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TORSION SIMPLIFIED: A FAILURE PLANE MODEL FOR DESIGN

OF SPANDREL BEAMS

Gary Klein, Gregory Lucier, Sami Rizkalla, Paul Zia and Harry Gleich

Biography: Gary Klein, FACI, is Executive Vice President and Senior Principal at Wiss,

Janney, Elstner Associates, Northbrook, Ill. A licensed structural engineer, he is a member of

ACI Committees 318, Structural Concrete Building Code; 342, Evaluation of Concrete

Bridges and Bridge Elements; and Joint ACI-ASCE Committee 445, Shear and Torsion.

ACI member, Gregory Lucier is Manager of the Constructed Facilities Laboratory and a

doctoral candidate in civil engineering at North Carolina State University in Raleigh, N.C.

Sami Rizkalla, FACI, is Distinguished Professor of Civil Engineering and Construction at

North Carolina State University. He is a member of ACI Committee 440, Fiber-Reinforced

Polymer Reinforcement; Joint ACI-ASCE Committee 423, Prestressed Concrete; and Joint

ACI-ASCE Committee 550, Precast Concrete Structures.

Paul Zia, ACI Past President and Honorary Member, is a Distinguished University Professor

Emeritus at North Carolina State University. He is a member of ACI Committees 363, High-

Strength Concrete; Joint ACI-ASCE Committee 423, Prestressed Concrete; ACI Committee

440, Fiber-Reinforced Polymer Reinforcement; Joint ACI-ASCE Committee 445, Shear and

Torsion; and the Concrete Research Council.

Harry Gleich, FACI, is Vice President of Engineering at Metromont Corporation, Greenville,

SC. A licensed structural engineer, he is a member ACI Committees 318-A, General,

Concrete and Construction; 362, Parking Structures; Joint ACI-ASCE 423, Prestressed

Concrete; Joint ACI-ASCE 533, Precast Panels; and Chairman of Joint ACI-ASCE 550,

Precast Concrete Structures.

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ABSTRACT

Based on an extensive program of analyses and tests, a simplified design procedure for

combined shear and torsion has been developed for precast spandrel beams. The design

model is based on a simple but realistic premise: beams resisting torsion must also resist

shear, and, as such, failure will occur along a plane inclined upward toward midspan. The

design methods are based on straightforward physical models of forces acting on the failure

plane. Torsion is divided into bending and twist components, and simple equations are used

to determine the required reinforcement for skew bending and to proportion the concrete

section for twist. The failure plane shear/torsion model, developed for slender precast

spandrels, should also be applicable to compact precast sections and to cast-in-place beams.

Further research and testing that examines the shear/torsion behavior of such members

promises to simplify their design and detailing as well.

Keywords: Torsion, spandrel beam, skew bending, twist

INTRODUCTION

Until recently, design for torsion was largely based on studies of beams tested in pure torsion.

Current ACI 318 design equations assume the face shell will separate from the core.

Accordingly, closed ties are required to resist torsional forces and the outer shell is neglected

in torsional strength equations. For precast beams loaded in torsion, complicated formulas are

used to account for the interaction of torsion with shear. This article describes the

development and application of a simplified model for design of concrete beams that resist

torsion. The design model is based on a simple but realistic premise: beams resisting torsion

must also resist shear, and, as such, failure will occur along a plane inclined upward toward

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midspan. By dividing torsion into bending and twist components that act on the inclined

plane, the design model provides a direct procedure for estimating the cracking load and for

proportioning the concrete section and reinforcement to resist the combined effects of shear

and torsion.

The authors recently completed a comprehensive research project for the Precast/Prestressed

Concrete Institute aimed at developing a rational design methodology for slender precast

spandrel beams, both prestressed and conventionally reinforced. (This research work for PCI

is described in References 1 and 2, and is hereafter referred to as the PCI Spandrel Research.)

The experimental program included testing of 16 full-sized spandrel beams to failure, and

explored alternative design and detailing approaches for the web steel. In particular, the

research program re-examined the need for closed ties in slender precast spandrel beams,

which are especially difficult to install. The research findings enabled the authors to develop

a torsion design approach based on forces acting on the failure plane, thus simplifying torsion

design and providing a direct link to the design approaches commonly used for bending and

shear.

