Composite Plastic Moment Capacity for Positive Bending
Transcript of Composite Plastic Moment Capacity for Positive Bending
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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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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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Composite Beam Design BS 5950-90 Composite Plastic Moment Capacity for Positive Bending
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
Beam Section Beam Elevation Beam Internal Forces
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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Location of the Plastic Neutral Axis in Steel Page 9 of 14
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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Location of the Plastic Neutral Axis in Steel Page 10 of 14
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
Beam Section Beam Elevation Beam Internal Forces
TF TTK T
TF B
TK B
TWeb
TC P
Plastic neutral axis (PNA)
CF T
yp
zp
Fstud
Beam Section Beam Elevation Beam Internal Forces
CK TTK T
TF B
TK B
TWeb
TC P
Plastic neutral axis (PNA)
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
Beam Section Beam Elevation Beam Internal Forces
CK T
CWeb
TF B
TK B
TWeb
TC P
Plastic neutral axis (PNA)
CF T
yp
zp
Fstud
Beam Section Beam Elevation Beam Internal Forces
CK T
CK B
TF B
TK B
CWeb
TC P
CF T
Plastic neutral axis (PNA)
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
Beam Section Beam Elevation Beam Internal Forces
CK T
CK B
TF B
CF B
CWeb
TC P
CF T
Plastic neutral axis (PNA)yp
zp
Fstud
Beam Section Beam Elevation Beam Internal Forces
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.