INVESTIGATION OF LONGITUDINAL ELASTIC WAVE PROPAGATION THROUGH INTERSECTING WELDED BARS

Transcription

1 8 Journal of Marine Science and Technology, Vol. 7, No. 1, (1999) INVESTIGATION OF LONGITUDINAL ELASTIC WAVE PROPAGATION THROUGH INTERSECTING WELDED BARS Ming-Te Liang* and Chiou-Jenn Chen** Keywords: Incident wave, Reflection, Transmission. ABSTRACT The main urose of this aer is to exress a general formulation for the theoretical analysis of longitudinal elastic wave roagation through a general and in articular the T and junctions. The bar joint is modeled as a rigid block. The bars are assumed to be all of the same thickness but may differ in width and may be of different materials. Elementary theory is alied to investigate the roagation of longitudinal wave. The resent study indicates that for a T junction there would be no transmission of the longitudinal elastic wave. For a junction there would be no transmission of the longitudinal elastic wave into the erendicular branches but the transmission into the horizontal bar was haened. INTRODUCTION Welding is used extensively in steel structures, sace shuttles, aircraft, shis, cars, locomotives and offshore latforms. When these structures are subjected to a sudden seismic or aerodynamic disturbance, the energy of longitudinal elastic wave should be absorbed by the structural joints if structural damage is to be revented. The fate of longitudinal elastic waves as they roagate through welded intersection joint is thus a very imortant consideration in structural design for dynamic loading conditions. Mandel et al. [1] have studied the roblem of stress-wave roagation through a rigid right-angle joint by using the method of characteristics, and also exerimentally verified their theoretical result. They found that tension and shear in the horizontal bar becomes Paer Received Jan. 5, Revised March 17, Acceted May, Author for Corresondence: Ming-Te Liang. *Associate Professor, Deartment of Harbor and River Engineering Taiwan Ocean University, Keelung, Taiwan, 0, R.O.C. **Teaching Assistant, Deartment of Harbor and River Engineering Taiwan Ocean University, Keelung, Taiwan, 0, R.O.C. shear and tension in the vertical bar. Desmond [] found that when a longitudinal stress wave iminges on a junction of three elastic bars where two bars are collinear and the third is noncollinear to the others, a longitudinal stress wave and a flexural (shear) wave are reflected back along the first bar, and stress waves of both tyes are transmitted into the second and third bars. Simha and Fourney [] have resented a general formulation for the analysis of stress wave roagation through a junction of rectangular bars, and the dynamic hotoelasticity measurement was used for their exerimental investigation. They concluded that a longitudinal stress wave in the horizontal bar is not transmitted into the erendicular branch. It is worth ointing out that there exist a lot of mistakes in the theoretical analysis of Simha and Fourney []. The theoretical derivation using the method of Lalace's transform is incorrect at there, such as Eqs. (8), (9), (10), (11), (1), (1) and Aendix A in the aer of Simha and Fourney []. The inverse Lalace transform is also not used in Simha and Fourney []. The exerimental results do not clarify the theoretical result. Wu and Lundberg [] dealt with harmonic elastic waves in a uniform bar with a straight semi-infinite inut section, a bend with constant radius of curvature and a straight semi-infinite outut section. The effects of rotary inertia and shear deformation were both neglected. They found that for a shar 99.9 bend, an incident extensional wave does not roduce a transmitted longitudinal wave at any frequency. For a shar right-angle bend, the energy flux of the transmitted extensional wave is u to % of the incident extensional wave, deending on the frequency. For a straight bar, the extensional wave is totally transmitted. In order to correct the formulas derived by Simha and Fourney [], this aer again resents a general formulation for the theoretical analysis of longitudinal elastic wave roagation around a general junction. First, the theoretical derivation is introduced. Then

