V. IS 800 2007 LSD - I Section - High Shear

Verify the design of an I section beam subject to high shear per the IS 800:2007 limit state method.

BIS. IS 800: General Construction in Steel - Code of Practice. 2007. Bureau of Indian Standards. New Delhi.

Details

The member is a 3 m long, simply-supported beam subject to the following forces:

compression force of 100 kN at the "roller" end, and
concentrated transverse loads of 350 kN at both of the one-third points acting along the minor axis of the section

The section used is a built-up I shape with 400 mm × 20 mm flanges, a web thickness of 8 mm, and an overall depth of 550 mm. The steel has a yield strength of 250 MPa, an ultimate tensile strength of 420 MPa, and a modulus of elasticity of 205×(10)³ MPa. The member is laterally supported along its length.

Validation

Section Properties

Depth of web, d = 550 - 2×20 = 510 mm

Cross-sectional area, A_x = 2(400 × 20) + 510 × 8 = 20,080 mm² = 200.8 cm²

Moment of inertia about major axis, $I_{z} = \frac{2 b_{f} t_{f}^{3}}{12} + 2 b_{f} t_{f} {(\frac{h}{2} - \frac{t_{f}}{2})}^{2} + \frac{t_{2} d^{3}}{12} = 21,336 {cm}^{4}$

Moment of inertia about minor axis, $I_{y} = \frac{2 t_{f} b_{f}^{3}}{12} + \frac{d t_{w}^{3}}{12} = 137 {cm}^{4}$

Radius of gyration about major axis, $r_{z} = \sqrt{I_{z} / A_{x}} = 247.7 mm$

Radius of gyration about minor axis, $r_{y} = \sqrt{I_{y} / A_{x}} = 103.1 mm$

Elastic section modulus about major axis, $Z_{e z} = \frac{I_{z}}{h / 2} = 4,409 {cm}^{3}$

Elastic section modulus about minor axis, $Z_{e y} = \frac{I_{z}}{b_{f} / 2} = 1,067 {cm}^{3}$

Plastic section modulus about major axis, $Z_{p z} = b_{f} t_{f} (h - t_{f}) + 0.25 t_{w} {(h - 2 t_{f})}^{2} = 4,760 {cm}^{3}$

Plastic section modulus about minor axis, $Z_{p y} = \frac{b_{f}^{2} t_{f}}{2} + 0.25 t_{w}^{2} (h - 2 t_{f}) = 1,608 {cm}^{3}$

Constants

γ_m0 = 1.1
γ_m1 = 1.25

Section Classification

ε = \sqrt{250 / f_{y}} = 1

b = b_f/2 = 200 mm

b / t_f = 10 < 9.4ε = 9.4 and b / t_f = 10 > 10.5ε = 10.5

Since the section is symmetric about both axes, the neutral axis is at mid-depth.

d_w / t_w = 63.75 < 84ε = 84

Hence, from Table 2 of IS 800:2007, the flange is compact and the web is plastic.

Therefore, the overall classification of the section is "compact".

h / b_f = 1.375 > 1.2 and t_f = 20 mm < 40 mm

Hence, from Table 10 of IS 800:2007, the buckling class about the major axis is "a" and the same about the minor axis is "b".

Check Slenderness Ratio

k_y = k_z = 1

The maximum allowable slenderness ratio is 180.

Slenderness ratio about the major axis, k_zL / r_z = 12.21 < 180

Slenderness ratio about the minor axis, k_yL / r_y = 29.10 < 180

So the slenderness is within the limit.

Axial Tension Capacity

Check for yielding of gross section:

T_dg = A_g × f_y / γ_m0 = 4,564 kN

Check for rupture of critical section:

α = 0.8

T_dn = α × A_net × f_u / γ_m1 = 5,398 kN

Block shear areas are not specified and therefore not checked.

No tension in the member, so critical ratio is 0.

