Stress distribution in elastic isotropic semi-space with concentrated vertical force

Transcription

1 Bulgrin Chemicl Communictions Volume Specil Issue pp. 4 9 Stress distribution in elstic isotropic semi-spce with concentrted verticl force L. B. Petrov Deprtment of Mechnics Todor Kbleshkov Universit of Trnsport Geo Milev str. 74 Sofi Bulgri Received: Jul 4 7 Revised: November 7 The distribution of stresses in elstic isotropic semi-spce in horizontl nd verticl direction under the effect of concentrted verticl force is investigted. A trnsition to line influences for stresses nd their determintion to rbitrr lod is performed. Anlsis nd comprison of the results obtined is mde. Kewords: Elstic isotropic semi-spce Stress distribution Line influences INTRODUCTION Elstic hlfplne is disk limited onl on one strightliner end nd spred to infinit on one side of this end. Such is the stress nd deformtion stte of disk loded on its contour the dimensions of which re too big in comprison with the length of the loding prt. The solution of infinite elstic hlfplne t uniforml distributed lod problem of Boussines with concentrted lod problem of Flmmnt nd t rbitrr distributed lod is discussed in []. Epressions re derived for the stresses t n rbitrr point of the hlfplne. Through limits trnsition of these problems the epressions for determintion of stresses under the effect of concentrted forces re obtined. A similr problem is encountered in the investigtion of bems of elstic foundtion long strips fundments etc. The purpose of the present work is to determine the distribution of the stresses in the elstic hlfplne in horizontl nd verticl direction under the effect of concentrted force to construct the influences for the stresses nd through the obtined influences to determine the stresses t rbitrr lod. STRESSES IN THE HALFPLANE Norml nd tngentil stresses re known from strength of mterils. In the cse of concentrted F const M Fig.. The epressions of the stresses in n rbitrr point of the hlfplne verticl force cting on the top edge the epressions of the stresses in n rbitrr point of the hlfplne re s presented in Figure: where re the coordintes of point t which the stresses should be determined is the intensit of lod euivlent to the force F distributed uniforml on the prt with length nd smmetricll locted bout is The introduced coordinte sstem O is in the middle of the distributed lod Fig.. As generl significtion of stresses S is used. The written epressions re pplied for investigtion of the elstic hlfplne loded with concentrted force with the chrcteristics m kn F. The intensit of the lod in the cses is m to coordinte with single vlue of the force. The nlsis of the distribution of stresses is mde with rbitrr nd different in this cse nd in prticulr multiple of n ccepted discretiztion step in the e s directions of the hlfplne. Clcultions re mde with compound progrms of PC. δ F F F. * To whom ll correspondence should be sent. E-mil: lbphr@bv.bg 4 Bulgrin Acdem of Sciences Union of Chemists in Bulgri

2 L. B. Petrov: Stress distribution in elstic isotropic semi-spce with concentrted verticl force Stresses distribution in verticl direction The solution of the stresses is mde t Tble. The solution of the stresses is mde t b step / b step. The results of the solutions re shown in Tble e-7-666e e e--666e-4-9e e- -46e e e e- -97e- -94e e e- 76e- -9e e-6-749e e- -947e-4-79e e-6-477e- -769e- -79e- -e-4-999e e-7-4e- -796e- -44e- -6e-4-69e e-7-496e- -444e e- -49e-4-4e e-7-97e- -4e e- -74e-4-644e e-7 -e- -49e- -9e- -4e-4-47e e-7-4e- -44e- -64e e- -e e-7 -e- -97e e- -767e- -79e e-7-74e-4-674e- 6 -e- -774e- -e e e- -946e- -6e e-4-479e- 7-6e- -49e- -7e e- -9e- -744e- -99e-7-644e-4-47e- 7-9e- -79e- -46e-4-4e- -4e- -94e- -e-7-69e-4-96e- -44e- -44e- -4e-4-797e- -796e- -774e e- -49e-4-964e-4-66e- -77e- -e e- -74e- -766e- -4e- -474e-4-74e-4 9-7e- -69e- -e- -776e- -7e- -69e- -79e- -96e e e- -46e- -764e- -67e e- -6e- -74e- -7e-4-69e-4-666e- -6e- -64e- -666e- -64e- -64e- -666e- -67e-4-64e e- -66e- -4e- -66e- -67e- -94e e- -74e-4-67e-4-47e- -9e- -47e- -77e- -766e- -69e- -6e- -69e-4-74e-4-49e- -4e- -4e- -e- -e- -4e- -4e- -97e-4-474e-4-64e- -97e-6-6e- -e- -6e- -e- -44e- -e-4-46e-4-9e- -e-6 -e- -9e- -767e- -4e e- -7e-4-47e-4-977e- -79e-6-67e- -497e- -46e- -496e- -767e- -779e-4-7e-4-7e- -64e-6-9e- -477e- -47e e- -49e- -74e-4-4e-4 4 -e- -74e-6-966e- -447e- -47e- -4e- -4e- -699e-4 -e-4 4 -e- -e-6-6e- -49e- -4e- -449e- -79e- -4e-4-999e-4-6e- -47e-6-696e- -444e- -447e- -467e- -94e- -46e-4-44e Fig.. Grphicl distribution of the stresses in the elstic hlfplne in verticl direction

