1/15M.Chrzanowski: Strength of Materials
SM1-04: Statics 3: Statically determined bar structures
STATICALLY DETERMINED PLANE BAR STURCTURES
(BEAMS)
2/15M.Chrzanowski: Strength of Materials
SM1-04: Statics 3: Statically determined bar structures
Superposition principle
To determine cross-sectional forces we need to know all external loadings including reaction of different type of supports. This can be easily done if we assume solely for this purpose that the body is non-deformable – as we did in Theoretical Mechanics. Therefore, we deal with linear systems and we can exploit superposition principle.
It says, in our case, that the effect of sum of different loadings is equal the sum of effects of individual loadings. This is a principal property of so called linear systems. As the result we will use linear algebra as already done considering the relation of M-Q-q.
Quite often it is worthwhile to apply this principle when we have to deal with complicated loadings, and therefore it is strongly recommended in practical use. It is much easer to avoid mistakes when considering simple loadings and summing up their effects than to deal with very complex ones.
3/15M.Chrzanowski: Strength of Materials
SM1-04: Statics 3: Statically determined bar structures
Superposition principle
An example of the use of superposition principle for determination of support reactions
2 kN
1 kN/m
4 MNm
2 m 2 m 4 m
RLRP
K
MK=0
Y
Y=0
+2·2
+
+4 +1·4·(2+2+4/2)+(2+2+4)·RP= 0 32+8RP=0 RP = - 4
4 kN
RL= 6 + RP RL= 6 – 4 = + 2+ RL - 2 = 0– RP- 1·4
2 kN
4/15M.Chrzanowski: Strength of Materials
SM1-04: Statics 3: Statically determined bar structures
Superposition principle
4 kNm
1 kN/m
RL1=[2·6]/8=1,5
RL3=[1·4 ·2]/8=1,0
RL2=-4/8= - 0,5 RP2=0,5
RR1=0,5
RP3=3,0
+
MK=0 Y=0
K
Y
RL= 2,0 RR = 4,0
0,5 kN
2 kN
+
1,5 kN
2 kN
1 kN
0,5 kN
0,5 kN
3,0 kN
+
+
= 4 kN
2 kN
1 kN/m
4 kNm
2 m 2 m 4 m
5/15M.Chrzanowski: Strength of Materials
SM1-04: Statics 3: Statically determined bar structures
Typical simple beams
Pin-pointed (or simply supported) beam
Beams
Semi-cantilever beam
Cantilever beam
6/15M.Chrzanowski: Strength of Materials
SM1-04: Statics 3: Statically determined bar structures
Fundamental assumptions used for drawing diagrams of cross sectional forces
Beams
1. Diagram of bending moment will appear always on the side of a beam which is subjected to the tension caused by this bending moment. Therefore, no sign is necessary.
