112. When a beam is fixed at one end and loaded at the other the deflection is sixteen times greater than when merely supported at both ends and loaded at the middle; but the deflection is much modified by fixing the beam as in Fig. 43; it increases with the distance AC, because the parts between A and G will be extended, and of course increase the deflection. And as it is impossible to fix a bar so that it will not be extended beyond the point of support, there is much irregularity in the results of experiments on beams fixed in this manner.

The forms of the equations [2], [3], [4], etc, remain the same, only a different constant quantity b must be used, which, if the beam be perfectly fixed at one end, is sixteen times greater than the constant a. It is, however, obtained directly from the following Table of Experiments.

Fig. 43.

Of The Stiffness Of Beams Supported At One End 53

* The demonstration is given in the ' Encyclopaedia Britannica,' art. Carpentry, and in Barlow's ' Essay on the Strength of Timber,' and also is evident from arts. 326 and 330 of Dr. Young's ' Natural Philosophy,' Vol.ii.

113. Table XII. - Experiments On The Stiffness Of Beams Fixed At One End

Kind of Wood.

Spec, grav.

Length in feet.

Breadth in inches.

Depth in inches.

Deflection in inches.

Weight producing the deflection in lbs.

Values of constant quantity b.

Authorities.

Dantzic oak ......

.854

4

2

2

2.5

112

.223

Beaufoy.

English oak .......

.922

4

2

2

1.176

112

.105

"

Ditto, another specimen ..

..

4

2

2

1.5

112

.1335

"

Riga fir...............

.537

4

2

2

1.34

112

.12

"

Pitch pine............

..

4

2

2

112

112

.099

" *

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

..

3

2

2

3 375

221

.313

Barlow.

114. Table XIII. - Experiments On Beams Fixed At One End. (Fincham.)

Length = 2 feet beyond Support. Scantling = 3 in. x 3 in.

Description of Timber.

Specific gravity.

Deflection.

Weight that broke the piece.

Values of b.

Remarks.

With 560 lbs.

With 1120 lbs.

With 1400 lbs.

With 1680 lbs.

in.

ins.

ins.

ins.

lbs.

All the specimens were dry.

Riga fir

Top

•605

•52

1.02

2.07

3.10

1848

•3761

,,

Butt

•668

•40

•80

1.50

2.87

2100

•2893

,,

,,

•821

•37

1.00

1.37

1.62

1820

•2676

Red pine

Top

•544

•63

1.42

2.68

..

1630

•4556

,,

Butt

•634

•60

1.07

1.95

..

1877

•4339

American spruce

Top

•504

•56

1.32

213

..

1518

•4050

Butt

•570

•50

•90

1.67

..

1786

•3616

Norway fir

Top

•464

•55

1.04

..

..

1370

•3978

,,

Butt

•506

•62

1.35

1.97

3.00

1860

•4484

Adriatic

Top

•467

•50

1.00

2.00

..

1454

•3616

,,

Butt

•493

•40

•70

1.40

..

1716

•2893

Yellow pine

Top

•406

•62

2.00

..

..

1260

•4484

,,

Butt

•493

•63

2.12

..

..

1390

•4556

Scotch spruce

Top

•389

•58

.,

1108

•4195

Butt

•440

•54

2.00

..

..

1230

•3906

Kowrie

Top

•626

•37

•75

1.12

1.62

1960

•2676

,,

Butt

•632

•50

•87

1.25

1.87

2100

•3616

Poona Top, Outside

•654

•46

•62

•90

1.40

2086

•3327

,,

Heart

•608

•62

..

..

..

1110

•4484

Poona Butt, Outside

•666

•37

•75

1.00

1.25

2338

•2676

,,

Heart

•646

•57

.80

..

..

1228

•4122

* In Colonel Beaufoy's experiments the mean result has been taken of the most regular of his experiments, and the deflection converted into inches. His experiments are published in Thomson's ' Annals of Philosophy,' vol ix., p. 274.

115. As beams fixed at one end are not frequently used in the construction of buildings, it will be unnecessary to repeat the rules in words at length.

"When the beam is fixed in a horizontal position,

(Lx(bxW/0.6)1/2)1/2=D, [11]

And 0.6D = B.

When the beam is inclined, and c is the angle of inclination,

(Lx(bxcos.cxW)1/2)1/2.=D [12]

And 0.6D=B.

When a solid cylinder is fixed at one end,

(Lx(l.7bxW1/2)1/2 = D. [13]

Where the value of b may be obtained from the preceding Table, and the deflection is calculated to be 1/40 of an inch per foot in length.

When the Weight is uniformly distributed over the Length of Beams Fixed at One End.

116. When the weight W is uniformly diffused over the length of the beam, the deflection is to the deflection of a beam loaded in the middle as .75:2; or as .375:1. Therefore, instead of W in the above equation substitute .375 W, and the scantlings for beams uniformly loaded may be obtained.