APPENDIX C STRESS WAVE ANALYSES OF WEIGHT-DROP TESTS

STRESS WAVE ANALYSES OF WEIGHT-DROP TESTS

Note: The computer source code presented in this appendix is a simplified version of a more complex program including plotting of the stress waves. The latter is included on a floppy diskette and presented in Appendix O along with instructions for running.



C.1 Initial Peak Stress upon Impact

Figure 3.7 shows the mechanical model for a steel rod subjected to the loading from a dropped weight. The top end of the steel rod is firmly clamped to a concrete beam at the ceiling and is assumed to be fixed. A metal plate is firmly attached to the bottom end to accept the impact of the weight.

Suppose a weight of mass M impacts on the bottom plate at an initial velocity of vi. Assuming that the weight and the bottom plate behave as rigid bodies and that they move downwards together immediately after the impact, the velocity of the bottom plate (and of the weight) vo immediately after the impact can be obtained, using the law of conservation of momentum, as follows: 

		vo = vi M / (M + mp)            
                                                                    (C1)
where mp is the mass of the metal plate. Since the bottom plate is fixed to the bottom end of the steel rod, the initial dynamic stress at time zero so is therefore: 

		so  = rcvo                        
                                                                    (C2)
where r and c are the density and "bar wave propagation velocity" of the steel rod respectively.



C.2 Successive Wave Sources from the Bottom Plate

Immediately after the impact, a stress wave, with the initial stress of so, is generated at the bottom plate and propagates upwards. This wave source is reflected at the fixed top end of the steel rod and then propagates downwards. At time t0 (a time variable) after the impact but before the first reflection at the bottom plate, we have the following equation for the equilibrium of the bottom plate and the weight by Newton's second law: 

	- (M + mp) (dv/dt0) = Ft0(0) = As - (M + mp)g
                                                                      (C3)
where v is the velocity of the bottom plate (with the weight) at time t0, s the stress wave source at the bottom of the steel rod with s = rcv, g the gravitational acceleration and A the cross-sectional area of steel rod. Here, Ft0(0), denotes the sum of forces in the steel rod acting on the bottom plate, where the superscript is the reflection number index and it indicates, in this instance, zero reflection at the bottom plate. Equation (C3) can be written as: 

		ds/dt0 - gs = -gsst 

                                                                      (C4)
where g = - rcA(M + mp)-1 and sst = (M + mp) gA-1. The above equation can be solved for the initial stress wave source generated at the bottom plate: 

	      st0(0) = exp(gt0)I(0) + sst        0 < t0 < 2t	

                                                                      (C5)
where I(0) is a stress constant with its superscript being the reflection number index at the bottom plate and t is the time for a wave to propagate the entire length of the steel rod with t = Lc-1. We find gt = - m(M + mp)-1, where m is the mass of the steel rod. The superscript (0) in st0(0) is the reflection number index which indicates, in this instance, zero-th reflection at the bottom plate. Since the initial stress at time zero of the initial wave source is so, the stress constant can be obtained as follows: 

		I(0) = so - sst 

                                                                   (C6)
A equation similar to equation (C5) was derived and discussed by J. Hopkinson (1972, note 2009), Taylor (1946) and Kolsky (1963) where the stress due to the weight of the drop-weight was ignored.

As seen from equation (C5), the highest value of the initial stress wave source occurs at t0 = 0, and is expressed as: 

		sH(0) = so 

                                                                    (C7)
The corresponding lowest value occurs at t0 = 2t and can be expressed as: 

	      sL(0) = exp(2gt)I(0) + sst        	

                                                                      (C8)
As stated earlier, the initial stress wave, represented by equation (C5), propagates upwards, is reflected at the top end of the steel rod and then propagates downwards. When this wave comes back to the bottom plate, it interacts with the bottom plate and a new (the 1st) stress wave source is generated. On the other hand, the old (the initial) wave continues its journey up and down the steel rod. In the following, the new wave source will be derived.

