7.2 Internal forces for 1D members (beam, beam as slab, column)

There is different evaluation according to type of member (beam, beam as slab, column). Each type of member has special settings which can affect the internal forces used for design of reinforcement.

7.2.1 Beams and beam as slab

Recalculated internal forces are dependent on

strength reduction factor for bending (see chapter "8.2 Design of reinforcement for 1D members (beams, beams as slab, columns)" for more information about used method and this factor) and for shear which is usually 0,75

capping of moment and shear forces (dependent on settings used in concrete setup or concrete member data

7.2.1.1 Strength reduction factors and internal forces

There are used two types of internal forces during design of reinforcement

Factored internal forces (X_u) – internal forces from statical analysis

Nominal internal forces (X_n) – internal forces with respect of strength reduction factor

Generally it is required to use strength reduction factor on side of the material strength, but we don’t know phi factor before design starts (see different method in chapter "8.2 Design of reinforcement for 1D members (beams, beams as slab, columns)"). That’s why the strength reduction factor is used on side of increasing of the load instead of the decreasing of the strength.

There are used different strength reduction factors for bending and for shear. The default limits are set in Concrete setup>Design strength (see chapter "4 Global setting ").

7.2.1.1.1 Bending, tension, compression

Strength reduction factor for bending is used according to used method described in chapter "8.2 Design of reinforcement for 1D members (beams, beams as slab, columns)". There are two limits for:

Tensioned controlled section - in case when strain of reinforcement is higher than 0,005

Compression controlled section - in case when strain of reinforcement is lower than 0,002

Transition zone – special zone where linear interpolation according to calculated strain in reinforcement between tension controlled section and compression controlled section is applied

7.2.1.1.2 Shear

Strength reduction factor used for shear is fixed and it is equal to 0,75 all the time.

7.2.1.2 Capping above support

The capping above the support is another effect which influences the values of internal forces used for design of reinforcement. There are two types of capping – moment and shear capping.

7.2.1.2.1 Moment capping

Moment capping reduces the peaks of the bending moment above the supports. The reducing depends on width of the supports where reducing is applied on. Width of the support is taken into account dependent on existence of Concrete member data.

CMD

Support width

YES

The comparison of the bending moments with and without capping for standard support and for column support is done in the following table.

Moment capping	NO	YES
Standard support
Column support

7.2.1.2.2 Shear force capping

Shear capping is another possibility of capping used for the shear force. There are two types of calculation of reduced shear force depending on settings in Concrete setup.

A) At the face (support/column) - shear force above the support is the same as shear force in the face of the support/column

B) At the effective depth from the face (support/column) – shear force is calculated at distance d (effective depth of cross-section) from the face of support/column

Type of shear capping	Evaluation of shear force
Without capping
At the face
At the effective depth

7.2.2 Column

Recalculation bending moment for column can be impacted only by slenderness effect (magnified moments). Magnified moments are calculated only in case that

slenderness ratio is bigger than lower limit of slenderness ratio and
slenderness ratio is lesser than upper limit of slenderness and
check box Use buckling data is ON in Concrete setup (see chapter "4.1 Concrete setup for 1D member") or in concrete member data (if it is defined on the member), see chapter "5.2 Member data 1D (beams, beams as slab, columns)"

Calculation of these magnified moments depends on it, if the member is sway or not. The following formulas are used for calculation:

Type of member	Clause in ACI 318-05	Equation
Non-sway	10.12.3	M_uy(z),rec = max (δ_nsy(z)⋅ M_uy(z); M_miny(z))
Sway	10.13.3	M_uy(z),rec = max (δ_nsy(z)⋅ M_nsy(z) + δ_sy(z)⋅ M_sy(z); M_miny(z))

where
δ_nsy(z)	moment magnification factor about y(z) axis of LCS for member braced against sidesway (non-sway member), to reflect effects of member curvature between ends of compression member, "7.2 Internal forces for 1D members (beam, beam as slab, column)" and "7.2 Internal forces for 1D members (beam, beam as slab, column)"
M_uy(z)	factored moment about y (z) axis of LCS of compression member. It can be calculated according to equation: M_uy(z) = M_nsy(z) + M_sy(z)
M_miny(z)	Minimum value of magnified moment about y (z) axis of LCS of compression member, see charter "7.2 Internal forces for 1D members (beam, beam as slab, column)"
M_nsy(z)	factored moment about y (z) axis of LCS of compression member due to loads that cause no appreciable sidesway
δ_sy(z)	moment magnification factor factor about y(z) axis of LCS for member not braced against sidesway (sway member), to reflect lateral drift resulting from lateral and gravity loads, "7.2 Internal forces for 1D members (beam, beam as slab, column)"
M_sy(z)	factored moment about y (z) axis of LCS of compression member due to loads causing appreciable sway

If the member will be sway or non-sway can be set in dialog Buckling and relative lengths > Buckling data or in dialog Buckling coefficients (member buckling data is defined, user input is selected and kyy(zz) = Code dependent ) via property Swaz zz or Sway zz

The loads cause appreciable sway (load caused factored moment M_sy(z)), if the load belong to load case, for which check box Loads cause appreciable sideway is ON in dialog for definition of load cases, see picture below

7.2.2.1 Magnified moments for non sway member

The calculation of magnified moments for non-sway member is calculated according to formula

M_uy(z),rec = max (δ_nsy(z)⋅ M_uy(z); M_miny(z)) = max (δ_nsy(z)⋅( M_nsy(z) + M_sy(z)); M_miny(z))