Check Axial Compression

Axial compression capacity of major axis:

The non-dimensional slenderness ratio, $λ_{z} = \sqrt{\frac{f_{y} {(k_{z} L / r_{z})}^{2}}{π^{2} E}} = 0.136$

For buckling class "a", the imperfection factor, α = 0.21

ϕ_{z} = 0.5 [1 + α (λ_{z} - 0.2) + λ_{z}^{2}] = 0.502

χ_{z} = \frac{1}{ϕ_{z} + \sqrt{ϕ_{z}^{2} - λ_{z}^{2}}} = 1.014

So, as per Cl. 7.1.2.1 of IS 800:2007, $f_{c d z} = min (\frac{χ_{z} f_{y}}{γ_{m 0}}, \frac{f_{y}}{γ_{m 0}}) = 227.3 MPa$

Therefore, axial compression capacity in the major axis: P_dz = A_xf_cdz = 4,564 kN

Axial compression capacity of minor axis:

The non-dimensional slenderness ratio, $λ_{y} = \sqrt{\frac{f_{y} {(k_{y} L / r_{y})}^{2}}{π^{2} E}} = 0.324$

For buckling class "b", the imperfection factor, α = 0.34

ϕ_{y} = 0.5 [1 + α (λ_{y} - 0.2) + λ_{y}^{2}] = 0.573

χ_{y} = \frac{1}{ϕ_{y} + \sqrt{ϕ_{y}^{2} - λ_{y}^{2}}} = 0.955

So, as per Cl. 7.1.2.1 of IS 800:2007, $f_{c d y} = min (\frac{χ_{y} f_{y}}{γ_{m 0}}, \frac{f_{y}}{γ_{m 0}}) = 217.1 MPa$

Therefore, axial compression capacity in the minor axis: P_dy = A_xf_cdy = 4,360 kN, which governs.

The critical ratio for axial compression = 100 / 4,360 = 0.023

Check Shear Capacity

Major Axis

As per Cl. 8.4.1.1 of IS 800:2007, the shear area, A_vz = 2×b_ft_f = 16,000 mm²

So, as per Cl. 8.4.1, $V_{p z} = \frac{A_{v z} f_{y w}}{\sqrt{3}} = 2,309 kN$

Hence, shear capacity along major axis: $V_{d z} = \frac{V_{p z}}{γ_{m 0}} = 2,099 kN$

Minor Axis

As per Cl. 8.4.1.1 of IS 800:2007, the shear area, A_vy = h×t_w = 4,400 mm²

So, as per Cl. 8.4.1, $V_{p y} = \frac{A_{v y} f_{y w}}{\sqrt{3}} = 635.1 kN$

Hence, shear capacity along major axis: $V_{d y} = \frac{V_{p y}}{γ_{m 0}} = 577.4 kN$

As per Cl. 8.4.2, d / t_w = 63.75 < 67ε, so resistance to shear buckling need not be checked.

No shear force along major axis (i.e., ratio = 0).

Critical shear ratio along minor axis: 350 / 577.4 = 0.606

Check Bending Capacity

Since the section has been classified as "compact", β_b = 1.

Major Axis Bending

The actual shear force along the major axis, V = 350 kN > 0.6V_d = 346.4 kN. Thus, the section is subjected to high shear.

M_{d z} = min (\frac{β_{b} Z_{p z} f_{y}}{γ_{m 0}}, \frac{1.2 Z_{e z} f_{y}}{γ_{m 0}}) = 1,082 kN·m

(Cl 8.2.1.1)

Per Cl 9.2.2(a) for "compact" sections:

β = {(2 \frac{V_{z}}{V_{d}} - 1)}^{2} = 0.045

The plastic section modulus excluding shear area, $Z_{f d z} = Z_{p z} - \frac{t_{w} h^{2}}{4} = 4,155 {cm}^{3}$

Plastic design strength of cross-section area excluding shear area, $M_{f d z} = \frac{Z_{f d z} f_{y}}{γ_{m 0}} = 944.4 kN·m$

M_dvz = M_dz - β(M_dz - M_fdz) = 1,082 - 0.045×(1,082 - 944.4) = 1,076 kN·m

(Cl. 8.2.1.3)

So, the governing bending strength with respect to the major axis is M_dz = M_dvz = 1,076 kN·m

Actual bending moment about the major axis is M_z = 350 kN·m.

