J-jEAM OF IfECrANGE JjAlt CliOSS-SECTlON tTA^OElJ. ANY SYSTEM OF IjOAI). I 4,9 
If a be made to tend towards infinity, we get the expressions : 
/ 1 
U = ^ 
2&E ^ TT Jo 10 
1 'A'' + fjb 
cosh II 
— I cosh u — — u siiili i( 
/r / j HU ur 
•A o“~V o ■ - cosh -- cos , 
sinh 2 h. + 2 u ' 1 , ], 
da 
ttIj ]a /X sinh 2 i(. + 2 v 
1 
■ , ?h/ oi.r 
sinh - - cos , da 
0 0 
1 
, . , cosh 1C 4 —^ }i sinh u 
Y _ E7;y/ _ b j " 1 V + fJL IX 
2 hE TT Jo 1C 
sinh 2 ic 2 u 
sinli , sin - da 
L// p 1 cosh V. 
nrlj ]q IX sinli 2 u + 2 ic 
1 »// • j 
cosh - sin , du 
0 0 
T) _ E 2 L 2 cosh w—H sinli a a . ii.>- 
■c — -; -T^rw—yw-cosli ^ sill ' da 
2b ttoJo sinh 2 « + 2 i 6 I h 
2 L?/ i'” wcosliK . mi . h.j: 
- 70 • , o —tat Sinh A sin , c/u 
ttA Jo sinh 2i< + 2« h h 
Q. = - 
2L 
a sinh u 
lljl . H.r 
7 1 • 1 o , cosh , sin — du 
irb Jo sinh 2i( + 2a h h 
2Li/ 1” 
, i/coslm^- . U1J . ux , 
A— 7 " • 1 o—rA 7 “ siidi sin — die 
ttO-J o sinh 2/7 + 2(7 h h 
rj -^W'/ 1 '-■UfiH !' , ail il,r 
^ — A-'’ o—TTr cosh - cos V da 
TT/n Jo snih 27 / + 277 h h 
2 L// p 77 cosh 77 
'irlr j 0 s 
+ i J J 
2L cosh 77 — 77 sinh a . mi a.r 
‘ • 1 , J, siiili ^ cos — da 
0 sinh 2 ic + 2 ic h 1 
(iU8). 
§ 38. Correction to he Applied in this Case to the Stretch alony the Edges as loe 
apgjroach the Points of Application of the Load. 
One of tbe must interesting ^joints about a problem of this kind is to find out at 
what distance from the region of loading the stretch parallel to the axis takes tlie 
vahm it should have on the uniform tension hypothesis. In practice all measurements 
of \ouugs modulus for bars are made by observing the stretch between two points 
marked on the outer surface of tlie liar. It is of importance to know tlie error 
introduced as we bring these points closer to the places where the stress is applied. 
Let us therefore see how the stretch dU/dx varies as we go away from the points 
of application of the load, keeping upon either edge of the beam. If v^e differentiate 
the expression (108) for U with regard to x and then write ij — h ~ y' and transform 
the expression as in § 10 , we get easily 
