m ... .fl . .. P. ' ..‘ ‘3’ O :89- :52“: ~ 2‘s4‘. a. ' J t v“ 0" p '1: o 5': '«I ‘ l U. 6.9 A... V «w? (IL ESIS LIBRARY Michigan State University AB ST RAC T DYNAMIC SHEAR IN SIMPLE SPAN BRIDGES AND EFFECT OF VEHICLE DAMPING by Dalsukhrai Parikh An analytical study is made of the dynamic shear at the support of simple span bridges traversed by single axle loads. A bridge, in this analysis, is represented by an elastic beam having a uniform dis- tribution of mass density and flexural rigidity. A load unit consists of a sprung mass connected to an unsprung mass through a linearly elastic spring and a viscous damping device. The analysis is carried out under the assumption that the deflection shape of the beam at any time is proportional to its static deflection curve due to its own weight and 'the weight of the moving load. Numerical results are obtained by use of a computing program written for use on the MISTIC (the old digital computer of Michigan State University). The variables considered in this study include three ”typical" bridges having span length of 20 ft. , 45 ft. and 70 ft. , the vehicle speed, surface unevenness, and viscous damping in the vehicle. Results indicate that the characteristics of dynamic shear at the support, in general, are qualitatively similar to those of the dynamic Dalsukhrai Parikh bending moments and deflections at mid-span. However, the magnitude of the dynamic increments of shear is somewhat smaller than that of the bending moments and deflections at mid-span. Vehicle damping is found to be very effective in reducing the dynamic response of the bridges having a wavy deck surface when the speed is such that resonance of vehicle motion takes place. DYNAMIC SHEAR IN SIMPLE SPAN BRIDGES AND EFFECT OF VEHICLE DAMPING BY Dals ukhrai Parikh A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Civil Engineering 1963 AC KNOW LEDG MENTS The author wishes to express his gratitude to his advisor Dr. R. K. L. Wen for his valuable guidance during all phases of the work. ii TAB LE OF CONTENTS ACKNOWLEDGMENTS ..................................... LIST OF TABLES ....................................... LIST OF FIGURES ....................................... I. INT RODUC TION ..................................... II. MATHEMATICAL ANALYSIS OF SYSTEM CONSIDERED . 1. III. NUMERICAL RESULTS ............................... 3. l. 3. 2. 3. 3. 3. 4. 3. IV. SUMMARY AND CONCLUDING REMARKS LIST OF REFERENCES FIGURES .NPEVE" PP!" 2 2 2. 2 CDKIO‘U'I 5. Physical System Considered Assumptions in Analysis Expression of Beam Deflection Derivation of the Governing Differential Equation of Motion Profile Variation Functions Dynamic Shear and Amplification Factor Reduction of Equations to Dimensionless Form Method of Solution Variables Considered in the Study Effect of Speed of Vehicle Effect of Sinusoidal Unevenness Effect of Vehicle Damping on Bridges Having a Smooth Surface Effect of Vehicle Damping on Bridges Having a Sinusoidal Surface Profile OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 10 12 16 l8 18 22 23 25 26 30 33 34 LIST OF TAB LES Number Page 1. Weight and Frequencies of "Typical" Bridges ...... 20 I 2. Characteristics of ”Typical Vehicle” ............. 20 3. Maximum Amplification Factors in Study of Effect of Vehicle Speed ............................... 21 4. Comparison of Actual and Expected Critical Values of Speed Parameters ........ . ....... . . . . . 21 iv LIST OF FIGURES Number Page 1. System Analyzed ............................... 34 2. Effect of Vehicle Speed (70-ft. Span) .............. 35 3. Effect of Sinusoidal Unevenness (20-ft. Span) ...... 36 4. Effect of Sinusoidal Unevenness (45-ft. Span) ...... 37 5. Effect of Sinusoidal Unevenness (70-ft. Span) ...... 38 6. Effect of Vehicle Damping (20-ft. Span, b = O) ...... 39 7. Effect of Vehicle Damping (45-ft. Span, b = 0) ...... 4O 8. Effect of Vehicle Damping (70-ft. Span, b = O) ...... 41 9. Effect of Vehicle Damping (20-ft. span, 6 = 0.001) .. 42 10. Effect of Vehicle Damping (45-ft. Span, 13 = 0. 001) .. 43 11. Effect of Vehicle Damping (70-ft. Span, E = 0. 001) .. 44 12. Effect of Vehicle Damping (20-ft. span, 6 = 0. 004) . . 45 13. Effect of Vehicle Damping (45-ft. Span, b = O. 004) . . 46 14. . Effect of Vehicle Damping (70-ft. Span, E = 0. 004) .. 