The critical bending ratio in the major axis is 350 / 1,076 = 0.325

Minor Axis Bending

There is no shear along the minor axis.

\frac{β_{b} Z_{p y} f_{y}}{γ_{m 0}} = 365.5 kN·m

\frac{1.2 Z_{e y} f_{y}}{γ_{m 0}} = 290.9 kN·m

So, the governing bending strength with respect to the minor axis is M_dy = 290.9 kN·m

No bending moment about the minor axis in beam, so critical ratio is 0.

Check Combined Interaction of Axial Compression and Bending

Section strength calculation as per Cl. 9.3.1:

n = N / N_d = 0.022 < 0.2

Hence, from Table 17:

α₁ = max(5n , 1) = 1

α₂ = 2

M_ndy = M_dy = 290.9 kN·m

(Cl. 9.3.1.2(c) )

1.11·M_dz·(1 - n) = 1,168 kN·m > M_dz

M_ndz = M_dz = 1,076 kN·m

{(\frac{M_{y}}{M_{n d y}})}^{α_{1}} + {(\frac{M_{z}}{M_{n d z}})}^{α_{2}} = {(\frac{0}{290.9})}^{1} + {(\frac{350}{1,076})}^{2} = 0.106

Overall member strength in bending and axial compression

n_y = F_x / P_dy = 0.023

n_z = F_x / P_dz = 0.022

K_{y} = min | \begin{array}{l} 1 + (λ_{y} - 0.2) n_{y} = 1 + (0.324 - 0.2) 0.023 = 1.003 \\ 1 + 0.8 n_{y} = 1 + 0.8 \cdot 0.023 = 1.018 \end{array}

K_{z} = min | \begin{array}{l} 1 + (λ_{z} - 0.2) n_{z} = 1 + (0.136 - 0.2) 0.022 = 0.999 \\ 1 + 0.8 n_{z} = 1 + 0.8 \cdot 0.022 = 1.018 \end{array}

\frac{P}{P_{d z}} + 0.6 K_{y} \frac{C_{m y} M_{y}}{M_{d y}} + K_{z} \frac{C_{m z} M_{z}}{M_{d z}} = 0.314

Results

Table 1.
Result Type	Reference	STAAD.Pro	Difference
Slenderness ratio	29.1	29.10	none
Tension capacity, gross yielding (kN)	4,564	4,564	none
Tension capacity, rupture of critical section (kN)	5,398	5,398	none
Ratio for tension	0	0	none
Axial compression capacity about major axis (kN)	4,564	4,564	none
Axial compression capacity about minor axis (kN)	4,360	4,360	none
Ratio for compression	0.023	0.023	none
Shear capacity in major axis (kN)	2,099	2,099	none
Ratio for shear in major axis	0	0	none
Shear capacity in minor axis (kN)	577.4	577.35	negligible
Ratio for shear in minor axis	0.606	0.606	none
Bending capacity about major axis (kN·m)	1,076	1,077	negligible
Ratio for bending about major axis	0.325	0.325	none
Bending capacity about minor axis (kN·m)	290.9	209.939	negligible
Ratio for bending about minor axis	0	0	none
Ratio per Cl. 9.3.1.1	0.106	0.106	none
Ratio per Eq.2 of Cl. 9.3.2.2	0.314	0.314	none

STAAD.Pro Input

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\09 Steel Design\India\IS 800 2007 LSD - I Section - High Shear.STD is typically installed with the program.

