Question: Why should we conduct the calculation of Chapter 4 Ultimate Bearing Capacity of Diagonal Section for Flexural Members 4.1 Introduction Question: Why should we conduct the calculation of shear strength for flexural members? Take a simply supported beam under symmetrical concentrated loads for example. Pure bending segment ：As the bending moment M is larger, the vertical crack occurs, then the failure happens along the perpendicular section.（the longitudinal reinforcements required, see previous chapter.） Shear-bending segment： As the combined actions of shear V and moment M, the diagonal crack occurs, then the failure happens along the diagonal section.（the transverse ties or bent-up bars required, see this chapter.） Web reinforcement is composed of transverse ties and bent-up bars

The bent-up bar is perpendicular to the diagonal crack, so it is the effective way to resist the crack. Existing problems In design of beams, transverse ties (stirrups) are usually used firstly, then bent-up bars are considered in some cases. Longitudinal reinforcements, web reinforcements and other detailing bars form skeleton of steel in a beam.

Rigid skeleton of steel Bent-up bars Skeleton bars Detailing bars Stirrups Longitudinal reinforcement 梁的钢筋构造 Rigid skeleton of steel

stirrups (transverse ties) bent-up bars web reinforcement longitudinal reinforcement web reinforcement some other detailing bars skeleton of steel

Notes: (1)As the ratio of a slab is so large that it usually has adequate bearing capacity in diagonal section. Therefore the bearing capacity in normal section is only needed in the design for a slab. No stirrups exist in the concrete slab except thick slab. (such as thick pile cap) （2）Diagonal section capacity includes shearing resistance and bending resistance in a diagonal section.

斜截面受剪承载力---由计算+构造满足。 本章内容 配箍筋或弯起筋 斜截面受弯承载力---由对纵筋和箍筋的构造要求 来满足。 Shear capacity of diagonal section --- met by calculating & detailing. (stirrups or bent-up bars needed) Bending capacity of diagonal section --- met by detailing only (longitudinal reinforcements and stirrups)

4.2 Diagonal Cracks，Shear Span Ratio & Failure Modes of Diagonal Section of Beams 1. Formation of Diagonal Cracks Fig.5.2 shows principal stress trajectories for a simply supported RC beam without web reinforcement under two symmetrical concentrated loads. ( solid lines principal tensile stress trajectories dotted lines principal compressive stress trajectories) Before diagonal crack occurs, the beam may be assumed as a homogeneous beam. The principal tension stress and the principal compression stress at any point can be determined by the formulas from Material Mechanics. The diagonal crack is along the direction of principal compressive stress trajectories.

Principal stress trajectories Principal tensile stress Principal compressive stress The angle of αis the angle of principal tensile stress with the longitudinal axis of the beam.

Principal tensile stress trajectories Principal compressive stress trajectories

Principal stress trajectories in a homogeneous beam

剪应力的分布 Distribution of shear stress at section For rectangle section, the maximum stress locates at the neutral axis, and zero stress at both top and bottom of the section.

Diagonal cracks The character of two types of diagonal cracks : When the principal tensile stress exceeds the tensile strength of concrete, diagonal cracks may be formed. Web-shear cracks Flexural-shear cracks The character of two types of diagonal cracks : 1）web-shear cracks, resulting from the larger shearing stress near the neutral axis, is wide in the middle and thin in both end of the crack, like a date pit. (usually occurring in thin-web beams) 2）flexural-shear cracks, developed as an extension of the flexural cracks. (usually occurring in normal beams)

flexural cracks web-shear cracks flexural cracks flexural-shear cracks

Beam under concentrated load ： 2. Shear Span Ratio Beam under concentrated load ： — calculated shear span ratio where is the shear span, is the effective depth. is the ratio of shear span and effective depth, it shows the position of the concentrated load. — generalized shear span ratio Shear span ratio is a very important parameter, which is closely related with the mode of the shear failure and shear strength of a beam.

