AN ANALYSIS OF ELASTO PLASTIC BAR CROSS SECTION STRESS STRAIN STATE IN A PURE BENDING

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1 AN ANALYSIS OF ELASTO PLASTIC BAR CROSS SECTION STRESS STRAIN STATE IN A PURE BENDING Eugedjus Dulnskas, Renata Zamblauskatė, Darus Zabulons 3 Vlnus Gedmnas Techncal Unversty, Saulėteko ave., LT-3 Vlnus, Lthuana. E-mal: skmml@vgtu.lt; verdas_@yahoo.com; 3 Darusz@vgtu.lt Abstract. Ths paper analyses a method of nonlnear bendng beam calculaton usng a rectangular stress block when the behavor of materals s non lnear. Also, geometrcal characterstcs of cross sectons have been nvestgated and presented here when the cross secton stress dagram s trangular and rectangular n tensle and compressve zones, and when the stress dagram s rectangular n the compressve zone and trangular n the compressve zone. Cross secton geometrcal characterstcs obtaned wth the help of the proposed method are consstent wth the known characterstcs. Therefore, the proposed method can be used for calculaton of a non lnear bendng bar cross secton. Keywords: rectangular cross secton nonlnear bendng, rectangular stress block, geometrc characterstcs of cross sectons. Introducton A lot of structural members n practce bend nonlnearly. They can be renforced concrete members, steel beams, etc. Nonlnear bendng has been nvestgated n many works. For example, (Yu and Zhang 996), (Сен- Венан 954), (Ржаницын 954), (Tmoshenko 947). There are a few smple calculaton methods for steel beams (Shackell and Welsh 95), (Roderck and Heyman 95). However, n general case, to calculate a nonlnear bendng member, the system of non lnear equatons must be solved numercally (Webster and Ellson 967). It s not convenent n practcal engneerng calculaton. A rectangular stress block s wdely used for calculaton of renforced concrete beams (EN 99--:4: E), (Hsu T.T.C. 993), (Dulnskas, and Zabulons 7.). The known methods whch use a rectangular stress block can be used when we know the stran at extreme edges,.e. when the cross secton s at the ultmate lmt state. However, a rectangular stress block can be used for all loadng stages n bendng untl the state of ultmate lmts when the cross secton materal behavor s non lnear. In present artcle the method for calculaton of a nonlnear bendng bar cross secton by usng rectangular stress bloc s gven. Also, rectangular cross secton geometrcal characterstcs have been nvestgated here. General dependences Let us consder a rectangular cross secton subject to bendng moment M. Ths cross secton wdth s b and depth s h (Fg ). Dstances between the top and the bottom of the cross secton and ts neutral axs are h and h respectvely. Let us make assumptons: the plane secton hypothess s vald stran and dsplacement are very small stress stran state s unaxal. Fg. A cross secton of a bar a and dstrbuton of stran through ts depth b 599

2 Fg. Cross secton stress dagram a, equvalent stress block b and averaged stress block c It s known that the stran n an arbtrary pont of the cross-secton at the dstance y from the neutral axs s y/r yκ. Where r s a curvature radus, y the dstance between the neutral axs and the pont under consderaton. The neutral axs curvature s κ /r. Then the stran at the top and the bottom of cross secton are h /r and h /r. Snce the hypothess of plane sectons s vald, the normal stresses σ n an arbtrary pont of the cross-secton s calculated usng the formula σ E Eyκ. Where E s the elastc modulus of materal. Let us wrte well known condtons of equlbrum h σ ( ) σ ( ), () N y da b y dy A h h M yσ ( yda ) yσ ( ydy ). () A h where N s the resultant of all nternal axal forces (Fg ) N N + N, where N and N are resultants of compressve and tensle zones (Fg ), M s ther bendng moment M N z + N z, where z and z are dstances between the resultants of nternal forces and the neutral axs (Fg ), The dstance between nternal forces s z z + z. The curvature κ of neutral axs s κ /r M/(EI). After ntegraton by substtuton of equatons () and () takng y r we get h σ ( ) σ ( ), (3) N y dy br d h h σ ( ) σ ( ). (4) M b y ydy br d h Accordng to the mean value theorem, we can wrte (3) n such a form N br σ ( ) d br σ ( ) d br σ + br σ (5) + where σ and σ are average stresses of tensle and compressve zones respectvely. Accordng to our coordnate systems, σ > and σ <. For more convenence, we ntroduce new parameters σ, m σ and σ σ, then (5) s as follows, m N br + br (6) σ σ, m, m Let us ntroduce new parameters: λ, η and λ, η for tensle and compressve zones, respectvly (Fg ). These coeffcents are wdely used for determnaton of the depth and wdth of the stress block of an equvalent rectangular n renforced concrete beams (EN 99-- :4: E), (Hsu T.T.C. 993), (Dulnskas and Zabulons 7.). Takng nto account these coeffcents we get br σ ( y) d brσ, m bσ, mh bη σ, maxλh (7) br σ ( y) d brσ, m bσ, mh bη σ, maxλh (8) where σ,max and σ,max are maxmal stresses of the tensle and compressve zone respectvely (Fg ). From (7) and (8) we get average stresses σ,m and σ,m. (9) σ, m σ, max η λ () σ, m σ, max η λ 6

