The Bending of Rectangular Deep Beams with Fixed at Both Ends under Uniform Load

Size: px

Start display at page:

Download "The Bending of Rectangular Deep Beams with Fixed at Both Ends under Uniform Load"

Nelson Phillips
5 years ago
Views:

1 Engineering,,, 8-9 doi:.6/eng..7 Pubised Onine December (ttp://.scirp.org/journa/eng) Te Bending of Rectanguar Deep Beams it Fied at Bot Ends under Uniform Load Abstract Ying-Jie Cen, Bao-Lian Fu, Gang Li, Jie Wu, Ming Bai, Xia Cen Department of Civi Engineering and Mecanics, Yansan University, Qinguangdao, Cina Senkan Qinguangdao Engineering & Tecnoogy Corporation, Qinguangdao, Cina E-mai: {dg, {9789, Received June 5, ; revised Juy 7, ; accepted August 9, Considering te effects of te beam section rotation, sear deformation of te adjacent section and transverse pressure, derived te ne equation of rectanguar section deep beams, and gives te basic soution of deep beams []. And discussed at te bending probems of deep rectanguar beams it fied at bot ends under uniform oad, based on te equations given in tis paper, appication of reciproca a, doing numerica cacuation in Matab patform, compare it te resuts of ANSYS finite eement anaysis []. Keyords: Te Bending, Rectanguar Section, Deep Beams, Te Basic Soution, Uniform Load, Fied at Bot Ends, Reciproca Metod. Te Bending of Rectanguar Deep Beams under Uniform Load.. Te Derivation of Ne Equations of te Rectanguar Deep Beams... Te Derivation of te Transverse Pressure Te eastic mecanics equations in direction is y () y Te y of straigt beam ic is in Figure is, and ten te Formua () cange into () According to te materia mecanics knoedge [], tere is Q J y () For te rectanguar cross-section as son in Figure, tere is And ten b J y () 6Q (5) b Cacuating te Formua () to get d C (6) Paying attention to Due to 6 Q b Q q q is te oad strengt of unit engt aong te direction. Put Formuae (7) and (8) into (6) to get en 6q b (a) (7) (8) d C (9) Figure. Straigt beam of rectanguar cross section. (b) Copyrigt SciRes.

2 Y.-J. CHEN ET AL. 8 q 6q b b Reduction is q q C, C b b q b At ast, getting: C () () q 8 b () After cacuation, e can get tat en,.... Te Derivation of te Moment M Introducing te concept of average corner. Making te is te average corner of cross section around ais, te is: ' d b u M () ' In te formua, u is te aia dispacement of te straigt beam. Because M () b Put Formua () into () to get Tere is M ' d b b u M u ' d () Using te Hooke s a to get u ' E Put Formua () into (5) to get d u d u q 8 E E d d b So te epression of te moment M b d M is (5) (6) d u' q 8 b E d d b (7) d q 8 EJ d d d 6 EJ q d 5Eb After tis, e estabis te reationsips beteen and... Te Derivation of According to te sear ooker a [], tere is u' ' 6Q Gb M (8) On bot sides of Formua (8) are by 6, b and doing definite integration beteen and to get: d u' 6 ' 6 d b b Q 6 d G b Cacuating te first part of Formua (9) to get u' 6 b d 6 ' ' u u d b (9) u ' d b Te Formua () is paid attention to get u' 6 d () b b Introducing te concept of average defection. is te average defection of various of points aong te eigt of straigt beam [5]. Te is 6Q Q b'd b Cacuating to get And ten ' d () '6 d () Putting Formuae () and () into (9) to get d () Q 6 b b G b Copyrigt SciRes.

3 8 Y.-J. CHEN ET AL. Given G E, after cacuating to get Q 5 Eb After tis, e estabis te reationsips beteen and Q. (), After tis, e estabis te reationsips beteen and. Putting Formua (7) into (9) to get d d q EJ q () d d Te Formua () is te equiibrium differentia equation of te deep Beams under uniform oad. After tat, te Formua () soud be anayed. Assuming tat q qq q ()... Te Estabisment of Equiibrium Differentia Equation According to te materia mecanics knoedge, tere is dq In te Formua (), q is te uniform oad and q q is te concentrated oad of te point of ic can be d (5) epressed as dm Q d (6) q P () In te Formua (), is te one-dimensiona d M Deta function of te point of and q is te concentrated coupes of te point of ic can be epressed q (7) d as Putting Formua (5) into (8) to get d dq M EJ q (8) d 5 d Putting Formua (5) into (8) to get d q (9) d M EJ After tis, e estabis te reationsips beteen and. Putting Formua (6) into (9) to get M d dq Q EJ () d d After tis, e estabis te reationsips beteen and. Putting Formua () into () to get Q d Q d 5 Eb () d J d 6 dq d 5b d 5 Eb d q M (5) Te contro equation of te deep beams under uniform oad q, concentrated oad P and concentrated coupes M is d d q EJ q P d d P d M d M d d.. Te Basic Soution of te Bending of Rectanguar Deep Beams Considering te boundary conditions of te deep rectanguar beam as son in Figure. Te deep rectanguar beams as son in Figure it bot ends simpy supported under te one-dimensiona Deta function ( ) is considered as te basic system. Te soution of te basic system is te basic soution of Figure. Rectanguar beam it different conditions. Copyrigt SciRes.

