The document presents calculations to determine the displacement of a simply supported beam subjected to a concentrated load at its mid-span. It first uses the direct stiffness method to derive equations for the beam's displacement (v2) and internal force (f1) in terms of the applied load (P), length of the beam (L), Young's modulus (E), shear modulus (G), moment of inertia (I), and shear correction factor (j). It then substitutes numerical values into the equations to calculate the displacement. It also presents the classical beam deflection formula which neglects the shear correction term and calculates the displacement using this formula, finding a 4.97% difference between the two results.

1. Note that the boundary conditions given by v1 ¼ 0 and f2 ¼ 0 have been included in
Eq. (4.3.34).
Using the second and third equations of Eq. (4.3.34) whose rows are associated
with the two unknowns, f1 and v2, we obtain
v2 ¼
PL3
ð4 þ jÞ
24EI
and f1 ¼
PL2
4EI
ð4:3:35Þ
As the beam is rectangular in cross section, the moment of inertia is
I ¼ bh3
=12
Substituting the numerical values for b and h, we obtain I as
I ¼ 0:26 106
m4
The shear correction factor is given by
j ¼
12EI
ksAGL2
and ks for a rectangular cross section is given by ks ¼ 5=6.
Substituting numerical values for E; I; G; L; and ks, we obtain
j ¼
12 207 109
0:26 106
5=6 0:025 0:05 80 109 0:22
¼ 0:1938
Substituting for P ¼ 10;000 N, L ¼ 0:2 m, and j ¼ 0:1938 into Eq. (4.3.35), we
obtain the displacement at the mid-span as
v2 ¼ 2:597 104
m ð4:3:36Þ
If we let l ¼ the whole length of the beam, then l ¼ 2L and we can substitute L ¼ l=2
into Eq. (4.3.35) to obtain the displacement in terms of the whole length of the beam as
v2 ¼
Pl3
ð4 þ jÞ
192EI
ð4:3:37Þ
For long slender beams with l about 10 or more times the beam depth, h, the transverse
shear correction term j is small and can be neglected. Therefore, Eq. (4.3.37) becomes
v2 ¼
Pl3
48EI
ð4:3:38Þ
Equation (4.3.38) is the classical beam deflection formula for a simply supported beam
subjected to a concentrated load at mid-span.
Using Eq. (4.3.38), the deflection is obtained as
v2 ¼ 2:474 104
m ð4:3:39Þ
Comparing the deflections obtained using the shear-correction factor with the deflec-
tion predicted using the beam-bending contribution only, we obtain
% change ¼
2:597 2:474
2:474
100 ¼ 4:97% difference
9
4.3 Examples of Beam Analysis Using the Direct Stiffness Method d 191
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