47 I. INTRODUCTION In recent years a great deal of work has been done on the subject of the dynamic behavior of simple span bridges under the influence of moving vehicles. The past work, however, was concerned with the maximum dynamic bending moment and deflection which generally occur at or near the mid-span of the bridge. Very little information is available for the maximum shear which generally occurs near the sup- port. The object of this investigation is to study analytically the maxi- mum dynamic shear in bridges traversed by heavy vehicles. Of course, in design work shear can be an important factor, particularly for short span bridges and for end bearing support. The dynamic response of a bridge-vehicle system is influenced by a large number of factors. For example, for the bridge, there are the weight and stiffness and the unevenness of the deck surface. For the vehicle, there are the weight, suspension characteristics, damping and its ”initial conditions” (vibrations at the moment when it enters the bridge). This investigation is limited, however, to only a few of the above factors. The factors considered in the problem are the speed of the vehicle, weight and frequency of the bridge and vehicle, deck surface profile of the bridge and damping in the vehicle. In an investigation (1) of the effect of sinusoidal unevenness on the dynamic response at mid-span of the bridge, it was reported that large response occurred for even very small amplitudes of the unevenness of the bridge surface profile. The large response was thought to be due chiefly to certain conditions of resonance or synchronization of the wavy surface with the vibrations of the vehicle and/or the bridge. In that study no damping was considered in the system. It is thought, how- ever, that the influence of damping under such conditions is likely to be most important. Since damping does exist to a substantial degree in vehicles, the results obtained with vehicle damping would be more realistic. In here, only damping in the vehicles is considered since it is believed that damping in bridges is relatively small. Damping in a vehicle may come from several sources such as the friction between the dry sliding surface of the leaf springs, shock absorbers, air resistance and the energy loss in the deformation changes of the tires. Damping resistance due to these various sources will, in turn, depend upon many factors such as the coefficient of friction be- tween dry surfaces, force in the spring, velocity of the vertical motion, and the tire pressure. It is recognized that a realistic model for vehicle damping probably should be considered together with characteristics of the suspension system. The composite system may have the following characteristics. First, there is a constant frictional force that ”locks" the spring. After the initial frictional force is overcome there exists a somewhat con- stant damping force a little smaller in magnitude accompanying the deformation of the spring. In this investigation, in order to simplify the analysis, it is assumed that all the vehicle damping could be lumped as a viscous damping (i. e. damping force proportional to velocity). The present investigation includes (i) an analysis of the physical system, (ii) the preparation of a computing program for use on the MISTIC (old computer Of Michigan State University) to obtain numerical results, and (iii) a study of the effects of certain parameters as men- tioned previously. The mathematical model for the bridge-vehicle system considered consists of a simply supported beam traversed by a single axle load unit moving at constant speed. The load is composed of a sprung mass connected by a spring and damping device to an unsprung mass. In the following chapters of this thesis, Chapter 11 contains a description of the mathematical model and its analysis. The variables considered in the study of effects of parameter and numerical results are described and discussed in Chapter III. The last chapter consists of a summary and concluding remarks. Notation The notations used in this thesis are explained in the text where they first appear. II. MATHEMATICAL ANALYSIS OF SYSTEM CONSIDERED In this chapter is presented the derivation of the differential equa- tions that govern the motion of a system consisting of a simply supported beam and a single axle load unit moving across the beam at a constant speed. Included in this chapter is a description of the physical system considered, the assumption made in the analysis, and a derivation of the expressions for shear and amplification factor for shear. The method of analysis used for this problem is essentially the same as that used in Ref. (2), except that the D'Alemberts principle is used herein for the derivation of the equations of motion. 2. 1. Physical System Considered: The system considered is shown in Fig. 1. It consists of a simply supported beam spanned between two rigid approaches and a single axle load unit moving across the beam with constant speed. The beam is assumed to be linearly elastic, of uniform mass distribution and cross- section. The load unit consists of a sprung mass connected to an un- sprung mass through a linearly elastic spring and a viscous damping device. 2. 