适筋梁破坏 超筋梁破坏 少筋梁破坏 剪压破坏 斜压破坏 斜拉破坏 3. Failure Modes of Diagonal Section Failure modes of Normal Section: determined by steel ratio 适筋梁破坏 超筋梁破坏 少筋梁破坏 Failure modes of Diagonal Section: determined by shear span ratio and stirrup ratio 剪压破坏 斜压破坏 斜拉破坏 （适箍梁） （超箍梁） （少箍梁） Under- reinforced beam Over-reinforced beam Scarce-reinforced beam Shear compression failure Diagonal compression failure Diagonal splitting failure

1. Diagonal compression failure 斜压破坏（超箍梁破坏） In case of happening：When and the web reinforcement is numerous, or for thin-web beam（T-section and I-section beam） Characteristics ：The principal compression stress in the concrete near the support exceeds the compressive strength of concrete and the concrete between the support and the concentrated load is crushed by the principal compression. However the stirrups haven’t yielded before failure. It usually occurs in the zone with small M and large V.

斜压破坏特点 斜裂缝首先在梁腹部出现。 梁是因为斜压柱体被压碎而破坏，故称为斜压破坏。 华南理工大学土木工程系 斜压破坏特点 F 斜裂缝首先在梁腹部出现。 梁是因为斜压柱体被压碎而破坏，故称为斜压破坏。 破坏时与斜裂缝相交的箍筋应力达不到屈服强度，梁的受剪承载力主要取决于混凝土斜压柱体的受压承载力。

2. 斜拉破坏（少箍梁破坏） Diagonal splitting failure 2. 斜拉破坏（少箍梁破坏） Diagonal splitting failure In case of happening：When and simultaneously the amount of web reinforcement is low. Characteristics：There is only one diagonal crack occurred with larger width. Along two sides of the crack, concrete member is split into two parts. The stirrups yield immediately after the crack. Failure process: Because of the shear stress rapidly increases in the residual concrete section, the principal tensile stress in the compression zone exceeds the tensile strength of the concrete. The diagonal crack extends rapidly upward and splits the member into two.

其斜截面受剪承载力主要取决于混凝土的抗拉强度。 斜拉破坏特点 F 斜裂缝一出现，即很快形成临界斜裂缝，并迅速延伸到集中荷载作用点处。 t sx stp 因腹筋数量过少，腹筋很快达到屈服，不能抑制斜裂缝的开展，梁斜向被拉裂成两部分而突然破坏。 这种破坏是混凝土在正应力和剪应力共同作用下发生的主拉应力破坏，故称为斜拉破坏。 其斜截面受剪承载力主要取决于混凝土的抗拉强度。

3. 剪压破坏（适箍梁破坏） Shear compression failure In case of happening：When is in the area of ,and the web reinforcements are properly set, or is large （λ >3），and the web reinforcements are not less. Characteristics：Many cracks occur during loading. When a critical diagonal crack occurs, concrete shear compressive zone gets small and small, and the concrete stress in this residual zone is increasing under normal stress and shear stress, until to exceed concrete strength under Bi-axial loads. Then concrete is crushed. The stirrups yield before the concrete is crushed. =

剪压破坏特点 弯剪段下边缘先出现初始垂直裂缝。 随着荷载的增加，这些初始垂直裂缝将大体上沿着主压应力轨迹向集中荷载作用点延伸。 F 弯剪段下边缘先出现初始垂直裂缝。 临界斜裂缝 随着荷载的增加，这些初始垂直裂缝将大体上沿着主压应力轨迹向集中荷载作用点延伸。 在几条斜裂缝中会形成一条主要的斜裂缝，这一斜裂缝被称为临界斜裂缝。 最后，与临界斜裂缝相交的箍筋达到屈服，剪压区混凝土在剪压复合应力作用下达到混凝土复合受力强度而破坏，梁丧失受剪承载力。

Diagonal compression failure Shear strength depends on compressive strength of concrete. Shear compression failure Shear strength depends on concrete strength under Bi-axial loads. Diagonal splitting failure Shear strength depends on tensile strength of concrete. Failure mode of diagonal section

Summary: All the three shear failures are belong to brittle failure. In contrast, shear compression failure is better than the other twos，in which the stirrup can yield and considerable cracks may develop before the member fails. In diagonal compression failure and diagonal splitting failure, the concrete or steel is not full used. They are should be avoided. The design equation of shear strength is based on the mode of shear compression failure. A concept is presented: strong column and weak beam, strong shear and weak bending.

diagonal compression failure Failure modes of Diagonal Section: Determined by shear span ratio and stirrup ratio Shear compression failure diagonal compression failure diagonal splitting failure To ensure the shear capacity of diagonal section by calculation. To avoid diagonal compression failure: The section size should be larger than a minimal area To avoid diagonal splitting failure: the minimum amount of transverse stirrups, stirrup diameter and stirrup space are limited.