The following design parameters are used:

The default value of ALPHA is used.
The default values of KY and KZ are used.
The default value of the maximum allowable slenderness ratio, MAIN, of 180 is used.
The beam type is simply supported, specified by CAN 2
The beam is laterally supported, specified by LAT 1
The default values of CMX, CMY, and CMZ of 0.9 are used.

STAAD PLANE
START JOB INFORMATION
ENGINEER DATE 23-Jun-19
END JOB INFORMATION
*****************************************************************************
*This problem has been created to validate the program calculated design
*results of laterally supported I section subjected to high shear force due to 
*applied axial compression and
*biaxial bending, designed using IS 800 2007 (LSD)
*****************************************************************************
INPUT WIDTH 79
SET SHEAR
UNIT METER KN
JOINT COORDINATES
1 0 0 0; 2 3 0 0;
MEMBER INCIDENCES
1 1 2;
DEFINE MATERIAL START
ISOTROPIC STEEL
E 2.05e+08
POISSON 0.3
DENSITY 76.8195
ALPHA 1.2e-05
DAMP 0.03
TYPE STEEL
STRENGTH RY 1.5 RT 1.2
END DEFINE MATERIAL
MEMBER PROPERTY AMERICAN
1 TAPERED 0.55 0.008 0.55 0.4 0.02
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 PINNED
2 FIXED BUT FX MX MY MZ
LOAD 1 LOADTYPE None  TITLE LOAD CASE 1
MEMBER LOAD
1 CON GY -350 1
1 CON GY -350 2
JOINT LOAD
2 FX -100
PERFORM ANALYSIS
SECTION 0.33 0.66
PRINT ANALYSIS RESULTS
PRINT MEMBER SECTION FORCES
PARAMETER 1
CODE IS800 LSD
CAN 2 ALL
FYLD 250000 ALL
TRACK 2 ALL
LAT 1 ALL
CHECK CODE ALL
FINISH