2 M.T. Liang & C.J. Chen: Investigation of Longitudinal Elastic Wave Proagation Through Intersecting Welded Bars 9 both T and geometrical cases are calculated for checking the theoretical analysis. Finally, the conclusions are made. THEORETICAL DERIVATION In this section, the structures of the derivation is divided into three arts: I. Transverse waves, II. Longitudinal waves, and III. The secial case of welded bars with T-geometry. As for arts I and II, the method of Lalace s transform is alied. As to art III, it is the alication of arts I and II. I. Transverse waves Consider a junction of three elastic lates as shown in Fig. 1. Assume that the steel late is elastic. For transverse waves, the following equations (Atkins and Hunter [5]) should be satisfied: y j x + ρ j A j y j j E j I j t = 0, (j=1,,), y j(x j,0)=0, t y j(x j,0)=0, (1) where y j, E j, I j, ρ j and A j reresent the dislacement, Young s modulus, moment of inertia, mass density and cross-sectional area, resectively, and x and t stand for sace and time coordinates, resectively. Let C j = E j ρ, A j = L j t j, and I j = t j L j,, where t j j is the thickness of the steel late, L j is width, and is wave seed. Then Eq. (1) can be written as y j x + y j j C j L j t = 0, y j(x j, 0) = 0, t y j(x j, 0) = 0 () The definition of the Lalace transform is L[y j (x j, t)] = y(x j, )= 0 e t y j (x j, t)dt () After alying the Lalace transform, Eq. () becomes d y j dx + j C j L y j = 0 () j Substituting β j = C j L and i = 1, the solution of Eq. () is j y j (x j,)=a j ()e (1+i)β j X j + B j ()e (1 i)β j X j + ()e (1 + i)β j X j + D j ()e (1 i)β j X j (5) For the boundedness of the solution, let () = D j () = 0. Thus, Eq. (5) becomes and y j (x j,)=a j ()e (1+i)β j X j + B j ()e (1 i)β j X j (6) The moments and shear forces are given by M j = E j I j d y j dx j Q j = E j I j d y j dx j x j = 0 x j = 0 Using Eqs.(7) and (8) yields A j ()= 1 E j I j β j B j ()= 1 E j I j β j Q j (7) (8) β j +(1 i) M j (9) Q j β j +(1+i) M j (10) There are three unknowns, X, Y and θ as shown in Fig., where X is the horizontal dislacement, Y is the vertical dislacement and θ is the rotational angle, which should satisfy the following equations: dy j = θ (11) dx j x j =0 y j x j =0 = Xsin θ j Ycos θ j θd j (1) where θ j is the bar angle, and d j is the moment arm of shear force Q j (see Fig. ). Substituting Eqs. (9) and (10) into Eqs. (1) and (11) yield Fig. 1. General sketch of a trifurcated welded bar. 1 E j I j β j Q j β j + M j = Xsin θ j Ycos θ j θd j (1)

3 10 Journal of Marine Science and Technology, Vol. 7, No. 1 (1999) Fig.. Free-body diagram and systems of coordinates of a trifurcated welding bar. For the boundedness of the solution, let C 1 (x j, ) = 0. Thus, U(x j,)=c (x j,)e x j (1) The Lalace transform of the joint boundary conditions is U j x j =0 = Xcos θ j + Ysin θ j () Substituting Eq. () into Eq. (1), we obtain U(x j,)=( Xcos θ j + Ysin θ j )e x j () The substitution of Eq. (16) into Eq. () yields φ 1 + φ R = X () 1 E j I j β j II. Longitudinal waves Q j β j +M j = θ (1) Similarly, the longitudinal waves should satisfy the following equations (Atkins and Hunter 1975, Simha and Fourney []): U j x 1 U j j t = 0 (15) with the initial conditions U j (x j, 0) = 0 and U j(x j,0) = 0, t where U j is the dislacement of longitudinal waves. As illustrated in Fig. 1, φ I and φ R indicate, resectively, the incident and reflected longitudinal waves in bar 1; φ and φ denote the transmitted longitudinal waves in bar and bar, resectively. Their relationshis to each other can be exressed as U 1 (x 1,t)=φ I t + x 1 C 1 + φ R t x 1 C 1 (16) U (x,t)=φ t x C (17) U (x,t)=φ t x (18) C Alying the Lalace transform to Eq. (15) gives d U j dx j U j = 0 (19) The solution of Eq. (19) is U(x j,)=c 1 (x j,)e x j + C (x j,)e x j (0) Inserting Eqs. (17) and (18) into Eq. () leads to φ j = Xcos θ j + Ysin θ j,(j =,) (5) The relationshi of stress and internal force can be exressed as U j T j = E j A j,(j =1,,) (6) x j where A j = L j t j, A j is area, and t j is thickness. Differentiating Eq. () with resect to x j and substituting the resultant into Eq. (6) yields T j = ρ j L j t j φ j,(j = 1,,) (7) Since the horizontal force, the vertical force and the moment (see Fig. ) are in equilibrium, we get Σ ( T jcos θ j Q j sin θ j )+m X = 0 (8) j = 1,, Σ ( T jsin θ j Q j cos θ j ) m Y = 0 (9) j = 1,, Σ ( M j + e j T j + d j Q j ) I θ = 0 (0) j = 1,, where e j is the moment arm of axial force T j, m = ρl is the mass of er unit length of the joint and I = 16ρt5 is the moment of inertia of the welded joint. Finally, the 15 unknowns can be solved from 15 equations obtained from Eqs. (1), (1), (), (5), (7), (8), (9) and (0). III. The secial case Now, consider the case of welded bars with T- geometry (see Fig. ). Assume each bar is constituted