2. Assumfiions in Analysis: In deriving the governing differential equations of motion for the system described above the following assumptions are made. 1. The usual beam theory is applicable to this problem. 2. The unsprung mass is in contact with the top surface of the beam at all times. 3. The deflection configuration of the beam at any time is propor- tional to the static deflection shape of the beam under its own weight and the weight of the moving load. 4. For several rapidly converging series only the first term of each series needs to be considered. 2. 3. Expression of Beam Deflection. In view of Assumption 3 of Art. 2. 2. , the dynamic deflection can be represented by the expression y = f(t) q; (x, t) 2.2.1 in which y is the deflection of the beam of any point on the neutral axis of the beam at any time measured from its unstressed position. See Fig. 1. L): (x, t) is a function proportional to the static deflection of the beam due to its own weight and the weight of the load. f(t) is a function of time referred to previously as a coefficient of proportionality. Henceforth it will be simply referred to as 'f.‘ The function 41 in series form can be expressed as oo : Z ' nTTX Ll) (x,t) n=l nn(t) Sin L 2. 2. in which 2m gL - 1 E nvx b nn(t) - K [E 4 sm L + 5 ] 2.2. n n T!" wherein 6 = l for O < x < L otherwise 6 = 0 K = coefficient of proportionality; when K = DEE} (id/'1, q. is equal to static deflection; W = the total weight of the load; x = the distance from left support of the beam to the load as shown in Fig. 1; L = length of the beam; m = mass of beam per unit length; g = gravitational acceleration. 2, 4. Derivation of the Governing Differential Equations of Motion: The governing differential equation of motion of a beam is obtained by applying D'Alemberts principle. The differential equation of motion of an ordinary beam subjected to a continuous load of intensity q is given by In the present problem the loading q consists of: 2 . 1 l. the inertia force of the beam which varies along the length of the beam and is given by the expression 2. the viscous damping force in the beam which also varies along the length of beam and is given by the expression 3 -C _x b at wherein cb = viscous damping coefficient of the beam 3. the weight of beam; expressed in terms of sine series it is given by 4m b9 l/n sin n1Tx 1r n=odd L 4. Concentrated instantaneous dynamic reaction P on the beam due to the load unit. P can be expressed as a continuous function p(x, t), along the beam by the use of Fourier series expansion and is given by the equation 2 (D - : _ z . nTIX . I'lTIX P 1SiannL Adding expressions 1 to 4 and substituting for q in equation 2. 4. l, the governing equation of motion of beam is given by 4 2 4m g 8 8 8 b . nTrx 8X4 mb ayz Cb a t 71' n :-0dd 1/n Sln L -p(x.t)=0 2.4.2 The instantaneous reaction is given by the following: p=k(ZS+Z‘w)+c(Z-w)t+m(g-th) 2.4-3 in which k = spring constant: ZS = initial static compression of the load spring w = w(x, t) evaluated at x = x, as shown in the Fig. 1. It is the vertical distance of the top face of the beam from the horizontal through its support at any time. c = viscous damping coefficient for load unit A15 0 noting that and hence y .2: fZJnn sin mix (from Eqs. 2. 2. 1 and 2. 2. 2) a4 4 4 1r -—§= fZTIn n‘rr4 sinnl:< 3x L fly : Z . mm + Z . nflx 3t ft nn51n L f (nn)t sm L By: 2 ,nTrx+2 Z ,nTrx Z ,nnx 4 2 f nn51n L ft (nn)t Sln +f (nn)ttsm 2. . 6t tt L L where the s mbols f f ( ) and( ) re resents 22 2-2; 922- and y t’ tt' nn t nn tt p at' at?" at aznn V respectively. form. Substituting Eqs. 2. 4. 3 and Z. 4. 4, Eq. 2. 4. 2 takes the following 4 4 n EIon '1; sin nix + mbfttznn sing-TIT?- L n17 Z . + 2 mb ft (nn)t Sln X I'lTTX + E ‘ mbf (nn)tt Sln L nTTX + f 2 s' Cb t nn in + cbf 2(nn)t sin nnx L m g - b 2 ,nTrX Z . n1Tx , nTrx - —- -—Z 4 1r n=odd1/n81n L L Sin L Sin L [k(ZS+Z-w)+c(Z -W)t+m(g-W )]=0 tt Taking only first term of series as stated in assumption 4 of Art. 2. 2 (i. e. n = l), the above equation takes the following form. 4 EIé‘Yn-E L + — + L + mefttn mb ftnt m [k(ZS+Z-w)+c(Z-w)t+m(g-wtt)=0 2.4.6 The differential equation of motion for the sprung mass of the moving loadisMZtt+k(Z-w)+c(Z -w)t=0 2.4.7 2. 5. Profile Variation Functions: In the above equations, W represents the profile of travel as evaluated at x. As seen from the Fig. 1. w=y+p=f¢u+p. 10 The derivatives ofw with respect to time are w = f + f + t t LP qJ1; pt w =f +2f +f + 2.5.1 and 121’. tt LP 1: th Ltht ptt The function p represents the profile variation and it may be arbitrarily prescribed. In here, only a simple profile variation of the sinusoidal type is considered. In this case p(x) is given by mTrx L p(x) = b sin in which b = amplitude of the sinusoidal profile and m = number of half sine waves along the span of the beam. . " ' . mTIX . . . . For the pomt x, p(x) = b 5111 and its derivatives With respect L to time are m 1T mTTX pt (x) = b L cos L mzflz p (x) = - p(x) 2. 