182. B = breadth in inches.

C = cohesive strength in lbs. per square inch, as in Tables II., III., IV., and XXVIII., or the crushing force, as in Tables XXI., XXII., and XXVIII.

D = depth in inches of rectangular, or = diameter of cylindrical beams.Synopsis Of The Formulae Most Useful For Estimatin 58 = deflection in inches.

L = length in feet.

S = area of cross section in square inches.

W = load in lbs.

a, c, e, etc, constant numbers to be found in the Tables.

Resistance To Tension

W = C S = the weight in lbs. that would tear asunder a beam the area of whose section is S.

* Rankine, 'Civil Engineering, 1869.'

Resistance To Cross Strains

Stiffness of Rectangular Beams supported at Both Ends and loaded in the Middle.

Resistance To Cross Strains 59

= aL3W/40 BD3, = deflection in inches.

WhenResistance To Cross Strains 60 is limited to 1/40 th of an inch to a foot we have

Resistance To Cross Strains 61

= depth in inches.

Resistance To Cross Strains 62

= ditto when the beam is inclined, c being the angle which it makes with the horizontal.

B = aL3W/D3 = breadth in inches. W =BD3/aL2 = weight in lbs. sustained by a beam without yielding more than 1/40 inch per foot.

W = BD3/aL2cos.c------- = ditto when the beam is inclined.

When W is uniformly distributed the deflection is only 5/8 ths of that caused by a central load. (Art. 111.)

When the beam is fixed at one end and loaded at the other, the deflection is sixteen times greater than when the beam is merely supported at the ends. (Art. 112.)

Strength of Rectangular Beams supported at Both Ends and loaded in the Middle.

W = cBD2/L = breaking weight in lbs.

W = cBD2/Lcos.c= ditto when the beam is inclined.

Calling W the load in the middle as in the two last formulae, we have for the breaking weight of- -

Beams fixed at both ends and loaded in the middle

W

X

1 1/2

Ditto fixed at one end and loaded at the other

W

X

1/4

Ditto supported at both ends and the load uniformly distributed............

w

X

2

Ditto ditto loaded at any point, m and n representing the segments into which the beam is divided by the load............

w

X

L

4 m n

The strength and stiffness of a cylindrical beam are to those of a square one as 10 is to 17. (Arts. 108 and 122.)

Resistance to Compression. Stiffness of Beams or Pillars above 30 diameters in lengtn.

w = D4e/L2= weight in lbs. for square pillars to resist flexure.

w = BT3e/L2 = ditto for rectangular pillars. w = D4e/1.7 L2= ditto for cylindrical pillars T being the least thickness in inches.

Strength of Beams or Pillars less than 30 diameters in length.

W = CS/1.1+L2/2.9Te = breaking weight in lbs.

The strength of cylindrical beams or pillars is to square ones as 10 is to 17.

Table XXVIII. - A Selection Of Constant Numbers For The Strength And Stiffness Of Beams And Pillars

Name of Timber.

Cohesive

Force per sq. inch in lbs.

Transverse Strains.

Compression.

a

c

e

c

Stiffness.

Strength.

Flexure.

Crushing per sq. in.

in lbs.

Ash ..........

16,800

.0105

675

1840

8,683

Beech....................................

11,500

.0128

519

1587

7,733

Elm ..........

14,400

.0212

338

1620

8,265

Fir, Riga...............................

12,600

.0114

359

2035

5,400

" Memel...........................

. .

.0089

515

2361

Larch....................................

8,900

.0126

300

1645

3,201

Mahogany............................

8,000

.0109

450

1921

8,198

Oak, English.......................

12,000

.0119

557

2068

6,484

" Dantzic.......................

.0105

486

2410

6,185

" Canada.......................

.009

589

..

4,231

Pine, American Red...........

10,000

.0148

447

2219

5,395

" " Yellow .....

.019

383

1930

5,375

Teak....................................

15,000

.0076

821

2614

10,081

The constants in the above Table are based on experiments made by the most eminent authorities. The specimens used by them were, however, of small scantling and of a quality superior to that which would be found throughout the whole substance of a large beam. Among the experiments made for the Britannia Bridge were two balks of American red pine selected from the scaffolding intended for the bridge. Each balk was 12 inches square and 15 feet long between the supports. The breaking weight in the middle, as deduced from experiments on small pieces, was 23 tons nearly, yet one of the balks broke with 13.24 tons, and the other with 14.82 tons.

A comparison of Hodgkinson's and Kirkaldy's experiments on pillars of wood will show a similar discrepancy.

It would therefore appear that the application of rules and general formulae to the designs of the carpenter requires considerable judgment and practical knowledge.

Form to be given to Beams exposed to a Transverse Strain, so that they may be of Uniform Strength.