Research Journal of Applied Sciences, Engineering and Technology 4(22): 4830-4834, 2012
ISSN: 2040-7467
© Maxwell Scientific Organization, 2012
Submitted: May 04, 2012
Accepted: June 08, 2012
Published: November 15, 2012
Relative Error of the Mechanics of Material Solution on Simply Supported
Beam under Uniform Load
Wen-jie Niu
College of Mechanics and Engineering Department, Liaoning Technical University,
Fuxin 123000, Liaoning Province, China
Abstract: The mechanics of material solution is a coarse analytical solution on the problem of simply
supported beam under uniform load. This paper intends to determine the relative error of the mechanics of
material solution. Solutions according to plane stress theory of elasticity theory are believed to be the true
solution. Results indicate that material mechanics solution σx seems applicable only when the ratio of beam
height to beam length is less than 0.1 and the ratio of distance between beam studied cross section and
midpoint in x direction to beam length is less than 0.49. When the ratio of beam height to beam length is
less than 0.1 and the ratio of distance between beam studied cross section and midpoint in x direction to
beam length is less than 0.49, maximum value of relative error for material mechanics solution σx is no
more than 12%. Material mechanics solution τxy is always correct for the problem of simply supported
beam under uniform load. Material mechanics solution σy is not applicable for the problem of simply
supported beam under uniform load.
Keywords: Beam cross section coordinate in x direction, material mechanics solution, plane stress theory
of theory of elasticity, ratio of beam height to beam length, relative error
INTRODUCTION
Not like suspension bridges (Niu, 2011; Niu and
Wang, 2011), beam bridges are the most simple of
structural forms being supported by an abutment at each
end of the deck. No moments are transferred throughout
the support hence their structural type is known as
simply supported. Beam bridges are often only used for
relatively short distances because, unlike truss bridges,
they have no built in supports. The only supports are
provided by piers (Wikipedia, 2012).
Material mechanics solution is originally for pure
bending beam. For the beam bending problem where
there is shear stress and compressive stress among the
beam longitude layers, material mechanics solution is
not accurate (Wang, 2008). However when the ratio of
beam height to beam length is less than 0.2, the material
mechanics solution is applicable to compute the normal
stress of beam (Wang, 2008).
Since theory of elasticity is an accurate solution
(Xia et al., 1997), this paper intends to discuss how
much accuracy the material mechanics solution can get.
A common problem-simply supported beam under
uniform load: In Fig. 1, it is a simply supported beam
under uniform load of intensity q. The beam length is L
with its hight as h and width as b. The coordinate
origin o is at the beam geometric center.
Fig. 1: Simply supported beam under uniform load5
Solutions according to mechanics of material:
According to mechanics of material (Wang, 2008), the
stress distribution in the beam of Fig. 1 is:

q  L2 2 
 x    x  y
2J  4



 y  0

2
   qx  h  y2 
xy


2J  4


(1)
where, σx is the normal stress between the crosssections of the beam, σy is the compressive normal
stress between the beam horizontal layers, τxy is the
shear stress acted on the beam cross-section:
4830
Res. J. Appl. Sci. Eng. Technol., 4(22): 4830-4834, 2012
J 
bh 3 h 3

12 12
(2)

 
y
h
h
L
Substituting (3) and (4) into (5) yields:
C xy  0
(11)
Substituting (6), (7) and (8) into (9) yields:
C x
    x e
 x m

 x e
3
5
6  1 2   2 3
      4  
5
2 4
 
4 2 
RESULTS
Relative error of
y :
Considering Eq. (10), the
relative error Cσy of material mechanics solution
 y is
always extremely large. Material mechanics solution
y
is not applicable for the problem in Fig. 1.
Relative error of
 xy :
Considering Eq. (11), the
relative error C xy of material mechanics solution
(5)
is always 0. Material mechanics solution
 xy
 xy
is always
correct for the problem in Fig. 1.
Relative error of
Define
x
L
(10)
(4)
Define relative error as:

C y  1
(12)
EVALUATION METHOD

    x e
C  x  x m
 x e


 y m   y e

C  y 
 y e


    xy e
C  xy m
xy

 xy e
(9)
(3)
Solutions according to plane stress theory of
elasticity theory: Since Eq. (3) according to material
mechanics is just an approximate solution, not the
correct solution, correct solution (4) according to plane
stress theory of theory of elasticity is proposed as below
(Xia et al., 1997):

y  y2 3 
6q  L2
2
 x   x e  3   x  y  q  4 2  
h 4
h  h 5


2

q  y  2 y 



1
1










 y
y e
2  h 
h 



6q  h2
 xy   xy e   3 x  y 2 
h 4


 x m   x e
 x e
y  y2 3 
6q  L2 2  6q  L2 2 
  x  y  3   x  y  q  4 2  
3 
h 4
h  h 5

 h 4
 
2
2



y y 3
6q L
  x2  y  q  4 2  
3 
h 4
h  h 5

J is the area moment of inertia.
Substitute (2) into (3) yields:


