CIVIL ENGINEERING FORMULAS

L

Load

x(l – x)wL^2

x(l – 2x^2 + x^3 )

• xwL

Shear

Moment

Elastic curve

(a)

w

xL

R=

R

R

wL

2 R=

wL

2

wL^2

8

wL^3

24 EI

5 wL^4

384 EI

wL^4

24 EI

L

2

1

2

1

2

1

2

x<^12

cL

L

Load

a+

(2c + b)

(x′ < a) (x′′ < c)

Shear

Moment

(b)

b

w

ac

R=

R 2

R 1

wb

2 L

R 2 =wb 2 L(2c + b)

R 1

w

R^12 w

R 1 – w(x – a)

w 2

x

(a < x <a+b)

(x

• 
  

a)

2

• 
  

x′ x′′

R 1 x′ R^1

x R(^1

a +

R 2 x′′

R 1 a R 2 c

L

Load

k(1 – k)PL

k(1 – k)(2 – k)

Shear

Moment

Elastic curve

(c)

P

R 2

R 1

R 1 = (l – k)PR 2 =Pk

PL^2

6 EI

1

2

3

2

k(1 – k^2 )

1 – k^2

3

PL^2

6 EI

PL k (^2) (1 – k) 2
3
3 EI = k
dmax 1 – k^2
3
PL^3
3 EI
PLkx′′(1 – k (^2) – x′′ (^2) )
3
6 EI
PL(1 – k)x′(2k – k (^2) – x′ (^2) )
3
6 EI
k <
(l – k)L
x′L
(x′ < k)
R 1 x′L R 2 x′′L
L
x′′L
(x′′ < (1 – k))
kL
32
FIGURE 2.3 Elastic-curve equations for prismatic beams: (a) Shears, moments, and deflections for full uniform load on a simply sup-
ported prismatic beam. (b) Shears and moments for uniform load over part of a simply supported prismatic beam. (c) Shears, moments,
and deflections for a concentrated load at any point of a simply supported prismatic beam.