Axial Force Member Design

CE A433 – Spring 2008T. Bart Quimby, P.E., Ph.D.

University of Alaska AnchorageCivil Engineering

Tension Members

• Members subject to axial tension include truss elements, diaphragm chords, and drag struts.

• The basic Design Inequality is:

ASD: ft < F’tTa/An < Ft CDCMCF

LRFD: Tu < T’nTu < KFFt CMCF An

An: Net Area

• Net Area accounts for loss of area due to holes and other cuts in the member.

• Net Area is the gross area less the area of any grain that is cut.

• There is no account for stagger.

Compression Members

• Members subject to axial compression include columns, studs, truss elements, diaphragm chords, and drag struts.

• The basic Design Inequality is:

ASD: fc < F’cPa/A < Fc CDCMCFCP

LRFD: Pu < P’nPu < KFFt CMCFCP A

A: Area

• In Buckling Region– A = Gross Area, Ag

• In Non-Buckling Regions (i.e. near ends in most cases)– A = Net Area, An

CP: Column Stability Factor

• Applies only to compressive stress, Fc

• Applies to both Sawn Lumber and Glulams

• Found in NDS 3.7.1– This factor accounts for

instability in laterally unsupported columns (i.e. column buckling)

– Different in each principle direction

More CP

• See NDS Equation 3.7-1• Column buckling is a function of the laterally unbraced

(buckling) length, le, and cross section properties (Moment of Inertia, Ie, and Area, A) and is different in each principle cross section direction.

• First check the slenderness ratio– le/d must not exceed 50

• Then compute CP

• Note that CP is a function of the member size!– This means that you must know the member size before computing

this factor– When designing, this dependency leads to iterative computations

Laterally Unbraced Length, lu

• This is the distance between locations of lateral support in the plane of buckling

• Most members have two principle directions and lu is frequently different in each direction.

Weak Axis Buckling

Strong Axis Buckling

le: Effective Length

• Effective length is a function of the laterally unbraced length and the end conditions.

• Most timber connections are considered to be pinned.

Effective Length CoefficientsFrom AISC Steel Construction Manual

Slenderness

• NDS 3.7.1.4

• The slenderness ratio le/d must not exceed 50– luy1/d1, luy2/d1

– lux1/d2

Computing CP

• NDS Equation 3.7-1

• Accounts for buckling and material strengths

Material Strength

Euler Strength

Combined Bending & Axial Force

NDS 3.9

Combined Axial Force and Bending

• Both axial force and bending create normal stresses on a cross section.

total, x,y = axial + bending-x + bending-y

x,y = P/A + Mxy/Ix + Myx/Iy

• The result is a planar equation across the section.

Allowable Composite Stress

• Note that each stress component has a DIFFERENT allowable stress, so the limiting value of the combined stress needs to be a composite of the individual allowable stresses.

axial < axial,allowable

bending-x < bending-x,allowable

bending-y < bending-y,allowable

combined < combined,allowable

Combining Allowable Stress

• These can be rewritten as the following ratios:

axial / axial,allowable < 1.00bending-x / bending-x,allowable < 1.00bending-y / bending-y,allowable < 1.00

• In each case, the ratio goes to 1.0 as the normal stress approaches it’s allowable

Interaction Equation

• Instead of finding a composite allowable stress, we can combine the stress ratios

axial / a,allow + bx / bx,allow + by / by,allow < 1.00

• Most combined stress and combined force equations used in structural engineering use this form.

Second Order Effects

• Secondary moments are created with axial force is applied to an already bent member.

• See text in A Beginner’s Guide to the Steel Construction Manual, section 10.3 for more explanation about second order effects.

• Second order effects are ignored in combined tension and bending

• Second order effects can be very significant in combined compression and bending

Bending & Axial Tension

• NDS 3.9.1

• Both interaction equations must be satisfied.

Bending & Axial Compression

• NDS 3.9.2

• Moments are “magnified” by the factors

1

1

1cE

cF

fM

2

1

2

2

1bE

b

cE

cF

fF

f

M