The purpose of this paper is to explain this simplified design model and to discuss its

potential application to concrete members other than slender precast spandrel beams. After a

brief review of classical torsion theory, the paper describes development of the failure plane

shear/torsion model and describes how the failure-plane principles may be applied to compact

sections and cast-in-place construction.

BRIEF REVIEW OF CLASSICAL TORSION THEORY

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In 1853, St. Venant developed an elastic solution for torsion of a member with a rectangular

cross-section. St. Venant showed that torsional stresses flow in a circulatory pattern as can be

seen in Figure 1(a). The maximum torsional shear stresses occur along the middle regions of

the long sides. The relationship between torque, T, and torsional shear stress, τ, is given by

(1)

where

x = smaller dimension of the rectangular section;

y = larger dimension of the rectangular section; and

α = coefficient that varies between 0.208( ⁄ ) and 0.333( ⁄ ) .

In 1903, Prandtl discovered an analogy between torsional stresses and the deflection of a

membrane under uniform loading. The direction and magnitude of the developed shear

stresses can be visualized by the contour lines of the deflected surface of a membrane, such

as a soap bubble, within a boundary having a shape matching that of the cross section. Shear

stresses are proportional to the slope of the membrane, and the total torque is proportional to

the area under the membrane. The shear stress contour lines and “soap bubble” membrane are

depicted in Figure 1(b).

THE COMPONENTS OF TORSION

Figure 2 shows the torsional forces acting on the end region of the precast spandrel beam.

Assuming an inclined failure plane, the factored torque, T (green arrow), is divided into two

components: 1) skew bending (blue arrow), and 2) twist (yellow arrow). Skew bending, also

referred to as plate bending for slender sections, is a bending moment about the 45-degree

failure plane and is equal to √ ⁄ . The second component of torsion, twist, is a torque acting

about a line perpendicular to the failure plane and is also equal to √ ⁄ . The vector sum of

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these components of torsion equal the applied torque, T, which acts about a line parallel to the

axis of the member.

Assuming a 45-degree incline failure plane, the circulatory shear stresses acting on the

section cut at 90 degrees are transformed in accordance with Mohr’s circle. A colored

contoured plot of the stresses acting normal to the 45-degree plane can be seen in Figure 3(a).

(Figure 3 was developed using a finite element model with the brick elements oriented at 45

degrees. Although unconventional, this technique facilitated plotting stress contours on the

inclined failure plane.) Shear stresses along the sides of the 90-degree section are transformed

into normal stresses that produce a bending couple (skew bending). As predicted by the soap

bubble analogy, shear stresses acting on the 45-degree inclined plane are concentrated at the

top and bottom of the section and produce a couple that resist the twist component of torsion.

These shear stresses are depicted in Figure 3(b).

Referring to Figure 2, the section modulus about Line a-a is equal to √ ⁄ . The tensile

stress, ft, due to bending about a-a is determined by dividing the bending moment, M, by the

section modulus, S:

√ ⁄

√ ⁄

(2)

Rearranging this result, the torsional resistance of plain concrete is given by

(3)

where is the tensile resistance of plain concrete. Note that this torque is exactly equal to

that determined from St. Venant’s equations for very slender sections ( ⁄ ) . This

equality is not coincidental; rather, it indicates that for slender sections, the torsional

resistance of a plane concrete section is equal to the plate bending resistance across an

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inclined failure plane. Out-of-plane shear stresses resist the twist component of torsion, but

transverse shear does not control in an unreinforced rectangular section.

COMBINED SHEAR AND TORSION

Practically speaking, pure torsion only exists in the laboratory. In actual structures, torsion is

always combined with shear. Referring to Figure 4, stresses due to torsion add to shear

stresses on the inside face of a spandrel beam, while torsional stresses counteract shear

stresses on the outside face.

In the experimental phase of the PCI Spandrel Beam Research, slender spandrel beams

consistently failed due to combined shear and torsion in the end region. An example of such a

failure can be seen in Figure 5. These experimental results are described in detail by Lucier et

al.1

Spalling of the concrete outside the perimeter of the transverse reinforcing is expected for

compact beams loaded in pure torsion, as observed by Mitchell and Collins.2 Face-shall

spalling was not observed in any of the spandrel beams tested under the PCI Spandrel

Research; rather, the failure plane was inclined upward toward midspan at an angle of

roughly 45 degrees, which is indicative of a classical shear failure. As can be seen in Figure 5,

the failure plane was skewed near the top and bottom of the section. The direction of the

skew is consistent with the out-of-plane shear stresses due to the twist component of torsion

(Figure 3(b)).

