Transcript 9 - unisa.it

Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Sample Problem 9.8
For the uniform beam and loading shown,
determine the reaction at each support and
the slope at end A.
SOLUTION:
• Release the “redundant” support at B, and find deformation.
• Apply reaction at B as an unknown load to force zero displacement at B.
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
9-1
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Sample Problem 9.8
• Distributed Loading:
 yB w  

w
x 4  2 Lx3  L3 x
24 EI

At point B, x  23 L
 yB w
4
3

w  2 
2 
3 2 

 L   2 L L   L  L 
24 EI  3 
3 
 3 
wL4
 0.01132
EI
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
9-2
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Sample Problem 9.8
• Redundant Reaction Loading:
At x  a,
Pa 2b 2
y
3EIL
For a  23 L and b  13 L
 yB R
2
R 2   L
 B  L  
3EIL  3   3 
2
RB L3
 0.01646
EI
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
9-3
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Sample Problem 9.8
• For compatibility with original supports, yB = 0
wL4
RB L3
0   yB w   yB R  0.01132
 0.01646
EI
EI
RB  0.688wL 
• From statics,
RA  0.271wL 
RC  0.0413wL 
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
9-4
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Sample Problem 9.8
Slope at end A,
wL3
wL3
 A w  
 0.04167
24 EI
EI
 A R


2
Pb L2  b2 0.0688wL  L  2  L  
wL3


  L      0.03398
6 EIL
6 EIL  3 
EI
 3  
wL3
wL3
 A   A w   A R  0.04167
 0.03398
EI
EI
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
wL3
 A  0.00769
EI
9-5
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Moment-Area Theorems
• Geometric properties of the elastic curve can
be used to determine deflection and slope.
• Consider a beam subjected to arbitrary loading,
d d 2 y M


dx dx 2 EI
D
xD
C
xC
 d 

 D  C 
M
dx
EI
xD

xC
M
dx
EI
• First Moment-Area Theorem:
 D C  area under (M/EI) diagram between
C and D.
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
9-6
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Moment-Area Theorems
• Tangents to the elastic curve at P and P’ intercept a
segment of length dt on the vertical through C.
dt  x1d  x1
tC D 
xD

xC
x1
M
dx
EI
M
dx = tangential deviation of C
EI
with respect to D
• Second Moment-Area Theorem:
The tangential deviation of C with respect to D
is equal to the first moment with respect to a
vertical axis through C of the area under the
(M/EI) diagram between C and D.
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
9-7
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Application to Cantilever Beams and Beams With
Symmetric Loadings
• Cantilever beam - Select tangent at A as the
reference.
with θ A  0,
D  D
A
yD  t D A
• Simply supported, symmetrically loaded
beam - select tangent at C as the reference.
with θC  0,
B  B C
yB  t B C
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
9-8
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Bending Moment Diagrams by Parts
• Determination of the change of slope and the
tangential deviation is simplified if the effect of
each load is evaluated separately.
• Construct a separate (M/EI) diagram for each
load.
- The change of slope, D/C, is obtained by
adding the areas under the diagrams.
- The tangential deviation, tD/C is obtained by
adding the first moments of the areas with
respect to a vertical axis through D.
• Bending moment diagram constructed from
individual loads is said to be drawn by parts.
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
9-9
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Sample Problem 9.11
SOLUTION:
• Determine the reactions at supports.
• Construct shear, bending moment and
(M/EI) diagrams.
For the prismatic beam shown, determine
• Taking the tangent at C as the
the slope and deflection at E.
reference, evaluate the slope and
tangential deviations at E.
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
9 - 10
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Sample Problem 9.11
SOLUTION:
• Determine the reactions at supports.
RB  RD  wa
• Construct shear, bending moment and
(M/EI) diagrams.
wa2  L 
wa2 L
A1  
 
2 EI  2 
4 EI
1  wa2 
wa3
a   
A2  


3  2 EI 
6 EI
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
9 - 11
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Sample Problem 9.11
• Slope at E:
 E  C   E
C
 E C
wa 2 L wa3
 A1  A2  

4 EI
6 EI
wa 2
3L  2a 
E  
12 EI
• Deflection at E:
yE  t E
C
 tD C
L
 
 3a    L 
  A1 a    A2     A1 
4
 4    4 
 
 wa3 L wa 2 L2 wa 4   wa 2 L2 
 


  

4
EI
16
EI
8
EI
16
EI

 

wa3
2 L  a 
yE  
8 EI
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
9 - 12
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Application of Moment-Area Theorems to Beams
With Unsymmetric Loadings
• Define reference tangent at support A. Evaluate A
by determining the tangential deviation at B with
respect to A.
A  
tB A
L
• The slope at other points is found with respect to
reference tangent.
D   A  D
A
• The deflection at D is found from the tangential
deviation at D.
EF HB

x
L
EF 
x
tB A
L
yD  ED  EF  t D A 
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
x
tB A
L
9 - 13
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Maximum Deflection
• Maximum deflection occurs at point K
where the tangent is horizontal.
tB A
A  
L
K  0   A K
K
A
A   A
• Point K may be determined by measuring
an area under the (M/EI) diagram equal
to -A .
• Obtain ymax by computing the first
moment with respect to the vertical axis
through A of the area between A and K.
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
9 - 14
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Use of Moment-Area Theorems With Statically
Indeterminate Beams
• Reactions at supports of statically indeterminate
beams are found by designating a redundant
constraint and treating it as an unknown load which
satisfies a displacement compatibility requirement.
• The (M/EI) diagram is drawn by parts. The
resulting tangential deviations are superposed and
related by the compatibility requirement.
• With reactions determined, the slope and deflection
are found from the moment-area method.
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
9 - 15