CIVIL ENGINEERING FORMULAS

33

P

P

P P

L

Load

(1 – 2k)L

PxL

x(3k – 3k^2 – x^2 )

k(3x′ – 3x′^2 – k^2 )

k(3 – 4k^2 )

Shear

Moment

Elastic curve

(e)

xL

(x < k)

PLk(1 – k)

2

2 EI

PL^3

6 EI

PL^3

24 EI

PkL

PL^3

6 EI

L

2

L′′

2

kL kL

R = P R = P

x′

(k < x′ < (1 – k))

P

P

R P P

P

P

P PPP

aL aL aL aL aL aL

(Forn an

even number)

(Forn an even number)

Load

maL

Shear

Moment

Elastic curve

(f)

L

R

W=np

(n + 1)(For n an

odd number)

PL

8

n(n + 2)

n + 1

PL

8

L

2

L

2

L

2

L

2

R = nP

a = 1

2 R = nP

1

2

1

n + 1

n(n + 2)

n + 1

PL^2

24 EI

PL^3 n(n + 2)

384 EI

(5n^2 + 10n + 6)

(n + 1)^3

PL (Forn an odd number)

3

384 EI

5 n^2 + 10n + 1

n + l

m(n – m + 1)

n + 1

PL

2

cL

P/2

P/2

P

L

Load

xPL

x(3 – 4x^2 )

Shear

Moment

Elastic curve

(d)

xL

PL^2

16 EI

PL^3

48 EI

PL

4

PL^3

48 EI

1

2

L

2

L

2

R =^12 P R =^12 P

x<^1

2

cL

FIGURE 2.3 Elastic-curve equations for prismatic beams: (d) Shears, moments, and deflections for a concentrated load at midspan of a simply supported

prismatic beam. (e) Shears, moments, and deflections for two equal concentrated loads on a simply supported prismatic beam. (f) Shears, moments, and deflec-

tions for several equal loads equally spaced on a simply supported prismatic beam. (Continued)