In both conventionally reinforced and pretensioned concrete beams, shear reinforcement is

provided to make up the difference between the factored shear force and the allowable

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concrete contribution to shear. The forces carried by individual stirrups are depicted in Fig.

6(a). Fig. 6(b) shows the forces developed in transverse and longitudinal reinforcement

provided for the skew bending component of torsion and also shows the corresponding

compression on the opposite web-face concrete. For a typical spandrel beam, the

reinforcement required for shear and the reinforcement required for the skew bending

component of torsion are additive on the inside face. Conversely, the compression due to

skew bending on the outside face counteracts the shear force.

FAILURE PLANE DESIGN APPROACH FOR SLENDER SPANDRELS

Based on the fundamental principles described above, a design procedure has been developed

for combined shear and torsion in slender spandrel beams. Details of this design approach are

described by Lucier et al.3 The key design steps are summarized in the following paragraphs.

Other steps, including determining loading demands and design for flexure follow standard

practice.

Design for Skew Bending Component of Torsion

Referring to Figure 6, longitudinal and transverse reinforcement must be provided on the

inside face to resist the plate bending component of torsion. Assuming equal amounts of

longitudinal and transverse reinforcement, the required reinforcement is given by

⁄ ⁄

(4)

where

Asv = area of vertical reinforcement for skew bending

Asl = area of longitudinal reinforcement for skew bending

s = spacing of reinforcement

= factored torque

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= strength reduction factor for flexure

= yield strength of mild steel

= effective depth from outer surface of web to centroid of combined horizontal and

vertical steel reinforcement of web; usually taken as web thickness less concrete cover

less diameter of innerface vertical steel bars

h = height of spandrel section

As illustrated in Figure 7, reinforcement should be provided to resist the potential skew

bending force about a line inclined downward toward midspan, opposite the primary

shear/torsion failure plane. Accordingly, the longitudinal reinforcement required by Equation

4 should also be provided on the outside face. The transverse reinforcement required by

Equation 4 need not be provided on the outside face if shear reinforcement is provided as

discussed below under Design for Shear.

Design for Shear

Shear and torsion interact on the failure surface. As shown in Figure 3, torsion acting on the

inclined failure plane creates a tension and compression couple along the sides of the inclined

section as well as transverse shear stresses concentrated at the top and bottom of the section.

These stresses interact with shear-induced tensile stresses across the inclined section, which,

before cracking, are distributed parabolically (maximum at mid-height) over the height of the

section. Thus, torsion and shear combine to increase diagonal tension near mid-height of the

inside face, but torsion and shear counteract each other on the outside face.

Analysis of the PCI Spandrel Research data indicates that, after stripping out strength

reduction factors and using actual material properties, the residual concrete contribution to

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shear was in excess of √ , where is the specified concrete compressive strength.

Furthermore, the as-tested strength of all spandrel beams, which were designed on the

principles described herein, exhibited strengths in combined shear and torsion that

substantially exceeded the factored design forces.

Accordingly, design for shear should follow standard practice for both of prestressed and

conventionally reinforced sections. The required shear reinforcement is given by

⁄

⁄ (5)

where

= area of vertical reinforcement required for shear

= factored shear force

= nominal concrete shear

= yield strength of mild steel

d = distance from extreme compression fiber to centroid of longitudinal reinforcement

On the inside face, half of the required shear reinforcement required by Equation 5 should be

added to the vertical plate bending reinforcement required by Equation 4. As illustrated in

Figure 6, vertical tension due to shear and plate bending add together on the inside face. On

the outside face, half of the shear reinforcement required by Equation 5 should be provided,

even though shear is counteracted by torsion (see Figure 6). Vertical reinforcement on the

outside face contributes to resisting outside face plate bending, as illustrated in Figure 7.

Design for Twist Component of Torsion

Evaluation of the twisting resistance is based on a well-accepted mechanism used to transfer

unbalanced moments from reinforced concrete slabs to columns. In the case of a column-slab

interface, ACI 318-11 section 11.11.7.2 requires that the shear stress "shall vary linearly

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about the centroid of the critical section."4 As illustrated in Figure 8, shear stresses at the 45°

failure plane are assumed to very linearly about the centroid of the section. This design model

was then calibrated to the experimental results. The resulting torsional strength is given by

√ (6)

where

Tnt = nominal twist resistance of section; and

h = height of the spandrel section.