STAAD.Pro Output

                       STAAD.PRO CODE CHECKING - IS-800-2007-LSD (V2.2)
                     ****************************************************
 |----------------------------------------------------------------------------------|
 |   Member Number:      1                                                            |
 |  Member Section:       TAPERED               (AISC SECTIONS)                     |
 |  Status: PASS  Ratio:   0.606  Critical Load Case:     1  Location:       0.00   |
 |  Critical Condition: Sec. 8.4                                                    |
 |  Critical Design Forces:  (Unit: KN    METE)                                     |
 |      FX:     100.000E+00 C     FY:     350.000E+00      FZ:       0.000E+00      |
 |      MX:       0.000E+00       MY:       0.000E+00      MZ:       0.000E+00      |
 |----------------------------------------------------------------------------------|
 |  Section Properties:    (Unit:  CM  )                                            |
 |  AXX:     200.800E+00          IZZ:     121.257E+03          RZZ:      24.574E+00|
 |  AYY:      44.000E+00          IYY:      21.336E+03          RYY:      10.308E+00|
 |  AZZ:     160.000E+00          IXX:     222.037E+00           CW:      14.981E+06|
 |  ZEZ:       4.409E+03          ZPZ:       4.760E+03                              |
 |  ZEY:       1.067E+03          ZPY:       1.608E+03                              |
 |----------------------------------------------------------------------------------|
 |  Slenderness Check:    (Unit:  KN    METE)                                       |
 |  Actual Length:        3.000E+00                                                 |
 |  Parameters:     LZ:        3.000E+00  LY:        3.000E+00                      |
 |                  KZ:            1.000  KY:            1.000                      |
 |  Actual Ratio:  29.10 Allowable Ratio: 180.00  LOAD:     1                       |
 |----------------------------------------------------------------------------------|
 |  Section Class:      Compact; Flange Class:      Compact; Web Class:      Plastic|
 |----------------------------------------------------------------------------------|
      STAAD PLANE                                              -- PAGE NO.    8
                       STAAD.PRO CODE CHECKING - IS-800-2007-LSD (V2.2)
                     ****************************************************
 |----------------------------------------------------------------------------------|
 |  Member Number:      1                                                           |
 |  Member Section:       TAPERED               (AISC SECTIONS)                     |
 |----------------------------------------------------------------------------------|
 |  Tension:    (Unit:  KN    METE)                                                 |
 |  Parameters:     FYLD:      250.000E+03       FU:      420.000E+03               |
 |                   NSF:            1.000    ALPHA:            0.800               |
 |  Yielding   :  Design Force:        0.000E+00     LC:     0                      |
 |                Capacity:            4.564E+03     As per:  Sec. 6.2              |
 |  Rupture    :  Design Force:        0.000E+00     LC:     0                      |
 |                Capacity:            5.398E+03     As per:  Sec. 6.3              |
 |----------------------------------------------------------------------------------|
 |  Compression:    (Unit:  KN    METE)                                             |
 |  Buckling Class:   Major: a    Minor: b    As per:Cl. 7.1.2.2                    |
 |  Major Axis:   Design Force:      100.000E+00    LC:     1  Loc:     0.000       |
 |                Capacity:            4.564E+03    As per:  Sec. 7.1.2             |
 |  Minor Axis:   Design Force:      100.000E+00    LC:     1  Loc:     0.000       |
 |                Capacity:            4.360E+03    As per:  Sec. 7.1.2             |
 |----------------------------------------------------------------------------------|
 |  Shear:    (Unit:  KN  )                                                         |
 |  Major Axis:   Design Force:        0.000E+00    LC:     0  Loc:     0.000       |
 |                Capacity:            2.099E+03    As per:  Sec. 8.4               |
 |  Minor Axis:   Design Force:      350.000E+00    LC:     1  Loc:     0.000       |
 |                Capacity:          577.350E+00    As per:  Sec. 8.4               |
 |----------------------------------------------------------------------------------|
 |  Bending:    (Unit:  KN    METE)                                                 |
 |  Parameters:   Laterally Supported      KX:   1.00  LX:   3.000E+00  Simp Suppr  |
 |  Major Axis:   Design Force:     -350.000E+00    LC:     1  Loc:     1.000       |
 |                Capacity:            1.077E+03    As per:  Sec. 9.2.2(a           |
 |  Minor Axis:   Design Force:        0.000E+00    LC:     0  Loc:     0.000       |
 |                Capacity:          290.939E+00    As per:  Sec. 8.2.1.2           |
 |----------------------------------------------------------------------------------|
 |  Combined Interaction:                                                           |
 |  Parameters:  PSI: 1.00   CMX:    0.900   CMY:    0.900   CMZ:    0.900          |
 |  Section Strength:    Ratio:     0.106    As per:  Sec. 9.3.1.1                  |
 |                       LC:            1    Loc:       1.000                       |
 |  Overall Member Strength (Bending+Compression):                                  |
 |  Equation 2:          Ratio:     0.314    As per:  Sec. 9.3.2.2   Ky :   1.0028  |
 |                       LC:            1    Loc:       1.000        Kz :   0.9986  |
 |----------------------------------------------------------------------------------|
 |  Checks               Ratio   Load Case No.      Location from Start( METE)      |
 |                                                                                  |
 |  Tension                0.000       0                       0.000E+00            |
 |  Compression            0.023       1                       0.000E+00            |
 |  Shear Major            0.000       0                       0.000E+00            |
 |  Shear Minor            0.606       1                       0.000E+00            |
 |  Bend Major             0.325       1                       1.000E+00            |
 |  Bend Minor             0.000       0                       0.000E+00            |
 |  Sec. 9.3.1.1           0.106       1                       1.000E+00            |
 |  Sec. 9.3.2.2(ii)       0.314       1                       1.000E+00            |
 |----------------------------------------------------------------------------------|