3 By puttng (9) and () n (6) we get N N+ N br( σ, m+ σ, m) br( σ η λ + σ η λ ), max, max () where N brσ, m and N brσ, m. In an analogous way, n accordance wth the mean value theorem, we rearrange (4) M br d + d σ ( ) σ ( ) ( σ ( ) ) ( σ ( ) ) br avg d + avg d () where avg (σ()) and avg (σ()) are the average value of the product of the stress and stran functons,.e. σ(), n tensle and compressve zones respectvely or n [,] nterval for avg ( ) and n [,] nterval for avg ( ). Let us desgnate that the average values of stress are such avg σ σ (3) ( ( ) ) m,, m avg σ σ (4) ( ( ) ) m,, m where σ,m and σ,m are defned n (5). Then we have M br σ, m, m d+ σ, m, m d br [ σ + σ ] (5), m, m, m, m Let us show that,m and,m are strans at the actng ponts of resultants n tensle and compressve zones respectvely. Let us desgnate: σ,m {σ,m, σ,m},,m {,m,,m}, {, }, and avg ( ) {avg ( ), avg ( )}. From (3) and (4) we have avg ( σ ( ) ) (6) m, σ m, From (5) we have br σ ( ) d brσ, m from where σ σ m, ( ) d (7) From () we have br σ ( ) d avg ( σ ( ) ) d avg σ from where ( ( ) ) avg ( σ ( ) ) σ ( ) d Put (7) and (8) n (6) we get m, (8) d σ ( ) σ ( ) d (9) If σ() s the materal stress stran dependence functon, then,m s coordnates of stress stran dagram resultants and hence,m s strans at the actng pont of resultants. From (5) we get M br [ σ m,, m + σ m,, m ] () br[ σ h + σ h ], m, m, m, m Snce the resultants of compressve and tensle zones must be equal, we can desgnate A σ,m h σ,m h and () we can wrte n such a form M Abz () where z s the dstance between the resultants of nternal forces z r + r r + z + z. (),, ( m m, m, m) Takng nto account coeffcents λ and λ, the expressons of z, z and z are such (Fg ): z h h,5λ h (,5λ ), (3) z h h,5λ h,5λ, (4) ( ) z h,5λ + h,5λ, (5) ( ) ( ) M z h(,5λ) + h(,5λ). (6) ba When coeffcents λ and λ are the same,.e. λ λ λ, then (6) s as follows z ( h h)(,5λ) h(,5λ). (7) Analyss of partcular cases In ths part we nvestgate partcular cases of beam bendng: lnear bendng when the stress stran dependence of materals s accordng to Hooke law, non lnear bendng when the stress stran dependence of materals s nonlnear Also, we wll be analyzng the cross sectons parameters n the above mentoned cases. Frst case: lnear bendng Let us consder the cross secton whose materal stress stran behavor s lnear,.e. n accordance wth the Hooke law. As t s well known, the dagrams of stresses are trangular n tensle and compressve zones (Fg 3). The resultant of the trangular stress dagram s,5h σ,max and an equvalent rectangular stress block resultant s η σ,max λ h where η {η,η } λ {λ,λ }, σ,max {σ,max, σ,max }, and h {h,h }. If the resultants of the trangular stress dagram and the rectangular stress block are equal, then we get h σ λη hσ (8) from where we have,5 max, max, λη,5. (9) 6