5. 2 tt L2 2. 6. Dynamic Shear and Amplification Factor: The dynamic axle load is obtained by treating the instantaneous reaction P between the axle load and the beam as statically applied force. It may be obtained directly from Eq. 2. 4. 3. P=k(Z-w)+c(Z-w)t-mwtt+Mg+mg 2.6.1 It is seen from the above equation that dynamic load differs from the static load by k (Z - w) + c (Z -w)t - mwtt. 11 The first term and the second term represent the spring force and the damping resistance of the load spring respectively. The last term is the effect of the unsprung mass. The shear due to dynamic axle load P at any section x is given by equations A P ‘ - : — < (Qd)x L when x - x = 315.33 when x 2 x 2. 6. 2 The dynamic shear force due to the inertia forces and damping forces of the beam at a section x is given by the equation X — + + + . . BLILLLX (m btty bet)dx f(mbytt cbyt) dx 2 6 3 0 By taking only the first term of the series in the Eqs. 2. 4. 4 and substituting them in Eq. 2. 6. 3, one obtains B- 1; Ii 1: w_x (Qd)X —-mb(fi)(fn)tt cos L -Cb(11)(fn)t cos L 2.6.4 The total dynamic shear at any section is the sum of the Eqs. 2. 6. 3 and 2. 6. 4, i. e. (Qd)X = (Qd)x + (Qd) x . 2. 6. 5 The dynamic shear at right support obtained by replacing x with L in Eqs. 2. 6. 5 is given by 12 L + — f + mb 1T(n)tt c D e'r'v. L b ; (fn)t 2. 6. 6 Amplification factor, in this report, is defined as the ratio of maximum dynamic shear at the right support of the beam to the maxi- mum static shear at the same support due to the load unit. The maxi- mum static shear is obviously equal to (M + m) g. Qd Hence the amplification factor = m 2. 6. 7 in which Qd is given by equation 2. 6. 6. 2. 7. 1. Reduction of Equations to Dimensionless Form: The governing differential equations of motion and various other expressions are converted now to dimensionless form. Let the following dimensionless variable and constants be introduced. 1' 2 E z m ‘3 = C L L 2M0) (.0 2 C _ b _ b (I) - 2g ft} p.b Zmbwb w 2 b TTVAt : — Z : A : 5, 2g h L 2 VT a : __b- B : :03. b 2L 32g = L y m/mb 13 Also the following relationship should be noted. ,. int Zn “b f :_Zg_(.q_>_:_5¢) 1' “2 T) 2 b 77 f -—2—& -1-[4> zen—7w +¢{Z(:)—T)Z TIT—T] 2 TT— 2 T1 T1". T) T T) - T) ' w b 2 inwhichf =2!- andf :_8__f_. 1' 31' 11' 81’2 Dividing the equations 2. 4. 6 and 2. 4. 7 by the quantity mbgL, in terms of dimensionless quantities one obtains: 1 =— +A +A +A +A +A (p11- A1(AZ¢T 34) 4&1 5g 6 7) 2 and gTT=B1¢T+BZ¢+B3g+B4gT+B5 2 in which A =l+2y-L-:-sin'r l -2 "i- i 39 LP. A2 3";- Ub+4¥L|J-n—Z'Sin1' -4‘{;‘ Sin‘l’ - a TWESInT 2 A--1—4:73—¢'T+2—T—T-tp'r 3 2- y 3 sm y 2 8111 a 77 77 n 4, +4y—TL); sin'r-Zy T7 sinT-Zarisinr 772 T a2 r) T) If 4 T , 4 1' . +a (31"V0' :7-2- up Sin 1’ - a [31" V0 7 Sin 1' 7. .7. .7. 14 45 . A =— I‘ 0' 8111‘? 4 (1 20’1‘ . AS— 2 51n1' a - , 64 - , 32 - . A6— -32yp1__rsm‘r-—a— {31“\/0' stm'r -a20Fp51nT l 2 :— —+ + A7 le (Y D] a -32 i B1_a\/;n 2. 7. 2. Dynamic Axle Load and Dynamic Shear: The dimensionless form of the dynamic axle load is obtained by dividing the dynamic axle load P by the total weight of load, i. e. P f3:——__ . . P' . E . . . . (M+m)g inwhich is given by the q 2 61 Dimensionless form of P excluding the effect of profile variation is given by -_ 201" i 4af31" V0" 31 P-1+r—+Y[§ n¢]+—_F+Y [éT-chn n L) 41 2 2n 4) 241 T .1 2“) If _ _1___ +(T72 -n)¢]-F+y[n¢'r1'( 2 Tl)¢T 2.72.). n x) 2n L) 4 +( T3 _ 1"; _ TZT + 7:1.)43] 2 T) TI T) If a sinusoidal profile variation is present the following terms should be added to the right-hand side of the equation. 01‘ a may ' 64abl" 0 pr -32[ -———]p- 2. y+F F+y F+y Dynamic shear at right-hand support in dimensionless form is obtained by dividing the equation 2. 6. 6 by the total weight of the load, i.e. (M +m)g: C3 = 2P'1:+ —1——[2a2¢ + 4ua¢ ] 2. d 71' 1' MPH) The amplification factor is numerically the same as the dimen- sionless dynamic shear given above. 2. 7. 3. ([1 Functions, 7’) Functions and Their Derivatives: 16mbg L 4 By taking f = T (1:) in the equation 2. 2.1 one obtains the following equations in dimensionless form. .7. 7. 7. l6 4 2 3 4 2 -2 q. 31“}(1/8-T+T)+E—(Y+P)T(1/Z-T) 24 3 3 3 ft -2 -3 17 - -2 -3 =— - — + - 111T 48 (1/8 31' +4T)+ 6(y PHI/21' 31’ +41") 1r2- TI'Z - -2 14472—8-T(T-1/2)+-1—2'(Y+r)(1/2-6T+12T) 2.7. The first term at right-hand side of the above equation is the contribution of the beam and second term is the contribution of the load unit. The T) functions in dimensionless form are obtained by taking k in the Eq. 2. 2. 3 to be equal to -—l— . 1’) functions in dimensionless form 8mb gL are given by _ l l . n—4fl+8(y+l")81n'r r) = l ( + F) cos i" 1' 8 Y 1 . and r) =-—(y+1")51n‘r. 2.7.8 TT 8 2. 