Denis Mitchell, McGill University Michael P. Collins, University of Toronto Shrinivas B. Bhide and Basile G. Rabbat, Portland Cement Association
530 k
545 k
585 k
5.00'
5.00'
2.43'
7.57'
0.5'
k
k -867
stirrup band
1 = 5.00'
2 = 5.00'
3 = 5.00'
8.132'
0k
13.7° -1075 k
stirrup band
-1
8.132' H J L -1653k -1653k
k
F stirrup band
1653 k
-1075 k
D
8.132'
0k
9
-538
B
K
-585 k
325 k
6.291'
I
1653k 1653k
18
0.75'
1653 k 47.0°
-1
0k
k
5.074'
15
403 k
0k
843 k -5
522 k -7 4 3.858' 4 k
G
E
-585 k
C
A
0k
250'
AASHTO LRFD Strut-and-Tie Model Design Examples
1.57' 2.43' 2.43' 1.57'
9 18
k
6.2
Direct all correspondence to: Shrinivas B. Bhide Portland Cement Association 5420 Old Orchard Road Skokie, Illinois 60077-1083 Voice: 847.972.9100 Fax: 847.972.9101 E-mail:
[email protected]
ENGINEERING BULLETIN EB231
AASHTO LRFD Strut-and-Tie Model Design Examples
Denis Mitchell, McGill University; Michael P. Collins, University of Toronto; and Shrinivas B. Bhide and Basile G. Rabbat, Portland Cement Association
5420 Old Orchard Road Skokie, Illinois 60077-1083 847.966.6200 Fax 847.966.9781 www.cement.org An organization of cement companies to improve and extend the uses of portland cement and concrete through market development, engineering, research, education, and public affairs work.
KEYWORDS: AASHTO LRFD Specifications, anchorage, bearing stress, B-region, bridge, compressive strut, concentrated load, concrete, crack control reinforcement, dapped end, deep beam, development length, D-region, equilibrium, factored load, footing, geometric discontinuity, hammerhead, limiting compressive stress, moving load, nodal region, node, pier, pile cap, posttensioned, prestressed concrete, prestressing steel, principal tensile strain, reinforcing steel, resistance factor, shear, St. Venant’s principle, stirrup, Strut-and-Tie Model, tension tie, transfer length, truss model, wall pier ABSTRACT: The strut-and-tie method (STM) prescribed in the AASHTO LRFD Specifications is explained. Disturbed regions of structures resulting from geometric or force discontinuities where STM must be used are identified. A step-by-step procedure for STM is provided. Five detailed design examples are also provided; they include: (1) Design of cap beam, (2) Design of footing, (3) Design of pile cap, (4) Design of dapped end region of girder, and (5) Design of hammerhead pier. REFERENCE: Denis Mitchell, Michael P. Collins, Shrinivas B. Bhide, and Basile G. Rabbat, AASHTO LRFD Strut-and-Tie Model Design Examples, EB231, Portland Cement Association, Skokie, Illinois, USA, 2004, 58 pages.
Portland Cement Association (“PCA”) is a not-for-profit organization and provides this publication solely for the continuing education of qualified professionals. THIS PUBLICATION SHOULD ONLY BE USED BY QUALIFIED PROFESSIONALS who possess all required license(s), who are competent to evaluate the significance and limitations of the information provided herein, and who accept total responsibility for the application of this information. OTHER READERS SHOULD OBTAIN ASSISTANCE FROM A QUALIFIED PROFESSIONAL BEFORE PROCEEDING.
Cover photos: I-25/I-40 “Big-I” Interchange, Albuquerque, New Mexico, winner of the 2002 PCA Bridge Design Awards Competition. (IMG15512)
PCA AND ITS MEMBERS MAKE NO EXPRESS OR IMPLIED WARRANTY WITH RESPECT TO THIS PUBLICATION OR ANY INFORMATION CONTAINED HEREIN. IN PARTICULAR, NO WARRANTY IS MADE OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. PCA AND ITS MEMBERS DISCLAIM ANY PRODUCT LIABILITY (INCLUDING WITHOUT LIMITATION ANY STRICT LIABILITY IN TORT) IN CONNECTION WITH THIS PUBLICATION OR ANY INFORMATION CONTAINED HEREIN.