4 M.T. Liang & C.J. Chen: Investigation of Longitudinal Elastic Wave Proagation Through Intersecting Welded Bars 11 Fig.. T-tye steel late dimensions. The late is 0.05m thick. Other dimensions are as follows: a = 1.m, b = d = 0.05m, c = e = 1.m. M 1 + M + M L(Q + Q Q 1 )+ 16ρL5 θ = 0 (6) Q1 + M 1 = Y Lθ (7) Fig.. Free-body diagram of the T-tye steel late. Q 1,, reresent the shear forces, T 1,, the axial forces and M 1,, the moment. θ 1 = 0, θ = 90, θ = 70, (see Fig. 1 for an exlanation of θ). Q + M = X Lθ (8) of the same material and has the same material roerties as listed in Table 1. Let coefficient α = (see Atkins and Hunter [5]), substituting the aroriate values (see Fig. ) into Eqs.(1) and (1) and Eqs. ()~(0) gives: Q + M = X Lθ (9) Q M1 θ = 0 (0) φ R + T 1 ρcl = φ I (1) Q + 1 M θ = 0 (1) φ + T ρcl =0 () Q + 1 M θ = 0 () φ + T ρcl =0 () T 1 Q + Q +8ρL X = 0 () Q 1 T + T +8ρL Y = 0 (5) φ Y = 0 () φ + Y = 0 () and φ R X = φ I (5)

5 1 Journal of Marine Science and Technology, Vol. 7, No. 1 (1999) Table 1. Physical characteristics of the steel late material unit weight density Young's modulus shear modulus Poisson's ratio roerties γ ρ E G v unit (kn/m ) (kg/m ) (Ga) (Ga) - values Fig. 5. -tye steel late dimensions. The late is 0.05m thick. Other dimensions are as follows: A = 1.m, B = 0.05m, C = 1.m, D = 0.05m, E = 1.m, F = 1.m. The incident wave φ I is known. Thus from Eqs. (1)~(5), the values of M 1,M,M,Q 1,Q,Q,T 1,T,T,φ R, φ,φ,x,y and θ can be determined and the following equations can thereby be derived (see Aendix) φ = φ (6) φ = 0 (7) Since φ = φ = 0, the inverse Lalace transform need not be used. We know that for a T junction there would be no transmission of the longitudinal stress wave in the vertical bar, because boundary conditions are satisfied the condition of comatibility and both bar and bar are symmetrical. Thus, the couling action is occurred and the infinitesimal values of transmitted longitudinal wave in the vertical bar does not exist. In this aer, we adot the bar with finite length as examle. The first single received from sensor was taken for analysis. The longitudinal wave is only to be discussed. As to the action time, we take the time when the first incident wave is not yet arrived at the end oint of bar. Thus, the interference due to the other reflection waves and diffraction waves can be avoided for analysis. Fig. 6. Free-body diagram of the -tye steel late. Q 1,,, reresent the shear forces, T 1,,, the axial forces and M 1,,, the moment θ 1 = 0, θ = 90, θ = 70, θ = 180, (see Fig. 1 for an exlanation of θ). EXTENDED APPLICATION OF THE MODEL In engineering ractice, multile bars are usually welded together. In this section, the T geometry case develoed above is extended to the geometry case (see Fig. 5 and Fig. 6). Extending Eqs. (1) and (1)and Eqs. ()~(0), the following equations can be added for j = : φ + 1 E I β T ρ C L t = 0 (8) Q β + M + Xsin θ + Ycos θ + θd = 0 (9)