8. Method of Solution and Description of Program The response of the beam could be obtained by solving Equations 2. 7. 2 and 2. 7. 3. These equations are solved numerically by using the B-method of integration (3), with B = 1/6 (i. e. linear acceleration method). Numerical results are obtained by a program written for use on the MISTIC (the old computer of Michigan State University). 17 The program prepared can handle various combinations of parameters defining the system. It can handle problems involving the speed of vehicle, weight and frequency of bridge and vehicle, initial conditions of oscillations of vehicle and bridge, deck waviness and damping in the vehicle as well as the bridge. III. NUMERICAL RESULTS In this chapter are considered the effects of vehicle speed, sinusoidal unevenness of the bridge deck and vehicle damping on the dynamic shear at the support of bridges. Also is presented a brief discussion of the parameters involved in this study. The numerical results presented here have been obtained by use of a digital computer as mentioned previously. The results presented in this chapter are described in terms of amplification factors for maximum shear at the right-hand support of the bridge. These factors represent the ratios of the maximum dynamic shear to the maximum static shear at the support. Results obtained are for the following conditions: (a) The bridge is at rest when vehicle enters onto the bridge, and (b) the sprung mass of the vehicle has no vertical motion. 3.1. Variables Considered in Study: 1. Bridge and Vehicle: In this study three "typical" bridges and a "typical" load are considered. The characteristics of the bridges and the load are given in Tables 1 and 2 respectively. (These data are taken from Ref. 4.) 2. Vehicle Speed: In this study the range of vehicle speed varies from 15 m. p. h. to 75 m. p. h. It may be noted that this range corresponds to a = 0. 05 18 19 to 0. 22 (for all three bridges). 3. Waviness of Deck Surface: The initial waviness of the deck surface is given by 171fo L p(x) = b sin in which m is the number of half-sine waves and b is the amplitude of the waves. The dimensionless parameter for profile variation is given by - b mn’x : A Z ' p p/l6 b 15: b 5111 L in which Ab = 2g/wbz, cob being the circular frequency of the bridge. The range of 13 considered in this problem is from 0 to . 004. Cor- responding to b = . 004, the values of actual amplitude of unevenness are . 008, . 05 and .125 inch for 20-ft. , 45-ft. and 70-ft. bridges respectively. In all cases considered m is taken as seven. 4. Vehicle Damping: The amount of vehicle damping specified by the parameter ‘3 _ c _ actual viscous damping coefficient ZMw "critical damping” coefficient It is well known that if {3 = l, the sprung load when disturbed will not oscillate. The values of (3 considered are 0, 0.2 and 0. 5. That is, the sprung load is either undamped or underdamped. Table 1. Weight and Frequencies of "Typical" Bridges Span Total weight :1 iniafrjftalc .. Tb/ZL n (ft.) (1b.) a a que“ Y (sec./ft.) (c. p. s.) 20 98, 000 12. 13 0. 00206 45 227, 000 5. 41 0.00206 70 385, 700 3. 19 0. 00224 Complete composite action is assumed between the concrete slab- and I-beams. Table 2. Characteristics of "Typical” Vehicle Front axle Rear axle Unsprung weight (1b.) Sprung weight (1b.) Spring constant (1b. /in. ) Frequency (c. p. s.) Axle spacing (ft. ) Gross vehicle weight 5, 200 26, 800 21, 700 2.81 3, 400 28, 600 26, 000 2.98 21 Table 3. Maximum Amplification Factors in Study of Effect of Vehicle Speed S Maximum Maximum Maximum pan deflection bending moment shear at right- (ft. ) . . at mid-span at mid-span hand support 20 1. 67 l. 61 l. 27 45 1. 24 1. 15 1. 15 70 l. 26 1. 30 1. 14 Table 4. Comparison of Actual and Expected Critical Values of Speed Parameter Span a a * a a * (ft. ) l ' 1 - 2 2 20 . 023 ---** . 143 .150 45 . 074 . 070 .143 .150 70 .127 ---** .143 .145 *. a and (.1 are the same as a and az respectively except that they are obtained from the results. **For the 20-ft. bridge, a is outside the range of parameter considered and for the 70-ft. bridge, (11 and (12 are too close to be separated distinctly. 22 3. 