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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