4 Y.-J. CHEN ET AL. 85 Figure. Fictitiousy basic system for deep beams of te rectanguar cross section. deep beams. Teoreticay, any straigt beam under transverse uniform oad q can be considered as te basic system, te soution of ic can be considered as te actua system. Hoever, te deep rectanguar beams it bot ends simpy supported under transverse uniform oad soud be cosen as te basic system, for te soution of ic is simpe. Te contro equation of te defection functions is d EJ (7) d In te Formua (7), is te one-dimensiona Deta function of te point of, ic is, ;, For any a, b ic satisfy te condition of a b, as te fooing properties b a b ) d ) f d f a b n ) d n n f f a For te straigt sender beam ic is not considered te searing deformation, in te Formua (7), is te transverse unit concentrated oad of te point of,. Hoever, for deep beams under te bending, is te one-dimensiona Deta function of te point of, ic dose not ave No mecanica significance is caed unit concentrated oad. And ten te rectanguar deep beam in Figure is te basic system, te soution of ic is te basic soution. For te researc, te function of and its derivative of Fourier coefficient are provided in Tabe. If i be taken as singe eavy trigonometric series, means tat sin m π Am. And ten m, i be spread out into a sine trigonometric series mπ mπ sin sin (8) m, Tabe. δ function and te fourier coefficients of its derivative it different orders. f() ( ) sine series m b m cosine series a a m sin mπ cosmπ mπ cos mπ mπ sinmπ ( ) m π sin mπ m π cosmπ ( ) m π cos mπ m π sinmπ ( ) Putting Formua (8) into (7) to get A m mπ sin (9) mπ EJ Te basic soution can be got easiy π, sin m, m mπ sin mπ EJ m () For cacuating te feibe cranksaft equation of te actua system, boundary corner epression ic is in te form of sine series of te actua system is provided as foo d m, mπ m d m, mπ d, mπ sin () EJ d, mπ sin EJ () And boundary corner epression ic is in te form of poynomia of te actua system is provided as foo d, d 6EJ d, d 6EJ () () Copyrigt SciRes.

5 86 Y.-J. CHEN ET AL.. Te Bending of Rectanguar Deep Beams it Bot Ends Simpy Supported, Fied at Bot Ends under Uniform Load.. Te Bending of Rectanguar Deep Beams it Bot Ends Simpy Supported Te actua system of rectanguar deep beams it bot ends simpy supported, fied at bot ends under uniform oad as son in Figure. Te corresponding contro equation is d d q EJ q (5) d d Te basic system as son in Figure, taking te eft boundary corner it poynomia form as foo d, d 6EJ (6) Taking te rigt boundary corner it poynomia form as foo d, d 6EJ (7) Considering te reciproca metod beteen te basic system and te actua system (as son in Figure ), to get te bucking ine equation of te actua system d q q, d d M M m,,5 m π EJ m mπ sin + M EJ q 5 π is taken a derivatives for to get (8) d q π d m,,5 m π EJ m mπ cos M EJ is taken tree derivatives for to get d q π d m,,5 π EJ m (9) (5) mπ cos dq q mπ cos (5) d m,,5 Putting Formuae (9)-(5) into () to get m,,5 m π EJ m J q M EJ b EJ q mπ π cos 5 m,,5 π mπ π cos m 6 q mπ cos 5 Eb m,,5 (5) By te eft of te beam boundary conditions to get te corner tat soud be, and ten m,,5 q π M m π EJ m EJ J q π 5b m,,5 π EJ m 6 q 5 Eb m,,5 Soving to Formua (5) to get (5) )(see beo). (5) q M O q MO o o (a) (b) Figure. Actua system of deep beams of te rectanguar cross section it to edges fied under uniformy distributed oad. M q J qj π m 5b 5 b m,,5 π m (5) Copyrigt SciRes.

