13. shear deformations. shear deformations...formalue for timoshenko shear coefficients. note: is...

Advanced Structural Analysis

Shear Deformations © Richard L Wood, 2017 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 (____________).



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:



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:



4) The two deformation modes of a beam are sketched below:



5) Where free body diagrams of a beam segment of _____ length are:



6) The total deformation of beam can be expressed as the summation of the _______________

_______________________ and the ______________________________________.



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.



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 _____________.



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.



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.



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.



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:



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:



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 ______________________________

_________________________________________________________________.



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. _________________________________________________________________.

13. shear deformations. shear deformations...formalue for timoshenko shear coefficients. note: is...

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