2. Effect of Speed of Vehicle In Fig. 2 are plotted the maximum amplification factors for the maximum shear at the right-hand support of a 70-ft. bridge as a func- tion of speed parameter a, The values of the vehicle speed are also indicated on the horizontal scale. It is observed that this curve is oscillary in nature and that both the amplitude and period of oscillations increase with increasing values of the speed parameter a. This general trend is similar to that ob- served in previous investigations of the dynamic bending moment and deflection at mid-span of the bridges under passage of moving vehicles. An explanation of this trend is given in Ref. 4. It is obviously also applicable to the present case of the dynamic shear of the bridge. The peaks of the curve are most significant because of its prac- tical importance. It is seen that in general the value increases with speed. For the range of the speed of vehicle speed considered in this problem (15 to 70 m. p. h.) the maximumamplification for shear ob- tained for 20-ft. , 45-ft. and 70-ft. bridges are given in Table 3. Also for comparison are given the maximum amplification factors for mid- span dynamic bending moment and deflection. These amplification factors are taken from Ref. 4, and they are for tandem axle vehicle. It is seen from Table 3 that the dynamic increment for shear of the 20-ft. span bridge is larger than those of the 45-ft. , and 70-ft. spans. However, it is small as compared to the dynamic increments 23 for bending moment and deflections. It must be pointed out, however, that the data for the bending moments and deflections were based on two axle vehicle, while the data for shear were obtained for a single axle vehicle. It is entirely possible that a larger dynamic increment would result had a two axle vehicle been used. 3. 3. Effect of Sinusoidal Unevenness: The results obtained on the effect of sinusoidal movements are presented in Figs. 3 through 5. In Fig. 3 are plotted the maximum amplification factor for shear of the 20-ft. bridge as a function of the speed parameter a. The curves presented are for amplitude parameter b = . 001 and . 004. For both curves, the number of half sinusoidal waves on the profile of the bridge is seven. Also, for comparison, is presented the curve for smooth surface profile. Similar curves are presented for the 45-ft. bridge in Fig. 4 and for the 70-ft. bridge in Fig. 5. The range of speed considered is from 15 to 70 m. p. h. It is noted that in the case of the 20-ft. bridge, the influence of the unevenness is quite small. The difference in peak value of ampli- fication factor for uneven surface with an amplitude parameter b = . 004 and that of the smooth surface is only 2 %. For the 45-ft. bridge and 70-ft. bridge, however, the effect of unevenness becomes more obvious. The deviation of the dynamic increments is appreciable in magnitude. The difference in peak values between the 24 amplification factor of smooth surface and uneven surface is 18% for the 45-ft. and 65% for the 70-ft. bridges. The large dynamic incre- ments in the case of the 45-ft. bridge and 70-ft. bridge may be due to the following two factors. (a) The actual magnitude of amplitude: For the same amplitude parameter b, the actual value of unevenness is larger for the 45-ft. , and 70-ft. bridge than that of the 20-ft. bridge. (The amplitude for the 20-ft. , 45-ft. and 70-ft. bridge with an ampli- tude parameter b = . 004 is 0. 008 in. , 0. 05 in. , and 0. 125 in. , re- spectively.) (b) The condition of synchronization: One might expect large dynamic increments when the ”period of the profile, ” defined to be the time required for a load to travel over the distance covered by a complete wave, coincides with either the natural period of vibration of the bridge or with that of the vehicle. The critical values of the speed parameters at which these conditions may be expected are ob- tained from the following formulas given and discussed in Ref. 1. When the period of profile Tp = TL, the natural period of the load, the value of the speed parameter is given by 01 = (1/m)Tb/ TL. When Tp = Tb, the natural period of the bridge, the value of the speed parameter is given (12 =1/m. In Table 4 are given the "expected" and actual values of the "critical" speed parameters. It is seen that they are in good agree- ment. Actual critical values of the speed parameters in Table 4 are 25 obtained from the results for the amplitude parameter b = . 004. The large response in case of the 70-ft. bridge might be due to the fact that the values of its two critical parameters are very close to each other. 3. 4. Effect of Vehicle Damping on Bridges Having a Smooth Surface: Results showing the effect of vehicle damping on the maximum response of the bridges having smooth surface profile are presented in Figs. 6 through 8. In Fig. 6 are plotted the maximum amplification factors for shear at the right-hand support of the 20-ft. bridge against the values of speed parameter a. Values of the damping parameter 8 considered are 0, 0.2 and 0.5. Similar curves for 45-ft. bridge are shown in Fig. 7 and for 70-ft. bridge in Fig. 8. It is seen that for all cases these curves are oscillatory in nature and the effect of damping is to decrease the sharpness of undulations of these curves. In the case of the 20—ft. span, the effect of damping in general, is to lower the peaks of the amplification factors. However, this is not so in the case of the 45—ft. bridge and 70-ft. bridge. For the 45— ft. bridge, it is seen that the maximum bridge response with vehicle damping may be smaller or greater than that without damping depend- ing upon the values of a. For large values of vehicle damping generally increases the structural response. It may be noted that the values of the amplification factors for B = . 