Stresses near a plate vertex due to a shear force on one of the edges

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1 Stree near a plate vertex due to a hear force on one of the edge P.C.J. Hoogenboom Delft Univerity of Technology, Faculty of Civil Engineering and Geocience, Delft, the Netherland A cloed form olution i preented for the tree near a rectangular vertex of linear elatic plate loaded by an evenly ditributed hear force on one of the edge. The force method and the Airy tre function have been applied in the derivation. Remarkable i that the tre in the vertex depend on the direction in which the vertex i approached. The conequence for tructural analyi are dicued. Key word: Elaticity theory, Airy tre function, biharmonic equation, ingular point, plain tre, plain train, polar coordinate 1 Introduction Many elementary ituation in tructural engineering have cloed form olution for the tree auming linear elatic material behaviour [Timohenko 1959, Timohenko 1971, Barber 1993]. For example the tree in a half pace due to a concentrated urface force. Thee olution are ued in code regulation and preliminary deign of many type of tructure. They have been derived before the introduction of finite element computer program and mathematical computer program. The hitorical derivation took many hour of accurate work [Melehko 003], however, to date they can be performed by modern mathematical oftware in le than a econd. Therefore, computer make it poible to explore many more problem for cloed form olution. One of thee problem i the ubject of thi paper. Conider a rectangular vertex of an infinitely large plate (Fig. 1). One of the plate edge i loaded by an evenly ditributed hear force. The plate thickne i t. The coordinate axe are directed along the plate edge. HERON Vol. 51 (006) No. 4 5

2 vertex y plate edge θ x r plate hear force Figure 1: Shear force loading on one edge of a plate near a rectangular vertex A hown in the following ection, the tre ditribution in the plate how a compreive tre of π σ xx = (1) t everywhere along the loaded edge and a tenile tre of π σ yy = () t everywhere along the not loaded edge. Thi i alo the larget principal tre σ 1. The mallet principal tre occur everywhere along the loaded edge. σ 1 = ( 16 ).06 t 4 π+ +π = t (3) The larget comparion tre according to the failure criterion of Von Mie i σ 1 VM = 3.34 t + 4 π = t, (4) which alo occur everywhere along the loaded edge. 6

3 The Force Method In elaticity theory the force method i uccefully applied to analye plate loaded in plain tre or plane train [Den Hartog 1987]. In thi method we look for an Airy tre function φ that fulfil the biharmonic equation φ φ φ + + = 0. (5) 4 4 x x y y When found, the plate force are derived by φ φ φ nxx =, n, yy = n xy = y x xy (6) where nxx = tσ xx, nyy = tσ yy, nxy = tσxy and t i the plate thickne. For the ituation analyed in thi paper the boundary condition are x = 0 φ φ = 0, = 0 y xy y = 0 φ φ = 0, = x xy (7) The following tre function fulfil the biharmonic equation (5) and the boundary condition (7). 1 1 ( )arctan x φ= xy x x y + π + y (8) It ha been obtained by trail and error, tarting with an aumption for the hear tre ditribution n xy and deriving the normal tree n xx and n yy boundary condition. from equilibrium and the 7

4 The tre function can alo be written in the cylinder coordinate r and θ uing x = rcoθand y = rin θ. 1 1 φ= r in θcoθ πin θ+θ (9) 3 Stre Reult Subtituting (8) in (6) the force field i found. x xy nxx = arctan y x + y 1 x xy nyy = π arctan y x + y x nxy = x y + (10) (11) (1) Thee are plotted in Figure, 3 and 4. The plate force are independent of the ditance r to the vertex. nxx nyy nxy 1 ( co in ) = π θ θ θ (13) = ( θ in θcoθ ) (14) = co θ (15) The principal tree σ 1 and σ are calculated by 1 ( ( n ) 1 xx nyy ( nxx nyy ) nxy ) 1 ( ( n ) 1 xx nyy ( nxx nyy ) nxy ) 1 σ 1 = + + +, t 4 1 σ = + +, (16) t 4 which can be evaluated to 8

5 ( co in 16 co ) ( co in 16 co ) σ 1 = θ π+ π π θ θ+ θ, t 4 4 σ = θ π π π θ θ+ θ. (17) t 4 4 Figure : Normal force nxx Figure 3: Normal force n yy Figure 4: Shear force n xy The Von Mie tre i calculated by 1 σ VM = nxx nxxnyy + nyy + 3nxy, (18) t 9

6 which can be evaluated to 1 σ VM = π πθ+ 4θ 6πcoθin θ+ 1 co θ. (19) t All tree are plotted in Figure 5. σ VM σt 1 σ yy + σ xy σ 1 σ yy 0 σ xx σ xy -1 σ θ Figure 5: Stre quantitie a a function of θ 4 Concluion and Remark A cloed form olution ha been found for the tree cloe to a rectangular vertex of linear elatic plate loaded by an evenly ditributed hear force on one of the edge. Thi cloed form olution wa earlier derived by E. Reiner and publihed in a two-page technical note (Reiner 1944). 30

7 A pointed out (Reiner 1944) the olution how that the tre tenor can be aymmetrical alo in the abence of body moment. (Including body moment would nowaday be referred to a a Coerat continuum.) After all, in the vertex the hear tree on perpendicular face are not the ame. However, thi ha no practical conequence. The very concept of tre break down when we trongly zoom in on the material. At a very large magnification we would ee individual molecule and the effect of force between them. It would be an unrealitic idealiation to require a valid tre tenor for the ingle molecule that reide in the plate vertex. In other word, it i not a problem if a plate olution doe not fulfil all governing equation in a finite number of point. Neither do we need to extent the concept of tre to include aymmetrical tre tenor. In the vertex the tre i not unique. It depend on the direction in which the vertex i approached. Thi ingularity cannot be computed eaily by the finite element method becaue jump are not included in the hape function of regular element. An extremely fine element meh can be ued to approximate the ingularity. The problem of thi paper i particularly uitable for bench marking finite element oftware becaue depite the ingularity none of the quantitie (diplacement, train, tree) go to infinity. Due to modern mathematical oftware the claical force method i very uitable to find cloed form olution of many a yet unolved problem in elaticity theory. For example, the problem of Figure 6 ha a coniderable a hear lag. It can be oberved in the bottom plate of a box-girder bridge or a core wall of a high-rie building. A far a the author know, the cloed form olution for that problem i till waiting to be found. hear force tenile tre hear force Figure 6: Shear force loading on the edge of a emi-infinite trip 31

8 Reference Barber, J.R. (1993) Elaticity, Kluwer Academic Publiher, Boton. Den Hartog, J.P. (1987) Advanced Strength of Material, Dover Publication Inc., New York. Melehko, V.V. (003) Selected topic in the hitory of the two-dimenional biharmonic problem, Appl. Mech. Rev. Vol. 56, No. 1. Reiner, E (1944) Note on the Theorem of the Symmetry of the Stre Tenor, Journal of Mathematic and Phyic, MIT Pre Cambridge Ma. (Current title: Studie in Applied Mathematic) Vol. 3, pp Timohenko, S.P., W. Woinowky-Krieger (1959) Theory of Plate and Shell, McGraw- Hill, New York. Timohenko, S.P., J.N. Goodier (1971) Theory of Elaticity, McGraw-Hill, New York. 3