At time t1 (a time variable) after the first but before the second reflection at the bottom plate, we have the following equation by Newton's second law: 

	- (M + mp) (dv/dt1) = Ft1(1) = Ft1(0) + RAs(t1)(0)  for 0 < t1 < 2t 

                                                                      (C9)
where R = Rt + Rb with Rt and Rb being the reflection coefficients at the top end and bottom end of the steel rod respectively. Rt is equal to 1 or 0 if the top end is fixed or free to move, and Rb is equal to 1 or 0 if the bottom plate is fixed on to the steel rod or free to move on it. Equation (C3) can be substituted into the above equation with s now as a function of t1, and equation (C5) can be substituted into the above equation with t0 replaced with t1. The following differential equation and the 1st stress wave source can be obtained (Yi 1992): 

		ds/dt1 - gs = -gsst(1 - R) + gR I(0) exp(gt1) 

                                                                      (C10) 


	      st1(1) = exp(gt1)[I(1) + I(0)(Rgt1)] + sst(1 - R)    for    0 < t1 < 2t	

                                                                      (C11)
where I(1) is the stress constant for the first reflection at the bottom plate. Since the downward movement of the bottom plate is continuous, the consecutive stress wave sources are also continuous, that is, s(t1=0)(1) = s (t0=2t)(0). The highest value of the 1st stress wave source sH(1) occurs at t1 = 0 or equivalently at t0 = 2t, i.e., sH(1) = s(t1=0)(1) = s(t0=2t)(0) = sL(0). In other words: 

	      sH(1) = sL(0) = exp(2gt)I(0) + sst        	

                                                                      (C12)
Also from equation (C11), we have (Yi 1993) 

		I(1) = sH(1) - sst (1 - R) 

                                                                      (C13)
Clearly, the stress wave sources expressed in equations (C5) and (C11) form a continuously declining curve with respect to time.

Further derivations for subsequent reflections at the bottom plate lead to the following general formulae for the determination of the highest and lowest values of successive stress wave sources: 

	 

		 sH(0) = so	

		 sH(1) =  sL(0)  = exp(2gt)I(0) + sst	


		 sH(n) = exp(2gt)[I(n-1) + B1(2Rgt) + B2(2Rgt)2 / 2! + ... + I(0)(2Rgt)n-1 / (n-1)!] + sst(1 - R)n-1	

		 sH(n+1) = exp(2gt)[I(n) + (I(n-1) + B1)(2Rgt) + (B1 + B2)(2Rgt)2 / 2! + ... + I(0)(2Rgt)n / n!] + sst(1 - R)n	


	 

		 I(n) = sH(n) - sst(1 - R)n

		 I(n+1) = sH(n+1) - sst(1 - R)n+1

		 sL(n-1) = sH(n)

	 

	                      for n > 1	           (C14)
where sH(n) and sL(n) are the highest and lowest values of the nth stress wave source after the nth but before the (n+1)th reflection at the bottom plate, I(n) is the stress constant corresponding to the stress source peak at the previous reflection (Yi 2011), and B1, B2, ... are constants obtained in a step by step evaluation following the above formulae. B1(1) = B2(1) = 0, B1(n+1) = I(n-1) + B1(n), B2(n+1) = B1(n) + B2(n) (Yi 2009-2015).



C.3 Stresses at the Top and Bottom Ends of the Steel Rod

At a certain location on the steel rod, the dynamic stress value at a certain time is the superposition of all the stress wave sources arriving at the location at this particular time. We define the peak or valley stress at the nth reflection as the sum of the highest or lowest values of all (from 0 to nth) arriving stress wave sources. It follows that the peak and valley stresses for the same reflection No. have a time span of 2t, but the valley stress for the nth reflection occurs at the same time as the peak stress for the (n+1)th reflection. We will consider two special locations, namely the top and bottom ends of the steel rod. Taking the top end as an example, at the first reflection the dynamic peak stress is (1 + Rt)so or 2so for Rt = 1; and the valley stress is (1 + Rt)sH(1). General formulas for the dynamic peak stress at the nth reflection at the top and bottom ends are expressed as follows: 