4 where A, σ,m h σ,max h/. We can defne the expresson 7/4bh as plastc secton modulus W pl 7/4 /6bh,75W γw, where γ 7/4,75 s the factor of the plastc stran of tensle materals. We can see that the obtaned value of plastc secton modulus s equal to a well known value. Fg. The lnear elastc cross secton stress dagram a and stran dagram b If the stress dagram and the stress block are equvalent, then ther coordnates must also be equal.e. /3h λ h. From where we have λ 3. (3) If we put the obtaned λ value n (9), we get Fg 3. The rectangular stress dagram of a cross secton η 3. (3) λ 4 When the stress dagrams for compressve and tensle zones are the same, the depth of compressve zones s h h/ and takng nto account (3) the rectangular stress block depth s λ h h/3. The dstance between nternal forces z s z z + z h (,5λ ) + h (,5λ ) /3h. Average stresses are as follows: σ,m 3/4σ,max. Accordng to (), A σ,m h η λ σ,max h, then A /3 3/4 σ,max h /σ,max h and M σ,max bh /6 σ,max W. Where W bh /6 s secton modulus. Second case: non lnear bendng Let us consder a cross secton whose stress dagram s rectangular n compressve and tensle zones (Fg 4). In ths case the stress dagram corresponds to the rectangular stress block, hence coeffcents η η λ λ and σ,m σ,max. If σ,max σ,max, then z /h and accordng to (), M σ,max /hbh/ σ,max W. Where W bh /4, and I h/ W bh 3 /8. Thrd case: non lnear bendng Let us consder a cross secton whose stress dagram s trangular n the compressve zone and rectangular n the tensle zone (Fg 5). If h h, then z /h(/ + /3) 7/h. Hence, accordng to () we get 7 M A bz bh 4 σ,,max (3) Fg 4. The rectangular stress dagram n the tensle zone and the rectangular stress dagram n the compressve zone Analyss leads us to conclude that the proposed dependences are vald for calculaton of the cross secton geometrc characterstcs when the stress stran dependence of materals s nonlnear. Conclusons The dependences of the cross-secton geometrcal characterstcs have been obtaned by usng the rectangular stress blocks for tensle and compressve zones for a bendng bar when the materal behavor s nonlnear. The performed analyss showed that the characterstcs of cross sectons obtaned by the proposed methods are dentcal wth the known characterstcs. Therefore, the proposed method can be used for calculaton of cross 6

5 sectons characterstcs when stress the stran dagram of materals s non lnear. References Atkočūnas, J.; Čžas, A. E. Netamprų konstrukcjų mechanka [Mechancs of nelastc structures]. Vlnus: Technka p. Dulnskas, E.-J.; Zabulons, D. 7. Analyss of equvalent substtuton of rectangular stress block for nonlnear stress dagram, Mechanka 6(68): EN 99--:4: E. Eurocode : Desgn of concrete structures Part : General rules and rules for buldngs. Brussels, European commttee for standardzaton p. Hsu, T.T.C. Unfed theory of renforced concrete. CRC Press. 993, 33 p. Roderck, J. W.; Heyman, J. 95. Extenson of the smple plastc theory to take account of the stran hardenng range, n Proceedngs of the Insttuton of Mechancal Engneers, 65: Shackell, K. K.; Welsh, J. H. 95. Plastc flexure of mld steel beams of rectangular cross secton, n Proceedngs of the Insttuton of Mechancal Engneers, 66:. Tmoshenko S. Strength of materals. Part II. Advanced theory and problems. D. Van Nostrand Company Inc. 947, 5 p. Webster, G. A.; Ellson, E. G Iteratve procedures for elastc, plastc and creep deformaton of beams, Journal of Mechancal Engneerng Scence 9: 7 4. do:.43/jmes_jour_967_9_9_ Yu, T. X.; Zhang, L. Plastc bendng. World Scentfc Publshng Company. 996, 554 p. Ржаницын, А. Р. Расчет сооружений с учетом пластических свойств материалов [A calculaton of structures takng nto account plastc propertes of materals]. Москва, Стройиздат, 954, 9 c. Сен-Венан, Б. Мемуар о кручении призм. Мемуар об изгибе призм. [Memor about prsms torson. Memor about prsms bendng]. Москва, Физматлит, 96, 58 c. 63