Composite Plastic Moment Capacity for Positive Bending

8/11/2019 Composite Plastic Moment Capacity for Positive Bending

1/14Technical Note E-CB-AISC-ASD89-001 Page 1 of 14

COMPUTERS AND STRUCTURES, INC., BERKELEY, CALIFORNIA SEPTEMBER 2002

COMPOSITE BEAM DESIGN BS 5950-90

Technical NoteComposite Plastic Moment Capacity for

Positive Bending

This Technical Note describes how the program calculates the positive bend-

ing moment capacity for a composite section assuming a plastic stress distri-

bution.

Overview

The plastic moment capacity of a composite cross-section is calculated in the

program on the following basis (BS 4.4.2):

Concrete is assumed to be stressed to a uniform compression of 0.45 fcu

over the full depth of concrete on the compression side of the plastic neu-

tral axis (PNA) (BS 4.4.2.a). Concrete is assumed to have no tensile

strength.

The structural steel member is assumed to be stressed to its design

strengthpyeither in tension or in compression for Class 1 (Plastic), Class 2

(Compact) and Class 3 (Semi-Compact) sections (BS 4.4.2.b). Class 4(Slender) sections are not designed by the program. For sections under the

influence of high shear, the web is ignored in calculating the plastic moment

capacity (BS 5.3.4).

The longitudinal reinforcement is ignored in the program for calculating

plastic moment capacities for both positive and negative moment. This is

conservative.

The effect of partial composite connections is considered in computing the

plastic moment capacity for positive moment.

Figure 1 illustrates a generic plastic stress distribution for positive bending.

Note that the concrete is stressed to 0.45 fcuand the steel is stressed to py.

The distances ypand ycare measured from the bottom of the beam bottom

flange (not cover plate) to the plastic neutral axis (PNA) and the bottom of

the concrete compression block, respectively. The illustrated plastic stress


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Composite Beam Design BS 5950-90 Composite Plastic Moment Capacity for Positive Bending

Overview Page 2 of 14

distribution is the basic distribution of stress used by the program when con-

sidering a plastic stress distribution for positive bending. Note that if the

metal deck ribs are parallel to the beam, the concrete in the ribs is also con-

sidered.

Figure 1: Generic Plastic Stress Distribution for Positive Bending

Figure 2 illustrates how the program idealizes a steel beam for calculating the

plastic stress distribution. Two different cases are shown, one for a rolled

section and the other for a user-defined section. The idealization for the rolledsection considers the fillets whereas the idealization for the user-defined sec-

tion assumes there are no fillets because none are specified in the section

definition. Although not shown in Figures 1 and 2, the deck type and orienta-

tion may be different on the left and right sides of the beam as shown in

Figure 2 of Technical Note Effective Width of the Concrete Slab Composite

Beam Design.

For a rolled steel section, the fillets are idealized as a rectangular block of

steel. The depth and width of this rectangular block are given by:

kdepth= k- T (Rolled)

The rectangular block, kwidth, is:

kwidth= (As- 2BT - tD) / 2kdepth (Rolled)

Beam Section Beam Elevation Plastic Stress

Distribution

CConc

CSteel

TSteel

0.45f cu

py

py

a

Plastic neutral axis (PNA)

yp

zp

yc
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Overview Page 3 of 14

Figure 2: Idealization of a Rolled Section and a User-Defined Section usedfor Calculating the Plastic Stress Distribution

B bot

kwidth

kwidth

t

B top

Dp

tc

Ttop

k

k

Dd

kdepth

kdepth

Tcp

Tbot

Idealization for Rolled Section

Bcp

Bbot

t

Btop

Dp

tc

Ttop

Dd

Tcp

Tbot

Idealization for User-Defined Section

Ds

Be

Be

Ds

Bcp


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Maximum Compressive Force in Concrete Page 4 of 14

For welded sections, the fillets are non-existent. However, for the purpose of

plastic moment capacity calculation, the depth and width of the rectangular

blocks of fillets are taken as the following. This definition of the fillets for

welded, user-defined sections allows them to be treated under the same

framework as the rolled sections.

kdepth = 0 (Welded)

kwidth = t (Welded)

The basic steps in computing the positive plastic moment capacity are as fol-

lows:

Determine the maximum compressive force that can be generated in con-

crete and steel for full and partial composite connection.