© Portland Cement Association 2004 First edition WARNING: Contact with wet (unhardened) concrete, mortar, cement, or cement mixtures can cause SKIN IRRITATION, SEVERE CHEMICAL BURNS (THIRD DEGREE), or SERIOUS EYE DAMAGE. Frequent exposure may be associated with irritant and/or allergic contact dermatitis. Wear waterproof gloves, a long-sleeved shirt, full-length trousers, and proper eye protection when working with these materials. If you have to stand in wet concrete, use waterproof boots that are high enough to keep concrete from flowing into them. Wash wet concrete, mortar, cement, or cement mixtures from your skin immediately. Flush eyes with clean water immediately after contact. Indirect contact through clothing can be as serious as direct contact, so promptly rinse out wet concrete, mortar, cement, or cement mixtures from clothing. Seek immediate medical attention if you have persistent or severe discomfort.
Printed in U.S.A.
ISBN 0-89312-241-6
All rights reserved. No part of this book may be reproduced in any form without permission in writing from the publisher, except by a reviewer who wishes to quote brief passages in a review written for inclusion in a magazine or newspaper. EB231 iii
Preface
In an effort to advance the state of the art, the AASHTO LRFD Bridge Design Specifications have introduced several new concepts and design methods. They include the use of limit states design principles, reliability-based factored load combinations and, for concrete structures, new shear design methods and the introduction of a general strut-and-tie design model. These new methods are significantly different from those included in the Standard Bridge Design Specifications. By 2007 all bridges receiving federal aid will have to be designed by the LRFD Bridge Design Specifications. States have begun the transition from the Standard Specifications to the LRFD Specifications. The purpose of this bulletin is to assist bridge engineers in the proper application of the new strut-and-tie model for the design of disturbed regions of bridge components. The emphasis is placed on “how to use” the specifications. This publication also should be a valuable aid to educators and students. While every attempt has been made to ensure the accuracy of the design examples presented, PCA would be grateful to any reader who brings any typographical or other errors to our attention. Other suggestions for improvement are also most welcome. Shri Bhide Engineered Structures Department
iv
Acknowledgement
The authors wish to thank the following individuals without whose help the publication of this bulletin would not have been possible: Jian Zhou and William Cook for preparing the drawings. Dale McFarlane, Diane Vanderlinde, Caron Johnsen, and Cheryl Taylor, PCA, for the word processing, layout, and formatting of the document, and David Bilow, PCA, for technical review.
v
Table of Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. Visualize Flow of Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2. Sketch an Idealized Strut-and-Tie-Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3. Select Area of Ties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4. Check Nodal Zone Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5. Check Strength of Struts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 6. Provide Adequate Anchorage for the Ties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Example 1 – Design of Cap Beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1 Example 2 – Design of Footing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1 Example 3 – Design of Pile Cap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1 Example 4 – Design of Dapped End Region of Girder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-1 Example 5 – Design of Hammerhead Pier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-1
vi
vii
Introduction The purpose of this bulletin is to provide bridge designers with representative examples illustrating the use of strut-and-tie models in accordance with the AASHTO LRFD Specifications (AASHTO 2004). Although the concept of modeling the flow of forces in structural concrete members using struts and ties was used in the early 1900’s (Ritter, 1899; Mörsch, 1909), design procedures using strut-and-tie models only recently have been codified (CSA, 1984; AASHTO, 1994, and ACI, 2002). In the design of reinforced and prestressed concrete elements, there are two types of regions: flexural (bending) regions (B-regions) and regions near discontinuities (D-regions) (Schlaich et al., 1987; Collins and Mitchell, 1986). Within flexural regions, it is accurate to assume that plane sections prior to loading remain plane after loading and that the shear stresses are distributed in a reasonably uniform manner over the effective web area. For the design of B-regions, a sectional design approach is used. In this approach it is not necessary to address how the forces are introduced into the member. Strut-and-tie models are used primarily to design regions near discontinuities or D-regions. These regions have a disturbed flow of stresses and hence plane sections do not remain plane and the shear stresses are not