Effect of Shear and Axial Effects on Bending Deformation

StevenVukazichSanJoseStateUniversity

The statically indeterminate steel moment frame on Line B of our CE 160 Lab Building is subjected to the seismic load found in Lab #10. This frame was analyzed in found to have the support reactions, moment, shear, and axial force diagrams shown below.

2 Í3

20’-0”

22’-0”

roof level

1st floor level

P = 2.37 k

P/2 = 1.185 k 2.155 k

W18x65

W14

x 8

2

W14

x 8

2

2.155 k P/2 = 1.185 k

Using the moment (MP), shear (VP), and axial force (FP) diagrams found in Lab #10; find the horizontal displacement, in inches, at the point where the load P is applied using the principle of virtual work.

Lab Exercise

RealMomentDiagram(Mp)

23.70 k-ft

23.70 k-ft

0 k-ft 0 k-ft

Moment diagram on compression Side of members

Real Moment Diagram (Mp)

1.185 k 1.185 k

-2.155 k

Positive Shear

Real Shear Diagram (Vp)

-2.155 k 2.155 k

-1.185 k

Tension positve

Real Axial Force Diagram (Fp)

2 Í3

20’-0”

22’-0”

roof level

1st floor level

1

(1)/2 = 0.5(1)/2 = 0.5

Need to find MQ, VQ, and FQ diagrams (use consistent sign convention)

Virtual System to Measure Horizontal Displacement at Point of Load Application

1 ∙ ! = !!!!!"

!

!!" + !!

!!!!!

!

!!" + !!

!!!!"

Internal work Internal work Internal work done by bending done by shear done by axial force

1 ∙ ! = 1!" !!!!!

!!" + 1

!!!!!!!!

!!" + 1

!" !!!!!

Bending Shear Axial force

The effect of bending, shear, and axial force internal work can be found by evaluating the following expression:

The appropriate section properties (I, A, As) for each steel W section can be found in the tables of section properties in the AISC Manual of Steel Construction attached to the lab package.

Note that the shear area for a W section is the web area:

!" = $%&

Cross Sectional Properties of Beam and Columns

Cross Section Properties for W18x65 Beam From the AISC Steel Manual

Cross Section Properties for W14x82 ColumnsFrom the AISC Steel Manual

The material properties, E and G, are related to Poisson’s ratio and can be found from:

E = 29,000 ksi for steel

ν = 0.3 (Poisson’s Ratio for steel)

Recall from CE112:

! = !2 1+ ! = 11,153.846 !"#

Material Properties of Beam and Columns

1 ∙ ! = 1!" !!!!!

!!" + 1

!!!!!!!!

!!" + 1

!" !!!!!

Can Use Table to Evaluate Product Integrals

Evaluate Internal Work terms for Bending, Shear, and Axial Force

Table to Evaluate Product Integrals

SAP 2000 Output Showing Joint Displacement