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75 
APPENDIX E 
flection of a uiform elastic beam on spring supports 
Ei. Fig.Ei represents a uniform 
elastic beam of length 2a supported 
On spring supvorts and subjected to 
a uniformly distributed load w 
suddenly applied per wit length of 
the beam. Under these conditions, 
the deflection of any section of the 
beam distant x from the centre at 
time t is taken as y, and the 
deflection of the ends of the beam 
at this time is denoted by—. The 
deflection of the centre of the beam 
relative to the ends of the beam is 
represented by 2 while the mass per 
unit length of the beam is denoted by 
Me 
Fig.E1. Deflection of uniform elastio 
beam on spring supports 
First, the beam is assumed to deflect in the constant shape 
y =f + Q cos L* ere eoe coo coo eve coe eee eee (£1) 
2a 
Equation Ei satisfies the necessary conditions that 
y= 
2 
Bending moment EI + =o 
at ends when x = + a See Cte Coe eee) 
The problem is now reduced to finding 5 and 7 as functions of time. The 
kinetic energy of the beam Q[ is 
(gee | = Qbre.. Telesis weraet' dacs ce ee 
"a 
Whilst the total potential erergy H of the system is 
need od | mQia sisint iy aioe!’ ¢-e'soeegylie’stoh MURteiatna wiatstemn RIP) 
-a 
where the first term represents the contribution of the two end springs and 
the second term is the elastic energy of bending of the bean. 
Thirdly, the 
virtual work of the applied load w due to any small variation $ y in the 
deflection is 
sr= | BiOTIRG elk eee, dS aka lp os ee ene 
-a 
