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Advanced Structural Analysis
Shear Deformations © Richard L Wood, 2017 Page 1 of 15
Shear Deformations
Lesson Objectives:
1) Define when shear deformation of members should be accounted for within analyses.
2) Derive the member stiffness modifications to account for shear deformations.
3) Compute the response of a structure to account for combined flexural and shear
deformations using Timoshenko Beam Theory.
Introduction:
1) In the previous structural analysis procedures, the focus has been only on ______________
and __________________________ deformations.
2) __________________________ deformations were not included.
3) When are __________________________ deformations typically no longer negligible?
a. _________________________________________________________________
_________________________________________________________________.
4) Before outlining how to account for _____________________ deformations, let’s review
the deformation modes of a beam.
Deformation Modes:
1) In the previous analysis techniques discussed in this course, ________________________
__________________________________________ theory was assumed which neglected
_______________________ deformations.
a. In _________________________________________________________ theory,
____________________________ remain ___________________________, and
_______________________________ to the neutral axis during bending.
b. Therefore ____________________________________________ are removed
from the theory.
c. _____________________ forces are recovered using equilibrium (____________).
Advanced Structural Analysis
Shear Deformations © Richard L Wood, 2017 Page 2 of 15
2) However in reality, the cross section of a beam behaves somewhat as:
3) This is particularly the case for deep beams, ___________________ in comparison to the
beam length and significant___________________________________________.
a. The fundamental assumption of the Timoshenko beam theory is ______________
_______________ remain ____________, but are no longer __________________
to the beam natural axis (as a result of shear deformation).
b. This beam deformation is sketched below:
Advanced Structural Analysis
Shear Deformations © Richard L Wood, 2017 Page 3 of 15
c. Therefore the implication of this assumption is reflected in the resultant shear
deformation.
i. The shear deformation, _____, is constant over the cross section.
ii. This is illustrated in the sketches below:
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Shear Deformations © Richard L Wood, 2017 Page 4 of 15
4) The two deformation modes of a beam are sketched below:
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Shear Deformations © Richard L Wood, 2017 Page 5 of 15
5) Where free body diagrams of a beam segment of _____ length are:
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Shear Deformations © Richard L Wood, 2017 Page 6 of 15
6) The total deformation of beam can be expressed as the summation of the _______________
_______________________ and the ______________________________________.
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Shear Deformations © Richard L Wood, 2017 Page 7 of 15
Equation for Shear Deformation:
1) Now let’s focus on how one can account for _________________________ deformations.
2) The relationship between the ___________________________________________ at a
cross section of the ______________________________________ and the ____________
_________________________ due to shear can be obtained from Figure 1.
a. Note this examines geometry for a section of differential length (______).
b. This equation can be expressed as:
Figure 1. ______________________________ on a differential length beam member1.
3) Using substitution, the slope of the elastic curve can be expressed.
a. Hooke’s Law:
1 Figure modified from: Kassimali, Aslam. (2012). Matrix Analysis of Structures. 2nd edition. Cengage Learning.
Advanced Structural Analysis
Shear Deformations © Richard L Wood, 2017 Page 8 of 15
b. Stress-force relationship:
c. Desired equation:
d. The new variable ______ denotes the ___________________________________
______________________________.
i. This depends on each shape of the cross-section and is must be determined
for each ____________________________________________________.
ii. Typically this is determined from ________________________________.
iii. Values of _____ for the simple cross sections include:
1. Rectangular cross section: _____________
2. Circular cross section: ________________
iv. These values can also be determined from literature. For example this is
noted within Cowper (1966).
1. The key figure is illustrated in Figure 2.
a. Note the _________________ is a function of the _______
____________________ and _______________________.
4) Integration of the above derived relationship will yield an expression for the shear
deformation.
Shear and Flexure Deformation:
1) The total deflection of a beam, however, is due to the combined effect of _____________
and ____________________________.
a. This can be determined by ____________________ of the __________________
or __________________ caused by ___________________ and _____________.
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Shear Deformations © Richard L Wood, 2017 Page 9 of 15
b. The __________________ deflection relationship can be expressed as:
2) Expressions for elements of the ____________________________________ for a beam
member due to a combined effect of _____________ and ___________ can be derived
using the direct integration approach.
3) To perform this derivation, let’s first construct a ________________ beam member of
length __________ to a unit valued displacement.
a. This is illustrated in Figure 3.
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Shear Deformations © Richard L Wood, 2017 Page 10 of 15
Figure 2. Formalue for Timoshenko shear coefficients. Note: is the Poisson ratio’s ratio and the neutral axis is denoted as a dotted line2.
2 Figure obtain from: Cowper, G.R. (1966). “The Shear Coefficient in Timoshenko’s Beam Theory”. Journal of Applied Mechanics. ASME. June: 335-340.
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Shear Deformations © Richard L Wood, 2017 Page 11 of 15
Figure 3. Determining the member stiffnesses for the _____ column (_______)3.
4) From taking a section cut at _____, one can write two equations using equilibrium:
5) Substitution of the equation for shear into the equation for ________________________
________________________________, one can write after integrating once:
3 Figure modified from: Kassimali, Aslam. (2012). Matrix Analysis of Structures. 2nd edition. Cengage Learning.
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Shear Deformations © Richard L Wood, 2017 Page 12 of 15
6) Substitution of the equation for moment into the equation for ______________________,
one can write two equations after integrating twice:
7) As noted from Figure 1, ____________________________ does not result in _________
_______________________ of the member’s cross section. Therefore this is uncoupled
from ___________ and ____ is only a result of _________________________________.
8) Therefore one can combine equations for the total deflection as the combined effect of __
________________ and ___________________________ deformation and express it as:
9) To determine the unknowns in the equation above, let’s examine the boundary
conditions:
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Shear Deformations © Richard L Wood, 2017 Page 13 of 15
10) Applying the boundary conditions, one can write:
11) Where _____ is introduced as a dimensionless flexural-to-shear stiffness ratio. This can
be evaluated using the equation:
12) The remaining coefficients for the first column can be found using equilibrium:
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Shear Deformations © Richard L Wood, 2017 Page 14 of 15
13) A similar approach can be applied to the other three columns and the complete beam
member stiffness matrix can be assembled due to combined ___________ and ________
_______________________.
14) If shear is not considered, the above beam member stiffness matrix reduces to _________
___________________________________.
a. This is based on the previous __________________________________ theory.
15) In the front tabulated values in the book, the fixed-end forces due to loading along the
member’s length account for _______________________________________________.
a. If ______________________________________________ are desired, a similar
procedure to the aforementioned methodology can be utilized.
Identification of When Beam Theory Does Not Apply:
1) A structure is made of many different elements:
a. __________________________________________________________________
b. __________________________________________________________________
c. __________________________________________________________________
d. __________________________________________________________________
2) The frame skeleton itself is comprised of _________________ and _________________.
a. Regions where beam theory applies include ______________________________
_________________________________________________________________.
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Shear Deformations © Richard L Wood, 2017 Page 15 of 15
i. These can be termed as ________________________________________.
b. Regions where classical beam theories do not apply are:
i. ___________________________________________________________.
ii. ___________________________________________________________.
iii. ___________________________________________________________.
iv. These can be termed as ________________________________________.
c. Refer to Schlaich, J., Schafer, K., and Jennewein, M. (1987). “Toward a
Consistent Design of Structural Concrete.” PCI Journal. May-June, Special
Report: 74-150.
d. A sketch is illustrated below:
3) Recall that beam theories are based on the assumptions of:
a. _________________________________________________________________.
i. ___________________________________________________________.
b. _________________________________________________________________.