6 Y.-J. CHEN ET AL. 87 Putting Formua (5) into (8) to get (55) (see beo)... Numerica Cacuation As a numerica cacuation eampe, e make tat te span of beam is m, te eigt of beam is, te idt of beam is b, dept-span ratio /.,.,,.8,.9, Eastic moduus E.6 Pa, Poisson s ration is. to cacuate te defection vaue of beams it different dept-span ratio at various points aong te y direction. In tis eampe, te beam is divided into ten copies aong te y direction (.,.,,. ), and tere is te.,.,.,.,,.8,.9... Finite Eement Simuation We use softare of ANSYS Programming to cacuate te defection vaue of beams aong te y direction. Te defection vaue of beams it different dept-span ratio at various points aong te y direction is ist in te Tabes -5. Considering te symmetry of te boundary, te uniform oading across te entire span and te symmetry of defection vaue of te beams, every tabe ony ist defection vaue of af of te beam... Anaysis of te Resuts Te defection vaue of beams it different dept-span ratio at various points aong te y direction is ist in te Tabes 6-8. Te numerica soution and finite eement cacuation vaues of te defection of beams it different deptspan ratio at various points of te / cross section are respectivey ist in te Tabes and. In tis paper, te error beteen ANSYS finite eement soution and te Tabe. Finite eement defection vaues at b of deep beam of =.. Layer / average d q q, dm M d q mπ 5 π sin m,,5 m π EJ m EJ q J π m,,5 π mπ 5b m 5 qj b (55) Copyrigt SciRes.

7 88 Y.-J. CHEN ET AL. Tabe. Finite eement defection vaues at b of deep beam of =.. Layer / average Tabe. Finite eement defection vaues at b of deep beam of =.5. Layer / Average Copyrigt SciRes.

8 Y.-J. CHEN ET AL. 89 Tabe 5. Finite eement defection vaues at b of deep beam of =.7. Layer / Average Tabe 6. Defection vaues at =.,.,.. /... tet ANSYS tet ANSYS tet ANSYS Tabe 7. Defection vaues at =.,.5,.6. /..5.6 tet ANSYS tet ANSYS tet ANSYS Copyrigt SciRes.

9 9 Y.-J. CHEN ET AL. Tabe 8. Defection vaues at =.7,.8, / tet ANSYS tet ANSYS tet ANSYS m ANSYS 5 /= (a) 6-8 m ANSYS 5 /=. /=.5 /= (c) 5-7 m ANSYS /=. /= (b) -9 m ANSYS 8 6 /=.7 /=.8 /= (d) Figure 5. Defection distribution curve at b it different dept-span ratios. numerica soution respectivey are.9%, 5.%, 5.65%, 6.%, 7.%, 7.6%, 8.6%, 8.5%, 8.7%, a of ic are in te range of aoabe error. We can kno te resut is correct, and considering te reciproca metod to sove te probem is rigt. Te defection distribution curve of beams it different dept-span ratio at various points aong te y direction and te distribution curve of te finite eement soution of te deep beam ( /.,.,.,.,,.8,.9 ) are respectivey ist in te Figure 5. Directy comparing it numerica resuts, and te to resuts can e fitting.. Concusions Considering te effects of te beam section rotation, sear deformation of te adjacent section and transverse pressure, derived te ne equation of rectanguar section deep beams, and gives te basic soution of deep beams. And e sove te eampe of te bending probems of Copyrigt SciRes.

10 Y.-J. CHEN ET AL. 9 deep rectanguar beams it bot ends simpy supported, fied at bot ends under uniform oad, based on te equations given in tis paper, appication of reciproca a, doing numerica cacuation in Matab patform, compare it te resuts of ANSYS finite eement anaysis.. References [] G. Q. Liu and K. Sun, Te Researc of Concrete Deep Beams, Tecnoogy Information, Vo., 8, pp [] X. Xu, Consistent Variation from Levinson Teory to Hig Times Warp Beam Teory, Engineering Mecanics, Vo. 5, No., 8, pp [] S. P. Timosenko and J. M. Gere, Strengt of Materias, Vannostrand Company, Ne York, 97, pp [] R. D. Mindin, Infuence of Rotary Inertia and Sear on Feura Motions of Isotropic, Eastic Pates, Journa of Appied Mecanics, Vo. 8, No., 95, pp. -8. [5] G. R. Coper, Te Sear Coefficient in Timosenko s Beam Teory, Journa of Appied Mecanics, Vo., No., 966, pp. 5-. doi:.5/.656 Copyrigt SciRes.

Instructional Objectives:

Instructional Objectives: Instructiona Objectives: At te end of tis esson, te students soud be abe to understand: Ways in wic eccentric oads appear in a weded joint. Genera procedure of designing a weded joint for eccentric oading.