strengt for concrete according to ti> = ACKNOWLEDGMENT DN 1045; and strengt reduction factor. Te autor wises to tank te staff of Law/Geoconsult nternational for teir assistance in preparing te manuscript. Also, gratitude is expressed to Klaus Mussger of Geoconsult, Austria, and Herbert H. Einstein of te Massacusetts nstitute of Tecnology, Cambridge, Massacusetts, for teir valuable contributions and review. REFERENCES 1. Deutsces nstitut flir Normung. Concrete and Reinforced Concrete Design and Construction. Standard Berlin, Federal Republic of Germany, Beut Verlag Gmb, Dec Deutsces nstitut rur Normung. Bemessung von Beton- und Stalbeton nac DN 1045, Ausgabe 978 (Design of Plain Concrete and Reinforced Concrete Members per DN 1045, 1978 ed.). Manual 220. Verlag Wilelm Ernst & Son, Berlin, Federal Republic of Germany, F. Leonardt. Vorlesungen uber Massivbau (Lectures on Concrete Design), Part 1, Belon Ka/ender (Concrete Almanac). Verlag Wilelm Ernst & Son, Berlin, Federal Republic of Germany, S. L. Paul, A. J. Hendron, E. J. Cording, G. E. Sgouros, and P. K. Saa. Design Recommendations for Concrete Tunnel Linings. Report UMTA-MA UMTA, U.S. Department of Transportation, Building Code Requirements for Structural Plain Concrete. AC 318.1, Committee 318, American Concrete nstitute, Detroit, Mic., Building Code Requirements for Structural Plain Concrete. AC Committee 322, American Concrete nstitute, Detroit, Mic., R. Gnilsen. Unreinforced Concrete Tunnel Lining -Design Concepts. Tecnical Bulletin 1. Law/Geoconsult nternational, nc., Atlanta, Ga., May Guidelines for Tunnel Lining Design. Underground Tecnology Researc Council, ASCE Tecnical Researc Council, Tecnical Committee on Tunnel Lining Design, Precast Concrete Tunnel Linings for Toronto Subway. Preprint ASCE, National Meeting on Transportation Engineering, Boston, Mass., H. H. Einstein and C. W. Scwartz. mproved Design of Tunnel Supports. Department of Civil Engineering, Massacusetts nstitute of Tecnology, Cambridge; UMTA, U.S. Department of Transportation, Building Code Requirements for Reinforced Concrete. AC Committee 318, American Concrete nstitute, Detroit, Mic.,

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