3 L. B. Petrov: Stress distribution in elstic isotropic semi-spce with concentrted verticl force Tble. The ordintes nd the kind of the line influences e-6-74e-4-674e- -49e-4-6e-6-46e- 947e- 767e- 79e e-6 -e- -97e- -697e-4-44e-6-69e- 64e- 9499e- 4e e-6-4e- -44e- -6e- -4e- -947e- 9e- 4e-4 477e e-6 -e- -49e- -e- -96e- -49e-4 696e- 74e-4 644e e-6-97e- -4e- -46e- -6e- -e e- 49e-4 e e-6-49e- -444e- -444e- -9e- -66e-4 44e-9 6e-4 69e e-6-4e- -796e- -79e- -9e- -666e-4 79e-9 e-4 999e- - -6e- -477e- -769e- -697e- -e-4 -e- 4744e-9 947e-4 79e- - -9e- -749e e- -447e-4-46e- 7977e-9 76e- 9e e e- -7e- -67e- 6e- 97e- 94e e e- -9e e e e e e e e e e- -9e e e e- -7e- -67e- -6e- -97e- -94e- -9e- -749e e- -447e-4-46e e-9-76e- -9e- -6e- -477e- -769e- -697e- -e-4 -e e-9-947e-4-79e- -776e-6-4e- -796e- -79e- -9e- -666e-4-79e-9 -e-4-999e- -969e-6-49e- -444e- -444e- -9e- -66e-4-44e-9-6e-4-69e e-6-97e- -4e- -46e- -6e- -e e- -49e-4 -e-4 4-4e-6 -e- -49e- -e- -96e- -49e-4-696e- -74e-4-644e-4-46e-6-4e- -44e- -6e- -4e- -947e- -9e- -4e-4-477e-4-4e-6 -e- -97e- -697e-4-44e-6-69e- -64e e- -4e e-6-74e-4-674e- -49e-4-6e-6-46e- -947e- -767e- -79e-4 4E- E E- -E E- -E E- -E -4E s s/ s E -E t t/ t -E -E- -6E -E- 6-7E Fig.. Grphicl distribution of the stresses in the is elstic hlfplne in horizontl direction From the results obtined it is seen tht the norml stresses in verticl direction hve big vlues in the intervl. The norml stresses in verticl direction hve ver big vlues in the intervl nd then grdull decrese. Тhe tngentil stresses hve big vlues in the intervl nd then grdull decrese. The biggest vlues of the norml nd the tngentil stresses in verticl direction re obtined t. The norml stresses in horizontl direction hve big vlues in the intervl 6 6. The norml stresses in the horizontl directions hve ver big vlues in the intervl nd then grdull decrese. Тhe tngentil stresses hve reltive big vlues in the intervl -4E- 4 4 nd then grdull decrese. The biggest vlues of the norml nd the tngentil stresses in horizontl direction re obtined t и. In verticl direction the norml stresses re bigger in comprison with the norml stresses. The results obtined form stresses line s stte in n elstic hlfplne. From the digrms of stress s distribution in verticl nd horizontl direction in selection rbitrr section of elstic hlfplne fter integrtion nlticl of numericl cn to mke verifiction of euilibrium of clculted stresses. STRESSES LINE INFLUENCES The epressions of the functions s stresses in horizontl directions cn to interpreter to construct the lines influences of the norml nd tngentil stresses in elstic hlfplne Fig. 4.

4 L. B. Petrov: Stress distribution in elstic isotropic semi-spce with concentrted verticl force const F 4 δ d d d d d d ' ' Fig. 4. The epressions of the functions s stresses in horizontl directions Here d is the length of the discretiztion step in horizontl direction i 4... successive situtions of the force F in the top edge of the elstic hlfplne. The ordintes nd the kind of the line influences for emple etc. re shown in Tble nd Fig.. The ordintes of stresses influences re reports with multipl of discrtiztion step suitble to the sitution of the forces F nd of the stress t the point to from which the line influence refers. For ech prticulr cse epressions re written for line influences of the stresses t the point of elstic hlfplne presented through stresses line sttes. For emple:. Grphicl line influences of the stresses t the selected points of the elstic hlfplne re shown in Fig Fig.. Influences of the stresses t the selected points of the elstic hlfplne From the line influences the stresses t unspecified point of elstic hlfplne t n rbitrr lod cn be determined. At the lod distributed b rbitrr lw it is known tht the rbitrr stress is determines with the epres- sion: S S d where re the limits strt end of the distributed lod is previousl determined level in verticl direction t the point t which the stresses re determined through line influences. At uniforml distributed lod const with length а smmetricll situted bout is tg d 4 tg d 99 7

5 L. B. Petrov: Stress distribution in elstic isotropic semi-spce with concentrted verticl force d 9 d d 7. At lod with length distributed b tringle lw with zero of tringle t the Fig. 6 the stresses determined through line influences re of the tpe: δ Fig. 6. Tpe of the stresses determined through line influences d ln ln d 7 d 9 rctg d 6 rctg d 4.

6 L. B. Petrov: Stress distribution in elstic isotropic semi-spce with concentrted verticl force The results obtined through line influences coincide ectl with the vlues of the stresses in the elstic isotropic hlfspce obtined from the solution under the effect of indicted distributed lods. CONCLUSIONS With more concentrted forces the digrms of stresses re superimposed. In similr w the stresses t different points of n elstic isotropic semispce with other cting lods cn be determined. REFERENCES. Ch. Vrbnov Theor of elsticit Technik Sofi 99.. T. Krustev T. Krmnski Guidnce for solution of problem of theor of elsticit stbilit nd dnmics of elstic sstems Technik Sofi