The corresponding peak transverse shear stress in the concrete is approximately √ .

First Cracking Load

The load likely causing initial cracking in the end region of a spandrel beam is often of

interest. The appearance of cracks can be problematic. The crack may be considered

aesthetically objectionable, and structural concerns may arise even though narrow diagonal

cracks do not necessarily indicate a structural deficiency.

The equations for tensile stress due to skew bending (Equation 3) and diagonal tension due to

shear can be combined and rearranged to predict the shear force at cracking, . Assuming a

limiting tensile stress of concrete of √ , is given by

[ √

( ⁄ )] (7)

where…

e = eccentricity; and

bw= thickness of web.

Comparison with PCI Spandrel Research experimental data indicates that Equation 7

provides a conservative estimate of the load at which cracking is first likely to appear. First

cracking loads can be increased by increasing web thickness, increasing the concrete strength,

or distributing prestressing strands through the height of the web. Prestressing strands

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concentrated only near the bottom of the section do not effectively control diagonal cracks

near the support.

Detailing Considerations

Failure for slender spandrel beams occurs along an inclined plane and is not accompanied by

face-shell spalling. As such, closed ties are not required to confine the core. In typical cases,

the concrete section can be proportioned to resist the twist component of torsion;

supplemental transverse reinforcement across the narrow top and bottom faces is not needed.

The PCI Spandrel Research verified that open web reinforcement is a safe and cost-effective

alternative to closed ties for slender spandrels.

DESIGN OF COMPACT SPANDRELS FOR COMBINED SHEAR AND TORSION

The principles described above, which were developed for slender spandrels, generally apply

to compact spandrels as well. A currently un-published test program included a few compact

spandrels. (The authors participated in an experimental research program for a precast

concrete producer that investigated shear/torsion response of compact precast spandrel

beams.) As shown in Figure 9, the shear/torsion failure in the end region is similar to that

observed in slender spandrels. However, compared to slender spandrels, the behavior of

compact spandrels is different in two important ways.

First, the expression for twist resistance (Equation 6) was developed for slender spandrels

assuming the shear stress varies linearly about the centroid of the critical section, as

illustrated in Figure 8. Using the approach described in ACI 318 commentary section

R1.11.7.2.2, the general equation for calculation of shear stress, vut, is:

√

⁄ ⁄

(8)

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For slender spandrels, the second term in the denominator is relatively small and can be

ignored. For compact spandrels, the second term is significant and contributes to twist

resistance. This difference can be accommodated by increasing the coefficient in Equation 6

as a function of the aspect ratio.

Second, closed ties are much more effective in compact spandrels because spiral cracks

across the top and bottom of the section cross several ties. Furthermore, the twist resistance

of the concrete section alone is likely insufficient for typical compact spandrels. In such cases,

transverse steel must be placed on all faces of the cross-section, as will probably be required

for all compact sections. However, face-shell spalling is still unlikely for compact sections,

except in rare cases where members are heavily reinforced with closed ties and are primarily

loaded in torsion. The experimental results available to date suggest that the ACI detailing

requirements for transverse torsional reinforcement can be relaxed for typical compact

spandrels beams.

DESIGN OF CAST-IN-PLACE SECTIONS FOR COMBINED

SHEAR AND TORSION

The principles described above can be applied to cast-in-place construction. However, for

typical cast-in-place members, the torsional demand is indeterminate. The design torque is

allowed to be reduced to the cracking torque due to redistribution of internal forces after

cracking.

Also, in the negative moment region of continuous cast-in-place beams, maximum shear and

torsion occur in a region of high bending moment. Accordingly, the interaction among

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flexure, shear, and torsion must be considered. In contrast, there is very little bending

moment near the supports of typical precast concrete spandrel beams.

In spite of these important differences, a design based on forces acting on an inclined failure

plane is possible. As is the case with precast construction, face-shell spalling is unlikely to

develop in practice in cast-in-place beams subjected to combined shear and torsion.