6 M.T. Liang & C.J. Chen: Investigation of Longitudinal Elastic Wave Proagation Through Intersecting Welded Bars 1 Q + M = X Lθ (60) Q + M = X Lθ (61) Q + M = Y Lθ (6) Q M1 θ = 0 (6) Q + 1 M θ = 0 (6) Q + 1 M θ = 0 (65) Fig. 7. Values of transmitted longitudinal elastic waves in the horizontal welded bar for the -tye steel late. Q + 1 M θ = 0 (66) 1 E I β Q β +M θ = 0 (50) φ + Xcos θ Ysin θ = 0 (51) Substitution of the aroriate values (see Fig. 5 and Fig. 6) into Eqs. (1) and (1), Eqs. ()~(0) and Eqs. (8)~(51) yields φ R + φ + T 1 ρcl = φ I (5) T ρcl = 0 (5) φ Y = 0 (67) φ + Y = 0 (68) φ X = 0 (69) φ R X = φ I (70) Since φ I is given, from Eqs. (5)~(70), the unknowns M 1,M,M,M,Q 1,Q,Q,Q,T 1,T,T,T,φ R,φ,φ, φ, X,Y and θ can be calculated. Here we are concerned with the values of φ,φ and φ for the transmitted longitudinal elastic waves. As before, the following equations can be derived (see Aendix): φ + φ + T ρcl = 0 (5) T ρcl = 0 (55) φ = φ (71) φ = 0 (7) φ = W()φ I (7) T 1 Q + Q T +8ρL X = 0 (56) Q 1 T + T Q +8ρL Y = 0 (57) M 1 + M +M + M L(Q + Q Q 1 Q ) + 16ρL5 θ = 0 (58) Q1 + M 1 = Y Lθ (59) where W()= + L C + (7) For convenience, the original ositive x 1 direction of φ I is reversed. Thus, Eq. (7) becomes φ T = W()φ I (75)

7 1 Journal of Marine Science and Technology, Vol. 7, No. 1 (1999) where Alying the convolution integral to Eq. (75) yields φ T = 0 t W(t t') dφ I(t') dt' (76) dt' W(t) L 1 [W()] (77) dφ δ(t) (78) dt τ = α (79) W(t)=X( t τ ) (80) and X( t τ )=X(x) (81) Using the Lalace transform on Eq. (81) gives X(s)= = 0 X (x)e sx dx αs 1 s 1 s + s + α = s 1 Q(s 1 ) (8) Alying the inverse Lalace transform to Eq. (8) gives X(x)= 1 u (πx) 1 e x Q(u)du 0 = 0.866e x (1 erf (9.987 x)) e 0.878x (1 erf (0.67 x)) Re [( i)W(0.881 x xi)] e 0.756x (6.1cos 0.87x sin 0.87x) (8) Finally, the substitution of Eq. (8) into Eq. (76) yields t φ T = X[(t t') / τ] dφ I(t') dt' (8) 0 dt' Following Achenback [6], we have a b h(x)δ (n) (x x')dx'= hn (x), x (a,b) 0, x (a,b) The x term in Eq. (8) is simly relaced by t: (85) X(t) = 0.866e t (1 erf (9.987 t)) e 0.878x (1 erf (0.67 t)) Re [( i)W(0.881 t ti)] e 0.756x (6.1cos 0.87t sin 0.87t) (86) Table. Values of transmitted longitudinal elastic waves in -tye steel late Action time The transformed The values of transmitted longitudinal elastic waves t(sec) action time t/τ(sec) (φ ) (φ ) (φ )

