Load Discontinuity in Shock Response and Its Remedies
It has been shown that a load discontinuity at the end of
an impulse will result in an extra impulse and hence an extra amplitude
distortion if a step-by-step integration method is employed to yield the
shock response. In order to overcome this difficulty, three remedies
are proposed to reduce the extra amplitude distortion. The first remedy
is to solve the momentum equation of motion instead of the force
equation of motion in the step-by-step solution of the shock response,
where an external momentum is used in the solution of the momentum
equation of motion. Since the external momentum is a resultant of the
time integration of external force, the problem of load discontinuity
will automatically disappear. The second remedy is to perform a single
small time step immediately upon termination of the applied impulse
while the other time steps can still be conducted by using the time step
determined from general considerations. This is because that the extra
impulse caused by a load discontinuity at the end of an impulse is
almost linearly proportional to the step size. Finally, the third remedy
is to use the average value of the two different values at the integration
point of the load discontinuity to replace the use of one of them for
loading input. The basic motivation of this remedy originates from the
concept of no loading input error associated with the integration point
of load discontinuity. The feasibility of the three remedies are
analytically explained and numerically illustrated.
[1] T. Belytschko, and T.J.R. Hughes, Computational methods for transient
analysis, Elsevier Science Publishers B.V., North-Holland, 1983.
[2] R.W. Clough, and J. Penzien, Dynamics of structures, McGraw-Hill, Inc.,
International Editions, 1993.
[3] A.N. Chopra, Dynamics of structures, Prentice Hall, Inc., International
Editions, 1997.
[4] N.M. Newmark, "A method of computation for structural dynamics,"
Journal of Engineering Mechanics Division, ASCE, vol. 85, pp. 67-94,
1959.
[5] S.Y. Chang, "A series of energy conserving algorithms for structural
dynamics," Journal of Chinese Institute of Engineers, vol. 19, no. 2, pp.
219-230, 1996.
[6] S.Y. Chang, "Improved numerical dissipation for explicit methods in
pseudodynamic Tests," Earthquake Engineering and Structural
Dynamics, vol. 26, 917-929, 1997.
[7] S.Y. Chang, "Analytical study of the superiority of the momentum
equations of motion for impulsive loads." Computers & Structures, Vol.
79, no.15, pp.1377-1394, 2001.
[8] S.Y. Chang, "Application of the momentum equations of motion to
pseudodynamic testing." Philosophical Transactions of the Royal Society,
Series A, vol. 359, no.1786, pp. 1801-1827, 2001.
[9] S.Y. Chang, "Explicit pseudodynamic algorithm with unconditional
stability." Journal of Engineering Mechanics, ASCE, vol. 128, no. 9, pp.
935-947, 2002.
[10] S.Y. Chang, "Improved explicit method for structural dynamics," Journal
of Engineering Mechanics, ASCE, vol. 133 no. 7, pp. 748-760, 2007.
[11] S.Y. Chang, "An explicit method with improved stability property,"
International Journal for Numerical Method in Engineering, vol. 77, no
8, pp. 1100-1120, 2009.
[12] S.Y. Chang, "A new family of explicit method for linear structural
dynamics," Computers & Structures, vol. 88, no.11-12, pp. 755-772,
2010.
[13] H.M. Hilber, T.J.R. Hughes, and R.L. Taylor, "Improved numerical
dissipation for time integration algorithms in structural dynamics,"
Earthquake Engineering and Structural Dynamics, vol. 5, pp. 283-292,
1977.
[14] H.M. Hilber, and T.J.R. Hughes, "Collocation, dissipation, and
ÔÇÿovershoot- for time integration schemes in structural dynamics,"
Earthquake Engineering and Structural Dynamics, vol. 6, pp. 99-118,
1978.
[15] J.C. Houbolt, "A recurrence matrix solution for the dynamic response of
elastic aircraft." Journal of the Aeronautical Sciences, vol. 17, pp.
540-550, 1950.
[16] K.J. Bathe, and E.L. Wilson, "Stability and accuracy analysis of direct
integration methods." Earthquake Engineering and Structural Dynamics,
vol. 1, pp. 283-291, 1973.
[17] K.K. Tamma, X. Zhou, and D. Sha, "A theory of development and design
of generalized integration operators for computational structural
dynamics," International Journal for Numerical Methods in Engineering,
vol. 50, pp. 1619-1664, 2001.
[18] S.Y. Chang, "Accuracy of time history analysis of impulses," Journal of
Structural Engineering, ASCE, vol. 129, no.3, pp. 357-372, 2003.
[1] T. Belytschko, and T.J.R. Hughes, Computational methods for transient
analysis, Elsevier Science Publishers B.V., North-Holland, 1983.