uniformly distributed over the effective shear area. For the design of these regions it is important to consider how the forces are introduced into the member. Discontinuities are caused by abrupt changes in cross-sectional dimensions (e.g., the presence of openings or dapped ends) or abrupt changes in applied forces (e.g., support reactions, large concentrated loads and post-tensioned anchorage zones). For flexural regions, the AASHTO LRFD Specifications (§5.8.1.1)* permit the use of either traditional sectional models (§5.8.3) or the strut-and-tie model (§5.6.3). For regions near significant discontinuities, the use of the strut-and tie model is required (§5.8.1.2). A concentrated load which causes more than 50% of the shear at the face of the support and is closer than 2d from the support face is a situation that requires a strut-and-tie model (§5.8.1.1). If we consider the case of a simply supported beam of depth h, subjected to a concentrated load applied at midspan, there will be three disturbed regions, one adjacent to each support and one centered at midspan. The regions near the supports will be about h long while the disturbed region near midspan will be about 2h long (St. Venant’s principle). When the distance between the applied load and the support is less than about 2h, the disturbed regions will overlap. For typical girders (without geometric discontinuities) where the span is often about 18h, the presence of the disturbed regions has very little influence on the overall behavior of the member and hence the localized influence of these regions normally is ignored in design. However, if the span of the beam is less than 4h, all of the beam will be a disturbed region and hence the overall behavior will be influenced strongly by the disturbed flow of stresses. For this case the strut-and-tie model is used for design. Figure 1 illustrates some examples of disturbed regions with the flow of stresses modeled with concrete compressive struts and the tension ties provided by reinforcement. The compressive struts are shown with dashed lines while the tension ties are indicated with thick solid lines. Figure 1(a) shows a simply supported B-region in which the principal compressive stress trajectories will be essentially parallel. This flow is modeled by diagonal struts. In the D-region near the support, the principal compressive stress trajectories fan out from the support. These fanning compressive stresses can be represented by a diagonal strut along the centerline of the fan. The strut-and-tie model uses straight-line compressive struts that are assumed to act along the center of the flow of the compressive stresses. Each vertical tension tie represents the tension forces in a number of stirrups over a certain
*
Article numbers of the AASHTO LRFD Specifications are preceded by the symbol “§” 1
length of the member. The length of this stirrup band usually is taken as between 0.8h to about 2h, depending on the geometry and loading. D-region with fanning compressive stresses
B-region with uniform compressive stresses
tension tie
(a) Simply supported beam
(b) Pier cap
(c) Deep beam
(d) Wall pier with concentrated loads
Figure 1. Examples of D-regions modeled with struts and ties.
Figure 1(b) shows a double-sided corbel, with the flow of the compressive stresses into the corbel modeled by concrete compressive struts. The tension tie along the top of the corbel is required for the equilibrium of the truss formed by the strut-and-tie model. Figure 1(c) shows a deep beam subjected to a concentrated load at midspan. The entire beam is a D-region, in accordance with St. Venant’s principle. The flow of forces into the beam is modeled using compressive struts from the applied load into the support reaction areas. The tension tie for such beams typically consists of several layers of reinforcing bars or post-tensioned tendons. Figure 1(d) illustrates the fanning compressive stresses resulting from concentrated forces being applied to the top of a wall pier and the resulting tension ties required for equilibrium.
2
The flow of compressive stresses can follow a curved path as shown in Figure 2(a) as the compressive struts widen between the loading and reaction plates. This curving of the compressive stresses causes tension perpendicular to the flow as shown by the inclined tension ties in Figure 2(b). Instead of using curved compressive struts, the AASHTO LRFD Specifications prescribe simple straight-line struts to model the flow of compression and additional uniformly distributed horizontal and vertical reinforcement (§5.6.3.6) to control cracking in the disturbed region (Figure 2[c]). It is noted that slabs and footings are exempt from the crack control reinforcement requirement (§5.6.3.6). For these types of members the diagonal compressive stresses are usually considerably lower in magnitude than those for regions such as dapped ended beams, hammerhead piers, or deep beams. crack control reinforcement
(a) Flow of compressive stresses
(b) Tensions due to flow of compressive stresses
(c) Assumption of straight compressive struts
Figure 2. Straight-line struts and required crack control reinforcement (adapted from Schlaich et al., 1987).
