09 Design Examples

Steel Structures I. Project

9.

Plate girder design (strength and deflections)

9.1.

Static analysis

Static analysis of the plate girder is described in section 8.2. The plate girder has to be checked for stresses arising from bending moment (maximum at the mid span) and shear forces (maximum at the supports). The combined effect of normal stresses from bending and shear stresses (including the effect of concentrated loading at the top flange) has to be checked (point A).

9.2. 9.2.1.

ULS check: strength Checking points on the cross-section

In a cross-section where both bending moment and shear force are present (e.g. point A on the girder), the state of normal (σ) and shear (τ) stresses is shown in the figure below. At the extreme fibre (point 1), the normal stress σ is maximum, while shear stress τ is equal to zero. At the connection between web and flange (point 2), both normal and shear stresses are present. For the sake of simplicity shear stresses are considered acting on the web only, with a constant distribution.

σ1 σ σ

τ1 τ

τ τ

τ

σ

35

τ

τ

Steel Structures I. Project 9.2.2.

Geometric characteristics of the cross-section

Moment of inertia of the cross-section can be determined using an engineering approach as:

Ix =

b f ⋅ h3 12

− ( b f − tw )

hw3 12

Elastic section moduli at points 1 and 2 are:

Wx1 =

Ix y1

Wx2 =

Ix y2

9.2.3.

Bending moment (normal stresses)

The general relationship for checking normal stresses under bending section 7.1.1 moment is: STAS 10108/0-78

σ=

eq. (7.1) STAS 10108/0-78

M ≤R Wn

In the case of this particular situation, this relationship can be written as:

σ max = σ 1 = 9.2.4.

M max ≤R Wx1

Shear force (shear stresses)

In the case of double T cross-sections, the following engineering section 7.1.1 approximation for checking shear stresses can be used: STAS 10108/0-78

τ=

T ≤ R f ( = 0.6 ⋅ R ) hw ⋅ tw

In the case of this particular situation, this relationship can be written as:

τ max =

Tmax ≤ Rf hw ⋅ tw

9.2.5.

Combined bending and shear (equivalent stress)

In the point A along the plate girder, the cross section shall be checked under the combined effect of bending moment, shear force and concentrated force P. The following stresses are computed in point 2 of the cross-section: ƒ Normal stress from bending moment MA:

σ2 = ƒ

Shear stresses from the shear force TA:

τ2 = 36

MA ≤R Wx2 TA ≤ Rf hw ⋅ tw

eq. (7.3) STAS 10108/0-78

Steel Structures I. Project ƒ

When stiffeners are not present under the concentrated force P, local stresses σl from the concentrated force P are present as well:

σl =

P ≤R z ⋅ tw

section 7.1.2 STAS 10108/0-78 eq. (7.4) STAS 10108/0-78

where

z = b "+ 2 ⋅ t f

σl

section 7.1.3 At the connection between the web and top flange of the plate girder, the STAS 10108/0-78 following check shall be performed: eq. (7.5) 2 2 2 STAS 10108/0-78 σ = σ + σ − σ ⋅ σ + 3τ ≤ 1.1R ech

9.3.

l

l

Check of welds between the web and flange

Welds between the web and flanges of the plate girder are realised using section 17 fillet welds. Chose a weld throat thickness a to comply with the minimum STAS 10108/0-78 and maximum weld throat thickness using indications from section 5.1 in this document. When concentrated forces are present at the top chord, and no transversal stiffeners are provided, the weld between the web and the flange is checked using the following relationship: eq. (17.2) STAS 10108/0-78

τ ech = τ 12 + τ 22 ≤ R sf where: τ1 is the shear stress at the web-flange interface produced by plate girder bending τ2 is the shear stress at the web-flange interface produced concentrated force P Rfs is the design shear strength of fillet weld (Rfs =0.7R), see Table 10 STAS 10108/0-78

τ

37

τ

τ1

Steel Structures I. Project The following relationships are used to determine τ1 and τ2:

τ1 =

T 2 ⋅ a ⋅ hw

eq. (17.1) STAS 10108/0-78

τ2 =

P 2⋅a ⋅ z

section 17.1.b STAS 10108/0-78

where: T - the maximum shear force along the girder a - weld throat thickness z - determined according to 9.2.5 in this document

9.4.

SLS check: deflections

Allowable deflection of the plate girder: fa = L/400 The following relation shall be fulfilled:

f ≤ fa where f is the deflection of the plate girder (determined with final dimensions of the cross-section from strength checks, using loads from the SLS combination). The deflection f can be determined using standard methods of statics, a computer program, or the following relationships (accounting for deflection due to bending only):

f = f1 + f 2 f1 =

5 q SLS ⋅ L4 ⋅ 384 E ⋅ I x

f2 =

SLS ⋅ L2 5 M max ⋅ 48 E ⋅ I x

where: f1 - deflection due to uniformly distributed self-weight, at the serviceability limit state [SLS] (nominal values) f2 - deflection due to concentrated forces P, at the serviceability limit state [SLS] (nominal values)

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Table 15 STAS 10108/0-78