Determine the size of concrete stress block, a,and the location of the bot-

tom of the stress block, yc.

Determine the location of the PNA in steel, yp.

Calculate the plastic moment capacity, Mp.

Maximum Compressive Force in Concrete

The program determines the location of the PNA by comparing the maximumpossible compressive force that can be developed in the concrete with the

maximum possible tensile force that can be developed in the steel section (in-

cluding the cover plate, if applicable). The depth of the stress block is deter-

mined from the concrete compressive force in plastic condition. The location

of the PNA and the depth of the compression block are heavily influenced by

the partial composite connection ratio PCC.

The maximum concrete force, Fconc,max, that can be generated in a composite

deck is calculated differently depending on whether the deck ribs are parallelor perpendicular to the beam. If the deck ribs are perpendicular to the beam,

Fconc,maxis calculated as follows (BS 5.4.4.1). Note that the maximum concrete

force has contribution from the left and right sides of the beam. Those contri-

butions are treated separately because they may be different.

Fconc,max = [0.45 fcuBe(Ds Dp)]left+ [0.45 fcuBe(Ds Dp)]right(BS 5.4.4.1)


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Maximum Compressive Force in Concrete Page 5 of 14

If the deck ribs are parallel to the beam, the contributions of the ribs as well

as the contributions from the slabs are considered. In such cases, Fconc,max is

calculated as follows (BS 5.4.4.1):

Fconc,max =right

r

pr

cecu

leftr

pr

cecu s

Db

tBfs

Db

tBf

++

+ 45.045.0

The maximum steel force, Fsteel,max, that can be generated in a composite

beam is calculated differently depending on whether there is cover plate or

not.

Fsteel,max = Aspy (with no cover plate) (BS 5.4.4.1)

Fsteel,max = Aspy + BcpTcppycp (with cover plate) (BS 5.4.4.1)

In the preceding expressions, As is the total area of steel section alone. For

welded sections,Asis computed from plate dimensions. For rolled sections,As

is given in the section definition.

In practical cases, especially when the shear connection between the slab and

the steel beam is partial, the force in the concrete will not attain Fconc,max, and

the force in the steel section will not attain Fsteel,max. Assuming that the partial

composite connection ratio is PCC, the maximum concrete force and total

steel tensile force will be equal to Fstud

, which is given by the following equa-

tion:

Fstud = PCCmin{Fconc,max, Fsteel,max}

The value of PCCranges between 0 and 1. For full composite connection, PCC

is 1 and Fstudis the minimum of maximum concrete force and maximum steel

tensile force. In such cases, if Fconc,maxis greater than Fsteel,max, ypwill be equal

to the full depth of the beam dand the depth of compression block will be

smaller than Ds. For full composite connection and if Fsteel,max is greater than

Fconc,max, ypwill be less than hand the depth of the compression block will beequal to Ds. For partial composite connection, yp is always less than D, and

the depth of the compression block is always less than Ds.

In full or partial composite connection, both the concrete compression force

and steel tensile force will always to be equal to Fstud. The location of the


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Depth of the Compression Block Page 6 of 14

plastic neutral axis, yp, depth of the compression block, a, and plastic mo-

ment capacity, Mp, are calculated from this condition.

Depth of the Compression Block

The required depth of the compression block, a, is the depth of the concretethat is required to develop the concrete compression force equal to Fstud. The

definition of Fstud is given in the previous section of this Technical Note. For

the calculation of the required depth of the compression block, it is assumed

that the concrete is stressed to a uniform compression of 0.45fcuover the full

depth of concrete on the compression side (BS 4.4.2.a) and concrete is as-

sumed to have no tensile strength. The longitudinal reinforcing bars are ig-

nored.