Accordingly, the current ACI design requirements for torsion that neglect the strength

contribution of concrete outside the core should be reconsidered. Although the detailing

requirements could be relaxed, requirements for closed ties should be retained because closed

ties enhance ductility. Ductility of cast-in-place beams enables redistribution of indeterminate

flexural and torsional forces, while such forces in precast beams are determinate.

CONCLUSIONS

In conclusion, design for combined shear and torsion based on forces acting on the inclined

failure plane has simplified the design and detailing of slender precast spandrel beams. The

design procedure has been verified by analyses and tests. The design methods are based on

straightforward physical models of forces acting on the failure plane that are comparable to

the physical models used for calculating bending and shear strength. By dividing torsion into

bending and twist components, simple equations are used to determine the reinforcement

required for skew bending and to proportion the concrete section for twist. An experimentally

verified equation for estimating the shear/torsion forces causing diagonal cracks has also been

developed.

Tests conducted on both slender and compact spandrel beams verified that face-shell spalling

is not associated with failures due to combined shear and torsion. Thus, the entire cross-

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section contributes to torsional strength, and closed ties are not required for slender spandrels

to resist torsion.

The failure plane shear/torsion model, developed for slender precast spandrels, can also be

applied to compact precast sections and cast-in-place beams. Further research and testing that

examines the shear/torsion behavior of such members promises to simplify their design and

detailing as well.

REFERENCES

1 Lucier, G.; Walter, C.; Rizkalla, S.; Zia, P.; and Klein, G. "Development of a Rational

Design Methodology for Precast Concrete Slender Spandrel Beams: Part 1 Experimental

Program," PCI Journal, V. 56, No. 2 (Spring 2011).

2. Collins, M. and Mitchell, D., "Shear and Torsion Design of Prestressed and Not

Prestressed Concrete Beams," PCI Journal, V. 25, No. 5, pp. 32-100.

3. Lucier, G.; Walter, C.; Rizkalla, S.; Zia, P.; and Klein, G. "Development of a Rational

Design Methodology for Precast Concrete Slender Spandrel Beams: Part 2 Analysis and

Design Guidelines," PCI Journal, V. 56, No. 4 (Fall 2011), pp. 106-133.

4. ACI Committee 318, "Building Code Requirements for Structural Concrete (ACI 318-11)

and Commentary," American Concrete Institute, Farmington Hills, Michigan, 2011, 503 pp.

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FIGURES

List of Figures:

Fig. 1 – Torsional stresses acting on a rectangular section.

Fig. 2 – Torsional forces acting on the end region of a precast spandrel beam.

Fig. 3 – Stresses due to torsion acting on 45-degree plane.

Fig. 4 – Shear, torsional and combined stresses acting on a precast spandrel beam.

Fig. 5 – Failure in end region of precast spandrel beam due to combined shear and torsion.

Fig. 6 – Stresses due to torsion acting on 45-degree plane.

Fig. 7 – Outside face skew bending.

Fig. 8 – Linear distribution of shear stresses on inclined failure plane.

Fig. 9 – Failure in end region of compact spandrel due to combined shear and torsion.

(a) “Flow” of shear stress around the section. (b) “Soap bubble” contour lines. Circulatory

shear stress is proportional to the slope of the

soap bubble.

Fig. 1 – Torsional stresses acting on a rectangular section.

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0

Fig. 2 – Torsional forces acting on the end region of a precast spandrel beam.

(a) Stresses normal to 45-degree plane (skew

bending)

(b) Out-of -plane shear stress (twist)

Fig. 3 – Stresses due to torsion acting on 45-degree plane.

Tension Compression Shear (-) Shear (+)

T T

Torque, T

Twist, Tt

Skew bending, Tb

a

a

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Fig. 4 – Shear, torsional and combined stresses acting on a precast spandrel beam.

(a) Inside face (b) Outside face

Fig. 5 – Failure in end region of precast spandrel beam due to combined shear and

torsion

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(a) Forces in stirrups provided for shear (b) Forces in transverse and longitudinal

reinforcement provided for the skew

bending component of torsion, and the

corresponding compression on the opposite

face

Fig. 6 – Forces acting on 45-degree plane

Fig. 7 – Outside face skew bending

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Fig. 8 – Linear distribution of shear stresses on inclined failure plane

Fig. 9 – Failure in end region of compact spandrel due to combined shear and torsion