4.3 Factors Affecting the Strength of Diagonal Section 1. Shear span ratio 2. Strength of concrete 3. Stirrup ratio and the strength of stirrup 4. Longitudinal steel ratio 5. Loading way 6. Section form Above items 1.,2.,3. are main factors affecting the shear strength of diagonal section.

4.3 Factors Affecting the Strength of Diagonal Section 1. Shear span ratio （剪跨比） With the increase of ，the shear failure mode changes. For , the ultimate shear strength tends to decrease with the increase of . For , the ratio has little influence on the ultimate shear strength. P85 Fig.4-17 diagonal compression failure Shear compression failure diagonal splitting failure

Different shear span ratio leads to a different failure pattern. diagonal compression failure 斜压破坏 Shear compression failure 剪压破坏 diagonal splitting failure 斜拉破坏

Shear strength against shear span ratio

4.3 Factors Affecting the Strength of Diagonal Section 2. concrete strength Any type of shear failures is due to that the concrete attains its compressive or tensile strength. The strength of concrete directly affects the shear strength of members. The shear strength of members increases with the concrete strength and the relation is nearly linear. （混凝土强度） Shear strength against cubic strength of concrete

The stirrup ratio is given by 3. Stirrup ratio and the strength of stirrup The stirrup ratio is given by where ——leg number of stirrup; ——area of one leg of stirrup; ——spacing of stirrup. With the increase of the ratio and yielding strength , the shear strength Vu tends to increase, and the relation is linear. （配箍率和箍筋强度） b Asv1 s Asv= 2Asv1

4.3 Factors Affecting the Strength of Diagonal Section 4. Longitudinal steel ratio The longitudinal bar can supply dowel action force, that it can resist some of shear forces in the cross section. The longitudinal bar can restrict the extending of diagonal cracks, and then enlarge the compression zone. With the increase of , the shear strength increases slightly. （纵筋配筋率） dowel action force

4.3 Factors Affecting the Strength of Diagonal Section 5. Loading way directly loading and indirectly loading Indirectly loading leads to decrease of the shear strength. （加载方式） compression zone secondary beam main beam In lab loading test, the beam is usually loaded directly on its top face, that is called “directly loading ” In practical project, such as cast-in-situ ribbed floor, the load of secondary beam is acted on the middle or bottom zone of main beam, that is called “indirectly loading ”

Indirectly loading leads to decrease of the shear strength.

4.3 Factors Affecting the Strength of Diagonal Section 6. Section form For T-section beam or I-section beam, the existence of flange is helpful to increase the capacity of the beams with web reinforcement about 5％－25％, however the effect is not obvious, and it is usually ignored in practical design. 1.,2.,3. are main factors affecting the shear strength of diagonal section. （截面形式）

4.4 Strength Calculation of Members under Shear 1. Mechanism of Shear Resistance （受剪机理，计算模型） Comb-shape arch mode with tie rod （带拉杆的梳形拱模型） Arch-truss mode （拱形桁架模型） Truss mode （桁架模型）

1. Comb arch mode with tie rod （带拉杆的梳形拱模型） （used for beam without web reinforcements） The longitudinal bar is tensile rod. After the concrete cracks, the compressive force is mainly transferred from the compress-shear zone (on top of the beam) to the support. At the same time. The concrete between the cracks transfer some other force by the bite force between the aggregates.

2. Arch-truss mode （拱形桁架模型） （used for beam with web reinforcements） The arch is compressive zone of concrete, the longitudinal bars are the lower chords, the stirrups are the tensile web members, and the concrete teeth are the compression inclined web members.

3. Truss mode （桁架模型） （used for beam with web reinforcements） It is hinged truss. The compressive zone of concrete forms the upper chord, the longitudinal bars are the lower chords, the stirrups are the vertical tensile rods, and concrete between cracks are the inclined compression rods.

华南理工大学土木工程系 板的一般构造 分布钢筋 板厚h≥60 混凝土强度等级一般为 C20 ~ C35 受力钢筋 间距 板宽 板跨