The main steps for design of a D-region can be summarized as follows: 1. 2. 3. 4. 5. 6.
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 the ties
The steps required to design a D-region, such as the deep beam shown in Figure 3, are given below: 1. Visualize Flow of Stresses Visualize the flow of the compressive stresses in the D-region, and idealize the flow with straight-line struts. Determine the locations of tension ties required for equilibrium. In sketching the truss, made up of struts and ties, make suitable assumptions for the positions of the centroids of the ties, allowing sufficient space for placement of the required reinforcement. 2. Sketch an Idealized Strut-and-Tie Model Sketch the strut-and-tie model in the form of a truss idealization. In modeling bridge girders subjected to moving loads, it is convenient to space the nodes of the truss such that the truck axle loads can be applied directly to the nodes. Determine the forces required in the struts and ties due to the applied factored loads. For moving loads one
3
should consider different positions of the axle loads to achieve the maximum design forces for a particular member. If the member being designed is statically indeterminate (e.g., a two-span continuous deep beam subjected to point loads in each span), a realistic strut-and-tie model also will be statically indeterminate. The support reactions for this indeterminate truss can be found from elastic analysis of the actual member, and the internal forces in the truss model then can be determined from statics. Alternatively, stiffness values can be assigned to the members of the truss and hence the forces in the members and the reactions of the truss can be determined by analyzing the statically indeterminate truss. For this analysis the stiffness of the ties can be taken as the axial stiffness of the reinforcing bars and prestressing steel (e.g., E s A st + E p A ps ) that the tie represents. The stiffness
of a strut may be taken as the cross-sectional area of the strut multiplied by the concrete modulus and the area of any compression steel multiplied by the steel modulus (i.e., E c A cs + E s A s + E p A ps ). nodal zone
0.85 φ f c'
_ φ fcu fc <
ε1
0.75 φ f c'
tension tie develop tension tie force over this length (a) Strut-and-tie model truss node
αs
tension tie
effective anchorage area
(b) Truss model
Figure 3. Strut-and-tie model and idealized truss model for deep beam.
3. Select Area of Ties
Choose the required area of reinforcement in the tension ties to ensure that the factored resistance in the ties equals or exceeds the required factored force in the ties (§5.6.3.2 and §5.6.3.4.1).
4
4. Check Nodal Zone Stresses
Compare the nodal zone stresses with the nodal zone stress limits. Because of the dimensions of the struts and the reinforcement making up the ties, the truss joint, or node, represents a nodal zone with finite dimensions (see Figure 3). The nodal zones serve to transfer the forces between the ties, the struts, the support reaction areas , CCC node CCT node CCT node CTT node (a) Deep beam
(b) Dapped end
lb lb sin θ + ha cos θ
θ
da
θ ha
0.5 ha lb sin θ + da cos θ
lb
(c) Strut anchored by bearing plate and reinforcing bars (CCT node) Stress limit = 0.75 φ fc'
(d) Strut anchored by bearing plate and strut (CCC node) Stress limit = 0.85 φ fc'
lb sin θ + ha cos θ θ ha
0.5 ha lb
lb
(e) Strut anchored by two tension ties (CTT node) Stress limit = 0.65 φ fc'
(f) Continuous beam support (CCC node) Stress limit = 0.85 φ fc'
Figure 4. Types of nodal zones and dimensions of struts.