Once the required depth of compression block is determined, the location ofthe bottom of the compression block, yc, is also determined. For simple cases

when the deck on the left and right sides of the beam have the same dimen-

sions,yccan be calculated as follows:

yc = D+ Ds- a

For simple cases when the deck on the left and right sides of the beam have

the same thicknesses and the same rib depths, the calculation of aand ycis

simple. This calculation is also simple when there is only one slab on either

the left or right side of the beam. However, the program considers the gen-

eral condition where the slabs on the left and right sides are different. In such

cases, the compression block may include part of the slab on either side or

both sides, full slab and part of the ribs on either side or both sides. Also note

that if the deck ribs are perpendicular to the beam, the ribs do not contribute

to the compression block. The deck ribs may orient differently, parallel or

perpendicular to the beam, on the two sides of the beam. Those geometric

variations make the calculation of aand yc. The program handles these gen-

eralities using an efficient iterative procedure. In the iterative procedure, the

program starts with a small value of aand progressively increases its value

until the compression in concrete based on the assumed compression block

becomes equal to Fstud. If the concrete decks are the same on both sides, or if

there is one concrete deck at either side, and if the block sizes are smaller

than the slab thickness, the iterative procedure will converge in a single step.


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Depth of the Compression Block Page 7 of 14

Figures 3 and 4 show the internal forces for the conditions where the com-

pression stress block lies in the slab and the deck rib, respectively, for a sim-

ple case where decks at the left and right sides are the same.

Figure 3: Rolled Steel Section with PNA in Concrete Slab Above MetalDeck, Positive Bending (For User-Defined Welded Sections,Ignore the Fillets)

Figure 4: Rolled Steel Section with PNA within Height of Metal Deck,

Positive Bending (For User-Defined Welded Sections, Ignore

the Fillets)

CC 1

Beam Section Beam Elevation Beam Internal Forces

CC 2

Bottom of the compression

block

Fstud

yc

a

CC 1


Bottom of the compressionblock

Fstud

yc

a

D


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Location of the Plastic Neutral Axis in Steel Page 8 of 14

Location of the Plastic Neutral Axis in Steel

The location of the PNA is located by the distance yp that is measured from

the bottom of the beam bottom flange (not cover plate) to the PNA. For steel

sections without cover plates, yp represents the depth of the tension zone of

the steel section under plastic condition. The calculation of ypinvolves finding

its value so that the total steel tension force becomes equal to Fstud, which is

also equal to the compression force in concrete. The definition of Fstudis given

previously in this Technical Note.

In determining the value of yp, it is assumed that the structural steel is

stressed to its design strength, py, either in tension or in compression for all

classes of sections, including Class 1 (Plastic), Class 2 (Compact), and Class 3

(Semi-Compact) (BS 4.4.2.b). Class 4 (Slender) sections are not designed for

composite beams. For sections under the influence of high shear, the web isignored in calculating the plastic moment capacity (BS 5.3.4).

The location of the PNA is heavily influenced by the partial composite connec-

tion ratio, PCC. If the PCCis 1 and Fconc,maxis greater than Fsteel,max, ypwill be

equal to the full depth of the beam D. If PCCis less than 1, or if PCCis 1 but

Fsteel,max is greater than Fconc,max, Fstudwill be less than Fsteel,max, and the PNA

will be below the top of the top flange. The location of the PNA can lie in any

of the six following general locations depending on the relative value of Fstud

and Fsteel,max. See Figures 5 to 10 for more details.

Within the beam top flange.

Within the beam top fillet (applies to rolled shapes from the program's

section database only).

Within the beam web.

Within the beam bottom fillet (applies to rolled shapes from the program's

section database only).

Within the beam bottom flange.

Within the cover plate (if one is specified).

Note it is very unlikely that the PNA would be below the beam web but there

is nothing in the program to prevent it. This condition would require a very

large beam bottom flange and/or cover plate and a small PCC.


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For typical composite beams with equal flange and moderate PCC, the PNA

would lie in the upper side of the web, in the top fillet, or in the top flange.

Adding a cover plate would drag the PNA down.

The program calculates the value of ypusing an efficient procedure. The pro-

gram starts with a value of yp equal to D and progressively decreases its

value until the total tensile force in the steel section (including the cover plate

if present) based on the assumed location of the PNA becomes equal to Fstud.

