Strut and Tie

8.1

Strut-and-Tie Model • Background • AASHTO LRFD Provisions • Design Example

Background z z

STM is a Truss Analogy Truss Analogy Used in Standard and LRFD Specifications Vs = [Asfy/s]d(cotθ) Vn = Vc + V s - AASHTO Standard Vs Æ 45º Truss - AASHTO LRFD Vs Æ Variable Angle Truss

8.2

8.3

STM in Codes z z z z z

CSA 23.3-84 OHBDC Third Edition, 1991 AASHTO LRFD - First Edition, 1994 CHBDC - 2000 ACI 318-02 Appendix A

8.4

Quiz z

z

A Three-Span Concrete Beam Is Built Monolithically, with Continuous Reinforcement Placed Only in the Bottom of the Beam How Will this Beam Perform Under Service Loads? and at Ultimate?

8.5

As Built

8.6

Under Service Loads - Uncracked Condition -

8.7

Under Service Loads - Cracked Condition -

8.8

Observations z

z

z

z

Reinforcement Becomes Active After Concrete Cracks Redistribution of Internal Stresses Occurs After Concrete Cracks After Cracking, Concrete Structures Behave the Way they Are Reinforced For Best Serviceability, the Reinforcement Must Follow the Flow of Elastic Tensile Stresses

8.9

8.10

Strut-and-Tie Model (STM) z

Valuable tool for the analysis and design of concrete members, especially for regions where the plane sections assumption of beam theory does not apply

Deep Beam Stress Trajectories

8.11

STM for D-Regions

Dapped Beam

Tee Beam

8.12

Past Practice z

D-Regions Designed Based On: » Experience » Empirical Rules » Rules of Thumb

8.13

Basic Description of the Strut-and-Tie Model z

z

z

z

A design tool for “disturbed” regions where the flow of stresses is non-uniform and the usual rules of analysis do not apply A rational approach to visualize the flow of forces at the strength limit state based on the variable-angle truss analogy A unified approach that considers all load effects simultaneously A highly flexible and conceptual method that recognizes that several possible solutions may exist for any problem

8.14

STM Basic Principle z

z

Concrete is Strong in Compression Æ Compression Struts Steel is Strong in Tension Æ Tension Ties

8.15

8.16

P

φ > P 2

P 2

8.17

P Strut

C

C

Fill Fill

C T P 2

Fill

Nodal Zones

C T Tie

P 2

8.18

P

C

A φ

T P 2

C > u

C

C

f c c

C

φ As fy > T

T P 2

Basic Concepts Visualize a truss-like system to transfer load to the supports where: • Compressive forces are resisted by concrete “struts” • Tensile forces are resisted by steel “ties” • Struts and ties meet at “nodes” For best serviceability, the model should follow the elastic flow of forces

8.19

Strut-and-Tie Model for Simple Span Beam8.20

Examples of Strut-and-Tie Models

8.21

Methods for Formulating Strut-and-Tie Models z

Stress trajectories from elastic analysis

z

Load path approach

z

Experimentally

Æ Standard models

8.22

Deep Beam Stress Trajectories

8.23

Examples of Strut-and-Tie Models

8.24

Examples of Strut-and-Tie Models

8.25

8.26

Procedures for Load Path Approach z z

z

z

Find reactions Subdivide loads and internal forces - Replace stresses with resultants - Replace asymmetrical stresses with couple and resultant Provide struts and ties to provide load path Locate ties using practical dimensions

STM from Tests - Dapped Beam

8.27

Dapped Beam

8.28

Types of Nodes (Schlaich et al. 1987)

CCC CCT CTT TTT C - Compression T - Tension

8.29

Assumptions z z z z z z

Ties yield before struts crush (for ductility) Reinforcement adequately anchored Forces in struts and ties are uniaxial Tension in concrete is neglected External forces applied at nodes Prestressing is a load

Equilibrium must be maintained

8.30

Strut-and-Tie Model Design Procedure

8.31

Examples of Good and Poor Strut-and-Tie Models

8.32

Factors Affecting Size of Strut

Width of the strut is affected by: • Location and distribution of reinforcement (tie) and its anchorage • Size and location of bearing

8.33

Strut-and-Tie vs. Traditional Analysis/Design Traditional section analysis/design z Linear strain over member depth z Uniform shear stress distribution z Not valid for D-regions Strut-and-tie z Regions with nonlinear strain distribution » Deep beams, pile caps » Brackets, beam ledges, P/T anchors » Shear span/member height < 2

8.34

8.35

V/bdfc’

a/d Source: Prestressed Concrete Structures by Collins & Mitchell

LRFD 5.2 - Definitions

8.36

Strut-and-Tie Model - A model used principally in regions of concentrated forces and geometric discontinuities to determine concrete proportions and reinforcement quantities and patterns based on assumed compression struts in the concrete, tensile ties in the reinforcement, and the geometry of nodes at their points of intersection

5.6.3.1 D-Regions

8.37

Strut-and-tie models may be used to determine internal force effects near supports and the points of application of concentrated loads at strength and extreme event limit states. The strut-and-tie model should be considered for the design of deep footings and pile caps or other situations in which the distance between the centers of applied load and the supporting reactions is less than about twice the member thickness.