and the loaded bearing surfaces. The nodal zones occur at the intersections of the truss elements and at the loading points and support reaction areas. It is important to allow for the transfer of forces without overstressing the 5
concrete in the nodal zones. In many practical cases it will be necessary to spread the tie reinforcement into several layers so that the nodal zone stress limit is not exceeded in the effective anchorage area (see Figure 3[b]). The integrity of the nodal zone is checked by comparing the normal stresses applied to the boundaries of the nodal zone with the specified nodal zone stress limits. The compressive strength of the nodal zone depends on the tensile straining from intersecting tension ties and on confinement due to the presence of transverse reinforcement. The nodal zone stress limits in the AASHTO specifications (§5.6.3.5) depend on the number of ties that are being anchored in the nodal zone. Figure 4 illustrates a number of typical nodal zone conditions and their corresponding stress limits depending on the presence of compressive struts (indicated by “C”) and tension ties (indicated by “T”). 5. Check Strength of Struts
Compare the factored resistance of the struts with the calculated factored loads in the strut members (§5.6.3.2 and §5.6.3.3). The nominal resistance of the strut is determined by multiplying the limiting compressive stress, f cu , by the effective cross-sectional area of the strut, A cs . The limiting compressive stress depends on the angle, α s , between the compressive strut and the tension tie and the tensile strain, ε s , in the tie where it crosses the strut. The limiting compressive stress is given as: f cu =
f c' ≤ 0.85f c' 0.8 + 170ε1
in which the principle tensile strain in the concrete, ε1 , is taken as:
ε1 = ε s + (ε s + 0.002 ) cot 2 α s where α s is the smallest angle between the compressive strut and the tension tie and ε s is the average tensile strain in the concrete in the direction of the tension tie. For a tension tie consisting of reinforcing bars, εs , can be taken as the tensile strain due to factored loads in the reinforcing bars. As shown in Figure 5, the limiting compressive stress, fcu, reduces significantly as the angle, α s , becomes smaller. A strain value of εs of 0.002 would correspond to a tie yielding in tension at the location where it crosses the centerline of the strut. If the tension force in the tie changes as it crosses the strut, then εs may be taken as the value of the strain where the tie meets the centerline of the strut. For example if the tie is developed within the width of the strut, the tensile strain in the tie might change from 0.002 at the inner edge of the strut to zero at the outer edge of the strut, resulting in an εs value of 0.001. For a tension tie consisting of prestressed steel, εs can be taken as zero until the precompression in the concrete due to the prestress is overcome. Figure 5 illustrates the influence of α s and εs on the limiting compressive stress, fcu. It can be seen that even struts at very shallow angles (e.g., 20°) can resist significant stresses ( 0.297f c' ) if the tie is prestressed.
6
1.0 0.9 0.8 0.7
εs = 0
fcu /f'c
0.6
εs = 0.001
0.5
εs = 0.002
0.4 0.3 0.2 0.1 0.0 0
10
20
30
40
50
60
70
80
90
αs Figure 5. Influence of α s and ε s on the limiting compressive stress in a strut.
Figure 6 shows two specific examples of calculating fcu for cases involving reinforcing bars as ties. For the two cases shown, if the strain εs is 0.002 at the inner edge of the strut, and it is conservatively assumed that this strain reduces to 0.001 at the centerline of the strut, then the limit on the compressive strength of the struts, fcu, changes from 0.68f c' for the case where α s equals 45o to 0.30f c' for the case where α s equals 25o (see § 5.6.3.3.3).
ε1 ε1 αs = 45o
α s = 2 5o
εs
εs
(a) αs = 45 o
(b) αs = 25o
fcu = 0.68 fc'
fcu = 0.30 fc'
Figure 6. Examples of determining fcu..
If the strut is anchored by a bearing area, the cross-sectional dimensions of the strut will be influenced by the length of the bearing area, the dimensions of the adjacent ties or struts, and the inclination of the strut (see Figures 4[c] and 4[d]). If the strut is anchored only by reinforcement, the effective dimensions of the strut are related to
7
the reinforcement details as shown in Figure 7. The strut bears against the longitudinal reinforcing bars which in turn are anchored by the stirrups. It is assumed that the effective width of the strut across the thickness of the member can extend a distance of up to six times the diameter of the longitudinal bar anchored by the stirrups (i.e., 6d ba ) (see § 5.6.3.3.2). X