6q  L2
 x   x m  3   x 2  y
h
4










0
 y
y m

2
      6qx  h  y 2 
xy
xy
3


m

h  4


C x 
(6)
(7)
 x : Fig. 2, 3, 4, 5, 6 and 7 presents
the relative error Cσx distribution. x/L means the cross
section of the studied beam in Fig. 1. y/h means the
vertical position of the studied point on the cross
section.
From the results of Fig. 2, 3, 4, 5, 6 and 7, one can
see that when h/L increases, the relative error of
material mechanics solution  x increases.
For the results in Fig. 2, 3 and 4 where h/L<0.5,
when x/L increases, Cσx increases. It means that the
(8)
material mechanics solution σx acted on the cross
sections near the beam support deviates more than the
material mechanics solution σx acted on the cross
sections near the midpoint of the beam (x = 0 in Fig. 1).
4831 Res. J. Appl. Sci. Eng. Technol., 4(22): 4830-4834, 2012
Fig. 2: Stress
x
distribution in x and y direction in Fig. 1 when
h
 0.01
L
Fig. 3: Stress
x
distribution in x and y direction in Fig. 1 when
h
 0.1
L
For the results in Fig. 2, 3 and 4, material
mechanics solution  x seems applicable whe h  0.1
When h  1.0 , the relative error of material
L
mechanics solution  x is extremely large. It can be
L
and x  0.49 . When h  0.1 and
L
L
x
,
 0.49
L
maximum
more than 30 times larger than the true solution
value.
value of Cσx is no more than 12%.
4832 Res. J. Appl. Sci. Eng. Technol., 4(22): 4830-4834, 2012
Fig. 4: Stress
x
distribution in x and y direction in Fig. 1 when h  0.5
L
x/L = 0
x/L = 0.1
x/L = 0.2
C
x/L = 0.3
x/L = 0.4
x/L = 0.45
x/L = 0
x/L = 0.1
x/L = 0.2
C
x
30
8
25
x/L = 0.3
x/L = 0.4
x/L = 0.45
x
6
20
15
4
h/L = 1.0
10
5
-0.6
-0.4
-0.2
x
Fig. 5: Stress
2
y/h
0
0.0
-5
0.2
0.4
h/L = 2.0
0.6
-0.6
-10
-0.4
-0.2
0
0.0
y/h
0.2
0.4
0.6
-2
distribution in x and y direction in Fig. 1
when h
 1.0
-4
L
x/L = 0
x/L = 0.1
x/L = 0.2
C
x/L = 0.3
x/L = 0.4
x/L = 0.45
Fig. 7: Stress
x
distribution in x and y direction in Fig. 1
when h  2.0
L
x
40
CONCLUSION
30
20
10
-0.6
-0.4
-0.2
0
0.0
y/h
0.2
0.4
0.6
-10
Fig. 6: Stress
x

Material mechanics solution
y
is not applicable

for the problem in Fig.1.
Material mechanics solution
 xy
is always correct
h/L = 1.5

-20
distribution in x and y direction in Fig. 1
when h  1.5
L
for the problem in Fig.1.
Material mechanics solution σx seems applicable
only when h  0.1 and x  0.49 . When h  0.1
L
L
L
x
and  0.49 , maximum value of relative error Cσx
L
is no more than 12%.
4833 Res. J. Appl. Sci. Eng. Technol., 4(22): 4830-4834, 2012
ACKNOWLEDGMENT
The author is supported by the Scientific Research
Starting Funds at Liaoning Technical University (No.
11-415). The financial help is greatly appreciated.
REFERENCES
Niu, W.J., 2011. Wire suspension bridge analyzed
considering suspenders deformation. Commun. Inf.
Sci. Manage. Eng., 1(3): 39-41.
Niu, W.J. and Z.Y. Wang, 2011. Application of
flexibility method to a chain suspension stiffening
beam bridge. Adv. Mat. Res., 255-260: 1220-1224.
Wang, S., 2008. Terial Mechanics. Xi'an, Northwestern
Polytechnical University Press, China.
Wikipedia, 2012. Beam Bridge. Retrieved from: http:/
/en.wikipedia. org/wiki/ Beam-bridge.
Xia, Z.G., L.P. Jiang and S.G. Tang, 1997. Eory of
Elasticity and Numerical Solutions. Shanghai
Tongji University Press, China.
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