2. We will make use of q-Q-M dependence:
)()(
xQdx
xdM)(xq
dx
dQ
3 . Shear and normal forces will be always marked with their sign according to the following convention:
Q
No
nn
oQ
N
7/15M.Chrzanowski: Strength of Materials
SM1-04: Statics 3: Statically determined bar structures
P
Pa/lPb/l
la b
M
)()(
xQdx
xdM)(xq
dx
dQ
P
P/2l/2 l/2
Pl/4
M
P/2 Q- P/2
+
P/2
Q
No
nn
oQ
N
q0
q=d
Q=c
Q=dx+e M=dx2/2+ex+f
M=cx+b
Typical beams
Pab/l
QPa/l+
-Pb/l
8/15M.Chrzanowski: Strength of Materials
SM1-04: Statics 3: Statically determined bar structures
P
)()(
xQdx
xdM )(xq
dx
dQ
M
ql/2P
P
a al
P
ql/2
q
l
l/2 l/2
ql2/8
Q+-
ql/2ql/2
M
Pa
PP +
-Q
Typical beams
q=d Q=dx+e M=dx2/2+ex+f
9/15M.Chrzanowski: Strength of Materials
SM1-04: Statics 3: Statically determined bar structures
M/lM/l
la b
M
M
Mb/l
Ma/l
QM/l -
QM/l -
M
M/l
M
M/l
M
QM/l +
)()(
xQdx
xdM )(xq
dx
dQ
MM
MM/l
M/l
10/15M.Chrzanowski: Strength of Materials
SM1-04: Statics 3: Statically determined bar structures
[1+(a/l)]2[ql/2]
l aM
Q
)()(
xQdx
xdM )(xq
dx
dQ
+-
M/lM/l
l a
M
Q
M
M
M/l -
P
P(1+a/l)Pa/l l a
M
QPa
PPa/l
+-
P(1+a/l)
q[1-(a/l)2][ql/2]
qaqa2/2
(l/2)[1-(a/l)2]
Typical beams
l a
M
M
M/l -
11/15M.Chrzanowski: Strength of Materials
SM1-04: Statics 3: Statically determined bar structures
[1+(a/l)]2[ql/2]
l aM
Q
)()(
xQdx
xdM
)(xqdx
dQ
+-
M/lM/l
l aM
Q
M
M
M/l -
P
P(1+a/l)Pa/l
l aM
QPa
PPa/l
+-
P(1+a/l)
q
qaqa2/2
(l/2)[1-(a/l)2]
P·a
P
M
q·a
q·a2/2
Q
No
n
Cantilever beam
12/15M.Chrzanowski: Strength of Materials
SM1-04: Statics 3: Statically determined bar structures
The concept of multiple co-linear beams
Gerber’s beams
13/15M.Chrzanowski: Strength of Materials
SM1-04: Statics 3: Statically determined bar structures
Q0, N0 M=0
Q0, N0 M=0
Gerber’s beams
14/15M.Chrzanowski: Strength of Materials
SM1-04: Statics 3: Statically determined bar structures
Formal definition: the set of aligned simple bars, hinged together and supported in the way which assures kinematical stability
Equilbrium equations
X
Y
K A B
X = 0
Y = 0MK = 0
MB = 0MA = 0
1 equation
2 equations
4 unknown horizontal reactions
4 unknown vertictal reactions2 equations
Structure is statically indetermined with respect to horizontal reactions
Number of unknown reactions:4 horizontal + 4 vertical
Gerber’s beams
15/15M.Chrzanowski: Strength of Materials
SM1-04: Statics 3: Statically determined bar structures
Number of unknown reactions to be found:1 horizontal reaction from 1 equilibrium equation (or we accept indeterminancy with respect to normal cross-sectional forces)(2 + n) remaining reactions (vertical reactions, moment reactions) from 2 equilibrium equations and n equations of vanishing bending moment in hingesNumber of hinges is determined from the second of the above condition: structures appears to be kinemtaically unstable if there are too many hinges, and hyper-stiff – if there are too little hinges). However, the location of hinges – even if their number is correct one – cannot be arbitrary!
GOOD!
WRONG!
WRONG!
HYPER-STIFF
HYPER-STIFF
UNSTABLE
UNSTABLE
Gerber’s beams
16/15M.Chrzanowski: Strength of Materials
SM1-04: Statics 3: Statically determined bar structures
Partitioning of a Gerber’s beam into series of simple beams allows for better understanding of structure’s work!
M
Gerber’s beams
17/15M.Chrzanowski: Strength of Materials
SM1-04: Statics 3: Statically determined bar structures
Gerber’s beams
SUPERPOSITION!
++
+
+
+=
18/15M.Chrzanowski: Strength of Materials
SM1-04: Statics 3: Statically determined bar structures
Gerber’s beams
SUPERPOSITION!
M
Q
++ + +
19/15M.Chrzanowski: Strength of Materials
SM1-04: Statics 3: Statically determined bar structures
stop
20/15M.Chrzanowski: Strength of Materials
SM1-04: Statics 3: Statically determined bar structures
Gerber’s beams