            Topp(n) = (1 + Rt)sH(n-1) + RbRt(1 + Rt)sH(n-2) + ... + (RbRt)n-1(1 + Rt)sH(0) = 

                    = RbRtTopp(n-1) + (1 + Rt) sH(n-1)         for n > 1  and 


            Bottomp(n) = sH(n) + Rt(1 + Rb)sH(n-1) + RtRb(1 + Rb)sH(n-2) + Rt2Rb(1 + Rb)sH(n-3) + 

                         + Rt3Rb2(1 + Rb)sH(n-4) + ... + Rt(RtRb)n-1(1 + Rb)sH(0) =

	                 = RtTopp(n) + sH(n)        for n > 0	

                                                            (C15)
where Topp(n) and Bottomp(n) are dynamic peak stresses at the nth reflection at the top and bottom ends respectively. Topp(0) = 0. sH(n) is shown in equation (C14). Similarly, the dynamic valley stresses Topv(n) and Bottomv(n) at the nth reflection at the top and bottom ends are expressed as follows: 

                Topv(n) = RbRtTopv(n-1) + (1 + Rt)sL(n-1) = 

                     = Rb Rt Topv(n-1) + (1 + Rt)sH(n)   for n > 1; and	 

                Bottomv(n) = RtTopv(n) + sL(n) = 

                     = Rt Topv(n) + sH(n+1)   for n > 0.  	              

                                                              (C16)
where Topv(0) = 0. The dynamic peak and valley stresses at the top and bottom ends of the steel rod in equation (C15) and (C16) can be computed as a function of the reflection number index n. These stresses can be plotted against (n) or time (t) to obtain the stress wave in a steel rod. As seen from equations (C14), (C15) and (C16), the dynamic stress at any reflection number (time) is essentially a function of the initial velocity of the bottom plate (with the weight) vo and the mass ratio of the steel rod to the drop weight (represented by gt). In the derivations, the reflection coefficients Rt and Rb are assumed constant for all reflections. This assumption is only realistic in special situations.

Taylor (1946) listed in a table the equivalent stresses Topp(n), Topv(n), Bottomp(n) and Bottomv(n) for up to the 2nd reflection at the bottom plate (n < 2) and third reflection at the top fixed end (n < 3), and plotted the values in a graph for up to the 3rd reflection at the bottom plate (n < 3). In the graph the classic harmonic wave for a weightless spring vibration system was also plotted. Although Taylor mentioned that the complete curves in the graph could be worked out by a mathmatician, the derivation method and formulas were not given and the step by step general formulas such as those in equations (C14), (C15) and (C16) were not derived (energy loss “R” at the fixed ends was not considered). He probably felt that the unconventional stress wave superposition method could not be proven at that time due to energy loss in experiments and it could be rejected by the journal reviewers (2009 notes).



C.4 Stress Discontinuity due to Wave Reflections

As shown previously, at a specific location, the dynamic valley stress for the nth reflection occurs at exactly the same time as the dynamic peak stress for the (n+1)th reflection. Therefore, there is a stress discontinuity at each reflection. The difference between these two stress values can be derived from equations (C15) and (C16): 


	          Topp(n+1) - Topv(n) = RbRt[Topp(n) - Topv(n-1)] = ... 

                     = (RbRt)n[Topp(1) - Topv(0)] = 

                     = (RbRt)n(1 + Rt)sH(0) = 

                     = (RbRt)n (1 + Rt) so               

	                                                      (C18) 

	          Bottomp(n+1) - Bottomv(n) = Rt[Topp(n+1) - Topv(n)] = 

                     = Rt (RbRt)n (1 + Rt) so               

	                                                      (C19)
The stress discontinuity is generally a function of the reflection coefficients Rb and Rt, and the reflection number n. Since the coefficients are normally less than one, the difference diminishes with the elapse of time. However, if the top and the bottom ends of the steel rod are fixed (i.e. complete reflections, Rb = Rt = 1), the differences for both top and bottom locations are always 2so. The above analysis did neglect the minimal non elastic wave energy loss in steel rod (Yi 2015).



C.5 Input File and Computer Program (scanned pages)