CIVIL ENGINEERING FORMULAS

38 CHAPTER TWO

Shear

Moment

(p)

R=V =P

=Pl

=Px

=

Mmax(at fixed end)

∆max(at free end)

∆x

Mx

R

P

V

x

l

Mmax

Pl^3

3 EI

= P

6 EI

(2l^3 – 3l^2 x + x^3 )

Shear

Moment

(q)

R=V

Mmax(at center and ends)

∆max(at center)

∆x

R

P

V

x

l

V

R

Mmax

Mmax Mmax

=P

2

=(4P x – l)

8

=

(3l – 4x)

Pl^3

192 EI

= Px

2

48 EI

=Pl

8

l

4

l

2

l

2

Mxwhen x <l

2

FIGURE 2.3 Elastic-curve equations for prismatic beams: (p) Cantilever beam—con-

centrated load at free end. (Continued)

FIGURE 2.3 Elastic-curve equations for prismatic beams: (q) Beam fixed at both

ends—concentrated load at center. (Continued)