8 M.T. Liang & C.J. Chen: Investigation of Longitudinal Elastic Wave Proagation Through Intersecting Welded Bars 15 Substituting the t values into Eq. (86) and using Table 1 and the aroriate formulas in Abramowitz and Stegum [7] yields Table. From Table, Fig. 7 can be lotted. It is obvious that there is no transmission into the vertical welding bars. However, the maximum values of the longitudinal elastic wave transmitted into the horizontal bar are about 99% (See Table and Fig. 7). This result is very close to the result obtained by Wu and Lundberg[]. CONUSIONS The incorrect formulas, such as Eqs.(8), (9), (10), (11), (1) and (1), derived by Simha and Fourney [] now have been corrected as Eqs. (9), (10), (11), (1), (1) and (1) in this aer,reectively. When a horizontal force is alied to the horizontal bar in a T geometry configuration (Figs. & ), the longitudinal elastic wave is not transmitted along the vertical bars. This henomenon is due to the boundary conditions which are satisfying the condition of comatibility and both bar and bar symmetrical. Thus, the couling action is haened and the infinitesimal values of transmitted longitudinal wave in vertical bar is not existent. In the case of welded bars with geometry (Figs. 5 & 6), again there is no transmission into the vertical welding bars. However, the maximum values of the longitudinal elastic wave transmitted into the horizontal bar are about 99%. This result is very close to the result obtained by Wu and Lundberg[]. In this aer, we aly the theory to illustrating the bar with finite length. The henomena of longitudinal wave roagation is merely to be discussed. Since the roagation velocity of longitudinal waves is larger than the transverse waves, the longitudinal waves certainly arrive at the end oint of bar before the transverse waves. Furthermore, the longitudinal waves result in both the reflection and refraction waves in the bar. Therefore, if we want to consider the situation of transverse wave roagation in the bar with finite length, the more detail investigation is needed. The values for the transmitted longitudinal waves φ,φ and φ are derived out in the resent aer. Using the same method it would also be ossible to derive values for other unknowns including φ R (the reflected wave) and φ j where j refers to the j th welded bar at a junction at angle θ j to the incident wave. φ = Y It follows from Eq. () that φ = Y From Eqs. (A-1) and (A-), we acquire φ = φ (A-1) (A-) (A-) It is worth noticing that Eq. (A-) is Eq. (6). The substitution of Eq. (A-) into Eq. () yields T =ρcl φ Similarly, we rocure T = ρcl φ T 1 =ρcl (φ I φ R ) (A-) (A-5) (A-6) From Eqs. (0), (1), and (), we obtain, resectively, Q 1 = αp Q = αp Q = αp θ θ θ 1 M1 (A-7) 1 M (A-8) 1 M (A-9) Using Eqs. (A-1)~(A-) and Eqs. (A-8) and (A-9) and substituting the values of Q 1,Q,Q,Y and X, into Eqs. (7), (8) and (9), we get M 1 = M = αp αp φ + L + L + 1 θ (A-10) 1 θ φ I φ R (A-11) APPENDIX The derivation rocess of Eqs. (6), (7), (71), (7), (7) and (7) is described as follows: From Eq. (), we obtain M = αp L + 1 θ + φ I + φ R (A-1)

9 16 Journal of Marine Science and Technology, Vol. 7, No. 1 (1999) The substitution of Eqs. (A-10), (A-11) and (A-1) into Eq. (6) leads to θ = EL 6L +7L L 1 + Putting Eq. (A-1) into Eq. (5) yields φ R = 6ρL P 6ρL P 1 ρcpl +ρcpl +ρl 5 P (A-1) +8EL +8EL φ φ I (A-1) The substitution of Eq. (A-1) into Eq. (5) yields φ = 0 (A-15) It is worthy to oint out that Eq. (A-15) is Eq. (7). In the same manner, we can rove Eqs. (71)~(7)as follow where φ = φ φ = 0 φ = W()φ I W()= + L C + (A-16) (A-17) (A-18) (A-19) It is deserving to oint out that Eqs. (A-16)~ (A-19) is Eqs. (71)~(7), resectively. REFERENCES 1. Mandel, J.A., Mathur, R.K. and Chang, Y.C., Stress Wave at Rigid Right Angle Joint, Journal of the Engineering Mechanics Division, ASCE, 97(), (1971).. Desmond, T.P., Theoretical and Exerimental Investi- gation of Stress Waves at a Junction of Three Bars, Journal of Alied Mechanics, ASME, 8, (1981).. Simha, K.R.Y. and Fourney, W.L., Investigation of Stress Wave Proagation Through Intersecting Bars, Journal of Alied Mechanics, ASME, 51, 5-5 (198).. Wu, C.M. and Lundberg, B., Reflection and Transmission of the Energy of Harmonic Elastic Waves in a Bent Bar, Journal of Sound and Vibration, 190(), (1996). 5. Atkins, K.J. and Hunter, S.C., The Proagation of Longitudinal Elastic Waves around Right Angled Corners in Rods of Square Cross-section, Quarterly Journal of Mechanics and Alied Mathematics, 8, 5-60 (1975). 6. Achenback. J.D., Wave Proagation in Elastic Solids, rd Ed., North Holland Publishing Comany, New York (1980). 7. Abramowitz, M. and Stegum, I.A., Handbook of Mathematical Function, Dover Publication, Inc. New York (195).!"#T! "#$%!"#$!"#$%&"'()*+!"#$%&'()*+,-./01!"#$%&'() * +, -./0!"#$%&!" '()&*T!"#$%&'()*+,T!"#$!"#$%&' ()*+#,-./0!1!"#$%&'()#*+,-./01!"#$%&'()*+,-$%.'!"#$%&'()*+,-!"./!"#$%&99%!"#$%&'()!!"#$%#$&#'