5.8.1.1 D-Regions Components in which the distance from the point of zero shear to the face of the support is less than 2d, or components for which a load causing more than ½ of the shear at a support is closer than 2d from the face of the support, may be considered to be deep components for which the provisions of Article 5.6.3 and the detailing requirements of Article 5.13.2.3 apply.

8.38

Strength Limit State for STM Pr = ϕ Pn

8.39

(5.6.3.2-1)

where: Pr = Factored resistance Pn = Nominal resistance of strut or tie

ϕ = Resistance factor for tension or compression (5.5.4.2)

Strength of Struts LRFD 5.6.3.3 Unreinforced strut: Pn = fcu Acs

(5.6.3.3.1-1)

Reinforced strut: Pn = fcu Acs + fy Ass

(5.6.3.3.4-1)

where: ϕ = 0.70 for compression in strut-and-tie models (LRFD 5.5.4.2.1) Acs= effective cross-sectional area of strut (LRFD 5.6.3.3.2) Ass= area of reinforcement in the strut

8.40

STM for Deep Beam LRFD Fig. C5.6.3.2-1

8.41

8.42 Effective Cross-Sectional Area of Strut, Acs

LRFD 5.6.3.3.2 Determined by considering available concrete area and anchorage conditions. When anchored by reinforcement, strut may extend from the anchored bar.

a) Strut Anchored by Reinforcement

C-T-T Node

8.43 Effective Cross-Sectional Area of Strut, Acs

LRFD 5.6.3.3.2

b) Strut Anchored by Bearing and Reinforcement

C-C-T Node

Effective Cross-Sectional Area of Strut, A8.44 cs LRFD 5.6.3.3.2

c) Strut Anchored by Bearing and Strut

C-C-C Node

Limiting Compressive Stress in Strut LRFD 5.6.3.3.3 fcu

fc′ ≤ 0.85fc′ = 0.8 + 170 ε1

where: ε1 = ε s + (ε s + 0.002 ) cot 2 α s fcu = the limiting compressiv e stress α s = the smallest angle between the compressiv e strut and adjoining tension ties (DEG) ε s = the tensile strain in the concrete in the direction of the tension tie (IN/IN)

8.45

Strength of Tie LRFD 5.6.3.4.1 Pn = Ast fy + Aps ( fpe + fy ) where Ast = Total area of longitudinal mild steel reinforcement on the tie Aps = Area of prestressing steel fy = Yield strength of mild steel longitudinal reinforcement fpe = Stress in prestressing steel due to prestress after losses

8.46

Development of Ties

Critical Section =x

If x < ld Æ fs = fy (x/ld)

8.47

Development of Ties (ACI 318)

8.48

Limiting Stresses for STM Elements LRFD 5.6.3.3 - 5.6.3.5

Element

Limiting Stress

ϕ

1 - CCC Node

0.85 fc’

0.70

2 - CCT Node

0.75fc’

0.70

3 - CTT or TTT Node

0.65fc’

0.70

fcu

0.70

fy or (fpe + fy)

0.90 or 1.00

4 - Strut 5 - Tie

8.49

Crack Control Reinforcement LRFD 5.6.3.6 z

z z

z

Provide orthogonal grid of reinforcement near each face of D-Region Maximum Bar Spacing = 12 in. Ratio As / Ag ≥ 0.003 in each of the orthogonal directions Crack control reinforcement, located within tie, considered as part of tie

8.50

Summary 1. 2. 3. 4. 5. 6.

8.51

Visualize flow of stresses Sketch an idealized strut-and-tie model Select area of ties Check nodal zone stresses Check strength of struts Provide adequate anchorage for ties

8.52

Strut-and-Tie Model

8.53

Strut-and-Tie Model

8.54

Design Examples 1. Two Column Bent Cap 2. Spread Footing 3. Pile Cap 4. Dapped-End Beam 5. Hammerhead Pier

8.55