It follows, that magnified moments depends on:

moment magnification factor for member braced against sidesway (non-sway member), value δ_nsy(z)
total value of factored bending moment (sum of factored moment due to loads that cause no appreciable sidesway and factored moment due to loads that cause appreciable sidesway) at calculated section of the column
minimum value of magnified moments (value M_miny(z) )

7.2.2.1.1 Moment magnification factor for member braced against sidesway

This value is calculated according to equation 10-9 in ACI 318-05

where
	factor relating actual moment diagram to an equivalent uniform moment diagram calculated according to clause 10.12.3.1 in ACI 318-05 if M_max,y(z) ≤ M_2y(z) (without transverse load) if M_max,y(z) > M_2y(z) (with transverse load)
	smaller value of factored end moment on a compression member about y(z) axis of LCS of compression member. The value is positive if member is bent in single curvature, and negative if bent in double curvature
	larger factored end moment on compression member about y(z) axis of LCS of compression member .The value is usually positive, but the value is negative only in case that both end moments are negative too
M_max,y(z)	the maximum value of factored bending moment at whole length of the column
Pu	factored axial force
P_c	Critical buckling load calculated according to equation 10-10 in ACI 318-05
	stiffness for calculation the critical column load, see chapter 4.1.4.2 Equation for calculation EI for calculation Pc
l_uy(z)	System length (unsupported length) of member about y(z) axis of LCS, see chapter 6 Calculation of slenderness
k_y(z)	Effective length factor about y(z) axis of LCS of compression member, see chapter 6 Calculation of slenderness

If the value the calculation of magnified moment finishes with error 918 (The compression member is instable, because Pu ≥ 0,75Pc)

7.2.2.1.2 Minimum value of magnified moment

This value is calculated according to equation 10-14 in ACI 318-05

where
Pu	factored axial force
	The dimension of cross-section of compression member in direction of z(y) axis of LCS. For different shape of cross-section as than rectangular shape, the dimensions of circumscribed rectangular is taken into account

7.2.2.2 Magnified moments for sway member

The calculation of magnified moments for sway member is calculated according to formula

M_uy(z),rec = max (δ_nsy(z)⋅M_nsy(z) + δ_sy(z)⋅ M_sy(z); M_miny(z))

It follows, that magnified moments depends on:

moment magnification factor for member braced against sidesway (non-sway member), value δ_nsy(z)
moment magnification factor for member not braced against sidesway (sway member), value δ_sy(z)
factored moment due to loads that cause no appreciable sidesway at calculated section, value M_nsy(z)
factored moment due to loads causing appreciable sway, see chapter value M_sy(z)
minimum value of magnified moments (value M_miny(z) )

7.2.2.2.1 Moment magnification factor for member braced against sidesway

This value is calculated according to clause 10.13.5 in ACI 318-05 and calculation of this factor depends on condition below

Condition

Moment magnification factor

If the value the calculation of magnified moment finishes with error 918 (The compression member is instable, because Pu ≥ 0,75Pc)

where
	factor relating actual moment diagram to an equivalent uniform moment diagram calculated according to clause 10.12.3.1 in ACI 318-05 if M_max,y(z) ≤ M_2y(z) (without transverse load) if M_max,y(z) > M_2y(z) (with transverse load)
	smaller value of factored end moment on a compression member about y(z) axis of LCS of compression member. The value is positive if member is bent in single curvature, and negative if bent in double curvature
	larger factored end moment on compression member about y(z) axis of LCS of compression member. If Else
M_max,y(z)	the maximum value of factored bending moment at whole length of the column
	factored end moment (at head of compression member) about y (z) axis of LCS of compression member due to loads causing appreciable sway
	factored end moment (at head of compression member) about y (z) axis of LCS of compression member due to loads that cause no appreciable sidesway
	factored end moment (at foot of compression member) about y (z) axis of LCS of compression member due to loads causing appreciable sway
	factored end moment (at foot of compression member) about y (z) axis of LCS of compression member due to loads that cause no appreciable sidesway
δ_sy(z)	moment magnification factor about y(z) axis of LCS for member not braced against sidesway (sway member), to reflect lateral drift resulting from lateral and gravity loads, "7.2 Internal forces for 1D members (beam, beam as slab, column)"
Pu	factored axial force
P_c	Critical buckling load calculated according to equation 10-10 in ACI 318-05
	stiffness for calculation the critical column load, see chapter 4.1.4.2 Equation for calculation EI for calculation Pc
l_uy(z)	System length (unsupported length) of member about y(z) axis of LCS, see chapter 6 Calculation of slenderness
k_y(z)	Effective length factor about y(z) axis of LCS of compression member, see chapter 6 Calculation of slenderness
r_y(z)	radius of gyration of cross section about y(z) axis of LCS of compression member,
f_c’	specified compressive strength of concrete
A_g	gross area of concrete section

7.2.2.2.2 Moment magnification factor for member not braced against sidesway

This value is calculated according to equation 10-18 in ACI 318-05, but there is made some simplification of the calculation, where this value is calculated only for individual compression member. It follows:

If the value δ_sy(z) ≥ 2.5 and bending moment due to loads causing appreciable sway is non-zero (M_sy(z)≠0), the compression member is instable (clause 10.13.6c in ACI 318-05) and the calculation finishes with error 917 (The compression member is instable, because delta_s ≥ 2.5)

In the formula above is expression ∑P_u (the summation for all the factored vertical loads in a story) substituted by value P_u (factored force at individual compression member ) and expression ∑P_c (the summation critical buckling load for all sway resisting columns in a story) is substituted by value P_c (critical buckling of individual compression member)

7.2.2.2.3 Minimum value of magnified moment

This value is calculated the same as for non-sway member, see chapter "7.2 Internal forces for 1D members (beam, beam as slab, column)"