In that procedure, if the location of the PNA is known to lie in any one of the

six general locations described previously, the value of yp is determined di-

rectly. That means the value of ypcan be obtained by at best six trials. The

details of the expressions for different cases are given as follows:

If Fstud= Fsteel,max, then,

yp= D,

else if Fstud(Fsteel,max2 TtopBtoppy) then,

yp= D(Fsteel,maxFstud) / (2 TtopBtoppy),

else if FstudFsteel,max2 (TtopBtop +kdepthkwidth)py then,

yp= DTtop(Fsteel,max2 TtopBtoppy Fstud)/ (2 kwidthpy),

else if FstudFsteel,max2 (TtopBtop+ kdepthkwidth+ 2 t d)py then,

yp= DTtopkdepthy

studywidthdepthytoptopsteel

tp

FpkkpBTF

2

22max, ,

else if FstudFsteel,max2 (TtopBtop+ kdepthkwidth+ t d + kdepthbwidth)pythen,

[Fsteel,max2(TtopBtop+ kepthkwidth+ t d)Fstud]yp= D Ttopkdepthd [2 kwidthpy],


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else if FstudFsteel,max2 (TtopBtop+2 kdepthkwidth+ t d + Tbot Bbot)pythen,

yp= DTtopkdepthdkdepth[Fsteel,max2(TtopBtop+ 2kepthkwidth+ t d)pyFstud]

[2 Bbotpy],

else,

[Fsteel,maxFstud] 2(TtopBtop+ 2kepthkwidth+ t d + TbotBbot)pyyp= [2 Bcppycp]

[2 Bcppycp]

Figures 5 through 10 show the internal forces for the conditions where the

PNA lies in the six general locations of the steel sections. Those locations

were described previously in this section of this Technical Note. In the figures,

the rolled sections and welded sections are treated under uniform framework,

even though there is no fillet in the welded section. For welded sections, the

depth of the fillets should be considered as zero in all expressions. Also, Fig-

ures 6 and 8 should be ignored for welded sections.


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Figure 5: Rolled Steel Section with PNA within Beam Top Flange,Positive Bending (For User-Defined Welded Sections, Ignorethe Fillets)

Figure 6: Rolled Steel Section with PNA within Beam Top Fillet, Positive

Bending (This Case Does Not Apply for Welded Sections)

Fstud


TF TTK T

TF B

TK B

TWeb

TC P


CF T

yp

zp

Fstud


CK TTK T

TF B

TK B

TWeb

TC P


CF T

yp

zp


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Figure 7: Rolled Steel Section with PNA within Beam Web, Positive

Bending (For User-Defined Welded Sections, Ignore the Fillets)

Figure 8: Rolled Steel Section with PNA within Beam Bottom Fillet, Posi-

tive Bending (This Case Does Not Apply for Welded Sections)

Fstud


CK T

CWeb

TF B

TK B

TWeb

TC P


CF T

yp

zp

Fstud


CK T

CK B

TF B

TK B

CWeb

TC P

CF T


yp

zp


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Figure 9: Rolled Steel Section with PNA within Beam Bottom Flange,Positive Bending (For User-Defined Welded Sections, Ignorethe Fillets)

Figure 10: Rolled Steel Section with PNA within Cover Plate, PositiveBending (For User-Defined Welded Sections, Ignore theFillets)

Fstud


CK T

CK B

TF B

CF B

CWeb

TC P

CF T

Plastic neutral axis (PNA)yp

zp

Fstud


CK T

CK B

CCP

CF B

CWeb

TC P

CF T

Plastic neutral axis (PNA)yp

zp


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Plastic Moment Capacity for Positive Bending

After the depth of the compression block and the location of the PNA are

known, the forces in all individual elements are computed using the design

basis described in the Overview section of this Technical Note. In addition, the

centroid of tension and compression forces can be determined. The plastic

moment capacity is determined using statics.

If the shear is high, the web of the steel section is ignored in computing the

plastic moment capacity. In general, the forces in the following individual

elements are considered.

Concrete slab above the metal deck (left)

Concrete slab above the metal deck (right)

Concrete ribs in the metal deck (left)

Concrete ribs in the metal deck (right)

Steel in the beam top flange

Steel in the beam top fillet

Steel in the beam web

Steel in the bottom fillet

Steel in the bottom flange

Steel in the cover plate

Depending on the size of the concrete compression block, some of the forces

in concrete can be zero, because concrete tensile strength is assumed to be

zero. Also, depending on the location of the PNA, some of the forces in any of

the six elements can be compressive and some can be tensile. However, the

element in which the PNA will lie has been split into two parts: one involving

tension and the other part involving compression.

Because the total axial force over the whole composite section is zero, the

moment can be computed using any axis. The program uses the bottom of

the bottom flange as the reference axis for calculating the plastic moment ca-

pacity.

Composite Plastic Moment Capacity for Positive Bending

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