[2] R.W. Clough, and J. Penzien, Dynamics of structures, McGraw-Hill, Inc.,
International Editions, 1993.
[3] A.N. Chopra, Dynamics of structures, Prentice Hall, Inc., International
Editions, 1997.
[4] N.M. Newmark, "A method of computation for structural dynamics,"
Journal of Engineering Mechanics Division, ASCE, vol. 85, pp. 67-94,
1959.
[5] S.Y. Chang, "A series of energy conserving algorithms for structural
dynamics," Journal of Chinese Institute of Engineers, vol. 19, no. 2, pp.
219-230, 1996.
[6] S.Y. Chang, "Improved numerical dissipation for explicit methods in
pseudodynamic Tests," Earthquake Engineering and Structural
Dynamics, vol. 26, 917-929, 1997.
[7] S.Y. Chang, "Analytical study of the superiority of the momentum
equations of motion for impulsive loads." Computers & Structures, Vol.
79, no.15, pp.1377-1394, 2001.
[8] S.Y. Chang, "Application of the momentum equations of motion to
pseudodynamic testing." Philosophical Transactions of the Royal Society,
Series A, vol. 359, no.1786, pp. 1801-1827, 2001.
[9] S.Y. Chang, "Explicit pseudodynamic algorithm with unconditional
stability." Journal of Engineering Mechanics, ASCE, vol. 128, no. 9, pp.
935-947, 2002.
[10] S.Y. Chang, "Improved explicit method for structural dynamics," Journal
of Engineering Mechanics, ASCE, vol. 133 no. 7, pp. 748-760, 2007.
[11] S.Y. Chang, "An explicit method with improved stability property,"
International Journal for Numerical Method in Engineering, vol. 77, no
8, pp. 1100-1120, 2009.
[12] S.Y. Chang, "A new family of explicit method for linear structural
dynamics," Computers & Structures, vol. 88, no.11-12, pp. 755-772,
2010.
[13] H.M. Hilber, T.J.R. Hughes, and R.L. Taylor, "Improved numerical
dissipation for time integration algorithms in structural dynamics,"
Earthquake Engineering and Structural Dynamics, vol. 5, pp. 283-292,
1977.
[14] H.M. Hilber, and T.J.R. Hughes, "Collocation, dissipation, and
ÔÇÿovershoot- for time integration schemes in structural dynamics,"
Earthquake Engineering and Structural Dynamics, vol. 6, pp. 99-118,
1978.
[15] J.C. Houbolt, "A recurrence matrix solution for the dynamic response of
elastic aircraft." Journal of the Aeronautical Sciences, vol. 17, pp.
540-550, 1950.
[16] K.J. Bathe, and E.L. Wilson, "Stability and accuracy analysis of direct
integration methods." Earthquake Engineering and Structural Dynamics,
vol. 1, pp. 283-291, 1973.
[17] K.K. Tamma, X. Zhou, and D. Sha, "A theory of development and design
of generalized integration operators for computational structural
dynamics," International Journal for Numerical Methods in Engineering,
vol. 50, pp. 1619-1664, 2001.
[18] S.Y. Chang, "Accuracy of time history analysis of impulses," Journal of
Structural Engineering, ASCE, vol. 129, no.3, pp. 357-372, 2003.
@article{"International Journal of Architectural, Civil and Construction Sciences:62921", author = "Shuenn-Yih Chang and Chiu-Li Huang", title = "Load Discontinuity in Shock Response and Its Remedies", abstract = "It has been shown that a load discontinuity at the end of
an impulse will result in an extra impulse and hence an extra amplitude
distortion if a step-by-step integration method is employed to yield the
shock response. In order to overcome this difficulty, three remedies
are proposed to reduce the extra amplitude distortion. The first remedy
is to solve the momentum equation of motion instead of the force
equation of motion in the step-by-step solution of the shock response,
where an external momentum is used in the solution of the momentum
equation of motion. Since the external momentum is a resultant of the
time integration of external force, the problem of load discontinuity
will automatically disappear. The second remedy is to perform a single
small time step immediately upon termination of the applied impulse
while the other time steps can still be conducted by using the time step
determined from general considerations. This is because that the extra
impulse caused by a load discontinuity at the end of an impulse is
almost linearly proportional to the step size. Finally, the third remedy
is to use the average value of the two different values at the integration
point of the load discontinuity to replace the use of one of them for
loading input. The basic motivation of this remedy originates from the
concept of no loading input error associated with the integration point
of load discontinuity. The feasibility of the three remedies are
analytically explained and numerically illustrated.", keywords = "Dynamic analysis, load discontinuity, shock response,step-by-step integration", volume = "5", number = "7", pages = "310-6", }