CIVIL ENGINEERING FORMULAS

P

P

Load

Shear

Moment

Elastic curve

(g)

R 1 =L′P

L

L′

xPL′

xL x′L′

Px′L′

PL′

L

R 2 = P

R 2

R 1

L + L′

L

2(L + L′) – L′x′(1 + x′)

L

3

PL′(L + L′)

2

PL′Lx(1 – x (^2) ) 3 EI
2
6 EI
PL′^2 (1 – x′)
6 EI
dmax =PL′L
2
9 3 EI
P
PL
Load
Shear
Moment
Elastic curve
(h)
R = P
L P
PL
xL
PxL
R
PL (2 – 3x + x (^3) )
3
3 EI
PL^3
3 EI
LL′
Load
(L^2 – L′^2 )
xL
wL′x′
wL′
Shear
Moment
Elastic curve
(i)
w w
R 1 =
R 1
R 2
w
2 L
R 1 =wLx
R 2 =w
2 L
w(L (^2) – L′ (^2) )
2 L
w(L + L′) (^2) (L – L′) 2
8 L^2
wL^2 x
24 EI
wL′
24 EI
wL′^2
2 x′
wL′^22
2
L
2 1 –
(L^2 – L′^2 – xL^2 )
L^2 (1 – 2x^2 +x^2 ) –2L′^2 (1 – x^2 )
(4L′^2 – L^3 + 3L′^3 )
wx
2
L′^2
L^2
L1 –
(L + L′)^2
x′L′
L′^2
L^2
34
FIGURE 2.3 Elastic-curve equations for prismatic beams: (g) Shears, moments, and deflections for a concentrated load on a beam overhang. (h) Shears,
moments, and deflections for a concentrated load on the end of a prismatic cantilever. (i) Shears, moments, and deflections for a uniform load over the full
length of a beam with overhang. (Continued)