5 are generally larger than those for 0:02. 26 For the 70-ft. bridge the influence of vehicle damping on the bridge response is qualitatively very similar to the case of the 45-ft. bridge. In this case, the response for {3 = . 5 is generally larger than that for )3 = 0. This increase in dynamic response particularly for large values of a in the case of 45-ft. and 70-ft. bridges is somewhat unexpected. A detailed and careful rechecking of the analysis and programming was made and no error could be found. Hence, it may be noted that large vehicle damping is not effective in reducing the response of the smooth surface bridges of comparative longer spans. Reduction in response in case of short span bridge is significant be- cause the values of the amplification factors are generally larger than those for the longer spans. 3. 5. Effect of Vehicle Damping on Bridges Having Sinusoidal Surface Profile: Results obtained for the effect of vehicle damping on the dynamic shear of the bridge having a sinusoidal profile are presented in Figs. 9 through 14. In Fig. 9 are plotted the curves showing the maximum amplification factors for shear of the 20-ft. bridge as a function of speed parameter a. The number of half sinusoidal waves on the surface profile of the bridge is seven. The value of the amplitude parameter b specifying the surface unevenness of the bridge is 0. 001. The values of the damping parameter {3 considered are 0. 0, 0. 2 and O. 5. In Figs. 10 and 11 are shown similar curves for the 45-ft. and 70-ft. bridge, respectively. 27 For the 20-ft. bridge, the effect of vehicle damping, in general, is to lower the peaks of amplification factors. In the case of the 45-ft. bridge, as explained in Art. 3. 3, the large response seen at a = . 07 is due to the synchronization of the period of the deck profile with the period of the vehicle. The large response seen at a = . 15 is due to the synchronization of the period of the deck profile with the period of the bridge. It is seen in the figure that vehicle damping reduces sharply the amplification factor at a = . 07. The effect of vehicle damping at a = . 15 is, however, rather small. At other speeds, the effect of vehicle damping is insignificant for 5 = . 02. For B = . 05 appreciable increase in response is seen at higher speeds. The above results may be explained as follows. The effect of damping force on a system acted upon by a periodic disturbing force having the same period as that of the system, is to reduce sharply the dynamic response of the system and to keep it under certain maxi- mum. Therefore, one might expect vehicle damping to reduce con- siderably the dynamic response of a vehicle when Tp = TV and thereby to reduce the dynamic response of a bridge. The second condition of resonance T = Tb is essentially independent of the characteristics of the vehicle. Hence the effect of vehicle damping is expected to be insignificant. The increase in response at high speeds for large damping is similar to the observations made in the case of bridges with a smooth surface profile. 28 For the 70—ft. bridge, the period of the bridge and the period of the vehicle are close to each other. Hence the peak response at a = . 1425 is due to both critical conditions existing simultaneously. It is seen that the damping did reduce the maximum response at a =. 1425. However, the reduction is not as much as in the case of the 45-ft. bridge at a = . 07. This may be partly due to the fact that the second condition of resonance that exists concurrently with the first condition of resonance in the case of the 70-ft. bridge does not exist in the case of the 45-ft. bridge. In Figs. 12 to 14 are presented the results for the larger ampli- tude parameter b = . 004. In Fig. 12 are plotted the amplification factors for the shear of the 20-ft. bridge with a sinusoidal surface profile. The number of half sine waves on the bridge surface is again seven. Also the values of the damping parameter considered are 0. 0, 0. 2 and 0. 5. Similar curves are presented for the 45-ft. bridge in Fig. 13 and for the 70-ft. bridge in Fig. 14. The general trends indicated by these data are very similar to those observed in the pre- vious case of b = 0. 001. In the case of the 20—ft. bridge, the effect of vehicle damping is to reduce the peak of amplification factors. Reduction of the response depends on the value of damping. For B -' . 5 amplification factor is reduced from 1. 24 to l. 17. For the 45-ft. bridge the effect of the first resonance is more pronounced at a =.07. It is seen that vehicle damping sharply reduces the peak amplification 29 factor for the first resonance condition, while it increases the re- sponse at other values of a. For the 70-ft. bridge, the vehicle damping substantially reduces the response at a = 0. 15. As explained earlier, the effect of second condition of resonance could be seen at a = . 1425. As in the case of the 45-ft. bridge the effect of damping on the response at lower values of speed is practically negligible. However, at higher speeds damping increases the bridge response appreciably. The preceding observations indicate that the most important effect of vehicle damping is to reduce very effectively the large dynamic responses due to the synchronization of the period of deck profile with the period of the vehicle. IV. SUMMARY AND CONCLUDING REMARKS The purpose of this investigation has been to study analytically the maximum dynamic shear in simple span bridges. The effect of vehicle speed, sinusoidal unevenness of the deck surface profile and vehicle damping are studied for three "typical” bridges with span lengths of 20-ft. , 45-ft. and 70-ft. The characteristics of the curves of the amplification factors for maximum shear are found to be qualitatively similar to those for bending moment and deflection at mid span. As expected, the shear response for the 20-ft. bridge with a smooth sur- face is found to be larger than that of the other two bridges. Results indicate that the values of the maximum amplification factors for dynamic shear are somewhat smaller than those for dynamic bending moment and deflections. Therefore, it seems that fma design point of view, amplification factors based on maximum bending moment should cover the dynamic effects in shear. In the case of bridges with sinusoidal surface profile, at certain critical speeds, large responses were obtained for small amplitude of unevenness. These large responses were obtained at a speed when the "period of the profile” is either equal to the period of vehicle or to the period of the bridge or both. Further, the amplification factors increase with the increased amplitude of unevenness. The effect of vehicle damping in the case of 20—ft. bridge is to 30 31 reduce the maximum response of the bridge. This is important in view of the fact that such responses are larger and hence more sig- nificant for the short span bridges. Vehicle damping is not seen to be effective in reducing the response of the 45-ft. and the 70-ft. bridges. In fact, when damping is large ((3 = 0.50) the response of the bridge increased considerably. However, it must be remembered that the vehicle damping is effective in reducing the amplification factors so far as vehicle vibration are concerned. The most important effect of vehicle damping is seen in this study to be the sharp reduction of the dynamic response of the bridges with wavy deck surfaces when the vehicle speed is such that the resonance of the vehicle motion takes place. It might also be pointed out that the model of the viscous damp- ing used herein is only a convenient approximation of the real damping characteristics that exist in actual vehicles. If the amount of damping is small, such approximation probably is quite realistic. For large amount of damping such as those used in this study (8 = 0. 20 and 8 = 0. 50), it is possible that the model may not sufficiently accurately represent the actual vehicle system. In this connection, it would seem desirable as future study, to consider a more realistic damping mechanism for the vehicle, such as constant frictional forces as dis- cussed in the Chapter I, Introduction. Another worthwhile extension of this study would be a consideration of the effect vehicle damping 32 on the bridge response if the vehicle enters the bridge with an "initial vertical oscillation. " LIST OF REFERENCES Levya, J. Toledo., Velestos, A. 5., "Effects of Roadway Unevenness on Dynamic Response of Simple Span Highway Bridges, ” Highway Bridge Impact Investigation, Eighth Progress Report, University of Illinois, 1958. Wen, R. K., ”Dynamic Response of Beams Traversed by Two- Axle Loads, ” Journal of the Engineering Mechanics Division, Proceedings A. S. C. E. , October, 1960. Nevy'ark, N. M. , "A Method of Computation for Structural Dynamics, " Journal of Engineering Mechanics Division, A. S.C.E., Vol. 85, No. EM 3, July, 1959. Wen, R. K. and Velestos, A. S. , "Dynamic Behavior of Simple- Span Highway Bridges, " Highway Research Board, Washington, D. C. , Bulletin 315, 1962. 33 34 mmma magnum pmumgmcou 889W. \\\\mm 4 .3...“ 35 nouoamumnm poomm .teaqg 10} 101323 uoii'eognduxv a: -. am. 2. S. 2 2. 2. 8. 8. mo. £98 3 om 2. on om as 3.0 o H WK /\ H H < S a. 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